PORE-SCALE SIMULATION OF NMR RESPONSE IN POROUS MEDIA

Transcription

1 PORE-SCAE SIMUATION OF NMR RESPONSE IN POROUS MEDIA A DISSERTATION SUBMITTED TO THE DEPARTMENT OF EARTH SCIENCE AND ENGINEERING, IMPERIA COEGE ONDON IN PARTIA FUFIMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHIOSOPHY Olumide Adegbenga Talabi September, 8 i

2 ABSTRACT Rock properties are usuall predicted using 3D images of the rock s microstructure. While single-phase rock properties can be computed directl on these images, twophase properties are usuall predicted using networks etracted from these images. To make accurate predictions with networks, the must be topologicall similar to the porous medium of interest. In this work, NMR response is simulated using a random walk method. The simulations were performed on 3D images obtained from micro-ct scanning and in topologicall equivalent networks etracted from these images using a maimal ball algorithm. A comparison of the NMR simulations on a 3D image and an etracted network helps to ascertain if the network is representative of the underling 3D image. Single-phase NMR simulations are performed on 3D images and etracted networks for different porous media including sand packs, poorl consolidated sandstones, consolidated sandstones and carbonates, and are compared successfull with eperimental measurements. The algorithm developed for the simulation of NMR response in networks was validated using a tuned Berea network that reproduced eperimental capillar pressure data in Bentheimer sandstone. Simulation results of the sand packs and poorl consolidated media show that the T distributions of the networks are narrower than those of the corresponding micro-ct images and eperimental data. This is attributed to the loss of some fine details of the pore structure in the network etraction algorithm. The algorithm developed for singlephase NMR response in networks was etended to two-phase fluids in order to stud the effect of wettabilit on simulated NMR response in networks. While NMR behaviour is influenced pore structure, wettabilit and phase saturation, it is not possible to determine each of these influences uniquel. The ultimate goal is to have a two-phase simulator that predicts relative permeabilit, electrical resistivit, NMR response and capillar pressure which will be used to determine the wettabilit of a porous medium. i

3 ACKNOWEDGMENTS I would like to epress m utmost gratitude to GOD almight, who has been m help in ages past and m hope in ears to come. M sincere gratitude also goes to m supervisor, Professor Martin. J. Blunt, who has alwas been a source of inspiration for me, I consider mself privileged to be one of his research students. Words are inadequate to show m appreciation for his support, guidance and encouragement during this work. I would like to thank the Department of Earth Science and Engineering, Imperial College ondon for awarding me a Janet Watson scholarship to undertake this research work. The financial supports of Universities UK and Petroleum Technolog Development Fund (PTDF) of Nigeria are also acknowledged. M appreciation also goes to the members of the Imperial College consortium on Pore- Scale Modelling (BHP, Eni, JOGMEC, Saudi Aramco, Schlumberger, Shell, Statoil, Total, the U.K. Department of Trade and Industr and the EPSRC) for their technical and financial support. M sincere gratitude goes to Prof Arne Graue and Dr Matthew Jackson for agreeing to be m eaminers. I would like to thank Saif Alsaari, Dr Stefan Iglauer and Saleh Al-Mansoori for their tremendous work on the eperimental data (NMR response, permeabilit and formation factor) used in this work. I also acknowledge the support of Reslab for allowing us to conduct some of the eperimental measurements in their laboratories. I am also grateful to Randir Singh of the Department of Materials for his assistance in the micro-ct scanning. M gratitude also goes to Martin Fernø and Haldis Riskedal of the Department of Phsics and Technolog, Universit of Bergen, Norwa for the collaboration to work on carbonates, especiall the provision of eperimental data of Bentheimer sandstone and Edward limestones. M appreciation also goes to Dr Hu Dong for developing the network etraction method used in this work; his assistance in the computation of micro-ct image properties (permeabilit and formation factor) is also acknowledged. I also thank Numerical Rocks AS, for providing the technical resources to compute these micro-ct properties. I am also grateful to Dr Pål-Eric Øren (Numerical Rocks) for providing the Fontainebleau network data used in this work, the invaluable insights gained during discussions with him over the ears is appreciated. ii

4 I would also like to epress m gratitude to m other colleagues in the Pore-Scale Modeling group, Branko Bijeljic, Nasiru Idowu, Ran Qi and Chris Pentland for their valuable suggestions and assistance during this work. Finall, I would like to dedicate this work to m famil, for their support, encouragement, understanding and companionship during these past three ears and to all m friends too numerous to mention. iii

5 CONTENTS ABSTRACT...i ACKNOWEDGEMENTS......ii CONTENTS..... iv IST OF TABES......vii IST OF FIGURES... viii Introduction... iterature review 3. Methods of generating microstructures..3. Network models..6.3 Network etraction Prediction of transport properties.. 3. Nuclear Magnetic Resonance Basics of NMR response Spin Spin packets ongitudinal relaation processes Transverse relaation processes Relaation mechanisms Surface relaation Bulk relaation Diffusive relaation Relaation due to all mechanisms 3.3 Modelling of NMR response 3.3. The reaction-diffusion equation Modelling surface relaation Modelling bulk relaation Modelling diffusive relaation Modelling all relaation mechanisms Simulation of NMR response in 3D images Geometr of 3D images 9 4. Initialiation of walkers in 3D images..9 iv

6 4.3 Diffusion of walkers in 3D images Calculation grain surface areas Validation of simulation algorithm Simulation results Simulation of NMR response of single-phase fluids in networks Geometr of network elements Initialiation of walkers in networks Diffusion of walkers in networks Relaation of walkers in networks Validation of simulation algorithm Simulation of NMR response of two-phase fluids in networks Initialiation of walkers in the fluid phases Diffusion of walkers in the network Single-phase simulation results for sand packs Sand preparation Sand properties NMR measurement Micro-CT imaging Eperimental results Simulation results Simulation parameters Surface relaivit Magnetiation spectrum Comparisons of other single-phase properties Pore-sie distributions Single-phase simulation results for sandstones Fontainebleau sandstone Poorl consolidated sandstone Berea sandstone Bentheimer sandstone NMR measurement Simulation parameters Simulation results.75 v

7 9. Single-phase simulation results for carbonates Heterogeneous carbonate Edward limestone Homogeneous carbonate Two-phase simulation results..9. Sand packs 9. Fontainebleau network 4.3 Poorl consolidated sandstone network (S) Berea PBM Network Berea MB Network..6 Bentheimer network..7 Carbonate (C) network Comparison with eperimental results 7. Conclusions and recommendations for future work. Conclusions... Recommendations for future work.. 3 References.4 Appendi... 3 A- Simulation code manual. 3 A- Diffusion in micro-ct images A-3 T inversion from magnetiation deca 48 A-4 Flow chart for the simulation of NMR response in networks.. 53 vi

8 IST OF TABES Table 7-: Micro-CT images and etracted networks properties..53 Table 7-: Comparison of eperimental data with simulation results... 6 Table 8-: Comparison of the number of elements and average coordination number in each network etracted using the dilation method (Øren and Bakke,, 3) and maimal ball (Dong, 7).. 66 Table 8-: Rock and fluid properties used in the simulation 66 Table 8-3: Comparison of the fractional surface area of the smallest sides (Sides ), middle sides (Sides ), longest sides (Sides 3) of all the triangular elements and square sides (Sq. sides) in the network 69 Table 8-4: Micro-CT and etracted networks properties for sandstone S and Berea sandstones, the microstructure properties of Berea PBM network.. 73 Table 8-5: Comparison of mean transverse relaation time T, permeabilit and formation factor of micro-ct images and etracted networks. 73 Table 9-: Comparison of mean transverse relaation time T, permeabilit and formation factor of micro-ct image and etracted network 8 vii

9 IST OF FIGURES Figure -: (a) 3D Micro-CT image of porous media (silica sphere pack), imaged at the Australian National Universit ( (b) Corresponding -D cross sectional image.. 4 Figure -: Comparison of micro-ct image (a) and process based image (b) of Fontainebleau sandstone. (Øren and Bakke, ).6 Figure -3: A principal MB absorbs the clusters of smaller neighbours to form multiclusters, maimal balls connected to two clusters define a throat... Figure -4: (a) Micro-CT image of a sandstone and (b) topologicall equivalent network etracted from this image using the maimal ball algorithm.. Figure 3-: (a) In longitudinal relaation processes, the net magnetiation increases eponentiall to the initial value M M o with the time constant T. (b) For transverse relaation processes...7 Figure 3-: Magnetiation decas faster in small pores than in large ones...9 Figure 3-3: A comparison of the magnetiation deca of a fluid in a spherical pore obtained b random walk method and the analtical solution.. 6 Figure 4-: (a) Micro-CT image in gra scale is converted to a binar format (b) where pores are clearl distinguished from grains... 8 Figure 4-: Figure showing the twent-si voels that a walker can diffuse into, if the walker is in voel X...3 Figure 4-3: Comparison of the magnetiation deca of a cubic pore with its equivalent discretied image... 3 Figure 4-4: (a) Micro-CT image of sandstone S with a voel sie of 3 (b) voel sie 3 and (c) voel sie Figure 4-5: (a) The magnetiation decas of 3, 3 and 3 3 voel images, S_, S_ and S_3 respectivel 33 Figure 4-6: Comparison of the magnetiation deca of the 3 3 voel image of S using.5 million and million walkers 34 Figure 5-: The geometr of a tpical triangular element in a network 37 Figure 5-: (a) Throats are connected to pores at different locations within the pore section...38 viii

10 Figure 5-3: A triangular shaped throat connected to the two ends of triangular shaped pores...39 Figure 5-4: The three faces available for relaing walkers.. 39 Figure 6-: (a) The curved areas occupied b water in corners are replaced b (b) equivalent triangles. (c) Uninvaded element.43 Figure 6-: (a) The oil laers formed in corners are replaced b (b) equivalent rectangles Figure 7-: Grain sie distributions of the F4 and V6 sand packs..48 Figure 7-: 3D micro-ct sections of (a) V6A, (b) V6B, (c) V6C, (d) F4A, (e) F4B and (f) F4C....5 Figure 7-3: (a) The eperimental magnetiation deca of two samples (V6X and V6Y) of the V6 sand packs... 5 Figure 7-4: (a) The eperimental magnetiation deca of two samples (F4X and F4Y) of F4 sand packs...5 Figure 7-5: (a) Sections of the micro-ct image of V6A sand pack and (b) F4A sand pack Figure 7-6: (a) Comparison of the eperimental magnetiation deca of F4Y with the micro-ct image and etracted network for F4A 56 Figure 7-7: (a) Comparison of the eperimental magnetiation deca of F4Y with the micro-ct image and etracted network for F4B 57 Figure 7-8: (a) Comparison of the eperimental magnetiation deca of F4Y with the micro-ct image and etracted network for F4C 58 Figure 7-9: (a) Comparison of the eperimental magnetiation deca of V6Y with the micro-ct image and etracted network for V6A...59 Figure 7-: (a) Comparison of the eperimental magnetiation deca of V6Y with the micro-ct image and etracted network for V6B...6 Figure 7-: (a) Comparison of the eperimental magnetiation deca of V6Y with the micro-ct image and etracted network for V6C...6 Figure 7-: Comparison of inscribed radius distribution of network elements of F4A with the pore sie distribution of F4 sand pack Figure 8-: The pore spaces in a 8 3 voels sub region of a reconstructed Fontainebleau sandstone (right) of porosit.8 and a micro-ct image of an actual Fontainebleau sandstone (left) with similar porosit Figure 8-: (a) Comparison of the magnetiation deca of the reconstructed microstructure of Fontainebleau sandstone..67 i

11 Figure 8-3: (a) 3D image of sandstone S. (b) Network etracted from the image...7 Figure 8-4: (a) Magnetiation decas of the micro-ct image and etracted network of sandstone S Figure 8-5: (a) 3D micro-ct image of Berea sandstone. (b) Network etracted from the image in (a) using maimal ball method Figure 8-6: (a) Comparison of the magnetiation decas of the micro-ct image of Berea sandstone, the network etracted using a dilation method (Øren and Bakke;, 3) from reconstructed Berea sandstone (erdahl et al., ) and network etracted from the micro-ct image 7 Figure 8-7: Comparison of the eperimental capillar pressure data of the Bentheimer sandstone with simulation results of the tuned Berea network. 75 Figure 8-8: (a) Comparison of magnetiation decas of the eperimental data of Bentheimer sandstone with simulation results of the tuned network 76 Figure 9.: (a) 3D micro-ct image of carbonate C. (b) Network etracted from the image in (a) using the maimal ball method Figure 9-a: Comparison of the magnetiation decas of the micro-ct image of carbonate C and the etracted network...79 Figure 9-b: Comparison of the inverted T distribution of the magnetiation decas of carbonate C shown in Figure 9-a Figure 9-3: Comparison of the measured capillar pressure curve of MB3 with simulation results and the original Berea network. Berea network elements properties were tuned to give a capillar pressure Figure 9-4: (a) Comparison of magnetiation decas of the eperimental data of MB3 with simulation results of the tuned network. (b) The inverted T distribution from the magnetiation decas in (a) 83 Figure 9-5: Comparison of the measured capillar pressure curve of MB with simulation results of the tuned network Figure 9-6a: Comparison of the magnetiation decas of the eperimental data of MB3 with simulation results of the tuned network. 84 Figure 9-6b: Comparison of the inverted T distribution of the magnetiation decas of MB shown in Figure 9-6a...85 Figure 9-7: Comparison of the measured capillar pressure curve of carbonate C with simulation results of the tuned network 86

