THE INFLUENCE OF MORPHOLOGY ON PHYSICAL PROPERTIES OF RESERVOIR ROCKS by C.H. ARNS Dipl. Phys., University of Technology, Aachen, 1996

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1 THE INFLUENCE OF MORPHOLOGY ON PHYSICAL PROPERTIES OF RESERVOIR ROCKS by C.H. ARNS Dipl. Phys., University of Technology, Aachen, 1996 A thesis submitted in fulfilment of the requirements for the degree of Doctor of Philosophy in Petroleum Engineering School of Petroleum Engineering The University of New South Wales 22

2 iii DECLARATION I hereby declare that this submission is my own work and that, to the best of my knowledge and belief, it contains no material previously published or written by another person nor material which to a substantial extent has been accepted for the award of any other degree or diploma of a university or other institute of higher learning, except where due acknowledgement is made in the text. C.H. Arns

3 v PUBLICATIONS ARISING FROM THIS THESIS [1] C. H. Arns, M. A. Knackstedt, A. P. Roberts, and V. W. Pinczewski; Morphology, co-continuity and conductive properties of anisotropic polymer blends, Macromolecules, 32: , [2] C. H. Arns, M. A. Knackstedt, W. V. Pinczewski, and K. R. Mecke; Euler- Poincarè characteristics of classes of disordered media, Physical Review E, 63:31112, 21. [3] C. H. Arns, M. A. Knackstedt, W. V. Pinczewski, and K. R. Mecke; Characterisation of irregular spatial structures by parallel sets and integral geometric measures, Physical Review E, submitted, 2. [4] C. H. Arns, M. A. Knackstedt, W. V. Pinczewski, and K. R. Mecke; Physical properties of porous media from a single tomographic image, Physical Review E, submitted, 22. [5] C. H. Arns, M. A. Knackstedt, W. V. Pinczewski, and W. B. Lindquist; Accurate estimation of transport properties from microtomographic images, Geophys. Res. Lett., 28: , 21. [6] C. H. Arns, M. A. Knackstedt, W. V. Pinczewski, and E. J. Garboczi; Computation of linear elastic properties from microtomographic images: Methodology and agreement between theory and experiment, Geophysics, accepted, 21. [7] C. H. Arns, M. A. Knackstedt, W. V. Pinczewski; Accurate V p :V s relationships for dry sandstones, Geophys. Res. Lett., accepted, 21.

4 vii ABSTRACT We consider the structural and physical properties of complex model morphologies and microstructures obtained by Xray-CT imaging. The Minkowski functionals, a family of statistical measures based on the Euler-Poincaré characteristic of n-dimensional space, are shown to be sensitive measures of the morphology of disordered structures. Analytic results for the Boolean model are given and used to devise a reconstruction scheme, which allows one to accurately reconstruct a complex Boolean structure given at any phase fraction for all other phase fractions. The percolation thresholds of either phase are obtained with good accuracy. From the reconstructions one can subsequently predict property curves for the material across all phase fractions from a single 3D image. We illustrate this for transport and mechanical properties of complex Boolean systems and for experimental sandstone samples. By extending the Minkowski functionals to parallel surfaces using operations from mathematical morphology, a powerful discrimination of structure is obtained. Further the sensitivity of the Minkowski functionals under experimental conditions is analysed. Accurate calculations of conductive and elastic properties directly from tomographic images are achieved by estimating and minimising several sources of numerical error. Simulations of electrical conductivity and linear elastic properties on microtomographic images of Fontainebleau sandstone are in excellent agreement with experimental measurements over a wide range of porosity. The results show the feasibility of combining digitised images with transport and elasticity calculations to accurately predict physical properties of individual material morphologies. We show that measurements of properties based on microtomographic images are more accurate than those based on conventional theories for disordered materials. We study the elastic behaviour of model clean and cemented sandstones. Results are in excellent agreement with available experimental data, and are compared to conventional theoretical and empirical laws. A new predictive empirical method is given for predicting the elastic moduli of sandstone morphologies. The method gives an excellent match to numerical and experimental data.

5 ACKNOWLEDGEMENTS The work for this thesis was undertaken at the School of Petroleum Engineering, University of New South Wales in collaboration with the Department of Applied Mathematics, Research School of Physical Sciences and Engineering, Australian National University. I thank both schools for having provided a stimulating working environment. In particular I thank Nino, Rachel, and Jennifer, who kept my time spend off topic to a bare minimum. My faith in academia was greatly increased by my supervisors. I gratefully acknowledge the courage of Val Pinczewski in supporting basic science in an increasingly commercial environment, and the enthusiastic supervision of Mark Knackstedt. His ideas and contributions set this work on a firm basis. I couldn t have asked for better supervisors. Much of the work contained in this thesis was carried out in collaboration with other researchers. Individual contributions up to chapter 5 can be seen on the publication page (iii). Chapter 6 is based on collaborative research with Mark Knackstedt. I m indepted to Klaus Mecke for setting me on the morphological path, and to Anthony Roberts and Ed Garboczi to getting me started with the numerical part of this thesis. I also like to thank Henry Salisch and Yujin Zhang for introducing me to formation evaluation and Michel Kagan for help in mathematics. Further I would like to thank my fellow researchers for providing a challenging multicultural and intellectual environment. Thanks guys. Particular to Yujin, Mofazzal, Dilip, Mark, Adrian, Rob, Tim, Vincent, Michel, and Izzy (who kept my spirits alive with the occasional game of chess). I thank the UNSW and ANU supercomputing facilities for generous allocation of computer time and data storage; Russell Standish, Ben Evans, and particular Dave Singleton were always at hand to explain matters when I got stuck.

6 x For their financial support I thank the Australian Petroleum Cooperative Research Council (APCRC), the School of Petroleum Engineering, the Faculty of Engineering at UNSW, and the Department of Applied Mathematics at ANU. I remain grateful to Henry Salisch and Jürgen Wohlenberg. Without them I would never have gone to Australia, would never have started this thesis, and would never have met my wife. Finally I would like to thank Ji-Youn, who I met when starting this thesis. Her ongoing support and care during these years was invaluable we are now a happily married couple.

7 DEDICATION I wish to dedicate this work to my parents and my wife; for their guidance, support, and unconditional love.

8 CONTENTS CHAPTER 1 INTRODUCTION Microstructure Morphology of disordered materials Physical properties Geophysical applications MODEL COMPOSITES AND EXPERIMENTAL DATA Basic models of irregular spatial structures Particle based models Gaussian models Voronoi models Experimental data Fontainebleau sandstone Berea sandstone Cross-bedded sandstone Termite nest MORPHOLOGY OF DIGITISED DISORDERED MATERIALS Minkowski functionals Mathematical background Boolean model Computational aspects Discretisation effects Comparison of MFs for different model morphologies Boolean grain models Gaussian models Voronoi models Application to a cross-bedded sandstone Development of Boolean reconstruction methods

9 xiv Minkowski measures of a Boolean composite Computational aspects Derivation of local morphological measures Error in morphological measures Generation of equivalent Boolean models Prediction of percolation thresholds Application to sandstone images Characterisation of structures using the geometry of parallel sets Mathematical background Characterisation of model systems Discrimination of model composites Application to experimental data Sensitivity of Minkowski functionals Sampling strategies Effects of distortions on images Discussion CONDUCTIVITY OF RESERVOIR ROCKS Background Bounds on the effective conductivity Effective medium approximations Empirical equations Numerical solution Accurate image-based prediction of DC conductivity Numerical simulation and error analysis Formation factor of Fontainebleau sandstone Other sandstones Comparison to theoretical formula Comparison to empirical models DC conductivity of equivalent stochastic reconstructions Heterogeneous Boolean mixture Fontainebleau sandstone Cross-bedded sandstone LINEAR ELASTIC PROPERTIES OF RESERVOIR ROCKS Background Bounds on elastic moduli Effective medium approximations Empirical equations

10 xv Gassmann s relations Structure based derivation of effective moduli Elastic properties of Fontainebleau sandstone Choice of representative image volume Property prediction Fluid substitution Comparison to experiment Comparison to theoretical formula Comparison to empirical models Conclusion Elastic properties of Boolean and Gaussian reconstructions Heterogeneous Boolean mixture Fontainebleau sandstone Cross-bedded sandstone GEOPHYSICAL APPLICATIONS: VELOCITY-POROSITY RELATIONSHIPS Methods Structural model for clean sandstone Structural model for sandstone having two mineral phases Elastic simulation Accurate V p : V s relationships for dry sandstones Poisson s ratio for dry sandstones Implications Conclusion Accurate velocity model for clean consolidated sandstones Numerical results Comparison to experiment Comparison to velocity-porosity models Proposed velocity-porosity model for clean sandstones Summary Accurate velocity model for two-mineral cemented sands Numerical results Comparison to models New velocity-porosity model for clay-bearing sands Summary

11 xvi BIBLIOGRAPHY 224 APPENDIX A EUCLIDEAN DISTANCE TRANSFORMATION (EDT) 237 B CALCULATION OF EQUIVALENT BOOLEAN MODELS 239 B.1 Local measures of oriented prisms B.2 Local measures of spheroids B.3 Derivation of a Boolean model based on morphological opening

12 TABLES Table 2.1 Volume fractions and variability of the Fontainebleau samples. Given are the nominal porosity plug of the core plugs, from which the samples were extracted, the porosities 48 of the 48 3 samples used in this thesis, as well as the absolute and relative standard deviation of the slice porosity sl along the coordinate axes of the samples extracted from the core plugs (z cylindrical axis) Volume fractions and variability of the two cross-bedded sandstone 3 3 subsamples. Given are the porosities 3 of the 3 3 samples used in this thesis, as well as the absolute and relative standard deviation of the slice porosity sl along the coordinate axes of the samples extracted from the core plugs (z cylindrical axis) Local contributions to the global Minkowski measures for the 22 isotropic configurations and their mapping to the 256 vertex configurations following the order shown in Fig. 1. L gives the configuration in Fig. 3.2 which matches the isotropic configuration number (IC (3D) 21 ). N notes the multiplicity of the configuration Comparison of the local Minkowski measures in 2 and 3 dimensions for spherical grains in the continuum against the discretised spheres used in the simulations. We use * to denote the continuum values as compared to the discretised. The radii r i in the last three columns are equivalent sphere radii based on the measures V i and Eqn for the intrinsic volumes of continuum spheres Parameters of the models for the cross-bedded sandstone in microns. The original sandstone image has resolution at 1µm per pixel. For the IOS model, the equivalent sphere radius is in the r c column... 52

13 xviii 3.4 Parameters for the mixture models shown in Fig. 2.3.[a]-[f] together with the (exact) predictions of the Minkowski measures of the average local grains given by Eqn. B.1. The probability to place the second particle p 2 is given by p 2 = 1 p 1. All individual grains are convex and it follows directly V 3 = Local Minkowski measures for systems of Poisson distributed cubes at λ = 4 (L = 2) and λ = 8 (L = 4). The first row (labeled exact) gives the analytical values for the local V ν : the subsequent rows give the relative error (Eqn. 3.21, in [%]!) of the prediction of V im ν from the measurement of v ν and use of Eqn for different ρ. ρ in the table is the analytic Poisson density from Eqn and ρ image gives the density measured from the images Local Minkowski measures of systems of Poisson distributed spheres at r = 4 (top) and r = 8 (bottom) on a 2 3 lattice. The theoretical values of the single grains in the first lines are for discretised spheres Local morphological measures of the 5-mix system of 1% sticks (4x1x1), 1% plates (2x2x1), 4% cubes (8x8x8), 2% each of rectangular prisms of size (1x5x2) and (16x8x4) on a 5 3 lattice. The theoretical values of the mixture are given in the first line, the other rows give the relative error in the predictions from the simulations taken at different particle fractions f Match of the equivalent Boolean model to the 5-mix system. The norm of the relative errors in the local measures (Eqn. 3.21) are given. To accurately match the morphology one must consider a two-particle system Comparison of the percolation thresholds between the complex 5-mix model and its matching Boolean model defined in Tab Local Minkowksi measures of the four Fontainebleau sandstone samples of size 48 3 given in section (V 3 = 1 for compact grains). 62

14 xix 3.11 Parameters for the Boolean models of the Fontainebleau sandstone and errors of the morphological matches. The first IOS model (IOS C ) was matched using the void-void correlation function. The other models are matched using the Boolean reconstruction. IOS is generalised to more than one sphere, all spheroids are randomly oriented (ROS). Note the large errors E Vi in the local measures for the 1- particle IOS model; the cubic equation (B.3) has no real solution. For all two-particle models one of the particles is very small and has a width of the order of a few voxels. a, b, c note the length of the half-axes with the exception of the OSC model, where a notes the median of the radius distribution G(r c ) = 1/2 (see appendix B.3) Variability σ z ( sl ) [%] of the pore volume per slice along the x,y,zdirections of the Fontainebleau samples and the z-directions of its reconstructions (see also Tab. 2.1, the values are from a single realisation at 48 3 ) Parameters for the Boolean models of the cross-bedded sandstone. The first IOS model (IOS C ) was matched using the void-void correlation function. The other models are matched using the Boolean reconstruction. IOS is generalised to more than one sphere, all spheroids are randomly oriented (ROS). a, b, c note the length of the half-axes with the exception of the OSC model, where a notes the median of the radius distribution G(r c ) = 1/2 (see appendix B.3) Variability σ( sl ) [%] of the pore volume per slice along different directions of the 3 3 cross-bedded sandstone samples and its reconstructions (see also Tab. 2.2). The reconstruction reflecting heterogeneity best is the OSC model Integral error of the Minkowski measures over fraction (Eqn. (3.28)) for the models defined in Table I of Ref. [131]. The last two columns show the error in the CDF from Ref. [131] Integral error of the Minkowski measures over fraction (Eqn. (3.28)) for the models given in Table II of Ref. [131] Integral error of the Mfs for the models of the tomographic image of the termite mound calculated according to Eqn. (3.28) compared to the relative error in the 2-point correlation function Ep (2) and parameters of the matching models N c, with c [, 1] being the level cut parameter, for the termite nest in mm. The original tomographic data set image has resolution at 1mm per voxel. For the IOS model, the equivalent sphere radius is in the r c column

15 xx 3.18 Integral error of the Mfs for the models of the tomographic image of the Fontainebleau sandstones calculated according to Eqn. (3.28) Derivation of a representative sample size for the v 3 measure, assuming a target standard deviation of 1%. Here l notes the distance between the original seeds in the Voronoi model, while ξ is the correlation length derived from the 2-point correlation function Decomposition of the 3D vertex configurations into two 2D vertex configurations parallel to the x-y plane Speedup of the FEM calculations by exploring additional symmetries in the local stiffness matrices. Three different model structures are considered. The IOS model is resolved at radii r = L/1, where L is the lattice size of the cubic lattice, with a sphere volume fraction of 8%. The Gaussian model is a 1-level-cut resolved at 5% volume fraction, and the Voronoi a foam-like structure at 1% volume fraction, generated by 5 initial Poisson distributed seeds. The elastic moduli of quartz were used for the solid, vacuum as porous phase and timings taken on an alpha ev67 running at 667MHz. For compilation the native f95 compiler was used. The timings of the general code are noted as t gen, the timings using bulk symmetry as t sym, and using bulk symmetry combined with further optimisations for vacuum phases as t vac. Memory use is given for the general code (in Mbytes). The versions exploring the symmetry in the local stiffness matrices need an additional 1%. For lattices larger than 2 3 memory access becomes non-local and memory bandwidth limits speedup for the serial version of the code. As comparison the timings of the OpenMP version of the code are given in the last section of the table for the same hardware, but using 4 processors. This code version is not fully optimised yet (t gen should approach 12 sec/it.) Material properties for the minerals and fluids used in the simulations presented in this chapter [92]. K is the bulk modulus µ the shear modulus, and ρ the density. The water conditions are chosen to allow for comparison with experimental data [54]

16 xxi 6.2 Effective Poisson ratio for the mixed lithologies at zero porosity. We report the ratio of the mineral phases along with the order of placement, mineral 1 then mineral 2 (M 1 : M 2 ). The prediction of the framework model ν f, interstitial model ν I and the arithmetic average ν av are reported Moduli in GP a for the mixed lithologies at zero porosity. The subscript f indicates data for the framework model and I for the interstitial model. We also give the upper and lower Hashin-Shtrickman bounds for the two phase system (subscripts l and u respectively). All the numerical data sits within the bounds validating the predictions of the numerical simulation. For the bulk modulus and shear modulus of the quartz:dolomite systems we observe little dependence on the microstructural model. The shear modulus of the quartz:clay system does exhibit the strongest dependence Dry moduli for the mixed lithologies at zero porosity in GP a. The comparison with the Voigt-Ruess-Hill average and predictions based on the SCA and DEM are satisfactory for most systems. Only the shear modulus data for the quartz:clay system where the contrast between the two phases is more than 7 : 1 gives a poorer prediction. The error in this case is up to 1%. The predictions of the time average equation [17] are poor

17 FIGURES Figure 2.1 Illustration of the choices of rotation of particles and the resulting isotropic and anisotropic composites. [a] Randomly oriented ellipses. [b-c] Lattice oriented ellipses and rectangles. [d] Partially and [e] fully aligned ellipses. [f] Mixture of grains with different shapes and rotational freedoms. [g] Randomly oriented and [h] partially (p a = 1) 2 as well as [i] fully aligned ellipses with a range of half-axes The interface of models of Poisson distributed particles. Top: [a] oriented cubes of sidelength λ = 8 and [b] overlapping spheres of radius r = 8. Bottom: overlapping spheroids with an half-axis range of r = 4..2; [c] randomly oriented, [d] fully aligned. The volume fraction of the particle phase is =.2. All models are discretised on a 2 3 lattice The interface of mixtures of oriented rectangular blocks on a 1 3 lattice [a] plates, [b] sticks, [c] cubes and sticks, [d] cubes and plates, [e] cubes and cubes, [f] cubes and cubes. All models are shown at a total particle fraction of =.2. For the exact parameters of the model see Tab Slices through three 3D models of porous media with two solid phases. Top: IOS on IOS, middle: two IOS phases distributed at the same time, bottom: parallel surfaces on IOS. The ratio of the two solid phases is held constant at 2:1. Fractions of the phases are left: 6:3:1, middle: 53:27:2, right: 4:2:4. The models are discretised on a 12 3 lattice The interface of Gaussian models of periodicity t = 1 for volume fractions of =.25 and =.75 from left to right. Top: 1-levelcut. Middle: 2-level-cut. Bottom: intersection of two 2-level-cuts. The models are discretised on a 2 3 lattice

18 xxiv 2.6 Calculation of the discrete Voronoi tesselation of 5 Poisson distributed points (pixels) on a 2 2 lattice. Each point [a] is grown concentrically [b-g] until all space is covered [h]. The different labels of each grain are visualised by different colours. Only background pixels (white) can be updated with a new label Visualisation of the set of polyhedra boundary voxels (see text) resulting from the Voronoi tesselation. [a] and [c] represent a Voronoi network of the set of initial Poisson points in two and three dimensions. [b] shows the initial interface build by the union of the surfaces of all Voronoi cells The interface of Voronoi models with 1 seeds on a 2 3 lattice. First row: facet models for volume fractions of [a] =.26 and [b] =.74. Second row: edge models for volume fractions of [c] =.6 and [d] = Generalisation of the Voronoi models using correlated points. [a] Poisson distributed points; [b] a normalised Gaussian field, [c] the Euclidean distance from the centre of the structure, and [d] the function sin(x) 2 is used to transform the uniformly distributed points. All structures are generated using 2 seeds Right: Compilation of the four 45 2 slices. Left: Final first 48 2 slice of fb15, after removing overlap, and recombination Pore space images of a central 12 3 piece of each of the four 48 3 samples, each taken from the centre. From top left to bottom right: fb7.5 ( = 7.4%), fb13 ( = 12.6%), fb15 ( = 19.6%), fb22 ( = 19.%). Here notes the porosity of the 12 3 subsamples Variations in porosity and 2-point correlation functions of the 4 Fontainebleau sandstone samples (48 3 subvolumes). [a-c]: porosity variation along the x,y,z axes of the core plugs. [d]: 2-point correlation functions Pore space images of the Berea sandstone sample. Top: [a] the first, [b] middle, and [c] last slice of the 216 slices from which the full analysed section of the core plug was selected. The bulk porosity of the analysed section is 12.1%. Bottom: Two of its eighteen disjoint 18 3 subpieces. Units are in voxels (resolution 1 µm) Left: porosity variation along the cylindrical axis of the Berea sandstone core plug for the extracted subvolume. Right: 2-point correlation function of the former

19 2.15 The interface of sections of the sized segmented Xray- CT scan of a cross-bedded sandstone. [a]: 15 3 (112), [b]: 153 (213), [c]: 1 3 (214), [d]: 13 (335), [e]: 63 (211), [f]: 63 (221). The subscripts note the position in the sandstone sample matrix. Top: 2x2x4, middle: 3x3x6, bottom: 5x5x1. Samples [e,f] are disjoint (non-overlapping) xxv touching neighbours Variations in porosity and 2-point correlation functions of the crossbedded sandstone data. [a-c]: porosity variation along the x,y,z axes for the extracted subvolume. The cylindrical axis of the core plug coincides with the z-direction and the volume is subdivided into 2 samples of size 3 3 for the x,y directions to compare the same slice volumes. [d]: 2-point correlation functions of the two 3 3 subsamples The binary interface of the termite nest after segmentation at a resolution of 1 mm. The total image size is A high resolution colour image can be seen at sjr Left: porosity variation along the z-direction of the termite nest. Right: 2-point correlation function of the sample Configuration resulting in an ambiguous geometry on a square lattice. The ambiguity at the central vertex [a] can be resolved in two ways: [b] the configuration is continuous in the white phase, or [c] continuous in the black phase Catalog of filling patterns of a unit cell giving rise to distinct configurations. Some configurations with exchanged phases are not shown; 22 isotropic configurations exist, the rest can be generated by rotations. Isotropic configurations (A )-(G ) are generated by inverting the phases of configurations (A)-(G) and (N). (D ) and (G ) are shown as an illustration Minkowski measures over fraction: comparison of theoretical predictions (lines) with numerical simulations (symbols) for Poisson distributed cubes at different sizes: (a) λ = 1, (b) λ = 2, (c) λ = 4, and (d) λ = 8. The measures are scaled as v 1 λ v 1, v 2 λ 2 v 2, and v 3 λ 3 v

20 xxvi 3.4 Minkowski measures over fraction: comparison of theoretical predictions (lines, highly connected neighbourhoods) with numerical simulations (symbols, both neighbourhoods) for Poisson distributed spheres of different radius: (a) r = 4, (b) r = 8, (c) r = 12, and (d) r = 16. The measures are scaled as v 1 r v 1, v 2 r 2 v 2, and v 3 r 3 v The configuration probabilities P (IC (3D) 21 ) = P (L) over fraction of Poisson distributed cubes and spheres for the ambiguous configurations in Fig. 3.2 (see also Tab ). The letters correspond to the specific configurations L. Data for the cube model is given by open squares and for the IOS by solid circles. The configurational probabilities decrease with increasing size. In the figure we show data for sphere radii: r = (4, 8, 12, 16) and for the cube sidelength: l = (1, 2, 4, 8) Correlation length over pore fraction derived from the two-point correlation functions in real space; IOS: identical overlapping spheres, ROS: randomly oriented spheroids, three different Gaussian, and two Voronoi tesselation models Minkowski measures over fraction for Poisson distributed overlapping spheroids of half-axes r = 4..2, with r = 12, compared to fully aligned overlapping spheroids of the same size distribution and IOS of r = 12. The measures are scaled as v 1 r v 1, v 2 r 2 v 2, and v 3 r 3 v Minkowski measures over fraction for Gaussian 1-level-cut models of periodicity t = Minkowski measures over fraction for Gaussian 2-level-cut models of periodicity t = Minkowski measures over fraction for Gaussian intersection models of periodicity t = Minkowski measures over fraction for Voronoi facet models of 1 seeds on a 2 3 lattice Minkowski measures over fraction for Voronoi edge models of 1 seeds on a 2 3 lattice Normalised 2-point correlation functions S(r) of the two 3 3 sandstone samples and the averaged best fits for the model structures.. 5

21 3.14 Minkowski measures for a cross-bedded sandstone. From top to bottom: v 1 (a,d), v 2 (b,e), and v 3 (c,f). In the left column we show the data points for the different blocks of sizes 6 3, 1 3, 15 3, and 3 3. One obtains consistency in the morphological measures down to scales of 1 3. Right: comparison of the MFs for the sandstone xxvii data to the three stochastic models based on a window size of Minkowski measures over particle fraction for the Poisson mixture models shown in Fig The lines give the analytic predictions based on Eqn while the data points are based on the simulation results. The grains chosen and local Minkowski are defined in Tab [a] plates, [b] sticks, [c] cubes and sticks, [d] cubes and plates, [e] cubes and cubes with p 1 = 1/2 [f] cubes and cubes with p 1 = 1/64. As in Fig. 3.3 all global measures are scaled with corresponding local measures, so they can be shown in one plot Scaling of the standard deviation σ(l) of the local measures V ν for Boolean models with L. The plots are for cubes of λ = 8 (left) and spheres of r = 4 (right). Data points are from right to left for L = 15, 2, 3, 4, 6 and L = 8. We show the numerical results match the theoretical prediction that σ(l) L 1.5. One obtains a very low standard deviation in the local measures from images measured at large L. Top: f =.2, middle: f =.5, bottom: f =.8. Each data point represents at least 1 realisations Visual comparisons of reconstructions to a 24 3 subset of the original Fontainebleau sandstone sample fb15 ( = 18.2%). [a] Fontainebleau sandstone, [b] IOS C, [c] ROS (1), [d] IOS (2), [e] ROS (2), and [f] OSC Minkowski measures over fraction for each of the four Fontainebleau sandstone samples compared to the matching Boolean models. [a] (left) IOS C, [b] (middle) OSC, and [c] (right) ROS (2). For v 2 and v 3 the measures for the highly connected neighbourhood are given Comparison in the variation of the porosity distribution along the z- axis of the Fontainebleau samples and reconstructions. Sample size is the same as in Fig

22 xxviii 3.2 Minkowski measures over fraction for a cross-bedded sandstone compared to the matching Boolean models. [a] (left) IOS model matched using the void-void auto-correlation function as well as Boolean reconstructions with one or two spheres. [b] (middle) Boolean models of rectangular lattice oriented bars. [c] (right) Boolean models of randomly oriented spheroids and the OSC model. For v 2 and v 3 the measures for the highly connected neighbourhood are given Visual comparisons of reconstructions to a 15 3 subset of the original cross-bedded sandstone sample ( = 18%). [a] Cross-bedded sandstone, [b] IOS C, [c] ROS (2), [d] OSC. The domain size is comparable to the 24 3 Fontainebleau datasets shown in Fig Comparison in the variation of the porosity distribution along the x-axis of the cross-bedded sandstone and reconstructions. Sample size is the same as in Fig Illustration of the differences in the erosion/dilation operation on a convex set and a non-convex set. In [a] we start from a collection of convex shapes (point pattern) and dilate first to (r ɛ) obtaining the black pattern and then further to (r) =.5 resulting in the grey pattern. In [b] beginning from the non-convex (grey) pattern at (r) =.5, we obtain a different morphology at (r ɛ). Enhanced connectivity between overlapping sphere centers is one obvious qualitative difference. Domain size is , with r = 46.5 and ɛ = 15.5 in pixel units. The black phase on the left side has a fraction of =.134 and 34 components (periodic boundaries). In contrast the black phase on the right side has a fraction of =.156 and 2 components Boolean Models: Minkowski functionals of different Poisson processes starting at =.5. The point process and IOS have exactly the same particle centres for each of the realisations, such that the processes have the same Poisson density at =.5. In case of the former the process is started with 777 points at. In contrast the spheres of the IOS models have a radius of r = 12. The spheroids of the spheroidal models have half axes of average length r = 12 with r [4, 2]. See also Fig

23 xxix 3.25 Gaussian models: Minkowski functionals of erosion/dilation operations over phase fraction for GRF models having kernel g(r) as in Eqn. 2.8 with r c = 2, ξ = 4, d = 3 in pixels, and periodicity t = 1 on a grid. The erosion/dilation operations are started at an initial fraction of =.5. The models are shown in Fig Voronoi models: Morphological measures (MFs and Mfs) over phase fraction for Voronoi facet and foam models. Compared are the densification process against the erosion/dilation process started at an initial fraction of =.5. The Voronoi cells are generated by 1 seeds periodically mapped on a 2 3 lattice. See also Fig Reconstruction of an experimental Gaussian model (N 1 ): Minkowski functionals of the erosion/dilation process for the experimental data set and the corresponding Mfs for the matching model data sets defined in Table I of [131]. exp corresponds to the experimental image, and the lines give the best fit to the model morphology Reconstruction of an experimental Gaussian model (N ): Minkowski functionals of the erosion/dilation process for the experimental data set and the corresponding Mfs for the matching model data sets defined in Table II of [131] [a] Two-point correlation function of the termite nest and the best fits for the three models. [b] Correlation function measured along the three axes and averaged (S r ). Data along the x-axis is given in number of voxels Termite Nest: Minkowski functionals of the erosion/dilation process over fraction. Note, that here no scaling factor is applied (l = 1) to honour the natural length scale of the experimental data set Fontainebleau sandstone, sample fb7.5: Minkowski functionals over fraction ( = 8.29%) Fontainebleau sandstone, sample fb13: Minkowski functionals over fraction ( = 12.9%) Fontainebleau sandstone, sample fb15: Minkowski functionals over fraction ( = 17.7%) Fontainebleau sandstone, sample fb22: Minkowski functionals over fraction ( = 21.%)

24 xxx 3.35 Effect of increasing the sampling volume on the normalised standard deviation of the Euler characteristic. [a-b] cubes of λ = 1, [c-d] spheres of r = 12, and [e-f] Voronoi tube models having 1 seeds. All models are discretised on 2 3 lattices and the sampled volume is variied by varying the number of slices sampled (n) for a single measure. Left: comparison of the v 3 measures for the average of 1 realisations at 2 3 to 1 realisations of pairs of slices (2 2 2). The standard errors are very small and the measures coincide. Right: Scaling of the standard deviation of v 3 for the three models. The standard deviations scales N, where N is the number of vertices. The normalisation is chosen, such that σ(v 3 )/ max(v 3 ) min(v 3 ) =.1 for f =.5, and the corresponding N given in the plots The effect of increasing slice distance s on v 3 for three different Boolean grain models, lattice oriented cubes of l = 1 [a-b] and l = 8 [c-d] as well as IOS of r = 8 [e-f]. Two different normalisations are used, volume (left), pronouncing curve regions where the information loss is small (i.e. insets in [c,e] for dilute fractions of particles) and number of vertices (right), pronouncing the asymptotic behaviour for s. For the Boolean cubes of l = 1 all slices are uncorrelated and the measures v 3 scale with s [a] or stay constant [b] Variation of the zero of the Euler characteristic (p c f(v 3 = )) as function of the sampling distance s between parallel slices in units of a characteristic length (Ξ = l, 2r) The effect of increasing slice distance s on v 3 for three different Gaussian models, 1-level cuts (top), 2-level cuts (middle), and intersection models (bottom). Two different normalisations are used, volume (left), pronouncing curve regions where the information loss is small ([a]) and number of vertices (right), pronouncing the asymptotic behaviour for s ([d,f]). The different Gaussian models are generated using the same kernel and length parameters r c, d, and ξ (see chap ). While the topological measures of the 1-level-cuts are not much affected even at s = 4 (Ξ 7), for the same slice distance the topological information for the interface pronounced 2-level-cut (Ξ < 4 for <.3) and intersection (Ξ < 5 for <.3) models is almost totally lost

25 xxxi 3.39 Left: a X-ray density map (grey-scale) from a X-ray CT image of paper. Right: corresponding binary image. The image is taken from Fig. 3-4 of [51]. The total image size is (2.33 mm) 2 and the extracted regions shown have a size of (4 µm) SEM images of sideviews of paper at a resolution of 1 nm. The thickness of the paper is around 1 µm. The extend and shape of the individual fibres is clearly visible [Ray Roberts, unpublished] Model of partially aligned ellipsoids demonstrating the effects of resolution limitation by applying a Gaussian blur (details follow in later subsections). [a] Original model of aligned ellipsoids. [b] Resegmented image after applying Gaussian blur. [c] Cut through original model. [d] Corresponding signal after blurring. [e] Final slice. The originally needle-like particular structure is lost in the process Illustration of finite resolution for an IOS model, simulated by an isotropic Gaussian blur with σ = σ x = σ y = σ z in units of the sphere radius (r = 12). The corresponding Minkowski measures are shown below. From left to right: slices through the blurred signal and resegmented data for =.25 (left), =.5 (middle), and =.75 (right). For σ see below Influence of finite resolution on the global Minkowski for the IOS model (r = 12). The sphere radius is used to scale σ. For larger σ all morphological measures of the resegmented structures approach those derived for the single cut Gaussian field (compare to Fig. 3.8) Illustration of finite resolution for a Gaussian 1-level-cut, simulated by an isotropic Gaussian blur with σ = σ x = σ y = σ z in units of the periodicity of the Gaussian (L/t = 1). The corresponding Minkowski measures are shown below. From left to right: slices through the blurred signal and resegmented data for =.25 (left), =.5 (middle), and =.75 (right). For σ see below Influence of finite resolution on the global Minkowski measures for a Gaussian 1-level-cut of t = 2. L/t is used to scale σ. The morphology is already Gaussian and the Gaussian instrument function just changes the scale of the model

26 xxxii 3.46 Illustration of finite resolution for a Voronoi tube model (1 seeds on a 2 3 lattice), simulated by an isotropic Gaussian blur with σ = σ x = σ y = σ z in units of the average seed distance of the Voronoi model (l = 43). The corresponding Minkowski measures are shown below. From left to right: slices through the blurred signal and resegmented data for =.6 (left), =.1 (middle), and =.9 (right). For σ see below Influence of finite resolution on the Voronoi tube model shown above. The length scale used for σ is the average seed distance (l = 43). For larger σ all morphological measures of the resegmented structures become Gaussian Illustration of the effects of interfacial speckle noise ( salt and pepper ) for three different structures. From left to right: Slices through an IOS ( =.25), 1-level-cut ( =.5), and Voronoi model ( =.1). For each model the probability p of speckle noise increases from left to right and top to bottom, where p =, p =.1%, p = 1.%, and p = 1%. The corresponding Minkowski measures are shown below Influence of interfacial speckle noise of probability p to switch the phase of a voxel bordering the interface, on the global Minkowski measures of three different structures. Left: IOS of r = 12, middle: a Gaussian 1-level-cut model, right: a Voronoi tube model. The v 3 measure of the IOS model is much more affected by interfacial noise than of the smoother other models Comparison of the results for conductivity for cell sizes at nearly full core size 4 3 with data measured at 24 3, 16 3, and This illustrates that finite size errors at 12 3 are small. We also show the statistical fluctuations in the measurement of the conductivity for the 12 3 data sets binned as described in the text The discretisation errors for the conductivity for each binned set of porosity. In all cases the fit to Eqn is excellent DC conductivity of the original data at 12 3 and 5.7µm resolution and the results for poorer resolutions. The continuum value fitted by matching to Eqn and shown in Fig. 4.2 is dramatically different to the measurements made directly on the image for all L and illustrates the importance of considering discretisation effects

27 xxxiii 4.4 Comparison of the continuum prediction of the conductivity simulations for the Fontainebleau sandstone with experimental data. The lines indicate best fits to the data Continuum prediction of the conductivity simulations for all sandstones described in chapter 2. All simulations are run for : 1 contrast and the differences in the sandstones are due purely to structural effects Comparison of the continuum prediction of the conductivity simulations for the sandstones to theoretical formula. The Hashin- Shtrikman upper bound is not predictive. Also plotted is the DEM prediction for spherical inclusions Comparison of the unbiased continuum prediction of the conductivity simulations for the sandstone to experimental data and Archie s law Estimation of the inclusion fraction for the Fontainebleau (12 3 ) and cross-bedded sandstone (1 3 ) samples Comparison of the continuum prediction of the conductivity simulations for the sandstone with experimental data and the general Archie s law. The porosity scale is corrected to mimic effective porosity (see Fig. 4.8) Effective conductivity over particle fraction f for the original 5- particle mixture given in Tab. 3.7 compared to its equivalent 2-grain Boolean model defined in Table 3.8. The equivalent model defined from the single 3-D image at f = 5% matches the full f-conductance curve for both phases. [a] conductivity grain:pore contrasts 1 : and : 1, [b] conductivity contrasts 1 :.1 and.1 : 1. Note that the simulation data for finite contrast slightly violates the upper bound, which is a computational artefact Comparison between the prediction for conductance of the matching Boolean and Gaussian models and the Fontainebleau sandstone data. [a] fb7.5, [b] fb13, [c] fb15, and [d] fb22. The Gaussian model gives the worst match to the data over the full fractional range. Of the Boolean models, the OSC reconstruction gives a poor match to the data for the low porosity sample. The ROS (2) model gives the best match, but the simple IOS C model performs well

28 xxxiv 4.12 Comparison of the running average (4 points) of the inclusion fractions over total porosity for the reconstructions to the Fontainebleau data sets on a logarithmic scale. The OSC models significantly underestimates the inclusion fraction, while the Gaussian overestimates consistently. None of the models is able to match the inclusion fraction of the Fontainebleau sandstone. For the parameters of the Boolean models see Tab Comparison between the prediction for the effective conductivity of the matching Boolean reconstructions and the cross-bedded sandstone. None of the Boolean reconstructions is able to match the correlated heterogeneity of the sandstone. The model performing best is the ROS (2) model Comparison of the running average (1 points) of the inclusion fractions over porosity for the reconstructions to the cross-bedded sandstone on a logarithmic scale. All models significantly underestimate the inclusion fraction. Of the Boolean reconstructions, the ROS (2) model performs best. For the parameters of the Boolean models see Tab Numerical prediction and error analysis of the bulk modulus of water saturated Fontainebleau sandstone (each modulus is scaled separately, see 5.2). [a] Raw data at different domain sizes. [b] Binned data of [a] with standard error, no finite size effects are present. The standard error for the endpoints and larger grids varies because of the limited number of samples of these fractions/sizes. [c] Raw data at different resolutions and constant domain size (.32 mm 3 ). [d] Binned data of [c] with standard error One observes strong resolution/discretisation effects. [e] Discretisation/resolution scaling of the data for different porosities, the porosities of the bins at different resolution are interpolated to the same fractions using natural spline interpolation beforehand. [f] Final prediction at ideal resolution with scaled standard deviation, compared to the original data at maximal resolution ( = 5.68 µm, split into the four core plugs). The intercept with the y-axis is given by K Quartz = 37GPa ([a-d,f]). 151

29 xxxv 5.2 Scaling of the elastic properties of water saturated Fontainebleau sandstone. The sandstone matrix is simulated as isotropic quartz (K = 37GPa, µ = 44GPa) and the water is assumed to be clean (K = 2.2GPa, µ = GPa). All moduli decrease with increasing resolution as do the wave velocities V p and V s, while the Poisson ratio increases with resolution. The continuous lines indicate the binned data at the various resolutions, while the symbols indicate the individual simulations at a domain size of 12 3 voxels, with different symbols to separate the four different cores. The binned and scaled data is given with standard error and scaled standard error Comparison of the results of the simulations for dry, water-saturated and oil-saturated to Gassmann prediction based on the dry rock data. In [a] we give the numerical predictions for the dry rock data (squares) and show a best fit to the data points (solid line). We use this fit and Eqn. 5.3 to predict the values of the water- and oilsaturated curves (dotted curves). We show the numerical predictions for the water- and oil-saturated results. The fit to the Gassmann s equations is excellent, further indicating the ability of this methodology to quantitatively predict geophysical properties. In [b] we show that the shear modulus is independent of the pore fluid as predicted by Gassmann Comparison of the continuum prediction of the elastic simulations for the digitised images under dry and the water saturated (4 MPa) conditions to experimental data (Han, [54]). The figures give predictions for the [a] dry and [b] wet bulk modulus, and [c] the shear modulus Comparison of the continuum prediction of the elastic simulations for the digitised images under dry and the water saturated (4 MPa) conditions to effective medium approximations and bounds. The figures give predictions for the [a] dry and [b] wet bulk modulus, and [c] the shear modulus Comparison of the results of the simulations (squares and dashed line) for water-saturated sandstone to experimental data (circles) and the empirical equations of Wyllie (Eqn. 5.2) and Raymer (Eqn. 5.22). The fit of the numerical data compares well to either empirical equation

30 xxxvi 5.7 Comparison of the simulation results to the empirical equation of Nur [11] and of Roberts and Garboczi [123] for the dry case. Under both water-saturated and dry conditions the fit of the empirical equations to experiment and numerical data is excellent and comparable to the data obtained from images Effective elastic moduli over fraction for the 5-particle mixture given in Tab. 3.7 (p. 57) compared to a mixture of 2 lattice rectangles matched to have the same average local measures. The reconstruction for the particle phase being modeled as the solid (moduli increasing with particle fraction f) is very accurate. The results are also compared to the Hashin-Shtrikman bounds (Eqns ). The upper bounds are not predictive for the chosen contrast and the lower bounds zero Derivation of the Fontainebleau data at the fixed resolution of 5.68µm but for periodic boundary conditions (simulating an infinite size sample). [a-b]: Effective moduli for two water saturated model structures generated with periodic and non-periodic boundary conditions. [cd]: Comparison of the periodic data with the non-periodic corrected data. [e-f]: Prediction of the elastic moduli of a water saturated periodic ( infinite ) Fontainebleau sandstone sample using the simulations for the non-periodic tomographic image. The correction is specific to the given resolution and lattice size Image based calculations of the water saturated elastic moduli of the four Fontainebleau sandstone samples compared to Boolean and Gaussian reconstructions. [a-b]: fb7.5, [c-d]: fb13, [e-f]: fb15, and [gh]: fb22. The Fontainebleau sandstone data is corrected to represent a periodic infinite sample Running average (4 points) of the fractions of particles in suspension over total porosity for the reconstructions to the Fontainebleau data sets. For each model the reconstructions to the four Fontainebleau sandstones are combined in one plot, thus resulting in 4 points per fraction. The 2-particle and OSC models show a significant effect, as one of the particles in the Boolean reconstruction is very small (see Tab. 3.11) Image based calculations of the elastic moduli of the water-saturated cross-bedded sandstone samples compared to the Boolean reconstructions given on page 68. The ROS (2) is giving the closest match to the original sandstone

31 xxxvii 5.13 Running average (1 points) of the fractions of particles in suspension over total porosity for the cross-bedded sandstone and its Boolean reconstructions (see Tab on page 68) Images of the pore space of an IOS model. 3D pore space images of the model microstructure are shown at porosities of [a] 1%, [b] 2%, and [c] 3%. 2D slices through the 3D data sets are shown at [d] = 1%, [e] 2%, and [f] 3%. The image size is that of the simulations (12) 3 and the sphere sizes are Slices through the framework two phase mineral model at the resolution used in the elastic simulations (12 3 ). The top row is at a M 1 : M 2 ratio of 1:1, the second row at a ratio of 2:1 and the bottom row at a ratio of 4:1. The volume fractions of the three (M 1 : M 2 : ) phases are [a]: 45:45:1, [b]: 4:4:2, [c]: 3:3:4, [d]: 6:3:1, [e]: 53:27:2, [f]: 4:2:4, [g]: 72:18:1, [h]: 64:16:2, [i]: 48:12: Slices through the interstitial two phase mineral model at the resolution used in the elastic simulations (12 3 ). The top row is at a M 1 : M 2 ratio of 1:1, the second row at a ratio of 2:1 and the bottom row at a ratio of 4:1. The volume fractions of the three (M 1 : M 2 : ) phases are the same as in Fig The Poisson s ratio of the IOS model as a function of porosity. In [a] we show results for the clean dry quartz and feldspar systems. Standard error bars are shown and are of the order of the size of the data point. The experimental data of [54] for clean quartz sandstones is also shown. The fit of Eqn. 6.5 to the data is excellent. The Poisson ratio for mixtures of quartz/clay and quartz/dolomite are shown for [b] 1 : 1 ratio, [c] 2 : 1 ratio, and [d] 4 : 1 ratio. The filled symbols give the prediction of the interstitial model and the open symbols the prediction of the framework model. We show in [b] the characteristic standard errors in the numerical calculation for systems of mixed mineralogy. While they remain small for <.3, they can diverge for some models at higher porosities. The flow diagram still converges to a fixed point at c.5 and seems independent of the cement deposition model chosen. Agreement with empirical Eqn. 6.5 is good for both models at all mineral ratios Comparison of data from [54] and [39] for clay-bearing sands to the prediction of Eqn

32 xxxviii 6.6 Comparison of the conventional model against the empirical nonlinear model based on the V p /V s ratio in terms of fluid saturation predictions. Left: water/oil mixture; right: water/gas mixture Results of the simulation for the variation of the [a] bulk modulus, [b] shear modulus, [c] V p, and [d] V p /V s ratio as a function of porosity for the single phase IOS model with quartz and feldspar as the mineral phase Results of the simulation for the variation of the [a] bulk modulus, [b] shear modulus, [c] V p, and [d] V p /V s ratio as a function of porosity for the single phase IOS model with quartz and feldspar as the mineral phase Comparison of the results of the simulations for dry and watersaturated IOS model to Gassmann prediction based on the dry rock data with their standard errors. In [a] we give the numerical predictions for the dry rock data (squares) and show a best fit to the data points (solid line). We use this fit and Eqn. 5.3 to predict the values of the water-saturated curves (dotted curves). We also show the numerical predictions for the water-saturated results. The fit to the Gassmann s equations is excellent. In [b] we show that the shear modulus is independent of the pore fluid (Eqn. 5.31) Comparison of the results of the simulations for water-saturated sandstone to the dry velocity data of [54] for clean sandstones Comparison of the results of the simulations for water-saturated sandstone to the empirical velocity-porosity equations derived by [54] for clean sandstones. The match is excellent for all Comparison of the results of the simulations for water-saturated sandstone to the empirical equations of [33] and [54]. The numerical data is slightly below the two empirical curves, but exhibits the expected linear relationship between V p and V s. In (b) we show the raw data of [54] for clean sandstones and note that the numerical fit gives the best match to the data Comparison of the simulation results to the range of theories used to predict the moduli of dry porous rocks. [a-b] give predictions for quartz while [c-d] give predictions for feldspar. The theories all overestimate the data for all porosities. The SCA gives the best theoretical fit to the data as expected from [15]

33 xxxix 6.14 Comparison of the simulation data to the SCA theory for Poisson s ratio. While the SCA does give the observed limiting behaviour, ν.2, the prediction of the theory is not generally correct for all Comparison of the simulation results on dry monominerallic sandstone microstructures to the empirical models of 5.27 [73] and 5.25 [11]. The match to the [73] model is superior for the shear modulus. Neither model provides a good match to the bulk modulus data Comparison of the simulation results on dry monominerallic sandstone microstructures to the empirical model of [5] given in Eqn Agreement is excellent Comparison of the simulation results to the range of theories used to predict the moduli of water saturated rock. [a-b] give predictions for quartz while [c-d] give predictions for feldspar. The HS bounds are quite broad for the bulk modulus and neither bound is predictive. The lower bound is zero for the shear modulus. The SCA and DEM both overestimate the data for all porosities. The SCA gives the best theoretical fit to the data as expected from [15] Comparison of the results of the simulations for water-saturated [a] Quartz and [b] Feldspar sands to the empirical equations of Wyllie (Eqn. 5.2) and Raymer (Eqns ). The Raymer Equation is satisfactory except at high and gives a better prediction than the Wyllie equation. The prediction of both empirical models for the model quartzose sands is poorer than for feldspathic sands Comparison of the results of the simulations for water-saturated [a] Quartz and [b] Feldspar sands to the empirical equation of [11]. For the bulk modulus the model accurately describes the quartzose sand but not the feldspathic sands. For the shear modulus the feldspathic sand is best matched Comparison between the prediction of the proposed empirical model, Eqn , the critical porosity model Eqn , and the simulation results for dry [a] bulk modulus, [b] shear modulus, [c] V p, and [d] V p /V s ratio as a function of porosity for the single phase IOS model with quartz and feldspar as the mineral phase. [e-h] give the predictions for the water-saturated case derived from the dry data using Gassmann s relations

34 xl 6.21 Comparison of the prediction of the proposed empirical equations ( ) and the critical porosity model [11] to the experimental data [54] for clean dry quartz sands [a-b] and to the empirical velocity-porosity equations for the clean water-saturated sands [c-d] The effective mineral modulus of the mixtures of quartz and clay or dolomite. The circles are the bulk modulus data, and the squares the shear modulus. The filled symbols represent numerical data for the framework model and the open symbols the interstitial model. The Hashin-Shtrickman bounds are also shown. All the numerical data lies within the bounds Comparison of the results of the simulation for the variation of the bulk modulus, shear modulus, V p, and V p /V s ratio for model cemented sands at 1:1 mineral:mineral ratio. The lines give the prediction of the framework model and the data points the prediction of the interstitial model. The standard error is shown for the data points associated with the interstitial model; for K, µ and V p the standard error is always smaller than the data points. The error for the V p /V s ratio is small for porosities <.2 but begins to increase rapidly at higher porosity. The moduli-porosity relationships are non-linear with a limiting porosity (where modulus ) of.5. The quartz:dolomite systems exhibit little dependence on microstructure. The quartz:clay systems do exhibit structural dependencies. For example, the framework model for quartz:clay exhibits a larger bulk modulus than the interstitial Q:C. The difference, which is reflected in the moduli of the solid phase (see Table 6.3), is likely to be due to the better interconnectivity of the quartz phase for the framework model where quartz has the higher modulus. The enhanced connectivity of quartz is responsible for the interstitial Clay:Quartz system exhibiting higher modulus than the framework C:Q system. Similar results are observed for V p. The V p /V s ratio data, although noisy at higher, seems to be similar for almost all and independent of structure

35 xli 6.24 Comparison of the results of the simulation for the variation of the bulk modulus, shear modulus and V p /V s ratio for the model cemented sands at a 2:1 mineral:mineral ratio. The lines give the prediction of the framework model and the data points the prediction of the interstitial model. As in the 1:1 model (Fig. 6.23), the moduli-porosity relationships are non-linear with a limiting porosity of.5. The quartz:dolomite systems exhibit little dependence on microstructure. The quartz:clay systems again exhibit structural dependencies as discussed in Fig The V p /V s ratio data is again independent of microstructure Comparison of the results of the simulation for the variation of the bulk modulus, shear modulus and V p /V s ratio for the model cemented sands at a 4:1 mineral:mineral ratio. The lines give the prediction of the framework model and the data points the prediction of the interstitial model. There is little effect of the microstructure on bulk and shear modulus. Similar results are seen for V p. The V p /V s ratio data is both independent of and exhibits a fixed point at V p /V s 1.63 for Comparison of the results of the simulation for the variation of the water-saturated bulk modulus, shear modulus and V p /V s ratio for the model cemented sands at a 1:1 mineral:mineral ratio. The lines give the prediction of the framework model and the data points the prediction of the interstitial model. Standard errors are now smaller than the data points for all measures across all. The moduli exhibit in general a non-linear modulus- relationship. The quartz:clay framework model exhibits a larger modulus than the quartz:clay interstitial model, but the differences are not as large as observed in the dry system (Fig a) The V p /V s ratio data seems to be similar for almost all <.2. At higher porosities the curves begin to diverge Comparison of the results of the simulation for the variation of the water saturated bulk modulus, shear modulus and V p /V s ratio for the model cemented sands at a 2:1 mineral:mineral ratio. The lines give the prediction of the framework model and the data points the prediction of the interstitial model Comparison of the results of the simulation for the variation of the water saturated bulk modulus, shear modulus and V p /V s ratio for the model cemented sands at a 4:1 mineral:mineral ratio

36 xlii 6.29 Comparison of the results of the simulations for dry and watersaturated model morphologies at a 2:1 mineral ratio. [a-b] give data for the framework model and [c-d] for the interstitial model. In [a] and [c] dry data is given by open symbols and wet by closed symbols. The solid line give the best fit to the dry data. Gassmann s relation Eqn. 5.3 is used to predict the water-saturated bulk modulus based on the fit to the dry rock data and this result is given by the dashed lines. We see that the numerical predictions for the water-saturated results match the prediction of Gassmann s equations for all. In [b] and [d] we show that the shear modulus is independent of the pore fluid as predicted by Eqn Comparison of the data of [54] to simulation for clean sand and shaley sands with a clay content of X c = 33%. The two simulation results both qualitatively match the observed effect of increased clay content on the elastic properties and quantitatively match the data from [54] for different clay contents Comparison of the data of [39] for high porosity Oseberg sandstone at Clay volume fractions C = 1 25% to simulation for clean sand and shaley sands with a clay solid fraction X c = 33%. The experimental data lies mostly within the bounds of the two simulated results. Previous models gave a reasonable match to the velocities, but a poor prediction of the V p /V s ratio [39] Comparison of the empirical equations of [54], Eqns to simulation data. The simulation data matches very well for all V p data but overestimates the empirical prediction for V s. The numerical prediction for clean sandstone match the empirical results well Comparison of the results of the simulations for water-saturated sandstone to the empirical equations of [33] and [54]. The numerical data for intermediate Quartz:Clay fractions is in excellent agreement with the two empirical curves. All numerical data sets exhibit the expected linear relationship between V p and V s. In (b) we illustrate the linear fit given in Eqns to the numerical data Error in the predictions of the effective mineral moduli for the four models. In [a] and [b] we give the error for the systems where quartz (stiffer) forms the preferred backbone and [c-d] where clay is the more continuous phase. We observe that the SCA and the HS average give the smallest errors in general. The DEM works well only in the weaker system. The VRH estimate is overall the poorest

37 xliii 6.35 HS coefficients as a function of clay content for the bulk (closed symbols) and shear (open symbols) modulus for systems where the weaker clay phase (circles) and the stiffer quartz phase (squares) is the most interconnected phase. Both exhibit a dependency on the clay concentration as was observed experimentally by [163]. Linear best fits to the bulk (dotted line) and shear (dashed line) are also shown Comparison of the simulation results on model dry cemented sands with mixed mineralogy to the empirical models given in Eqns [73, 11] where we use the K, µ derived from the simulation (Table 2) in the equations. The first column gives data for quartz:cement and quartz:dolomite systems at a volume ratio of 1:1. The second column gives data for all sand:cement systems considered at a ratio of 2:1/1:2, and the third column systems at a ratio of 4:1/1:4. [a-c] gives the prediction of the bulk modulus for the framework cement model, [d-f] the bulk modulus for the interstitial model, [g-i] the shear modulus of the framework model and [j-l] the shear modulus for the interstitial model. One observes that in most cases there is a significant curvature in the modulus porosity relationship which is not captured by the linear equation of [11]. Neither the equation of [73] nor [11] fit the bulk modulus data well. The match to the [73] model is superior for the shear modulus and gives a good fit for nearly all systems. Only at the highest clay content does the model of [11] work best. This result is similar to that observed for monominerallic systems Comparison of the simulation results on model cemented sands to the empirical model of [7] given in Eqn Agreement is good and much improved on the estimate of most empirical models where (V p /V s )() = (V p /V s ) mineral. At a 4:1 ratio the limiting value for (V p /V s ) may be greater than 1.63 for the interstitial model. However, the error bars on the data become very large at higher so it is difficult to make quantitative conclusions. Closed symbols are data for the interstitial model and open symbols data for the framework model. There is in general little dependence on model microstructure. 218

38 xliv 6.38 Comparison of the results of the simulations for water-saturated shaley sands to the equations of Wyllie (Eqn. 5.2) and Raymer (Eqns ). The left hand column gives data for the framework model, and the right hand column data for the interstitial model. The mineral ratios are [a-b] 1:1, [c-d] 2:1, and [e-f] 4:1. The Raymer equation gives a better prediction than the Wyllie equation, but in many cases underestimates the value of V p for a given Comparison of the data of [54] to the prediction of [a] Eqn for the shear modulus and [b] Eqn Comparison of the data of [39] to the prediction of [a] the Krief model for the shear modulus and [b] Eqn The fit is good for the shear modulus and satisfactory for the V p /V s ratio. In the latter case the fit is superior to the models of [73, 11] (V p /V s )() = (V p /V s ) mineral and the model of [39] (V p /V s )() Comparison of the empirical fit of [54] for clay cemented sandstones, Equations (lines), to the empirical model derived from numerical data, Equations and Gassmann s relations (data points) A.1 Euclidean distance maps of the particle phase and inverse phase of a two-dimensional IOS model with a particle fraction of =.7. [a] original structure, [b] EDM of particle phase, [c] EDM of inverse phase.238 B.1 Illustration of the influence of rotation on the average of the 1D projections of the ellipsoid ( V 1 ) B.2 Probability density of the discretised spheres used for the reconstruction of the sandstone samples (the radius distribution is not evenly spaced). [a] fb7.5, [b] fb13, [c] fb15, [d] fb22, and [e] cross-bedded sandstone B.3 Radius distribution functions of the discretised spheres used for the reconstruction of the sandstone samples

39 CHAPTER 1 INTRODUCTION Can more oil and gas be recovered from reservoirs? Can one accurately monitor fluid content during reservoir flooding? Can one optimise properties of engineered composites materials? How does one compare and define the structure of complex, disordered materials? These questions are of enormous interest to a number of scientific disciplines and industrial communities. For example, in rock physics one is concerned with deriving the physical properties of interest (porosity, permeability, oil phase saturation) from a number of properties (sonic velocity, electrical self-potential, resistivity, natural γ-ray, γ-γ-response, epithermal and thermal neutron, NMR T 2 -relaxation,...) which can be directly measured. In well logging acoustic logs, electrical logs, formation density logs, and NMR logs, are all related to the single quantity, porosity. Clearly the relationships between these measures depend on environment, rock microstructure, and fluid phase distribution. The correlations used to estimate porosity, permeability, etc. are often based on empirical laws and for a specific environment. In some cases the use of general relationships can lead to a poor interpretation of measurements. There is little scientific basis to support the interpretation of geophysical properties. A major drawback is that there is little work on accurately characterising the structure of sedimentary rocks. In fact, successful answers to all the questions listed above, require an ability to characterise the microstructure of a complex material and the prediction of a number of physical properties based on this characterisation. This in turn will enable one to develop accurate relationships to predict and cross-correlate properties for real materials. The thesis describes research work related to this long-term aim. We consider, in particular, studies in microstructure modelling, descriptions of complex morphology and transport and elastic properties of materials. We also consider specific applications to geophysical systems.