12 Figure 9-8a: Comparison of the magnetiation decas of the eperimental data of carbonate C with simulation results of the tuned network 86 Figure 9-8b: Comparison of the inverted T distributions of the magnetiation decas of carbonate C shown in Figure 9-8a 87 Figure 9-9: Comparison of the measured capillar pressure curve of carbonate C3 with simulation results of the tuned network 87 Figure 9-: (a) Comparison of magnetiation decas of the eperimental data of carbonate C3 with simulation results of the tuned network 88 Figure 9-: (a) Comparison of pore sie distributions estimated from the T distributions for carbonates C and C3 with the original Berea network 89 Figure -: (a) Comparison of two-phase drainage magnetiation decas at different water saturations for the F4A sand pack network... 9 Figure -: (a) Comparison of two-phase waterflooding magnetiation decas at contact angles 3 o -4 o (moderatel water-wet), 7 o -8 o (intermediate-wet) and o - o (oil-wet) at an intermediate water saturation....9 Figure -3: (a) Comparison of two-phase drainage magnetiation decas at different water saturations for the F4B sand pack network Figure -4: (a) Comparison of two-phase waterflooding magnetiation decas at contact angles 3 o -4 o (moderatel water-wet), 7 o -8 o (intermediate-wet) and o - o (oil-wet) at an intermediate water saturation...94 Figure -5: (a) Comparison of two-phase drainage magnetiation decas at different water saturations for the F4C sand pack network...95 Figure -6: (a) Comparison of two-phase waterflooding magnetiation decas at contact angles 3 o -4 o (moderatel water-wet), 7 o -8 o (intermediate-wet) and o - o (oil-wet) at an intermediate water saturation...96 Figure -7: (a) Comparison of two-phase drainage magnetiation decas at different water saturations for the V6A sand pack network Figure -8: (a) Comparison of two-phase waterflooding magnetiation decas at contact angles 3 o -4 o (moderatel water-wet), 7 o -8 o (intermediate-wet) and o - o (oil-wet) at an intermediate water saturation Figure -9: (a) Comparison of two-phase drainage magnetiation decas at different water saturations for the V6B sand pack network Figure -: (a) Comparison of two-phase waterflooding magnetiation decas at contact angles 3 o -4 o (moderatel water-wet), 7 o -8 o (intermediate-wet) and o - o (oil-wet) at an intermediate water saturation i

13 Figure -: (a) Comparison of two-phase drainage magnetiation decas at different water saturations for the V6C sand pack network.. Figure -: (a) Comparison of two-phase waterflooding magnetiation decas at contact angles 3 o -4 o (moderatel water-wet), 7 o -8 o (intermediate-wet) and o - o (oil-wet) at an intermediate water saturation..... Figure -3: (a) Comparison of the magnetiation decas due to bulk and surface relaations of oil and water phases in F4A network 3 Figure -4: (a) Comparison of two-phase drainage magnetiation decas at different water saturations for the Fontainebleau network....4 Figure -5: (a) Comparison of two-phase waterflooding magnetiation decas at contact angles 3 o -4 o (moderatel water-wet), 7 o -8 o (intermediate-wet) and o - o (oil-wet) at an intermediate water saturation Figure -6: (a) Comparison of two-phase drainage magnetiation decas at different water saturations for the poorl consolidated sandstone....6 Figure -7: (a) Comparison of two-phase waterflooding magnetiation decas at contact angles 3 o -4 o (moderatel water-wet), 7 o -8 o (intermediate-wet) and o - o (oil-wet) at an intermediate water saturation.7 Figure -8: (a) Comparison of two-phase drainage magnetiation decas at different water saturations for the network obtained from reconstructed microstructure of Berea sandstone, Berea PBM.. 8 Figure -9: (a) Comparison of two-phase waterflooding magnetiation decas at contact angles 3 o -4 o (moderatel water-wet), 7 o -8 o (intermediate-wet) and o - o (oil-wet) at an intermediate water saturation Figure -: (a) Comparison of two-phase drainage magnetiation decas at different water saturations for the network etracted using the maimal ball algorithm from the micro-ct image of a Berea sandstone.. Figure -: (a) Comparison of two-phase waterflooding magnetiation decas at contact angles 3 o -4 o (moderatel water-wet), 7 o -8 o (intermediate-wet) and o - o (oil-wet) at an intermediate water saturation..... Figure -: (a) Comparison of two-phase drainage magnetiation decas at different water saturations for the network obtained b tuning the Berea PBM network to match the eperimental capillar pressure data. Figure -3: (a) Comparison of two-phase waterflooding magnetiation decas at contact angles 3 o -4 o (moderatel water-wet), 7 o -8 o (intermediate-wet) and o - o (oil-wet) at an intermediate water saturation ii

14 Figure -4: (a) Comparison of two-phase drainage magnetiation decas at different water saturations for the network etracted from the micro-ct image of the carbonate C, using the maimal ball algorithm Figure -5: (a) Comparison of two-phase waterflooding magnetiation decas at contact angles 3 o -4 o (moderatel water-wet), 7 o -8 o (intermediate-wet) and o - o (oil-wet) at an intermediate water saturation Figure -6: During waterflooding oil laers are sandwiched between water in corners and water at the centre of the pore..6 Figure -7: Eperimental T distributions of a mied-wet North Sea sandstone at water saturations (S w. and.7)...9 Figure -8: Simulated T distributions of Berea PBM network at contact angles in the range 7 8 (mied-wet) and saturations (S w. and.9)..9 Figure -9: Eperimental T distributions of a mied-wet North Sea sandstone at water saturations (S w.39 and.76) 9 Figure -3: Simulated T distributions of Berea PBM network at contact angles in the range 7 8 (mied-wet) and saturations (S w.4 and.76) 9 Figure -3: Simulated magnetiation decas of the Berea PBM network at miedwet condition for different water saturations... Figure -3: Simulated magnetiation decas of the Berea PBM network at oil-wet condition for different water saturations. Figure -33: Eperimental T distributions of an oil-wet North Sea sandstone at water saturations (S w. and.38) Figure -34: Simulated T distributions of Berea PBM network at contact angles in the range (oil-wet) and saturations (S w. and.38)... Figure -35: Eperimental T distributions of an oil-wet North Sea sandstone at water saturations (S w.4 and.85).. Figure -36: Simulated T distributions of Berea PBM network at contact angles in the range (oil-wet) and saturations (S w.4 and.85). Figure A.. Comparison of the T distributions inverted using Win DXP software with that obtained using the curvature smoothing method.5 iii

15 . Introduction To make accurate predictions using network models, the wettabilit of the porous medium must be known, especiall if comparison with eperimental measurements is desired. NMR response provides information about pore sie distribution and wettabilit, which if combined with other wettabilit dependent properties like relative permeabilit, capillar pressure and electrical resistivit can be used to pin down the wettabilit of a porous medium. As a preliminar step to the simulation of NMR response of two-phase fluid in networks to investigate the effect of wettabilit, NMR response of single-phase fluid is simulated in networks etracted from the micro-ct images of different porous media ranging from sand packs to carbonates. One of the uses of NMR response is that it can be used to validate how representative a network is, to the micro-ct image or the microstructure from which it was etracted. The starting point in the prediction of transport properties of reservoir rocks is the generation of 3D images; a detailed literature review of the different methods used in obtaining these images is described in Chapter. A review of the different algorithms used in etracting networks from these 3D images is also discussed along with different mathematical methods used for the computation of transport properties in 3D images and networks. Fundamentals of NMR response and the modelling of the different relaation mechanisms are discussed in Chapter 3, along with a detailed description of the random walk method. In Chapter 4, the algorithm used for the simulation of NMR response of single-phase fluid in micro-ct images is described and validated. Chapter 5 describes the algorithm developed for the simulation of NMR response of single-phase fluids in networks. The single-phase NMR simulation algorithm is etended to two-phase fluids in Chapter 6. In order to further validate the algorithms developed for the simulation of NMR response in 3D images and networks, sand packs were studied because of their homogeneit.

16 A detailed comparison of NMR simulation results of sand packs with eperimental data is discussed in Chapter 7. These comparisons are further etended to poorl consolidated and consolidated sandstones in Chapter 8. In chapter 9, networks are generated for carbonates from their eperimental capillar pressure b tuning the properties of a Berea network to match the measured capillar pressure. NMR response was then simulated in these carbonates networks and are compared with eperimental data. Two-phase NMR simulation results for different porous media and the effect of wettabilit are discussed in Chapter. Finall in chapter, the important findings of this work are summaried and recommendations for future work are discussed. The following articles have been either based on material presented in this thesis or used b the simulation code developed in this work. Talabi, O., AlSaari, S., Iglauer, S., and M. J. Blunt, Pore-Scale simulation of NMR response, Submitted to Journal of Petroleum Science and Engineering, 8 Talabi, O., AlSaari, S., Blunt, M. J., Dong, H. and X. Zhao, Predictive Pore- Scale Modeling: From Three-Dimensional Images to Multiphase Flow Simulations. Proceedings of the 8 SPE Annual Technical Conference and Ehibition, SPE 5535, Denver, Colorado, USA, -4 September, 8. Talabi, O., AlSaari, S., Fernø, M. A., Riskedal, H., Graue, A. and M. J. Blunt, Pore-scale simulation of NMR response in carbonates. International Smposium of the Societ of Core Analsts held in Abu Dhabi, UAE 9 October- November, 8

17 . iterature Review Rock flow properties cannot be predicted without an accurate 3D representation of the rock microstructure. This is most commonl achieved using one of the following three methods: () Eperimental - (Halett, 995; Arns et al., 4), () Statistical which involves stochastic microstructural modelling, also known as image reconstruction procedure, (Adler et al., 99; iang et al., 998; Okabe and Blunt, 4) and (3) Process- or phsics-based method, which simulates the rock forming processes (Brant et al., 993, Bakke and Øren, 997; Øren and Bakke, 3; Jin et al., 3).. Methods of generating 3D images of microstructures Eperimental (Micro-CT imaging) The eperimental 3D representation of a rock microstructure involves the use of X-ra computed tomograph (micro-ct). This is alread an established and rapidl evolving technolog which is of significant importance in the image analsis of porous media. It was initiall developed for widespread use in the field of medicine; however its use has been etended to a wide variet of fields which include, archaeolog, geophsics, oceanograph, materials science and astrophsics. X-ra computed tomograph is a quick and non destructive three-dimensional visualiation tool used for viewing the internal structure of objects as determined b the variations in densit and atomic composition. It is used efficientl for qualitative and quantitative analsis of internal structures of reservoir rocks especiall if those structures are marked b sufficientl great differences in atomic composition and/or densit as shown in Figure -. Densit transitions usuall correspond to boundaries between materials or phases. 3

18 The densit contrast which is usuall observed between fluid phases and grains in reservoir rocks is the basis for the current widespread application of X-ra computed tomograph (CT) in the field of petroleum geolog, rock mechanics and soil science (Peters et al., 996). The term microtomograph (micro-ct) is often used in imaging reservoir rocks in order to indicate that the piel sies of the cross-section are in the µm range. (a) (b) Figure -: (a) 3D Micro-CT image of porous media (silica sphere pack), imaged at the Australian National Universit ( (b) Corresponding -D cross sectional image. Until recentl, direct imaging of 3D microstructures using X-ra computed microtomograph (Dunsmuir et al., 99; Coles et al., 994, 996; Spanne et al., 994; Halett, 995; Coker et al., 996), was often time consuming, epensive and not readil applicable to damaged cores and drill cuttings. Hence, it was quite challenging to obtain reliable images of the 3D pore structure. Even with modern machines, the resolution is limited to a few microns which ecludes small pores in tight sandstones and carbonates. This limitation has led to the development of statistical methods based on high resolution D images. Statistical method In the past, information about the microstructure of porous materials was often limited to two-dimensional thin section images. Such D images have formed the basis for the reconstruction techniques developed to generate 3D microstructures statisticall (Joshi, 974; Quiblier, 984; Adler et al., 99, 99; Roberts, 997; Halett, 997; Yeong and Torquato, 998a, b; Manswart and Hilfer, 998; Okabe and Blunt, 5). 4

19 The reconstruction techniques use the spatial information of D thin section images b measuring statistical properties, such as porosit, correlation and lineal path functions on the D image. 3D microstructures are then generated such that the measured statistical properties of the D thin section images are preserved. Detailed quantitative comparisons of statisticall generated microstructures with X-ra tomographic images of sedimentar rocks have shown that statistical reconstructions ma differ significantl from the original sample in their geometric connectivit especiall for low porosit materials, (Halett, 997; Biswal et al., 999; Manswart et al., ; Øren and Bakke, 3). This has led into more sophisticated statistical techniques, such as multiple point statistics. These methods preserve higher-order information, describing the statistical relation between multiple spatial locations. The replicate tpical patterns of the void space seen in thin sections thereb capturing the large-scale features of the pore space geometr. The application of multiple point statistics to reconstruct 3D images using D image data has successfull been used to reconstruct long-range connectivit of sandstone and carbonate samples (Okabe and Blunt, 4; Okabe and Blunt, 5). Process-based reconstruction method A different approach to the stochastic generation of 3D pore structures is the processbased reconstruction method (Bakke and Øren, 997), where the main rock forming processes; sedimentation, compaction and diagenesis are simulated. Input data into the simulation are petrographical measurements on BSE images of thin sections such as grain sies and shapes, porosit, amount of quart cement, cla and other diagenetic minerals. This method has been used to reconstruct the 3D microstructure of Fontainebleau sandstone (Øren and Bakke, ). Transport properties computed in reconstructed microstructures had been compared successfull with those computed on equivalent micro-ct images of similar porosit, Figure - shows a comparison of a reconstructed microstructure with a micro-ct image of Fontainebleau sandstone. In a similar approach to the process-based reconstruction method, Jin et al., (3) used a phsics-based depositional model to reconstruct natural sedimentar rock, and generated 3D images of the pore space at an arbitrar resolution. 5

20 This model provided a detailed microstructure of the rock thereb making it possible to calculate the stead state velocit field in the single-phase fluid flow. The model was used to anale unconsolidated rocks whose micro-tomographic images cannot be obtained. The effect of compaction and various stles of cementation on the microstructure and permeabilit of reservoir rock were also studied using this model. (a) (b) Figure -: Comparison of a micro-ct image (a) and process based image (b) of Fontainebleau sandstone (Øren and Bakke, ). Among the three methods discussed for the generation of 3D representation of the rock microstructure, X-ra computed microtomograph (micro-ct) despite its limitations, is the most promising since it is a direct measurement of the actual microstructure. Process-based reconstructions ma be problematic for rocks whose depositional and diagenetic histor is comple or unknown such as man carbonates, thus statistical methods are useful in overcoming these limitations of the process-based method.. Network models The use of network models to stud multiphase flow in porous media was pioneered b Fatt (956a). These models were based on the idea that pore space ma be represented as an interconnected network of capillar tubes whose radii represent the dimensions of the pores within a porous medium. 6

21 Fatt s initial network was a 5 5 lattice of resistors which was used to compute relative permeabilit and capillar pressure for drainage-tpe displacement. Although the results of this initial work were qualitative because the networks were regular and two-dimensional which is not representative of a real porous medium, it is still the basis of the current use of networks for the prediction of transport properties. Fatt s initial network assumed that the bonds intersections do not possess an volume of their own. Chatis and Dullien (977) later focused on the assumptions made b Fatt (956a). The used a 3D model that consisted of sites which are connected to each other b bonds. The sites correspond to pore bodies while the interconnecting bonds are analogous to pore throats connecting the pore bodies. Most researchers nowadas attribute the majorit of the porosit to the pore bodies. The basis of these network models is that the void space of a porous medium can be represented b a graph of connected sites. The interconnecting pathwas of networks make them structurall similar to porous media. A network of pores and throats preserves the essential topolog of the pore space it represents. In addition, if sies are assigned to the pores and throats in accordance with the sies of the pore bodies and throats in the porous medium, then macroscopic properties of the resulting network should accuratel reflect the properties of the porous medium. For accurate prediction of properties of the porous medium, a network model should replicate both the geometr and topolog of pore space as much as possible, so that flow through the network is equivalent to flow through the actual porous medium. Direct replication of a porous medium with network models has proven elusive because the pore space in real granular media is complicated. The pioneering networks of Fatt (956a) and Chatis and Dullien (977) were based on a regular lattice; however the pore space in a real porous medium is complicated and ma not be adequatel described b such regular lattice. Chatis and Dullien, (977) compared results from these regular cubic networks with eperimental mercur intrusion data from sandstones and noted that regular networks combined with circular capillaries did not ield a realistic description of real sandstones. 7

22 In order to obtain a network that best describes the complicated structure of a given rock sample, the coordination number of the rock which is tpicall between 3 and 8 (Jerauld and Salter, 99) can be matched b removing throats from a regular lattice of higher coordination number. An alternative wa of constructing a representative network is to randoml distribute points within the model area and then construct a network from the triangulation of these points. Jerauld et al. (984a) constructed random networks from Voroni diagrams; these networks had an initial coordination number of more than 5. This was significantl reduced to a coordination number of 6 b removing the longest throats. ittle difference was observed in the comparison of the relative permeabilit of the networks constructed b reducing coordination numbers and networks based on a regular lattice. All the methods of constructing networks fail to capture an of the statistics associated with real porous media, thereb necessitating the need for other network construction methods. Brant et al. (99; 993a), pioneered the use of networks derived from a real porous medium. The porous medium was formed from a dense random packing of equall sied spheres. The spatial coordinates of ever sphere was measured thereb enabling the microstructure of the medium to be completel determined; a network model which replicates the pore space was then etracted. Relative permeabilities were then predicted from the networks which were successfull compared with eperimental values measured on sand packs, bead packs and a simple sandstone. In contrast, using a statisticall equivalent but uncorrelated network, the permeabilit was over predicted b a factor of two (Brant et al., 993b). This result emphasies the significance of using geologicall realistic networks when predicting flow behaviour. Other transport properties such as electrical conductivit and capillar pressures have also been compared successfull with eperimental results from sand packs, bead packs and a simple sandstone. It is unlikel that most porous media will comprise of equall-sied grains resulting in a constant coordination number, as such the application of the method used for network etraction b Brant et al. (99; 993a; 993b), to more comple porous media is limited. 8

23 .3 Network etraction In order to etract a representative network from a real porous medium, an accurate 3D representation of the microstructure is required. This is most commonl achieved using an of the three methods described earlier. A common method of etracting networks from microstructures involves defining the skeleton of the pore space as a set of points (voels) at equal distances from two or more points of the solid grain (Serra, 98; Adler, 99). This skeleton ma be regarded as the centre line of the pore network; this line contains points where two or more lines meet. These points are the network nodes which are connected to other nodes through the pore throats. Skeleton etraction from binaried images is often carried out b means of thinning algorithms (Hilditch, 969), which are based on deleting or preserving the analsed piel according to the rules for the binar - and - pattern of its surrounding neighbour piels. indquist et al. (996; 999; ) etended a similar thinning algorithm (ee et al., 994) in which the pore space is eroded until onl the centrelines remain in order to anale geometric properties such as connectivit and tortuosit of skeletons generated from various micro-ct imaged rock samples. An alternative method of skeleton etraction from binar images is the dilation method (Bakke and Oren, 997) which is the inverse function of the thinning method. In the dilation method, each grain is numbered, and is subjected to ultimate dilation, the border surface between two ultimate dilated grains is a grain-to-grain connection surface. This surface contains the pore network skeleton. The skeleton of the pore network is thus found where two or more such dilated grain border surfaces meet. This skeleton is used as a basis for mapping of pore throats between nodes. For each pore throat skeleton voel, the pore throat wall is mapped in a plane normal to the local direction of the pore throat. The throat radius is measured with a rotating vector which measures the distance from the skeleton voel to the pore throat wall in this plane. The area, perimeter and inscribed radius for the narrowest constriction of the pore throat are recorded. The volume of each pore bod is calculated using an average radius from the geometrical centre point which is defined b the skeleton node. The geometr of the pore bodies are analsed in a manner similar to that of the pore throats. 9

24 An alternative method of generating networks from 3D images without the need to first etract a skeleton is the maimal ball method (Silin et al., 3; Silin and Patek, 6; Al-Kharusi and Blunt, 7; Dong, 7). This method involves finding the largest inscribed spheres (maimal ball) that touch grain surfaces. Maimal balls (MB) are the basic elements used to define the pore space and detect the geometrical changes and connectivit. A maimal ball is defined as a set of voels assembling a largest sphere, as such, a maimal ball must touch the grain surface and it cannot be a subset of an other maimal ball. Ever maimal ball possesses at least one voel that is not contained in an other maimal ball, the aggregate of all maimal balls defines the void space in a rock image. Figure -3: A principal MB absorbs the clusters of smaller neighbours to form multiclusters; maimal balls connected to two clusters define a throat. Pore-throat chains (white arrows) define the topolog of the pore space. The maimal balls fill the entire void space measuring the local apertures in pore central spaces and irregular corners. To define the topolog, maimal balls are merged into clusters. Two tpes of clusters are defined: single and multiclusters. In a single cluster, a principal maimal ball absorbs all its direct smaller neighbours in its domain. Multiclusters are etensions of single clusters; the principal maimal ball absorbs the clusters of smaller neighbours in its domain. The maimal balls are then sorted into interconnected clusters, the common principal maimal ball of each cluster defines a pore.

25 If a maimal ball is connected to two clusters, then it defines a throat. As shown in Figure -3, clusters A and B meet at the throat node indicated; this is referred to as a pore-throat chain. These chains are used to define the topolog of the pore space as shown in Figure -4. The radius, volume and shape factor of the pores and throats are then calculated. The inscribed radius of an element corresponds to the radius of its principal maimal ball. Their volumes are found b counting the number of pore voels associated with each element. Shape factor, G is calculated from: V G (.) A s where V and are the volume and length of the element respectivel, surface area of the element. A s is the grain (a) (b) Figure -4: (a) Micro-CT image of a sandstone and (b) topologicall equivalent network etracted from this image using the maimal ball algorithm..4 Prediction of transport properties Permeabilit, formation factor and NMR response can be predicted in micro-ct images and etracted networks. Multiphase properties such as capillar pressure and relative permeabilit can onl be predicted in networks.

26 Permeabilit Permeabilit in the 3D images is computed as the mean of the directional permeabilities. These are calculated b appling a macroscopic pressure gradient ( p p ) / in the reference direction. Q i ( p p ) ki A (.) µ Q i is the macroscopic flu obtained from the macroscopic pressure gradient, µ is the viscosit of the fluid and A is the cross sectional area perpendicular to the reference direction. The fluid flow is assumed to be governed b the stead state Stokes equations for an incompressible Newtonian fluid subject to a no slip boundar condition at the solid wall, (Øren et al., ). Permeabilit is also calculated in the networks using equation (.), pressure gradient on the network and ( p p ) / in this case, is the applied macroscopic Q i is the total single-phase flow rate through the network which is found b solving for pressure everwhere and imposing mass conservation at ever pore (Valvatne and Blunt, 4). Formation factor Formation factor is calculated in the 3D images as the harmonic mean of the directional formation factors. The directional formation factors F i are defined as the inverse of the macroscopic electrical conductance, σ i in a given i-direction: F i σ w (.3) σ i where σ w is the bulk electrical conductance of the fluid that fills the pore space. σ i is computed from a linear relation between the total electric flu, Q i, and the applied potential gradient.