40 2 Introduction 1.1 Microstructure Simplified representations of material structure are still often used. Examples include dilute spheres, cylindrical tubes, or periodic networks. The advantage is that their physical properties can be evaluated directly (i.e. [71, 16]). Most natural materials exhibit a more complex structure however. More realistic descriptions of complex materials are required. A simple model of a disordered granular structure widely used in theoretical and numerical analysis is the identical overlapping sphere (IOS) model [164, 157, 156]. In chapter two we consider three general distinct classes of microstructure, particle based composites, structures generated by level-cuts through Gaussian fields, and models based on a Voronoi tesselation of space. The models are chosen to represent a wide range of composite structure. Particle models with different grain shapes and mixtures of grains are employed. Also introduced are data sets from Xray-CT, comprising Fontainebleau, Berea, and a cross-bedded sandstone, as well as a termite nest. 1.2 Morphology of disordered materials In chapter three we consider ways of defining the structure of complex materials. Previous approaches on characterising morphology were mainly based on statistical concepts employing the n-point statistical correlation functions. In practice commonly only lower order morphological information (volume fraction and two-point correlation function) is used to characterise the morphology of rocks. It is well known, that a match between the two-point correlation function of an experimental data set and a model structure does not ensure a good match between the two systems [2, 1]. Higher order correlation functions can be considered, but their calculation is very time consuming. Other useful two-point characterizations of microstructure include the chord-length distribution function [145, 131] (and the related lineal-path function [171]) and the pore size distribution function [137]. However reconstructions of experimental data sets based on these characterisations have been shown to give a poor representation of the connectivity of the systems [84]. Functions that may provide more complete information about connectivity [62] are unfortunately too complex to incorporate into reconstruction schemes [84]. Here we introduce the Minkowski functionals, a family of statistical measures based on the Euler-Poincaré characteristic of n-dimensional space, as sensitive measures of morphology. Their direct interpretation in terms of n-dimensional intrinsic volumes (volume, surface area, mean curvature, and Euler characteristic) will often allow a better understanding of the structure characterised. The Minkowski functionals are additive measures, embody information from every order of the

41 Physical properties 3 correlation function, and can be calculated with a single sweep of the lattice. In the chapter we discuss four aspects: characterisation, reconstruction, discrimination, and representative measurement of structure. First the Minkowski functionals are used to characterise different classes of structure. Next the analytic Boolean model is employed to derive a reconstruction scheme for Boolean mixtures of particles and the technique applied to sandstone. Third, the Minkowski functionals are extended via concepts of mathematical morphology to sensitively discriminate between different morphologys. Finally the experimental derivation of the Minkowski functionals is discussed in terms of representative sampling, resolution limitations, and signal/noise. 1.3 Physical properties We consider the conductive and elastic properties of disordered materials in chapters four and five. As direct measurement of a 3D structure at resolutions down to a few microns is now readily available from synchrotron and X-ray computed microtomography [38, 147] and laser confocal microscopy [43], one can solve numerically for the effective physical properties of the phase distributions given by the images and compare to theoretical calculations and laboratory measurements. This has been done previously by [147, 139, 9] for transport properties. In all these studies however the numerical prediction of electrical conductivity underestimated the experimental results by 3 1%. In this work accurate calculations of DC conductivity and elastic moduli are achieved by minimising statistical, finite size, and discretisation errors. In chapter five we present for the first time a study of the elastic moduli of sandstone directly calculated from tomographic images. Both the DC conductivity and the elastic moduli are in excellent agreement with experimental data. We consider the predictions derived from tomographic images to more conventional methods. Effective medium approximations are widely used to describe the physical properties of complex media. They are limited by the fact that the microstructure corresponding to a particular model is not precisely known; hence agreement or disagreement with data can neither confirm nor reject a particular model. A more complicated approach is to consider the variational bounds for the properties of random composites [58, 14, 13, 158]. The large differences between upper and lower bounds however severely limit their predictive power. In particular, for porous materials the lower bound for transport and elastic properties vanishes and the upper bound may not provide a good estimate. We compare our numerical calculations to these theories and demonstrate, that all calculations are within

42 4 Introduction bounds. Common effective medium theories (SCA and DEM) however don t give a good agreement with experimental or numerical data. The direct numerical calculation of effective properties from tomographic images or for model structures is still experimentally challenging. In chapter three a reconstruction scheme based on the Boolean model is devised. In chapters four and five we compare the conductivity and elastic moduli of the reconstructed images to the original structures and find excellent agreement for Boolean structures, and good agreement for the sandstones. The reconstructions are derived from a single image at a given porosity and are used to predict the full porosity-property curve. The reconstruction method is superior to common methods in terms of matching the physical properties of the sandstones. 1.4 Geophysical applications We use computer simulation to derive the elastic properties of model clean (monomineralic) consolidated sandstones and cemented consolidated sands composed of two mineral phases. The model morphologies are based on overlapping permeable spherical grains of a mineral phase. For cemented sands we vary the distribution of the second mineral phase and consider two models; a structural or framework cement/sand mix, and a dispersed or interstitial cement/sand mixture. We consider model clay/quartz and dolomitic/quartz cemented sands. The effective solid moduli of the mineral mixtures show a small dependence on microstructure. Results are compared to currently employed theoretical and empirical relationships for deriving moduli-porosity estimates. We note that the Krief empirical relationship is particularly successful at describing the numerical data for dry shear modulus and develop an empirical method ([4]) giving a good prediction of the Poisson s ratio data for dry rock. We find that in all cases Gassmann s relations accurately map between the dry and fluid saturated states. We find that the Raymer equation gives the best modulus porosity model (< 1% error) for the water-saturated cases. Based on these results we propose an accurate empirical method which enables one to derive the full velocity-porosity relationship for clean (monomineralic) and cemented (multimineralic) consolidated sands solely based on the knowledge of the mineral constituents and proportion of each mineral phase present. Using the empirical Krief equation for the shear modulus of a dry sandstone and the empirical equation of [4] for the Poissons ratio of a dry porous rock, the model for the dry

43 Geophysical applications 5 rock is given by µ dry () = µ (1 ) m(), (1.1) ν dry () = a() + (1 (2) 3/2 ) ν, (1.2) where µ and ν are the effective shear modulus and Poisson s ratio of the mixed mineral constituent. µ and ν are derived from an estimate of the mixed solid moduli based on either the average or weighted Hashin-Shtrickman bounds. Gassmann s equations are subsequently used to predict the fluid saturated states. Comparisons of this deterministic empirical method to several experimental studies show that the proposed empirical method gives an excellent match to data.

44 CHAPTER 2 MODEL COMPOSITES AND EXPERIMENTAL DATA In this chapter various models of n-phase composites are introduced and descriptions of a few experimental tomographic data sets are given. These together form the data base for the numerical experiments performed in the following chapters of this thesis. The microstructural models were chosen to represent a wide range of possible composite structures, ranging from particulate media to amorphous structures, and both open and closed cell foams. The experimental tomographic data sets include three different sandstones; Fontainebleau, Berea, and a cross-bedded sandstone sample from South Australia. These three data sets cover a wide variety of sandstone morphologies. Fontainebleau is widely considered the benchmark homogeneous sandstone by researchers studying sedimentary rock. The cross-bedded sandstone is a more heterogeneous sample, but more realistic to applications for the Petroleum industry. Additionally, a termite nest is analysed. The next two sections describe the generation of a set of model composite media and the experimental tomographic images of the sandstones and termite nest. 2.1 Basic models of irregular spatial structures Three basic classes of model composites will be introduced, divided into different classes by the densification process utilised. The first class is generated by a Poisson densification process, where particles or grains of different shape are placed on the lattice in an independent process. A range of grain shapes are considered. Particulate models generated this way are simple and commonly studied in literature. They constitute a good system to test the algorithms applied and developed in this thesis, and they constitute in many cases, including ceramic powders [123], wood composites [162], and sedimentary rock [14, 9], a good model microstructure [158]. A second class is generated by taking level-cuts through Gaussian fields. This class can represent structures developing from spinodal decomposition as well as foams and some rocks. The third class is based on the Voronoi tesselation of a

45 8 Model composites and experimental data distribution of initial particles, which in the easiest case are points distributed by a Poisson process. The initial particles are grown until they are space-filling and the n-th order contacts used in a second densification process, leading to facet like polyhedral structures, as well as networks of cylindrical tubes. The three classes of microstructure span a wide range of composites and form a good basis for analysing general composite material behaviour and detailed below Particle based models Various problems in science or industry lead naturally to particle or grain based models of disordered materials. The Boolean model, which is generated by the gradual build up of a phase via the overlap of permeable grains, although idealised, is often used to describe the morphology of complex materials [158]; examples in the literature include ceramic powders [123], wood composites [162], paper [7], sedimentary rock [14, 9] and hydrating cement-based materials [44]. A prominent example of a Boolean model often used in theoretical and numerical analysis is the identical overlapping sphere (IOS) model [164, 157, 156]. In this thesis a range of Boolean models with different grain shapes and mixtures of grains are employed. The grain shapes considered include ellipses and rectangles in two dimensions and spheroids and rectangular bars in three dimensions. In the following, three cartesian coordinate systems are introduced. One is the coordinate system of the particle Σ p = {O p, x p, y p, z p }, where the principal axes of the grain coincide with the axes x p, y p, z p of the coordinate system and the origin O p is the center of gravity of the particle. The second is a local coordinate system Σ l = {O l, x l, y l, z l }, having the same origin (O p = O l ), but being rotated against Σ p by an angle α around a rotational axis g through the origin. Here α and g are chosen, such that Σ l can be transformed to the (third) lattice coordinate system Σ = {O, x, y, z} by a simple translation (x l x, y l y, z l z). In two dimensions the rotation around the rotational axis g reduces to a simple rotation around the origin of Σ p. Thus apart from its particular shape and probability to occur, any grain is characterised by orientation and position of its local coordinate system Σ p to the lattice coordinate system Σ, which are chosen randomly. By restricting the choices of α (and g in 3D), anisotropic Boolean grain models are generated. All particle models are generated with periodic boundary conditions. The process and choices of α are first illustrated in two dimensions (Fig. 2.1). For α uniformly distributed on [; 2π) the resulting structures are isotropic (Fig. 2.1.a). By only allowing α {; π 2 }1 structures result, where all grains are 1 Only two angles need to be considered because of the symmetry of the particles used.

46 Basic models of irregular spatial structures 9 [a] [b] [c] [d] [e] [f] [g] [h] [i] Figure 2.1: Illustration of the choices of rotation of particles and the resulting isotropic and anisotropic composites. [a] Randomly oriented ellipses. [b-c] Lattice oriented ellipses and rectangles. [d] Partially and [e] fully aligned ellipses. [f] Mixture of grains with different shapes and rotational freedoms. [g] Randomly oriented and [h] partially (p a = 1 ) as well as [i] fully aligned ellipses with a range of half-axes. 2 aligned to lattice vectors. If all rotation angles have the same probability to occur, measurements made along the coordinate axes will still show an isotropic system (Fig. 2.1.[b-c]). These systems are termed (lattice) oriented and are particularly important for rectangular particles as there are no discretisation effects, as long as integer values for the sidelength of the rectangles (in lattice units) are used. Yet another choice is to restrict α to a smaller interval α [ (1 p a ) π; (1 p 2 a) π ], with 2 p a [; 1]. In this case anisotropic systems with partially aligned particles result, which become fully aligned for the choice p a = 1 (Fig. 2.1.[d-e]). By applying these choices for each particle separately a range of complex mixtures of grains can be generated (Fig. 2.1.f). Mixtures of particles with different aspect ratios may be defined by specifying size and valid rotation angles for each constituing grain (Fig. 2.1.f) or by allowing a range of uniformly distributed half-axes a, b [h min ; h max ] (Fig. 2.1.[g-i]).

47 1 Model composites and experimental data In three dimensions there are three rotational degrees of freedom. In the following [ π 2 ; π 2 ] notes the angle of the rotational axis g with the x p, z p -plane, θ [ π; π] is the angle between g and z 2 2 p, and α [ π; π ] the angle of rotation 2 2 around g. The transformation from Σ p to Σ l can best be written using a rotation matrix which has to be calculated once for each particle. It may be written as [26] l 1 m 1 n 1 [T R ] rot = [T R ] α x l [T R ] y l [T R ] θ z l [T R ] y l [T R ] α x l = l 2 m 2 n 2, (2.1) l 3 m 3 n 3 with the rotational matrices [T R ] α x l = [T R ] y l = [T R ] θ z l = 1 cos(α) sin(α) sin(α) cos(α) cos() sin() 1, sin() cos() cos(θ) sin(θ) sin(θ) cos(θ), 1, (2.2) thus giving the transformed coordinates as x p = l 1 x l + m 1 y l + n 1 z l, (2.3) y p = l 2 x l + m 2 y l + n 2 z l, z p = l 3 x l + m 3 y l + n 3 z l. The grains are discretised in the transformed coordinate system, as the rotation of discretised particles is less accurate. By restricting the choices of α,, and θ anisotropic structures again can be generated. Lattice oriented particles result by only considering rotations leading to Σ p such that x p x x p y x p z. Partial alignment is reached by requiring α,, θ [ (1 p a ) π; (1 p 2 a) π], where p 2 a [; 1]; analog to the 2D case only one parameter is used to restrict the range of allowed angles. The extremes are randomly oriented particles (p a = ) with α,, θ uniformly distributed on [ π; π] and fully aligned ones (p 2 2 a = 1) having α = = θ =. A set of models generated this way is shown in Fig As in two dimensions, by considering multiple independent Poisson processes running parallel and placing different kinds of particles, complex mixtures of grains can be generated. In the implementation of the program we take a vector of particles, each particle being characterised by its size, shape, rotational freedom, and

48 Basic models of irregular spatial structures 11 [a] [b] [c] [d] Figure 2.2: The interface of models of Poisson distributed particles. Top: [a] oriented cubes of sidelength λ = 8 and [b] overlapping spheres of radius r = 8. Bottom: overlapping spheroids with an half-axis range of r = 4..2; [c] randomly oriented, [d] fully aligned. The volume fraction of the particle phase is =.2. All models are discretised on a 2 3 lattice. probability to be placed (resulting in its Poisson density). Thus particles of different kind can be intermixed easily (see Fig. 2.3), which is used in chapter 3 to generate equivalent Boolean models of structures. For many important porous media a single solid granular phase is not realistic. Examples include sintered ceramics and as an example of geophysical materials shaly sands. Here shaly notes a solid phase given by a mixture of clay and quartz. One can choose a number of ways to generate n-phase systems, depending on the system to model and the distribution of the solids (or fluids). Often used are grain packing algorithms based on an initial hard particle mix with additional cementation or densification methods. The best example is the grain consolidation model, which has been often used to describe the morphology of sedimentary rock [13, 142]. Alternatively one can try to mimic the complex diagenetic process following the primary grain sedimentation [113, 23, 112]. This approach includes various consolidation mechanisms with several fitting parameters [155]. A more simplistic but

49 12 Model composites and experimental data [a] [b] [c] [d] [e] [f] Figure 2.3: The interface of mixtures of oriented rectangular blocks on a 1 3 lattice [a] plates, [b] sticks, [c] cubes and sticks, [d] cubes and plates, [e] cubes and cubes, [f] cubes and cubes. All models are shown at a total particle fraction of =.2. For the exact parameters of the model see Tab. 3.4.

50 Basic models of irregular spatial structures 13 [a] [b] [c] [d] [e] [f] [g] [h] [i] Figure 2.4: Slices through three 3D models of porous media with two solid phases. Top: IOS on IOS, middle: two IOS phases distributed at the same time, bottom: parallel surfaces on IOS. The ratio of the two solid phases is held constant at 2:1. Fractions of the phases are left: 6:3:1, middle: 53:27:2, right: 4:2:4. The models are discretised on a 12 3 lattice. useful model is to consider only a few idealised shale distributions. Keeping the complexity of n-phase materials to a minimum, two different ways of generating n-phase materials are considered. The first way is to extend the granular approach by placing particles of different phase. One can sequentially place each phase (Fig. 2.4.[a-c]), or place each grain (of arbitrary phase) sequentially according to its Poisson density (Fig. 2.4.[d-f]). The second approach employs a simple geometrical overgrowth algorithm based on parallel surfaces. 2 Particles can be covered with a parallel layer with the thickness determined by the volume fraction desired. In this thesis structures with up to three phases are considered, in particular the models shown in Fig. 2.4 are used to simulate some possible simple quartz/clay distributions. These simple models may be thought of as representing some of the salient features of dispersed clay, structural clay, or clay overgrowth. 2 Parallel surfaces are implemented using the Euclidean distance transformation (EDT, see appendix A). A formal mathematical treatment of parallel surfaces is provided in chapter 3.3.

51 14 Model composites and experimental data Gaussian models A class of material that is not in general well described by particulate models is that of amorphous composites. Model random materials have been described by level cuts of a superposition of random plane waves; the levelled-wave model [18, 152]. Originally developed to describe the morphologies associated with spinodal decomposition [31], and later to describe the structure of bicontinous microemulsions [17], the levelled wave model accounts for many features observed in real disordered materials [18] including polymer blends [72] and foams [124]. In the original scheme due to Cahn [31] one associates an interface between two distinct phases (e.g., pore/matrix) with a level set (or contour) of a random standing wave y(r), composed of N sinusoids with fixed wavelength λ = 2π/k but random directions k n, phase constants Φ n and amplitudes A n, y(r) = 1 A n cos(k k n r + Φ i ). (2.4) N n As y(r) is as positive as often as it is negative, a 5/5 (isometric) blend coincides with the zero set of y(r). If a distribution of wavelengths is allowed, the function y(r) is just a Gaussian random field (GRF). For an isotropic GRF using the fieldfield correlation function g(r) and spectral density ρ(k) this can be written as [12] g(r ij ) = y(r i )y(r j ) = 4πk 2 ρ(k) sin(k r i r j ), (2.5) k r i r j where ρ(k) is chosen so that g() = 1. The function g(r) further has the property lim r g(r) [12]. Now g(r) can be specified either in Fourier space by choosing a kernel ρ(k) or directly by rewriting Eqn. (2.4) for an isotropic GRF as y(r) = 2 N N cos(k i ˆk i r + Φ i ), (2.6) i=1 where Φ i is a uniform deviate on [, 2π) and ˆk i is uniformly distributed on a unit sphere. The magnitudes of the wave vectors k i are distributed on [, ) with a spectral density ρ(k) related to g(r) by a Fourier transform. This method is most useful when spikes in the spectral density have to be resolved [129]. In contrast the Fourier summation technique is applicable if periodic structures are desired, and is used throughout this thesis. Then the field-field correlation function of a T-periodic GRF with a maximum wavenumber K = 2πN/T is given as N N N y K (r) = c lmn e iklmn r, l= N m= N n= N k lmn = 2π (li + mj + nk). (2.7) T

52 Basic models of irregular spatial structures 15 The complex coefficients c lmn are determined by a set of physical and statistical conditions on g(r) and y K (r). Requiring y to be real gives c l,m,n = c l, m, n, and y = results in c =. Discretization requires c lmn = for k lmn = k lmn K. To generate Gaussian statistics the coefficients c lmn are taken as random independent variables with Gaussian distributions such that R(c lmn ) = I(c lmn ) = as well as (R(c lmn )) 2 = (I(c lmn )) 2 = 1ρ ( 2π ) 3. 2 Kk lmn T In the following chapters Gaussian models with three different kernels are used, characterised by their Fourier-transform pairs [153, 91, 125, 131]. The first model is parameterised by a correlation length ξ, domain scale d and cut-off scale r c, and is the standard model adapted; g(r) = e r/ξ (r c /ξ)e r/rc sin2πr/d 1 (r c /ξ) 2πr/d, (2.8) π 2 (ξ r c ) 1 ξ 4 d 4 ρ(k) = [d 2 + ξ 2 (kd 2π) 2 ][d 2 + ξ 2 (kd + 2π) 2 ] (2.9) π 2 (ξ r c ) 1 r 4 cd 4 [d 2 + r 2 c(kd 2π) 2 ][d 2 + r 2 c(kd + 2π) 2 ]. If no specific comment about the kernel of the model is made, this is the model used. The other Gaussian kernels are only employed once each and introduced when applied. The resultant morphologies are characterised by an undulating interface of consistent curvature and exhibits two similar phase structures (see Fig. 2.5.[a-b]). Cahn s approach was extended [17] to a pair of cuts through the same random wave at levels α, β, α y(r) < β. The volume between the pair of interfaces generated by the two cuts is considered to be one phase, while the two regions contiguous to this (y(r) < α; y(r) > β) are defined as the second phase. The 1-levelcut can be recovered by setting β =. In the current algorithmic implementation the macroscopic volume fractions of the two phases are specified, and the positions of the level cuts found numerical by a bisection algorithm on the digitised structure. Mathematical structures so defined exhibit a wide range of morphologies. symmetric 2-level cut model α = β exhibits a ribbon- or sheet-like structure, and is characterised by a high degree of interconnectivity even at low volume fractions (see Fig. 2.5.[c-d]). The One may choose any number of β and α for a given phase fraction. The freedom in choosing the position of the level cuts (for a chosen volume fraction) allows one to model an even larger variety of microstructures [125]. More general models can also be developed based on this approach. Intersection and union sets of any number of fields can be generated [121, 129] or additional level cuts γ, δ,... introduced to generate n-phase composites. Realisations of three Gaussian models, a 1-level cut, 2-level cut, and an intersection of two 2-level cuts, are shown in Fig. 2.5.

53 16 Model composites and experimental data [a] [b] [c] [d] [e] Figure 2.5: The interface of Gaussian models of periodicity t = 1 for volume fractions of =.25 and =.75 from left to right. Top: 1-level-cut. Middle: 2-level-cut. Bottom: intersection of two 2-level-cuts. The models are discretised on a 2 3 lattice. [f]

54 Basic models of irregular spatial structures 17 The statistics of the material are completely determined by the specification of the level-cut parameters and the function g(r). Defining phase 1 as the region α y(r) β and phase 2 as the remainder, the volume fraction of phase 1 is given by p = p β p α, where p γ = 1 2π γ e t2 /2 dt, γ = α, β. (2.1) Berk showed that the two-point correlation function is given by p (2) (r) = p g(r) dt α2 [exp( 2π 1 t t ) 2exp( α2 2αβ + β 2 ). + exp( β2 )] (2.11) 2(1 t 2 ) 1 + t With g(r) defined by Eqn. (2.8) the specific surface S v = 4 p (2) ()/ r reads 2 S v = p 2 1 4π + 2π (e 2 α2 + e 1 2 β2 ) 2 6d r c ɛ. (2.12) A concise way of expressing the level cut field [131] is to define a variable c where p α = c 2 c 2 p, p β = c 2 + (1 c )p. (2.13) 2 Setting c = and c = 1 corresponds to the one- and two-level cut models. Other models considered can be formed from intersection and union sets of n primary GRF models [131] Voronoi models The third class of model morphologies, which include random cellular solids [5] and foam-like structures [15, 129], are constructed using a Voronoi tesselation to propagate the labels of initial kernels to space covering cells. The edges or surfaces of these cells are in a second step used as the initial structure, on which parallel layers are grown to vary phase fractions. In the following the process is explained in detail. A simple set of initial grains or kernels are Poisson distributed points of a given density. Between each pair of this set of points the bisecting planes are constructed. The tiles formed by the intersections of the bisecting planes between a given Poisson point and all of its neighbours generate convex polyhedra. Within each polyhedra every point is closer to the given Poisson point than to any other and the polyhedra are space filling. This is called the Voronoi tesselation of space. Computationally this procedure is implemented by calculating the Euclidean distance transform (EDT, see appendix A) of the Poisson points, while at the same

55 18 Model composites and experimental data [a] [b] [c] [d] [e] [f] [g] [h] Figure 2.6: Calculation of the discrete Voronoi tesselation of 5 Poisson distributed points (pixels) on a 2 2 lattice. Each point [a] is grown concentrically [b-g] until all space is covered [h]. The different labels of each grain are visualised by different colours. Only background pixels (white) can be updated with a new label. time advancing a different label for each kernel with it. In this particular case it amounts to a concentrically growing of the initial points until all space is covered. Only voxels of the background phase can be updated with a new label in each growth step. A 2D visualisation of the advancing labels for Poisson distributed points is given in Fig The resultant structure is the Voronoi diagram of the initial set of points. On the voxelated image the boundaries of the polyhedra are found by searching for all voxels having a neighbouring voxel with a different label. In two dimensions the set of these boundary voxels represents a Voronoi network (Fig. 2.7.a). In three dimensions choosing all boundary voxels results in a structure similar to that of a closed-cell foam Fig. 2.7.b). Limiting the choice to edges of polyhedra, which are intersections of at least three polyhedra walls, gives the 3D Vornoi network (Fig. 2.7.c). 3 The morphology in Fig. 2.7.[b-c] are reminiscent of open and closed cell foams. In three dimensions the choice of the boundary set (polyhedral walls or the Voronoi network) decides, whether the model is approximating a closed cell foam, or a network of open-foam-like connected cylinders. The network has by construction no dangling ends. At fractions below ca. 1% this may be a good model of foams (see i.e. figures 2 & 4 in [15]) and of aerogels [129]. By thickening the boundary sets defined in Fig. 2.7, a large range of phase fractions can be probed (see Fig. 2.8). Apart from phase fractions the only parameter of the model is the density of initial 3 The definition of the contact faces of different polyhedras is not exact on a discretised image. It can be made slightly more precise by applying the test for different labels to the vertices of the lattice.

56 Basic models of irregular spatial structures 19 [a] [b] [c] Figure 2.7: Visualisation of the set of polyhedra boundary voxels (see text) resulting from the Voronoi tesselation. [a] and [c] represent a Voronoi network of the set of initial Poisson points in two and three dimensions. [b] shows the initial interface build by the union of the surfaces of all Voronoi cells. [a] [b] [c] [d] Figure 2.8: The interface of Voronoi models with 1 seeds on a 2 3 lattice. First row: facet models for volume fractions of [a] =.26 and [b] =.74. Second row: edge models for volume fractions of [c] =.6 and [d] =.5.

57 2 Model composites and experimental data [a] [b] [c] [d] Figure 2.9: Generalisation of the Voronoi models using correlated points. [a] Poisson distributed points; [b] a normalised Gaussian field, [c] the Euclidean distance from the centre of the structure, and [d] the function sin(x) 2 is used to transform the uniformly distributed points. All structures are generated using 2 seeds. Poisson distributed points (seeds). This controls the domain size and the resolution of the discretised sample. The models are generated with periodic boundaries throughout this thesis. This ensures, that the statistical properties of boundary cells are the same as of those in the interior. In later chapters of this thesis only Voronoi models based on Poisson distributed points are employed. However, more general models can be introduced simply by considering as initial seeds correlated points, i.e. requiring a minimum distance between the seeds or use non-uniform distributions of points (see Fig. 2.9). 2.2 Experimental data In this thesis a number of morphologies derived from experimental images are considered. Direct measurement of a 3D structure is now available via micro X-ray computed microtomography [38, 147, 13]. These techniques provide the opportunity to experimentally measure the complex morphology of a range of materials in three dimensions at resolutions down to 5 µm and lower. Using this technology data sets for three types of rock, Fontainebleau sandstone, Berea sandstone and a crossbedded sandstone were acquired. Additionally a termite nest was available. X-ray

58 Experimental data 21 computed tomographic images of porous media are grey scale images, usually with a bimodal population apparent, one mode corresponding to the signal from the void space, the second to the signal from the grain space. A simple thresholding based on matching a predetermined bulk measurement (phase fraction) is often used to segment a tomographic image. In practice this may not be a reasonable method due to the peak overlap in the intensity histogram or to uncertainty about the agreement between a bulk measurement and a measurement on a smaller subvolume. For these reasons, each grey scale image was thresholded using a kriging-based thresholding method [111] to give a binary pore-solid image [8, 81]. The images are also corrected for noise by re-identifying the phase type of all isolated grain and void voxel clusters. These isolated voxels would otherwise have a strong effect on many morphological measures Fontainebleau sandstone The tomographic data of the Fontainebleau sandstone was obtained from four 4.52 mm diameter cylindrical core samples with bulk porosities = 7.5%, 13%, 15%, and 22%. In the following they are sometines named fb, i.e. fb13 for the 13% sample of Fontainebleau sandstone. A 2.91 mm length of each core was imaged [4, 38, 147]. The reconstructed images have a resolution of 5.68 µm resulting in imaged sections. Due to limitations in computational hardware at the time the image was initial processed and segmented [79, 8] the original image was divided into 4 subsamples of size by Lindquist, overlapping by 5 pixels on the inner sides. The kriging was carried out on these subsamples, which then were recombined to a binary image (see Fig. 2.1). Finally from each of the recombined binarised cylindrical plugs a (centered) 48 3 cubic subset was extracted for analysis corresponding to a volume of 2.3 mm 3. From each of these final 48 3 samples a central 12 3 subsample is illustrated in Fig The samples show the variability observed in the Fontainebleau cores. Moreover, the porous fraction of the central 12 3 section of the core fb15 is 19.6%, which does not match the macroscopic porosity (Fig [c]), and is higher than 19.% for the central section of the fb22 sample (Fig [d]). This illustrates a possible strong variability in the porosity observed at a local scale. Fig illustrates the porosity variation for per slice along the x,y,z directions of the 48 3 samples as well as the 2- point void-void correlation functions p (2) (r). Each of the slices represents a volume of.42 mm 3. Fontainebleau is considered very homogeneous, and considered a benchmark of a homogeneous rock in the petroleum industry, but the plot of the porosity variations shows considerable variability on the local grain scale. A statistical summary of average and standard deviations is compiled in table 2.1.

59 22 Model composites and experimental data [a] [b] Figure 2.1: Right: Compilation of the four 45 2 slices. Left: Final first 48 2 slice of fb15, after removing overlap, and recombination. [a] [b] [c] [d] Figure 2.11: Pore space images of a central 12 3 piece of each of the four 48 3 samples, each taken from the centre. From top left to bottom right: fb7.5 ( = 7.4%), fb13 ( = 12.6%), fb15 ( = 19.6%), fb22 ( = 19.%). Here notes the porosity of the 12 3 subsamples.

60 Experimental data 23 The standard deviation σ( slice ) of the data increases with fraction. The normalised deviation as one possible measure of heterogeneity σ( slice )/ slice shows fb22 as the most homogeneous, and fb7.5 as the most heterogeneous sample..3 fb7.5 fb13 fb15 fb22.3 fb7.5 fb13 fb15 fb slice slice.1.1 [a] slice x [mm] fb7.5 fb13 fb15 fb22 [b] p (2) (r) y [mm] fb7.5 fb13 fb15 fb22.5 [c] z [mm] [d] r [mm] Figure 2.12: Variations in porosity and 2-point correlation functions of the 4 Fontainebleau sandstone samples (48 3 subvolumes). [a-c]: porosity variation along the x,y,z axes of the core plugs. [d]: 2-point correlation functions. Table 2.1: Volume fractions and variability of the Fontainebleau samples. Given are the nominal porosity plug of the core plugs, from which the samples were extracted, the porosities 48 of the 48 3 samples used in this thesis, as well as the absolute and relative standard deviation of the slice porosity sl along the coordinate axes of the samples extracted from the core plugs (z cylindrical axis). σ x( sl ) σ y( sl ) σ z( sl ) sl Sample plug 48 σ x ( sl ) sl σ y ( sl ) sl σ z ( sl ) fb % %.91 11% % fb13 13% % % % fb15 15% % % % fb22 22% % % %

61 24 Model composites and experimental data [a] [b] [c] [d] [e] Figure 2.13: Pore space images of the Berea sandstone sample. Top: [a] the first, [b] middle, and [c] last slice of the 216 slices from which the full analysed section of the core plug was selected. The bulk porosity of the analysed section is 12.1%. Bottom: Two of its eighteen disjoint 18 3 subpieces. Units are in voxels (resolution 1 µm) Berea sandstone A cylindrical core plug of Berea sandstone 5 mm in diameter and 2.4 mm in length was imaged using Xray-CT imaging at 1 µm resolution. 4 Of the image acquired a subvolume representing a volume of 22.7 mm 3 was selected for further analysis (see Fig. 2.13). The bulk porosity of this subvolume is 12.1%. In the petroleum industry Berea sandstone is also considered as a benchmark homogeneous sandstone. On the imaging scale used it is not homogeneous as shown in Fig Characterising the heterogeneity in the same way as for the Fontainbleau samples, one gets for the the standard deviation σ( slice ) =.14, or normalised σ( slice )/ slice = 11.3%. This is despite the fact, that the sampling volume per slice for the Berea sample is.15 mm 3 compared to.42 mm 3 for the Fontainebleau. Thus the Berea sandstone seems significantly more heterogeneous 4 Raw data from the University of Texas.

62 Experimental data 25 than the Fontainebleau sandstone (see table 2.1) berea, 324x324x216 berea Slice.12 p (2) (r) [a] z [mm] [b] r [mm] Figure 2.14: Left: porosity variation along the cylindrical axis of the Berea sandstone core plug for the extracted subvolume. Right: 2-point correlation function of the former Cross-bedded sandstone A image of a cross-bedded sandstone was obtained at 1 µm resolution via micro-ct imaging. 5 The image was converted to a binary image in the same way as the other rock samples. Of the cylindrical sample a subset was selected, having a bulk porosity of 13.7% and representing a volume of 54 mm 3. For further analysis this volume was subsampled into 2 disjoint cubes of size 3 3, 16 disjoint cubes of size 15 3, 54 disjoint cubes of size 1 3, and 25 disjoint cubes of size 6 3. Examples of the subvolumes are shown in Fig Cross-bedding geologically implies a higher energy depositional environment and therefore stronger heterogeneity of the microstructure. This behaviour is observed in Fig In particular, strong heterogeneity perpendicular to the primary bedding plane can be seen. The porosity per slice over the cylindrical axis ranges between 5% and 21% for the cross-bedded as compared to between 9% and 16% for the homogeneous industry standard Berea sandstone. The volume per slice is.9 mm 3 and is comparable to the volume per slice of the Berea sandstone sample given above. The standard deviations of the slice fractions along the cylindrical axis of the core plug are given by σ z ( slice ) =.28, or normalised σ z ( slice )/ slice = 2.4%. A more detailed statistical summary analog to the Fontainebleau samples, but for the 2 touching 3 3 subvolumes of the cross-bedded sandstone, is given in Tab Cleary the sample is the most heterogeneous sandstone sample in this study. 5 Raw data from the University of Texas.

63 26 Model composites and experimental data [a] [b] [c] [d] [e] Figure 2.15: The interface of sections of the sized segmented Xray- CT scan of a cross-bedded sandstone. [a]: 15 3 (112), [b]: 153 (213), [c]: 13 (214), [d]: 1 3 (335), [e]: 63 (211), [f]: 63 (221). The subscripts note the position in the sandstone sample matrix. Top: 2x2x4, middle: 3x3x6, bottom: 5x5x1. Samples [e,f] are disjoint (non-overlapping) touching neighbours. [f]

64 Experimental data , z = , z = 31 6 slice slice , z = , z = [c] x [mm] [c] y [mm].2 xbed, 3x3x6.15 xbed, 3 3, #1 xbed, 3 3, #2 Slice.15 p (2) (r) [c] z [mm] [d] r [mm] Figure 2.16: Variations in porosity and 2-point correlation functions of the crossbedded sandstone data. [a-c]: porosity variation along the x,y,z axes for the extracted subvolume. The cylindrical axis of the core plug coincides with the z-direction and the volume is subdivided into 2 samples of size 3 3 for the x,y directions to compare the same slice volumes. [d]: 2-point correlation functions of the two 3 3 subsamples. Table 2.2: Volume fractions and variability of the two cross-bedded sandstone 3 3 subsamples. Given are the porosities 3 of the 3 3 samples used in this thesis, as well as the absolute and relative standard deviation of the slice porosity sl along the coordinate axes of the samples extracted from the core plugs (z cylindrical axis). σ x( sl ) σ y( sl ) σ z( sl ) sl Sample 3 σ x ( sl ) sl σ y ( sl ) sl σ z ( sl ) # % % % # % % %

65 28 Model composites and experimental data Termite nest The termite nest is an entirely different experimental data set compared to the rock samples introduced above. Here the process of generating a porous sample is merely a biological one 6 as compared to primarily physical and chemical processes (diagenesis) of the samples above. The sample was acquired using X-ray micro-ct imaging at a resolution of 1 mm and has a size of mm 3 or 14 cm 3. The segmented image had a porosity of 46%. No independent porosity measure of the sample was available. Fig illustrates the resultant binary image of the termite nest. Compared to the size of the features the sample is rather small. This Figure 2.17: The binary interface of the termite nest after segmentation at a resolution of 1 mm. The total image size is A high resolution colour image can be seen at sjr. can be deduced both from Fig a by simple observation and from Fig b by looking at p (2) (r) for the largest radius r obtainable from the image. The 2-point void-void correlation function does neither cross nor smoothly approach its value for r, given by p (2) (r ) = 2 =.212. Thus quantitative results derived from this image are questionable and a larger sample would be needed..7.5 termite nest termite nest, 65x65x Slice.5 p (2) (r).3.4 [a] z [mm] [b] r [mm] Figure 2.18: Left: porosity variation along the z-direction of the termite nest. Right: 2-point correlation function of the sample. 6 A termite nest is generated by the solidification of the excrements of the termites.

66 CHAPTER 3 MORPHOLOGY OF DIGITISED DISORDERED MATERIALS The structure of a disordered material an oil bearing rock, a piece of paper, or a polymer composite is a remarkably incoherent concept. Despite this, scientists and engineers are asked to predict the properties of a disordered material based on the structure of its constituent components. A major shortcoming in the understanding of processes involving complex materials has been an inability to accurately characterise microstructure. The specification of structure requires topological as well as geometric descriptors to characterise the connectivity and the shape of the spatial configuration. In oil recovery from petroleum reservoir rocks, for example, recovery depends crucially on the topology of the pore space and on the mean curvature of the surfaces where immiscible phases meet at a contact angle. To determine accurate flow models and to devise intelligent recovery strategies requires an accurate characterization of reservoir rocks in terms of topology and geometry. To date, the toolkit used to quantify complex structures has been primarily that of the statistical physicist. Complete characterization of the effective morphology however requires knowledge of an infinite set of n-point statistical correlation functions. In practice only lower order morphological information is available; common methods [69, 65, 78] are based on matching the first two moments (volume fraction and two-point correlation function) of the binary phase function to a random model. It is widely recognised that although the two-point correlation function of a reference and a reconstructed system is in good agreement, this does not ensure that the structures of the two systems will match well and attempts to reconstruct materials from experimentally measured two-point information have not been very successful [2, 1]. Two point information is non-unique and does not capture many important features of the microstructure. Other useful two-point characterizations of microstructure include the chordlength distribution function [145, 131] (and the related lineal-path function [171]) and the pore size distribution function [137]. However reconstructions of experimental data sets based on these characterisations have been shown to give a poor representation of the connectivity of the systems [84]. Functions that may provide

67 3 Morphology of digitised disordered materials more complete information about connectivity [62] are unfortunately too complex to incorporate into reconstruction schemes [84, 159]. Incorporation of three- and four-point information may lead to a better estimation of structure, but their measurement is very complex and therefore including them in a reconstruction algorithm makes the reconstruction computationally very expensive and the implementation of such an algorithm a formidable task. There is a need for morphological measures which include higher-order correlations, but are fast and reliable for characterizing the morphology of a structure. Statistical measures which are sensitive to the morphology of structures have been extensively investigated in other fields such as image analysis and pattern recognition [145, 132, 136]. Integral geometry [136] is often used and provides alternative methods and tools for measuring spatial structure. A family of measures, the Minkowski functionals (MFs) in particular seem to be promising measures for describing the morphology of complex materials. These measures embody information from every order of the correlation functions, are numerically robust even for small samples, are independent of statistical assumptions on the distribution of phases, and yield global as well as local morphological information. The MFs are additive measures allowing one to calculate these measures effectively by simply summing over local contributions [98, 1]. The MFs characterise not only the connectivity, but the shape and content of spatial figures. In three dimensions the functionals V i are related to the familiar measures of volume fraction, surface area S, integral mean curvature H, and Euler-Poincaré characteristic X V = V, V 1 = S/6, V 2 = H/3π, V 3 = X. (3.1) The morphological measures are useful order parameters to describe spatial patterns quantitatively and provide for a comparison between experiment and theory. In particular, MFs have been used to distinguish different classes of disordered media [7], to quantitatively resolve differences between the turbulent and regular Turing patterns from chemical reaction-diffusion systems [97] and to show that the hole distribution in thin films are inconsistent with the concept of spinodal decomposition, but consistent with a nucleation scenario [66]. MFs have also been used to discriminate between different cosmological models of the early universe [11]. In this chapter we consider these integral geometric measures in the context of disordered materials. In the first section we review concepts in mathematical morphology and the Minkowski measures are formally introduced as additive statistical measures of the morphology of disordered structures. We show that in the context of Boolean models, systems generated by the gradual build up of a phase via the overlap of permeable grains, integral geometry provides powerful formulae.