27 Q i ( Φ I Φ O ) A σ i (.4) The potential applied to and outlet face Φ I is applied to and inlet face, Ω O separated from Ω I of area A, and potential Φ O is Ω I b a distance (Øren et al., ). Formation factor is calculated in the network b a similar approach invoking Ohm s law to find the overall conductance (Valvatne and Blunt, 4). Capillar pressure and relative permeabilit Capillar pressure is computed in the network from the maimum local displacement pressure of the invading phase as a function of saturation, (Valvatne and Blunt, 4). The relative permeabilit, k ri is given b: Q im k ri (.5) Qis where Q im is the total flow rate in multiphase conditions with the same imposed pressure drop in equation (.) while and Blunt, 4). Q is is the total single-phase flow rate (Valvatne NMR Response NMR response is predicted in 3D images and networks b a random walk method, (Ramakrishnan et al., 998; Øren et al., ). This involves the simulation of the Brownian motion of a diffusing particle called the walker. These walkers diffuse within the pore space and when the come in contact with a solid surface, the are killed with a finite killing probabilit. The life-time distribution of the walkers is the magnetiation deca which can be inverted into T distribution b using a curvaturesmoothing regulariation method (Chen et al., 999). 3

28 Toumelin et al., (3) simulated NMR response of two-phase fluids in 3D bimodal pack of spheres which replicates the comple pore structure (micro porosit and macro porosit) geometr of carbonate formations. Each grain in the model was defined as a sphere inscribed into a concentric cubic cell. If the sphere does not completel fill its cubic unit, then the complimentar volume is filled with fluids. Fied blobs centred in the macropore space was included in the model to represent partial saturations of the immiscible nonwetting oil phase, while the microporosit was onl filled with irreducible water. The simulations were performed in the presence of constant magnetic gradients to replicate actual well-logging conditions that include the effect of Carr-Purcell-Meiboom-Gill (CPMG) pulse sequences at a microscopic level. Magnetiation deca is simulated in each phase and the total deca of the two phases is given b a weighted average of the magnetiation deca of each phase. 4

29 3. Nuclear Magnetic Resonance 3. Basics of NMR response Nuclear Magnetic Resonance (NMR) is a phsical phenomenon based upon the quantum magnetic properties of an atom's nucleus. It is observed when the nuclei of certain atoms are aligned with an applied constant magnetic field and then perturbed using an alternating orthogonal magnetic field. Magnetic nuclei absorb energ when placed in a magnetic field of a strength specific to the identit of the nuclei. When the absorption occurs, the nucleus is described as being in resonance, different atoms within a molecule resonate at different frequencies at a given field strength. The observation of the resonance frequencies of a molecule allows the determination of the structural information about the molecule. This resulting response to the perturbing magnetic field is the phenomenon that is eploited in NMR spectroscop and magnetic resonance imaging. Onl nuclei with a propert called spin eperience this resonance phenomenon. 3.. Spin Spin is a fundamental propert of nature like electrical charge or mass. Spin comes in multiples of and can be positive or negative. Protons, electrons, and neutrons possess spin. Individual unpaired electrons, protons, and neutrons each possess a spin of. Two or more particles with spins having opposite signs can pair up to eliminate the observable manifestations of spin. In nuclear magnetic resonance, it is unpaired nuclear spins that are of importance. The spin propert possessed b a proton enables it to be considered as a small magnetic field which will cause the nucleus to produce an NMR signal. The most commonl measured nuclei are H (Hdrogen ; the most receptive isotope at natural abundance) and 3 C (Carbon 3), although nuclei from isotopes of man other elements can also be observed. When placed in a magnetic field of strength B, a particle with a net spin can absorb a photon, of frequenc ν (Purcell et al., 946; Bloch et al., 946). 5

30 The frequenc ν (often referred to as the armor frequenc) depends on the gromagnetic ratio, γ of the particle and the magnetic field strength B. ν γb (3.) For hdrogen ( H ), γ 4.58 MH / T (Hornak, 997). 3.. Spin packets It is cumbersome to describe NMR on a microscopic scale. A macroscopic picture is more convenient, (Hornak, 997). The first step in developing the macroscopic picture is to define the spin packet. A spin packet is a group of spins eperiencing the same magnetic field strength. At an instant in time, the magnetic field due to the spins in each spin packet can be represented b a magnetiation vector. The vector sum of the magnetiation vectors from all of the spin packets is the net magnetiation. The eternal magnetic field and the net magnetiation vector at equilibrium are both along the -ais ongitudinal relaation processes At equilibrium, the net magnetiation vector lies along the direction of the applied magnetic field B o and is called the equilibrium magnetiation M. In this configuration, the Z component of magnetiation M equal M o. o M is referred to as the longitudinal magnetiation. It is possible to change the net magnetiation b eposing the nuclear spin sstem to energ of a frequenc equal to the energ difference between the spin states. If enough energ is put into the sstem, it is possible to saturate the spin sstem and make M as shown in Figure 3-a. The time constant which describes how M returns to its equilibrium value is called the spin lattice relaation time T (Bloembergen et al., 948). The equation governing this behaviour as a function of the time t after its displacement is: M t /T M ( e ) (3.) o 6

31 The spin-lattice relaation time, T is the time to reduce the difference between the longitudinal magnetiation M and its equilibrium value b a factor of e. (a) (b) Figure 3-: (a) In longitudinal relaation processes, the net magnetiation increases eponentiall to the initial value transverse relaation processes, the net magnetiation ero from its the initial value M with the time constant T. M M o with the time constant T. (b) For M decreases eponentiall to 3..4 Transverse relaation processes If the net magnetiation is placed in the XY plane, it will rotate about the Z ais at a frequenc equal to the frequenc of the photon which would cause a transition between the two energ levels of the spin. This frequenc is called the armor frequenc. In addition to the rotation, the net magnetiation starts to dephase because each of the spin packets making it up is eperiencing a slightl different magnetic field and rotates at its own armor frequenc. The longer the elapsed time, the greater the phase difference. The time constant which describes the return to equilibrium of the transverse magnetiation, relaation time, T. M is called the spin-spin relaation time or transverse M o t /T M e (3.3) 7

32 T is alwas less than or equal to T. The net magnetiation in the XY plane decreases to ero while the longitudinal magnetiation increases to M o along the -ais. 3. Relaation mechanisms There are three main mechanisms that cause both longitudinal and transverse relaation during NMR measurements; these are surface relaation, bulk relaation and diffusive relaation. 3.. Surface relaation Surface relaation occurs as a result of Brownian motion which causes fluid molecules to diffuse substantial distance during NMR measurements. From the equation for diffusion; X 6D t (3.4) where X is the mean squared distance a molecule diffuses in time t, and D is the molecular diffusion coefficient which is a propert of the fluid and is dependent on temperature and viscosit. During the random motion of molecules in a fluid, the protons in these molecules collide with the pore surfaces and consequentl rela thereb contributing to both the transverse and longitudinal relaation. The rate at which these protons rela upon contact with a pore surface is dependent on the surface relaivit of the pore. Surface relaivit is thus defined as the abilit of a surface to rela the magnetiation of the protons in the molecules of a given fluid and is often determined from diffusion eperiments and is denoted b ρ (for T relaation) and ρ (for T relaation). Not all surfaces are effective in relaing proton spins; paramagnetic ions such as iron, manganese, nickel and chromium are powerful relaers and tend to control the rate of relaation when the are present. 8

33 Carbonates are known to have lower rates of surface relaivit than sandstones. Surface relaivit also depends on temperature (Godefro et al., ) and the tpe of fluid in the pore; surface relaivities to water are higher than surface relaivit to oil (ooestijn and Hofman, 5). The relaation time of protons will be short if their molecules are contained in small pores because their diffusion is restricted b the pore sies. Correspondingl, the relaation time will be long in large pores because the protons spend an appreciable period diffusing substantial distances before eventuall relaing as shown in Figure 3-. Magnetiation Magnetiation Time (s) Time (s) Figure 3-: Magnetiation decas faster in small pores than in large ones The relaation time will also be high if onl a small amount of surface area is available to rela the protons of a large volume of liquid. As such, relaation rates, (/T s and / T s ) are dependent on the intrinsic surface relaivit of the pore surface and its surface area-to-volume ratio (Brownstein and Tarr, 979; oren and Robinson, 97), and ρ S (3.5) T s V ρ S (3.6) T s V where S is the surface area and V is the volume of the pore. 9

34 3.. Bulk relaation Relaation in bulk reservoir fluids is mainl as a result of fluctuating local magnetic fields that occur due to the random motion of neighbouring spins. Bulk relaation is important when a fluid is prevented from coming in contact with a solid surface, for eample a droplet of oil or gas at the centre of a pore without having an contact with the surface. Bulk relaation is a propert of the fluid alone and is independent of an propert of the formation in which it resides; it does however depend on fluid temperature and viscosit. Whenever bulk relaation is dominant, longitudinal relaation time T is equal to transverse relaation time T Diffusive relaation The Carr-Purcell-Meiboom-Gill (CPMG) sequence is often used for transverse relaation measurements; an radio frequenc pulse at the armor frequenc causes the net proton magnetiation to undergo pure rotation. The CPMG sequence starts with a pulse producing a 9 o rotation, so that an longitudinal magnetiation now becomes transverse and produces a precession signal detected b the radio-frequenc coil (Meiboom and Gill, 958). This signal decas ver quickl if the magnetic field is ver inhomogeneous and in consequence the precession of individual protons quickl dephase. A subsequent pulse produces a 8 o rotation which reverses the dephasing, and produces a second signal called the spin echo. Each subsequent 8 o pulse produces an additional spin echo, and the sequence of amplitudes of successive spin echoes is recorded. The sequence of these spin echo amplitudes is affected b inhomogeneities in the permanent magnetic field when protons are diffusing. When water or hdrocarbon molecules diffuse, the refocusing is no longer a perfect mirror image of the dephasing, the molecules have probabl diffused into a region of different magnetic field strength and in consequence their protons now precess at different frequenc. The global relaation rate due to diffusion in such magnetic field gradients is given as (Kleinberg and Vinegar, 996);

35 T D γ G TE D (3.7) where D is the molecular diffusion coefficient, G is the magnetic gradient strength in T/m, T E is the time between spin echoes in the CPMG sequence and γ is the gromagnetic ratio of the proton Relaation due to all mechanisms In summar the relaation mechanisms described all act in parallel and as such their rates add up, (Bloembergen et al., 948). For transverse relaation; + + (3.8) T TS TB TD where / T is the surface contribution, / T B is the bulk contribution and / T D is the S diffusion in field gradient contribution (Kenon, 997). The corresponding equation for longitudinal relaation is, + (3.9) T B T S T 3.3 Modelling of NMR response In order to model NMR response, each of the relaation mechanisms - surface relaation, bulk relaation and relaation due to diffusion in a magnetic field gradient must all be modelled independentl since the all occur independentl and simultaneousl. The reaction-diffusion equation is the basis for the modelling of surface relaation and bulk relaation mechanisms of NMR response.

36 3.3. The reaction-diffusion equation Senturia and Robinson (97) proposed that the NMR response of a porous medium assuming no magnetic gradient can be modelled b a reaction-diffusion equation. This diffusion model is based on a random-walk picture in which the protons in the molecules of a liquid at the surface of a confining region eperience an enhanced relaation probabilit per unit time. This reaction-diffusion equation is; [ M ] D t [ M ] [ M ] T b (3.) with initial and boundar conditions, [M] [M o ] at t and Dn. [ M ] ρ[ M ] where [M] is the magnetiation per unit volume, ρ is the surface relaivit and D is the diffusion coefficient. The magnetiation profile can be obtained b solving the differential equation. The total magnetiation M ( t) at a given time can be calculated b integrating the magnetiation over the whole volume. If the structure of the porous medium is known then a direct solution of equation (3.) can be used to estimate the magnetiation deca. The reaction-diffusion equation (3.) can be solved b finite-difference method or the random walk method. The finite difference approach ma require large computer memor and long computation time especiall if the porous medium has a complicated geometr. For simpl shaped pores, such as a sphere, a clinder, cube, or rectangular parallelepiped, analtical solutions can be derived for the magnetiation deca. An analtical solution for the magnetiation deca of a fluid in a spherical pore has been proposed b Crank (975). The magnetiation deca as a function of time in a spherical pore of radius r is given as: M ( t) M ep t Dt ( β n / r ) ( Bn + ) TB n Bn 6 ep (3.)

37 where β is the root of β cot + and ρr / D. In the fast diffusion limit n n β n where ( << ), the magnetiation follows a single eponential deca (Senturia and Robinson, 97), i.e. ( ) ep t M t M (3.) T M is the initial transverse magnetiation at time t and the deca rate T is given b, T T B ρs + V (3.3) which is a combination of the bulk and surface relaation mechanisms. S/V for a sphere is 3/r, equation (3.3) is also valid for other simpl shaped pores. From equation (3.3), for a spherical pore, we have; T 3ρ + (3.4) T r B the surface area to volume ratio (S/V) is a measure of pore sie, thus the transverse relaation time, T is directl proportional to pore sie. In realit, a porous medium will comprise of several interconnected pores of different sies and shapes, as such, a multieponential epression representing a distribution of T constants is much more suitable for describing the magnetiation deca of a real porous medium (Kenon, 997). M () t M n t ai ep (3.5) i Ti a is the volume fraction of pores of sie i that deca with relaation timet. Usuall i a logarithmic mean is used to determine the mean transverse relaation time, T. 3 i lm

38 T lm ai logti ep (3.6) ai 3.3. Modelling surface relaation The reaction-diffusion equation (3.) was solved b a random walk method (Ramakrishnan et al. 998). This method is based on simulating the Brownian motion of diffusing particles called the random walkers. Initiall, the walkers are placed randoml in the pore space. At each time step, the walkers diffuse from their initial position [ () t, () t, () t ] to a new position [ ( t + t) ( t + t), ( t + t) ], on the surface of a sphere with radius, s centred on their initial positions. The time it takes to diffuse this distance s is given b; s t (3.7) 6D The new position of the walkers after each time step is given b; ( t t) ( t) ssinφ cosθ + + (3.8) ( t t) ( t) ssinφ sinθ + + (3.9) ( t t) ( t) s cosφ + + (3.) the angle θ is randoml selected in the range ( θ π ) and φ is randoml selected in the range ( φ π). If the walker encounters a solid surface, it is killed with a finite killing probabilit δ. The killing probabilit δ is related to the surface relaivit (Bergman et al. 995) b; ρs δ (3.) 3D 4

39 If the walker survives, it simpl bounces off the surface and returns to its previous position and time is advanced b the time step given in equation (3.7). Since the walkers move with equal probabilit in all directions, this algorithm is valid because there is no magnetic gradient in the sstem. At each time step, a record of the fraction of the walkers alive P () t is kept. N t P( t) (3.) N N t is the number of walkers alive at time t and N is the initial number of walkers at time t. ( t) P is analogous to the normalied amplitude due to surface relaation mechanism in NMR measurements, as such: () t ( t) M P (3.3) M The random walk solution method can be validated with the analtical solution for the magnetiation deca of a fluid in a spherical pore. The analtical solution for the magnetiation deca as a result of surface relaation mechanism alone of a fluid in a spherical pore in fast diffusion limit ( << ) is given b: M M ( t) ep 3ρt r (3.4) A direct substitution of the surface relaivit, ρ and the radius, r of the spherical pore into equation (3.4) gives the magnetiation deca as a function of time. For the random walk solution, the diffusion coefficent of the fluid contained in the pore is required in addition to the radius and surface relaivit of the pore. Using a spherical pore of radius 5µm, surface relaivit of µm/s, a fluid diffusion coefficient of.5-9 m /s and an initial number of, walkers, the magnetiation deca is simulated using the random walk solution method. The ratio of the number of walkers alive to the initial number of walkers, P(t) was recorded at each time interval. 5

40 Normalied Amplitude. Analtical Random Walk Time (s) Figure 3-3. A comparison of the magnetiation deca of a fluid in a spherical pore obtained b random walk method and the analtical solution. A plot of this ratio P(t) against time (t) is the magnetiation deca due to surface relaation mechanism alone. The dimensionless constant defined as ρr / D, in this analtical solution above is.4. From Figure 3-3, it is evident that the random walk solution is consistent with the analtical solution thereb validating the random walk solution method. A further advantage of the random walk solution method is that it is not onl limited to simple geometrical shapes, it can be used to simulate magnetiation decas in highl complicated geometries as we have in the comple pore spaces of 3D images Modelling bulk relaation Bulk relaation is modelled b multipling the magnetiation deca due to surface relaation mechanism and relaation due to diffusion in magnetic gradients b the eponential deca; b () t t ep (3.5) TB 6

41 3.3.4 Modelling diffusive relaation The additional deca of the amplitudes of spin echoes as a result of diffusion in magnetic field gradients depends on the time interval between these spin echoes t. This additional deca can be epressed as a function of time and as a multiplicative factor, (Kenon, 99); g ( S t) () t ep γ G D ( τ ) (3.6) where t is time, D is the molecular self-diffusion coefficient of the fluid, γ is the gromagnetic ratio of the proton and G S is the spatial gradient of the magnetic field intensit. This deca becomes rapid as the spacing of the echoes t increases. When the echo spacing is ver short, relaation due to diffusion in magnetic field gradients is negligible. In order to incorporate this relaation mechanism in NMR simulations, the eperimental values of the spatial gradient of the magnetic field G S and the echo spacing τ must be known. This is required if comparison of simulation results with eperimental data is desired Modelling relaation due to all mechanisms Since all the relaation mechanisms occur simultaneousl and independentl, the total magnetiation amplitude as a function of time is given as (Kenon, 99); ( t) M [ P( t) g( t) b( t) ] M (3.7) where P(t), g(t) and b(t) are given b equations (3.), (3.6) and (3.5) respectivel. 7

42 4. Simulation of NMR response in 3D images The capabilit of the random walk method to simulate magnetiation deca in comple pore geometries has enabled the pore scale simulation of NMR response in discretied images of reservoir rocks. Simulation of magnetiation deca using the random walk method is significant when surface relaation mechanism is dominant because the random walk simulation method onl simulates the surface relaation mechanism of the NMR response. Bulk relaation and relaation due to diffusion in magnetic gradients are independent of the diffusion paths of the random walkers. Hence, if these relaation mechanisms are dominant, then NMR response ma not offer good insight about the pore sie distribution of the porous medium. Discretied images obtained from micro-ct imaging have to be processed to enable the grains to be clearl differentiated from pores. A suitable method of achieving this is to convert the image in gra scale into binar format and denote the pores as and the grains as or vice versa as shown in Figure 4-. In order to ensure that the discretied image is representative of the whole sample from which it was taken, its porosit can be compared to the porosit of the parent sample. Figure 4-: (a) Micro-CT image in gra scale is converted to a binar format (b) where pores are clearl distinguished from grains. 8

43 4. Geometr of the micro-ct image The geometr of the pore spaces in micro-ct images are quite comple. These compleities are captured b the resolution of the micro-ct images. The inputs into the simulation code are the resolution of the image and the binaried image where and represents the pore and grain voels respectivel. The porosit of this binaried image is determined b simpl counting the number of pore voels and dividing b the total number of voels in the image. 4. Initialiation of walkers in the micro-ct image Each pore voel is assigned a unique identification number; equal numbers of walkers are then placed randoml in each pore voel so as to ensure a uniform coverage of the entire pore space of the image. Assigning a high number of walkers to each pore voel can be computationall epensive; for eample, a 3 3 voel image of porosit 33% will have 9 million pore voels and if walker is assigned to each pore voel, we will have 9 million walkers. This will result in longer simulation time. Alternativel, we can limit the number of walkers to be assigned to the pore voels - in this case the walkers are assigned to randoml selected pore voels. 4.3 Diffusion of walkers in the micro-ct image The unit distance diffused b the walkers at each time step is chosen as a function of the resolution of the image: s. R where R is the resolution of the image with s substituted in equation (3.7) to determine the time step, t. This will ensure that walkers take significant number of steps before encountering a grain surface. Each walker can either diffuse within its respective pore voel or into an of the neighbouring pore voels. A walker in a pore voel which is not at the boundar of the image has 6 other voels surrounding it, as shown in Figure 4-. These surrounding voels ma include both grain and pore voels. A walker in pore voel X shown in Figure 4- can diffuse into an of the neighbouring pore voels depending on its respective, and -coordinates. When a walker diffuses out of its voel, the new voel it enters is determined based on its, and -coordinates as shown in Appendi A-. 9

44 X Figure 4-: Figure showing the twent-si voels that a walker can diffuse into, if the walker is in voel X. (a) Shows the voels in the negative -ais i.e. ( < ), (b) shows the voels surrounding the reference voel X, i.e. ( < < ) while (c) shows the voels in the positive -ais, i.e. ( > ) where is the resolution of the image. This new voel is checked whether it is a pore voel or a grain voel. If it is a pore voel, the walker diffuses into that voel, and if it is a grain voel, the killing probabilit of the walker is calculated from equation (3.) to determine if the walker survives or not. If the walker survives, it is returned to its previous voel and position. The fraction of walkers alive at a given time interval P(t) is then recorded against their respective times. This is the normalied magnetiation deca due to surface relaation alone. The relevant eperimental and fluid parameters are substituted along with P(t) into equation (3.7) to generate the magnetiation deca due to all relaation mechanisms. When a walker in an pore voel diffuses out of the entire image, a new walker is placed in a randoml selected pore voel on the opposite face of the image. This will lead to a conservation of the total number of walkers in the image at an given time. 4.4 Calculation of grain surface areas Magnetiation deca is highl influenced b the surface area available to rela the protons (walkers). Magnetiation will deca more rapidl if we have a high surface area to rela a given volume of fluid than a small area to rela the same volume of fluid. Hence, it is necessar to calculate the total grain surface area in a micro-ct image. This will assist in providing suitable eplanation for differences in the simulation results of an arra of micro-ct images. 3

45 A voel has si surfaces each with an area of R the adjoining si voels to each pore voels are checked if the are grain voels. All faces of the grain voels adjacent to pore voels are summed and multiplied b the surface area of each side of the voel as shown in equation (4.). W T N G R (4.) where W T is the total grain surface area, N G is the total number of the faces of grain voels adjacent to pore voels and R is the resolution of the image. 4.5 Validation of the simulation algorithm The algorithm for the simulation of magnetiation deca in micro-ct images was validated b comparing the magnetiation deca of a cubic pore with its equivalent discretied image. For this cubic pore with dimension d, we assign walkers randoml within the pore, a walker encounters a solid surface when an of its, or coordinate is negative or greater than the dimension of the cube. Simultaneousl, we discretie the pore space into voels with the surfaces as the grain voels. For a cubic pore of length 48µm, we can discretie the pore into voels. The pore voels are actuall voels; the remaining voels are the boundar grain voels. The resolution of this discretied image is µm, so that it is of the same pore dimension as the cubic pore. The magnetiation deca of this cubic pore and its equivalent discretied image was simulated b using fluid with a diffusion coefficient of. -9 m /s, pore surface relaivit of 5-6 m/s, time step of 3.3µs and, walkers as shown in Figure 4-3. From this figure, it is shown that the magnetiation deca of the cubic pore is consistent with that of its equivalent discretied image thereb validating the algorithm for the simulation of magnetiation deca in 3D images. 3

46 Normalied Amplitude. Discretied Image Cubic Pore..5.5 Time (s) Figure 4-3: Comparison of the magnetiation deca of a cubic pore with its equivalent discretied image. 4.6 Simulation Results Poorl consolidated sandstone, S was used to investigate the effect of sample sie and number of walkers on magnetiation deca. This sandstone has a porosit of 4.6%, resolution of 5µm and a voel sie of 3 3. The magnetiation decas of three voel sies, 3, 3 and 3 3 shown in Figure 4-4 were simulated..5mm (a) (b) (c) Figure 4-4: (a) Micro-CT image of sandstone S with a voel sie of 3 (b) voel sie 3 and (c) voel sie

47 The 3, 3 voel images were cut from the central subsections of the original image of voel sie 3 3. From the comparisons of the simulated magnetiation decas shown in Figure 4-5, it can be observed that the magnetiation of the 3 and 3 3 images decas faster than that of the 3 image. This is due to slightl microstructural differences in these images; however the decas of the 3 and 3 3 images are consistent with each other. Normalied Amplitude. S_ S_ S_ Time (ms) (a) S_ S_ S_3 Frequenc.3.. T (ms) (b) Figure 4-5: (a) The magnetiation decas of 3, 3 and 3 3 voel images, S_, S_ and S_3 respectivel and (b) their corresponding T distributions. 33

48 The T distribution of the 3 voel image is narrower than those of the other images. This is as a result of the small sie of the image thereb preventing it from having a wide variation of pore sies. The largest image that can sample a wider range of pore sie has the broadest T distribution. The algorithm for the inversion of T distributions from magnetiation decas is described in Appendi A-3. The effect of the number of walkers on magnetiation deca was also investigated on the 3 3 voel image. Normalied Amplitude..5 Million Million. 3 Time (ms) Figure 4-6: Comparison of the magnetiation deca of the 3 3 voel image of S using.5 million and million walkers. The magnetiation deca obtained using.5 million walkers is consistent with that obtained using million walkers as shown in Figure 4-6; thus the number of walkers does not reall have a significant effect on the magnetiation deca. However, sufficient walkers must be used so as to ensure that the walkers diffuse through the entire pore space. 34

49 5. Simulation of NMR response of single-phase fluids in networks The random walk method is etended to simulate magnetiation deca in networks. This is of significant importance because it will enable a suitable comparison with the magnetiation deca of the micro-ct image from which the network was etracted. It also allows the simulation of NMR response for multiphase fluid distributions see chapter 6. Magnetiation deca has been used to validate the reconstructed microstructure of Fontainebleau sandstone b comparing the simulation results of this reconstructed microstructure with that of a micro-ct image of similar porosit (Øren et al., ). Magnetiation deca is primaril influenced b the surface areas available to rela the proton spins, the volume of the fluid and surface relaivit of the porous medium. However, just because two pores have the same surface area, surface relaivit and volume does not necessaril mean that the will have the same magnetiation deca. The will have the same magnetiation deca if the have the same relative paths available for the protons to diffuse through, which are similar to the paths available for fluid flow. If a network is a good representation of the micro- CT image from which it was etracted, then the two magnetiation decas and T distributions must be similar. 5. Geometr of network elements The volume V, inscribed radius r and shape factor, G of each network element (pore or throat) was used to determine its length. From the equation relating the crosssectional area A, with the shape factor G and inscribed radius r of a pore (Mason and Morrow, 99): and the volume given b: r A (5.) 4 G V A (5.) 35