68 Minkowski functionals 31 Analytic results for these Boolean grain models are given and the computational algorithm verified. The evolution of the measures with density for the range of disordered microstructure models defined in chapter 2 is computed. The morphological parameters for the cross-bedded sandstone are compared to three common stochastic model systems of porous media. In the second section it is shown that for any Boolean system, one can, from a single image at any particle fraction, define a set of morphological parameters which allows one to accurately reconstruct the medium for all other phase fractions. The percolation thresholds of either phase are obtained with good accuracy. We use this method to reconstruct the morphology of the sandstone samples defined in chapter 2. In the third section parallel surfaces are formally introduced and a characterisation method involving the Minkowski functionals combined with parallel surfaces is demonstrated to be a very powerful discriminator of microstructure. The method is used to determine the accuracy of model reconstructions of random systems and experimental data sets for the termite nest and Fontainebleau sandstone samples. In the final section of this chapter we consider the effects of distortions (drift, noise, and blurring) on the morphological properties of complex random models. Images obtained from the experimental techniques described in chapter 2 include distortions due to diffraction, absorption, and sample drift. The question is asked, how critically these distortions effect image quality. Defining a length scale based on the 2-point correlation function, we consider how distortion at different scales can lead to quantitative errors in morphological measures. 3.1 Minkowski functionals In this section a family of statistical measures based on the Euler-Poincaré characteristic of n-dimensional space, which are sensitive to the morphology of disordered structures, is introduced. These measures embody information from every order of the correlation function but can be calculated simply by summing over local contributions. The evolution of the measures with density is computed for a range of disordered microstructural models; Boolean grain models, amorphous microstructures and cellular and foam-like structures. Analytic results for the Boolean grain models are given and the computational algorithm verified. Computational results for the different microstructures exhibit a range of qualitative behaviour. A length scale is derived based on the two-point correlation functions to allow qualitative comparison between the different structures. We compute the morphological parameters for the experimental microstructure of a sandstone sample and compare to three common stochastic model systems for porous media. None of the statistical

69 32 Morphology of digitised disordered materials models are able to accurately reproduce the morphology of the sandstone Mathematical background In this study we consider the Minkowski functionals of digitised representations of complex media at various volume fractions. We consider a two-component medium filling a cubic volume V = L d. A digitised set Q = i Q i of either component can be described by a collection of voxels Q i or compact (closed and bounded) convex sets. In order to characterise Q in a morphological way we describe some basic facts from combinatorial integral geometry [52, 1]. The convex ring R denotes the class of all subsets A of the Euclidean space IR d, which can be represented in terms of a finite union of bounded closed convex sets. Digitised spatial configurations Q are members of A and belong to the convex ring R. The Euler characteristic χ is introduced as an additive functional over R, so that for A, B R, and χ(a B) = χ(a) + χ(b) χ(a B) (3.2) χ(a) = { 1, convex A,, A =. (3.3) We note that this functional χ coincides with the Euler-Poincaré characteristic in algebraic topology. The Minkowski functionals over R are now defined through W ν (A) = χ(a E ν )dµ(e ν ). (3.4) Here, E ν is an ν dimensional plane in IR d, dµ(e ν ) denotes its kinematical density normalised so that for a d-dimensional ball B d (r) with radius r, W ν (B d (r)) = ω d r d ν ; ω d = π d/2 /Γ(1 + d/2) is the volume of the unit ball. From definition (3.4) it is clear that the Minkowski functionals inherit additivity from χ. For lattice configurations Q, i.e., configurations sampled as unions of voxels Q i it is convenient to renormalise the Minkowski functionals by setting V ν (Q) = W ν(q) ω ν (3.5) so that V ν (Q i ) = 1 for a single cube (voxel) Q i. Note that according to the definition given by Eqn. (3.4) the Minkowski functionals V ν can be considered as Euler- Poincaré characteristics χ for lower-dimensional planar intersections of the spatial configuration Q. The Minkowski functionals in three dimensions are related to familiar geometric quantities, for instance, the surface area 6V 1 and integral mean

70 Minkowski functionals 33 curvature 3πV 2 of the surface exposed by a coverage with volume V = V and Euler characteristic X = V 3. Two general properties that a functional V(Q) should possess in order to be a morphological measure are motion invariance and continuity, since the shape of a domain does not depend on its location and orientation, and should be approximately given by an inscribed polygon. In many cases it is important that a domain can be decomposed into parts such as a digitised set Q = i Q i into a collection of voxels Q i. Therefore, we require the additivity relation (3.2) as a third property of a morphological functional V(Q). In three dimensional space examples of such measures include volume and surface area of a domain Q. In two dimensions they include the boundary length and area. A remarkable theorem in integral geometry is the completeness of the Minkowski functionals [52]. The theorem asserts that any additive, continuous and motion invariant functional V(A) on subsets A IR d, A R, is a linear combination of the d + 1 Minkowski functionals: V(A) = d c ν V ν (A), (3.6) ν= with real coefficients c ν independent of A. The d+1 Minkowski functionals therefore are the complete set of morphological measures. The continuity of the functionals V ν allows the definition of integrals of the curvature function to be evaluated for surfaces with singular edges; i.e, the Minkowski functionals generalise curvatures as differential geometric quantities to singular edges [96, 1, 11, 98]. Therefore it is straightforward to apply the notion of morphological measures even to patterns consisting of individual lattice grains (voxel-based images). Let us now consider a three-dimensional cubic lattice Λ (3) a Z 3, where a denotes the lattice constant. At each point x (3) i Z 3 one can fix a three-dimensional cube K i (voxel) of edge length a with x (3) i as midpoint, where i = 1,... L 3 numerates the lattice sites inside a cubic window of size L. Let E (ν) Λ (3) denote a ν- dimensional planar sublattice, i.e. a ν-dimensional hyper-plane cutting Λ (3). The Euler characteristic χ can be defined by χ(k i ) = 1, χ( ) = and for non-empty intersections χ( i K i ) = 1. Since χ is an additive functional one obtains for the union N i=1k i of N voxels N χ( i=1 N 1 K i ) = χ( = i i=1 N 1 K i ) + χ(k N ) χ( χ(k i ) i<j i=1 + ( 1) N+1 χ(k 1 K 2... K N ). K i K N ) χ(k i K j ) +... (3.7)

71 34 Morphology of digitised disordered materials Generally the statistics of a heterogeneous medium in a domain Ω with volume Ω is specified by a sequence of correlation functions ρ (n) ( r 1,..., r n ) of the voxels K i with the homogeneous density ρ ρ (1) ( r 1 ) (porosity). The Euler characteristic embodies information from every order n of the correlation functions ρ (n) ( r 1,..., r n ) due to its dependence on multiple intersections of voxels. The simple measure of χ on a disordered medium [7] however only gives one higher order information at the local scale (scale of the voxel). By applying erosion/dilation operations (section 3.3) one can probe larger distances of ρ (n). We note that this functional χ coincides with the Euler characteristic of algebraic topology. The normalised Minkowski functionals v ν (A) = V ν (A)/L 3 of a structure A = N i=1k i on a three-dimensional cubic lattice of size L 3 can be defined by [11, 1] v ν (A) = ( 3 ν lν ) L 3 E ν χ(a E ν ), (3.8) and for ν =, 1, 2 and v 3 (A) = l 3 χ(a)/l 3. Here, l denotes a dimensionless scaling factor which takes into account the size of typical homogeneous domains of the structure A with respect to the lattice constant a. The sum runs over all positions (induced by lattice translations and rotations) of the ν-dimensional hyper-planes E ν. Obviously, the Minkowski measures inherit the dependence on the n-th order correlation functions ρ (n) ( r 1,..., r n ) for the Euler characteristic χ. Therefore, one obtains for the mean values of the MFs: v ν ({ ρ (n) }) = ( 1) n+1 n=1 n! n n v ν [ K i ] ρ (n) ( r 1,..., r n ) (3.9) i=1 r i Ω where the voxel K i is centered at position r i. If the density correlation functions ρ (n) ( r 1,..., r n ) were independent of position, the integrals in Eqn. (3.9) could be performed using the fundamental kinematic formula yielding analytic mean values for v ν (ρ) which depend only on the porosity ρ [1]. But this is not the case for most physical applications and the Minkowski functionals depend in an integral way on the short-range behaviour of all correlation functions ρ (n). From definition (3.8) it is clear that the Minkowski functionals inherit additivity from χ and that they are related to familiar geometric quantities of the structure A, namely the number v (A) of voxels normalised by the total number of lattice sites L 3, the normalised number 6v 1 (A)/l of boundary plaquettes (pairs of neighboured black and white voxels), and the normalised signed number 12v 2 (A)/l 2 of edges where concave (convex) edges are counted as negative (positive). In particular, for a j-dimensional cube K (j) = 23 j i=1 K i with center x (j) generated by the i=1

72 Minkowski functionals 35 intersection of 2 3 j neighboured lattice cubes K i (voxels) one obtains v ν (K (j) ) = ν!j! l ν 6(ν + j 3)! L, 3 j ν 3 (3.1) 3 and v ν (K (j) ) = for ν < 3 j. One should note, that K (j) is not a voxel K but part of its boundary K and that the midpoints x (j) are not lattice points on Λ, except for x (3). Repeating the definition (3.8) and using Morses theorem [1] in successive dimensions one obtains the decomposition of the morphological measures v ν in terms of edge and corner contributions [11, 1] v ν (A) = 3 µ=3 ν ( 1) 3 ν+µ ν!µ! χ(a x (µ) ), (3.11) 3!(ν + µ 3)! x (µ) where the sum runs over all midpoints x (µ) of the µ-dimensional boundary cubes K (µ), i.e., of the plaquettes, edges, and corners of voxels K i. For all voxel-based images one must specify the local neighbourhood of the voxels. Since one must consider topologically closed sets, voxels may be connected only by a single point, leading to some ambiguity in the measure. On a square lattice, diagonally connected pixels have only one point in common in the absence of a preferred connectivity of either phase, the interface can be considered to be curved equally toward either medium. For example, consider the configuration consisting of diagonal cells on a square lattice (Fig. 3.1). The probability of either phase being connected depends on the physics of the situation. In the presence of strongly preferred connectivity of one phase the neighbourhood for the phase of preferred continuity will have eight neighbours (a pixel is connected to nearest and next-nearest neighbours) and the other phase will have four neighbours (only nearest neighbour connections). This leads to a duality of measures. Similarly on a cubic lattice one may have six or twenty six neighbours depending on the absence [a] [b] [c] Figure 3.1: Configuration resulting in an ambiguous geometry on a square lattice. The ambiguity at the central vertex [a] can be resolved in two ways: [b] the configuration is continuous in the white phase, or [c] continuous in the black phase.

73 36 Morphology of digitised disordered materials or presence of preferred connectivity. The connectivity of either phase can be varied continuously by defining the probability α of a common edge to be connected in one phase and probability β = 1 α in the other phase [64]. If neither phase has preferred connectivity we set α = β = 1/ Boolean model A model well known in stochastic geometry is the Boolean model [1]. This model generates random structures by overlapping grains such as spheres or cubes each with arbitrary location and orientation (see also section 2.1.1). The normalised mean values v ν (ρ) = V ν (Q) /V of the Minkowski functionals for Poisson distributed lattice grains of density ρ (in units of a 3 and a is the lattice constant) are [1] v (ρ) = 1 e ρv v 1 (ρ) = e ρv (1 e ρv 1 ) (3.12) v (8) 2 (ρ) = e ρv ( 1 + 2e ρv 1 e ρ(2v 1+V 2 ) ) v (26) 3 (ρ) = e ρv (1 3e ρv 1 + 3e ρ(2v 1+V 2 ) e ρ(3v 1+3V 2 +V 3 ) ), where V ν (K) are the morphological measures of the individual grains K. For Poisson distributed cubes of sidelength λ, q = 1 p = e ρ being the probability that a cube is not placed at a lattice site, and V = λ 3, V 1 = λ 2, V 2 = λ, and V 3 = 1 this becomes v (ρ) = 1 q λ3 v 1 (ρ) = q λ3 (1 q λ2 ) (3.13) v (8) 2 (ρ) = q λ3 ( q λ2 q 2λ2 +λ ) v (26) 3 (ρ) = q λ3 (1 3 q λ2 + 3 q 2λ2 +λ q 3λ2 +3λ+1 ). The numbers in brackets specify the different neighbourhoods of the cubes. In contrast to the measures v and v 1, the measures v 2 and v 3 depend on the definition of the local neighbourhood of a grain (see Fig. 3.1 and discussion). As shown by Mecke [146], one can relate the measures of v 2 (ρ) and v 3 (ρ) for different neighbourhoods: v (4) 2 (ρ) = v (8) 2 (ρ) + δv 2 (3.14) v (6) 3 (ρ) = v (26) 3 (ρ) 3δv 2 + δv 3,

74 Minkowski functionals 37 with the two correction terms for Poisson distributed cubes given by: δv 2 = 2 q λ3 +2λ 2 λ (1 q λ ) 2 (3.15) δv 3 = q λ3 +3λ 2 λ 1 [ q 4λ 2 (8 p 2 q 2) + 24 q +12 q 3λ (1 + p) 6 q 2λ (1 + 4 p) + 8 p q λ 24 q λ+1 ] 4 q λ3 +3λ 2 3λ+1. The derivation of theoretical results for the other Boolean grain models is fully analogous. If using spheres as grains, the Minkowski measures of the single grain in the continuum become V = 4 3 πr3, V 1 = πr 2, V 2 = 2r, V 3 = 1, (3.16) leading to the global measures for Poisson distributed or identical overlapping spheres (IOS) using equation (3.12) Computational aspects The MFs are obtained directly from any image made up of discrete voxels. For example, the volume fraction of a phase is trivially obtained by dividing the number of voxels of that phase by the total number of voxels. The other functionals are obtained by considering the interface associated with the vertices of each voxel or the Voronoi cell of the lattice [64]. In three dimensions each vertex of the lattice is shared by eight neighbouring cubes; there are therefore 2 8 = 256 possible configurations. The configuration IC (3D) 256 of a particular vertex can be evaluated quickly using a masked sum over the neighboring voxels of the vertex, IC (3D) 256 = 7 2 i phase i, phase i {, 1}, (3.17) i= where the sum is taken over only one phase. For voxels of equal side length, the local Minkowski measures are rotationally invariant and the 256 configurations reduce to 22 (see Fig. 3.2). The mapping and the local contributions to the global Minkowski functionals are given in Tab The various patterns and their resultant MFs have been derived elsewhere [64, 146] for general α. The global measures for each configuration are obtained then by a configuration count over all vertices on any voxelated structure normalised by the total number of vertices. On a 5 MHz Alpha microprocessor we calculate all MFs on a 5 3 image in 4 seconds using 2 MB of memory. Decompressing and loading an image of that size ( 12 MB uncompressed for the storage scheme used) can be slower, thus this task might be I/O bound on some machines. The execution time scales linearly with the volume of the image.

75 38 Morphology of digitised disordered materials Table 3.1: Local contributions to the global Minkowski measures for the 22 isotropic configurations and their mapping to the 256 vertex configurations following the order shown in Fig. 1. L gives the configuration in Fig. 3.2 which matches the isotropic configuration number (IC (3D) 21 ). N notes the multiplicity of the configuration. IC (3D) 21 N L 8V 24V 1 24V (4) 2 24V (8) 2 8V (6) 3 8V (26) 3 Configuration IC (3D) 256 defined by Eqn N 1 8 A , 2, 4, 8, 16, 32, 64, B , 5, 1, 12, 17, 34, 48, 68, 8,136,16, C , 9, 18, 2, 33, 4, 65, 72, 96,13,132, D , 36, 66, E , 11, 13, 14, 19, 21, 35, 42, 49, 5, 69, 76, 81, 84,112,138,14,162,168,176,196,2,28, F , 26, 28, 37, 38, 44, 52, 56, 67, 7, 74, 82, 88, 98,1,131,133,137,145,152,161,164,193, G , 41, 73, 97,14,134,146, I , 51, 85,17,24, M , 43, 77,113,142,178,212, H , 29, 39, 46, 53, 58, 71, 78, 83, 92,114,116, 139,141,163,172,177,184,197,22,29,216,226, L , 45, 54, 57, 75, 86, 89, 99,11,16,18,12, 135,147,149,154,156,166,169,18,198,21,21, J , 9,12,153,165, K , G ,19,121,151,158,182,214, F , 62, 91, 94,13,11,118,122,124,155,157,167, 173,181,185,188,199,23,211,217,218,227,229, E , 47, 55, 59, 79, 87, 93,115,117,143,171,174, 179,186,25,26,213,22,234,236,241,242,244, D ,189,219, C ,123,125,159,183,19,215,222,235,237,246, B , 95,119,175,187,27,221,238,243,245,25, A ,191,223,239,247,251,253, N 8 255

76 Minkowski functionals 39 (A) (B) (C) (D) (E) (F) (G) (H) (I) (J) (K) (L) (M) (N) (D ) (G ) Figure 3.2: Catalog of filling patterns of a unit cell giving rise to distinct configurations. Some configurations with exchanged phases are not shown; 22 isotropic configurations exist, the rest can be generated by rotations. Isotropic configurations (A )-(G ) are generated by inverting the phases of configurations (A)-(G) and (N). (D ) and (G ) are shown as an illustration. The algorithm for calculating the Minkowski measures was validated against the theoretical predictions given by Eqns for model systems of Poisson distributed cubes of equal sidelength (Fig. 2.2) as well as Poisson distributed spheres of identical radius. All computational data are based on a minimum of 5 realisations on a 2 3 lattice. We measure the MFs over the full range of the volume fraction in steps of =.2 for each model. The models are mapped periodically therefore edge effects can be ignored. As can be seen in Fig. 3.3 the numerical data matches the theoretical Eqns prediction for cubes of varying sidelength. For cubes with λ = 1 the result reduces to random filled lattices as simulated by Jernot et al. [68]. To match the predictions for spheres one must be careful with the definition of the local measures of each spherical grain. One must use the local Minkowski measures of the digital sphere and not a continuum sphere of equivalent radius (using

77 4 Morphology of digitised disordered materials.4.2 (a) (b) λ i v i (c) v 1 v 2, 4 neighbors v 2, 8 neighbors v 3, 6 neighbors v 3, 26 neighbors v 1 v 2, 4 neighbors v 2, 8 neighbors v 3, 6 neighbors v 3, 26 neighbors λ=1 λ=2 (d) λ i v i.2 λ=4 λ= Figure 3.3: Minkowski measures over fraction: comparison of theoretical predictions (lines) with numerical simulations (symbols) for Poisson distributed cubes at different sizes: (a) λ = 1, (b) λ = 2, (c) λ = 4, and (d) λ = 8. The measures are scaled as v 1 λ v 1, v 2 λ 2 v 2, and v 3 λ 3 v 3. Eqn directly). We define the digital sphere of radius r by all voxels radiating from the central voxel that are separated by a Euclidean distance l < r. discretised 1-dimensional radius for continuum spheres is given by r 1d = r c.5, because the spheres are centered at a vertex. The The properties of the discretised sphere are given in Tab Using these values for the local measures in Eqns we obtain excellent agreement with theory (see Fig. 3.4). The symmetry of the Euler characteristic for dimensions d > 1 was also checked. Interchanging phases correctly mirrors for all binary models. v (4) 2 () v (8) 2 (1 ) (3.18) v (6) 3 () v (26) 3 (1 ) Discretisation effects Here we discuss the effect of the choice of preferred continuity on the measure of the MFs. In particular we consider the effect on the Euler characteristic. In

78 Minkowski functionals 41 Table 3.2: Comparison of the local Minkowski measures in 2 and 3 dimensions for spherical grains in the continuum against the discretised spheres used in the simulations. We use * to denote the continuum values as compared to the discretised. The radii r i in the last three columns are equivalent sphere radii based on the measures V i and Eqn for the intrinsic volumes of continuum spheres. r V1 V 1 V2 V 2 V3 V 3 r 1 r 2 r (a) (b) r i v i.2.4 v 1 v 2, 8 neighbors v 3, 26 neighbors v 1 v 2, 4 neighbors v 2, 8 neighbors v 3, 6 neighbors v 3, 26 neighbors r=4 r=8.2 (c) (d) r i v i.2 r=12 r= Figure 3.4: Minkowski measures over fraction: comparison of theoretical predictions (lines, highly connected neighbourhoods) with numerical simulations (symbols, both neighbourhoods) for Poisson distributed spheres of different radius: (a) r = 4, (b) r = 8, (c) r = 12, and (d) r = 16. The measures are scaled as v 1 r v 1, v 2 r 2 v 2, and v 3 r 3 v 3.

79 42 Morphology of digitised disordered materials Table I of [64] and Table IX of [146] they describe the local contribution to the Euler characteristic in three dimensions of the 22 possible local configurations of the unit cell. The eight (ambiguous) configurations that are dependent on preferred continuity are configurations C,D,F,G,C,D,F and G of Fig To consider the discretisation effects in the measure of the Euler characteristic, we compare the contribution of these configurations to both Poisson distributed cubes at different λ and to the IOS model at different r (see Fig. 3.5). One observes that for the Poisson cube model, the probabilities P (IC (3D) 21 ) = P (L) for these configurations are only considerable for λ = 2 and 4, while for larger λ the effects are minimal. This is mirrored in Fig. 3.3, where we observed the convergence of the curves for 6 (26) neighbourhoods for large λ. The reason for the fast convergence for the different neighbourhoods is that the configurations giving rise to the ambiguous configurations only occur for cubes that are exactly adjacent. Any overlap of two adjacent cubes surface will eliminate the possibility of generating these configurations. Thus we find that the contribution of these configurations scales approximately as the cube volume l 3 over most of the fractional range. For the IOS model the contributions do decrease, but more slowly, as r 1. The contribution of these ambiguous configurations to models exhibiting large local curvature is therefore considerable even at higher resolutions. The choice of preferred continuity should be considered carefully for Boolean models with curved surfaces. In the rest of this section the Minkowski functionals of the different microstructural models introduced in chapter 2 (particle-, GRF-, and Voronoi models) are presented, and we qualitatively compare the morphology of the models Comparison of MFs for different model morphologies The model microstructures are generated at roughly the same length scales. The MFs are in general size-dependent. For example, the Euler characteristic per unit volume for a sphere pack of radius r = 4 cannot be directly compared to results for a sphere pack of radius r = 16. A dimensionless measure such as Euler characteristic per particle is needed to compare results. In Fig. 3.4 we showed the MFs for the IOS model at different radii. By scaling the measures v 1 r v 1, v 2 r 2 v 2 and v 3 r 3 v 3, we see that the curves are very similar, therefore scaling of a Boolean model by particle size is appropriate. However for general models the definition of a particle size is problematic. For example, the 1- and 2-level cut Gaussian models generated from the same field exhibit very different correlation functions; the correlation function for the 2-level cut model decaying far more rapidly than in the single level-cut case. This point is clear from visual inspection of Fig We therefore choose our model systems to

80 Minkowski functionals 43 P(L) (C) (G ) P(L) P(L) P(L) (D) (F) (G) (F ) (D ) (C ) Figure 3.5: The configuration probabilities P (IC (3D) 21 ) = P (L) over fraction of Poisson distributed cubes and spheres for the ambiguous configurations in Fig. 3.2 (see also Tab ). The letters correspond to the specific configurations L. Data for the cube model is given by open squares and for the IOS by solid circles. The configurational probabilities decrease with increasing size. In the figure we show data for sphere radii: r = (4, 8, 12, 16) and for the cube sidelength: l = (1, 2, 4, 8).

81 44 Morphology of digitised disordered materials 2 15 IOS (r=12) ROS (r=4..2) 1 level cut (t=2) 2 level cut (t=1) intersection (t=1) Voronoi facets (1 seeds) Voronoi tubes (1 seeds) ξ [a] Figure 3.6: Correlation length over pore fraction derived from the two-point correlation functions in real space; IOS: identical overlapping spheres, ROS: randomly oriented spheroids, three different Gaussian, and two Voronoi tesselation models. have similar correlation length, ξ, defined by the decay of the envelope of the spatial two-point correlation function. In some cases, for example the 2-level-cut Gaussian, the correlation length at low volume fractions is very small, so direct comparison is difficult. In Fig. 3.6 we plot the correlation length ξ for the range of models considered in this paper for different volume fractions. The choice of correlation length ξ, domain scale d and cut-off scale r c for the Gaussian fields as well as the density of sites for the Voronoi models were made to allow for as close as possible a match to the Boolean grain models for r = 12 across a range of volume fraction. This allows us to form a basis for a semi-quantitative comparison of the MFs across the range of models Boolean grain models First we discuss the MFs for Boolean grain models shown in Figs. 3.3 and 3.4. Other than the discretisation effects discussed above, there is little difference when comparing the data for Poisson distributed cubes and spheres. For cubes of sidelength l = 1 the measures show a higher symmetry; Eqn is satisfied without interchanging phases. We compare the sensitivity of the Minkowski measures

82 Minkowski functionals Random OS Aligned OS IOS v 1 rv 1, r 2 v 2, r 3 v 3.1. v 2, 4 neighbors v 2, 8 neighbors.1 v 3, 6 neighbors v 3, 26 neighbors Figure 3.7: Minkowski measures over fraction for Poisson distributed overlapping spheroids of half-axes r = 4..2, with r = 12, compared to fully aligned overlapping spheroids of the same size distribution and IOS of r = 12. The measures are scaled as v 1 r v 1, v 2 r 2 v 2, and v 3 r 3 v 3. to deviations in form or alignment. To do this we consider packs of overlapping spheroids. We generate randomly oriented and fully aligned overlapping spheroids of uniformly distributed half-axes r = 4,.., 2 having r = 12 and compare to the IOS model with r = 12. As Fig. 3.7 shows, the Minkowski measures for these systems are quite similar. In fact, the different continuity rules have a far stronger effect than the differences in size or alignment. It may be non-trivial to distinguish the measures for these different systems Gaussian models The Gaussian 1-level cut model results in symmetric MFs around =.5 (see Fig. 3.8). The integral mean curvature is much less than for the Boolean grain models. For small and large isolated elliptical inclusions are present and the Euler characteristic is positive. In the regime.2 to.8 the interface becomes predominantly hyperbolic (v 3 < ) and both phases are continuous. The relative smoothness (small local curvature) of the interface when compared to the sphere pack model leads to a low probability for obtaining ambiguous configurations. Accordingly the measures for the two different neighbourhoods converge.

83 46 Morphology of digitised disordered materials.3 1v 1, 1v 2, 1v v 1 v 2, 4 neighbors v 2, 8 neighbors v 3, 6 neighbors v 3, 26 neighbors Figure 3.8: Minkowski measures over fraction for Gaussian 1-level-cut models of periodicity t = v 1 v 2, 4 neighbors v 2, 8 neighbors v 3, 6 neighbors v 3, 26 neighbors 1v 1, 1v 2, 1v Figure 3.9: Minkowski measures over fraction for Gaussian 2-level-cut models of periodicity t = 2.

84 Minkowski functionals v 1 v 2, 4 neighbors v 2, 8 neighbors v 3, 6 neighbors v 3, 26 neighbors.4 1v 1, 1v 2, 1v Figure 3.1: Minkowski measures over fraction for Gaussian intersection models of periodicity t = 2. For the symmetric two-level cut Gaussians of small the surface to volume ratio is much larger than for the previous systems. Particularly at low volume fractions the morphology exhibits a sheet-like phase, and the surface to volume ratio quickly reaches a maximum (Fig. 3.9). Further densification is associated with a thickening of the sheets and the surface to volume ratio drops. Also for small, discretisation effects are important and the different neighbourhoods play an important role. The Euler characteristic is very large and becomes negative for small, once the sheet-like phase connects, implying a strongly bicontinuous structure, as is evident in Fig The intersection set of two 2-level cut Gaussians has a distinct signature (Fig. 3.1) when compared to the 1- and 2-level cut models. The strong Gaussian curvature feature at small may be useful as a signature of this structure when compared with the other models. 1 In the mathematical model the sheets can be infinitely thin, and the surface to volume ratio becomes constant in the limit of. This behaviour can t be reproduced, but only be approached by a discretised structure (finite resolution), thus the v 1 measure exhibits a maximum.

85 48 Morphology of digitised disordered materials Voronoi models For the Voronoi facet model, the connected polyhedra phase always percolates, and the other phase is made up of disconnected inclusions. Accordingly, the curvature never changes sign the holes remain convex for all (Fig. 3.11). Similarly, the Euler characteristic always remains positive, while the surface area due to the construction of the fields starts at maximum and decreases with thickening facet boundaries. Small ambiguities arise at the intersections of the facets, again giving rise to a separation of the MFs based on the choice of continuity v 1, 5v 2, 25v 3..2 v 1 v 2, 4 neighbors v 2, 8 neighbors v 3, 6 neighbors v 3, 26 neighbors Figure 3.11: Minkowski measures over fraction for Voronoi facet models of 1 seeds on a 2 3 lattice. For the Voronoi cylinder model, both the network of cylinders and the background phase percolate over a wide range of. Here the surface area first grows with increasing cylinder radius, and then decreases again, after the cylinders begin to merge (Fig. 3.12). The curvature decreases with increasing dilation, and becomes negative, once individual edges of the Voronoi cells begin to generate isolated inclusions of the background phase. The Euler characteristic is initially negative, implying that both phases percolate, and then increases almost linearly due to closure of loops, finally becoming positive.

86 Minkowski functionals v 1, 1v 2, 1v v 1 v 2, 4 neighbors v 2, 8 neighbors v 3, 6 neighbors v 3, 26 neighbors Figure 3.12: Minkowski measures over fraction for Voronoi edge models of 1 seeds on a 2 3 lattice Application to a cross-bedded sandstone A number of statistical models have been proposed for reconstructing porous media from statistical information [69, 1, 121, 122, 117, 115, 28]. These methods, based on different underlying model microstructures, are generated in such a manner that they match the observed two-point statistical properties of the rock. In section it was seen, that the cross-bedded sandstone is a very heterogeneous sample (Fig. 2.15). Due to this fact, by choosing subsamples of the full data set, will give one a good spread of data across a range of porosities. to compare the characterisation of the sandstone samples with. It is therefore possible We compare the morphological measures for the sandstone to three standard stochastic models used to generate realisations of sedimentary rock microstructures from two-point information. 2 The first is based on the Boolean sphere model [115], the second on the 1-level cut model [69, 1], and the third on the intersection set model [122]. We compare the match to the sandstone morphology based on the 2-point voidvoid correlation and based on the Minkowski functionals. The quantities used to 2 One can derive more complex model systems which incorporate other two-point correlation information (e.g., chord distribution functions [131, 145]. Recent work [84] has shown that these measures give a poor representation of connectivity. Preliminary results on MFs of these more complex models indicate that this is reflected in a poor match to the v 2 measure. Further discussion on stochastic models is given in section 3.2.

87 5 Morphology of digitised disordered materials Cross bedded sandstone Identical overlapping spheres Gaussian 1 level cut Gaussian intersection.8 Cross bedded sandstone Identical overlapping spheres Gaussian 1 level cut Gaussian intersection.6.6 S(r) S(r) [a] r [µm] [b] r [µm] Figure 3.13: Normalised 2-point correlation functions S(r) of the two 3 3 sandstone samples and the averaged best fits for the model structures. characterise the microstructure of the stochastic models are the volume fraction, the surface to volume ratio S/V and p (2) (r), the two-point correlation function. Note that = p (2) (), and S/V = 4 dp(2) () dr. The correlation function for the phase external to the spheres of radius r in the IOS model is p (2) (r) = p (ν(r)) for r < 2r and p (2) (r) = 2 for r > 2r where ν(r) = 1+ 3r 4r r3, and S/V = 3ln/r 16r 3. To generate matching GRF models we employ the field-field correlation function given by Eqn. 2.8, which is characterised by a correlation length ξ, domain scale d and a cut-off scale r c. The three length scale parameters are obtained by a best fit procedure to minimise the non-linear least squares error [121], Ep (2) = M i=1 [p(2) fit (r i) p (2) exp(r i )] 2 M i=1 [p(2) fit (r, (3.19) i) p 2 exp] 2 and a downhill simplex method is used to find the best parameter set [18]. One of the parameters is eliminated by using the S/V ratio. We generate the stochastic models that best match the two 3 3 samples and compare the normalised 2-point correlation functions S(r) = p 2(r) p 2 of these models in Fig Values of r p p 2 for the IOS model, and d, ξ and r c are summarised in Tab. 3.3 along with the error estimate E(p (2) ). The fit based on the match to the 2-point correlation function is worst for the sphere model, and more than a factor two better for the Gaussian 1-levelcut and intersection set. The sphere model however has only one free parameter as compared to three for the Gaussian models. We now calculate the Minkowski functionals of the three stochastic models and compare with the sandstone morphology (Fig. 3.14). The measures are compared for a sample size of 1 3. The values for the samples were binned in steps of porosity =.2. None of the models

88 Minkowski functionals (a) 2. (d) v 1 [mm 1 ] samples of samples of samples of samples of 6 3 v 1 [mm 1 ] sandstone, 1 3 IOS 1 level cut intersection (b) 25 (e) 2 2 v 2 [mm 2 ] 15 1 v 2 [mm 2 ] (c) 1 (f) v 3 [mm 3 ] v 3 [mm 3 ] Figure 3.14: Minkowski measures for a cross-bedded sandstone. From top to bottom: v 1 (a,d), v 2 (b,e), and v 3 (c,f). In the left column we show the data points for the different blocks of sizes 6 3, 1 3, 15 3, and 3 3. One obtains consistency in the morphological measures down to scales of 1 3. Right: comparison of the MFs for the sandstone data to the three stochastic models based on a window size of 1 3.

89 52 Morphology of digitised disordered materials satisfactorily match the experimental data. The surface area (v 1 ) is matched best across all by the intersection model. Integral mean curvature v 2 is best matched by the IOS model which honours the granular structure of the sedimentary rock. The topology v 3 is described well by both the IOS and 1-cut models. The intersection model is particularly poor for this model over the full range of. The model which best captures the characteristics of the sandstone is the IOS model. Even though the other models do reasonably well at the fraction where the 2-point correlation functions were matched, they fail to describe the structure across a range of phase fractions. Different transport and mechanical processes will depend more strongly on the agreement with specific morphological measures. For example, single phase flow and conductivity will be most strongly affected by surface-to-volume (related to average constriction size) and the topology. A model which accurately describes these measures may still yield good agreement with experiment. However, multiphase flow properties depend crucially on the curvature of the surfaces where immiscible phases meet. For these processes a model that also accurately matches v 2 will be required. In this case the IOS model is the best of the three candidates as a reconstructed data set. Recently, a model for describing sandstone morphology has been developed based on a full process-based sedimentation, compaction and diagenesis model [113]. This model honours both the shape of the original grains and the geological formation processes and therefore may provide a more accurate description of pore space morphology of sedimentary rock. 3.2 Development of Boolean reconstruction methods The Boolean model is well known in stochastic geometry [1]. Structure is generated by the gradual build up of a phase via the overlap of permeable particles distributed by a Poisson process. For this model analytic expressions for the Minkowski functionals have been given (Eqn. 3.12), providing a compact description of morphology for all phase fractions. Although idealised, the model is often used to reconstruct the morphology of complex materials [158]; examples in the litera- Table 3.3: Parameters of the models for the cross-bedded sandstone in microns. The original sandstone image has resolution at 1µm per pixel. For the IOS model, the equivalent sphere radius is in the r c column. Model r c [µm] ξ[µm] d[µm] Ep (2) Spheres e 3 1-level-cut e 4 intersection e 4

90 Development of Boolean reconstruction methods 53 ture include ceramic powders [123], wood composites [162], paper [7], sedimentary rock [14, 9] and hydrating cement-based materials [44]. In many of these studies the grain size of the matching Boolean model is chosen by either matching two-point information or by choosing some averaged grain size. The latter is often difficult since for most real materials one has poor knowledge of the full distribution of grain sizes and shapes. In this section we show that one can accurately reconstruct complex Boolean systems from analysis of morphological measures given by integral geometry. As discussed in the introduction to this chapter, integral geometry provides powerful formulae to characterise structure. For the Boolean grain model in particular one can directly relate local morphological properties of the grains to global morphological measures via Eqn The global measures, in turn, can be derived from a tomographic image. Therefore one can, from a single 3-D image at any phase fraction, derive an equivalent local Boolean grain ensemble. This in turn, allows one to accurately reconstruct the medium for all particulate phase fractions f. The accuracy of the method is demonstrated for a complex Boolean mixture of grains and then applied to the (non-boolean) cross-bedded and Fontainebleau sandstone data sets. The physical properties of Boolean reconstructions are analysed in chapters Minkowski measures of a Boolean composite In section 3.2 the evolution of the global Minkowski measures of Boolean composites generated by a single grain was accurately characterised by the individual Minkowski measures of the single grain (Eqn. 3.12). This result holds even for mixtures of grains, each following a Poisson process, where V i is now replaced by proper averaged values. For mixtures of polyhedra one can replace the quantities V ν of a single grain by averages over an ensemble of grains, weighted by the probability p j of their occurrence V ν = n p j V νj for ν =,..., d, (3.2) j=1 where p j = ρ j /ρ, is the ratio of the densities of the Poisson process. In order to evaluate the influence of curvature on the physical properties of real systems, and because of their wide use as models of composites, spheroidal grains need also be considered. As discussed briefly in section 3.1, one must carefully define local measures of spherical grains. For the more general case of discrete (voxelated) spheroids one must consider both discretisation errors and projections of the grains onto the planar axes. A detailed discussion of the derivation of equivalent local measures for spheroidal systems is given in appendix B.

91 54 Morphology of digitised disordered materials Computational aspects In the preceding section (3.1) predictions for a Poisson process of a single grain were used to validate the algorithm for calculation of the Minkowski functionals (see Figs ). Here we validate the algorithm for any complex mixture of grains using Eqn and Eqn The models used for this test are illustrated in Fig. 2.3 and the local sizes defined in Tab Table 3.4: Parameters for the mixture models shown in Fig. 2.3.[a]-[f] together with the (exact) predictions of the Minkowski measures of the average local grains given by Eqn. B.1. The probability to place the second particle p 2 is given by p 2 = 1 p 1. All individual grains are convex and it follows directly V 3 = 1. Mixture mod. λ 11 λ 12 λ 13 λ 21 λ 22 λ 23 p 1 V V 1 V 2 plates /3 17/3 sticks /3 1/3 cubes/sticks /2 8 29/6 8/3 cubes/plates / /3 cubes/cubes / cubes/cubes /64 575/8 127/16 71/32 For the calculation of the Minkowski functionals 5 realisations of each of these models were generated on a 2 3 lattice with a fractional spacing of =.2. The comparison is given in Fig The agreement is excellent Derivation of local morphological measures To date we have used the knowledge of the local grain morphology to predict global morphological measures of a Boolean process [98, 7]. Here it is shown that one can also do the inverse; from a single snapshot of a complex Boolean process one can derive equivalent local grain morphology of the process. For all Boolean models made up of compact grains one has V 3 = 1. As one can evaluate the four global measures v ν from a single 3D image, Eqn can be used to directly solve for V, V 1, V 2 and Poisson density ρ. The method is first illustrated for Boolean samples made up of monodisperse cubic grains Homogeneous Boolean mixtures We generate realisations of Poisson distributed cubes of different length (λ = 1,..., 16) for a range of phase fractions at lattice sizes of 2 3 and 4 3. From the 3 For this section the index describing the local neighbourhood of the grain has been dropped and for all reconstructions the highly connected neighbourhoods of the global measures v (8) 2 and v (26) 3 of the particle phase are chosen.

92 Development of Boolean reconstruction methods v 1 * <V 2 >, v 2 * <V 1 >, v 3 * <V > v 1 v 2 (4) v 2 (8) v 3 (6) v 3 (26) theory v 1 * <V 2 >, v 2 * <V 1 >, v 3 * <V >..5 v 1 v 2 (4) v 2 (8) v 3 (6) v 3 (26) theory [a] [b] v 1 * <V 2 >, v 2 * <V 1 >, v 3 * <V > v 1 v 2 (4) v 2 (8) v 3 (6) v 3 (26) theory v 1 * <V 2 >, v 2 * <V 1 >, v 3 * <V >.5. v 1, sim v 2 (4) v 2 (8) v 3 (6) v 3 (26) theory.4 [c] [d] v 1 * <V 2 >, v 2 * <V 1 >, v 3 * <V > v 1 v 2 (4) v 2 (8) v 3 (6) v 3 (26) theory v 1 * <V 2 >, v 2 * <V 1 >, v 3 * <V >..1 v 1 v 2 (4) v 2 (8) v 3 (6) v 3 (26) theory [e] [f] Figure 3.15: Minkowski measures over particle fraction for the Poisson mixture models shown in Fig The lines give the analytic predictions based on Eqn while the data points are based on the simulation results. The grains chosen and local Minkowski are defined in Tab [a] plates, [b] sticks, [c] cubes and sticks, [d] cubes and plates, [e] cubes and cubes with p 1 = 1/2 [f] cubes and cubes with p 1 = 1/64. As in Fig. 3.3 all global measures are scaled with corresponding local measures, so they can be shown in one plot.

93 56 Morphology of digitised disordered materials image we determine the global measures v ν for different values of f and from Eqn evaluate the equivalent local measures for that image, ρ im and Vν im. In Tab. 3.5 we compare ρ im, Vν im to the exact values for two specific systems, using the norm of the relative error im Vν V ν E Vν = = V ν. (3.21) V ν V ν In all cases the densities and the local measures vary little from the analytic results. Table 3.5: Local Minkowski measures for systems of Poisson distributed cubes at λ = 4 (L = 2) and λ = 8 (L = 4). The first row (labeled exact) gives the analytical values for the local V ν : the subsequent rows give the relative error (Eqn. 3.21, in [%]!) of the prediction of Vν im from the measurement of v ν and use of Eqn for different ρ. ρ in the table is the analytic Poisson density from Eqn and ρ image gives the density measured from the images. λ = 4 V V 1 V 2 Exact f E [%] E 1 [%] E 2 [%] ρ[1 3 ] ρ im [1 3 ] λ = 8 V V 1 V 2 Exact f E [%] E 1 [%] E 2 [%] ρ[1 3 ] ρ im [1 3 ] Remarkably, even at high volume fractions where most grains overlap and resolving the local grain ensemble is difficult, the predictions of the local measures are still good (< 2% error). We further consider a system of identical Poisson distributed spheres. described in section 3.1 (see Tab. 3.2 and appendix B.2), the exact local measures of discretised spheres of radius r do not coincide with their continuum counterparts. Exact values for the local V ν are given in Tab. 3.6 along with their prediction from single digitised images at different f. As with the systems of cubes the resultant local measures V im ν are in excellent agreement over all f. As

94 Development of Boolean reconstruction methods 57 Table 3.6: Local Minkowski measures of systems of Poisson distributed spheres at r = 4 (top) and r = 8 (bottom) on a 2 3 lattice. The theoretical values of the single grains in the first lines are for discretised spheres. r=4 V V 1 V 2 Exact f E [%] E 1 [%] E 2 [%] ρ[1 3 ] ρ im [1 3 ] r=8 Exact f E [%] E 1 [%] E 2 [%] ρ[1 4 ] ρ im [1 4 ] Heterogeneous Boolean mixtures We now generate a very heterogeneous mixture of five polyhedra (5-mix): a system made up of sticks (4x1x1), plates (2x2x1), cubes (8x8x8), and two other rectangular prisms (1x5x2) and (16x8x4) in the ratio 1% : 1% : 4% : 2% : 2%, aligning the anisotropic grains with equal probability along each coordinate axis (i.e. the sticks occur with equal probability in the configurations (4x1x1, 1x4x1, and 1x1x4). The error in measures obtained from a single snapshot of the mixture at L = 5 to the analytic local measures derived from Eqn. 3.2 are given in Tab. 3.7 for different f. The error in the local measures is again very small despite the complexity of the mixture. This indicates that one can accurately define the equivalent local grain ensemble from a single 3D snapshot of any complex material. Table 3.7: Local morphological measures of the 5-mix system of 1% sticks (4x1x1), 1% plates (2x2x1), 4% cubes (8x8x8), 2% each of rectangular prisms of size (1x5x2) and (16x8x4) on a 5 3 lattice. The theoretical values of the mixture are given in the first line, the other rows give the relative error in the predictions from the simulations taken at different particle fractions f. V V 1 V 2 Analytic f E [%] E 1 [%] E 2 [%] ρ im

95 58 Morphology of digitised disordered materials Error in morphological measures In Tables we note that at high particle densities, where most particles overlap, the estimation of the local measures V ν is less accurate. At higher phase fractions one may need to consider larger sampling volumes (L) to obtain accurate predictions. Since we wish to accurately characterise the local V ν from a single 3D image, we quantify the standard deviation σ(l) of the measures Vν im as a function of image size L and analyse the scaling of the standard deviation. Analytically it has been shown that σ(l) L 3/2 [99]. We therefore plot σ(l) as a function of L 3/2 for Boolean models of cubes (λ = 8) and spheres (r = 4) at three different particle fractions f in Fig We note that the standard deviations increase with f but observe the analytical scaling behaviour for all local measures. Based on the results we conclude that for reasonably large images, L 4, about 5 times larger than the grain size in the simulations, σ(l) is 1%. At these domain sizes one would therefore expect to accurately define an equivalent grain ensemble. Accurate reconstruction techniques for complex morphologies have long been sought. Given that microcomputed tomographic imaging techniques can now produce images at scales of > 5 3 voxels one should be able to define equivalent local grain information from experimental images Generation of equivalent Boolean models Having accurately derived the equivalent grain ensemble defined by Vν im from a single image at any f one can use this to reconstruct statistically equivalent complex materials. In the simplest case one might derive an equivalent rectangular prism with sidelengths λ 1, λ 2 and λ 3 that match the observed Vν im. For example, a Boolean model with a grain defined by a rectangular prism with (λ 1 = 17.47, λ 2 = 6.516, λ 3 = 3.341) is equivalent to the heterogeneous 5-mix system at =.5 (V im = 371.1, V im 1 = 63.26, V im 2 = 8.967). However, to generate an equivalent voxelated Boolean model (on a lattice) we must generate mixtures of structures with integral λ ji. The best match of a single equivalent Boolean voxelated grain to the 5-mix system is now (λ 1 = 23, λ 2 = 4, λ 3 = 4). This gives a poor match to the V im ν (Tab. 3.8). To get a better match to the V im ν we must increase the degrees of freedom. We therefore generate the best two-grain match to the observed V im ν. To do this we consider the local measures of possible combinations of two-grains with λ [1; 5], λ 1 λ 2 λ 3. The equivalent two-boolean-grain match (BG (2) ) is defined in Tab The morphology of the BG (2) model now closely matches the original 5-mix system.

96 Development of Boolean reconstruction methods σ [%] 2 V V 1 V 2 line fit σ [%] 3 2 V V 1 V 2 line fit 1 l=8, p=.2 1 r=4, p= σ [%] V V 1 V 2 line fit σ [%] 4 2 V V 1 V 2 line fit 1 l=8, p=.5 r=4, p= σ [%] V V 1 V 2 line fit σ [%] 1 5 V V 1 V 2 line fit 2 l=8, p=.8 r=4, p= L L 1.5 Figure 3.16: Scaling of the standard deviation σ(l) of the local measures V ν for Boolean models with L. The plots are for cubes of λ = 8 (left) and spheres of r = 4 (right). Data points are from right to left for L = 15, 2, 3, 4, 6 and L = 8. We show the numerical results match the theoretical prediction that σ(l) L 1.5. One obtains a very low standard deviation in the local measures from images measured at large L. Top: f =.2, middle: f =.5, bottom: f =.8. Each data point represents at least 1 realisations.

97 6 Morphology of digitised disordered materials Table 3.8: Match of the equivalent Boolean model to the 5-mix system. The norm of the relative errors in the local measures (Eqn. 3.21) are given. To accurately match the morphology one must consider a two-particle system. Model Mixture E Vν BG (1) E V =.86% Single Boolean E V1 = 5.43% grain E V2 = 15.2% BG (2) 84.2% : E V =.26% Two Boolean 15.8% : E V1 =.4% grains E V2 =.41% Having defined an equivalent voxelated Boolean grain ensemble, the question remains; does this equivalent system exhibit similar physical properties to the original system? To investigate this we compare the percolation of the original 5-mix and equivalent BG (2) Boolean system Prediction of percolation thresholds An important test for any microstructural model used to describe transport and mechanical properties of random materials is the ability to estimate the percolation threshold p c as a function of volume fraction. Rough bounds on the percolation threshold of polydisperse systems have been derived [11] based on excluded volume arguments, where.84 v c.295 for continuum percolation in 3D. Accurate estimation of the percolation thresholds for arbitrary complex materials is primarily based on numerical simulations coupled with the knowledge of the size, shape and orientation of the grains in the original ensemble [46]. However, for most real random materials (e.g., cements, sedimentary rocks, etc.) one has no knowledge of the full distribution of grain sizes and shapes. Here we show that the equivalent grain model derived from the single 3D snapshot allows us to obtain a good estimate of p c of the original model. We vary the density of the original 5-mix system defined in Tab. 3.7 and the equivalent BG (2) ensemble given in Tab. 3.8 and numerically determine the percolation thresholds for both phases of the two models. For the determination of the percolation threshold p c of either phase we use finite size scaling techniques [148], measuring the size dependent p c (L) for various L and using the ansatz: p c = p c (L) + al b. The number of realisations was dependent on the sample volume; in general the number of realisations was chosen to obtain p c (L) to within a standard error of 1 4. As an example, for a large lattice (L = 4), 3 realisations were required, compared to realisations for L = 1. Results are summarised in

98 Development of Boolean reconstruction methods 61 Table 3.9: Comparison of the percolation thresholds between the complex 5-mix model and its matching Boolean model defined in Tab Phase p original c Particulate Inverse p stochastic c Tab The error in the predictions of p c for the equivalent stochastic model is.3% for the particulate phase and 1% for the inverse phase. This ability to closely predict p c of either phase from one 3D image at any phase fraction underlines the power of a morphological characterisation and reconstruction based on the integral geometric measures for Boolean models. In the following chapters we will show that the transport and mechanical properties of the original 5-mix and stochastic BG (2) also agree well Application to sandstone images Having illustrated the ability of the method to reconstruct complex Boolean models, we now attempt to accurately reconstruct experimental 3D microtomographic images of sandstone cores. We note that the microstructure of a sandstone is a result of a complex physical process which can include consolidation, compaction and cementation of an original grain packing. More complex and realistic models of sandstones have been derived [151, 82, 1, 113]. These methods require however, the simulation of the generating process including primary grain sedimentation followed by a diagenetic process such as compaction and cementation. This process is both computationally expensive and requires several fitting parameters. Reconstructing the microstructure of sandstones by a simplistic Boolean model may therefore not lead to the excellent match observed in the previous section. However, a Boolean sphere pack has been proposed as a model which gives a reasonable representation of consolidated sandstone and yields good qualitative information on structure/property relationships. Moreover, other work [14, 9, 4, 155] has shown that the Boolean sphere model gives a very good match to Fontainebleau sandstone data. In the remainder of this section the Boolean reconstruction method is applied to the cross-bedded and Fontainebleau sandstone samples. Grain ensembles will be determined, which match the global morphological Minkowski measures of the rock images. We compare the reconstructions of three different Boolean models; a sphere pack with matching 2-point information, the equivalent Boolean ensemble

99 62 Morphology of digitised disordered materials defined by local morphological measures, and a recent model based on the probability density of covering spheres [155]. Differences between the sandstone samples and the reconstructions are discussed Fontainebleau sandstone The Fontainebleau data sets were described in section Here we derive a number of equivalent Boolean ensembles based on different morphological measures. For the Boolean IOS model a matching single sphere size is first derived by a fit to the two-point correlation function introduced in section (IOS C model, see also Tab. 3.11). The sphere sizes are derived for each of the four images of the Fontainebleau sandstone. A single equivalent sphere pack can also be determined by using the Boolean reconstruction technique (Eqn. 3.12) based on the local grain measures V ν (IOS (1) model). 4 The local grain measures of the Fontainebleau sandstone samples are summarised in Tab. 3.1, and the best matching IOS (1) models in Tab As was observed previously for the heterogeneous 5-mix system with a single prismatic grain, a single Boolean sphere does not give a good match to the data (Tab. 3.11). To gain some degrees of freedom, we attempt to fit a single equivalent spheroidal grain (ROS (1) model) to the local V ν. We however see only a slight improvement in the prediction. This inability to match may be due to the finite resolution with which the spheroids and their rotations are resolved. For each possible spheroid the average local measures have to be derived at 18 rotations per axis for each spheroidal size (see appendix B.2) due to limits in computational time the half-axes of the spheroids were resolved at a = b = c =.1 voxel and the maximal half-axis length set to a = 2, resulting in of the order of 1 14 vertex evaluations. It is therefore necessary to use two grains to accurately match the local measures. This is done first for two spheres (IOS (2) ), as here the local measures neither have to be averaged over rotations nor are there different half-axis lengths, and the sphere radii were resolved at up to r =.1 voxel. This leads to 4 All model structures based on the Boolean reconstruction method outlined in this section carry a numeric index noting the number of different grains used for the reconstruction. Table 3.1: Local Minkowksi measures of the four Fontainebleau sandstone samples of size 48 3 given in section (V 3 = 1 for compact grains). Sample V [1 3 mm 3 ] V 1 [1 3 mm 2 ] V 2 [mm] fb fb fb fb

100 Development of Boolean reconstruction methods 63 Table 3.11: Parameters for the Boolean models of the Fontainebleau sandstone and errors of the morphological matches. The first IOS model (IOS C ) was matched using the void-void correlation function. The other models are matched using the Boolean reconstruction. IOS is generalised to more than one sphere, all spheroids are randomly oriented (ROS). Note the large errors E Vi in the local measures for the 1-particle IOS model; the cubic equation (B.3) has no real solution. For all two-particle models one of the particles is very small and has a width of the order of a few voxels. a, b, c note the length of the half-axes with the exception of the OSC model, where a notes the median of the radius distribution G(r c ) = 1/2 (see appendix B.3). Model Core p a [µm] b [µm] c [µm] E V E V1 E V2 IOS C fb fb fb fb IOS (1) fb fb fb fb IOS (2) fb fb fb fb ROS (1) fb fb fb fb ROS (2) fb fb fb fb OSC fb fb fb fb

101 64 Morphology of digitised disordered materials a better match, but the error in V 2 is still considerable. We then derive the best two-particle match for Boolean spheroids (ROS (2) model). The match to the local V ν of the experimental image is now excellent (Tab. 3.11). It is interesting to note that the unique information obtained from the integral geometric measures leads to a quite complex equivalent stochastic model for the Fontainebleau sandstone. The ROS (2) model is composed of two very different particle sizes (Tab. 3.11). Alternative methods to generate a Boolean ensemble exist for model microstructures based on spheres. Very recently Thovert et al. [155] introduced a reconstruction technique based on a Boolean sphere pack for sandstone where the sphere size distribution is defined by the probability density of the covering radius for spheres (OSC model). This method is employed to generate a third equivalent Boolean model for the four sandstone images. Details of the analysis and the resultant distributions of covering radius are given in appendix B.3. The median of the radius distributions (G(r c ) = 1/2) is given in Tab This median is comparable to the size of the larger particles in the two-particle Boolean reconstructions. The original sandstone microstructure for the sample fb15 and various reconstructions via Boolean models are illustrated in Fig Visual inspection suggests that the ROS (2) and OSC model more closely resemble the original microtomographic image. We now compare the global Minkowski functionals of the models. An analysis of the full 48 3 cubic subsets would only give a single value for the porosity of each of the four samples and would provide little data to compare to stochastic models. However, as shown in section 2.2.1, the samples are reasonably heterogeneous in the pore volume fraction. Due to this natural heterogeneity and by appropriately choosing different window sizes on the image it is possible to measure morphological parameters for the sandstone images across a range of pore volume fractions. This gives us a more comprehensive data set with which to compare experimental images to equivalent Boolean models. For the Fontainebleau samples we consider cubic blocks of 48 3, 24 3 and This provided in all cases a good spread of porosities across different sampling volumes. The measured morphological properties v ν () are summarised in Fig We note with interest that the fourth Fontainebleau data set fb22 exhibits very different measures to the first 3 sets. This could indicate the potential presence of heterogeneity, as seen in Fig. 5.(c) in Thovert et al. [155]. Another explanation would be the existence or different weighting of another diagenetic process for this sample (i.e. dissolution or a lower energy environment leading to smaller grain sizes and better sorting). As indicated by the relative error in the local Minkowski measures (Tab. 3.11) the ROS (2) model almost exactly matches the local measures of the sandstone sam-

102 Development of Boolean reconstruction methods 65 [a] [b] [c] [d] [e] [f] Figure 3.17: Visual comparisons of reconstructions to a 24 3 subset of the original Fontainebleau sandstone sample fb15 ( = 18.2%). [a] Fontainebleau sandstone, [b] IOS C, [c] ROS (1), [d] IOS (2), [e] ROS (2), and [f] OSC.