50 For an network element having a shape factor between and 3/36 (shape factor of an equilateral triangle), we determine the three half angles β, β, β 3 of the triangular cross-section of that element. For a given shape factor, the corner half angles β can take a range of values (Patek and Silin, ) where β < β < β 3 ; first β is chosen randoml within the allowed range: and (- 3G ) acos 4π β,min atan cos + (5.3) (- 3G ) acos β,ma atan cos (5.4) 3 3 with β given b: β tanβ + 4G β + asin sin β tan β 4G (5.5) and β 3 determined from: β / β β (5.6) 3 π For each network element with the shape factor of a triangle, we determine its three geometrical properties, these are the half angles β, β, β 3, the height H and the base length of the triangular cross section. For a network element having the shape factor of a square, we divide its volume b its cross sectional area using equations (5.) and (5.) in order to determine its length. 5. Initialiation of walkers in networks We define a walker densit, this is the number of walkers to be assigned to a given volume of the element. This will ensure a uniform distribution of walkers in all the network elements at initial time, t. A walker is assigned two data sets, the first is its positional data, which includes the element tpe (throat or pore) that the walker is embedded in, the inde number (which differentiates each network element), and the, and -coordinate of the walker. 36

51 All clindrical elements in the network are replaced with square shaped elements. This will reduce the compleit of the simulation procedure without having an impact on the magnetiation deca. This is due to the fact that circular shaped elements constitute onl a small proportion of the entire network elements. For triangularl shaped network elements, we place the walkers within the two rectangular sections shown in Figure 5- b generating their coordinates (,, ) randoml. Rectangular Section Section β 3 H - ais β β B 3 Figure 5-: The geometr of a tpical triangular element in a network. When the -coordinate of a walker is less than, the walker is in rectangular section, the arctangent of (/) is then determined, if it is less than β, the walker is placed in that location (i.e. its, and values are retained) if not, we determine another random location (,, ) for the walker until we get an arctangent of (/) that is less than the angle β. Similarl if the -coordinate of the walker is greater than, that means that the walker is now in rectangular section, we determine the arctangent of (/( B3 )) in this case, if it is greater than the angle β, that means that the random walker is outside the triangle and as such another random location is determined for the walker. This approach enables the walkers to be uniforml distributed within the triangular cross section. In all these cases the -coordinate is determined randoml based on the length of the network element. 37

52 To place walkers randoml within a square cross section, we place them randoml in the, and -ais where there limits are the dimension of the square cross section and the length of the network element determined in the geometr section. 5.3 Diffusion of walkers in networks It is assumed that a pore with a coordination number (number of throats connected to the pore) n, has half of the throats connected to the pore at, and the other half connected to the opposite face at as shown in Figure 5-. A walker in this pore will diffuse into an of the connected throats (pore-to-throat diffusion), if the - coordinate of the walker is negative or greater than the length of the pore; the walker then diffuses into a randoml selected throat connected to that pore. The -coordinate of the walker in this new throat is determined from its previous value to ensure continuit along the diffusion paths. The shape factor of this new throat is then checked to determine its cross section since it will influence how the walker will be placed randoml in its XY plane. Pores (a) Throats (b) Figure 5-: (a) Throats are connected to pores at different locations within the pore section (b) in the simulation algorithm. It is assumed that half of the throats connected to a pore at one end, while the remaining throats are connected at the other end. The positional data of this walker in a new throat now becomes: throat inde,, coordinates (determined randoml) and -coordinate (determined from its previous value). 38

53 This pore-to-throat diffusion of a walker is also valid for throat-to-pore diffusion ecept that a throat is now connected to onl one pore each at its opposite ends as shown in Figure 5-3. Throat Pore Pore Figure 5-3: A triangular shaped throat connected to the two ends of triangular shaped pores. 5.4 Relaation of walkers in networks For all the network elements there are 3 sides available for relaing the walkers in a triangular element as shown in Figure 5-4, and 4 sides for a square element. A walker encounters a solid surface when an of the following conditions are satisfied. β 3 Face B B Face β B 3 Face 3 Figure 5-4: The three faces available for relaing walkers in triangular elements. β 39

54 Face A walker hits face when its -coordinate falls within the interval < < B3, its - coordinate within the interval < < H and the tangent of (/( B3 )) is greater than the angle β. A record of the number of walkers that deca on this face is kept as D. Face A walker hits face when its -coordinate falls within the interval < <, its - coordinate fall within the interval < < H and the tangent of / is greater than the angle β. A record of the number of walkers that deca on this face is kept as D. Face 3 Since face 3 has the largest length we epect the majorit of the walkers to die on this face. A walker hits face 3 when its -coordinate is negative. A record of the number of walkers that deca on this face is kept as D 3. Square Shape: A walker will hit an of the four sides of the square cross-section when an of the or -coordinate of the walker is negative or greater than the dimension of the cross section. We also keep a record of the number of walkers surfaces of this square cross-section. D S that die on an of the Relaation: When a walker comes in contact with a solid surface, a random number is generated to determine if the killing probabilit is honored or not. If the walker survives, it is returned to its previous position. The fraction of walkers alive at selected time intervals is then recorded. The bulk relaation component is then applied to determine the magnetiation deca due to surface and bulk relaation mechanisms. A flow chart showing the algorithm for the simulation of NMR response of single-phase fluid in a network is described in Appendi A-4. 4

55 5.5 Validation of the simulation algorithm To validate the simulation algorithm, the surface area of each of the sides of the triangular cross section is determined. For each triangular element, the smallest side is B, the longest side is B 3, with B as the middle value as shown in Figure 5-4. Their respective surface areas, A, A and A 3 are determined from: A B (5.7) A B (5.8) A 3 3 B (5.9) where is the length of the network element determined from equations (5.) and (5.). For the square cross section we also determine the total surface area of that particular element. A S 4B (5.) where B is the dimension of the square cross section. The areas of each side of all the triangular elements are summed as: n A T A i (5.) i n A T A i (5.) i n A T 3 A3 i (5.3) i where A T, A T, A T3, are the total surface areas of sides B, B and B 3, respectivel while n is the number of triangularl shaped network elements. Similarl for square shaped elements, the total surface area A TS, is given as: m A TS A Si i (5.4) where m is the number of square shaped network elements. These total surface areas are then epressed as fractions. 4

56 F F A T A, (5.5) AT + AT + AT 3 + ATS A T A, (5.6) AT + AT + AT 3 + ATS F A T 3 A3, (5.7) AT + AT + AT 3 + ATS F AS A T + A T A TS + A T 3 + A TS, (5.8) The fraction of walkers F D, F D and F D3, that decas on sides B, B and B 3, respectivel are determined as: D FD, (5.9) D + D + D + D 3 S D FD, (5.) D + D + D + D 3 S D3 FD3 (5.) D + D + D + D 3 S F DS DS D + D + D 3 + D S (5.) It is epected that FA FD, FA FD, FA3 FD3 and FAS FDS because the deca of walkers depends to a large etent on the surface areas available to rela them. Thus, higher surface areas are epected to rela a higher number of walkers and vice versa. This comparison also helps to ascertain if the walkers full diffuse through the network and encounter the entire surface area. 4

57 6. Simulation of NMR response of two-phase fluids in networks Comparison of the magnetiation deca in a network with that of the micro-ct image from which the network was etracted, helps to check if the network is representative of the image. Such successful comparisons are the basis for the etension of the algorithm to model magnetiation deca of two-phase fluid in networks. For singlephase simulations, we onl have to deal with the contact of the diffusing protons (walkers) with the grain surface. On the other hand for two-phase simulations we have to deal with contacts with solid surfaces and fluid interfaces. Unlike single-phase flow where the fluid completel saturates the network, for two-phase situations, the saturation and configuration of each fluid phase in each element are required. From predictive pore scale modelling, the fluid saturations and configurations in each element can readil be determined (Valvatne and Blunt, 4). 6. Initialiation of walkers in the fluid phases At an given water saturation during drainage, some elements would have been invaded b oil, while the others are uninvaded. Oil Oil Water (a) (b) (c) Figure 6-: (a) The curved areas occupied b water in corners are replaced b (b) equivalent triangles. (c) Uninvaded element. 43

58 Invaded polgonal elements will have two phases as shown in Figure 6-. Water retained in corner i of a polgonal element will have area ( θ + β ) A i given b: cosθ r cos r i π A i R + θ r + β i (6.) sin β i In order to reduce the compleit of taking into consideration the curved fluid interface at each corner during the diffusion of walkers both in the oil and water phase, the water in these corners are assumed to have a triangular shape as shown in Figure 6-b while preserving the corner areas in equation (6.). We assign specified number of walkers, N w in the water phase and as water walkers and in the oil phase oil walkers. N o in the oil phase. We refer to walkers in the water phase For an uninvaded element, we assign water walkers based on the method described for the initialiation of walkers for a single-phase fluid in the previous section. In an invaded element, we assign oil walkers to the polgonal section occupied b the oil phase, while water walkers are assigned to the water in the corners of the element. Similarl at an given water saturation during imbibition, uninvaded triangular shaped elements will have the fluid configuration shown in Figure 6-a while invaded triangular shaped elements can have the fluid configuration shown in Figure 6-c or have oil laers in corners as shown in Figure 6-a. Water Water Figure 6-: (a) The oil laers formed in corners are replaced b (b) equivalent rectangles. 44

59 The oil laers formed in corners of polgonal elements are represented b polgonal sections. Oil walkers are assigned to the oil laers and water walkers are assigned to the sections occupied b water. 6. Diffusion of walkers in the network Since water and oil have different diffusion coefficients, their respective walkers diffuse at different speeds. At each time step t, the oil walkers diffuse from their initial position, [ () t, () t ( t) ] to a new position, [ ( t + t) ( t + t), ( t + t) ] o o, covering a distance of ε o defined b: o o,, o o ε 6 t (6.) The new position of the oil walkers after each time step is given b: o D o o ( t t) ( t) ε sinφ cosθ + + (6.3) o o ( t t) ( t) ε sinφ sinθ o o + + (6.4) o ( t t) ( t) ε cosφ o o + + (6.5) Similarl the water walkers diffuse from their initial position [ () t () t ( t) ] new position [ ( t + t) ( t + t), ( t + t) ] w o, to a w w,, w w, covering a distance of w w ε defined b: ε 6 t (6.6) The new position of the water walkers after each time step is given b: w D w w ( t t) ( t) ε sinφ cosθ + + (6.7) w w ( t t) ( t) ε sinφ sinθ w w + + (6.8) w ( t t) ( t) ε cosφ w w + + (6.9) w 45

60 D o and D w are the diffusion coefficients of the oil and water phase respectivel, the angle θ is randoml selected in the range ( θ π ) and φ is randoml selected in the range ( φ π). If an oil walker encounters a solid surface, it is killed with a finite killing probabilit, δ o defined b: ρ ε o o δ o (6.) 3Do Similarl when a water walker encounters a solid surface, it is killed with a finite killing probabilit, δ w also defined b: ρ ε w w δ w (6.) 3D w where ρ o and ρ w are the surface relaivities of the porous medium to oil and water and oil respectivel. The relaivit of a given surface to oil is much less than that of water, the ratio ρ / ρ is often in the range. to.4 (ooestijn and Hofman, 5). o w If an walker survives upon contact with a solid surface, it simpl bounces off the surface and returns to its previous position and time is advanced b the unit time step t. When a walker hits a fluid interface, we place the walker randoml in its respective phase within the element it is embedded in, because of negligible relaivit at the fluid interface (Toumelin et al., 3). When ( t + t) or ( t t) o w + of a walker in a pore is negative or greater than the length of its element, the walker diffuses into one of the connected throats with the same fluid phase as that of the walker. If the neighbouring element does not contain the walker s phase, then the walker has encountered a fluid interface and it is put back into a random position within the element. A similar thing occurs for diffusion of walkers from throats to pores. At each time step, a record of the fraction of the walkers alive in each fluid phase is kept. N P ot o ( t) (6.) N o 46

61 P o () t and ( t) respectivel while N P wt w ( t) (6.3) N w P w are the fraction of walkers that are alive in oil and water phase N ot and N wt are the actual number of walkers alive in the water and oil phase respectivel. Equations (6.) and (6.3) are the normalied magnetiation deca due to surface relaation in the oil and water phase respectivel. The total magnetiation deca as a result of both surface and bulk relaations in each fluid phase is obtained b incorporating into equations (6.) and (6.3) the bulk relaation component. M o () t and () t respectivel while M M o w t ep (6.4) () t P () o t TBo t ep (6.5) () t P () w t TBw M w are the total magnetiations at time (t) in oil and water phase T Bo and T Bw are the bulk relaation time of oil and water respectivel. The combined magnetiation M ( t) of the two phases is given b: M () t S w M w ( t) + H S M ( t) S w + H o o S o o o (6.6) where S o and S w are the oil and water saturations respectivel, H o in equation (6.6) is the hdrogen inde of the oil-phase (Toumelin et al., 7). The hdrogen inde of a fluid is a measure of its hdrogen content per unit volume. Hdrogen inde of oil is dependent on its specific gravit and is usuall determined from; H o. 6 ρ (6.7) o where ρ o is the specific gravit of the oil (Smolen, 996). 47

62 7. Single-phase simulation results for sand packs In this chapter, we compare the simulated NMR response on sand packs with eperimental measurements. The eperiments were performed in the Petrophsics aborator at Imperial College b m colleagues, Dr Stefan Iglauer, Saif AlSaari and Saleh Al-Mansoori. 7. Sand preparation Quart sands, V6 (eavenseat sand, WBB Minerals, UK) and Ottawa F4 (unground silica, US Silica Compan) were used in this stud. The grain sie distributions of the sands were determined b sieving them for 8 minutes using British standard meshes on an electric shaker. The mass fraction of grains that passed through each sieve sie is plotted against their respective sieve sie as shown in Figure F4 V6 Mass Fraction Sieve sie (µm) Figure 7-: Grain sie distributions of the F4 and V6 sand packs The sands were then prepared for the NMR eperiments b carefull pouring into a cm length of a thermo-plastic heat shrink sleeve of diameter 3.8 cm with fitted plastic end caps with one piece of circular filter paper to ensure a tight fit. The sleeve was filled with sand to about two thirds of its length, the sample was then tapped and vibrated using an electric vibrator to ensure compaction, the sleeve was filled to the top and the compaction process is repeated. 48

63 The sand pack sample is then placed in an oven at 8ºC for minutes. It is allowed to cool and placed in an oven at ºC to seal the end caps after which the sample is weighed. Two samples each of V6 and F4 sand packs were made for the NMR eperiments to ensure good reproducibilit of our packing procedures and the measurements were repeated thrice. 7. Sand properties The porosit of F4 sand pack is 35.4% ±.3% while that of V6 is 37.% ±.%. Formation factors were determined from electrical resistivit measurements. The eperimental formation factor is 4.8 and 5. for the V6 and F4 sand packs respectivel. Permeabilities of the sand packs were measured on similar but longer columns of cm diameter and m length; the values are 4.D ± 4. D for F4 and 3.D ±.3D for V NMR measurement As mentioned previousl, sand packs 3.8 cm in diameter and cm in length saturated in brine were used for NMR measurement. The brine was a miture of deionied water, (5wt% NaCl and wt% KCl solution). Potassium Chloride (KCl) was used to prevent the sand from swelling, while Sodium Chloride (NaCl) was added to increase the ionic strength of the water to levels found in oil reservoirs. The densit of the brine was measured to be 35 kg/m 3. NMR relaation measurements were performed using a MARAN bench top spectrometer (Resonance Instruments) at a temperature of 38K operating at MH. The Carr-Purcell-Meiboom-Gill (CPMG) sequence was used for transverse relaation measurements ( T -measurements) with an inter-echo spacing of µs. The short inter-echo time was selected to minimie relaation due to diffusion in internal field gradients. A single data point was acquired at the center of each echo, with 3, data points being collected. The T relaation time distributions were obtained from the magnetiation decas b using the curvaturesmoothing regulariation method (Chen et al., 999). 49

64 7.4 Micro-CT imaging In order to ensure the consistenc of our simulation results, si small sand pack samples, three for each of sand pack tpe, F4A, F4B and F4C for the F4 sand pack and V6A, V6B and V6C for V6 sand pack were made for micro-ct imaging as shown in Figure 7-. The samples are.65 cm in diameter and 4 cm in length. The were packed in a similar manner as the plugs used in the eperiments described above; analsis of the resultant images (see below) confirms that the have similar porosit as the plugs used in the eperiments. The micro-ct imaging was performed on a commercial XMT unit (Phoeni X-ra Sstems and Services GmbH). Scanned images of voels, with resolutions ranging between 8 µm and µm were collected for each sample with the images cropped digitall to remove the edge artifacts. (a) (b) (c) mm (d) (e) (f) Figure 7-: 3D micro-ct sections of (a) V6A, (b) V6B, (c) V6C, (d) F4A, (e) F4B and (f) F4C. 5

65 7.5 Eperimental results The eperimental results shown in Figures 7-3 and 7-4 demonstrate reproducibilit between different packs of the same sand. The magnetiation of V6 decas faster than F4 and hence the mean transverse relaation time, T for F4 is higher than that of V6. Normalied Amplitude. V6X V6Y..3 3 Time (ms) (a) V6X V6Y. Frequenc. T (ms) (b) Figure 7-3: (a) The eperimental magnetiation deca of two samples (V6X and V6Y) of the V6 sand packs. (b) The inverted T distribution of the magnetiation decas in (a). 5

66 The chemical compositions of the sands are similar; the difference in their magnetiation decas arises from their different pore sie distributions: the faster deca of the V6 sand implies smaller pores this is consistent with the smaller grain sie distribution shown in Figure 7-. Normalied Amplitude. F4X F4Y. 3 Time (ms) (a).3 F4X F4Y Frequenc.. T (ms) (b) Figure 7-4: (a) The eperimental magnetiation deca of two samples (F4X and F4Y) of F4 sand packs. (b) The inverted T distributions of the magnetiation decas in (a). 5

67 In the T distributions, frequenc is plotted against T values. The differences between the magnetiation decas of V6 and F4 sand packs are as a result of their grain shapes and sies which are responsible for their different pore sie distributions. The micro-ct images shown in Figure 7- indicate that the grains of the F4 sands are larger, more spherical and have smooth surfaces than those of V6. The grains of the V6 sand pack on the other hand, have rough surfaces with a wider variation of sies, as shown b the grain sie distributions in Figure Simulation results NMR Simulations were carried out on the si micro-ct images, F4A, F4B and F4C for the F4 sand pack and V6A, V6B and V6C for V6. Simulations were also performed on networks etracted from these images. The networks were etracted from the micro-ct images b using a maimal ball algorithm, (Silin et al., 3; Silin and Patek, 6; Al-Kharusi and Blunt, 7; Dong, 7). Figure 7-5 shows eample images and networks for the two sands. The micro-ct images and etracted networks cover a rock volume of 3 3 mm 3. Table 7- shows the resolution, voel sie, porosit, total grain surface area per unit volume of each micro-ct image. Sample Resolution (µm) Voel Sie Porosit (%) Surface Area (m /m 3 ) Pores Throats Coordination Number F4A ,77,46, F4B ,93,33 3,6 4.8 F4C ,76,434 3, V6A ,67 3,35 7, V6B ,9 3,5 9,44 5. V6C ,59 3,58 9, 4.9 Table 7-: Micro-CT images and etracted networks properties 53

68 Table 7- also shows the number of network elements (pores and throats) and coordination number of the networks etracted from each micro-ct image. The porosit of the packs inferred from the images is similar to that obtained b direct measurement on larger packs, although we tend to under-estimate the porosit of the F4 sand (33% from the images as opposed to 35% measured directl.) mm (a) (b) (c) (d) Figure 7-5: (a) Sections of the micro-ct image of V6A sand pack and (b) F4A sand pack. The resolution of both images is µm, (the dark areas are the pore spaces and the white represent grains). (c) Equivalent network etracted from V6A. (d) Network etracted from F4A. 54

69 7.6. Simulation Parameters The diffusion coefficient and the bulk relaation time of brine used in the simulation were determined from the correlations (Vinegar, 995): and T D( ). brine 98( η) (7.) 3 T 3 T ( ). B brine 98( η) (7.) where D ( brine) is the diffusion coefficient of the brine in m /s, T is the temperature of the brine in K, η is the viscosit in Pa.s and T B( brine) is the bulk relaivit of brine in seconds. Since the eperiment was performed at 38K, this corresponds to a diffusion coefficient of.7-9 m /s and bulk relaivit of 3. seconds using the eperimental brine viscosit of. Pa.s. Relaation due to diffusion in magnetic gradients is neglected in the simulations because of the small inter echo spacing used in the NMR eperiments, hence onl surface and bulk relaations were accounted for Surface Relaivit The surface relaivit (4-6 m/s) used in our simulations was obtained b adjusting the value until a match was obtained with the magnetiation deca of the eperimental data. The same surface relaivit was used to match the eperimental data for the two sand packs since the are made up of the same mineral (quart) and the have similar chemical composition. The relativel high value of surface relaivit used, compared to literature values in the range 9 to 46 µm/s for sandstones (Roberts et al., 995) and.89 to 3.6 µm/s for silica sands (Hinedi et al., 997) is due to two reasons. First, the sand packs used in this work has minute proportions of iron oide within the range.% to.6% along with some other paramagnetic substances. Second, and more significantl, the overall surface deca is given b the apparent surface area multiplied b the surface relaivit. In our work the surface area is determined at the resolution of the micro-ct image, around µm, whereas in other work (Hinedi et al., 997) this is determined using nitrogen absorption method, which probes the surface at a much higher resolution, resulting in a larger measured surface area per unit volume. Hence to obtain the same magnetiation, a lower apparent surface relaivit was obtained. 55

70 7.6.3 Magnetiation Spectrum F4 Sand Packs The simulated magnetiation decas and T distributions of the F4 samples (F4A, F4B and F4C) are compared with the eperimental data of F4Y as shown in Figures 7-6, 7-7 and 7-8. Normalied Amplitude. F4A Eperimental Micro-CT Simulation Network Simulation. 3 Time (ms) Frequenc F4A (a) Eperimental Micro-CT Simulation Network Simulation. (b) Figure 7-6: (a) Comparison of the eperimental magnetiation deca of F4Y with the micro-ct image and etracted network for F4A. (b) Comparison of the inverted T distributions from the magnetiation decas in (a). T (ms) 56

71 Normalied Amplitude. F4B Eperimental Micro-CT Simulation Network Simulation. 3 Time (ms) (a).5.4 F4B Eperimental Micro-CT Simulation Network Simulation Frequenc.3.. T (ms) (b) Figure 7-7: (a) Comparison of the eperimental magnetiation deca of F4Y with the micro-ct image and etracted network for F4B. (b) Comparison of the inverted T distributions from the magnetiation decas in (a). 57

72 Normalied Amplitude. F4C Eperimental Micro-CT Simulation Network Simulation. 3 Time (ms).5.4 F4C (a) Eperimental Micro-CT Simulation Network Simulation Frequenc.3.. T (ms) Figure 7-8: (a) Comparison of the eperimental magnetiation deca of F4Y with the micro-ct image and etracted network for F4C. (b) Comparison of the inverted T distributions from the magnetiation decas in (a). (b) In all cases there is good agreement between the eperimental measurement and the simulations on the micro-ct image and the network. However in the F4C sample, the network results tend to give a more rapid deca, indicating a lower estimated average pore sie. This is due to the more constricted diffusion in slit-like triangular elements that allows a low mean-free-path before encountering the grain. 58

73 V6 Sand Packs The simulated magnetiation decas and T distributions of the V6 samples, (V6A, V6B and V6C) are compared with the eperimental data from V6Y as shown in Figures 7-9, 7-, and 7-. Normalied Amplitude. V6A Eperimental Micro-CT Simulation Network Simulation. 3 Time (ms) (a).5.4 V6A Eperimental Micro-CT Simulation Network Simulation Frequenc.3.. T (ms) (b) Figure 7-9: (a) Comparison of the eperimental magnetiation deca of V6Y with the micro-ct image and etracted network for V6A. (b) Comparison of the inverted T distributions from the magnetiation decas in (a). 59

74 Normalied Amplitude. V6B Eperimental Micro-CT Simulation Network Simulation. 3 Time (ms) (a).5.4 V6B Eperimental Micro-CT Simulation Network Simulation Frequenc.3.. T (ms) (b) Figure 7-: (a) Comparison of the eperimental magnetiation deca of V6Y with the micro-ct image and etracted network for V6B. (b) Comparison of the inverted T distributions from the magnetiation decas in (a). 6

75 Normalied Amplitude. V6C Eperimental Micro-CT Simulation Network Simulation. 3 Time (ms) (a).5.4 V6C Eperimental Micro-CT Simulation Network Simulation Frequenc.3.. T (ms) (b) Figure 7-: (a) Comparison of the eperimental magnetiation deca of V6Y with the micro-ct image and etracted network for V6C. (b) Comparison of the inverted T distributions from the magnetiation decas in (a). 6

76 The magnetiation of the networks representing V6B and V6C deca faster than the network from V6A; this is as a result of an increase in the number of network elements etracted from their respective micro-ct images. These results again suggest that we under-estimate the mean-free-path of a random walker in the networks. The main peak is narrower for all the networks than the eperiment and micro-ct images, impling that some features of full ranges of pore sies and shape are lost in the etraction algorithm Comparisons of other single-phase properties Permeabilities and formation factors of the networks (Bakke and Øren, 997; Øren and Bakke, 3) were computed and compared with those of the micro-ct images (Øren et al., ; Knackstedt et al., 4) and eperimental values. We also computed the mean transverse relaation times T lm for the micro-ct images, networks and eperimental data using equation (3.6). Table 7- shows the comparisons of eperimental, image and network estimates of meant, permeabilit and formation factor. Sample F4A Mean T (ms) Permeabilit (D) * Eperiment Micro-CT Network Eperiment Micro-CT Network Formation Factor Eperiment Micro-CT Network F4B F4C V6A V6B V6C * D m Table 7-: Comparison of eperimental data with simulation results 6