103 66 Morphology of digitised disordered materials IOS C 4 OSC 4 ROS (2) v 1 [mm 1 ] fb7.5, 12 3 fb13, 12 3 fb15, 12 3 fb22, 12 3 all, 24 3 all, v 1 [mm 1 ] fb7.5, 12 3 fb13, 12 3 fb15, 12 3 fb22, v 1 [mm 1 ] fb7.5, 12 3 fb13, 12 3 fb15, 12 3 fb22, 12 3 all, 24 3 all, v 2 [mm 2 ] 3 2 v 2 [mm 2 ] 3 2 v 2 [mm 2 ] IOS C 1 OSC 1 ROS (2) v 3 [mm 3 ] v 3 [mm 3 ] v 3 [mm 3 ] IOS C 4 OSC 4 ROS (2) Figure 3.18: Minkowski measures over fraction for each of the four Fontainebleau sandstone samples compared to the matching Boolean models. [a] (left) IOS C, [b] (middle) OSC, and [c] (right) ROS (2). For v 2 and v 3 the measures for the highly connected neighbourhood are given. ples. The IOS C and OSC model perform poorly, particularly for v 2, where IOS C overestimates, and OSC underestimates the image data (see Fig. 3.18). Another measure of the accuracy of the reconstruction is based on a comparison of the variation of the pore volume fraction per slice. This gives one a measure of the heterogeneity of the sample. The two best models based on the Boolean reconstruction are compared to the common IOS C model and the OSC model in It is difficult to distinguish which reconstruction gives the best match. The models ROS (2) and IOS C both do well, while the OSC model gives enhanced heterogeneity. An attempt to quantify the differences is given in Tab by measuring the variability of the pore volume per slice for the Fontainebleau sandstone and its reconstructions. No firm conclusions can be drawn, as the measures itself vary too strongly (see the column for the x,y,z-variability of the Fontainebleau data in Tab The OSC model is too heterogeneous and gives the worst match, while ROS (2) and IOS C give

104 Development of Boolean reconstruction methods 67 reasonable agreement..3.2 Fontainebleau.3.2 IOS C Slice.1 Slice z [mm] ROS (2) z [mm] OSC Slice.1 Slice z [mm] z [mm] Figure 3.19: Comparison in the variation of the porosity distribution along the z- axis of the Fontainebleau samples and reconstructions. Sample size is the same as in Fig Table 3.12: Variability σ z ( sl ) [%] of the pore volume per slice along the x,y,zdirections of the Fontainebleau samples and the z-directions of its reconstructions (see also Tab. 2.1, the values are from a single realisation at 48 3 ). Sample Fontainebleau IOS C ROS (2) OSC fb7.5.78/.91/ fb13.76/.95/ fb /.95/ fb22 1.2/1.13/

105 68 Morphology of digitised disordered materials Cross-bedded sandstone We adopt a similar procedure to analyse the cross-bedded sandstone data set described in section Two cubic blocks of 3 3, 16 blocks at 15 3, 54 blocks at 1 3 and 25 blocks at 6 3 were obtained from the full sample volume. This provided a spread of porosities across different sampling volumes. With decreasing sampling volume the variability of the measures increases, but the values are consistent with the data for the larger volumes, suggesting that for the smaller blocks a meaningful average is determined. The global morphological measures v ν (f) of the solid fraction of the sandstone are v =.863, v 1 = 1.82 mm 1, v 2 = 15.5 mm 2, and v 3 = 2.73 mm 3. An IOS C model microstructure is generated by matching two-point correlation information as described in the previous section. The equivalent local grain measures V ν for the first 3 3 sample of the sandstone are V =.464 mm 3, V 1 =.34 mm 2, and V 2 =.199 mm. Attempts to fit a single spherical grain to these values are again not successful (Tab. 3.13). The best two-particle matches for Boolean spheroids are given in Tab The IOS (2) ensemble gives an excellent match to the average volume of the grains, but less so for average surface area and mean curvature. The ROS (2) system leads to an excellent match of all V ν of the experimental image. The equivalent OSC model is also given. Results for the morphological measures are summarised in Tab and Fig Again the IOS C model and the OSC model don t give a good match to the local grain measures V ν, leading to a poor match of the global v ν. In particular, IOS C overestimates v 1 and v 2, while OSC underestimates v 1, v 2, and v 3. ROS (2) closely Table 3.13: Parameters for the Boolean models of the cross-bedded sandstone. The first IOS model (IOS C ) was matched using the void-void correlation function. The other models are matched using the Boolean reconstruction. IOS is generalised to more than one sphere, all spheroids are randomly oriented (ROS). a, b, c note the length of the half-axes with the exception of the OSC model, where a notes the median of the radius distribution G(r c ) = 1/2 (see appendix B.3). Model p a[µm] b[µm] c[µm] E V E V1 E V2 IOS C IOS (1) IOS (2) ROS (1) ROS (2) OSC

106 Development of Boolean reconstruction methods IOS 2. OSC 2. ROS v 1 [mm 1 ] samples of samples of samples of samples of 6 3 v 1 [mm 1 ] samples of samples of samples of samples of 6 3 v 1 [mm 1 ] samples of samples of samples of samples of IOS C IOS (1) IOS (2) 25 2 OSC 25 2 ROS (1) ROS (2) v 2 [mm 2 ] 15 1 v 2 [mm 2 ] 15 1 v 2 [mm 2 ] IOS 5 OSC 5 ROS v 3 [mm 3 ] v 3 [mm 3 ] v 3 [mm 3 ] 1 IOS 1 OSC 1 ROS Figure 3.2: Minkowski measures over fraction for a cross-bedded sandstone compared to the matching Boolean models. [a] (left) IOS model matched using the void-void auto-correlation function as well as Boolean reconstructions with one or two spheres. [b] (middle) Boolean models of rectangular lattice oriented bars. [c] (right) Boolean models of randomly oriented spheroids and the OSC model. For v 2 and v 3 the measures for the highly connected neighbourhood are given. represents the curved interface of the original structure. This is consistent with a visual comparison of the cross-bedded sandstone to its Boolean reconstructions given in Fig We also compare the porosity variation of the original image and its reconstructions. In Fig the porosity variation along the x-axes of the two 3 3 cross-bedded sandstone samples are compared to the porosity variations of the reconstructions. The reconstructed images based on the two-grain models exhibit heterogeneity which is not observed in the IOS model (see Fig and Tab. 3.14). The OSC model gives a good match to the heterogeneity of the cross-bedded sandstone along this direction. This is different to the Fontainebleau sandstone match discussed previously, where OSC overpredicts heterogeneity and a higher hetero-

107 7 Morphology of digitised disordered materials [a] [b] [c] [d] Figure 3.21: Visual comparisons of reconstructions to a 15 3 subset of the original cross-bedded sandstone sample ( = 18%). [a] Cross-bedded sandstone, [b] IOS C, [c] ROS (2), [d] OSC. The domain size is comparable to the 24 3 Fontainebleau datasets shown in Fig geneity may be an inherent feature of the OSC model. However, the x-axis of the cross-bedded sandstone is the direction least affected by long-range heterogeneity (see Fig. 2.16). None of the isotropic reconstructed images are as heterogeneous as the original rock. This is not surprising. Cross-bedding is a result of a complex geological deposition process in a fluvial environment, where i.e. flow channel geometries change over time, and long-range heterogeneity and correlations result. The Boolean process does not mimic the complex physical processes which lead to the observed structure of a cross-bedded sandstone. In this section we have shown that for the Boolean model of random composite media one can define a set of measures from a single image at any phase fraction which allows one to accurately reconstruct the medium for all other phase fractions. The accuracy of the reconstruction is illustrated by accurately predicting p c for the reconstructed data sets. In following chapters it will be shown, that this leads to accurate predictions of property curves for both conductive and linear elastic prop-

108 Development of Boolean reconstruction methods Cross bedded sandstone.3.2 IOS C Slice.1 Slice x [mm] ROS (2) x [mm] OSC Slice.1 Slice x [mm] x [mm] Figure 3.22: Comparison in the variation of the porosity distribution along the x- axis of the cross-bedded sandstone and reconstructions. Sample size is the same as in Fig Table 3.14: Variability σ( sl ) [%] of the pore volume per slice along different directions of the 3 3 cross-bedded sandstone samples and its reconstructions (see also Tab. 2.2). The reconstruction reflecting heterogeneity best is the OSC model. Sandstone IOS C ROS (2) OSC x: 1.51/1.85 y: 1.71/ ± ± ±.8 z: 3.1/2.3

109 72 Morphology of digitised disordered materials erties across all phase fractions. We have illustrated the method for experimental sandstone images and showed that, in general, the Boolean reconstruction based on integral geometric measures gives a better match to the morphology than the commonly used IOS C and the recently developed OSC model. Moreover, unlike the IOS C and OSC models, the current method is not limited to spherical inclusions, but can be used to generate more complex inclusion shapes. However, the morphological reconstruction of sandstones with Boolean models is not giving as accurate a match, due to a poor representation of sandstone by Boolean models. A development that will further highlight the utility of the proposed technique is based on conditioning the equivalent ensemble to better mimic the local morphology of the medium. This could be done by conditioning to local curvature measures or chord-length distribution measurements which will limit the choice of grain shape to use in the equivalent Boolean ensemble. In oil recovery from petroleum reservoir rocks, an area of particular interest to the authors, recovery depends crucially on the mean curvature of the surfaces where immiscible phases meet at a contact angle. Conditioning an equivalent Boolean ensemble to exhibit the same distribution of local mean curvature should lead to excellent prediction of multiphase flow properties on reconstructed images. The current work is based on deriving equivalent local measures from an image for Boolean grain models; Eqn Extension of the methodology to more general systems; hard-sphere mixtures, soft sphere models and models based on Gaussian random fields representing a wider range of complex materials including ceramics, composite materials and fibrous media is now being considered. 3.3 Characterisation of structures using the geometry of parallel sets The statistical characterization and modeling of disordered microstructures is a central problem in several applied fields. Accurate modeling relies on the availability of good microstructural models, which in turn relies on accurate statistical characterization. Complete characterization of the effective morphology however requires knowledge of an infinite set of n-point statistical correlation functions. As discussed previously, in practice only lower order morphological information is available; common methods [69] are based on matching the first two moments (volume fraction and two-point correlation function) of the binary phase function to a random model. It is widely recognised that although the two-point correlation function of a reference and a reconstructed system is in good agreement, this does not ensure that the structures of the two systems will match well. Other useful characterizations of microstructure include the chord-length dis-

110 Characterisation of structures using the geometry of parallel sets 73 tribution function [145, 131] (and the related lineal-path function [171]) and the pore size distribution function [137]. However reconstructions of experimental data sets based on these characterisations have been shown to give a poor representation of the connectivity of the systems [84]. Functions that may provide more complete information about connectivity [62] are unfortunately too complex to incorporate into reconstruction schemes [84]. Incorporation of three- and four-point information may lead to a better estimation of structure, but their measurement is very complex and it is not clear how to incorporate the information within reconstruction algorithms. There is a need for statistical measures which include higher-order correlations, but are fast and reliable for characterizing the morphology of a structure. In the first section we introduced the MFs as a complimentary measure of material morphology. While strong differences were observed between classes of models, in some cases MF s were unable to discriminate measures for the same model class (e.g. Boolean models in Fig. 3.7). In this section we show that a more powerful discriminator of morphology is based on the evolution of the MFs during erosion and dilation operations. This methology has been used previously to discriminate between different cosmological models of the early universe [11]. In this cases, the MFs of parallel bodies of a point pattern (a convex set) was considered. The point pattern was dilated to obtain detailed morphological information where the parallel distance ɛ was used as the diagnostic parameter. In the current section we use the evolution of the Minkowski functions (Mfs) during erosion and dilation operations on complex non-convex morphologies. It is shown that this gives a discriminating signature of structure. Further we consider the same range of disordered morphologies as in section 3.2 and use the method to determine the accuracy of model reconstructions of random systems. We also consider the morphology of two experimental systems and use the method to optimally match the reconstructed model morphology Mathematical background The parallel body of a structure A is defined as the set of all points x with distances d(a, x) less than ɛ to A A ɛ = { x; d(a, x) ɛ}. (3.22) Here, d(a, x) = Min ( x y ; y A) denotes the minimal Euclidean distance where x is the standard norm in a d-dimensional Euclidean space. Changing ɛ corresponds to dilation ɛ > and erosion ɛ < of the spatial structure. For negative values of ɛ the parallel body is defined formally by A ɛ< = { x; d(a c, x) ɛ} c, i.e.,

111 74 Morphology of digitised disordered materials as the complement set of all points with distances less than ɛ to the complement A c of A. Alternatively, one may define the parallel body A ɛ = x A B ɛ ( x) (3.23) by the union of all spheres B ɛ ( x) of radius ɛ and centers x A inside of A. For ɛ < the parallel body is given by the set of all centers x so that the union A = x Aɛ B ɛ ( x) equals A. Mathematically the erosion/dilation operators can also be introduced in terms of Minkowski addition and subtraction operators. The Minkowski addition C = A B of two sets A and B consists of all points c = a + b which can be written as a sum of points a A and b B contained in A and B. A dilation A E of a set A by a structural element E can then be written as A E = X E(ɛ), where ɛ notes the characteristic size of the structuring element. If we take as the structuring element a d-dimensional sphere of radius ɛ, and grow the radius of the sphere to run through all possible radii until the structure becomes a one-phase medium, we can write this as an operation A ɛ = A ɛb with a unit sphere B multiplied by the actual dilation radius. Analogous to the Minkowski addition the Minkowski subtraction A B may be defined as consisting of all points c such that to each point b B one can find the points a A with c = a b. If no such point exists, the subtraction A B = is empty. In other words, the difference is given by A B = { x R d, x B A} or, alternatively, by the Minkowski addition A B = (A c ( B)) c with the complement A c where B denotes the inverted set. One should note that dilation and erosion cannot be exchanged, i.e., that (A ɛ ) ɛ A which is clearly visible in Figure Although the Minkowski addition is commutative, associative and distributive with the union as second operation, i.e., (A B) C = (A C) (B C), these relations do not hold for intersections but only a monotonic relation (A B) C (A C) (B C). The equality is valid, if A B and C are convex grains. Also one finds that A B A ( B) and that (A B) B A is only an subset of A. Measuring the Minkowski functions v ν (ɛ) of a body one obtains detailed morphological information with ɛ used as a diagnostic parameter. For a convex body K in d-dimensional Euclidean space the d-dimensional volume v (ɛ) of its parallel body K ɛ is given by the MFs. Using Steiners formula [136, 52] d ( ) d V (K ɛ ) = V (K Bɛ d ) = ω ν V ν (K)ɛ ν (3.24) ν with ω = 1, ω 1 = 2, ω 2 = π, and ω 3 = 4π/3 one finds a polynomial in ɛ where the coefficients are given by the local MFs V ν (K) of the grain K. In general, one finds ν=

112 Characterisation of structures using the geometry of parallel sets 75 Figure 3.23: Illustration of the differences in the erosion/dilation operation on a convex set and a non-convex set. In [a] we start from a collection of convex shapes (point pattern) and dilate first to (r ɛ) obtaining the black pattern and then further to (r) =.5 resulting in the grey pattern. In [b] beginning from the nonconvex (grey) pattern at (r) =.5, we obtain a different morphology at (r ɛ). Enhanced connectivity between overlapping sphere centers is one obvious qualitative difference. Domain size is , with r = 46.5 and ɛ = 15.5 in pixel units. The black phase on the left side has a fraction of =.134 and 34 components (periodic boundaries). In contrast the black phase on the right side has a fraction of =.156 and 2 components. for the Minkowski functionals of the parallel body K ɛ d ν ( ) d ν ων+µ V ν (K ɛ ) = V ν+µ (K)ɛ µ µ ω ν. (3.25) µ= Thus, if one knows the MFs v ν (K) = v ν (ɛ = ) of a convex structure K, one knows the MFs v ν (ɛ) for arbitrary ɛ, and conversely, if one measures the parallel volume v (ɛ) of a convex shape K, for instance, one can infer its MFs v ν (K). This is not the case for irregular non-convex shapes A. In this case the Minkowski functions v ν (ɛ) of the parallel body A ɛ contains more information about the spatial structure than just the MFs v ν (A) = v ν (ɛ = ) at zero dilation. particular, applying Eqn. 3.9 (page 34), contributions from every order n are contributing for distances ɛ. The parallel body includes all points of distance less than ɛ to A which depends in turn on non-local properties of A such as narrow throats or bottle necks. This is illustrated in Fig Thus the parallel surface A ɛ contains spatial information about the embedding of the surface A in space in addition to the MFs of the grains A. Parallel bodies may therefore be used to describe the morphology of irregular, non-convex patterns beyond the additive MFs. One should keep in mind that v ν (ɛ) is not unique and still depends on the shape of K so that often the notation v ν (ɛ; A) is used. In particular, one should note that v ν (ɛ; A ɛ ) v ν (ɛ + ɛ ; A). In

113 76 Morphology of digitised disordered materials A direct implementation of Equation (3.23) is computationally very inefficient as it requires one to place spheres of different size along all voxels at the interface at each erosion/dilation step. The algorithm developed instead utilises the definition given by Eqn. (3.22) using an Euclidean Distance Transform (EDT, see appendix A, i.e., a mapping giving the Euclidean distance (1, 2, 3, 2,..) of each voxel in either phase to its nearest interface. From the EDT the erosion/dilation is trivially implemented by stepping through the different Euclidean distances from the interface Characterisation of model systems To illustrate the method we consider the evolution of the Mfs v ν (ɛ) for the different model media beginning from an image A (ɛ = ) at the symmetric phase fraction of =.5. In order to introduce a common scale we always consider the morphological measures v ν () as function of the porosity = v (ɛ) with < < 1 instead of ɛ with the possible range < ɛ <. Results for the Boolean grain models are given in Fig where we define as the phase fraction of particles and choose the particles to have preferred connectivity. We consider first the differences in the measures for the two systems illustrated in Fig. 3.23; the erosion/dilation operation on a Poisson point pattern A = { x i } shown in Fig a and the operation on the IOS model A = i B( x i ) at =.5 shown in Fig b. For >.5 the configurations of the two processes are identical; differences are only due to numerical discretisation effects and the use of different EDMs for the two stochastic processes. But direct comparison for <.5 shows distinct features, especially in the topological measure v 3 (ɛ). It has been shown previously [96] that a negative value of v 3 is indicative of the bicontinuity of the phases. For the Poisson point pattern one finds v 3 > for <.2 in close agreement with the percolation threshold of the IOS model. In contrast, v 3 remains negative for smaller values of for the data set eroded off A = { x i } with = 5%. This reflects the structure noted in Fig where interconnections between overlapping spheres is enhanced. In section 3.2 we calculated the MFs for a range of Boolean grain models at different porosity [7]. Only very small differences between measures were observed for overlapping grains such as uniform and random spheroid packs. Therefore, the Mfs v ν (ɛ) are similar for >.5 because the dilated configurations ( i K i ) ɛ can 5 This is not an exact implementation of parallel bodies the distance is measured between voxel centres, while in principle the distance to the surface should be used, which would require triangulations or the analysis of vertex configurations, resulting in a more complex algorithm. For the purpose of discrimination of morphology, at large enough resolution, this approximation is reasonable.

114 Characterisation of structures using the geometry of parallel sets ε [a] Poisson point process IOS random spheroids aligned spheroids 1 v 1 [a 1 ] v 2 [a 2 ] v 3 [a 3 ] Figure 3.24: Boolean Models: Minkowski functionals of different Poisson processes starting at =.5. The point process and IOS have exactly the same particle centres for each of the realisations, such that the processes have the same Poisson density at =.5. In case of the former the process is started with 777 points at. In contrast the spheres of the IOS models have a radius of r = 12. The spheroids of the spheroidal models have half axes of average length r = 12 with r [4, 2]. See also Fig still be resembled by overlapping convex bodies i (K i ) ɛ (see the previous discussion of the distributive Minkowski addition). In contrast the Mfs v ν (ɛ) for uniform sphere packs have a distinct signature to spheroids with a distribution of radii for <.5 as illustrated in Fig Preferential alignment has little effect on the measures; however, measuring the Mfs along different axes would clearly allow one to distinguish isotropic systems from those preferentially aligned. The curves for different GRF models are given in Fig The 1-level cut model is symmetric around =.5 for v 1. The asymmetry for the two other measures is due to our choice of preferred connectivity of one phase. For the two cut models we define as the fraction of the phase where α y(r) β and choose this phase to have preferred connectivity. The curves for different GRF models exhibit distinct behavior and coincide only for very small or large porosity since

115 78 Morphology of digitised disordered materials 2.5 ε [a] 1 1 level cut 2 level cut intersection 1 v 1 [a 1 ] v 2 [a 2 ]..1 1 v 3 [a 3 ] Figure 3.25: Gaussian models: Minkowski functionals of erosion/dilation operations over phase fraction for GRF models having kernel g(r) as in Eqn. 2.8 with r c = 2, ξ = 4, d = 3 in pixels, and periodicity t = 1 on a grid. The erosion/dilation operations are started at an initial fraction of =.5. The models are shown in Fig the morphology of the remaining holes for ɛ 1 are similar. The difference in v 1 for the models is mainly due to the different surface area at the initial porosity =.5. One should note particularly the pronounced trough in the connectivity measure, v 3 (), at.21 for the two-level-cut model which indicates a net-like structure, i.e., a highly inter-connected skeleton structure. The curves for the Voronoi models are given in Fig Here, the porosity describes the fraction of the facet-phase which has preferred connectivity. Construction of the Voronoi models A = i (A i ) ρ is defined by parallel bodies (A i ) ρ of radius ρ to a set of two-dimensional polyhedra A i = P i (facet model) and one-dimensional lines A i = L i (cylinder model), respectively. It is instructive to compare the MFs v ν (ρ) for the original model A ρ = i (A i ) ρ to the Mfs v ν (ɛ) = v ν ((A ρ ) ɛ ) based on erosion and dilation from the Voronoi models at radius ρ with = v ( ρ) = 5%.

116 Characterisation of structures using the geometry of parallel sets Voronoi facets Voronoi tubes/cylinders.3 MF (densification process) Mf (erosion/dilation operation) ε [a] 1 1 v 1 [a 1 ] MF (densification process) Mf (erosion/dilation operation).5 MF (densification process) Mf (erosion/dilation operation) 1 v 2 [a 2 ].5. 1 v 3 [a 3 ] Figure 3.26: Voronoi models: Morphological measures (MFs and Mfs) over phase fraction for Voronoi facet and foam models. Compared are the densification process against the erosion/dilation process started at an initial fraction of =.5. The Voronoi cells are generated by 1 seeds periodically mapped on a 2 3 lattice. See also Fig For the variation in = v (ρ) of the original facet model A ρ, the connected polyhedra phase always percolates, and the other phase is made up of disconnected inclusions. Therefore the system remains convex v 3 (ρ) > for all. Accordingly, the connectivity v 3 (ɛ) = v 3 ((A ρ ) ɛ ) for the dilated configurations (ɛ > ) with = v ((A ρ ) ɛ ) >.5 is positive too which is shown in Figure In contrast, the behavior of v 3 (ɛ) for the erosion (ɛ < ) of the facet phase A ρ with = 5% exhibits a negative value at 15% implying a bicontinuous system at quite substantial facet phase fractions. This bicontinuity is due to the formation of holes in the facet phase and continuity of the nodes where the facets edges meet and is a consequence of the non-convex shape of the original structure. The curves for the Voronoi cylinder model does not differ significantly indicating that the open-cell foam model maintains a convex-like structure.

117 8 Morphology of digitised disordered materials Discrimination of model composites To illustrate the powerful discrimination of morphology which comes from measuring the Mfs of parallel bodies we follow Roberts [131] and select two model images and generate a range of equivalent model morphologies via matching first and second order statistical properties. This method was used in Ref. [131] to illustrate the relatively strong signature provided by the chord distribution function (CDF). We illustrate the strong discrimination which is possible when using Mfs for these systems. In Ref. [131] GRF systems are considered; the single cut model N (c=) (see Eqn. 2.13) and the symmetric two-cut model N (c=)1 (Berk s original model [17]). Other models considered can be formed from any number n of intersection I n and union U n sets of the GRF models. One then considers a specific GRF model with known statistical properties as the experimental data set. Reconstructions are generated by matching the volume fractions and by minimising the difference in the 2-point correlation functions Ep (2) = M i=1 [p(2) fit (r i) p (2) exp(r i )] 2 M i=1 [p(2) fit (r. (3.26) i) p 2 exp] 2 where M is the number of experimental points to be fitted. Numerical integration is used to find the best fit p (2) (r) for a number of different GRF models. The minimisation algorithm defines r c, ξ and d for each model. All the models are able to provide an excellent fit to the data based on matching the two-point information. In Ref. [131] the CDF is used to attempt to better discriminate the best matching model morphology. We use the Mfs v ν (ɛ) to measure the best morphological fit to the data. In the first case the experimental data set was based on a symmetric two-cut model (N 1 ) at = 2% obtained from the field-field correlation function [131]: g(r) = e ( r/l ) 2 ; ρ(k) = l3 /2)2 e(kl (4π) 2/3 (3.27) with l = 2µm. Attempts to match the experimental data set were made for models N, N 1, U1 2, I1. 2 The resultant values of r c, ξ and d for four different models are given in Table I of Ref. [131]. We show in Fig the Mfs for the parallel sets of each model morphology and the original experimental data set. As in the examples discussed previously we consider the morphological measures v ν () as a function of the porosity = v (ɛ) with < < 1.

118 Characterisation of structures using the geometry of parallel sets ε [a] exp I 1 2 N N 1 U v 1 [a 1 ] v 2 [a 2 ].1. 1 v 3 [a 3 ] Figure 3.27: Reconstruction of an experimental Gaussian model (N 1 ): Minkowski functionals of the erosion/dilation process for the experimental data set and the corresponding Mfs for the matching model data sets defined in Table I of [131]. exp corresponds to the experimental image, and the lines give the best fit to the model morphology. In Tab we quantify the error in the measure of the MFs for the different model morphologies by employing a relative least squares error: 1 E i = ( v i()) 2 d 1 (v i()) 2 d. (3.28) Since the porosity values are not identical for all ɛ, linear interpolation is used to do the resampling. We also show in Tab the errors Eρ 1 and Eρ 2 in the chord distribution function (CDF) measure from Ref. [131]. We observe that the Mfs provide a far stronger signature of morphology than the CDF. While the CDF does discriminate between some models, it is impossible to choose between model N 1 and U 1 as the best reconstruction for the two-level cut model. For all four measures of the Mfs the actual model morphology is more than two orders of magnitude more accurate than any other. It is important to note that we could discriminate morphology from one- or two-dimensional information (v 1 and v 2 ) as readily as

119 82 Morphology of digitised disordered materials Table 3.15: Integral error of the Minkowski measures over fraction (Eqn. (3.28)) for the models defined in Table I of Ref. [131]. The last two columns show the error in the CDF from Ref. [131]. M E 1 E 2 E 3 Eρ 1 Eρ 2 N N I U from v 3. Therefore the method can be used to discriminate morphology from twodimensional micrographs of the material or from 1D chord length measurements. In the second case the experimental image was a 1-level cut model (N ) and attempts to match this data set were made for models N, U 2, I. 2 Again the Mfs give a very strong signature of morphology and allow one to discriminate the correct morphology (see Fig and Tab. 3.16). 2 1 exp I 2 N U ε [a] 1 v 1 [a 1 ] v 2 [a 2 ] v 3 [a 3 ] Figure 3.28: Reconstruction of an experimental Gaussian model (N ): Minkowski functionals of the erosion/dilation process for the experimental data set and the corresponding Mfs for the matching model data sets defined in Table II of [131].

120 Characterisation of structures using the geometry of parallel sets 83 Table 3.16: Integral error of the Minkowski measures over fraction (Eqn. (3.28)) for the models given in Table II of Ref. [131]. M E 1 E 2 E 3 Eρ 1 Eρ 2 N I U The Mfs appear to provide a strong signature of composite morphology and hence may provide a method for selecting an accurate reconstruction of a disordered material. In the next section we attempt to do this for the experimental tomographic images of a termite nest and the Fontainebleau sandstone data sets Application to experimental data Termite nest The morphology of the termite nest (Fig. 2.17) is reminiscent of the structure of the GRF fields. The GRF model is therefore a natural choice for describing the morphology of this system. We first attempt to match the simple GRF models (N and N 1 ) and the IOS model to this structure to ascertain which model gives the most realistic representation. As in the previous section we match p (2) to determine r for the IOS model and d, ξ and r c for the GRF models. The p (2) for the experimental image and the model fits are shown in Fig a. From visual inspection of Fig one might expect an oscillatory correlation function indicating a regular domain size d. The experimental p (2) exhibits no.5.5 p (2) (r).4.3 Sr IOS 1 level cut 2 level cut p (2) (r).4.3 Sx Sy Sz Sr r [mm] r [mm] Figure 3.29: [a] Two-point correlation function of the termite nest and the best fits for the three models. [b] Correlation function measured along the three axes and averaged (S r ). Data along the x-axis is given in number of voxels.

121 84 Morphology of digitised disordered materials 1.1 ε [mm] 5 termite nest IOS 1 level cut 2 level cut v 1 [mm 1 ] v 2 [mm 2 ].1 v 3 [mm 3 ] Figure 3.3: Termite Nest: Minkowski functionals of the erosion/dilation process over fraction. Note, that here no scaling factor is applied (l = 1) to honour the natural length scale of the experimental data set. apparent domain size. We plot in Fig b the correlation function along the three axes. We note that the deviation is quite strong indicating either anisotropy or more probably, poor sampling due to the small size of the image. We therefore choose to fit the model p (2) (r) out to only r = 11 mm. As before, based on p (2) (r) data alone, one cannot discriminate between the goodness of the model fits. We first consider the match of the Mfs v i (ɛ) to the experimental data set for the IOS model and the one- and two-level-cut GRF models (N and N 1 ) in Fig The parameters of the models, r for IOS and d, ξ and r c for the GRF models, are reported in Tab along with the error E(p (2) ) and corresponding error estimates defined by Eqn The measure clearly discriminates the GRF models as the best model for the termite nest. It is difficult to discern which of the two simple GRF models gives the best fit to the experimental image. Since the volume fraction of the GRF models is a function of both α and β there are a number of choices which correspond to a given volume fraction. Any combination

122 Characterisation of structures using the geometry of parallel sets 85 Table 3.17: Integral error of the Mfs for the models of the tomographic image of the termite mound calculated according to Eqn. (3.28) compared to the relative error in the 2-point correlation function Ep (2) and parameters of the matching models N c, with c [, 1] being the level cut parameter, for the termite nest in mm. The original tomographic data set image has resolution at 1mm per voxel. For the IOS model, the equivalent sphere radius is in the r c column. Model r c [mm] ξ[mm] d[mm] Ep (2) E 1 E 2 E 3 IOS N N N N N N N N of (α, β) in Berk s model can be used to generate a structure with the same as the experimental data set. We therefore match the morphology of the experimental image to asymmetric (α β) GRF models with a range of c as defined by Eqn. (2.13). Also summarised in Tab are the values of d, ξ and r c that best match the experimental image along with the error estimate E(p (2) ) and the error in the Mfs for the range of asymetric GRF models. The best fit is for the asymetric twolevel-cut model with small c, c [.1,.125]. Unfortunately the discrimination of the fit is not comparable to that seen for our model systems. We believe this is primarily due to the limited size of the experimental data set. As can be seen in Figs , the size of the image is mm, while the typical feature size is of the order of 1 mm Fontainebleau sandstone In the previous section (3.2) three different Boolean reconstructions of the Fontainebleau samples were derived. One system gave an excellent match to the global Minkowski measures of the sandstone samples. Here the three different Boolean ensembles (IOS C, ROS (2), and OSC) and a fourth stochastic model for Fontainebleau sandstone, based on a Gaussian reconstruction given by 5 intersecting 1-level-cut Gaussian models [121], are compared for the accuracy of reconstruction by evaluating the Mfs. The Gaussian kernel used for the intersection reconstruction

123 86 Morphology of digitised disordered materials is given by the Fourier transform pair g(x) and ρ(k) = F (k)/(4πk) with g(x) = e x/ξ sin 2πx/d (1 + x/ξ) 2πx/d F (k) = d ( tan 1 c + tan 1 c + + c + c ± = ξ 2π ( ) 2π d ± k where ξ = 51.9 µm and d = 272 µm [121]. (3.29) + c ), (3.3) 1 + c c 2, (3.31) To preserve the variability of the data and allow a direct comparison to the Fontainebleau sandstone samples, all reconstructions were generated at the same resolution and domain size; the 48 3 (2.3 mm 3 ) Fontainebleau samples are used. The comparison of the Mf data from the tomographic images to the reconstructions is given in Figs Large jumps are observed in the Mf data for some morphologies, particularly near the original image porosity. These jumps are due primarily to discretisation effects. Unlike the termite nest, which has limited size, the Fontainebleau sandstone exhibits much greater detail. While homogeneous, we have resolution limitations leading to discretisation errors in ɛ. All images contain features on the scale of a small number of voxels. When one erodes/dilates by a small ɛ, the step size is large and the effect on the morphological measures is large. Discretisation errors in ɛ therefore happen for all images; less so if feature sizes are voxel size. We note peaks around the original image and reconstruction porosity ( ), particularly for the Boolean model. For structures made up of particles, the erosion of the particle fraction creates peaks in the Mfs and particular v 3. This effect is particularly pronounced for spheres. At ɛ = r, where r is the radius of the spheres, only dense clusters of particles are not eroded, i.e. a system of dilute spheres would be completely eroded. Such a strong contribution at ɛ = r would be expected for every sphere size (see also [155]). This is also observed in Figs , showing one peak for the single particle matches, and two peaks for the two-particle reconstructions. For more irregular particles, the 2-spheroid matches, the peaks become less pronounced. The same applies to the OSC model, where a continuous distribution of sphere sizes is present and this effect is smeared out. The integral error of the Mfs (Eqn. 3.28) for the four models is given in Tab We note that the Gaussian and ROS (2) models do best while IOS C and OSC perform poorly. It is surprising that despite the large errors in the Mfs of the Boolean reconstructions due to the peaks at and 1 the ROS (2) model is still good. Note however, that the Boolean models were matched individually to the four Fontainebleau sandstone samples. The Gaussian model of reference [121]

124 Characterisation of structures using the geometry of parallel sets fb7.5 Gaussian IOS C OSC ROS (2) 4 3 ε [mm] v 1 [mm 1 ] 2 1 fb7.5 Gaussian IOS C OSC ROS (2) [a] [b] fb7.5 Gaussian IOS C OSC ROS (2) 4 2 fb7.5 Gaussian IOS C OSC ROS (2) v 2 [mm 2 ] v 3 [mm 3 ] 2 2 [c] [d] Figure 3.31: Fontainebleau sandstone, sample fb7.5: Minkowski functionals over fraction ( = 8.29%) fb13 Gaussian IOS C OSC ROS (2) 4 3 ε [mm] v 1 [mm 1 ] 2 1 fb13 Gaussian IOS C OSC ROS (2) [a] [b] fb13 Gaussian IOS C OSC ROS (2) 5 fb13 Gaussian IOS C OSC ROS (2) v 2 [mm 2 ] v 3 [mm 3 ] 2 [c] [d] Figure 3.32: Fontainebleau sandstone, sample fb13: Minkowski functionals over fraction ( = 12.9%).

125 88 Morphology of digitised disordered materials fb15 Gaussian IOS C OSC ROS (2) 3 ε [mm] v 1 [mm 1 ] 2 1 fb15 Gaussian IOS C OSC ROS (2) [a] [b] fb15 Gaussian IOS C OSC ROS (2) 4 2 fb15 Gaussian IOS C OSC ROS (2) v 2 [mm 2 ] v 3 [mm 3 ] 2 2 [c] [d] Figure 3.33: Fontainebleau sandstone, sample fb15: Minkowski functionals over fraction ( = 17.7%) fb22 Gaussian IOS C OSC ROS (2) 4 3 ε [mm] v 1 [mm 1 ] 2 1 fb22 Gaussian IOS C OSC ROS (2) [a] [b] fb22 Gaussian IOS C OSC ROS (2) 5 fb22 Gaussian IOS C OSC ROS (2) v 2 [mm 2 ] v 3 [mm 3 ] 2 5 [c] [d] Figure 3.34: Fontainebleau sandstone, sample fb22: Minkowski functionals over fraction ( = 21.%).

126 Characterisation of structures using the geometry of parallel sets 89 Table 3.18: Integral error of the Mfs for the models of the tomographic image of the Fontainebleau sandstones calculated according to Eqn. (3.28). Model Sample E 1 E 2 E 3 Gaussian fb fb fb fb IOS C fb fb fb fb OSC fb fb fb fb ROS (2) fb fb fb fb was matched to a single sample at = 15.4% of Fontainebleau sandstone from another study. Thus the reconstruction using the Gaussian scheme does remarkably well. The OSC model gives the worst match to the Mfs of the Fontainebleau sandstone, performs however significantly better for the higher porosity samples. It is also remarkable, that the qualitative behaviour of most curves (e.g. the v 2 curves in Figs ) are matched by the different models, even though none of the reconstruction schemes is actually based on the parallel surface information. In this section new measures for the characterisation of the morphology of disordered systems based on the evolution of the Minkowski functions (Mfs) during erosion and dilation operations on non-convex morphologies were derived. The method can be used to discriminate morphology from 1D (chord-length), 2D (micrographs), and 3D (tomographs) data sets. It is shown that the use of parallel sets and Mfs leads to a very accurate discrimination of morphology. The application to experimental data of Fontainebleau sandstone shows that, while the reconstruction models considered work reasonable, none stands out as brilliant and more realistic morphologies need to be considered to get a good match as in subsection This is left as future work, where we will consider process based models as developed in [112].

127 9 Morphology of digitised disordered materials 3.4 Sensitivity of Minkowski functionals Quantitative studies of 3D structure rely on the acquisition of representative and accurate 3D images of the features of the structure. Common techniques used include serial sectioning, X-ray CT scanning, or Laser Scanning Confocal Microscopy (LSCM). Although all methods give true volume information, all imaging methods give rise to errors in measurements and/or distortions. Any imaging technique features a true or maximal resolution and the actual resolution of the measurement, and is determined by instrument parameters (for example magnification, focus, image depth, acquisition time). For digital recordings of images a voxel size has to be chosen. If this voxel size is greater than or equal to the resolution of the experimental acquisition method, we speak of an ideal voxel size. For example a structure digitised on a cubic lattice with lattice constant a = 2 µm using an acquisition technique having a resolution of 8 µm contains redundant information, and still only offers a resolution of 8 µm. Apart from choosing an appropriate resolution (and thus voxel size) to image a structure, one has to choose a representative volume (or sections) of the structure to be sampled. If the sampling strategy chosen at an ideal voxel size leads to representative morphological measures with standard deviations below some desired threshold, one may define the structure as ideally sampled. In practice, for sectioning techniques, ideal sampling is not easily achieved. For example in serial sectioning, the polished slices itself offer high optical resolution in-plane and may be sampled at any desired spacing, while the slice distance of any pair of parallel slices is determined by the sectioning/polishing process, and can be less then ideal. Distortions that can plaque acquisition techniques are discussed in general. In Xray-CT one must consider effects of noise, drift and distortions due to absorption. In LCSM one must consider attenuation losses with image depth, bleaching effects and distortions due to diffraction. These distortion effects can be corrected to some degree by the application of sophisticated pre- and post-processing steps tailored to the acquisition technique [132]. It is a very complex task to directly define all possible distortions for any specific technique. A correct treatment of the distortions of a particular imaging technique would require a full simulation of the instrument function. For example, in (cone-beam) Xray-CT imaging, one would need to consider the effect of noise in the X-ray density data from 2D projections and this effect on the reconstructed real-space image obtained after a Radon transform. In this section, rather than considering distortions associated with a specific technique, we consider generic distortions. These distortions will, when appropriate, be related to specific experimental techniques.

128 Sensitivity of Minkowski functionals 91 In the preceding sections it has been shown, that the Minkowski functionals give a good measure of structure. Here we use the MF s to quantify the morphological differences between model microstructures and distorted images of the same microstructure. Three effects relevant to different imaging techniques are considered; representative sampling, resolution limitations, and distortions by noise. The section is organised as follows. Firstly, sampling procedures are discussed using the Minkowski measures. Secondly, the simulation of different types of distortions on images is introduced, namely resolution limitations, sample drift, and noise. Finally the effect of these distortions is analysed. It emerges, that the Minkowski measures are a sensitive probe of structure and high image quality or resolution may in some cases be needed to preserve the morphology of complex materials Sampling strategies Sampling procedures are important for any digital recording of an image, as the structure is only exactly known at a limited number of points. In this section we consider two important aspects of sampling. We define a representative sample size for a given structure and the minimum sampling distance required, which still results in no loss of information Representative sampling Consider a disordered structure discretised with ideal resolution on a cubic lattice of size 2 3 with lattice constant a, where the feature size of the structure Ξ a. Three different models are considered, cubes of λ = 1, IOS with r = 12, and a Voronoi tube model with 1 seeds. In Fig [a,c,e] we show that the same mean morphological measures result whether measuring v 3 on pairs of slices of size or on volumetric images of 2 3. In the right column of Fig we show the standard deviations of the v 3 measures. All standard deviations scale 1/ N, where N notes the number of vertices sampled. As the standard deviation scales with N for all (isotropic) structures one can obtain the same quality of morphological information from a 1 3 volumetric structure as from a pair of planar images of size (or even a line of vertices ). In fact, if the feature size Ξ is large (e.g. Ξ 1), the volumetric sample may not be representative. One would obtain better estimates and averaging from a pair of planar images than for a volume image. If we define an (arbitrary) acceptable standard deviation in a measure of v 3 from a single image to be 1%, we ask the question. How large an image (in number of voxels N, or size l = 3 N) is required to obtain this acceptable measurements

129 92 Morphology of digitised disordered materials.5.2 N = [a] v 3 / max(v 3 ) min(v 3 ).5 n = 2 n = f.5 [b] σ(v 3 ) / max(v 3 ) min(v 3 ).1.5 n = 2 n = 4 n = 8 n = 16 n = 32 n = f.2 N = 9.15 v 3 / max(v 3 ) min(v 3 ).5 n = 2 n = 2 σ(v 3 ) / max(v 3 ) min(v 3 ).1.5 n = 2 n = 4 n = 8 n = 16 n = 32 n = 64 [c] f.5 [d] f v 3 / max(v 3 ) min(v 3 ).5 n = 2 n = 2 σ(v 3 ) / max(v 3 ) min(v 3 ) n = 2 n = 4 n = 8 n = 16 n = 32 n = 64.5 N = 36 [e] f [f] f Figure 3.35: Effect of increasing the sampling volume on the normalised standard deviation of the Euler characteristic. [a-b] cubes of λ = 1, [c-d] spheres of r = 12, and [e-f] Voronoi tube models having 1 seeds. All models are discretised on 2 3 lattices and the sampled volume is variied by varying the number of slices sampled (n) for a single measure. Left: comparison of the v 3 measures for the average of 1 realisations at 2 3 to 1 realisations of pairs of slices (2 2 2). The standard errors are very small and the measures coincide. Right: Scaling of the standard deviation of v 3 for the three models. The standard deviations scales N, where N is the number of vertices. The normalisation is chosen, such that σ(v 3 )/ max(v 3 ) min(v 3 ) =.1 for f =.5, and the corresponding N given in the plots.

130 Sensitivity of Minkowski functionals 93 Table 3.19: Derivation of a representative sample size for the v 3 measure, assuming a target standard deviation of 1%. Here l notes the distance between the original seeds in the Voronoi model, while ξ is the correlation length derived from the 2-point correlation function. Model Ξ N (l) 3 Cubes λ = (6) 3 Spheres r = 12 9 (97) 3 Voronoi tubes l = 43,(ξ 15) 36 (71) 3 of morphology? We choose the standard deviation at the symmetric phase fraction of 5% to answer this question. Results are summarised in Tab We find that the cell size l required to obtain reasonable measures of v 3 is l xξ, x [5; 1]. This result is in agreement with current thinking [123, 112] Anisotropic sampling In serial sectioning one often does not obtain isotropic sampling. The information in the plane of the polished section offers excellent (optical) resolution and can thus be sampled with a very small lattice constant. On the other hand the mechanical polishing process doesn t offer this resolution and the out-of-plane sampling distance might be larger. Here we consider the effect of anisotropic sampling; sampling at slice distances s larger than in-plane affect the morphological measures. We compare an image acquired this way to an image measured by a tomographic method with the same resolution as the in-lane image. We consider a periodic digitised morphology of size L 3 with voxel size a 3 and measure the Minkowski functionals of the original structure (N 3 voxels) and then at increasing slice distances s = na in one plane ( N3 voxels). For increasing slice distance s = na, n IN, n where IN is the set of natural numbers, the sampling in-plane remains ideal, however out-of-plane the increased spacing s will lead to a loss of information. lower-dimensional measures remain accurate as they can be measured in-plane, while the quality of the topological information (v 3 ) will deteriorate with sampling distance. When the distance between slices na is at the scale of the largest feature size the slices become uncorrelated and no further information loss can occur. The information loss occurring with larger sampling distance can be quantified by considering the local contributions to the global Minkowski measures (3.1.3). Defining the sectioned plane as the x,y-plane and the z-direction as the out-of-plane axis, the set of 3D vertex configurations IC (3D) 256 containing a certain 2D vertex configuration All

131 94 Morphology of digitised disordered materials IC (2D) 16, is given by the union of the two sets {IC (3D) 256 IC (3D) 256 = IC (2D) 16, + 16i; i =,..., 15}, upper slice, (3.32) {IC (3D) 256 IC (3D) 256 = 16IC (2D) 16, + i; i =,..., 15}, lower slice, (3.33) and expanded in Tab Using Tab. 3.2 one can calculate the probability for the occurrence of the IC (2D) 16 configurations when given the probabilities for the IC (3D) 256 configurations. We are interested in calculating the limiting case of total loss of the 3D topological information. Consider two 2D vertex configurations IC (2D) 16, and IC (2D) 16,, parallel to each other and at a distance s larger than the largest feature size (thus uncorrelated), recombined to a 3D vertex configuration IC (3D) 256,. This configuration is given by IC (3D) 256 = IC (2D) 16, + 24 IC (2D) 16,. (3.34) The 2D vertex configurations are independent and it follows for the configuration probability of the recombined vertex P (IC (3D) 256, ) = P (IC(2D) 16, ) P (IC(2D) 16, ). (3.35) The Minkowski measure v 3, for two uncorrelated parallel slices at distance s Ξ is then given by v 3, = 255 IC (3D) 256, = P (IC (3D) 256, ) V 3(IC (3D) 256, ). (3.36) If continuity between the two recombined slices is assumed for distances s = na; n > 1, the resulting Minkowski measures are normalised with the slice distance, as the vertex now refers to a larger volume (V = sa 2 ). In this normalisation the v 3 measure remains constant if no information loss occurrs and scales with s when the slices are uncorrelated. An alternative normalisation is to use as normalisation V = a 3 even for s > a. This effectively compresses the information density in z-direction and thus leads to an increase in the v 3 measures, until s is larger than the largest feature size. At that s the v 3 measures become constant. We generate the Minkowski functionals numerically for different s; for each microstructure a minimum of 1 realisations over the whole fractional range with a spacing of f =.2 was run on a 2 3 grid. Each run consists of calculating the Minkowksi functionals on the full 3D structure, but also on all disjoint sets of slices of distance , for each direction separately. The effect of increasing the slice distance is illustrated in Fig for three Boolean models, lattice oriented grains of l = 1, lattice oriented grains of l = 8, and discretised spheres of r = 8. Continuity between consecutive slices of distance s is assumed, thus the Minkowski

132 Sensitivity of Minkowski functionals [a].5 [b] 1 v 3 [a 2 s 1 ] s=1 s=2 s=4 s=8 s=16 s=32 s= f 1 v 3 [a 3 ] s=1 s=2 s=4 s=8 s=16 s=32 s= f 1 v 3 [a 2 s 1 ] 1 v 3 [a 2 s 1 ] [c] f [e].1.2 s=1 s=2 s=4 s=8 s=16 s=32 s=64 s=1 s=2 s=4 s=8 s=16 s=32 s= f 1 v 3 [a 3 ] 1 v 3 [a 3 ] [d] s=1 s=2 s=4 s=8 s=16 s=32 s= f [f] s=1 s=2 s=4 s=8 s=16 s=32 s= f Figure 3.36: The effect of increasing slice distance s on v 3 for three different Boolean grain models, lattice oriented cubes of l = 1 [a-b] and l = 8 [c-d] as well as IOS of r = 8 [e-f]. Two different normalisations are used, volume (left), pronouncing curve regions where the information loss is small (i.e. insets in [c,e] for dilute fractions of particles) and number of vertices (right), pronouncing the asymptotic behaviour for s. For the Boolean cubes of l = 1 all slices are uncorrelated and the measures v 3 scale with s [a] or stay constant [b].

133 96 Morphology of digitised disordered materials Table 3.2: Decomposition of the 3D vertex configurations into two 2D vertex configurations parallel to the x-y plane. IC 16 IC ,16,32,48,64,8,96,112,128,144,16,176,192,28,224,24 1 1,16-31,33,49,65,81,97,113,129,145,161,177,193,29,225, ,18,32-47,5,66,82,98,114,13,146,162,178,194,21,226, ,19,35,48-63,67,83,99,115,131,147,163,179,195,211,227, ,2,36,52,64-79,84,1,116,132,148,164,18,196,212,228, ,21,37,53,69,8-95,11,117,133,149,165,181,197,213,229, ,22,38,54,7,86,96-111,118,134,15,166,182,198,214,23, ,23,39,55,71,87,13, ,135,151,167,183,199,215,231, ,24,4,56,72,88,14,12, ,152,168,184,2,216,232, ,25,41,57,73,89,15,121,137, ,169,185,21,217,233, ,26,42,58,74,9,16,122,138,154,16-175,186,22,218,234, ,27,43,59,75,91,17,123,139,155,171, ,23,219,235, ,28,44,6,76,92,18,124,14,156,172,188,192-27,22,236, ,29,45,61,77,93,19,125,141,157,173,189,25,28-223,237, ,3,46,62,78,94,11,126,142,158,174,19,26,222, , ,31,47,63,79,95,111,127,143,159,175,191,27,223,239, measures of the lattice oriented cubes (λ = 1) scale with s (Fig a) or collapse to one curve (Fig b) under the different normalisations, as all slices are uncorrelated. In contrast for λ = 8 at dilute particle fractions (f <.1) for s < λ the Minkowski functionals can be measured accurately (Fig c), as few particles are close to each other and the extrapolation between slices works perfectly for non-overlapping lattice oriented rectangles having a minimum distance of s. For increasing s, particles which are separated in the original image, are now counted as one particle, and the true measure is underestimated. At slice distances larger than the particle size, the limiting behaviour of Eqn is reached (Fig d), and the v 3 curves collapse for s λ. A similar behaviour is observed for the IOS model in Figs [e-f]. It is important to realise, that the minima and maxima, and particularly the roots of v 3, as shown in Fig. 3.36, shift until s is greater than the largest feature size. The first root of v 3, at small particle fraction, is thought to be a good indicator of the percolation threshold of the particulate phase [146]. We quantify in Fig the range over which the percolation threshold of the solid phase p c f(v 3 = ) varies depending on the sampling distance s. For the Boolean cube system (l = 8) p c shifts by 7%, while for the IOS model (r = 8) the shift is 3%. In both cases the effect of varying s is significant. For s larger than particle size p c no longer shifts. The analysis is also carried out for three Gaussian models, a 1-level-cut, 2-

134 Sensitivity of Minkowski functionals 97 level-cut, and intersection model and depicted in Fig All Gaussian models are generated using the same length parameters (r c, d, and ξ), while the domainsize was set to t = 1 for the 1-level-cut, and t = 2 for the 2-level-cut and intersection models. Recall however, that the characteristic length Ξ of the models do differ; see Fig Therefore we observe that smooth 1-level-cut models are insensitive up to a slice distance of s = 4 (Ξ 7) for the chosen discretisation. The topological information of the 2-level-cuts (Ξ < 4 for <.3) and intersection models (Ξ < 5 for <.3) is poorly predicted at Ξ = 4. In general we observe that the v 3 measures for all models are not strongly affected for s < 1Ξ Effects of distortions on images Another source of error in measurements of morphology is distortion. Image distortions are relevant in all 3D imaging techniques. For LCSM for example, attenuation leads to a weaker signal with increasing image depth, and thus to a poorer signal/noise (S/N) ratio. For Xray-CT imaging, the sampling is isotropic: the sample is rotated and projections measured at different angles. Image distortions can be caused by limitations on spot size, acquisition times (S/N), and X-ray source fluctuations. The particular distortions considered here are Gaussian blurring of the signal, sample drift, and boundary noise. We choose the latter since many fil Cubes, l = 8 Spheres, r = 8 f(v 3 =) ~ p c s/ξ Figure 3.37: Variation of the zero of the Euler characteristic (p c f(v 3 = )) as function of the sampling distance s between parallel slices in units of a characteristic length (Ξ = l, 2r).

135 98 Morphology of digitised disordered materials 1 4 v 3 [a 3 ] [a] s=1 s=2 s=4 s=8 s=16 s=32 s= v 3 [a 3 ] s=1 s=2 s=4 s=8 s=16 s=32 s=64 [b] 1 v 3 [a 3 ] [c] s=1 s=2 s=4 s=8 s=16 s=32 s=64 1 v 3 [a 3 ] s=1 s=2 s=4 s=8 s=16 s=32 s=64 [d] 1 v 3 [a 3 ] [e] s=1 s=2 s=4 s=8 s=16 s=32 s= v 3 [a 3 ] s=1 s=2 s=4 s=8 s=16 s=32 s= [f] Figure 3.38: The effect of increasing slice distance s on v 3 for three different Gaussian models, 1-level cuts (top), 2-level cuts (middle), and intersection models (bottom). Two different normalisations are used, volume (left), pronouncing curve regions where the information loss is small ([a]) and number of vertices (right), pronouncing the asymptotic behaviour for s ([d,f]). The different Gaussian models are generated using the same kernel and length parameters r c, d, and ξ (see chap ). While the topological measures of the 1-level-cuts are not much affected even at s = 4 (Ξ 7), for the same slice distance the topological information for the interface pronounced 2-level-cut (Ξ < 4 for <.3) and intersection (Ξ < 5 for <.3) models is almost totally lost.

136 Sensitivity of Minkowski functionals 99 tering techniques exist to remove image noise in bulk phases, but noise at a phase boundary is harder to filter. Motivation for this study was provided by a recent paper published in TAPPI Journal [51], where researchers used X-ray CT to image paper structure [51]. The instrument resolution claimed was 8 µm and the voxel size was 2.28 µm. Figures 3-4 of this paper, taken orthogonal to the sheet axis, are reproduced in Fig Paper fibres are normally > 1 mm in length, 1-2 µm in width, and along the sheet axis (shown in Fig. 3.39) fibres are often compressed down to < 5 µm. An image resolution of 8 µm may not provide sufficient resolution. In these figures the white fibre phase is typically around 2-3 pixels wide in thinner regions and more than 1 pixels for thicker regions. This corresponds to a highly variable fibre thickness of 6-25 µm which is not observed normally. The images look rather noisy and it is interesting to ask, how confident one can be in the morphology derived from these images. In Fig. 3.4 we show an SEM image of paper along the same axis as shown in Fig The image is very different. One clearly observes a large number of ellipsoidal fibres in cross-section which exhibit intrafibre porosity, and lay anisotropic fibres which lie along the image plane. The fibres exhibit smooth boundaries. We illustrate that the discrepancy in the images shown in Fig and Fig. 3.4 is a consequence of insufficient resolution. We first describe some mathematical and computational background, before discussing this problem further. Figure 3.39: Left: a X-ray density map (grey-scale) from a X-ray CT image of paper. Right: corresponding binary image. The image is taken from Fig. 3-4 of [51]. The total image size is (2.33 mm) 2 and the extracted regions shown have a size of (4 µm) 2.

137 1 Morphology of digitised disordered materials [a] [b] Figure 3.4: SEM images of sideviews of paper at a resolution of 1 nm. The thickness of the paper is around 1 µm. The extend and shape of the individual fibres is clearly visible [Ray Roberts, unpublished] Mathematical and computational background The mathematical equivalent to the distortion of an optical measurement is to convolute the original signal with a response function(s) representing the pointspread functions, filters, the lens system; in short, all objects involved in the projection, including the detector. Here we consider the distortion of a perfect original signal and simulate the measured response. response function is defined as g(t) h(t) g( x) h( x) The convolution of a signal with a g(τ)h(t τ)dτ (3.37) g( α)h( x α)d α, (3.38) with τ and α being variables for integration. The convolution theorem 1 (f g)(x)e ixy dx = 2πF (y)g(y) (3.39) 2π allows one to evaluate the measured response employing Fourier transforms. All model structures considered are periodic and standard libraries are used to carry out the transforms. For many tomographic imaging techniques one should consider a Gaussian three-dimensional blur. A folding of the original signal S o with a Gaussian apparatus function S ag of the instrument results in the recorded image S exp ( x) = S ag ( x) S o ( x), (3.4) [ S ag ( x) exp 1 ( )] x 2 + y2 + z2, S 2 σx 2 σy 2 σz 2 ag ( x)d x = 1. (3.41) Drift of a sample over time implies that the signal emitted by a point of the sample over time is recorded at different positions, or formulated differently, the

138 Sensitivity of Minkowski functionals 11 recorded signal at a sensor of the imaging system is a time weighted average of the signals emitted by many sample points. We simulate the simplest case, where the same path is taken by every point of the image. A folding of the original signal S o with the time weighted path of the sample S ad results in that recorded image S exp ; S exp ( x) = S ad ( x) S o ( x) = t( x) t t where t notes the total time interval of the measurement. S o ( x), (3.42) Any technique to acquire images experimentally will be subject to experimental noise to some degree. Many types of noise occur; noise can be uncorrelated or correlated (in space and time) and follow various intensity distributions. It s presence and distribution over time and space as well as it s transformation by the instrument function make the simulation of noise computational complex. Likewise, sophisticated algorithms exist to remove noise, from acquired images (i.e. [85, 86, 48, 116]). In our study noise is limited to a simple interfacial noise function. In this section only noise distributed uniformly in intensity will be considered and is applied fully decoupled in time. Noise is applied as last operation selectively only on the interface of the resegmented signal as speckle noise. Voxels of phase 1 (white) at the boundary to phase 2 are flipped to phase 2 (black) with probability p p pepper, and voxels of phase 2 at the boundary to phase 1 are flipped to phase 1 with probability p s salt. Here we use p = p p = p s to preserve the initial fractions and drop the index Results We first show that the image in Fig may be strongly distorted. We show in Fig [a,c] an image of model fibres; the model consists of partially aligned prolate ellipsoids of half-axes a = b = 4 and c = 8 on a 32 3 lattice. It is convoluted with an isotropic Gaussian blur of σ x = σ y = σ z = σ = 4 voxel (see Eqns ). Thus model resolution limitations are of the scale of a fibre width. The blurred signal is resegmented in Fig [b,e] using static thresholding. The distortion of the original image by the Gaussian instrument function causes two effects. First, the morphology of the original structure is strongly distorted and components are seen to merge. Secondly, as the distortion function is isotropic, intensity is smeared out in all directions. It is now difficult to resolve individual fibres. This effectively reduces the aspect ratio of the resegmented structure. Above effect may have occurred in the experimental images shown in Fig (compare with Fig. 3.4). We now carry out a systematic analysis of the effect of distortions on image morphology as measured with the Minkowski functionals. Blurring at different

139 12 Morphology of digitised disordered materials [a] [b] [c] [d] [e] Figure 3.41: Model of partially aligned ellipsoids demonstrating the effects of resolution limitation by applying a Gaussian blur (details follow in later subsections). [a] Original model of aligned ellipsoids. [b] Resegmented image after applying Gaussian blur. [c] Cut through original model. [d] Corresponding signal after blurring. [e] Final slice. The originally needle-like particular structure is lost in the process. scales to a characteristic length of the structure is considered and the effect on the morphology measured. Consider the influence of isotropic blurring on the morphological measures of an IOS model of r = 12. Slices through the morphologies for the original system and systems subjected to blurring by σ (σ =.5r,.1r,.5r, 1.r) are shown in Fig With increasing σ the structures smear out. Resegmentation is performed using a simple thresholding technique; the cutoff is chosen so the phase fraction remains constant. The resultant structures are smoother and have less components and features. Information loss clearly occurrs when the image resolution approaches the size of the feature (σ r, r). We show all measures in Fig The effect 2 on the Minkowski functionals is similar to that observed in section The general effect of the Gaussian blur is to reduce the magnitude of v 1 and v 2 as seen in Fig This is consistent with smoothing of surfaces. For the topological measure (v 3 ) of the Boolean IOS model, we observe that for larger sphere fractions the amplitude of the v 3 measure first becomes larger before reducing in magnitude

140 Sensitivity of Minkowski functionals 13 Figure 3.42: Illustration of finite resolution for an IOS model, simulated by an isotropic Gaussian blur with σ = σ x = σ y = σ z in units of the sphere radius (r = 12). The corresponding Minkowski measures are shown below. From left to right: slices through the blurred signal and resegmented data for =.25 (left), =.5 (middle), and =.75 (right). For σ see below v 1 [a 1 ] v 2 [a 2 ] f f 1. 1 v 3 [a 3 ]. 1. original σ=.5 σ=.1 σ=.5 σ= f Figure 3.43: Influence of finite resolution on the global Minkowski for the IOS model (r = 12). The sphere radius is used to scale σ. For larger σ all morphological measures of the resegmented structures approach those derived for the single cut Gaussian field (compare to Fig. 3.8).

141 14 Morphology of digitised disordered materials (topological density). This seems to indicate that the structure becomes more bicontinuous first, before the blurring becomes so strong, that components of both phases merge. With increasing σ the large scale features of the Gaussian instrument function dominate and the qualitative behaviour of the measures approaches the characteristics observed for 1-level-cut Gaussian models (Fig. 3.8). It is surprising that the morphological measures are already dramatically affected by the smallest Gaussian blur, even though there are no differences visible in the segmented image (compare images for original and σ =.5r in Fig. 3.42). Recall that in section 3.1 the choice of Boolean spheres/spheroids etc. had little effect on v i (Fig. 3.7), but here a blur of less than 1 voxel is leading to strong variations in v i, in particular v 3. This is consistent with the observation in Fig. 3.7, where local continuity rules have a stronger effect on the morphological measures then a moderate change in particle shape. We next consider the influence of finite resolution on the morphological measures of a Gaussian 1-level-cut (see Fig. 2.5.[a-b]). Slices through the models for the original structure and four different σ are shown in Fig Here σ is scaled by the domain size of the Gaussian model (t = 2). Again with increasing σ the structures smear out and resegmentation leads to smoother structures having less components as shown in Fig At σ = 1% of the domain size the measures are still very similar to the original image. Resolution is less critical for the morphological measures of this morphology. Finally we consider the influence of finite resolution on the morphological measures of a Voronoi edge model (see Fig. 2.8.[c-d]). Slices through the models used in this analysis for four different σ are shown in Fig Here σ is scaled by the average point distance l of the Voronoi model. Again with increasing σ the structures smear out and resegmentation leads to smoother structures. The effect on the MFs (Fig. 3.47) is quite pronounced at σ/l = 1% the measures have changed markedly. At σ/l = 5% the morphological is no longer reminiscent of the original structure. For foam morphologies one must be careful with imaging resolution. Consider now a second type of distortion; noise at the interface of a two phase structure. This is simulated by applying speckle noise ( salt and pepper ) to voxels bordering a phase boundary only. This situation might arise for images filtered for noise in bulk phases, or for phase boundaries which oscillate slightly (i.e. fluid interfaces in porous rocks). By limiting speckle noise to the phase boundaries only, the length scale on which it arises is the dimension of a single voxel. Slices through the structures at different noise probabilities and their corresponding morphological measures are shown in Figures 3.48 and For all structures the addition

142 Sensitivity of Minkowski functionals 15 Figure 3.44: Illustration of finite resolution for a Gaussian 1-level-cut, simulated by an isotropic Gaussian blur with σ = σ x = σ y = σ z in units of the periodicity of the Gaussian (L/t = 1). The corresponding Minkowski measures are shown below. From left to right: slices through the blurred signal and resegmented data for =.25 (left), =.5 (middle), and =.75 (right). For σ see below v 1 [a 1 ].2 1 v 2 [a 2 ] f f 1. 1 v 3 [a 3 ]. 1. original σ=.1 σ=.5 σ=.75 σ= f Figure 3.45: Influence of finite resolution on the global Minkowski measures for a Gaussian 1-level-cut of t = 2. L/t is used to scale σ. The morphology is already Gaussian and the Gaussian instrument function just changes the scale of the model.

143 16 Morphology of digitised disordered materials Figure 3.46: Illustration of finite resolution for a Voronoi tube model (1 seeds on a 2 3 lattice), simulated by an isotropic Gaussian blur with σ = σ x = σ y = σ z in units of the average seed distance of the Voronoi model (l = 43). The corresponding Minkowski measures are shown below. From left to right: slices through the blurred signal and resegmented data for =.6 (left), =.1 (middle), and =.9 (right). For σ see below v 1 [a 1 ].1 1 v 2 [a 2 ] f f.5 1 v 3 [a 3 ]..5 original σ=.5 σ=.7 σ=.1 σ= f Figure 3.47: Influence of finite resolution on the Voronoi tube model shown above. The length scale used for σ is the average seed distance (l = 43). For larger σ all morphological measures of the resegmented structures become Gaussian.

144 Sensitivity of Minkowski functionals 17 Figure 3.48: Illustration of the effects of interfacial speckle noise ( salt and pepper ) for three different structures. From left to right: Slices through an IOS ( =.25), 1-level-cut ( =.5), and Voronoi model ( =.1). For each model the probability p of speckle noise increases from left to right and top to bottom, where p =, p =.1%, p = 1.%, and p = 1%. The corresponding Minkowski measures are shown below v 1 [a 1 ] v 1 [a 1 ] v 1 [a 1 ] v 2 [a 2 ] original p =.1% p = 1% p = 1% 1 v 2 [a 2 ] original p =.1% p = 1% p = 1% 1 v 2 [a 2 ] original p =.1% p = 1% p = 1% v 3 [a 3 ] v 3 [a 3 ] v 3 [a 3 ] f f f Figure 3.49: Influence of interfacial speckle noise of probability p to switch the phase of a voxel bordering the interface, on the global Minkowski measures of three different structures. Left: IOS of r = 12, middle: a Gaussian 1-level-cut model, right: a Voronoi tube model. The v 3 measure of the IOS model is much more affected by interfacial noise than of the smoother other models.

145 18 Morphology of digitised disordered materials of noise to the interface makes the apparent structures rougher and therefore increases the v 1 measures ( S/V ). The effect on the v 2 measures depends on the local neighbourhood of the grains. Many of the voxels exchanged on the interface lead to ambiguous configurations on the cubic lattice (see Fig. 3.1). Depicted in Fig are the highly connected 8- and 26-neighbourhoods of the black phase for the local vertex configurations. This leads to a decrease of the v (8) 2 measures with increasing noise. For the low connected neighbourhood the trend would reverse and v (4) 2 increases. It is likely that at some probability α of connectivity of one phase the curvature would not change with increasing noise. However this α will depend on fraction. For v 3 both neighbourhoods show the same qualitative behaviour with noise; we observe that small noise at the interface has little affect on v 3. For high noise probabilities (p = 1%) voxels of opposite phase at the boundary may both flip with probability p 2 and create small isolated islands on the interface; this totally changes the character of the v 3 curves. In general the noise probability should be lower than p = 1% to preserve the topological information (v 3 ). Above this interface noise level the measures rapidly deteriorate. A kriging method to smooth interfaces may allow one to remove noise. Of great importance, particularly if one wishes to derive material properties directly from tomographic images, will be discerning the effect of noise/drift etc. on properties. This was until recently a computationally intractable problem, which we hope to address in the near future.

146 Sensitivity of Minkowski functionals Discussion In this chapter the morphology of the microstructures introduced in the previous chapter were analysed and the Minkowksi functionals introduced as basic measures of digitised spatial structures. The functionals are calculated with a single sweep of the lattice and can be included into reconstruction schemes. The Boolean reconstruction technique was used to accurately reconstruct Boolean composites and also applied to the sandstone samples. This is one possible generic model of sandstone for which one can analytically find a matching grain model. It would be desirable to derive relationships for more complex non-boolean models; this will allow one to carry out comparisons, and we aim to further pursue this work. In addition, the Minkowski measures were combined with parallel surfaces to arrive at a sensitive method to discriminate between morphologies. Here it would be challenging to consider the inverse problem: devise a reconstruction scheme based on these measures. Finally the sensitivity of the Minkowski functionals was analysed. It was seen, that finite resolution and noise can be a severe constraint to accurate measurement of real morphological properties of a sample. The measures introduced in this chapter might be used in a context of image quality control. Geological systems rarely are composed of 2 phases only. The morphological analysis described here can easily be extended to n-phase systems. A relatively simple example one may study in a geological context is the morphological analysis of different types of inclusions in a rock. These different inclusions can hold various pore fluids, which can help in the analysis of the diagenesis of rocks. These inclusions can also have a profound impact on the physical response of the rock to logging measurements (i.e. NMR). A project in describing morphology for geophysical applications might require the study of n-phase morphological measures.

147 CHAPTER 4 CONDUCTIVITY OF RESERVOIR ROCKS Having considered a range of material morphologies and derived their morphological measures, we now turn our attention to predicting the physical properties of materials and the potential relationship between structure and measured properties. In particular we consider the fundamental question of whether it is possible to characterise the physical properties of the composite media on the basis of these morphological descriptors. Discovering accurate relationships between structure and properties is a long-standing problem in a range of disciplines. In geophysics, an area we consider at length in later parts of the thesis, understanding the interaction between rock, pore space and fluids and how they control rock properties is crucial to better interpretation of measurements. Expressions that relate the physical properties of rocks to the observable quantities in exploration (i.e. seismics, well logging, and core analysis), form the basis for reservoir assessment and monitoring procedures. In this chapter the emphasis will be on transport properties, in particular electrical conductivity. Past theoretical attempts to relate conductance to a description of microstructure of disordered materials have been limited. Most methods either ignore structural details or use oversimplified representations of structure to allow for transport properties to be evaluated. In particular, electrical transport properties have been empirically correlated to the volume fractions i of the component phases [3, 12, 49, 134] in a variety of fields. All correlations have the form σ m. All explicit morphological information describing the disordered material is ignored in these methods. Empirical relations therefore tend to be more useful for correlating data than for predicting properties. Lacking a rigorous connection with microstructure, these results offer neither predictive nor interpretive power. Simplified representations of the material structure (e.g. dilute spheres, cylindrical tubes and periodic networks) for which the properties can be evaluated are also used. These models often incorporate physically undefinable parameters (e.g. shape factors) which are estimated using fits with experimental data. Their application to general random composites is limited [133]. Effective medium theories have also been

148 112 Conductivity of reservoir rocks developed. They are limited by the fact that the microstructure corresponding to a particular formula is not precisely known; hence agreement or disagreement with data can neither confirm nor reject a particular model. A more complicated approach is to consider the variational bounds for the properties of random composites [58, 14, 13, 158]. The large differences between upper and lower bounds however severely limit their predictive power. In particular, for porous materials the lower bound for transport and elastic properties vanishes and the upper bound may not provide a good estimate. Another approach is to computationally solve for the properties from digitised images of the microstructure. As shown in chapter 2 modern imaging techniques (e.g., X-ray CT, laser confocal microscopy) now allow one to observe the full complexity of real disordered morphologies in 3D [38, 147, 41, 13]. In parallel, computational techniques have progressed to the point where material properties such as diffusivity, elasticity and conductivity can be routinely calculated on large three dimensional digitised images containing up to (1 3 ) voxels. With the development of these experimental and computational methods one can now directly simulate the effective transport properties of the imaged material. In this chapter we show that one can accurately predict transport properties directly from digitised images by estimating and minimising several sources of numerical error. Simulation of electrical conductivity on microtomographic images of Fontainebleau sandstone are in excellent agreement with experimental measurements over a wide range of porosity. The results show the feasibility of combining digitised images with transport calculations to accurately predict petrophysical properties of individual rock morphologies. However, relating behaviour of the two-phase material from tomographic images at porosity to other phase fractions remains hopelessly complicated and imaging disordered materials across a full range of is both expensive and time consuming. The bottleneck in elucidating structure/property relationships across the full range of is still limited by the capacity to generate realistic microstructures at will. In section 3.2 (52ff.) the Boolean model was introduced as a predictive model of morphology. It was shown, that from a single tomographic image the Minkowski measures of a set of individual grains can be found, which though not uniquely, describe the morphology of the structure for all. Here it will be shown that Boolean reconstructions of the sandstone samples can be used to match the effective transport properties of the original rock samples over a range of. The chapter is organised as follows. Firstly, the physical background together with theoretical, empirical and numerical approaches to the problem of deriving the effective conductivity of a structure are introduced. Secondly, the application of the numerical solution is detailed. Issues like discretisation, finite size scaling, and

149 Background 113 statistical error are discussed. The accuracy of the method is shown by calculating the DC conductivity of the sandstones and comparison to literature. Finally, the Boolean reconstruction method is used to derive an equivalent model of the image at a given, and predict the physical properties of the structure for the full range of. The analysis is tested using model Boolean systems and sandstone samples. 4.1 Background In this section we consider the calculation of the effective DC conductivity of composite n-phase materials in steady state. The charge conservation equation j + ρ t = (4.1) reduces to j = ( ρe ) = 2 =, (4.2) with the internal boundary conditions at regions of phases i, j in contact having conductivities σ i and σ j i = j, σ i i n = σ j j n. (4.3) The effective conductivity is then given by a macroscopic form of Ohm s law [127] σ eff = σ. (4.4) To predict the DC conductivity of mixtures one needs to specify the volume fractions, the DC conductivities of the constituents, and their geometric arrangement. For the complex internal boundaries of random composites an analytical solution to this problem becomes intractable. In the following we describe methods reducing the complex information given by irregular boundaries to tractable amounts. These are statistical bounds, effective medium theories, and empirical equations. Further, direct numerical solutions are introduced Bounds on the effective conductivity By reducing the geometrical information on the composite material, it is possible to analytical solve for the upper and lower bounds on the physical properties of a composite material. The simplest bounds using the phase fractions of the constituents f i and their DC properties σ i only are the Wiener bounds [165], which amounts to assuming a structure of parallel sheets, where notes transport along the sheets, and notes transport perpendicular to the sheets σ = n f i σ i, i=1 1 σ = n i=1 f i σ i, (4.5)

150 114 Conductivity of reservoir rocks with σ σ eff σ. More rigorous bounds which still require only information of phase fractions f i are the Hashin-Shtrikman bounds [58], which for two phases are given by σ ± HS = σ 1 + f 2 (σ 2 σ 1 ) 1 + f 1 3σ 1. (4.6) A general form of the bounds for n-phase materials was given by Berryman [21]. Higher order bounds including information from the 3-point correlation function have been derived [14, 14]. However, their application is difficult due to the need to calculate certain microstructure parameters involving the 3-point correlation functions. The computational difficulties associated with this approach result in these bounds in general not being applied. An alternative method to derive microstructure-property relationships are effective medium theories Effective medium approximations Effective medium theory is based on the Maxwell-Garnett-Wagner theory for calculating the conductivity of mixtures of dielectrics. Usually a specific inclusion shape is assumed, for which the dilute effect on the DC conductivity can be analytically calculated Self consistent approximation (SCA) In the coherent potential or self consistent approximation (SCA) the host medium for the embedding of an inclusion is chosen to be the effective medium itself [27, 77, 17, 21]. This choice of the embedding medium leads to a minimisation of the effects of the perturbation caused by the inclusion. SCA treats host and inclusion materials symmetrically with no preferred host material and finite percolation thresholds for any material. For a solid of σ s saturated with a fluid having σ f at a porosity of and aligned spheroidal inclusions the SCA equations are (1 )(σ s σ eff,i ) 1 + Λ i (σ s σ eff,i ) + (σ f σ eff,i ) 1 + Λ i (σ f σ eff,i ) =, i = 1, 2, 3, (4.7) where Λ i = L i /σ eff,i are shape factors, and L i are the depolarisation factors along the principal axes of the inclusions [141, 25]. For isotropic structures Eqns. 4.7 decouple. Otherwise the L i become a function of the conductivity contrast, and the depolarisation factors have to be calculated for spheroids in an anisotropic

151 Background 115 matrix [25]. The L i become L i = [ 3 ] 1 b i 1 2 σeff,i i=1 (s + b 2 i /σ ds, eff,i)r s (4.8) 3 R s = (s + b 2 i /σ eff,i), (4.9) i=1 where b i are the half-axes of the spheroid, and Eqns. 4.7 have to be solved numerically. For this work the equations were solved with a Newton-Rhapson method using a binomial expansion as initial guess [116, 3]. elliptic integrals reduce to simpler expressions [75]. For aligned ellipsoids the Starting with isotropic constituents and noting α = σ eff,2 /σ eff,1 the depolarisation factors for prolate spheroids (b 1 = b 3 < b 2 / α) of eccentricity e = α(a/b) 2 1 are L 2 = (1 + e2 ) e 3 (e tan 1 e), L 1/3 = 1 2 (1 L 2), (4.1) and oblate spheroids (b 1 = b 3 > b 2 / α) of eccentricity e = 1 α(a/b) 2 give L 2 = (1 e2 ) 2e 3 (log 1 + e 1 e 2e), L 1/3 = 1 2 (1 L 2). (4.11) In the case of aligned ellipsoids or spheres, distributions of different ellipsoid/sphere sizes were considered [143, 12, 144]. SCA describes a medium where both phases only percolate over the limited range 1 2. It works well at sufficient distance from the critical points (or for 3 3 low conductivity contrast). However, the porous fraction of rocks usually percolates at very small volume fractions, while the rock matrix is a poor conductor, thus SCA does not work so well for rocks Differential effective medium theory (DEM) In the DEM theory the host phase percolates over the full range of fractions. The original work of Hanai [56, 55] and Bruggeman [27] considers spherical inclusions. In the embedding scheme the inclusions are added to the host material in infinitesimal amounts and a differential equation results. For a fluid saturated rock it is natural to start with the fluid phase as host, as the percolation thresholds of the pore space is usually very low [3]. Noting σ s and σ f as the conductivities of the solid and fluid respectively, one obtains for spherical inclusions the differential equation dσ eff 3σ eff = d σ s σ eff σ s + 2σ eff, (4.12)

152 116 Conductivity of reservoir rocks which by integrating from = 1 to results in = ( σf σ eff ) 1/3 σ s σ eff σ s σ f. (4.13) This is the Hanai-Bruggeman equation [56, 27]. More general inclusions were considered in the works of [143, 12, 144]. For an isolating rock matrix σ s = and isotropic but randomly distributed spheroidal grains one has dσ σ = md, m = i=1 1 L i. (4.14) The L i (Eqn. 4.8 without iteration) satisfy L i 1, i = 1, 2, 3 and L 1 +L 2 +L 3 = 1. This leads to m in Eqn having a minimum for spherical grains with m = 3/2 [12]. The effective medium theories discussed so far have one shortcoming SCA implies a percolation threshold of p = 1/3 for either phase, while DEM has one phase which always conducts, and a second phase which never conducts. Schwartz [141] proposed a way to combine the two methods by starting the integration of the DEM equation at a fixed porosity towards zero porosity, and applying SCA for [ ; 1]. This combines the merits of both approaches, but it is not clear how to choose and the combined method still does not give a low but finite percolation threshold. This deficiency was addressed by McLachlan [94, 95], who proposed an improved self-consistent equation (GEM), which incorporates the percolation threshold c and two scaling exponents s and t (1 ) σ 1/s s σs 1/s σ 1/s eff + ( 1 c 1)σ 1/s eff + σ 1/t s σ 1/t f σ 1/t eff + ( 1 c 1)σ 1/t eff For t = s = 1 this reduces to the Bruggeman symmetric equation. =. (4.15) If t and s are used as fitting parameters, generally good agreement with experimental data is found [17, 95] Empirical equations As we are only discussing clean sandstones in this chapter, the most common empirical equation is Archie s equation [3], who suggested that the conductivity of the fluid saturated rock is proportional to the fluid conductivity σ rock σ f. Introducing the formation resistivity factor F as F = σ f /σ rock this relationship is usually written as F = a m. (4.16)

153 Background 117 Based on multiple measurements of cores of sandstone, Archie found empirically that m is dependent on lithology and the degree of cementation. In his original paper a = 1. For unconsolidated sandstones m 1.3 and m 2. for well cemented sandstones. Archie s equation plays a special role in well log analysis. The porosity of a formation can be obtained indirectly from different logging devices, the sonic log (acoustic), the formation density log (induced gamma ray), neutron log, or in clean formations also resistivity logs. Even though neutron and formation density logs respond to total porosity, sonic logs more to evenly distributed porosity, and electrical logs to effective porosity, they can be calibrated in a way that they give the same porosity in clean water-bearing sands (sandstone matrix scale for predominantly sandstone reservoirs). Other calibration scales, i.e. limestone, exist. 1 This allows to predict the fluid saturation of a formation of the same type, using the electrical logs, as the density of water and oil are similar. To do so the Archie parameters a and m have to be known. The quantities a and m play the roles of fitting parameters and are routinely derived from core measurements. A comparison with Eqn gives a = 1, m = 1.5 for the DEM with spherical grains. In general m defined by DEM would be a function of grain shape Numerical solution An alternative method, which incorporates full structural information, is to derive the solution to the problem of effective conductivity for disordered materials numerically. One of the simplest numerical methods is to choose a finite mesh of points and solve a discretised version of the Laplace equation with the given boundary conditions. For this work this is also the most convenient way, as the images are naturally discretised. To solve the Laplace equation with the given boundary conditions (Eqns ), an industry standard finite difference (FD) code developed at NIST 2 6-neighbour finite difference scheme [88, 44, 16]. is employed using a conjugate gradient solver and a The three dimensional voxel microstructure is first converted into a network of resistors by connecting each pair of adjacent voxels of resistance ρ i and ρ j by a bond resistor of resistance ρ b = 2(ρ i +ρ j ). A potential gradient is applied across the sample and the system is relaxed using a conjugate gradient technique to evaluate the field. A detailed description of the original program is given in [44]. A discussion of alternative numerical methods can be found in [16]. A number of changes were made to the original (F77) program developed at 1 The porosity referred to by these calibration techniques is the effective porosity, which is required for all reservoir computations. 2 National Institute of Standards and Technology, Gaithersburg, MD USA.

154 118 Conductivity of reservoir rocks NIST. Firstly it was translated to Fortran (F95); this allowed the use of data parallel constructs and the labelling scheme employed in the F77 version was dropped, leading to a significant speedup. Secondly, non-periodic boundary conditions were included in the code via the procedure suggested in [44], using fixed voltages at the inlet and outlet. In case of non-periodic boundary conditions a potential gradient was applied along all the axial directions of the image, and each case solved for the effective conductivity along the potential gradient. Open boundary conditions are assumed on the four faces parallel to the direction of the gradient. Only rectangular samples are considered and the inlet and outlet are taken to be faces perpendicular to the potential gradient. For periodic boundary conditions (and periodic structures only) the potential gradient was applied diagonally and σ eff derived from a single solution of the Laplace equation. For systems near p c at 1 : contrast, a 6-neighbourhood percolation test was run first to ensure connectivity of the conducting phase. Simulations are carried out only for the conductive fraction percolating in the direction of the applied potential gradient. In the following section we give further details of the numerical technique; in particular estimating and minimising sources of error. 4.2 Accurate image-based prediction of DC conductivity As direct measurement of a 3D structure at resolutions down to a few microns is now readily available from synchrotron and x-ray computed microtomography [38, 147] and laser confocal microscopy [43], one can make a direct comparison between theoretical calculations and laboratory measurements. This has been done previously [147, 139, 9, 78]. In all these studies the numerical prediction of electrical conductivity underestimated the experimental results by 3 1%. In this section we extend their work by showing that the calculation of transport properties from digitised images is greatly improved by minimising three sources of error: finite size effects, discretisation errors and statistical fluctuations. This has not been done in past studies because of limitations in computer memory and speed. The present simulations of electrical conductivity on digitised images of Fontainebleau sandstone are in excellent agreement with experimental measurements over the full range of. This shows the feasibility of combining digitised images with transport calculations to accurately predict petrophysical properties of individual rock morphologies. The tomographic data used in this analysis are obtained from the suite of Fontainebleau sandstone described in This provides the ideal experimental system for this study. Firstly a considerable amount of experimental data [67, 42] is available for Fontainebleau. Secondly, Fontainebleau is homogenous, made up

155 Image-based prediction of DC conductivity 119 of a single mineral, quartz, does not contain clay, and only displays intergranular porosity. Modelling the system as a simple two phase material would be expected to give a good match to experimental data Numerical simulation and error analysis We assign to the matrix phase of the sandstone a conductivity σ quartz = and to the (fluid-filled) pore phase a normalised conductivity σ f = 1. The formation factor, the ratio of the measured conductance of the fluid filled rock σ eff to that of the fluid itself, given by F = σ f /σ eff, is reported. In order to obtain accurate numerical results it is necessary to estimate and minimise three sources of error: finite size effects, statistical fluctuations and discretisation errors. This has generally not been done in the past owing to limitations in computational speed. However, if one wishes to compare computations directly to experiment, such error analyses must be carried out [123]. We discuss the three sources of error separately. The size of the system compared to some statistical length scale (e.g., correlation length, average grain size) controls how many equivalent pores/grains lie within a single computational cell. The full core has thousands of pores/grains within it. Errors will occur if we use too few pores in the cell to simulate the full core. We wish to choose the system size which has an acceptable finite size error but is small enough to be computationally feasible. We first simulated a large core section (4 3 ) on a distributed memory supercomputer. We compare data measured on this full core scale to predictions made on smaller subsets of the original image: 24 3, 16 3 and 12 3 in Fig The conductivity of the pore space measured on cells at 12 3 are in good agreement with the larger cells. This is in agreement with section 3.2, where it was shown, that the morphological measures for the Fontainebleau sandstone samples don t show noticable finite size effects (Fig. 3.18). The ability to use smaller block sizes to estimate properties is exciting. Use of the 12 3 blocks gives one a large ensemble of samples, 64 per core, provides a good spread of porosities from each core and is computationally feasible on a common workstation. Examples of digital images of some Fontainebleau sandstone sub-blocks at 12 3 are given in chapter 2 (Fig. 2.11). Secondly we discuss statistical uncertainties. To estimate errors we bin the measurements made on the computational cells at 12 3 as a function of porosity with bin sizes =.25. The error bars shown in Fig. 4.1 reflect twice the standard error (S.E. = Σ/ N), where Σ is the standard deviation). There is a 95% chance that the true result lies between the indicated standard error bars. The results are accurate for the binned data at this system size.

156 12 Conductivity of reservoir rocks.8.6 Binned data at 12 3 Individual data at 12 3 Individual data at 16 3 Individual data at 24 3 Individual data at 4 3 σ eff / σ Figure 4.1: Comparison of the results for conductivity for cell sizes at nearly full core size 4 3 with data measured at 24 3, 16 3, and This illustrates that finite size errors at 12 3 are small. We also show the statistical fluctuations in the measurement of the conductivity for the 12 3 data sets binned as described in the text. We finally consider the discretisation error; the error due to the use of discrete voxels to represent continuum objects. Errors due to discretisation can include for example, inaccurate description of curved grain boundaries and closing of narrow pores. It has been previously shown [126] that one can accurately estimate the continuum value of the conductivity by varying the discretisation used in the finite difference scheme and assuming: σ = σ( ) + a, (4.17) where σ( ) is the measurement of the conductivity at resolution and σ( ) = σ is the continuum result. To measure this effect we generate realisations of the original tomographic data sets at integer multiples of the resolution of the original image (i.e., n 5.7µm = 11.4µm, 17.1µm,..). To generate the images at poorer resolution we bin voxel clusters of sizes n 3, n = 2, 3, 4 using a simple majority rule. In Fig. 4.2 we illustrate the strong effect of finite resolution on the prediction of σ( ). The scaling of the discretisation error with along with the eventual

157 Image-based prediction of DC conductivity 121 continuum values σ derived using Eqn are also shown. It is important to note that even at a resolution of 5.7µm, the predictions σ( ) differ from the continuum σ value by up to 25% (see Fig. 4.2). The discretisation error at all resolutions and the resultant continuum prediction are illustrated in Fig Formation factor of Fontainebleau sandstone In Fig. 4.4 we compare the computed continuum formation factor F of the fluid saturated Fontainebleau sandstone with experimentally measured values [67, 42]. The predicted F are in good agreement with the experimental data for all porosities. Generally we see an overestimation of the measured formation factor at lower porosities but at intermediate and high porosities the agreement is good. The overestimation we observe in the conductivity at lowest porosity values may be due to insufficient sampling volume at these porosities. At the experimental scale (cm 3 ) lower porosity samples will still conduct, while at the scale of the computational cell (< mm 3 ) some subvolumes no longer percolate. The averaged data for F presented in Fig. 4.4 includes the contributions from subvolumes which do not conduct (1/F = ). By eliminating these subvolumes the curve matches the experimental data more closely for low. This highlights potential problems in using small computational volumes for tight (low porosity) rocks. Moreover, at these porosities, close to the percolation threshold of the pore space, the correlation length increases substantially and data obtained on smaller subvolumes may no longer be representative. The results in Fig. 4.4 strongly contrasts with previous calculations of F from microtomographic images [147, 139, 9]. In these papers there was a consistent overestimation of F by 3 1%. However in these studies discretisation errors were not considered. As illustrated in Fig. 4.2 an overestimation of F is consistent with a discretisation error. In [139] simulations on a core at =.148 at a resolution of 7.5µm observed 65% error between experiment and numerical prediction. In [9] ( =.152, resolution=7.5µm) the error observed was 3%. In [147] ( =.197, resolution=1µm) errors of up to 1% were reported. The increasing error with poorer resolution is consistent with the results summarised in Fig The overestimation of F noted in the previous work has been attributed to the presence of channels at scales smaller than the resolution of the image [139, 9]. Unfortunately this can only be proven by obtaining 3D images of sandstones at higher resolutions. Attempts to measure pore size distributions from 2D data and chord length distributions have led to predictions of small channel sizes [43]. However, measurements based on the chord length distribution are biased towards small pores as the distribution decays exponentially [145]. Measurement of the pore geometry on the 3D samples considered here [8, 81] show a mean pore constriction

158 122 Conductivity of reservoir rocks σ eff / σ [µm] Figure 4.2: The discretisation errors for the conductivity for each binned set of porosity. In all cases the fit to Eqn is excellent..8.6 "Continuum" ( >) =5.7 µm =11.4µm =17.1µm =22.8µm σ eff / σ Figure 4.3: DC conductivity of the original data at 12 3 and 5.7µm resolution and the results for poorer resolutions. The continuum value fitted by matching to Eqn and shown in Fig. 4.2 is dramatically different to the measurements made directly on the image for all L and illustrates the importance of considering discretisation effects.

159 Image-based prediction of DC conductivity Experiment Simulation.6 1 / F Figure 4.4: Comparison of the continuum prediction of the conductivity simulations for the Fontainebleau sandstone with experimental data. The lines indicate best fits to the data. size of 2µm but also a significant number of pore constrictions of size close to the image resolution. Whether a large number of smaller channels would be observed at higher resolutions remains an important open question. However, by considering discretisation errors, the potential error due to channels not being resolved, along with other discretisation effects may be scaled out and an accurate continuum value obtained Other sandstones The same analysis described for the Fontainebleau sandstone samples was also carried out for the other sandstones introduced in chapter 2. The cross-bedded sandstone sample was subdivided into 54 samples of size 1 3, and the DC calculations carried out on these samples as well as coarse grained samples at 5 3, 33 3, and 25 3 resolution. The Berea sandstone sample was subdivided into 27 samples of size For the Berea data set, in particular, due to the small number of samples, the statistics are not very good and the results have to be considered with caution. The results are given in Fig We note that the cross-bedded sandstone (more highly consolidated) exhibits a much reduced conductivity at the same to Fontainebleau.

160 124 Conductivity of reservoir rocks The cross-bedded sandstone sample displays stronger heterogeneity in morphological properties (e.g. -variation in Fig. 2.16). This large difference in F reflects the importance of pore-scale heterogeneity in determining the flow properties of sedimentary rocks. It also highlights the problem of defining a representative rock sample; although both rock samples, the curves differ substantially..1.8 Fontainebleau (exp) Fontainebleau (tomo) Xbed (tomo) Berea (tomo).6 1 / F Figure 4.5: Continuum prediction of the conductivity simulations for all sandstones described in chapter 2. All simulations are run for : 1 contrast and the differences in the sandstones are due purely to structural effects Comparison to theoretical formula In Fig. 4.6 we compare the theoretical predictions discussed in section 4.1 to the simulation and experimental data for the sandstones. The Hashin-Shtrikman bounds are not predictive. Neither is the DEM for spherical inclusions. The SCA prediction is not shown, as it predicts a finite percolation threshold of = 1 and 3 therefore predicts F = for all porosity shown in the figure Comparison to empirical models Clean sandstones are expected to give a good match to the Archie equation (4.16) as this equation was originally empirically derived for sandstone samples. It is not always clear, whether laboratory data is recorded over effective porosity ( eff ) or total porosity ( t ). A recording over total porosity means, that there exists a low

161 Image-based prediction of DC conductivity Fontainebleau (exp) Fontainebleau (tomo) Xbed (tomo) Berea (tomo) DEM, L=1/3 HS + bound 1 / F Figure 4.6: Comparison of the continuum prediction of the conductivity simulations for the sandstones to theoretical formula. The Hashin-Shtrikman upper bound is not predictive. Also plotted is the DEM prediction for spherical inclusions. but finite percolation threshold, which can result in the Archie parameters a and m being skewed in a fit to accomodate for the finite percolation threshold high values in a and m would result. The simulation data is unbiased (non-conducting samples are included) and recorded over total porosity. For the plot of F versus total porosity we show a simple one parameter fit to the original Archie equation in Fig. 4.7, setting a = 1. Excellent agreement between the cementation exponent m of the Archie equation for the experimental measurements on Fontainebleau sandstone to the numerical simulation is observed. The highly cemented and more heterogeneous cross-bedded sandstone shows a significantly higher m. It is widely accepted that increased cementation leads to higher values of m [138]. Doveton [37, 138] showed, that the bedding structure influences the magnitude of m and found higher values of m for cross-bedded (m 2.5) and ripple-bedded (m 2.2) zones of sandstone, consistent with the present study. To convert the total fraction of the pore space t to its effective fraction eff is not trivial. The samples used in the laboratory are usually an order of magnitude larger than the microtomographic images used in this study. One might define the effective porosity of the image as the fraction which percolates between any two opposite faces of the image. However, this measure is dependent on sample size. Another approach is, to consider the inclusion fraction i, and define as

162 126 Conductivity of reservoir rocks.1.8 Fontainebleau (experiment) Archie (exp), m=1.8 Fontainebleau (tomo) Archie (fb tomo), m=1.79 Xbed (tomo) Archie (xb tomo), m= / F t Figure 4.7: Comparison of the unbiased continuum prediction of the conductivity simulations for the sandstone to experimental data and Archie s law. inclusions any pore space not accessible from any boundary. eff is estimated by subtracting from t the inclusion fraction i, given at sample sizes corresponding to the conductivity calculations. This approach gives the fraction accessible from any boundary and at the finite cell size will provide an upper estimate of effective porosity; pores connected to the boundary which may not connect macroscopically will be counted as effective porosity. The i ( t ) curves are given in Fig. 4.8 and fits to the equation of the form i = 1 C, C = C 1 t + C 2 2 t are shown. We also show experimental data from Fredrich et al. derived from density measurements (total ) and He-Porosimetry (effective ). Our data from images agrees with this data except at smaller. Using these fits we generate the Formation factor/effective porosity curves for the sandstone samples in Fig. 4.9, along with fits to the general Archie equation (4.16). A reasonable fit to the general Archie equation is achieved particularly at smaller. After applying the porosity correction one obtains (a = 1.47, m = 2.2) for the tomographic images of the Fontainebleau sandstone, and (a = 2.2, m = 2.62) for the tomographic images of the cross-bedded sandstone. Again the cross-bedded sandstone shows higher a and m in agreement with observed

163 DC conductivity of equivalent stochastic reconstructions Individual inclusion fractions Running average i = 1 C, C= 3.2 t +7. t 2 Fredrich data Individual inclusions Running average i = 1 C, C = 22.4 t t 2.1 inclusion inclusion.1.1 [a] total [b] total Figure 4.8: Estimation of the inclusion fraction for the Fontainebleau (12 3 ) and cross-bedded sandstone (1 3 ) samples. effects of increased cementation and heterogeneity ([138] and references therein). 4.3 DC conductivity of equivalent stochastic reconstructions In section 3.2 it was shown how to derive the local morphological measures V ν (equivalent grain ensemble) of a Boolean model from a single image at any particle phase fraction f, leading to a match of the global morphology. While one can construct an equivalent Boolean model with matching global morphological measures, the question remains; does this equivalent system exhibit similar physical properties to the original system? In this section the DC conductivity of Boolean reconstructions of both the heterogeneous Boolean 5-mixture (p. 57) and reconstructions corresponding to the Fontainebleau and cross-bedded sandstone samples are calculated, and compared to the predictions for the original model and tomographic images respectively Heterogeneous Boolean mixture We compare the effective conductivity σ e of the 5-mix system with the equivalent BG 2 model at both infinite contrast σ 1 : σ 2 = : 1 and 1 :, and at finite contrast σ 1 : σ 2 = 1 : 1 and 1 : 1, where σ 1 is the conductance of the grain phase. In Fig. 4.1 we summarise the results. The match of the equivalent Boolean model to the original 5-mix system is excellent in all cases. Even at infinite contrast, where accurate matches are notoriously difficult, the result is excellent. For comparison, conventional two-point Hashin-Shtrickman bounds [57] are shown in the figure. The match of the reconstruction is much improved over bounds. It is remarkable that from the equivalent BG (2) reconstruction one can derive the conductance curve

164 128 Conductivity of reservoir rocks.1.8 Fontainebleau (experiment) Archie (exp), a=.86, m=1.71 Fontainebleau (tomo) Archie (fb tomo), a=1.47, m=2.2 Xbed (tomo) Archie (xb tomo), a=2.2, m= / F eff Figure 4.9: Comparison of the continuum prediction of the conductivity simulations for the sandstone with experimental data and the general Archie s law. The porosity scale is corrected to mimic effective porosity (see Fig. 4.8). across all phase fractions; recall that the reconstruction was generated from a single image of the original heterogeneous 5-mix system. Clearly the morphology of the equivalent Boolean model captures the important structural aspects affecting the conductivity of the two-phase medium in the original image. Calculation of the full conductivity curve via reconstruction and numerical simulation over a range of phase fractions still requires significant computational resources. A much simpler estimation of the conductivity can be derived from the exact solution for the microscopic conductivity σ m of a Bethe lattice [15, 6, 133] with the same p c as the complex material. The value of p c for a Bethe lattice of coordination number z is p c = 1/(z 1) and the exact solution of the microscopic electrical conductance σ m on the Bethe lattice is given in [149] as well as a set of explicit formulas for binary mixtures (σ 1 = 1, σ 2 = ) used here (i) p p c, (ii) p

165 DC conductivity of equivalent stochastic reconstructions Original model Boolean match HS upper bound Bethe approximation.8 Original model Boolean match HS Bounds.6.6 σ eff [a.u.].4 σ eff [a.u.] [a] f [b] f Figure 4.1: Effective conductivity over particle fraction f for the original 5-particle mixture given in Tab. 3.7 compared to its equivalent 2-grain Boolean model defined in Table 3.8. The equivalent model defined from the single 3-D image at f = 5% matches the full f-conductance curve for both phases. [a] conductivity grain:pore contrasts 1 : and : 1, [b] conductivity contrasts 1 :.1 and.1 : 1. Note that the simulation data for finite contrast slightly violates the upper bound, which is a computational artefact. close to 1, (iii) p close to p c, and (iv) conductivity at high coordination number z σ i =, σ ii z(z 2) z 1 σ 1 [ σ iii.761 2z z 2 σ 1 1 c { 1 + (1 + p c ) n=2 }] p n 1 c 1 + p 2n 1 c (4.18) (4.19) ( ) 2 p pc, (4.2) p c 4 σ iv zσ 1 G n, where (4.21) n= G = p, G 1 =, G 2 = p2 c p 2 pc, G 3 = p3 c p 5 2 pc [p(2p 1) + 3 p c], G 4 = p4 c p 6 2 pc [ 3c 2 p 2pc + p (1 3p+ 3p 2 ) pc 2p p pc 2 1 ], p 2 and c = 1 p, p = p p c. For small p c (and thus high coordination number) the true solution can be approximated by an interpolation between the different

166 13 Conductivity of reservoir rocks approximations p p c p σ = p c σ iii + (1 p p c )σ iv 1 p p c 1.1. (1 p 2 )σ iv + p 2 p σ ii p c > 1.1 (4.22) From the average conductivity of the network σ the microscopic conductance σ m is calculated as σ m = z 1 σ. (4.23) z 2 2z For site based disorder one equates the microscopic conductance to the macroscopic effective prediction of the material σ eff via: σ eff = σ m p, (4.24) where only a fraction p is conductive on the inlet and outlet of the domain. To calculate the Bethe approximations the percolation thresholds given in Tab. 3.9 are used. The match of this empirical fit to the data shown in Fig. 4.1.a is excellent Fontainebleau sandstone The microstructure of a sandstone is a result of a complex physical process which can include consolidation, compaction and cementation of an original grain packing. More complex and realistic models of sandstones have been derived; these include the grain consolidation model [115, 13, 166, 142] and process-based approaches [1, 113]. These methods require however, the simulation of the generating process including primary grain sedimentation followed by a diagenetic process such as compaction and cementation. This process is both computationally expensive and requires several fitting parameters. Modelling the microstructure of sandstones by a simplistic Boolean model may therefore not lead to the same excellent match observed for the stochastic reconstruction of a true Boolean structure in the previous section. However, a Boolean sphere pack has been proposed as a model which gives a reasonable representation of consolidated sandstone and yields good qualitative information on structure/property relationships. Moreover, other work [14, 9, 6] has shown that the Boolean sphere model gives a very good match to Fontainebleau sandstone. In this section the DC conductivity of the Boolean reconstructions of the Fontainebleau sandstone derived in section 3.2 are analysed and compared to the IOS model based on the 2-point void-void correlation function, the 5-level Gaussian intersection model [128], and a method of Thovert et al. based on the covering radius [155]. In the conductivity simulations we apply the potential gradient across the x, y and z axes. Non-periodic boundary conditions and fixed voltages are

167 DC conductivity of equivalent stochastic reconstructions 131 used for all systems in this section. We consider an infinite conductivity contrast: grain : pore = : 1 mimicking a measurement of the electrical conductivity of a fluid saturated sample. The conductivity is calculated on the Fontainebleau sandstone samples and its reconstructions at a scale of The predictions of the model reconstructions are shown in Figs Of the reconstructions the Gaussian model, the only non-boolean model, performs poorest. Of the Boolean models the model performing best is the ROS (2) model. However, the other one- and two-particle models still give a reasonable match, including the original IOS C model, which was simply matched to the 2-point correlation function (see also Tab. 3.11). The OSC reconstruction consistently overestimates the conductivity of the Fontainebleau samples and works better in the intermediate porosity range (Fig b-c)..2.4 σ eff / σ.1 fb7.5 IOS (1) IOS (2) ROS (1) ROS (2) IOS C OSC Gaussian σ eff / σ.3.2 fb13 IOS (1) IOS (2) ROS (1) ROS (2) IOS C OSC Gaussian.1 [a] [b] σ eff / σ.6.4 fb15 IOS (1) IOS (2) ROS (1) ROS (2) IOS C OSC Gaussian σ eff / σ.8.6 fb22 IOS (1) IOS (2) ROS (1) ROS (2) IOS C OSC Gaussian.2.4 [c] [d] Figure 4.11: Comparison between the prediction for conductance of the matching Boolean and Gaussian models and the Fontainebleau sandstone data. [a] fb7.5, [b] fb13, [c] fb15, and [d] fb22. The Gaussian model gives the worst match to the data over the full fractional range. Of the Boolean models, the OSC reconstruction gives a poor match to the data for the low porosity sample. The ROS (2) model gives the best match, but the simple IOS C model performs well. It was shown previously in section 4.1.3, that it is important to make a difference between effective and total porosity, when analysing the conductivity of a

168 132 Conductivity of reservoir rocks sample. 3 Pore space not connected to the inlet and outlet does not contribute to transport. The reconstruction methods discussed in chapter 3 ensure, that the morphological information contained in global morphological descriptors is matched by the reconstructed systems. The Boolean model is generated by the gradual build up of permeable particles, and one can obtain morphologies, where solid grains are completely immersed in void or pore space, clearly an aphysical situation for a rock. Rocks can t have material in a state of suspension, but Boolean models do. One also obtains a percentage of isolated inclusions. As transport properties are dominated by the effective porosity, we consider whether the poorer performance of some of the reconstructions can be related to differences in the isolated pore-space fraction between the reconstructions and the original data set. Such an analysis is carried out in Fig For each of the model reconstructions and the original sandstone a running average of the inclusion fraction over total porosity is presented. It can.1 inclusion.1 Fontainebleau IOS (1) IOS (2) ROS (1) ROS (2) IOS C OSC Gaussian total Figure 4.12: Comparison of the running average (4 points) of the inclusion fractions over total porosity for the reconstructions to the Fontainebleau data sets on a logarithmic scale. The OSC models significantly underestimates the inclusion fraction, while the Gaussian overestimates consistently. None of the models is able to match the inclusion fraction of the Fontainebleau sandstone. For the parameters of the Boolean models see Tab immediately be seen, that the Gaussian reconstruction features a larger discon- 3 One may make further distinctions for the transport backbone, which are not made here.

169 DC conductivity of equivalent stochastic reconstructions 133 nected fraction than the original sample and any other model. This is consistent with the result that it underestimates conductivity over the full range of fractions. The OSC model on the other hand shows a consistent underestimation of the inclusion fraction, and constantly overestimates the conductivity of the Fontainebleau sandstone. None of the reconstructions is able to predict the inclusion fraction over the full fractional range, and a crossover with the inclusion fraction of the original Fontainebleau sandstone data occurs. From Fig it appears, that the Boolean models IOS (1) and ROS (1) might tend to underestimate conductivity at higher fractions, while being better models then the two-particle models at lower fractions. This is consistent with the data of the DC-simulations given in Fig Cross-bedded sandstone We carried out a similar analysis on the cross-bedded sandstone sample and its reconstructions. The DC properties of the cross-bedded sandstone and its isotropic sphere and spheroid based Boolean reconstructions are shown in Fig For the parameters of the Boolean models see Tab cross bedded sandstone IOS (1) IOS (2) ROS (1) ROS (2) IOS C OSC σ eff / σ f Figure 4.13: Comparison between the prediction for the effective conductivity of the matching Boolean reconstructions and the cross-bedded sandstone. None of the Boolean reconstructions is able to match the correlated heterogeneity of the sandstone. The model performing best is the ROS (2) model. For the heterogeneous cross-bedded sandstone none of the Boolean recon-

170 134 Conductivity of reservoir rocks structions can predict the conductivity of the original sample, and all models overestimate the DC-conductivity. This differences are largest for the OSC model, while the ROS (2) model gives the best prediction. We again analyse the fraction of disconnected inclusions for the sandstone and the stochastic reconstructions. The results are depicted in Fig inclusion.1 Xbed IOS (1) IOS (2) ROS (1) ROS (2) IOS C OSC total Figure 4.14: Comparison of the running average (1 points) of the inclusion fractions over porosity for the reconstructions to the cross-bedded sandstone on a logarithmic scale. All models significantly underestimate the inclusion fraction. Of the Boolean reconstructions, the ROS (2) model performs best. For the parameters of the Boolean models see Tab It is immediately apparent that none of the reconstructions can mimic the large fraction of inclusions observed in the cross-bedded sandstone data set. The inclusion fraction is in all cases significantly underestimated, leading to an overestimation of the DC-conductivity. Of the Boolean reconstructions the ROS (2) model again performs best, while the OSC model performs worst. The results for the two sandstones show, that the consideration of global morphological descriptors as compared to the 2-point correlation function or the OSC model leads to Boolean models showing a better match to the DC properties of sandstone. Nevertheless, the Boolean process cannot mimic the true diagenetic processes that occur in rocks and can give aphysical characteristics (e.g., grains suspended in a pore space). The Boolean model is not able to match the function giving the fraction of isolated porosity with total porosity of the sandstones. This

171 DC conductivity of equivalent stochastic reconstructions 135 behaviour might be used as a condition parameter to improve the fit. However, the simple statistical description of the Boolean model would be lost. A possible way is, to start with a Boolean model as an initial structure, and then use i.e. simulated annealing, basing the cost-function on topological measures, but removing i.e. solid material in suspension, or in excess fractions of inclusions. Another method is to have a more realistic model structure; for example hard shell or soft shell models could be derived using equations similar to Eqns , 3.2 to match the morphology.

172 CHAPTER 5 LINEAR ELASTIC PROPERTIES OF RESERVOIR ROCKS In previous chapters we described a range of material morphologies, considered their morphological measures, and compared their transport properties based on morphological descriptors. In this chapter we consider the elastic properties of model morphologies and morphologies obtained from tomographic images. In particular we consider this problem in the context of geophysics (and petroleum reservoir monitoring). We also compare the elastic properties of reconstructed morphologies defined in chapter 3 to the original images. Discovering accurate relationships between pore structure and elastic properties of porous rocks is a long standing problem in geophysics. Understanding the interaction between rock, pore space and fluids and how they control rock properties is crucial to better interpretation of geophysical measurements. Expressions that relate elastic moduli to porosity, pore-fluid compressibility and fluid saturation form the basis for reservoir assessment and monitoring procedures. For example, they are used to infer porosity from well logs, where matrix and fluid properties are assumed to be known, or in reservoir monitoring one might have two or more 3D seismic surveys ( 4D seismics ) and one wishes to correlate changes in the seismics surveys to a change in fluid type and saturation. Properties of porous rocks depend primarily on the morphology of the pore space and solid phase(s). Relevant aspects of the rock structure studied in geophysics include porosity, pore shape and size and the type and frequency of interconnections between pore and solid regions. These features, which unfortunately, lack precise definition, comprise the morphology of the rock. Accurately predicting elastic properties from microstructural information requires; (i) an accurate quantitative description of the complex microstructure of the medium, (ii) the elastic properties of each constituent, and (iii) the ability to solve for mechanical properties on large three-dimensional grids. In the absence of a full structural characterisation, past attempts to relate the elastic properties of a rock to its porosity have been limited to empirical relationships [54], effective medium theories [19], rigorous bounding methods [58, 13] and simple deterministic models [168, 118]. None is

173 138 Linear elastic properties of reservoir rocks entirely satisfactory. Typically, empirical formulae are obtained statistically from experimental data sets. They provide a simple and convenient, but deceptive form of summarizing extensive experimental data. Lacking a rigorous connection with microstructure, these formulae do not offer predictive or interpretive power, seldom carry physical insight, and can often fail when applied to a wider range of rock types. In effective medium theory, the microstructure corresponding to a specific model is not realistic; agreement or disagreement with data can neither confirm nor reject a particular model. A clear advantage of bounds is that they incorporate microstructural information and can be applied to arbitrarily complex structures. Bounds are extremely useful if the two constituent materials have similar properties. For two-phase materials like porous sedimentary rocks the bounds are quite far apart due to the large contrast in elastic properties between pore fluid and rock matrix. This severely limits their predictive power. Simple deterministic models attempt to find a meaningful explanation to experimental observations. The best known example is Wyllie s equation [168]. This equation is based on the observation that for clean sandstones the compressional wave velocity has a strong linear correlation with porosity. Raymer [118] modified this formula by suggesting different laws for different porosity ranges. Nur also used this method in suggesting a critical porosity model [11]. These models work for certain classes of rock types, but do not have general applicability. Due to differences in basic interpretations, in choice of reference states and methods, different methods can strongly disagree. Exploring modulus-porosity relationships in a more controlled environment will lead to a better understanding and enhanced predictive power. In this chapter we do this by computational solving the equations of elasticity directly on digitised microstructures. Computer memory and processing speed now allow the handling of large three-dimensional data sets as well as the processing of many samples needed for statistical reasons in the case of disordered structures. The input to our study of modulus-porosity relationships comes from the digitised models of microstructure [123] and the tomographic images discussed in chapter 2. We will consider reconstructed morphologies based on both the Boolean and Gaussian reconstructions of the tomographic data as discussed in chapter 3. Comparing different reconstructed media to data directly obtained from tomographic images will enable one to decide on the best model structures. The plan of this chapter is as follows. (1) The physical basis of linear elastic properties of composite media is given, and bounding methods, effective medium approximations, empirical equations, and Gassmann s relations introduced.

174 Background 139 (2) As direct method of calculating the elastic properties for a given structure the finite element method (FEM) is discussed. We discuss in particular the need to estimate and minimise several sources of numerical error to obtain accurate predictions of properties. (3) Elastic property-porosity relationships are derived directly from microtomographic images of the suite of Fontainebleau sandstone samples introduced in section Results are in excellent agreement with experimental measurements over a wide range of porosity. Further the elastic properties of the digitised images are considered under dry, water-, and oil-saturated conditions. The observed change in the elastic properties due to fluid substitution is in excellent agreement with the exact Gassmann s equations. This shows both the accuracy and the feasibility of combining microtomographic images with elastic calculations to accurately predict petrophysical properties of individual rock morphologies. The numerical predictions are further compared to various empirical, effective medium and rigorous approximations used to relate the elastic properties of rocks to porosity under different saturations. (4) The elastic properties of Boolean and Gaussian reconstructions (see section 3.2) are compared to their originals for the Fontainebleau and cross-bedded sandstone images. 5.1 Background This section gives a short introduction into the physics of linear elastic materials [76, 138, 92]. The fundamental basis for the description of elastic wave propagation is the theory of elasticity. Given the stress-tensor σ ik and the straintensor ɛ ik, the equation of state for linear elastic materials is given by a generalised form of Hooke s law σ ik = C iklm ɛ lm, (5.1) where C iklm represents the elastic stiffness tensor or tensor of elasticity. Alternatively the strains can be expressed in terms of stresses using ɛ ik = D iklm σ lm, (5.2) where D iklm is the tensor of compliances. The fourth-rank tensors C iklm and D iklm are inverse tensors related by [92] C ijkl D klmn = I ijmn = 1 2 (δ imδ jn + δ in δ jm ), (5.3)

175 14 Linear elastic properties of reservoir rocks where δ ij is the Kronecker symbol (δ ij = if i j, δ ij = 1 if i = j). For linear elastic materials the elastic energy is independent from the strainhistory. The symmetry of both stress and strain tensor results in, at most, 21 of the 81 components being independent [61, 138]. For isotropic materials only 2 components are independent and the stress-strain relationship has the simple form σ ik = λδ iklm ɛ lm + 2µɛ ik, (5.4) where λ, µ are Lamè constants and δ iklm is the Kronecker symbol. For this special case the notation is simplified by introducing the bulk modulus K defined as the ratio of hydrostatic stress σ to volumetric strain [92] σ = 1 3 (σ xx + σ yy + σ zz ) = K(ɛ xx + ɛ yy + ɛ zz ), (5.5) and shear modulus µ, defined as the ratio of shear stress to shear strain σ ij = 2µɛ ij, i j. (5.6) The Young modulus E, defined as the ratio of extensional stress to extensional strain in an uniaxial stress state σ zz = Eɛ zz, σ xx = σ yy = σ xy = σ xz = σ yz =, (5.7) and the Poisson ratio ν, defined as minus the ratio of lateral strain to axial strain in an uniaxial stress state ν = ɛ xx ɛ zz, σ xx = σ yy = σ xy = σ xz = σ yz =, (5.8) can be calculated from bulk and shear moduli using relationships for the elastic constants in isotropic materials E = ν = 9Kµ 3K + µ, (5.9) 3K 2µ 2(3K + µ). (5.1) The units of the elastic moduli are in force/area and for rocks are in the GPa range, while the Poisson s ratio is dimensionless. From the moduli the velocities of the two body waves can be calculated. These measurements are made on geophysical materials, i.e. in well logging using a dipole sonic or array sonic log. If the density ρ of the material is known, the compressional or longitudinal wave velocity V p and the shear or transverse wave velocity V s are K + 4/3µ µ V p =, V s = ρ ρ. (5.11)

176 Background 141 To predict the effective elastic moduli of mixtures one needs the effective moduli of each constituent together with their phase fractions and microstructure. Full knowledge of both the properties of the phases and the structure allows an accurate prediction, which, because of irregular boundaries, needs to be derived numerically. In the absence of full structural information the best approaches are effective medium theories, rigorous bounding methods, or empirical equations Bounds on elastic moduli Bounds are rigourously based on microstructural information. Noting as K i and µ i the bulk and shear moduli of the constituents, and as f i their volume fraction, the simplest bounds are given by the Voigt upper bounds [161] K V = n f i K i, µ V = i=1 n f i µ i, (5.12) i=1 which assume all constituents to have the same strain (isostrain average), and the Reuss lower bounds [119] 1 K R = n i=1 f i K i, 1 µ R = n i=1 f i µ i, (5.13) which assume all constituents to have the same stress (isostress average). isotropic rocks the fundamental model assumption is based on a separation of the individual rock components into parallel sheets [138]. The most narrow bounds on the properties of a two phase composite without specifying any geometric information beyond porosity are the Hashin-Shtrikman bounds [59]. Higher order bounds can be derived [14], but the microstructural information needed to evaluate the results is not easy to obtain. From a specification of the volume fraction and constituent moduli one can obtain rigorous upper and lower bounds on the elastic moduli of any composite material. For a fluid saturated rock with fluid moduli K f, µ f, solid moduli K s, µ s, and porosity, the so-called Hashin-Shtrickman bounds are given by K ± HS = K s + (K f K s ) 1 + (1 )(K s + 4µ, 1 3 s) (5.14) µ ± HS = µ s +. (µ f µ s ) 1 + 2(1 )(Ks+2µs) 5µ s(k s+ 4 3 µs) (5.15) Upper and lower bounds are computed by interchanging the moduli of the solid and fluid components. In the case where one phase has zero elastic moduli, the lower bound becomes zero, and so only the upper bound is meaningful. A more general For

177 142 Linear elastic properties of reservoir rocks form of the bounds applicable to n phases was given by Berryman [21]. When one of the constituents is a phase with zero bulk modulus, the Hashin-Shtrickman bounds coincide with the Voigt upper bound [161] and the Reuss lower bound [119]. Otherwise the Hashin-Shtrickman bounds are more narrow and therefore preferred Effective medium approximations A different approach is to assume a specific inclusion (pore) shape, for which the dilute effect on the elastic properties can be analytically calculated. Various effective medium approaches [57] can then be used to systematically but approximately extend these exact results to higher porosity systems. Certain microstructures have been shown to correspond to these theories but the physical structures are not necessarily realistic [83]. A commonly-used effective medium theory, the differential (DEM) method is constructed by incrementally adding inclusions of one phase into the second phase with known constituent properties. DEM does not treat each constituent symmetrically, but defines a preferred host material. KDEM () at some porosity value is known. One From the composite host medium then treats K DEM () as the composite host medium and KDEM ( + d) as the effective constant after a small proportion d/(1 ) of the composite host has been replaced by inclusions of the second phase. For a solid matrix host, the coupled system of ordinary differential equations for the moduli is given by [2] (1 ) d d [K DEM()] = (K f KDEM)P, (5.16) (1 ) d d [µ DEM()] = (µ f µ DEM)Q, (5.17) with initial conditions K DEM () = K s and µ DEM () = µ s and where P and Q are shape dependent geometric factors for inclusions of the second phase in a background medium with effective moduli KDEM and µ DEM, given in several texts (see e.g., Table of [92]). In the present work the geometric factors for spherical grains (pores) are used. The equations are solved using a variation of the ODE solver given in [116]. In the self consistent approximation (SCA) of [63] and [29] the host medium is assumed to be the composite itself. The equations of elasticity are solved for a spherical inclusion embedded in a medium of unknown effective moduli. effective moduli are then found by treating KSC, µ SC as tunable quantities. The result is given in a general form [19] by: The (K f K SC)P fi + (1 )(K s K SC)P si =, (5.18) (µ f µ SC)Q fi + (1 )(µ s µ SC)Q si =. (5.19)

178 Background 143 In the present work the geometric factors for spherical grains and pores are used. The indices to P and Q note the inclusions of fluid fi and solid si into a background medium of effective moduli KSC and µ SC. As for the DEM equations the solution for the effective bulk moduli is found iteratively. The SCA produces a single formula in which all components are treated equally, with no material distinguished as the host to others. Such a symmetric formula has been thought to be more appropriate in complex aggregates like granular rocks and was shown [15] to accurately predict the mechanical behaviour of a sintered glass bead sample Empirical equations Experimental measurements have often shown that relatively simple empirical relationships can be used to describe the properties of sedimentary rocks. Measurements by Wyllie et al. [168, 169, 167] revealed that a monotonic relationship can be found between the sonic velocity and porosity in sedimentary rocks when they have relatively uniform mineralogy and are fluid saturated. They approximated these relationships with the expression 1 V p = + 1, (5.2) V pf V ps where V p, V pf and V ps are the p-wave sonic velocities of the saturated rock, the pore fluid, and the mineral material making up the rock respectively. The interpretation of this expression is that the total transit time is the sum of the transit time of the elastic wave in the mineral plus the transit time in the pore fluid. It is therefore referred to often as the time-averaged equation. Raymer et al. [118] suggested improvements to Wyllie s empirical equation as follows V p = (1 ) 2 V ps + V pf, <.37, (5.21) 1 (1 ) = +, >.47; (5.22) ρvp 2 ρ s Vps 2 ρ f Vpf 2 1 = ,.47 > >.37, (5.23) V p.1 V 37.1 V 47 where V 37 is calculated from the low porosity formula at =.37 and V 47 is calculated from the high porosity formula at =.47. Nur et al. [19, 11] have championed the idea that the moduli of rocks should trend betweeen the mineral grain modulus at low porosity to a value for a mineralpore suspension at some limiting high porosity. The idea is based on the observation that for most porous materials there is a critical porosity c that separates the mechanical behaviour into two distinct domains. For porosities lower than a critical

179 144 Linear elastic properties of reservoir rocks volume fraction c the mineral grains are load bearing, whereas at porosities greater than c the material falls apart. One can then modify a theoretical model by incorporating percolation behaviour at any desired c by redefining the endpoint porosity. The simplest modified model is based on a modified Voigt equation, where the original Voigt upper bound for a property P is given by P () = P 1 +P 2 (1 ). This empirical model [11] is then given by K dry = K (1 c ), (5.24) µ dry = µ (1 c ). (5.25) The critical porosity for sandstones was found empirically in [19] to be c =.4. A more general relationship for a porous elastic solid is given by K dry = K (1 β), where β is Biot s coefficient. Rather than using a linear relationship for β() as implied by Eqns , Krief et al. [73] derived an empirical relation for β() based on the experimental data of Raymer et al. [118] as (1 β) = (1 ) m() where m() = 3/(1 ). It combines the two mechanical domains, load bearing for < c and suspension for > c, in a single function. They also used the empirical result of [114] which assumed that the dry rock Poisson s ratio is approximately equal to the mineral Poisson s ratio or µ dry /K dry = µ /K, which leads to K dry = K (1 ) m(), (5.26) µ dry = µ (1 ) m(). (5.27) Mavko and Mukerji [93] suggested to modify the Krief model such that rock properties tend towards a suspension at high porosity, rather than the fluid properties. They showed that this change makes the Krief model and modified Voigt identical. Recently Roberts and Garboczi [123] developed empirical equations for the elastic properties of overlapping sphere packs under dry conditions E dry = E (1.652 )2.23, (5.28) ν dry =.14 (1.5 )1.22 (ν.14), (5.29) where E and ν are the Young s modulus and the Poisson ratio of the spherical grains Gassmann s relations One of the most common problems in the analysis of rock cores and seismic data is the accurate prediction of seismic velocities in rocks saturated with one

180 Structure based derivation of effective moduli 145 fluid from rocks saturated with a second fluid or from dry rock velocities. low-frequency Gassmann s equations [47, 22] relate the bulk and shear moduli of a saturated porous medium to the moduli of the same medium in a drained (dry) state. The effective bulk modulus K sat of the saturated rock is given by: K sat K K sat = K dry K K dry + The K f (K K f ), (5.3) where is the porosity and K, K dry and K f are the bulk moduli of the mineral material, the dry rock and the pore fluid, respectively. Gassmann s equations show that the shear modulus is mechanically independent of the properties of any fluid present in the pore space, µ dry = µ sat. (5.31) Gassmann s equations assume that the porous medium contains only one type of solid constituent with a homogeneous mineral modulus, that the pore space is statistically isotropic, and is valid under quasistatic conditions; only at sufficiently low frequencies such that the induced pore pressures are equilibrated throughout the pore space. This limit coincides with the conditions simulated with the finiteelement approach. 5.2 Structure based derivation of effective moduli Calculating the elastic properties of composite materials requires, except for the simplest structures, the application of numerical methods. The input to our simulation is voxelated discretised images. The method we implemented to perform this type of calculations is based on a FEM code for linear elastic problems developed at NIST 1 [45, 44]. The existence of an elastic potential for linear elastic responses allows the formulation of a variational principle [44]. The elastic energy stored in a voxel is given by En = 1 d 3 rɛ ik C iklm ɛ lm, (5.32) 2 with the total energy given as the sum over all voxels. To minimise En various partial derivatives must equal zero, and the simultaneous solution of the resulting system of equations is achieved by using an iterative (conjugate gradient) solver. The full matrix is never stored but instead the matrix vector product carried out by using local stiffness matrices. Each voxel is taken to be a tri-linear finite element. A strain is applied, with the average stress or the average elastic energy giving the effective elastic moduli. The image is assumed to have periodic boundary conditions. A detailed description of the original program is given in [44]. 1 National Institute of Standards and Technology, Gaithersburg, MD USA.

181 146 Linear elastic properties of reservoir rocks Some important changes were implemented to the original program. In this thesis only composites of constituents with isotropic elastic tensors are considered, and we consider primarily porous media filled with either a fluid or vacuum phase. This allowed considerable speedup by exploring additional symmetries in the local stiffness matrices. In the code we implemented an extra local stiffness matrix for internal phases only. For voxels within this internal bulk phase, that is voxels completely surrounded by the same (isotropic) material, the local stiffness matrix is more symmetric. The calculations per voxel in the interior voxels reduced from 243 multiplications and 648 summations to 24 multiplications and 156 summations per voxel. Particularly for computations of relative compact phase distributions, where the number of boundary voxels is relatively small, the change leads to a significant speedup. Neither the results nor convergence behaviour were affected by these alterations. For structures with a phase that has no modulus, e.g. simulation of an unconfined porous material, zero elements are encountered and calculations in the pore space phase(s) can be ignored altogether, providing a further speedup. Additional speedup can be obtained by reordering the matrices for (internal) boundary and non-boundary elements. The speedup reached for different structures and lattices is documented in Tab. 5.1 for a Compaq ES4 (alpha ev67, 4 667MHz, 8MB cache, 4GB main memory), using one node (1-4 processors). The speedup will be larger on hardware, where the floating point performance is not as strong. For larger fractions of phases with zero moduli the speedup achievable is in principle linear, but is not realised here, as the storage scheme used is still a dense matrix. With these speedups implemented, larger samples (usually 12 3, but up to 3 3 ) can be considered. 2 Convergence checks in the iterative solver were carried out every 25 iterations and the simulation terminated, if three consecutive tests showed a monotonic decrease in the elastic energy, and at the same time a relative tolerance on the variations of all 12 average stresses and strains between any of the last three tests (thus 24 conditions, as relative differences to the current results are considered) was met. Otherwise the simulations were allowed to proceed until either a maximum number of 5 iterations was reached, or no further improvement within the precision model was possible. The latter case never occurred while the former only occurrs close to the percolation threshold (of the solid). As an example for the IOS model with solid spheres of r = 12 discretised on a 12 3 lattice, 5 iterations were reached when porosity approached 6% 2 The simulations were performed on a state of the art workstation (Compaq XP1), a SGI Powerchallenge, a DEC alpha cluster, and later a supercomputer (46 ES4 nodes), using of the order of 3 years computer time (alpha ev67 running flat out) in total for all simulations in this chapter. For larger grids ( 2 3 ) an OpenMP version of the code on the supercomputer was used.

182 Structure based derivation of effective moduli 147 Table 5.1: Speedup of the FEM calculations by exploring additional symmetries in the local stiffness matrices. Three different model structures are considered. The IOS model is resolved at radii r = L/1, where L is the lattice size of the cubic lattice, with a sphere volume fraction of 8%. The Gaussian model is a 1- level-cut resolved at 5% volume fraction, and the Voronoi a foam-like structure at 1% volume fraction, generated by 5 initial Poisson distributed seeds. The elastic moduli of quartz were used for the solid, vacuum as porous phase and timings taken on an alpha ev67 running at 667MHz. For compilation the native f95 compiler was used. The timings of the general code are noted as t gen, the timings using bulk symmetry as t sym, and using bulk symmetry combined with further optimisations for vacuum phases as t vac. Memory use is given for the general code (in Mbytes). The versions exploring the symmetry in the local stiffness matrices need an additional 1%. For lattices larger than 2 3 memory access becomes non-local and memory bandwidth limits speedup for the serial version of the code. As comparison the timings of the OpenMP version of the code are given in the last section of the table for the same hardware, but using 4 processors. This code version is not fully optimised yet (t gen should approach 12 sec/it.). Model L Mem. [MB] Iterations t gen [sec/it.] t gen /t sym t gen /t vac IOS = 2% Gaussian = 5% Voronoi = 9% IOS Gaussian Voronoi

183 148 Linear elastic properties of reservoir rocks using the moduli of quartz for the spheres. The relative error on all average stress and strain components was set to 1 4, which together with the test for monotonic convergence proved to be a good termination criterion. When applying the FEM method to digital images, at least three sources of error need to be considered. These are finite size errors, discretisation errors, and statistical fluctuations. Finite size errors result from having a sample of finite domain size, where the largest length scales are of the order of the domain size. Samples at this scale are not representative any more, and the data becomes noisy. Discretisation errors are caused by finite resolution of the image, and the representation of the image by cubic voxels. Its influence can be analysed by varying the resolution of the model or experimental data set and the discretisation error reduced by a proper scaling technique. Statistical fluctuations are present in many experimental data sets. To produce acceptable uncertainties one must carefully choose the number of samples over which to average the properties. Large experimental data sets may be subsampled or many realisations of model reconstructions analysed to quantify this statistical error. 5.3 Elastic properties of Fontainebleau sandstone In this section we compare experimental data on sandstone to numerical predictions based on the tomographic data introduced in section Of the four original core plugs, the four centered 48 3 voxel subsets were selected for analysis, corresponding to a volume of 2.5 mm 3. Fontainebleau sandstone provides the ideal experimental system for this study. First, experimental data is available for the elastic properties of Fontainebleau sandstone over a range of porosity [54]. Second, Fontainebleau is homogeneous, made up of a single mineral, quartz, does not contain clay, and the structure of the sandstone is quite simple as it only displays intergranular porosity. Therefore modelling the system as a simple two phase material would be expected to give a good match to experimental data Choice of representative image volume The question of how the results depend on the total volume imaged has to be addressed. In previous work [147, 139, 9] calculation of transport properties on microtomographic images have been performed on either the full image or on a few subsets of the imaged data. For example Auzerais et al. [9] found that cubes of size of much greater than 1 mm 3 were required to estimate fluid permeability with acceptable accuracy. The analysis of only a small number of subsets however gives one little data to compare to experiment. The subsequent requirement to image

184 Elastic properties of Fontainebleau sandstone 149 many samples to obtain data across the full range of is both expensive and time consuming. In recent work [81], the distributions of the flow relevant geometrical properties (e.g., pore size distribution, throat size, etc.) was measured on the same set of Fontainebleau sandstone cores considered here. From the center of each core a voxel image (3.9mm 3 ) was extracted and geometrical properties compared to the full core of twenty times the volume. Even at the small scale (3.9mm 3 ), roughly ten grains on a side, the comparison showed good agreement for most blocks. The prospect of using smaller block sizes is an exciting one. The combination of choosing smaller window sizes on the imaged core and the natural heterogeneity of the rock allow us to derive the elastic properties for the rock for a large range of from a smaller number of core samples. A first test of the dependence of the digitised data on image volume requires ensuring that the geometrical and the topological descriptors of the image volume are consistent. An appropriate tool to do this are the Minkowski measures, which are depicted in Fig as a function of porosity for the original image at 48 3 and for cubic subsets of the image at scales of 24 3 and Variability of the measures increases with decreasing window size but the values are consistent with the data for the larger volumes, suggesting that for the smaller blocks a meaningful average is obtained. At low, near the percolation threshold of the pore space, the scatter is greatest, but still acceptable at the scale of Alternatively the representative cell size can be found by considering the two point correlation function the probability of finding two end points of a segment of length l within the same phase. Defining a correlation length ξ as the first zero of this two point function, we find that ξ 13µm. This is consistent with values reported in the literature [112]. For our system size of 12 3 at 5.68 µm per voxel, this implies the cell spans more than five times the correlation length and good averaging behaviour may be expected. Previous microtomographic work [9] found that for a system of similar size (112 3 at 7.5µm resolution), the averaging of the porosity was acceptable. One must carefully choose the number of samples over which the results need to be averaged to produce acceptable uncertainties. Use of the 12 3 samples gives one a large ensemble of samples, 64 per core, provides a good spread of porosities from each core, and is computationally realisable on common workstations. Examples of snapshots of 12 3 subvolumes from the four cores are shown in Fig Further testing of the errors associated with the choice of the window size on the numerical computation of the elastic properties of the images is required and subsequently addressed.

185 15 Linear elastic properties of reservoir rocks Property prediction The prediction of the effective elastic moduli of the Fontainebleau samples was carried out using the numerical method explained in section 5.2. From Mavko [92] the elastic properties of the rock skeleton are taken to be the ones of quartz (K = 37GPa, µ = 44GPa, and ρ = 2.65g/cc). Water is modeled at T = 2 o C and 4 MPa pressure (K = 2.2GPa, µ = GPa, ρ = 1.g/cc), and the oil saturated case (3 API oil) at T = 2 o C and 25 bar pressure (K =.5GPa, µ = GPa). The density of it follows from the definition N AP I = ρ 131.5, (5.33) giving ρ =.88g/cm 3. The density at temperature T = 2 o C and pressure P = 25bar is given by an empirical fit [33] and gives ρ =.77g/cm 3, with an oil bulk modulus of K f =.5GPa. The choice of the water saturated condition is made to allow for comparison with the experimental data from [54]. Calculations of effective moduli from digitised data need to be done carefully. In order to obtain accurate numerical results it is necessary to estimate and minimise three sources of error: finite size effects, statistical fluctuations, and discretisation effects. This has generally not been done in the past owing to limitations in computational speed. However, if one wishes to compare computations directly to experiment such error analyses must be carried out [123]. In the following the three sources of error are discussed separately. In the previous section it was argued that at a length scale of 12 voxels the averaging of the porosity is good. To consider if this cell size gives one good averaging of the elastic properties, simulations at three different domain sizes, 12 3, 16 3, and 24 3 representing individual sample volumes of.32 mm 3,.75 mm 3, and 2.53 mm 3 were carried out (see Fig. 5.1.a). The scatter of the data with varying cell size is minimal and for a cell size of 12 3 voxels the errors are acceptably small. Secondly, statistical uncertainties are discussed. To estimate errors the measurements are binned as a function of porosity with bin sizes =.25. The error bars reflect twice the standard error (S.E. = σ/ N), where σ is the standard deviation. There is a 95% chance that the true result lies between the indicated standard error bars shown in Fig. 5.1.b. The results are accurate to within a few percent for most data points. Finally one has to consider discretisation and resolution errors. Images of rocks, conventionally derived using tomographic techniques (i.e. with a maximal resolution of 5 µm), are often measured at a resolution which may not resolve all features of the rock morphology. For example, transport properties would be

186 Elastic properties of Fontainebleau sandstone (.32mm 3 ) 16 3 (.75mm 3 ) 24 3 (2.53mm 3 ) (.32mm 3 ) 16 3 (.75mm 3 ) 24 3 (2.53mm 3 ) Bulk modulus [GPa] 25 Bulk modulus [GPa] [a] [b] = 22.7µm = 17.µm = 11.4µm = 5.7µm 3 = 22.7µm = 17.µm = 11.4µm = 5.7µm Bulk modulus [GPa] 25 Bulk modulus [GPa] [c] [d] K [GPa] Bulk modulus [GPa] 3 25 "Continuum" values fb7.5 fb13 fb15 fb [e] resolution [µm] [f] Figure 5.1: Numerical prediction and error analysis of the bulk modulus of water saturated Fontainebleau sandstone (each modulus is scaled separately, see 5.2). [a] Raw data at different domain sizes. [b] Binned data of [a] with standard error, no finite size effects are present. The standard error for the endpoints and larger grids varies because of the limited number of samples of these fractions/sizes. [c] Raw data at different resolutions and constant domain size (.32 mm 3 ). [d] Binned data of [c] with standard error One observes strong resolution/discretisation effects. [e] Discretisation/resolution scaling of the data for different porosities, the porosities of the bins at different resolution are interpolated to the same fractions using natural spline interpolation beforehand. [f] Final prediction at ideal resolution with scaled standard deviation, compared to the original data at maximal resolution ( = 5.68 µm, split into the four core plugs). The intercept with the y-axis is given by K Quartz = 37GPa ([a-d,f]).

187 152 Linear elastic properties of reservoir rocks 1.18 Young modulus [GPa] inf to to to 3 3 fb7.5 fb13 fb15 fb22 Poisson ratio ν inf to to to 3 3 fb7.5 fb13 fb15 fb Bulk modulus [GPa] 25 2 inf to to to 3 3 fb7.5 fb13 fb15 fb22 v s [km/s] inf to to to 3 3 fb7.5 fb13 fb15 fb Shear modulus [GPa] 3 2 inf to to to 3 3 fb7.5 fb13 fb15 fb22 v p [km/s] inf to to to 3 3 fb7.5 fb13 fb15 fb Figure 5.2: Scaling of the elastic properties of water saturated Fontainebleau sandstone. The sandstone matrix is simulated as isotropic quartz (K = 37GPa, µ = 44GPa) and the water is assumed to be clean (K = 2.2GPa, µ = GPa). All moduli decrease with increasing resolution as do the wave velocities V p and V s, while the Poisson ratio increases with resolution. The continuous lines indicate the binned data at the various resolutions, while the symbols indicate the individual simulations at a domain size of 12 3 voxels, with different symbols to separate the four different cores. The binned and scaled data is given with standard error and scaled standard error.

188 Elastic properties of Fontainebleau sandstone 153 affected by narrow throats between grains that are not resolved. This may effect macroscopic behaviour [43]. To overcome this problem, we solve the elastic equations at different resolutions and extrapolate to infinite resolution. To vary the resolution, realisations of the original tomographic data sets at integer multiplies of the resolution of the original image (i.e., n 5.7µm = 11.4µm, 17.1µm,... ) were generated using coarse graining by binning voxel clusters of sizes n 3, n = 2, 3, 4 via a simple majority rule. The predictions for the bulk modulus of water saturated sandstone at different resolutions are given in Fig. 5.1.c-d. It has been previously shown [123] that the variation of elastic properties P follows P ( ) = P + a, (5.34) where is the resolution (lattice constant), and P is identified as the continuum value corresponding to. To carry out the scaling given by Eqn. 5.34, data at different resolution but identical porosity is needed. Due to statistical randomness and coarse graining the binned data at different resolution usually doesn t fulfil this requirement exactly. In Fig. 5.1.d one observes that the data points at different resolution are more or less on top of each other. To bring the fractions of each bin to the same fraction for each resolution a natural spline interpolation is carried out and the curves are resampled such that the spacing of the original curves and the number of samples is approximately preserved (see also Fig. 5.1.f for the final points). Fig. 5.1.e illustrates the scaling behaviour of the discretisation error with for different porosities, while Fig. 5.1.f shows the bulk moduli of the initial individual samples at maximum resolution ( = 5.68 µm) along with the continuum values P derived using Eqn In some cases one may need to increase the resolution of the original image to obtain a good estimate of the discretisation error [89]. However, for this data set the resolution seems sufficient and the fit to Eqn is good. It is important to note that even at a resolution of = 5.68 µm, the predictions P ( ) differ from the continuum P value by up to 15%. The discretisation errors increase approximately linearly with porosity. The standard deviation of P in the continuum limit is estimated by application of Eqn to the standard deviations at the different lattice sizes and essentially the same as that observed for P ( = 5.68 µm). The scaling results for the other moduli, Poisson ratio, and velocities is given in Fig Importantly all results scale strongly with, highlighting the need to consider discretisation effects Fluid substitution Fontainebleau sandstone is both homogeneous and in the simulations uniform moduli are applied for the solid phase. One would therefore expect the numeri-

189 154 Linear elastic properties of reservoir rocks Bulk modulus [GPa] 25 Shear modulus [GPa] Quartz & water Quartz & air Quartz & dead oil (2 o C, 25bar) Gassmann prediction from Quartz/Air 15 Quartz & water Quartz & air Quartz & dead oil (2 o C, 25bar) [a] [b] Figure 5.3: Comparison of the results of the simulations for dry, water-saturated and oil-saturated to Gassmann prediction based on the dry rock data. In [a] we give the numerical predictions for the dry rock data (squares) and show a best fit to the data points (solid line). We use this fit and Eqn. 5.3 to predict the values of the water- and oil-saturated curves (dotted curves). We show the numerical predictions for the water- and oil-saturated results. The fit to the Gassmann s equations is excellent, further indicating the ability of this methodology to quantitatively predict geophysical properties. In [b] we show that the shear modulus is independent of the pore fluid as predicted by Gassmann. cal data for Fontainebleau to obey Gassmann s equations for different pore fluids. Comparison of the numerically predicted moduli of the Fontainebleau images under dry, water- and oil-saturated conditions to Gassmann s equations provides an important test of the accuracy of the numerical results. The results are summarised in Fig The numerical prediction for both the bulk and shear modulus are in excellent agreement with Gassmann s equations Comparison to experiment In Fig. 5.4 we compare the computed bulk and shear modulus data for dry and water saturated Fontainebleau sandstone with experimentally measured values [54]. The predicted velocities are in good agreement with the experimental data. Generally we see a slight overestimation of the measured moduli this may be due to the finite resolution of the images. Small cracks and pores may not be resolved by the tomographic images but could have a strong effect on the elastic properties. The presence of small cracks is thought to be enhanced for the dry samples [54]; this is in accord with our data. The results from the simulations might be considered to be a predictive upper bound for the bulk and shear velocity data given the finite resolution of the images.

190 Elastic properties of Fontainebleau sandstone Simulation from image Experiment 3 Simulation from image Experiment Hashin Shtrickman SCA DEM Bulk modulus [GPa] 25 Bulk modulus [GPa] [a] [a] Simulation from image Experiment 3 Simulation from image Experiment Hashin Shtrickman SCA DEM Bulk modulus [GPa] 25 Bulk modulus [GPa] [b] [b] Simulation from image Experiment 4 Simulation from image Experiment Hashin Shtrickman SCA DEM Shear modulus [GPa] 3 2 Shear modulus [GPa] [c] [c] Figure 5.4: Comparison of the continuum prediction of the elastic simulations for the digitised images under dry and the water saturated (4 MPa) conditions to experimental data (Han, [54]). The figures give predictions for the [a] dry and [b] wet bulk modulus, and [c] the shear modulus. Figure 5.5: Comparison of the continuum prediction of the elastic simulations for the digitised images under dry and the water saturated (4 MPa) conditions to effective medium approximations and bounds. The figures give predictions for the [a] dry and [b] wet bulk modulus, and [c] the shear modulus.

191 156 Linear elastic properties of reservoir rocks Comparison to theoretical formula We compare the three theories discussed in section (SCA, DEM, and Hashin-Shtrikman bounds) to our numerical predictions and experiment for both dry and water saturated rock in Fig We note that none of the theoretical methods gives a satisfactory fit to the experimental data; in contrast the numerical results are in excellent agreement. The SCA theory gives a much better fit to the experimental data than either the DEM or the Hashin-Shtrickman upper bound. This is in agreement with the observations of Berge et al. [15] that the SCA should more accurately predict the elastic properties of granular rocks. However, the numerical prediction is far superior to any theoretical estimate. This conclusion is in accord with recent results of [123], where it was shown that neither bounds, SCA nor DEM accurately predict the properties of complex granular materials Comparison to empirical models Two simple empirical models often used in well log analysis are the Wyllie equation (Eqn. 5.2) and the model of Raymer-Hunt-Gardener (Eqn. 5.22). A comparison of the predicted and numerically derived velocities for the water saturated Fontainebleau sandstone are summarised in Fig The Raymer equation gives a satisfactory match to the data, while the Wyllie equation prediction is poor Numerical prediction Experimental (Han, 1986) Wyllie Raymer 5 v p [km/s] Figure 5.6: Comparison of the results of the simulations (squares and dashed line) for water-saturated sandstone to experimental data (circles) and the empirical equations of Wyllie (Eqn. 5.2) and Raymer (Eqn. 5.22). The fit of the numerical data compares well to either empirical equation.

192 Elastic properties of Fontainebleau sandstone Simulation from image Experiment Critical porosity model IOS, equation from [159] 3 Simulation from image Experiment Critical porosity model Bulk modulus [GPa] 2 Bulk modulus [GPa] [a] [b] Figure 5.7: Comparison of the simulation results to the empirical equation of Nur [11] and of Roberts and Garboczi [123] for the dry case. Under both watersaturated and dry conditions the fit of the empirical equations to experiment and numerical data is excellent and comparable to the data obtained from images. The critical porosity model [11] (Eqns ) is known to be a good model for sandstone, and works very well for the Fontainebleau data set. Likewise the empirical equation of Roberts and Garboczi for dry sandstone [123] shows excellent agreement across the full range of, as it mimics the structure of sandstones (Fig. 5.7) Conclusion In this section we have shown for the first time that elastic property-porosity relationships can be derived directly from microtomographic images. We show that one can obtain a large ensemble of samples from a single tomographic image providing measures of elastic properties across a range of porosities. Predictions were given for a number of conditions: dry, water-saturated and oil-saturated. Results for dry and water-saturated conditions are in good agreement with experiment. The observed change in the elastic properties due to fluid substitution is consistent with the exact Gassmann s equations. This shows both the accuracy and the feasibility of combining microtomographic images with elastic calculations to accurately predict petrophysical properties of individual rock morphologies.

193 158 Linear elastic properties of reservoir rocks 5.4 Elastic properties of Boolean and Gaussian reconstructions V ν In section 3.2 it was shown, how to derive the local morphological measures (equivalent grain ensemble) of a Boolean model from a single image at any particle phase fraction f, leading to a match of the global morphology. In the previous chapter (4.3) it was shown, that this also leads to an excellent match of the DC properties for Boolean composites. Further, an application of this technique to the sandstone samples did lead to Boolean models which compared to other Boolean reconstruction techniques and a Gaussian reconstruction gave a better match of the transport properties of the original data sets and reconstructions. In this section an analysis analog to section 4.3 is carried out for the elastic properties of Boolean reconstructions. Both the heterogeneous Boolean 5-mixture (p. 57) and reconstructions corresponding to the Fontainebleau and cross-bedded sandstone samples are compared to the predictions for the original model and tomographic images respectively Heterogeneous Boolean mixture The heterogeneous mixture of an ensemble of 5 Boolean rectangular bars was reconstructed with a smaller ensemble of 2 Boolean rectangular bars, leading to the same global morphological Minkowski measures. The elastic properties are evaluated along the principal directions of the structures, coinciding with the lattice directions (see i.e. Fig. 2.3, p. 12, for 2-particle mixtures of this kind). In Fig. 5.8 the simulation results for the original 5-mix and the equivalent two particle match are shown. The agreement is excellent for both 1: and :1 contrasts. We also show the predictions of the Hashin-Shtrikman (HS) bounds (Eqns ). The upper bounds are not predictive for the chosen contrast and the lower bounds become zero. This makes the Boolean reconstruction a far better predictor for the elastic properties of a Boolean composite, for which only the global Minkowski measures have to be known at any phase fraction. This result is encouraging. The Boolean reconstruction not only matches the morphology of the complex 5-mix system, but also the mechanical properties. It highlights the ability to accurately reconstruct complex images using the Minkowski functionals and has led us to consider the match of the physical properties of the reconstructed sandstone data sets. 3 3 A similar analysis for the transport properties (percolation and DC conductivity) of the heterogeneous mixture is carried out in chapter 4 with similar results, see also [8].

194 Elastic properties of Boolean and Gaussian reconstructions Original model Boolean match HS bounds.6 K eff / K s.4.2 [a] f.8 Original model Boolean match HS bounds.6 G eff / G s.4.2 [b] f Figure 5.8: Effective elastic moduli over fraction for the 5-particle mixture given in Tab. 3.7 (p. 57) compared to a mixture of 2 lattice rectangles matched to have the same average local measures. The reconstruction for the particle phase being modeled as the solid (moduli increasing with particle fraction f) is very accurate. The results are also compared to the Hashin-Shtrikman bounds (Eqns ). The upper bounds are not predictive for the chosen contrast and the lower bounds zero.

195 16 Linear elastic properties of reservoir rocks Fontainebleau sandstone As pointed out in section the microstructure of a sandstone is a result of a complex physical process. Nevertheless a Boolean sphere pack has been proposed as a model of consolidated sandstone, and yields good qualitative information on structure/property relationships. For Fontainebleau sandstone in particular, other work [14, 9, 6] has shown that the Boolean sphere model gives a very good match to Fontainebleau sandstone. In this section the elastic properties of the Boolean reconstructions of the Fontainebleau sandstone derived in section 3.2 are analysed and compared to the IOS model based on the 2-point void-void correlation function, a highly optimised 5-level Gaussian intersection model [128], and a method of Thovert et al. based on the covering radius [155]. To be able to directly compare the elastic simulations carried out at a fixed discretisation, for which the Boolean reconstructions were determined, a numerical caveat has to be considered. The elastic simulation algorithm assumes periodic boundary conditions. This is not problematic, when doing a scaling analysis, as the effect of the boundary condition should scale out in the discretisation error. However, as we are comparing results of finite resolution now, it must be carefully considered. The Boolean and Gaussian reconstructions are all generated as periodic structures, and the boundary conditions are correct. The tomographic data however is not periodic, and the associated error at finite resolution needs to be estimated. This is done in the following by evaluating the elastic moduli for two reconstructions of the Fontainebleau samples, using the IOS model reconstruction based on the 2- point correlation function for the sample fb13 (r c = 65.8; µm, see 3.11), and the Gaussian reconstruction using an intersection of 5 level cuts. Both systems are generated with non-periodic boundaries (the 12 3 centre pieces of realisations on a 2 3 lattice), and directly compared to 12 3 systems generated with periodic boundaries. For each model and porosity 1 realisations on a 12 3 lattice were carried out. The solid phase was assumed to be quartz (K = 37GPa, µ = 44GPa) and the fluid phase water (K = 2.2GPa, µ = GPa). The results are shown in Fig. 5.9.a-b. The standard error of the elastic moduli of the reconstructed data sets is very small and not shown on the plots. The difference between the nonperiodic and periodic structures is predictable with the moduli calculated with periodic boundary conditions consistently smaller than for non-periodic boundary conditions. The difference scales approximately linearly with. We calculate the linear correction factor associated with the choice of boundary conditions and find K (5.68µm) periodic () = K(5.68µm) n.p. () (1.751), (5.35) G (5.68µm) periodic () = G(5.68µm) n.p. () (1.926). (5.36)

196 Elastic properties of Boolean and Gaussian reconstructions Bulk modulus [GPa] 3 2 IOS C, periodic IOS C, non periodic Gaussian, periodic Gaussian, non periodic Straight line fits Shear modulus [GPa] IOS C, periodic IOS C, non periodic Gaussian, periodic Gaussian, non periodic [a] Bulk modulus [GPa] IOS C, periodic IOS C, non periodic, corrected Gaussian, periodic Gaussian, non periodic, corrected [b] Shear modulus [GPa] IOS C, periodic IOS C, non periodic, corrected Gaussian, periodic Gaussian, non periodic, corrected [c] [d] Bulk modulus [GPa] 25 fb7.5 fb13 fb15 fb22 fb7.5, corrected fb13, corrected fb15, corrected fb22, corrected Shear modulus [GPa] 3 2 fb7.5 fb13 fb15 fb22 fb7.5, corrected fb13, corrected fb15, corrected fb22, corrected [e] [f] Figure 5.9: Derivation of the Fontainebleau data at the fixed resolution of 5.68µm but for periodic boundary conditions (simulating an infinite size sample). [a-b]: Effective moduli for two water saturated model structures generated with periodic and non-periodic boundary conditions. [c-d]: Comparison of the periodic data with the non-periodic corrected data. [e-f]: Prediction of the elastic moduli of a water saturated periodic ( infinite ) Fontainebleau sandstone sample using the simulations for the non-periodic tomographic image. The correction is specific to the given resolution and lattice size.

197 162 Linear elastic properties of reservoir rocks The predictions of the non-periodic models corrected via Eqns and the original periodic systems is given in Fig. 5.9.c-d. The use of Eqns for the Fontainebleau data is shown in Fig. 5.9.e-f. The correction is an estimate of the error made, based on the assumption, that the two models used are showing similar boundary effects than the sandstone data. It is assumed to be valid for the given resolution, lattice size, and phase moduli. In Fig. 5.1 the corrected Fontainebleau sandstone data (Eqns ) is compared to the reconstructions introduced in sections at the fixed resolution of 5.68µm, namely the IOS model based on the 2-point correlation function (IOS C ), the Boolean models based on the Minkowski functionals (IOS (1), IOS (2), ROS (1), ROS (2) ), the Gaussian 5-level intersection model ([121]), and the Boolean models based on the distribution of the covering radii (OSC, [155]). All models are reasonable predictions. Errors in predictions are at most ±5%. Consistently the 1-particle models based on the Boolean reconstruction perform very well. The IOS model based on the 2-point correlation function shows a much weaker match to the elastic moduli of the Fontainebleau sandstone than the IOS model based on the Boolean reconstruction. The only difference is in the radii chosen (see Tab.3.11), and thus in the discretisation of the model and would scale out in the continuum limit. The Boolean matches based on 2-particle reconstructions are surprisingly giving a poorer match than the 1-particle models, despite the fact, that the local morphology is matched better by these 2-particle models. However a sandstone is not generated by a Boolean process. This can lead to errors and is discussed and quantified below. In chapter 3.2 the Boolean reconstructions of the Fontainebleau were derived, and it was found, that for all 2-particle Boolean reconstructions (see Tab.3.11) one of the particles was very small. At the same time the probability to place a small particle was much larger than for a larger particle. The resulting structure exhibits a relatively rough surface due to the small particles and with increasing porosity a significant number of small (solid) particles are left in suspensed state. This is aphysical and, of course, during segmentation of the original CT image doesn t occur. It is an example why using a Boolean model to match a non-boolean structure is problematic. Suspended particles do not contribute to the stiffness of the reconstructed data sets; thus the 2-particle Boolean reconstructions underpredict the elastic moduli. To find all particles in suspension, a 6-neighbourhood invasion (burn algorithm [148]) from all boundaries of the domain is carried out for the solid phase. The procedure is repeated after shifting all boundaries on the periodic lattice by the vector L( 1 2, 1 2, 1 2 ) with x, y, z [; L]. The difference between the porous fraction before and after this process gives ss, the fraction of the solid phase in

198 Elastic properties of Boolean and Gaussian reconstructions Bulk modulus [GPa] fb7.5 IOS (1) IOS (2) ROS (1) ROS (2) IOS C OSC Gaussian Shear modulus [GPa] 35 3 fb7.5 IOS (1) IOS (2) ROS (1) ROS (2) IOS C OSC Gaussian [a] Bulk modulus [GPa] fb13 IOS (1) IOS (2) ROS (1) ROS (2) IOS C OSC Gaussian [b] Shear modulus [GPa] fb13 IOS (1) IOS (2) ROS (1) ROS (2) IOS C OSC Gaussian [c] Bulk modulus [GPa] fb15 IOS (1) IOS (2) ROS (1) ROS (2) IOS C OSC Gaussian [d] Shear modulus [GPa] fb15 IOS (1) IOS (2) ROS (1) ROS (2) IOS C OSC Gaussian [e] Bulk modulus [GPa] fb22 IOS (1) IOS (2) ROS (1) ROS (2) IOS C OSC Gaussian [f] Shear modulus [GPa] fb22 IOS (1) IOS (2) ROS (1) ROS (2) IOS C OSC Gaussian [g] [h] Figure 5.1: Image based calculations of the water saturated elastic moduli of the four Fontainebleau sandstone samples compared to Boolean and Gaussian reconstructions. [a-b]: fb7.5, [c-d]: fb13, [e-f]: fb15, and [g-h]: fb22. The Fontainebleau sandstone data is corrected to represent a periodic infinite sample.

199 164 Linear elastic properties of reservoir rocks Fontainebleau IOS (1) IOS (2) ROS (1) ROS (2) IOS C OSC Gaussian ss total Figure 5.11: Running average (4 points) of the fractions of particles in suspension over total porosity for the reconstructions to the Fontainebleau data sets. For each model the reconstructions to the four Fontainebleau sandstones are combined in one plot, thus resulting in 4 points per fraction. The 2-particle and OSC models show a significant effect, as one of the particles in the Boolean reconstruction is very small (see Tab. 3.11). suspension, assuming that no solid particle in suspension has any length larger than L/2. The results for the different models are depicted in Fig The same analysis was carried out for the 12 3 Fontainebleau samples itself. The non-periodicity of the sample can lead to suspended solid regions located on the boundary. The amount of solid in suspension purely due to the boundary is more than an order of magnitude smaller and is negligible at higher porosity ( > 15%). It is apparent from Fig. 5.11, that the 2-particle models indeed have a relatively large amount of solid ( ss ( =.2).2) in suspension, and a further unknown amount of solid (dangling branches, e.g.), which do not contribute much to the stiffness of the reconstructed morphology. This effect is an order of magnitude smaller for the 1-particle and Gaussian models. The Gaussian model overpredicts the elastic properties of the Fontainebleau sandstone for all samples. However, the model given in [128] was matched for a different Fontainebleau sample and the match is still very

200 Elastic properties of Boolean and Gaussian reconstructions 165 close. Closing the discussion it seems logical to filter out all particles or clusters of particles in suspension. This however would make the model non-boolean, and would require changing the statistics and generality of the model. This will be left for future work. Currently we are considering the development of reconstructions based on the Minkowski functionals for hard/soft-shell spheroidal systems Cross-bedded sandstone We carried out a similar analysis on the cross-bedded sandstone sample and its reconstructions. The sample (see 2.2.3) was subsampled into 54 pieces of 1 3, and the elastic moduli calculated for each. The elastic moduli of the reconstructed data sets were calculated for the same sample size, and 1 realisations for each fraction were run. The elastic moduli of the cross-bedded sandstone were assumed to be the ones of quartz (K = 37GPa, µ = 44GPa) while the porous space was taken to be water saturated (K = 2.2GPa, µ = ). The correction for periodicity was based on the IOS C model and is of the same order as the one for the Fontainebleau sandstone derived in the previous section K (1µm) periodic () = K(1µm) n.p. () (1.54), (5.37) G (1µm) periodic () = G(1µm) n.p. () (1.79). (5.38) The elastic moduli of the cross-bedded sandstone and its isotropic sphere and spheroid based Boolean reconstructions are shown in Fig For the parameters of the Boolean models see Tab Unlike the Boolean reconstructions of the Fontainebleau sandstone, the 2-particle models have a smaller contrast in particle size, and additionally their probability to be placed in the Poisson process is smaller. Thus, the particle fraction of small particles in suspension is significantly reduced (see Fig. 5.13). The results show a better performance of the 2-particle Boolean reconstructions compared to their 1-particle counterparts and clustering of all datapoints. The IOS model reconstructed using the 2-point correlation function (IOS C ) in this case performs better than the reconstruction of the same model based on the Minkowski functionals (IOS (1) ). Here the radius of the reconstructed spheres is 94µm for IOS C and 13µm for IOS (1). For the Fontainebleau sandstone this relation was vice versa. The 2-particle spheroidal Boolean match is more heterogeneous and matches the elastic properties of the cross-bedded sandstone better, with decreasing difference for larger fractions, as the fraction of solid phase in suspension ( ss ) starts to increase (Fig. 5.13). Compared to the one- and two- particle matches, the reconstruction based on the covering radius (OSC) does not as well. Here the fraction of particles in suspension is comparable to the model ROS (2), and

201 166 Linear elastic properties of reservoir rocks 4 Bulk modulus [GPa] 35 3 cross bedded sandstone IOS (1) IOS (2) ROS (1) ROS (2) IOS C OSC 25 [a] Shear modulus [GPa] cross bedded sandstone IOS (1) IOS (2) ROS (1) ROS (2) IOS C OSC 2 [b] Figure 5.12: Image based calculations of the elastic moduli of the water-saturated cross-bedded sandstone samples compared to the Boolean reconstructions given on page 68. The ROS (2) is giving the closest match to the original sandstone.

202 Elastic properties of Boolean and Gaussian reconstructions Cross bedded IOS (1) IOS (2) ROS (1) ROS (2) IOS C OSC ss total Figure 5.13: Running average (1 points) of the fractions of particles in suspension over total porosity for the cross-bedded sandstone and its Boolean reconstructions (see Tab on page 68). other structural factors (i.e. the Minkowski functionals) might play a role, which are matched better by ROS (2). The Boolean reconstructions to the two different experimental data sets, the Fontainebleau and cross-bedded sandstones, highlight the problem with using a Boolean reconstruction to reconstruct an inherently non-boolean morphology. To reduce the effect of suspensed particles, one would need to consider reconstruction algorithms with additional physical constraints, i.e. cluster growth, or conditional statistical reconstruction. A simple way to improve the performance of the current Boolean reconstructions would be to favor homogeneous particle size distributions which still accurately match the local measures. Another way to preserve the matching global morphology (Minkowski measures), is to carry out simulated annealing with the additional constraint of preserving morphology (Minkowski measures). This could be done by finding the inclusions, and merging particles/clusters with the matrix.

203 CHAPTER 6 GEOPHYSICAL APPLICATIONS: VELOCITY-POROSITY RELATIONSHIPS Petroleum reservoir engineers are confronted with the formidable task of having to monitor the fluid distribution of a reservoir over its lifetime. The only data source able to probe the full reservoir continuously at a constant sampling rate is seismic data. Because of this reason seismic data are used extensively in exploration (3D seismic) and increasingly in monitoring of petroleum reservoirs (4D seismic). Here 4D simply means, that a number of 3D seismic surveys are conducted over the lifetime of the reservoir. Because 3D seismics are carried out at acoustic wavelengths, their resolution is poor. Therefore a calibration against sonic logs acquired by well logging tools and against experimental measurements on cored sections of the wells is normally carried out. The seismic velocities depend on the lithology, fluid type and content of the formation, and thus accurate elastic modulus-porosity or velocity-porosity relationships are critical to the determination of lithology from seismic or sonic log data as well as for direct seismic identification of pore fluids. Because of the cost of seismic surveys, and the implications of changes in these relationships to fluid content, even small improvements in velocity-porosity relationships are significant. One important relationship to consider in geophysical logging is the ratio of 3K+4µ the compressional wave velocity V p = to the shear wave velocity V 3ρ s = µ, ρ where K, µ, ρ are the bulk modulus, shear modulus, and density. To measure elastic properties one must obtain two independent measurements. While compressional wave velocities are commonly measured via logging tools, in practice, the shear wave velocity is often not measured and must instead be estimated from the measured compressional wave velocity V p. A common approach is to estimate V s from empirical V p -V s relationships or models. Examples are the empirical correlations of Han [54] and Castagna [33] for saturated sandstones, and the critical porosity model of Nur [19, 11] and Krief s relation [73] for dry sandstones. The common empirical modulus/porosity relationships of Krief and Nur for dry sands exhibit the same dependence on porosity for both the bulk and shear modulus, trending

204 17 Geophysical applications: velocity-porosity relationships towards the value for the solid mineral in the limit of zero porosity and towards zero in the limit of a critical porosity, c. This is equivalent to assuming that the dry rock V p /V s or Poisson s ratio, ν, is independent of porosity,, and equal to the solid (V p /V s ) s or ν s, ] [ Vp V s dry rock = [ Vp ] V s mineral ν dry rock = ν mineral. (6.1) In general the prediction of elastic modulus-porosity relationships developed by geophysicists can be reduced to two simple elements. For each lithology, empirical relationships are established among the elastic moduli and porosity for a reference fluid (either dry or water-saturated rock) based on experimental data. Then Gassmann s relations are used to map between different pore fluids. Unfortunately experimental modulus-porosity data are usually strongly scattered. This scatter can be due to lithology (pore shape, size, degree of compaction) [87] as well as clay content and distribution [54]. These features, which unfortunately lack precise definition, comprise the microstructure of the rock. In order to predict properties or properly interpret experimental modulus-porosity relationships, it is necessary to have an accurate method of relating elastic properties to porosity, mineralogy and microstructure. Because of the natural randomness and complexity observed in sedimentary rocks, analytical methods for quantitatively relating microstructure and properties of sedimentary rock are generally not accurate. Although useful, they can only be used by smearing out some aspect of the complex random microstructure. Due to differences in basic interpretations, in choice of reference states and methods, different models often strongly disagree. Exploring velocity-to-porosity transformations in a more controlled environment will lead to a better understanding and enhanced predictive power. As demonstrated in the previous chapter, it is now possible to computationally solve for the elastic properties of digital models of complex microstructure, where the pore space structure, microstructure and mineralogy of the material is controlled, and one has the ability to average over a number of statistically identical samples. In this chapter numerical simulations are used to calculate the elastic properties of model rock morphologies in dry, water-, and oil-saturated states. These computational data, although based on idealised microstructure, allows us to quantitatively analyze the effects of porosity and the elastic properties of the mineral phase on the resultant elastic data for porous rock. We consider both clean sandstone morphologies and model clay-cemented sands composed of two mineral phases. Results are compared to currently employed theoretical and empirical relationships for deriving moduli-porosity estimates. We show that the common assumption for dry rocks (Eqn. 6.1) is not correct and present a new predictive formula for the Pois-

205 Methods 171 son ratio based on an empirical fit to the observed shape and limiting behaviour of the Poisson ratio. We further note that the Krief empirical relationship is particularly successful at describing the numerical data for dry shear modulus. We find that the Raymer equation gives the best modulus porosity model (< 1% error) for the water-saturated cases. In all cases Gassmann s relations accurately map between the dry and fluid saturated states. Based on these results we propose an accurate empirical method which enables one to derive the full velocity-porosity relationship for general multimineralic consolidated sands solely based on knowledge of the elastic properties of the mineral constituents and their phase fractions. The model for dry rock is given by µ dry () = µ (1 ) m(), (6.2) ν dry = a() + (1 (2) 3/2 ) ν, (6.3) where µ and ν are the effective shear modulus and Poisson s ratio of the mixed mineral constituent. µ and ν are derived from an estimate of the mixed solid moduli based on either the average or weighted Hashin-Shtrickman bounds. For clean sands µ and ν are simply the shear modulus and Poisson ratio of the solid phase. Gassmann s equations are subsequently used to predict the fluid saturated states. Comparisons of this deterministic empirical method to several experimental studies show that the proposed empirical method gives an excellent match to data. The chapter is organised as follows. First we define the model morphologies and the simulation method. Second we consider the behaviour of Poisson s ratio for dry rocks. An empirical equation giving an accurate prediction of the Poisson ratio is presented. The effect of errors in the prediction of Poisson ratio in for example reservoir monitoring of fluid contents is discussed. The third section details a framework for predicting the elastic properties of clean sandstones from the knowledge of the elastic properties of the mineral phase alone. Gassmann s equations are subsequently used to predict the fluid saturated states. Section four extends this methology to multimineralic sandstones. Through all sections the proposed empirical methods are compared to available experimental data and give an excellent match. 6.1 Methods In section 5.3 a finite-element method [45] was used to derive modulus/porosity relationships directly from microtomographic images for a suite of Fontainebleau sandstone - a clean, homogeneous, consolidated sandstone, composed of quartz grains with quartz cement and displaying only intergranular porosity. The numer-

206 172 Geophysical applications: velocity-porosity relationships ically derived properties were in excellent agreement with experimental measurements on Fontainebleau [54] over the entire range of the porosity (.5 < <.25) sampled. Unfortunately the number of data sets available for numerical study via microtomography is still limited. The same numerical property calculations can however be performed on model rock morphologies. This provides a convenient means of generating extensive property data for a range of porosity, mineralogy and microstructure Structural model for clean sandstone The simplest statistical model for a consolidated sandstone like Fontainebleau is based on overlapping spherical grains (IOS model) [155, 7] introduced in section (see Fig. 2.2.b and Fig. 2.4). In Fig. 5.7 we showed a comparison between the numerical predictions of the elastic modulus for the IOS model with quartz grains, numerical predictions derived from microtomographic images for Fontainebleau sandstone [5], and experimental data of [54] for Fontainebleau sandstone samples. 1 The match between the three data sets is good. This suggests that the IOS model gives a reasonable description of the microstructure of homogeneous sandstones. The model morphology is generated by placing solid (discretised) spheres at random points in a cubic cell and illustrated in Fig The space outside the spheres is the pore space with porosity. The pore phase is macroscopically 1 These numerical results are scaled to the continuum limit, while the results of the Boolean analysis given in Fig. 5.1 are not scaled, but matched at the given resolution. The difference between the IOS (1) and IOS C model would scale out in the continuum limit. [a] [b] [c] [d] [e] [f] Figure 6.1: Images of the pore space of an IOS model. 3D pore space images of the model microstructure are shown at porosities of [a] 1%, [b] 2%, and [c] 3%. 2D slices through the 3D data sets are shown at [d] = 1%, [e] 2%, and [f] 3%. The image size is that of the simulations (12) 3 and the sphere sizes are 12 3.

207 Methods 173 connected above porosities of 4% and the solid phase remains connected to very high porosity. In this chapter we consider the elastic properties for a very wide range of fractions % < < 5%. To generate the microstructure we choose solid spheres of (discretised) radius r = 12 voxels in a cubic box of size L = 12 voxels. It was shown in previous chapters that numerical errors are acceptably small at this scale (see section 5.3 for further discussion) Structural model for sandstone having two mineral phases Sandstones are rarely clean. They often contain minerals other than quartz, such as clay minerals, which can affect their elastic properties. Clay can be distributed in a number of ways in the rock framework depending on the conditions at deposition, on compaction, bioturbation and diagenesis. Most laboratory experiments and many models ignore the type and location of the clay. Most empirical relationships (e.g. Castagna, Batzle and Eastwood [32], Han et al. [54, 53]) and theoretical models [17] account for only the volume of clay present and ignore the distribution of the clay. An advantage of using a numerical model is one can consider complex multi-phase materials yet control the different phase distributions. Here we consider two model distributions of cemented sandstone composed of two mineral phases, termed framework model and interstitial model according to their mineral distributions outlined below. The two-mineral-phase framework model is based on the placement of spheres of mineral phase 1, and then overgrowth by placement of identical permeable spheres of mineral phase 2. By varying the density of the spheres we generate a range of phase fractions. Throughout the chapter we report the ratio of the mineral phases along with the order of placement, mineral 1 then mineral 2 (M 1 : M 2 ), and the porosity. For example, a quartz:clay framework model microstructure at a 2:1 ratio is generated by placing quartz spheres to a predetermined volume fraction, then allowing clay spheres to overgrow that system made up of quartz and porosity, until the required volume ratios of both mineral phases and porosity is reached. Boolean statistics (see e.g, [7]) enable one to generate model systems at the correct phase fractions by relating the density of permeable spheres directly to the volume fraction of the phase required. For example, if we place with probability ρ spheres of volume V s, we can relate the porosity of the final configuration simply to this probability via (see Eqn. 3.12) (ρ) = e ρvs. (6.4) Equations for multiphase materials can also be easily derived.

208 174 Geophysical applications: velocity-porosity relationships [a] [b] [c] [d] [e] [f] [g] [h] [i] Figure 6.2: Slices through the framework two phase mineral model at the resolution used in the elastic simulations (12 3 ). The top row is at a M 1 : M 2 ratio of 1:1, the second row at a ratio of 2:1 and the bottom row at a ratio of 4:1. The volume fractions of the three (M 1 : M 2 : ) phases are [a]: 45:45:1, [b]: 4:4:2, [c]: 3:3:4, [d]: 6:3:1, [e]: 53:27:2, [f]: 4:2:4, [g]: 72:18:1, [h]: 64:16:2, [i]: 48:12:4. We consider a number of variations of sand:cement mixtures over the porosity range < < 5%. We consider quartz as M 1 and both clay and dolomite as the second (overgrowing) M 2 phase at 1:1, 2:1 and 4:1 ratios. We also consider the opposite systems; clay or dolomite placed first, with quartz overgrowth again at volume ratios of 1:1, 2:1 and 4:1. Examples of the microstructure of the framework model at different mineral ratios and porosities are illustrated in Fig In the interstitial model we begin with an IOS model for the first mineral phase and allow overgrowth of the second mineral phase into the pore space by parallel surfaces. Here we place spheres of M 1 until we reach the required volume fraction f via Eqn. 6.4, where f = 1. The second mineral phase, M 2, is then allowed to grow via parallel surfaces originating from the grain/pore interface of the original spheres into the pore space. Examples of the model microstructure are given in Fig The model is similar to that considered previously by [39], where grain overgrowth of a hard-grain pack was considered; in the current model the original mineral grains are permeable. We again consider the 6 ratios of quartz:cement mixtures as described for the framework model for the full range of the porosity.

209 Methods 175 [a] [b] [c] [d] [e] [f] [g] [h] [i] Figure 6.3: Slices through the interstitial two phase mineral model at the resolution used in the elastic simulations (12 3 ). The top row is at a M 1 : M 2 ratio of 1:1, the second row at a ratio of 2:1 and the bottom row at a ratio of 4:1. The volume fractions of the three (M 1 : M 2 : ) phases are the same as in Fig Neither model is limited to sphere packs and can be extended to the consideration of any kind of spheroidal shape. Preliminary results on isotropic packings of more complex spheroidal particles yield similar results to those reported here Elastic simulation We use the finite element method introduced in section 5.2 to estimate the properties of the model system. A minimum of 1 independent samples were used to reduce statistical errors to the order of a few percent. We account for discretization errors as detailed previously and scale the elastic properties to the continuum limit. The models were discretised on lattices of 12 3, 6 3, 4 3, and 3 3. Simulations were primarily run on several nodes of a Compaq ES4 supercomputer over a period of months. The equivalent of two years of workstation time was required to complete the simulations described here. From [92] we assign to the rock skeleton values of the elastic properties of different minerals and fluids. The values are summarised in Table 6.1. The choice of the water saturated condition is made to allow for comparison with experimental data from [54].

210 176 Geophysical applications: velocity-porosity relationships Table 6.1: Material properties for the minerals and fluids used in the simulations presented in this chapter [92]. K is the bulk modulus µ the shear modulus, and ρ the density. The water conditions are chosen to allow for comparison with experimental data [54]. Material K [GPa] µ [GPa] ρ [g/cm 3 ] Quartz Feldspar Dolomite Clay Water,T = 2 o C, 4 MPa Dead oil,t = 2 o C, 25 MPa The ability to control the pore space structure, microstructure and mineralogy of the material within a numerical model as well as the ability to average over a large number of statistically identical samples allows one to generate effectively noiseless data which, although based on an idealised microstructure, will allow us to quantitatively analyze the effects of porosity and the elastic properties of the mineral phase. 6.2 Accurate V p : V s relationships for dry sandstones In this section numerical simulations are used to calculate the elastic properties of model rock morphologies to show that the assumptions leading to Eqn. 6.1 are not correct. The calculations show that ν is a non-linear function of and that it trends from the value for the solid mineral in the limit of zero porosity to a constant value of.2 at porosity c =.5, independent of mineral type. This limit is shown to be true for clean (monomineralic) sands and for model sands containing cements (multiple - mineral phases). We present a simple universal predictive equation for dry ν() which holds for all the systems considered irrespective of the number of mineral phases present or the Poisson s ratio of the solid mineral Poisson s ratio for dry sandstones Clean sandstones It is reasonable to expect that the Poisson s ratio for isotropic porous rocks should depend on the properties of the solid matrix, ν s and. One would write ν = g(ν s, ) where the dimensionless function g depends on both microstructure and mineralogy. It has been shown both numerically [36] and analytically [34, 154] that g has remarkable properties for two-dimensional dry porous materials composed of a

211 Accurate V p : V s relationships for dry sandstones 177 single solid mineral; as the solid fraction decreases to the percolation threshold of the solid phase ( ), the effective Poisson s ratio converges to a fixed value independent of the solid Poisson s ratio or g(ν s, ) ν c for. The proof is true only in two dimensions and is thought not to hold rigorously in three dimensions. Recent numerical experiments by [123] for a wide range of model structures however suggest that a limiting, although non-universal, value of ν may also be observed in threedimensions. The present calculations confirm that this is indeed the case. We have used the IOS model to study the behaviour of Poisson s ratio for a dry porous model rock morphology across a range of for two single mineral phases, quartz (q) and feldspar (f), which have very different ν s. The properties of these minerals were taken from tables in [92] (for quartz K q = 37. GPa, µ q = 44. GPa and νs q =.8 and for feldspar K f = 37.5 GPa, µ f = 15. GPa and νs f =.32). The results are plotted in Fig. 6.4.a. Notable is the lack of noise in the data sets when compared to typical experimental results (see e.g., [54]). The standard error in the data (shown in the Fig. 6.4.a) is extremely small < 1% for for most measurements on the monomineralic structure. One striking feature is immediately evident - the Poisson s ratio for the model morphology becomes independent of the Poisson s ratio for the mineral solid at a critical porosity. In both cases, despite very different values of mineral ν s, the flow diagram for ν(, ν s ) converges to a fixed point at c.5. The behaviour is very similar to that observed in two-dimensions where the critical porosity coincides with the geometric (percolation) threshold p c =.5 [36]. The percolation threshold for the solid phase of the IOS model in three dimensions is p c.8. It is not clear why the point of convergence of the Poisson s ratio in three-dimensions coincides with the two-dimensional threshold. The experimental data of Han [54] for clean quartz sandstones is also plotted. This experimental data follows the numerical prediction of the pure quartz curve for the range of porosities studied. Roberts and Garboczi [123] found that ν(ν s, ) can, to a reasonable approximation, be described as a linear function of and ν s ; ν = ν s + c (ν c ν s ). This equation, with c =.5 and ν c =.2, in contrast to their prediction that ν c =.14, gives an improved prediction for the dry rock Poisson s ratio to the common empirical estimate that ν = ν s. However, the plot in Fig. 6.4.a exhibits a clear non-linear behaviour. This behaviour is well approximated by the predictive non-linear empirical equation, { ν s + (2) 1.5 (.2 ν s ), ν s <.2 ν = (6.5).2 + (1 2) 1.5 (ν s.2), ν s >.2 This equation provides an excellent fit to the data of Fig. 6.4.a.

212 178 Geophysical applications: velocity-porosity relationships Poisson ratio ν Quartz Feldspar Experiment Eqn. 6.5 SCA Poisson ratio ν [a] Quartz/Dolomite (2:1) Quartz/Dolomite (1:2) Quartz/Clay (2:1) Quartz/Clay (1:2) Eqn. 6.5 [b] Quartz/Dolomite Quartz/Clay Eqn Quartz/dolomite Quartz/clay Dolomite/quartz Clay/quartz Eqn. 6.5 Poisson ratio ν.2 Poisson Ratio ν [c] [d] Figure 6.4: The Poisson s ratio of the IOS model as a function of porosity. In [a] we show results for the clean dry quartz and feldspar systems. Standard error bars are shown and are of the order of the size of the data point. The experimental data of [54] for clean quartz sandstones is also shown. The fit of Eqn. 6.5 to the data is excellent. The Poisson ratio for mixtures of quartz/clay and quartz/dolomite are shown for [b] 1 : 1 ratio, [c] 2 : 1 ratio, and [d] 4 : 1 ratio. The filled symbols give the prediction of the interstitial model and the open symbols the prediction of the framework model. We show in [b] the characteristic standard errors in the numerical calculation for systems of mixed mineralogy. While they remain small for <.3, they can diverge for some models at higher porosities. The flow diagram still converges to a fixed point at c.5 and seems independent of the cement deposition model chosen. Agreement with empirical Eqn. 6.5 is good for both models at all mineral ratios.

213 Accurate V p : V s relationships for dry sandstones Cemented sandstones Sandstones are rarely clean, frequently containing varying amounts of clay and cements, which can affect their elastic properties. These minerals can be distributed in a number of ways in the rock framework depending on the conditions at deposition, on compaction, bioturbation and diagenesis. One would expect the Poisson s ratio for cemented sandstones to depend on the properties and volume fractions f i of the i components of the solid matrix mineralogy, i.e. ν = g(, νs, i f i ). We consider the two models for cemented sands introduced in the previous section; a structural or framework cement/sand mix, and a dispersed or interstitial cement/sand mixture. The framework model is based on the placement of spheres of mineral phase 1, and then overgrowth by placement of identical permeable spheres of mineral phase 2. The interstitial model begins with an IOS model for the first mineral phase and allows overgrowth of the second mineral phase into the pore space by parallel surfaces. Fig. 6.2 and Fig. 6.3 show the morphological differences between the two cementation schemes. We consider systems with different ratios of quartz to cement, including 1 : 4, 1 : 2, 1 : 1, 2 : 1 and 4 : 1. The cementing minerals used in the simulation are dolomite and clay. The values of the elastic properties are for dolomite K d = 69.4 GPa, µ d = 51.6 GPa and νs d =.22, and for clay K c = 2.8 GPa, µ c = 6.9 GPa and νs c =.351 [92]. The effective solid Poisson s ratio for the two phase mineral mixtures, ν s, is determined numerically (see Tab. 6.2). For the quartz/dolomite matrix the effective mineral phase ν is well approximated by the arithmetic average. The quartz/clay systems exhibit a small ( 5%) variation with structure and the arithmetic average consistently underestimates ν s by up to 1%. For the fits to Eqn. 6.5 we use the Table 6.2: Effective Poisson ratio for the mixed lithologies at zero porosity. We report the ratio of the mineral phases along with the order of placement, mineral 1 then mineral 2 (M 1 : M 2 ). The prediction of the framework model ν f, interstitial model ν I and the arithmetic average ν av are reported. Ratio ν f ν I ν av ν f ν I ν av Quartz : Clay Clay : Quartz 1: : : Quartz : Dolomite Dolomite : Quartz 1: : :

214 18 Geophysical applications: velocity-porosity relationships average of the numerically computed values for ν s for the two model microstructures. Fig. 6.4.b shows computed Poisson s ratio for quartz/clay and quartz/dolomite systems at a 1 : 1 volume ratio. The data for the mixed mineralogy system is noisier than for monominerallic systems indicating larger numerical errors, but the trend in the data is still clear. In all cases, despite very different values of νs, i the flow diagram for ν(, fs, i νs) i converges to the fixed point ν.2 at c.5 the same limit observed for the monomineralic system. The empirical equation for clean sandstones, Eqn. 6.5, again provides a good fit for the mixed mineralogy system. Interestingly, the distribution of the phases (intergranular deposition versus surface deposition) has little effect on the computed Poisson s ratio. This is not generally observed for other mechanical properties. The result is consistent with measured data which indicate that the V p /V s ratio is mostly insensitive to the presence and distribution of clays [92]. Similar results are obtained for a wider range of quartz/clay and quartz/dolomite ratios. Fig. 6.4.[c-d] shows the behaviour for ratios of 2 : 1 and 4 : 1 for both depositional models. Again, in all cases, the flow diagram for ν(, fs, i νs) i converges to the fixed point ν.2 at c.5 and the fit of the empirical Eqn. 6.5 to the data is still good. Again, the choice of intergranular deposition or surface deposition has little effect on the Poissons ratio. The results for the interstitial 4 : 1 model are very noisy for porosities >.35 due to difficulties with numerical discretisation of the parallel surfaces. Data for the model up to =.35 however shows a clear trend towards ν.2 for all ν s chosen. Data for the framework model at the 4:1 ratio all converge to ν.2. Comparison of the prediction of Eqn. 6.5 to experimental data is shown in Fig We consider the Poisson s ratio data for shaley sandstones from [54] where we have binned the data into groups of clean, intermediate (< 15%) and higher (15 35%) clay content. We also show laboratory data sets clay cemented quartz sands from the Oseberg field [39]. The variation with clay content and porosity is well described by the model results and far superior to other predictions; ν = ν s [73, 19] and ν.4 [39]. We have extended the work to three mineral phases and continue to observe the same limiting behaviour for the Poisson s ratio. We believe that these results indicate that this behaviour of Poisson s ratio is universal for this morphology and that it may be expected to occur for most dry sands. c and ν c may differ however for different rock morphologies [123]. Again, this is consistent with the observation that c for the empirical critical porosity model depends on rock type [92].

215 Accurate V p : V s relationships for dry sandstones Poisson ratio ν.15.1 Han: Clay <15% Han: Clean Sandstone Han: Clay >15% Eqn. 6.5: Clean Sands Eqn. 6.5: Q:C = 4:1 Eqn. 6.5: Q:C = 2:1 Poisson ratio ν.15.1 Eqn. 6.5: Clean Sands Eqn. 6.5: Q:C = 4:1 Eqn. 6.5: Q:C = 2:1 Oseberg Data [a] [b] Figure 6.5: Comparison of data from [54] and [39] for clay-bearing sands to the prediction of Eqn Implications equation The Poisson ratio of the dry rock ν dry is directly related to [V p /V s ] dry by the ν = [ 1 Vp 2 ] 2 V s 1 dry [ ] 2. (6.6) Vp V s 1 dry A good estimation of [V p /V s ] dry is crucial as seismic and sonic data are commonly used to infer saturation in situ. We illustrate the consequence of a poor prediction of [V p /V s ] dry to saturation predictions. Consider a pure quartz sandstone at a porosity = 24%. The value of V p we obtain for this system is V p = 4.14 km/s. Assuming that the Poisson s ratio for this system ν( =.24) = ν s =.8 (Eqn. 6.1) we obtain V p /V s = Using Eqn. 6.5 we obtain ν( =.24).13 or V p /V s = This leads to a prediction for K dry = GPa via Eqn. 6.1 and K dry = GPa using Eqn The value we obtain via simulation for =.24 is K sim = 14.8 ±.2 GPa. The low-frequency Gassmann s equation (Eqn. 5.3) is then used to relate the bulk modulus of a saturated porous medium to this prediction of K dry for the same medium in a dry state. If the rock is in a partially saturated state one uses the effective fluid model, replacing the collection of phases with a single effective fluid in Eqn Assuming the multiple pore fluid phases are mixed at a fine scale the effective bulk modulus of the mixture of i fluids Kf is described well by the Reuss average (Eqn. 5.13). For an air/fluid mixture we use the Voigt average (Eqn. 5.12). Based on this one can relate measurements of K sat for a fluid saturated system to the moduli of the fluids. We illustrate in Fig. 6.6 the saturation prediction based on different measured Kf for partially saturated sandstone. We show predictions for both an oil/water mixture (K water = 2.2 GPa, K oil =.5 GPa) and an air/water

216 182 Geophysical applications: velocity-porosity relationships mixture. In Fig. 6.6.a we observe that a measurement of the effective bulk modulus Kf = 16 GPa for a quartz/water/oil system would lead to a prediction of S water = 92%, S oil = 8% using Eqn. 6.1 and a prediction of S water = 54%, S oil = 46% using Eqn Clearly the error in the prediction of K dry leads to a very poor estimation of the pore fluid saturation and underlines how crucial the accurate Eqn. 6.5 can be to reservoir fluid predictions S water and S oil.6.4 S w, conventional S o, conventional S w, ratio S o, ratio S water and S gas.6.4 S w, conventional S g, conventional S w, ratio S g, ratio.2.2 [a] K sat [GPa] [b] K sat [GPa] Figure 6.6: Comparison of the conventional model against the empirical nonlinear model based on the V p /V s ratio in terms of fluid saturation predictions. Left: water/oil mixture; right: water/gas mixture Conclusion Numerical simulations on model morphologies for sandstones containing any number of solid mineral phases show that the Poisson s ratio for the dry system trends in a non-linear manner from the effective solid Poisson s ratio in the limit of zero porosity to a constant value of.2 at a critical porosity of c =.5 which is independent of the solid Poisson s ratio. A simple empirical equation, ν = ν(ν s, ), fits the numerical data well, is consistent with experiment and provides a better estimate of dry rock Poisson s ratio than the commonly used estimate ν = ν s. Implications to reservoir fluid predictions are discussed. 6.3 Accurate velocity model for clean consolidated sandstones In this section we repeat the general modulus-porosity relationships for both the dry and water-saturated states for the IOS model with a single mineral phase (clean sandstone). We consider both model quartzose and feldspathic sands. Simulations are performed as discussed in section 6.1. Results are presented for dry, and then for water-saturated systems, and are compared to currently employed

217 Accurate velocity model for clean consolidated sandstones 183 theoretical and empirical relationships for deriving moduli-porosity estimates. We note that the Krief empirical relationship is particularly successful at describing the numerical data for dry shear modulus and Eqn. 6.5 gives a good prediction of the Poisson s ratio data for dry rock. We find that the Raymer equation gives the best modulus porosity model (< 1% error) for the water-saturated cases. In all cases Gassmann s relations accurately map between the dry and fluid saturated states. Based on these results we establish an accurate empirical method which enables one to derive the full modulus/porosity relationship for consolidated sands solely based on the knowledge of the mineral content of the rock. Match to numerical data and available experimental data for clean sands is excellent Numerical results In Fig. 6.7 we give results for the dry bulk modulus K, shear modulus µ, V p and V p /V s ratio for the IOS model as a function of porosity for two different simple (single mineral) phases, quartz and feldspar. The moduli of the rocks trends between the modulus of the mineral at low porosities to zero at higher porosities. Although the bulk modulus of quartz (37. GPa) and feldspar (37.5 GPa) are very similar, the curve for the bulk modulus of the porous systems vary appreciably. Moreover, while K feld > K quartz, the curve for the feldspar lies consistently below that for the quartz. The V p curve is approximately linear with porosity for both mineral phases and the V p /V s ratio exhibits a strong dependence on porosity and goes towards a fixed point (V p /V s ) 1.63 = (8/3) at higher porosities. This fixed point was discussed above and corresponds to a fixed point in the Poisson s ratio ν =.2. This behaviour is similar to the rigorous behavior observed in two-dimensions [36]. Again we observe little noise in the data (Fig. 6.7). The standard error is extremely small < 1% for most measurements. Only for V p /V s at high porosities does the standard error exceed a few percent. For systems at higher porosity the elastic properties become more dependent on small tenuous connections, which are increasingly difficult to resolve, and therefore larger statistical errors are observed. Importantly, the numerical data in the region of most interest, 5% < < 4%, exhibits much less noise than typical experimental data. The ability to control the pore space structure, microstructure and mineralogy of the material within a numerical model as well as the ability to average over a large number of statistically identical samples allows one to generate effectively noiseless data which, although based on an idealised microstructure, will allow us to quantitatively analyze the effects of porosity and the elastic properties of the mineral phase. In section we will use this data to test common empirical and theoretical models used to

218 184 Geophysical applications: velocity-porosity relationships 4 5 Bulk modulus [GPa] Quartz & air Feldspar & air Shear modulus [GPa] Quartz & air Feldspar & air [a] [b] Quartz & air Feldspar & air 1.9 Quartz & air Feldspar & air V p [km/s] 4. V p /V s [c] [d] Figure 6.7: Results of the simulation for the variation of the [a] bulk modulus, [b] shear modulus, [c] V p, and [d] V p /V s ratio as a function of porosity for the single phase IOS model with quartz and feldspar as the mineral phase. predict elastic properties of sedimentary rocks. In Fig. 6.8 we give results for the water saturated values of the bulk modulus K, shear modulus µ, V p and V p /V s ratio for the IOS model as a function of porosity for the two different single mineral phases. Again, despite the bulk modulus of quartz and feldspar being very similar, the curve for the bulk modulus of the watersaturated porous system varies appreciably. The bulk modulus of the model trends between the modulus of the mineral at zero porosity to values corresponding to a mineral-pore suspension at large. The shear modulus exhibits the same behaviour as the dry rock, going to zero at higher porosities. The V p curves are approximately linear with porosity for both mineral phases. The V p /V s ratio exhibits a strong dependence on mineralogy. The quartz data shows a slow linear increase with porosity with a slow divergence. The feldspar data exhibits a constant region for < <.3 with a strong divergence for >.3. We further test the agreement of the numerical results with Gassmann s re-

219 Accurate velocity model for clean consolidated sandstones Bulk modulus [GPa] Quartz & water Feldspar & water Shear modulus [GPa] Quartz & water Feldspar & water [a] [b] Quartz & water Feldspar & water 3.5 Quartz & water Feldspar & water 5. 3 V p [km/s] 4. V p /V s [c] [d] Figure 6.8: Results of the simulation for the variation of the [a] bulk modulus, [b] shear modulus, [c] V p, and [d] V p /V s ratio as a function of porosity for the single phase IOS model with quartz and feldspar as the mineral phase. lations. In section it was shown, that the Gassmann equations are obeyed by the numerical results on the Fontainebleau sandstone data sets, using quartz as mineral phase and three different fluids. Here we show that the numerical data for the IOS model obeys Gassmann s equations for two fluids and two different mineral phases, quartz and feldspar. The results are summarised in Fig The numerical prediction for both the bulk and shear modulus are in excellent agreement with Gassmann s equations. The match to Gassmann s relations provides us with a further verification of the accuracy of the numerical results Comparison to experiment Han [54] made a number of velocity measurements for dry clean quartz sandstones. The numerical data for the quartz IOS model is compared to these experiments in Fig The match of the numerical calculation to experiment indicates that the IOS model gives a reasonable representation of consolidated sandstone

220 186 Geophysical applications: velocity-porosity relationships Bulk Modulus [GPa] Dry Quartz Water saturated Quartz Dry Feldspar Water saturated Feldspar Predictions from dry data + Gassmann Shear Modulus [GPa] Dry Quartz Water Saturated Quartz Dry Feldspar Water Saturated Feldspar [a] [b] Figure 6.9: Comparison of the results of the simulations for dry and water-saturated IOS model to Gassmann prediction based on the dry rock data with their standard errors. In [a] we give the numerical predictions for the dry rock data (squares) and show a best fit to the data points (solid line). We use this fit and Eqn. 5.3 to predict the values of the water-saturated curves (dotted curves). We also show the numerical predictions for the water-saturated results. The fit to the Gassmann s equations is excellent. In [b] we show that the shear modulus is independent of the pore fluid (Eqn. 5.31). structure. For water-saturated clean sandstones different empirical equations were obtained by [54] to describe the velocities as a function of porosity, using least squares regression. The velocities were fitted as V p = [km/s], (6.7) V s = [km/s]. (6.8) Numerical Han data 3. Quartz & air Han Data V p [km/s] 4. V s [km/s] [a] [b] Figure 6.1: Comparison of the results of the simulations for water-saturated sandstone to the dry velocity data of [54] for clean sandstones.

221 Accurate velocity model for clean consolidated sandstones Han: V p = Numerical V p [km/s] V s [km/s] Han: V s = Numerical [a] [b] Figure 6.11: Comparison of the results of the simulations for water-saturated sandstone to the empirical velocity-porosity equations derived by [54] for clean sandstones. The match is excellent for all. Comparison of the numerical data for the quartz IOS model to these linear fits are shown in Fig The agreement is excellent. This result underlines the ability of the numerical model to accurately predict the elastic modulus of clean sands. There are a number of empirical equations used to relate V p to V s based on cross-plots of experimental data. Data for sandstones was compiled by [33] and a least-squares linear fit to these data is given by V s =.84V p.856 [km/s]. (6.9) Han [54] derived an equation for sandstones with lower clay volume (< 25%); the least-squares linear fit to these data is given by V s =.754V p.657 [km/s]. (6.1) The fit of the numerical data to the two empirical fits is good. The numerical data is slightly below the two empirical curves, but exhibits the expected linear relationship between V p and V s for the full range of data. The empirical fits (Eqns. 6.9 and 6.1) include however sandstone data with significant proportions of clay. To better compare the model data we plot in Fig b experimental data points for the saturated low frequency velocities of clean sandstones from [54]. The numerical prediction gives the best match to this experimental data set. We are unaware of any data for feldspathic sands. However [33] did note that feldspathic sands tend to exhibit higher V p /V s than quartzose sandstones. This is in agreement with the numerical results shown in Fig. 6.8.d.

222 188 Geophysical applications: velocity-porosity relationships Numerical: Fit V s =.789V p.695 Castagna et.al. 1993: V s =.842V p.8559 Han, 1986, Clay<25%: V s =.754V p Numerical Castagna Han Han Data: Clean Sandstones 5.5 V p [km/s] V p [km/s] [a] V s [km/s] [b] V s [km/s] Figure 6.12: Comparison of the results of the simulations for water-saturated sandstone to the empirical equations of [33] and [54]. The numerical data is slightly below the two empirical curves, but exhibits the expected linear relationship between V p and V s. In (b) we show the raw data of [54] for clean sandstones and note that the numerical fit gives the best match to the data Comparison to velocity-porosity models Theoretical models for dry rock A number of theoretical methods have been proposed for describing the properties of sedimentary rock and were introduced in sections We compare the three theories (HS-bounds, DEM, and SCA) to our numerical predictions in Fig We note that none of the theoretical methods gives a satisfactory fit to elastic simulations for the IOS model. The SCA theory, discussed in chapter 4, (see also e.g., [92]), gives a much better fit to the data than either the DEM or the Hashin-Shtrickman upper bound. This is in agreement with the observations of Berge et al. [15]. Also the SCA predicts a vanishing modulus for >.5 which is similar to the empirical critical porosity model [11] described in the next section. Interestingly, both effective medium theories based on spherical pores have exact critical points ν =.2 (V p /V s = 8/3) [45] as c. The SCA has a c =.5 and DEM has c = 1. We might expect therefore that the SCA would closely match the numerical data. We show in Fig that although the prediction of the SCA has the correct limiting behaviour, the match is not consistent for all Empirical relationships for dry rock Experimental measurements have often shown that relatively simple empirical relationships can be used to describe the properties of sedimentary rock. In this section we compare the empirical results of Krief [73] (Eqns ) and Nur

223 Accurate velocity model for clean consolidated sandstones Hashin Shtrickman SCA DEM Numerical 5 4 Bulk Modulus [GPa] 2 Shear modulus [GPa] [a] [b] Bulk Modulus [GPa] 2 Shear Modulus [GPa] [c] [d] Figure 6.13: Comparison of the simulation results to the range of theories used to predict the moduli of dry porous rocks. [a-b] give predictions for quartz while [c-d] give predictions for feldspar. The theories all overestimate the data for all porosities. The SCA gives the best theoretical fit to the data as expected from [15]..4.3 Quartz data Feldspar data SCA Prediction Dry Poisson Ratio Figure 6.14: Comparison of the simulation data to the SCA theory for Poisson s ratio. While the SCA does give the observed limiting behaviour, ν.2, the prediction of the theory is not generally correct for all.

224 19 Geophysical applications: velocity-porosity relationships Bulk modulus [GPa] 4 2 Quartz & air Nur model Krief Model Feldspar & air Nur Model Krief Model Shear modulus [GPa] [a] [b] Figure 6.15: Comparison of the simulation results on dry monominerallic sandstone microstructures to the empirical models of 5.27 [73] and 5.25 [11]. The match to the [73] model is superior for the shear modulus. Neither model provides a good match to the bulk modulus data. [19] (Eqns ) with the numerically data computed in the previous section. Comparison of both models with the numerical data is given in Fig The results are mixed. Neither empirical equation accurately describes the bulk modulus data. Quartz, K = 37.GP a, and feldspar, K = 37.5GP a, should exhibit very similar curves according to both Eqn and 5.24, but the numerical data shows very strong deviations for the two mineral systems. The result of [73], Eqn. 5.27, is very accurate for both systems for the shear modulus data however. The prediction of [11] for the shear modulus is good, but doesn t capture the curvature observed in the numerical data in Fig b. As discussed in section 6.2, the prediction of the models of [73] and [19] are extremely poor for the dry V p /V s ratio. Both models assume [V p /V s ] dry rock = [V p /V s ] mineral. This prediction is clearly a poor representation of the numerical data in Fig. 6.7.d. In the previous section ([4]) we studied this limiting behaviour of the Poisson s ratio for porous granular models and derived a very accurate empirical model for the Poisson s ratio of dry porous materials ν dry = a() + ν s (1 2) 3/2, (6.11) where a() = (2) 3/2 /5, for ν s <.2 and a() = 1 (1 2) 3/2 /5, for ν s >.2. The relationship for V p /V s reads V p /V s = ν 1 ν.5 = a() + ν s (1 2) 3/2 1 a() + ν s (1 2) 3/2.5. (6.12) We show in Fig.6.16 the prediction of this empirical model compared to the numerical data for both ν and for V p /V s ; the match is excellent.

225 Accurate velocity model for clean consolidated sandstones 191 Poisson ratio ν Quartz & air Feldspar & air Equation 6.12 V p /V s Quartz & air Feldspar & air Empirical Equation [a] [b] Figure 6.16: Comparison of the simulation results on dry monominerallic sandstone microstructures to the empirical model of [5] given in Eqn Agreement is excellent. 5 4 Hashin Shtrickman SCA DEM Numerical 5 4 Bulk Modulus [GPa] 3 2 Shear Modulus [GPa] [a] [b] Bulk Modulus [GPa] 2 Shear Modulus [GPa] [c] [d] Figure 6.17: Comparison of the simulation results to the range of theories used to predict the moduli of water saturated rock. [a-b] give predictions for quartz while [c-d] give predictions for feldspar. The HS bounds are quite broad for the bulk modulus and neither bound is predictive. The lower bound is zero for the shear modulus. The SCA and DEM both overestimate the data for all porosities. The SCA gives the best theoretical fit to the data as expected from [15].

226 192 Geophysical applications: velocity-porosity relationships Theoretical models for water-saturated rock We compare the three theories to our numerical predictions in Fig We note that none of the theoretical methods gives a satisfactory fit to the IOS model. Again the SCA theory gives a much better fit to the data than either the DEM or the Hashin-Shtrickman bounds. This is in agreement with the observations of Berge et al. [15] that the SCA should more accurately predict the elastic properties of granular media Empirical relationships for water-saturated rock In section it was pointed out that the relatively simple Wyllie equation (Eqn. 5.2) can be used for fluid-saturated sedimentary rocks of homogeneous mineralogy. An improvement on this equation was given by Raymer (Eqns ). A comparison of the numerically predicted velocities for the IOS model are summarised in Fig for both quartz and feldspar sands. The Raymer equation gives a better fit to the data than the Wyllie equation, and is within 1% error for <.37. For higher porosities the Raymer prediction does not match the numerical data. Nur et al. [19, 11] postulated that the moduli of saturated rocks should trend in a similar manner to that predicted in Eqn The moduli should trend linearly betweeen the mineral grain modulus at low porosity to a value for a mineral-pore suspension at the critical porosity. Comparison with the numerical Numerical Wyllie Equation Raymer Equation V p [km/s] V p [km/s] [a] [b] Figure 6.18: Comparison of the results of the simulations for water-saturated [a] Quartz and [b] Feldspar sands to the empirical equations of Wyllie (Eqn. 5.2) and Raymer (Eqns ). The Raymer Equation is satisfactory except at high and gives a better prediction than the Wyllie equation. The prediction of both empirical models for the model quartzose sands is poorer than for feldspathic sands.

227 Accurate velocity model for clean consolidated sandstones 193 Bulk Modulus [GPa] Quartz: Numerical Quartz: Nur Model Feldspar: Numerical Nur: Feldspar Shear modulus [GPa] [a] [b] Figure 6.19: Comparison of the results of the simulations for water-saturated [a] Quartz and [b] Feldspar sands to the empirical equation of [11]. For the bulk modulus the model accurately describes the quartzose sand but not the feldspathic sands. For the shear modulus the feldspathic sand is best matched. data is given in Fig The results are mixed. As for the dry rock data, the model poorly describes the bulk modulus data. The bulk modulus for the two mineral systems should exhibit very similar curves according to Eqn. 5.25, but the numerical data shows very strong deviations. The prediction of [11] for the shear modulus is satisfactory Proposed velocity-porosity model for clean sandstones Based on the results given in the previous subsections we propose an accurate empirical method which enables one to derive the full velocity-porosity relationship for monomineralic consolidated sands solely based on the knowledge of the mineral modulus. The method is based on the excellent match of the empirical Krief equation for the shear modulus of a dry sandstone and the empirical equation of [5] for the Poissons ratio of a dry porous rock µ dry () = µ s (1 ) m(), (6.13) ν dry () = a() + ν s (1 2) 3/2, (6.14) where µ s and ν s are the shear modulus and Poissons ratio of the pure solid mineral phase, m() = 3/(1 ), and a() = (2) 3/2 /5, for ν s <.2 and a() = 1 (1 2) 3/2 /5, for ν s >.2. Gassmann s equations are subsequently used to predict the saturated states for any fluid. We first compare the predictions of the proposed empirical model to the numerical data shown in section We also compare results to arguably the most utilised current empirical method; that of [19, 11] based on critical porosities

228 194 Geophysical applications: velocity-porosity relationships 4 5 Bulk modulus [GPa] Numerical: Quartz Numerical: Feldspar Eqns Eqns Shear modulus [GPa] [a] [b] V p [km/s] 3. V p /V s [c] [d] Bulk modulus [GPa] 2 Shear modulus [GPa] [e] [f] V p [km/s] V p /V s [g] [h] Figure 6.2: Comparison between the prediction of the proposed empirical model, Eqn , the critical porosity model Eqn , and the simulation results for dry [a] bulk modulus, [b] shear modulus, [c] V p, and [d] V p /V s ratio as a function of porosity for the single phase IOS model with quartz and feldspar as the mineral phase. [e-h] give the predictions for the water-saturated case derived from the dry data using Gassmann s relations.

229 Accurate velocity model for clean consolidated sandstones Han Data Nur Equation Empirical V p [km/s] 4. Han data Nur Equation Empirical Prediction V s [km/s] [a] [b] V p [km/s] 4. Han: V p = Nur Model Eqns V s [km/s] 2. Han: V s = Nur Model Empirical [c] [d] Figure 6.21: Comparison of the prediction of the proposed empirical equations ( ) and the critical porosity model [11] to the experimental data [54] for clean dry quartz sands [a-b] and to the empirical velocity-porosity equations for the clean water-saturated sands [c-d]. (Eqns and 5.25). Results are summarised in Figs The agreement to the new empirical method is excellent for all elastic properties, both dry and watersaturated, and is superior to the critical porosity model. Only at higher porosities do we begin to see significant deviations of the new empirical model from the numerical data. For porosities < 35% the match exhibits at most 2 3% error for the dry case and 5% error for the water saturated case. In Fig we compare the prediction of the empirical model to the dry data and the water saturated velocity-porosity relationship derived by Han [54] for clean sandstones and to the critical porosity model. The agreement of the new model proposed here to the experimental data of [54] is excellent and superior to Eqns and 5.25.

230 196 Geophysical applications: velocity-porosity relationships Summary The utility of data sets generated by numerical simulation has been shown. In particular, the lack of noise in the data has enabled us to quantitatively compare results with theoretical and empirical theories. The numerical data is also in agreement with the analytical Gassmann s relations. The prediction of the common theoretical models (bounds and effective medium theories) for both dry and water-saturated states are not accurate (Figs and 6.17). Models significantly overestimate the modulus data as a function of porosity. The prediction of the SCA for ν (or V p /V s ) does however show the correct limiting behaviour for large (Fig. 6.14). We compare a number of standard dry modulus-porosity models to the IOS model data. The bulk modulus-porosity relationships of both [73] and [11] fail to accurately describe the IOS model data (Figs a). The Krief equation Eqn does however yield a very good fit to the dry shear modulus data with < 2% error (Fig b). An accurate empirical ν or V p /V s (Eqns or 6.12) relationship for dry sandstone also matches the numerical data very well (Fig. 6.16). The Raymer equation (Eqns ) is the best modulus-porosity model for water saturated conditions. Errors of at most 1% are observed across the full range of (Fig. 6.18). The critical porosity model gives a poor description of the K sat results and a satisfactory fit to µ sat (Fig. 6.19). The choice of empirical modulus-porosity relations for different lithologies is crucial as they allow the prediction of porosity from sonic logs and from seismic measurements. Based on the data presented here we propose that for clean consolidated sandstones one use the empirical Krief equation for the shear modulus of a dry sandstone and the empirical equation of Arns et al. for the Poissons ratio of a dry porous rock. Gassmann s equations should be subsequently used to predict the fluid saturated states. Comparison of the prediction of the empirical method to numerical predictions and available experimental data is excellent.

231 Accurate velocity model for two-mineral cemented sands Accurate velocity model for two-mineral cemented sands The previous section was limited to clean (monomineralic) rock systems. Sandstones are rarely clean however. They often contain minerals other than quartz, such as clay minerals, which can affect their elastic properties. Clay can be distributed in a number of ways in the rock framework depending on the conditions at deposition, on compaction, bioturbation and diagenesis. Here we consider the two model distributions of cemented sandstone composed of two mineral phases discussed in section 6.1 and generate moduli-porosity relationships for both the dry and water-saturated states for the two model cemented sandstone morphologies. The elastic simulations are performed as discussed in section 6.1. Again, due to the ability to control the pore space structure, microstructure and mineralogy of the material within a numerical model, the resultant data sets exhibit very little noise. This in turn allows one to quantitatively analyze the effect of microstructure on the porosity-modulus relationship for cemented sands. The equivalent effective modulus for the solid multiphase system shows a small dependence on structure. Accurate predictions of the effective solid moduli are given by the average of the Hashin-Shtrickman bounds. We find that for all porosities, the choice of microstructural model has a minimal effect on the resultant modulus-porosity relationship. Agreement of the predictions with experimental data sets for consolidated sandstones is excellent. We find that in all multimineralic systems Gassmann s relations accurately map between the dry and fluid saturated states. Numerical results are compared to currently employed theoretical and empirical relationships for deriving moduli-porosity estimates for porous sandstones. We note that the Krief empirical relationship is again successful at describing the numerical data for dry shear modulus and Eqn. 6.5 gives a good prediction of the V p /V s ratio data for dry rock. Based on these results we verify that the empirical method derived in section 6.3 for clean sandstones can be used to derive the full modulus/porosity relationship for multimineralic cemented sands. Comparisons with several experimental studies show that the proposed empirical method gives an excellent match to data Numerical results Effective mineral modulus In all empirical equations one must define the elastic properties of the solid mineral phase. As we now consider rocks composed of two mineral phases, we first analyse the effect of the distribution of minerals on the resultant modulus of the effective mineral phase ( = ). Results are summarised in Tab. 6.3 and in Fig

232 198 Geophysical applications: velocity-porosity relationships We note that the microstructure in most cases does not have a very strong effect on the resultant effective modulus. The only case where microstructural effects are observed is for the shear modulus of the quartz:clay system, where we find a 1% difference in µ with microstructure. This large effect is most probably due to the large contrast between the two phases (µ quartz = 44., µ clay = 6.9). For all other systems the contrast in the modulus between the two phases is less than a factor of two and the resultant moduli can be considered essentially independent of structure. We note a change in the modulus at the 1:1 fraction for the quartz:clay system when one changes the order of the mineral placement. For example, µ f (Q:C=1:1) = 18.7, while µ f (C:Q=1:1) = Similar changes are observed in the interstitial model, where µ I (C:Q=1:1) > µ I (Q:C=1:1). The change is related to the stronger interconnectivity of the phases in the different model microstructures. For the framework model, we observe a higher interconnectivity of the phase that is placed first than the overgrown phase (see Fig. 6.2). For the interstitial model the second mineral phase, grown by parallel surfaces, forms the most highly interconnected phase (see Fig. 6.3). The change in modulus is directly related to this change in structure. When the stiffer phase (quartz) exhibits enhanced connectivity (Framework, Q:C=1:1 and Interstitial C:Q=1:1) the effective modulus is greater. 8 Bulk Modulus Bounds Shear Modulus Bounds 6 Modulus [GPa] Clay/Dolomite Fraction Figure 6.22: The effective mineral modulus of the mixtures of quartz and clay or dolomite. The circles are the bulk modulus data, and the squares the shear modulus. The filled symbols represent numerical data for the framework model and the open symbols the interstitial model. The Hashin-Shtrickman bounds are also shown. All the numerical data lies within the bounds.

233 Accurate velocity model for two-mineral cemented sands 199 Table 6.3: Moduli in GP a for the mixed lithologies at zero porosity. The subscript f indicates data for the framework model and I for the interstitial model. We also give the upper and lower Hashin-Shtrickman bounds for the two phase system (subscripts l and u respectively). All the numerical data sits within the bounds validating the predictions of the numerical simulation. For the bulk modulus and shear modulus of the quartz:dolomite systems we observe little dependence on the microstructural model. The shear modulus of the quartz:clay system does exhibit the strongest dependence. Minerals/ratio K l K f K I K u µ l µ f µ I µ u Quartz:Clay 4: : : Clay:Quartz 1: : : Quartz:Dolomite 4: : : Dolomite:Quartz 1: : : When the elastic properties of the individual components are similar, bounds can be used to predict the effective elastic properties of multiphase mixtures to high accuracy. We use bounds to test the accuracy of the numerical results. The data in Table 6.3 and Fig includes predictions for the Hashin-Shtrickman lower (K l,µ l ) and upper (K u,µ u ) bounds for comparison with the numerical data. All numerically derived data points lie within the bounds, even when the bounds are very tight (Q:D system), indicating the ability of numerical simulation to accurately predict geophysical properties Dry rock data In Figs we give results for the dry bulk modulus K, shear modulus µ, V p and V p /V s ratio for the model systems as a function of porosity for the two different model morphologies of cemented sands at various sand:cement ratios. The standard error is plotted to show the quality of the numerical data. For K, µ and V p the standard error is always smaller than the data points. The error for the

234 2 Geophysical applications: velocity-porosity relationships 6 5 Bulk modulus [GPa] Quartz/clay: Framework Clay/quartz: Framework Quartz/dolomite: Framework Dolomite/quartz: Framework Shear modulus [GPa] Quartz/clay: Interstitial Clay/quartz: Interstitial Quartz/dolomite: Interstitial Dolomite/quartz: Interstitial 1 1 [a] [b] Quartz/clay Clay/quartz Quartz/dolomite Dolomite/quartz Quartz/clay Clay/quartz Quartz/dolomite Dolomite/quartz V p [km/s] 4. V p / V s [c] [d] Figure 6.23: Comparison of the results of the simulation for the variation of the bulk modulus, shear modulus, V p, and V p /V s ratio for model cemented sands at 1:1 mineral:mineral ratio. The lines give the prediction of the framework model and the data points the prediction of the interstitial model. The standard error is shown for the data points associated with the interstitial model; for K, µ and V p the standard error is always smaller than the data points. The error for the V p /V s ratio is small for porosities <.2 but begins to increase rapidly at higher porosity. The moduli-porosity relationships are non-linear with a limiting porosity (where modulus ) of.5. The quartz:dolomite systems exhibit little dependence on microstructure. The quartz:clay systems do exhibit structural dependencies. For example, the framework model for quartz:clay exhibits a larger bulk modulus than the interstitial Q:C. The difference, which is reflected in the moduli of the solid phase (see Table 6.3), is likely to be due to the better interconnectivity of the quartz phase for the framework model where quartz has the higher modulus. The enhanced connectivity of quartz is responsible for the interstitial Clay:Quartz system exhibiting higher modulus than the framework C:Q system. Similar results are observed for V p. The V p /V s ratio data, although noisy at higher, seems to be similar for almost all and independent of structure.

235 Accurate velocity model for two-mineral cemented sands 21 V p /V s ratio is small for porosities <.2 but begins to increase rapidly at higher porosity. The trend in the curve remains clear however. 6 5 Bulk modulus [GPa] Quartz/clay: Framework Clay/quartz: Framework Quartz/dolomite: Framework Dolomite/quartz: Framework Shear modulus [GPa] Quartz/clay: Interstitial Clay/quartz: Interstitial Quartz/dolomite: Interstitial Dolomite/quartz: Interstitial 1 1 [a] [b] Quartz/clay Clay/quartz Quartz/dolomite Dolomite/quartz 1.8 Quartz/clay Clay/quartz Quartz/dolomite Dolomite/quartz V p [km/s] 4. V p / V s [c] [d] Figure 6.24: Comparison of the results of the simulation for the variation of the bulk modulus, shear modulus and V p /V s ratio for the model cemented sands at a 2:1 mineral:mineral ratio. The lines give the prediction of the framework model and the data points the prediction of the interstitial model. As in the 1:1 model (Fig. 6.23), the moduli-porosity relationships are non-linear with a limiting porosity of.5. The quartz:dolomite systems exhibit little dependence on microstructure. The quartz:clay systems again exhibit structural dependencies as discussed in Fig The V p /V s ratio data is again independent of microstructure. The moduli of the rocks trend between the modulus of the mineral at low porosities to zero at higher porosities. The fit to the modulus is in most cases strongly non-linear with a critical porosity, where the modulus vanishes,.5. In all cases the Quartz:Dolomite systems exhibit little dependence on microstructure. The moduli of the quartz:clay systems show a small structural dependence with but this difference can be related to the difference in moduli exhibited by the solid mineral mixtures (recall Table 6.3). We do not observe any indication of a transition from the load bearing domain to the suspension domain as is often

236 22 Geophysical applications: velocity-porosity relationships 7 5 Bulk modulus [GPa] Quartz/clay: Framework Clay/quartz: Framework Quartz/dolomite: Framework Dolomite/quartz: Framework Shear modulus [GPa] Quartz/clay: Interstitial Clay/quartz: Interstitial Quartz/dolomite: Interstitial Dolomite/quartz: Interstitial 1 1 [a] [b] Quartz/clay Clay/quartz Quartz/dolomite Dolomite/quartz 1.9 Quartz/clay Clay/quartz Quartz/dolomite Dolomite/quartz V p [km/s] V p / V s [c] [d] Figure 6.25: Comparison of the results of the simulation for the variation of the bulk modulus, shear modulus and V p /V s ratio for the model cemented sands at a 4:1 mineral:mineral ratio. The lines give the prediction of the framework model and the data points the prediction of the interstitial model. There is little effect of the microstructure on bulk and shear modulus. Similar results are seen for V p. The V p /V s ratio data is both independent of and exhibits a fixed point at V p /V s 1.63 for.5. observed experimentally for 4%. This is due to the choice of the, certainly idealised, microstructural model based on consolidated spheres; at 4% the consolidated solid phases remain macroscopically connected so no transition to a system where the mineral phase falls apart is observed. In Fig we observe that the V p curve is approximately linear with porosity. The V p /V s ratio exhibits a strong dependence on porosity and goes towards a fixed point (V p /V s ) 1.63 = (8/3) at higher porosities. This fixed point was noted previously [6] and corresponds to a fixed point in the Poisson s ratio ν =.2. The V p /V s data does exhibit appreciable noise compared to the moduli, particularly at higher porosities. At high continuous pathways for each mineral phase become more dependent on small tenuous connections. These connections, crucial to the

237 Accurate velocity model for two-mineral cemented sands 23 macroscopic elastic properties, are increasingly difficult to resolve, and therefore larger statistical errors can be observed. Importantly, the numerical data for V p /V s in the region of most interest, 5% < < 35% is of high quality Water Saturated Rock In Figs we give results for the water saturated values of the bulk modulus K, shear modulus µ, V p and V p /V s ratio for model clay bearing sands as a function of porosity. The standard error is plotted in all Figures. The standard 6 5 Bulk modulus [GPa] Quartz/clay: Framework Clay/quartz: Framework Quartz/dolomite: Framework Dolomite/quartz: Framework Shear modulus [GPa] Quartz/clay: Interstitial Clay/quartz: Interstitial Quartz/dolomite: Interstitial Dolomite/quartz: Interstitial 1 1 [a] [b] Quartz/clay Clay/quartz Quartz/dolomite Dolomite/quartz Quartz/clay Clay/quartz Quartz/dolomite Dolomite/quartz V p [km/s] 4. V p / V s [c] [d] Figure 6.26: Comparison of the results of the simulation for the variation of the water-saturated bulk modulus, shear modulus and V p /V s ratio for the model cemented sands at a 1:1 mineral:mineral ratio. The lines give the prediction of the framework model and the data points the prediction of the interstitial model. Standard errors are now smaller than the data points for all measures across all. The moduli exhibit in general a non-linear modulus- relationship. The quartz:clay framework model exhibits a larger modulus than the quartz:clay interstitial model, but the differences are not as large as observed in the dry system (Fig a) The V p /V s ratio data seems to be similar for almost all <.2. At higher porosities the curves begin to diverge.

238 24 Geophysical applications: velocity-porosity relationships error is for most measures smaller than the data points. Data for the moduli exhibit similar qualitative behaviour to that observed for dry rock. The bulk moduli of the models now trend non-linearly between the modulus of the mineral at zero porosity to values corresponding to a mineral-pore suspension at large. The shear moduli exhibit the same behaviour as the dry rock, going to zero at higher porosities. For most structures the dependence of the moduli on microstructure is minimal. The V p /V s ratio exhibits a stronger dependence on both microstructure and on mineralogy. For the 1:1 clay:quartz system (Fig. 6.26(d)) a difference between the framework and interstitial models is observed. At a 4:1 ratio (Fig. 6.28(d) the quartz:clay system shows an even larger difference is observed. In general, the V p /V s ratio data seems to be similar for almost all <.2. At higher porosities curves can strongly differ. 6 5 Bulk modulus [GPa] Quartz/clay: Framework Clay/quartz: Framework Quartz/dolomite: Framework Dolomite/quartz: Framework Shear modulus [GPa] Quartz/clay: Interstitial Clay/quartz: Interstitial Quartz/dolomite: Interstitial Dolomite/quartz: Interstitial 1 1 [a] [b] Quartz/clay Clay/quartz Quartz/dolomite Dolomite/quartz Quartz/clay Clay/quartz Quartz/dolomite Dolomite/quartz V p [km/s] 4. V p / V s [c] [d] Figure 6.27: Comparison of the results of the simulation for the variation of the water saturated bulk modulus, shear modulus and V p /V s ratio for the model cemented sands at a 2:1 mineral:mineral ratio. The lines give the prediction of the framework model and the data points the prediction of the interstitial model.

239 Accurate velocity model for two-mineral cemented sands Bulk modulus [GPa] Quartz/clay: Framework Clay/quartz: Framework Quartz/dolomite: Framework Dolomite/quartz: Framework Shear modulus [GPa] Quartz/clay: Interstitial Clay/quartz: Interstitial Quartz/dolomite: Interstitial Dolomite/quartz: Interstitial 1 1 [a] Quartz/clay Clay/quartz Quartz/dolomite Dolomite/quartz [b] Quartz/clay Clay/quartz Quartz/dolomite Dolomite/quartz V p [km/s] 4. V p / V s [c] [d] Figure 6.28: Comparison of the results of the simulation for the variation of the water saturated bulk modulus, shear modulus and V p /V s ratio for the model cemented sands at a 4:1 mineral:mineral ratio Verification of Gassmann s equation for multiple mineral phases The low-frequency Gassmann s equations (Eqns ) relate the bulk and shear moduli of a saturated porous medium to the moduli of the same medium in a drained (dry) state and assume an isotropic, monominerallic medium. We find that the numerical data for the multimineralic model also obeys Gassmann s equations for all ratios of minerals. In Fig we illustrate the data for the cemented sands at a 2:1 ratio for both the framework model and the interstitial model. The numerical prediction for both the bulk and shear modulus are in excellent agreement with Gassmann s equations Comparison with Experiment Han [54] tabulated extensive experimental data for the elastic properties of 69 different sandstone samples with varying porosity and clay volume fraction C. He reported both wet and dry velocities. We plot the dry data for shaley sand-

240 26 Geophysical applications: velocity-porosity relationships 6 5 Bulk modulus [GPa] Dry quartz/dolomite Wet quartz/dolomite Dry quartz/clay Wet quartz/clay Dry dolomite/quartz Wet dolomite/quartz Dry clay/quartz Wet clay/quartz Wet from dry (Gassmann) Shear modulus [GPa] Dry quartz/dolomite Wet quartz/dolomite Dry quartz/clay Wet quartz/clay Dry dolomite/quartz Wet dolomite/quartz Dry clay/quartz Wet clay/quartz 1 1 [a] [b] Bulk modulus [GPa] Dry quartz/dolomite Wet quartz/dolomite Dry quartz/clay Wet quartz/clay Dry dolomite/quartz Wet dolomite/quartz Dry clay/quartz Wet clay/quartz Wet from dry (Gassmann) Shear modulus [GPa] Dry quartz/dolomite Wet quartz/dolomite Dry quartz/clay Wet quartz/clay Dry dolomite/quartz Wet dolomite/quartz Dry clay/quartz Wet clay/quartz 1 1 [c] [d] Figure 6.29: Comparison of the results of the simulations for dry and watersaturated model morphologies at a 2:1 mineral ratio. [a-b] give data for the framework model and [c-d] for the interstitial model. In [a] and [c] dry data is given by open symbols and wet by closed symbols. The solid line give the best fit to the dry data. Gassmann s relation Eqn. 5.3 is used to predict the water-saturated bulk modulus based on the fit to the dry rock data and this result is given by the dashed lines. We see that the numerical predictions for the water-saturated results match the prediction of Gassmann s equations for all. In [b] and [d] we show that the shear modulus is independent of the pore fluid as predicted by Eqn

241 Accurate velocity model for two-mineral cemented sands 27 stones from [54] in Fig We classify the data from [54] into 3 groups based on clay content; clean (C = ), intermediate (C < 15%) and high (15 < C < 35%) clay content, where C is the clay volume fraction. We compare this to the simulations for clean quartz (previous section) and for clay cemented quartz sands with a quartz:clay volume ratio of 2:1, or X c = 33%. X c is defined as the volume fraction of clay as a percentage of total solids volume or X c = C/(C + Q), where Q is the volume fraction of quartz. Although not exactly comparable, the data for a 2:1 quartz:clay ratio, X c = 33%, should approximately match the experimental data for higher total clay content(15 < C < 35%); X c = 33% corresponds to C( = 5%) = 31% and C( = 25%) = 25%. The clean sandstone simulation data should, in turn, match the clean sandstone data, and the intermediate (C < 15%) data should lie somewhere between the two curves. As the difference between the elastic properties of the framework and interstitial models is minimal in the porosity range measured by [54], 3%, we compare the experimental data to the framework model. The simulation results in Fig. 6.3 indicate that the numerical curves give both the correct trend for the experimental data with increasing clay content and an excellent quantitative match to the clean experimental data. The data for intermediate clay content does lie between the two numerical derived curves and the higher clay content data is described satisfactorily by the numerical prediction for X c = 33%. Dvorkin [39] studied laboratory data sets of high porosity = 2 35% from the Oseberg field. The samples have quartz as the main component bound by quartz and clay cements. Clay volume fractions ranged from 1 25%. As the conventional model of [19] gave a poor match to the data, [39] developed a new model for clay cemented sands which gave a better match to this data. [39] modelled the Oseberg sandstone as hard-shell quartz grains with cement evenly deposited on the grain surface. Results of model calculations based on an effective medium type approach gave reasonable agreement with velocities (moduli), but poor match to the Poisson s ratio (V p /V s ratio). Since the interstitial model we generate is very similar to that studied by [39] (except that the mineral grains we consider are permeable), we compare the Oseberg data to the results of the numerical simulation for the interstitial model. We compare the experimental data to clean sandstone and to clay bearing sand with X c = 33%. These two curves should provide upper and lower bounds for the experimental data. Results are summarised in Fig The simulated model results do accurately bound the Oseberg data well for both moduli and the V p /V s ratio. The prediction for V p /V s gives in particular a far better match to the data than the model prediction of [39], V p /V s From the extensive experimental data and using least squares regression, dif-

242 28 Geophysical applications: velocity-porosity relationships Bulk Modulus [GPa] Han: Clean Sandstone Han: Clay Content 15% Han: Clay content 15 35% Simulation: Clean Sand Simulation: Clay Content X c =33% Shear modulus [GPa] V p [km/s] V p /V s Figure 6.3: Comparison of the data of [54] to simulation for clean sand and shaley sands with a clay content of X c = 33%. The two simulation results both qualitatively match the observed effect of increased clay content on the elastic properties and quantitatively match the data from [54] for different clay contents. ferent empirical equations were obtained by [54] to describe the velocities of water saturated sandstone samples at a confining pressure of 4MP a across a range of porosities 5% < < 3% and volume clay content C ranging from 5%: V p [km/s] = C, (6.15) V s [km/s] = C (6.16) We directly compare the predictions of these empirical equations to the numerical data. We first recast the empirical V p (, C), V s (, C) functions in terms of the clay solid fraction X c. Trivially C = X c (1 ), allowing one to define Eqns as functions of (, X c ). The comparison of these equations to numerical data is compiled in Fig The match for X c = 33% and 2% is excellent for V p but poorer for V s. The clean sandstone predictions are compared to the

243 Accurate velocity model for two-mineral cemented sands 29 Bulk Modulus [GPa] Oseberg Data Simulation: Clean Sand Simulation: Shaley Sand, X c =33% Shear Modulus [GPa] V p /V s Figure 6.31: Comparison of the data of [39] for high porosity Oseberg sandstone at Clay volume fractions C = 1 25% to simulation for clean sand and shaley sands with a clay solid fraction X c = 33%. The experimental data lies mostly within the bounds of the two simulated results. Previous models gave a reasonable match to the velocities, but a poor prediction of the V p /V s ratio [39]. experimentally derived equation for clean sandstone samples [54]; V p [km/s] = , (6.17) V s [km/s] = (6.18) The match of the numerical data for clean sandstones to these empirical equations is excellent for both V p and V s Simulation: Clean Sand Simulation: Shaly Sand, X c =2% Simulation: X c =33% Eqn Eqn. 6.15, X c =2% Eqn. 6.15, X c =33% 3. Simulation: Clean Sand Simulation: X c =2% Simulation; X c =33% Eqn Eqn. 6.16, X c =2% Eqn. 6.16, X c =33% V p [km/s] 4. V s [km/s] Figure 6.32: Comparison of the empirical equations of [54], Eqns to simulation data. The simulation data matches very well for all V p data but overestimates the empirical prediction for V s. The numerical prediction for clean sandstone match the empirical results well. Finally, there are a number of empirical equations used to relate V p to V s based on cross-plots of experimental data. Data for sandstones was compiled by [33] and a least-squares linear fit to these data is given by: V s =.84V p.856 [km/s]. (6.19)

244 21 Geophysical applications: velocity-porosity relationships [54] derived an equation for sandstones with different clay volumes; the least-squares linear fits to these data were given by: V s =.754V p.657 [km/s], C < 25%, (6.2) V s =.842V p 1.99 [km/s], C > 25%, (6.21) The mudrock line of [33] is nearly identical to Eqn A comparison of the numerical data for clean and clay-bearing sands at different clay volume fractions to these empirical equations is given in Fig a. The agreement with the empirical fits is very good. The numerical predictions all collapse to a straight line as is observed experimentally for sands and shaley sands. The numerical predictions show that increased clay volume leads to a slightly lower V s prediction for a given V p. This is in accord with empirical relationships and with experimental data. The simulations show that at higher porosities/lower velocities, the three curves collapse, and the effect of clay is minimised. The high clay content curve of Han (Eqn. 6.21), and the mudrock line of [33], still exhibit a lower V s than we observe numerically for a given V p. To compare high clay content systems at low velocities (high porosities) to the data of [33, 54] we plot in Fig b the numerical prediction for highest clay ratio considered: quartz:clay=1:4. This shows significantly lower V s predictions for most values of V p in agreement with the empirical Eqn We do however observe a strong non-linear trend in the data at this very high clay fraction. In Fig b we give the best linear fits to the numerical data for the higher quartz fractions: V s =.796V p.694 [km/s], clean sand, (6.22) V s =.756V p.611 [km/s], quartz:clay=2:1, (6.23) V s =.736V p.589 [km/s], quartz:clay=1:1. (6.24) In all cases the fit to the linear curves is good. The consistent match of the numerical data to experiment indicates that the cemented sandstone models considered here give a good representation of consolidated sandstone structure, and confirms the utility of data sets generated by numerical simulation Comparison to models Experimental measurements have often shown that relatively simple theoretical models and empirical relationships can be used to describe the properties of complex sedimentary rock. In this section we compare a selection of these results to the numerical data computed in the previous section.

245 Accurate velocity model for two-mineral cemented sands 211 V p [km/s] Numerical: Clean Sandstone Numerical: Quartz:Clay=2:1 Numerical: Quartz:Clay=1:1 Han, C<25%: V s =.754V p.657 Han, C>25%: V s =.842V p 1.99 Castagna: V s =.84V p.856 V p [km/s] Numerical: Clean Sandstone Numerical: Quartz:Clay=2:1 Numerical: Quartz:Clay=1:1 Numerical: Quartz:Clay =1:4 V s =.796V p.694: Clean V s =.756V p.611: Q:C=2:1 V s =.736V p.589: Q:C=1:1 2 2 [a] V s [km/s] [b] V s [km/s] Figure 6.33: Comparison of the results of the simulations for water-saturated sandstone to the empirical equations of [33] and [54]. The numerical data for intermediate Quartz:Clay fractions is in excellent agreement with the two empirical curves. All numerical data sets exhibit the expected linear relationship between V p and V s. In (b) we illustrate the linear fit given in Eqns to the numerical data Effective Mineral Modulus In all moduli-porosity relationships one must define the moduli of the solid mineral mixture. While the moduli of a pure mineral phase is known, the effective modulus for a mineral mixture is not. In this section we first compare the solid moduli derived numerically for mixed lithologies (Table 6.3) to two theoretical effective medium theories (self-consistent approximation or SCA, and differential effective medium or DEM) and to three empirical estimates; the Voigt-Reuss-Hill (VRH) estimate, the averaged Hashin-Shtrickman (HS) estimate and geometric average of velocities used by Xu and White [17]. The two effective medium theories are described in detail in section and discussions can be found in several texts (see e.g., [92]). The Voigt-Reuss-Hill (VRH) average estimate of the moduli is derived from knowledge of the pure mineral phase moduli and the relative fraction of each mineral phase. The VRH estimate is simply the arithmetic average of the Voigt upper bound (M = N i f i M i ) and the Reuss lower bound (M = N i f i /M i ) where f i and M i are the volume fraction and the moduli of the ith component. Similarly one can use the Hashin-Shtrickman (HS) average, the average of the upper and lower Hashin-Shtrickman bounds [92]. Xu and White [17] estimate the velocities of a solid mineral mixture in shaley sands by a Wyllie time average equation of the

246 212 Geophysical applications: velocity-porosity relationships quartz and clay mineral velocities and an arithmetic average of their densities by: 1 = 1 V clay 1 + V clay 1, V P 1 V P quartz 1 V P clay (6.25) 1 = 1 V clay 1 + V clay 1, V S 1 V Squartz 1 V Sclay (6.26) ρ = 1 V clay ρ quartz + V clay 1 1 ρ clay, (6.27) where subscript denotes the mineral properties. The moduli K XW = ρ V 2 P, µ XW = ρ VS 2 are then derived. We compare the predictions of the five methods in Table 6.4 to the numerical results. We give the predictions for Eqns only for shaley sands (quartz:clay) mixtures as this estimate was derived specifically for these systems. All predictions for the quartz:dolomite systems are within 2% of the numerical data, which is of the order of the standard error for the numerical results. The numerical data does show some significant differences to model predictions for the quartz:clay systems, where the contrast betweeen moduli is higher. In these cases we find that the SCA and DEM predictions and the VRH and HS average estimates are in some cases poor and more importantly no consistency with model morphologies is observed. The empirical method of [17] based on the Wyllie time average equation gives predictions that lie outside the Hashin-Shtrickman bounds given in Table 6.3, and not surprisingly performs poorly for the quartz:clay mixtures. We do not consider this model further. From the data in Tab. 6.3 it is difficult to discern a best model estimate to match the numerical data for either microstructure. If we define systems, not by morphology, but by the preferred connectivity or continuity of the stiffer or weaker phase, we may observe a more consistent trend in the data. As discussed previously, for the framework model, there is higher interconnectivity of the phase that is placed first than the overgrown phase (see Fig. 6.2). For the interstitial model the second mineral phase, grown by parallel surfaces, forms the most highly interconnected phase (see Fig. 6.3). We therefore plot separately the data for the models where quartz and clay respectively preferentially form the backbone of the structure as a function of clay content. We plot the relative error in prediction of the four models for the bulk and shear modulus, where Error=(M model M sim )/M sim. The case where the quartz (stiffer) phase is preferentially connected is given in Fig [ab] and where the clay (weaker) phase is preferentially connected in Fig [c-d]. From the results we see that the SCA and the averaged HS bounds perform the best in general. HS bounds require only knowledge of the volume fractions of the mineral components. To better predict the effective moduli of two phase quartz/clay mineral

247 Accurate velocity model for two-mineral cemented sands 213 Table 6.4: Dry moduli for the mixed lithologies at zero porosity in GP a. The comparison with the Voigt-Ruess-Hill average and predictions based on the SCA and DEM are satisfactory for most systems. Only the shear modulus data for the quartz:clay system where the contrast between the two phases is more than 7 : 1 gives a poorer prediction. The error in this case is up to 1%. The predictions of the time average equation [17] are poor. Minerals/ratio K f K I K HS K VRH K SCA K DEM K XW Quartz:Clay 4: : : Clay:Quartz 1: : : Quartz:Dolomite 4: : : Dolomite:Quartz 1: : : Minerals/ratio µ f µ I µ HS µ VRH µ SCA µ DEM µ XW Quartz:Clay 4: : : Clay:Quartz 1: : : Quartz:Dolomite 4: : : Dolomite:Quartz 1: : :

248 214 Geophysical applications: velocity-porosity relationships Error in Bulk Modulus HS Average VRH average SCM DEM Error in Shear Modulus [a] Clay Fraction.2 [b] Clay Fraction.2 Error in Bulk Modulus.1 Error in Shear Modulus.1 [c] Clay Fraction [d] Clay Fraction Figure 6.34: Error in the predictions of the effective mineral moduli for the four models. In [a] and [b] we give the error for the systems where quartz (stiffer) forms the preferred backbone and [c-d] where clay is the more continuous phase. We observe that the SCA and the HS average give the smallest errors in general. The DEM works well only in the weaker system. The VRH estimate is overall the poorest. mixtures one can employ higher order bounds. Unfortunately these bounds require information in the form of three-point statistical correlations and is not easily obtained experimentally. One can use a method described very recently by [163] to improve predictions based on a primitive knowledge of the phase distribution, the weighted Hashin-Shtrickman (HS) average. It is well known that if the stiffer component forms the primary backbone of the two-component material, the real value of the effective moduli is closer to the upper HS bound and if the weaker material forms the more highly connected backbone, the effective moduli is closer to the lower HS bound. This was observed in Fig. 6.22, for example. Wang et al. [163] gave estimates for effective moduli from the HS lower and upper bounds (K l, µ l and K u, µ u ) based on the equations: K eff = ak u + (1 a)k l, (6.28) µ eff = aµ u + (1 a)µ l, (6.29)

249 Accurate velocity model for two-mineral cemented sands 215 where a and b are empirical weights defined as HS coefficients. [163] determined a and b experimentally by performing measurements on an epoxy/clay mixture, where the epoxy (softer material) was the continuous phase and the clay (stiffer material) was dispersed within the epoxy resin. They found that the HS coefficients were small (a, b <<.5) and exhibited a clear dependency on the epoxy concentration a = c and b =.33.2c. The low values were observed since the softer epoxy material formed the continuous phase. We use this method to obtain better estimates of the effective moduli of the mixed mineral systems. We plot the HS coefficients a and b in Fig for model systems in the two cases; first where the quartz and second the clay phases form the primary backbone of the solid mixture. Both the enhanced continuity of the weaker/stiffer phase and the clay concentration have an effect on the resultant values of the HS coefficients. We fit linear curves to the data in Fig The qualitative shift in the coefficients with clay content noted by [163] is consistent with our numerical data. The HS coefficients for the moduli are similar for bulk and shear moduli and can be approximated by a, b.7 (1 c);.2 < c <.8 for the case where the clay phase is more continuous phase and a, b (1.8 c);.2 < c <.8 when the quartz is the more continuous phase. We note that the epoxy/clay mixtures in [163] give lower values for a, b, but in their experiments the stiffer phase was completely unconsolidated. In general, the average HS prediction (a = b = 1) will 2 still give a good prediction; however if preferred continuity of the stiffer/weaker phase can be ascertained, one may choose to weight the bounds to higher/lower values. We use the simple average HS prediction for all estimates in the remainder of this paper Dry Porous Rock In the previous section (6.3) we compared the numerical results to a number of theoretical methods including Hashin-Shtrickman bounds, and effective medium theories (SCA and DEM). We found for the monomineralic systems that none of the theoretical methods gives a satisfactory fit to the data. The same conclusions were obtained for model cemented sands considered here although we do not show the data (for representative curves see Figs of the previous section). In all cases for dry and water-saturated systems, bounds and the SCA and DEM overestimate the numerical data. The SCA theory gives the best fit to the data as noted previously [15]. In Fig we compare the commonly used empirical equations of Krief [73] (Eqns ) and Nur [19] (Eqns ) to the numerical data. The results are mixed. Neither empirical equation accurately describes the bulk modulus data.

250 216 Geophysical applications: velocity-porosity relationships 1 Hashin Shtrickman Coefficients Clay Concentration Figure 6.35: HS coefficients as a function of clay content for the bulk (closed symbols) and shear (open symbols) modulus for systems where the weaker clay phase (circles) and the stiffer quartz phase (squares) is the most interconnected phase. Both exhibit a dependency on the clay concentration as was observed experimentally by [163]. Linear best fits to the bulk (dotted line) and shear (dashed line) are also shown. The result of [73], Eqn. 5.27, is very accurate for most clay mixtures for the shear modulus data however. The prediction of [11] for the shear modulus is generally not as good as it doesn t capture the non-linear behaviour evident in the numerical data in Fig The data at the highest clay fractions (clay:quartz data in Fig. 6.36(kl)) gives a linear behaviour and a better fit to the model of [19]. In general though the equation of [73] gives the best match to the numerical data. The prediction of the models of [73] and [19] are extremely poor for the dry V p /V s ratio. Both models assume [V p /V s ] dry rock = [V p /V s ] mineral. This prediction is clearly a poor representation of the numerical data in Fig d. We described in section 6.2 the accurate empirical model for the Poisson s ratio of dry porous materials ν dry = a() + ν s (1 2) 3/2, (6.3) where a() = (2) 3/2 /5, for ν s <.2 and a() = 1 (1 2) 3/2 /5, for ν s >.2. The relationship for V p /V s reads: V p ν 1 = V s ν.5 = a() + ν s (1 2) 3/2 1 a() + ν s (1 2) 3/2.5 (6.31) We show in Fig the prediction of this empirical model compared to the

251 Accurate velocity model for two-mineral cemented sands Bulk modulus [GPa] Quartz/dolomite Quartz/clay Dolomite/quartz Clay/quartz Nur Krief Bulk modulus [GPa] Bulk modulus [GPa] [a] [b] [c] Bulk modulus [GPa] 3 2 Bulk modulus [GPa] Bulk modulus [GPa] [d] [e] [f] Shear modulus [GPa] Quartz/dolomite Quartz/clay Dolomite/quartz Clay/quartz Nur Model Krief Model Shear modulus [GPa] Shear modulus [GPa] [g] [h] [i] Shear modulus [GPa] 3 2 Shear modulus [GPa] 3 2 Shear modulus [GPa] [j] [k] [l] Figure 6.36: Comparison of the simulation results on model dry cemented sands with mixed mineralogy to the empirical models given in Eqns [73, 11] where we use the K, µ derived from the simulation (Table 2) in the equations. The first column gives data for quartz:cement and quartz:dolomite systems at a volume ratio of 1:1. The second column gives data for all sand:cement systems considered at a ratio of 2:1/1:2, and the third column systems at a ratio of 4:1/1:4. [a-c] gives the prediction of the bulk modulus for the framework cement model, [d-f] the bulk modulus for the interstitial model, [g-i] the shear modulus of the framework model and [j-l] the shear modulus for the interstitial model. One observes that in most cases there is a significant curvature in the modulus porosity relationship which is not captured by the linear equation of [11]. Neither the equation of [73] nor [11] fit the bulk modulus data well. The match to the [73] model is superior for the shear modulus and gives a good fit for nearly all systems. Only at the highest clay content does the model of [11] work best. This result is similar to that observed for monominerallic systems.

252 218 Geophysical applications: velocity-porosity relationships 1.7 Quartz/Clay (1:1) Quartz/Dolomite (1:1) 1.8 Quartz/Dolomite (2:1) Quartz/Dolomite (1:2) Quartz/Clay (2:1) Quartz/Clay (1:2) V p /V s 1.6 V p /V s [a] [b] Quartz/Dolomite (4:1) Quartz/Dolomite (1:4) Quartz/Clay (4:1) Quartz/Clay (1:4) V p /V s [c] Figure 6.37: Comparison of the simulation results on model cemented sands to the empirical model of [7] given in Eqn Agreement is good and much improved on the estimate of most empirical models where (V p /V s )() = (V p /V s ) mineral. At a 4:1 ratio the limiting value for (V p /V s ) may be greater than 1.63 for the interstitial model. However, the error bars on the data become very large at higher so it is difficult to make quantitative conclusions. Closed symbols are data for the interstitial model and open symbols data for the framework model. There is in general little dependence on model microstructure. numerical data for V p /V s ; the match is good for all systems and clearly superior to assuming [V p /V s ] dry rock = [V p /V s ] mineral Water-saturated Rock As in the previous section (6.3), we compare the numerical results to the frequently used empirical equations for fluid-saturated rocks, the Wyllie time-average equation (Eqn. 5.2) and the Raymer-Hunt-Gardner equation (Eqns ). A comparison of the numerically predicted velocities for the model sands is given in Fig The Raymer equation gives in almost all cases a better fit to the data than the Wyllie equation, but consistently underestimates the numerical data for V p for a given. The Raymer equation predicts a transition from the load bearing domain

253 Accurate velocity model for two-mineral cemented sands 219 to the suspension domain as is often observed experimentally for 37%. This is again not observed in our numerical data due to the choice of the microstructural model; a model based on permeable spheres. For this model, even at 4%, the solid phases remain macroscopically connected so no transition to a system, where the mineral phase no longer transmits a load, is observed. Nur et al. [19, 11] postulated that the moduli of saturated rocks should trend linearly betweeen the mineral grain modulus at low porosity to a value for a mineral-pore suspension at the critical porosity. From the data in Fig we observe a general non-linear behaviour; only for the highest clay contents do we observe a linear trend New velocity-porosity model for clay-bearing sands Based on the results given in subsections and and the empirical method derived for clean sandstone described in the previous section, we propose an extension of that empirical method for clean sandstone to cemented sands composed of multiple mineral phases. This relationship enables one to derive the full velocityporosity relationship for these cemented sands solely based on the knowledge of the individual mineral moduli and the volume fractions of the mineral phases. method is based on the use of the empirical Krief equation for the shear modulus of a dry sandstone and the empirical equation of [7] for the Poisson s ratio of a dry porous rock: V p /V s = The µ dry () = µ (1 ) m(), (6.32) ν 1 ν.5 = a() + ν (1 2) 3/2 1 (6.33) a() + ν (1 2) 3/2.5 where µ and ν are the effective shear modulus and Poisson s ratio of the mixed mineral constituent. µ and ν are derived from an estimate of the mixed solid moduli based on either an average of the Hashin-Shtrickman bounds or via a weighted average. In the latter case one would require information on the spatial arrangements of the weakest/stiffest phase to define the appropriate coefficients. We use the average HS prediction for all the following comparisons to experiment. Gassmann s equations are subsequently used to predict the fluid saturated states Comparison to Experiment In this subsection we compare the prediction of the proposed empirical model Eqns to experimental data. We consider the dry shear modulus and the Poisson s ratio data for shaley sandstones from [54] in Fig where we have binned the data of [54] into groups

254 22 Geophysical applications: velocity-porosity relationships V p [km/s] Quartz/dolomite Quartz/clay Dolomite/quartz Clay/quartz Raymer Equation Wyllie Equation V p [km/s] Quartz/dolomite Quartz/clay Dolomite/quartz Clay/quartz [a] Quartz/dolomite Quartz/clay Dolomite/quartz Clay/quartz [b] Quartz/dolomite Quartz/clay Dolomite/quartz Clay/quartz V p [km/s] 4. V p [km/s] [c] Quartz/dolomite Quartz/clay Dolomite/quartz Clay/quartz [d] Quartz/dolomite Quartz/clay Dolomite/quartz Clay/quartz V p [km/s] 4. V p [km/s] [e] [f] Figure 6.38: Comparison of the results of the simulations for water-saturated shaley sands to the equations of Wyllie (Eqn. 5.2) and Raymer (Eqns ). The left hand column gives data for the framework model, and the right hand column data for the interstitial model. The mineral ratios are [a-b] 1:1, [c-d] 2:1, and [e-f] 4:1. The Raymer equation gives a better prediction than the Wyllie equation, but in many cases underestimates the value of V p for a given.

255 Accurate velocity model for two-mineral cemented sands 221 of clean, intermediate (< 15%) and higher (15 35%) clay content. We use the averaged HS estimate to define K and µ. In Figure 6.39.a the prediction for the shear modulus via Eqn for clean sandstones, sands with 2% clay content and sands with 35% clay content is given. We observe an excellent match to the experimental data. We also compare in Fig b the prediction of Eqn to the experimental data for different clay contents. The fit is consistent with the experimental data, which shows an increasing V p /V s with. The prediction is particularly good when compared to the prediction of both the model of [73] and [11]; the latter assuming that V p /V s is constant and equals the value of the mineral. Clearly the use of Eqn for the shear modulus and the use of Eqn leads to a much better match to the experimental data of [54] than other predictive equations. [a] Shear modulus [GPa] Eqn (32); Clean Sandstone Eqn. (32); 2% Clay Eqn (32); 35% Clay Han: Clean Sandstone Han: Clay Content 15% Han: 15 35% Clay [b] V p /V s Experiment: Clay <15% Experiment: Clay >15% Equation 33: Pure Quartz Equation 33: Quartz + 2% Clay Equation 33: Quartz + 33% Clay Experiment: Clean Sandstone Figure 6.39: Comparison of the data of [54] to the prediction of [a] Eqn for the shear modulus and [b] Eqn We compare the Oseberg data [39] to the prediction of the empirical model. We use the averaged HS estimate to define K and µ. As the experimental data was based on clay contents ranging from 1% to 25%, we compare the experimental data to the empirical model for a clean sandstone and a clay bearing sand with X c = 33%. These two curves should provide upper (X c = ) and lower bounds for the experimental data. Results are summarised in Fig The empirical model prediction provides reasonable bounds for the experimental shear modulus data. The fit is only satisfactory for the V p /V s ratio, but is superior to the models of [73, 11] (V p /V s )() = (V p /V s ) mineral and the model of [39] (V p /V s )() From the empirical equations for the shear modulus and V p /V s, we derive the porosity dependence of the bulk modulus. Gassmann s equations are then used to predict the fluid saturated moduli and velocities. We compare this empirical prediction for water-saturated clay-cemented sands to the empirical V p (, C), V s (, C) relationships derived by [54] from fits to extensive experimental

256 222 Geophysical applications: velocity-porosity relationships [a] Shear modulus [GPa] Eqn (32): 1% Quartz Eqn (32): 7:3 Quartz/Clay Oseberg Data (Dvorkin 1996) [b] V p /V s Oseberg Data MSA model Clean MSA Model: Clay = 3% Krief/Nur Models Figure 6.4: Comparison of the data of [39] to the prediction of [a] the Krief model for the shear modulus and [b] Eqn The fit is good for the shear modulus and satisfactory for the V p /V s ratio. In the latter case the fit is superior to the models of [73, 11] (V p /V s )() = (V p /V s ) mineral and the model of [39] (V p /V s )() data summarised by Equations The data is given in Fig The match is excellent Han: Eqn(8) Clean Han, Eqn(6) X c =33% Han, Eqn(6), X c =67% Model: Clean Sandstone Model: X c =33% Model: X c =67% 3.5 Han: Eqn(9) Clean Han: Eqn(7) X c =33% Han: Eqn(7) X c =67% Model: Clean Sandstone Model: X c =33% Model: X c =67% V p [km/s] 4. V s [km/s] [a] [b] Figure 6.41: Comparison of the empirical fit of [54] for clay cemented sandstones, Equations (lines), to the empirical model derived from numerical data, Equations and Gassmann s relations (data points) Summary Here systems made up of multiple mineral phases are considered. The utility of data sets generated by numerical simulation is again illustrated. Lack of noise in the data and the ability to control the pore space structure, microstructure and mineralogy of the material within the numer-

257 Accurate velocity model for two-mineral cemented sands 223 ical model has enabled us to quantitatively compare results with theoretical and empirical theories. The effective mineral modulus of the solid mixture exhibits only small structural dependencies. We find that in general the prediction of the average of the Hashin-Shtrickman bounds gives a reasonable match to the numerical data. For all porous systems we observe that the choice of microstructural model has a minimal effect on the resultant modulus-porosity relationship. This is in accord with many previous models which ignore the location of the clay phase. Gassmann s equations are verified for the porous materials made up of multiple solid constituents. We compare numerical predictions to experimental data for clay bearing consolidated sands. The numerical curves give both the correct trend for the experimental data with increasing clay content and give a good quantitative match to experimental data. We compare a number of standard dry modulus-porosity models to the numerical data. The bulk modulus-porosity relationships of both [73] and [11] fail to accurately describe the numerical data (Figs a). The Krief equation (Eqn. 6.32) does however yield a very good fit to the dry shear modulus data (Fig [g-l]). An accurate empirical ν or V p /V s (Eqn. 6.33) relationship for dry sands also matches the numerical data very well (Fig. 6.37). The Raymer equation Eqn is the best modulus-porosity model for water saturated conditions, but it consistently underestimates the V p / data. The Wyllie equation gives a poorer description of the V p / results. Based on the data presented here, we propose that for cemented multiminerallic consolidated sandstones one use the empirical Krief equation for the shear modulus of a dry sandstone Eqn and our empirical equation for the Poisson s ratio of a dry porous rock (Eqn. 6.33). The effective mineral moduli are derived from estimates of the bulk and shear moduli of a rock with multiple mineral constituents based on either the average or weighted Hashin-Shtrickman bounds. Gassmann s equations should be subsequently used to predict the fluid saturated states. Comparison of the prediction of the empirical method to numerical predictions and available experimental data is excellent.

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270 APPENDIX A EUCLIDEAN DISTANCE TRANSFORMATION (EDT) A distance transform converts a binary digital image, consisting of feature and non-feature pixels, into an image where all non-feature pixels have a value corresponding to the distance to the nearest feature pixel [24]. They are important tools in mathematical morphology in actual implementations of morphological operators, simply because of computational efficiency. The morphological operations of erosion/dilation with a sphere as structuring element lead to implementations of parallel surfaces and Voronoi tesselations, both employed in this thesis. The EDT is used in the context of microstructure generation to implement overgrowth of binary composites with an additional phase and to carry out the digital Voronoi tesselations needed for the Voronoi models (chap. 2). It is further applied in the context of microstructure characterisation with parallel surfaces (chap. 3.3), and as means of spatial analysis of NMR T 1 and T 2 images. In previous work [35, 24] approximations to the Euclidean distance map (EDM) have been employed due to the large computational overheads associated with calculating the map. This is not necessary anymore, as both fast algorithms and increased computational power are available to calculate the exact Euclidean distance map [135]. Often the map is used rather as a order structure and the Euclidean distance transform is used instead. Here the actual square root needed to calculate the distance does not have to be evaluated, and an integer map of radii squared suffices. In this work fast algorithms for calculating the EDT exactly have been developed. There are two different flavours. The first algorithm is based on Saito s algorithm number three [135] with extensions to include periodic boundary conditions. It is fast and scales well algorithm four in the same paper scales linearly but I didn t work out, how to extend it to periodic boundaries. Tests for integer overflow were added. The second algorithm was developed from scratch and while being slower has some other merits explained below. For the latter the exact Euclidean distance for each voxel is calculated by growing (discretised) spheres at each voxel x i of the image, until an interfacial voxel of the structure A is reached; the corresponding Euclidean distance d is returned.

271 238 Euclidean distance transformation (EDT) The digital sphere of radius r is given by all voxels radiating from the central voxel that are separated by a Euclidean distance d < r with the sphere center defined as the center of the voxel. The algorithm speed is dramatically accelerated by noting that the Euclidean distance to A from any voxel x j neighboring voxel x i will be given by d(a, x i ) d(a, x j ) 1. One need not step through all possible radii for each voxel, but instead can use d(a, x j ) = d(a, x i ) 1 as an initial guess. In order to quickly find this radius in the (sorted) list of all possible radii and it s associated list of x,y,z-offsets, a strongly monotonic vector of radii realisable on the grid with an associated index list was generated from that list and this sorted list traversed with a hunt algorithm [116] to return the index of this radius. Apart from scaling with the lattice size the routine scales with roughly the size of a shell of the sphere. Only a few sphere sizes at each voxel apart from the first one have to be tried that way. Speed depends strongly on the size of the maximum inscribed radii. The algorithm additionally provides the possibility to detect quenching points with the interface and can easily propagate labels of convex bodies for both periodic and non-periodic structures. 1 Using the definition of Serra [145], that a skeleton point is found, when the maximal inscribed sphere touches the interface at at least two points, the algorithm can be extended to calculate the EDM and skeleton simultaneously. Apart from the latter both algorithms give as result an identical exact integer map of the Euclidean distance squared (EDT). Examples of the 2D Euclidean distance maps for the particle and inverse phase of the identical overlapping sphere (circle) model are given in Fig. A.1. [a] [b] [c] Figure A.1: Euclidean distance maps of the particle phase and inverse phase of a two-dimensional IOS model with a particle fraction of =.7. [a] original structure, [b] EDM of particle phase, [c] EDM of inverse phase. The parallel surfaces are then defined by thresholding the Euclidean distance map (or EDT) at a desired radius (radius squared). 1 This feature is used to generate the Voronoi models in section