77 The V6 sand packs have a much higher surface area and a smaller grain sie distribution than the F4 sand packs, leading to more pores and throats per unit volume, a faster magnetiation deca (lower T ) and lower permeabilit. The predictions of permeabilit are good, although the permeabilit of the F4 pack is slightl over-estimated, while that of the V6 sand is under-estimated Pore sie distributions If bulk relaation and relaation due to magnetic gradients are negligible and there is no diffusion between elements, the pore sie (V/S) distribution can be estimated from equation (3.4). V S ρ (7.3) T The eperimental pore sie distribution of the F4Y sand pack using equation (7.3) is compared with the distribution of inscribed radius for the pores and throats in F4A network in Figure Eperimental Network Pores Network Throats Frequenc.5..5 Pore Sie (µm) Figure 7-: Comparison of inscribed radius distribution of network elements of F4A with the pore sie distribution of F4 sand pack inferred from the NMR response. 63

78 The inscribed radius distribution of the network elements is wider than the eperimental pore sie distribution obtained from the T distribution of the F4 sand pack. This is because the pore sie distribution from T distribution can combine several throats to a given pore and see them as a single pore; it can even combine several pores together and see them as a single pore, while in the network, the elements are clearl distinguished from each other. However, the range of the pore sie distribution obtained from the T distribution lies within the range of the inscribed radius distribution of the pores. The maimum distance that a walker will diffuse before bulk relaation is significant is approimatel ( 6DT ) B µm for our eperiments. This is of order a tpical pore-to-pore distance and hence some protons will sample more than one network element before decaing. The T distribution gives an indication of pore sie distribution but it is not a direct and accurate wa to find it. 64

79 8. Single-phase simulation results for sandstones Having obtained consistencies in the comparisons of the simulation results from networks and micro-ct images for sand packs, we also compared the results from a poorl consolidated sandstone (S) and consolidated sandstones (Fontainebleau, Berea and Bentheimer). One advantage of these comparisons is that we can determine how consistent are the etracted networks with the micro-ct image or reconstructed microstructures from which the were etracted. These comparisons can serve as a validation for pore-network models and the methods used to generate networks. 8. Fontainebleau sandstone Øren et. al., () reconstructed the three dimensional (3D) microstructure of Fontainebleau sandstone samples of different porosities b a process-based reconstruction method. The process-based reconstruction method consists of simulating the results of the main rock-forming processes; sedimentation, compaction and diagenesis. The reconstructions were performed in a discrete manner on a cubic grid of sie 3 3 voels with porosities ranging from.3 to.3. Magnetiation deca was used to validate these reconstructed microstructures b comparing their decas with those of micro-ct images of similar porosit. The comparisons in Figure 8- show that the reconstructions faithfull reproduce the angular and irregular pore shapes observed in the true microstructure. NMR simulations were performed on a reconstructed microstructure of Fontainebleau sandstone of porosit 3.6% and a network etracted from the microstructure using a dilation method (Øren and Bakke,, 3). Simulations were also performed on a network etracted using a maimal ball algorithm (Dong, 7). The microstructure has a resolution of 7.5µm with a cubic grid sie of 3 3 voels. The number elements in the networks etracted using the two network etraction methods are shown in Table

80 Figure 8-: The pore spaces in a 8 3 voels sub region of a reconstructed Fontainebleau sandstone (right) of porosit.8 and a micro-ct image of an actual Fontainebleau sandstone (left) with similar porosit. The resolution of both images is 5.7µm (Øren et. al., ). Network Pores Throats Dilation Method 4,997 8,9 Maimal Ball 3, 6, Coordination number Table 8-: Comparison of the number of elements and average coordination number in each network etracted using the dilation method (Øren and Bakke,, 3) and the maimal ball algorithm (Dong, 7). Diffusion coefficient (m /s) Bulk relaivit (s) Surface relaivit (m/s) Number of walkers,, Table 8-: Rock and fluid properties used in the simulations. 66

81 The diffusion coefficient and bulk relaivit used in the simulations shown in Table 8- are tpical values for brine; these are the same values used for the NMR simulations in the sand packs. The surface relaivit value used is a tpical value for Fontainebleau sandstone (Hurlimann et al., 994). Normalied Amplitude Time (ms) (a) Microstructure Network (Dilation Method) Network (Maimal Ball) Microstructure Network (Dilation Method) Network (Maimal Ball) Frequenc.3.. T (ms) (b) Figure 8-: (a) Comparison of the magnetiation deca of the reconstructed microstructure of Fontainebleau sandstone, the network etracted using a dilation method (Oren and Bakke;, 3) and the network etracted using a maimal ball algorithm (Dong, 7). (b) Comparison of the inverted T distributions of the magnetiation decas in (a). 67

82 The magnetiation deca of the network etracted using the dilation method is ver similar to that of the reconstructed microstructure, as shown in Figure 8- indicating, that this network is an ideal representation of the microstructure. On the other hand, the magnetiation of the network etracted using the maimal ball algorithm decas faster than the reconstructed microstructure. This is because the maimal ball algorithm seems to slightl over estimate the grain surface area in the networks. The total grain surface area of the reconstructed microstructure is.55-4 m. The total surface areas of the networks calculated using equations (5. 5.4) are.55-4 m and.74-4 m for the networks etracted using the dilation method and maimal ball respectivel. The total grain surface area in the microstructure is epected to be higher than the networks because the rough surfaces of the pores in the microstructures are replaced b smooth geometrical shapes in the networks. The mean transverse relaation times, is 57ms for the microstructure and network etracted using the dilation method while a value of 493ms was obtained for the maimal ball network. In order to further validate the results of the magnetiation deca obtained in the network, we compared the ratios of the respective areas of the sides of triangular and square shaped elements in the network as given in equations ( ) with the ratio of walkers that deca on their respective sides given in equations (5.9 5.) as shown in Table 8-3. T lm The comparisons in Table 8-3 show that the walkers deca on each side of the triangular elements based on their respective ratios as epected. The minor differences observed are as a result of the random nature of the walkers diffusion. The difference for the square shaped elements is quite significant, because of the small fraction of network elements that are made of square shaped sections. Hence, the number of walkers that deca on these square shaped sections depends on how accessible the walkers are; to these elements since the ma be scattered across the network or ma be clustered around a particular region. The maimal ball method over-estimates the surface area due to the creation of spurious connections between pores. This leads to the creation of slit-like elements in which walkers deca rapidl. 68

83 Sides Sides Sides Sides 3 Sq. Sides Fractional Areas Walkers deca Difference (±%) (a) 83.3 Sides Sides Sides Sides 3 Sq. Sides Fractional Areas Walkers deca Difference (±%) (b) Table 8-3: Comparison of the fractional surface area of the smallest sides (Sides ), middle sides (Sides ), longest sides (Sides 3) of all the triangular elements and square sides (Sq. sides) in the network with the fraction of walkers that deca on these respective sides for the network etracted from dilation method. (b) Comparisons of the maimal ball network. 8. Poorl consolidated sandstone (S) NMR simulations were also performed on the micro-ct image and etracted network of a poorl consolidated sandstone, S. The porosit of the micro-ct image is 6.9% while its resolution is 9.µm; as shown in Figure 8-3. The micro-ct image data were converted into binar format where and represent the pore and grain voels respectivel. A central cubic section of voel sie 3 3 cropped from the micro-ct image was used for the simulations. The maimal ball algorithm was used to etract a network of 3,7 pores and 7,58 throats with an average coordination number of 4.7. A surface relaivit of 5µm/s was used in the simulations; the fluid properties used are shown in Table

84 (a) (b) Figure 8-3: (a) 3-D image of sandstone S and (b) network etracted from the image using the maimal ball algorithm. Normalied Amplitude. Sandstone S Micro-CT Simulation Network Simulation Time (ms) Sandstone S (a) Micro-CT Simulation Network Simulation Frequenc.3.. T (ms) (b) Figure 8-4: (a) Magnetiation decas of the micro-ct image and etracted network of Sandstone S. (b) The inverted T distributions from the magnetiation decas in (a). 7

85 The decas shown in Figure 8-4 indicate that the magnetiation in the network decas slightl faster than the micro-ct image. The same trend was observed in the comparisons of the magnetiation decas of the networks etracted from the reconstructed microstructure of Fontainebleau sandstone shown in Figure 8-a. However, the inverted T distributions shown in Figure 8-4(b) indicate that the two distributions are similar having almost the same T value at their peaks. 8.3 Berea sandstone Magnetiation deca was simulated on Berea sandstone of porosit 9.6%. The simulations were carried out on a cubic grid sie of 3 3 voels cropped from the micro-ct image shown in Figure 8-5(a); the resolution of the image is 5.345µm. The maimal ball algorithm was used to etract a network of 3, pores and 5,669 throats with an average coordination number of 3.4 from the cubic grid. The simulation results were compared with that from a network of,349 pores and 6,46 throats etracted from reconstructed Berea sandstone (erdahl et al., ) using the dilation method (Øren and Bakke;, 3). A surface relaivit of 5µm/s was used in the simulations; the fluid properties used are shown in Table 8-. A summar of the micro-ct and etracted networks properties for Sandstone S and Berea sandstone is shown in Table 8-4. Permeabilit and formation factors are calculated in the networks and micro-ct images of sandstone S and Berea sandstones are shown in Table 8-5. (a) (b) (c) Figure 8-5: (a) 3D micro-ct image of the Berea sandstone. (b) Network etracted from the micro-ct image in (a) using the maimal ball method. (c) Network etracted from a reconstructed microstructure of Berea sandstone (erdahl et al., ). 7

86 Normalied Amplitude. Berea Micro-CT PBM Network MB Network. 3 Time (ms) (a).5.4 Berea Micro-CT PBM Network MB Network Frequenc.3.. T (ms) (b) Figure 8-6: (a) Comparison of the magnetiation decas of the micro-ct image of Berea sandstone, the network etracted using a dilation method (Øren and Bakke;, 3) from reconstructed Berea sandstone (erdahl et al., ) and a network etracted from the micro-ct image using a maimal ball algorithm (Dong, 7). (b) Comparison of the inverted T distributions of the magnetiation decas in 7(a). 7

87 Sample Resolution (µm) Voel Sie Porosit (%) Surface Area (m /m 3 ) Pores Throats Coordination Number Sandstone S ,4 3,7 7, Berea (MB) ,96 3, 5, Berea (PBM) ,349 6,46 4. Table 8-4: Micro-CT and etracted networks properties for sandstone S and Berea sandstone, the microstructure properties of Berea PBM network are unavailable. Sample Mean T (ms) Permeabilit (md) Formation Factor Micro CT Network Micro CT Network Micro CT Network Sandstone S Berea (MB) Berea (PBM) Table 8-5: Comparison of mean transverse relaation time T, permeabilit and formation factor of micro-ct images and etracted networks. The magnetiation of both networks deca faster than the micro-ct image as shown in Figure 8-6(a). The PBM network was etracted from a reconstructed microstructure of Berea sandstone and not from the micro-ct image shown in Figure 8-5. Also the PBM network was etracted from a microstructure of porosit 8.3% which is slightl lower than the porosit of 9.6% in both the micro-ct image and the network etracted using the maimal ball method. These could result in the differences observed in their respective magnetiation decas since lower porosit images and networks deca faster than higher porosities. 73

88 However, the magnetiation deca of the PBM network is much closer to that of the micro-ct image than the deca of the network etracted using the maimal ball algorithm. The T distributions shown in Figure 8-6(b) indicate that the distribution of the PBM network and the micro-ct image are somewhat similar inferring similar pore sie distributions. However, the distribution of the maimal ball network is narrower with a higher peak. 8.4 Bentheimer sandstone In order to further validate the method used in simulating NMR response in networks, simulations results were compared with eperimental data for Bentheimer sandstone performed b colleagues at the Department of Phsics and Technolog, Universit of Bergen. Unlike previous analsis where networks were etracted from either a micro- CT image or a reconstructed microstructure, we used the network etracted from a reconstructed microstructure of Berea sandstone (PBM Network) as alread discussed in section 8.3. The properties (pore and throat sie distribution) of this network were tuned to match the mercur injection capillar pressure data of Bentheimer sandstone as shown in Figure 8-7, an approach full described in Valvatne (3). The resulting network was then used for the simulation of NMR response which was compared with eperimental values. The Bentheimer sandstone used in these comparisons have porosities and permeabilities in the range 3% ± 4% and,4md ± 8mD respectivel NMR measurements Bentheimer sandstone plugs of diameter 3.73cm and length 6.cm were used for the NMR measurements. The plugs were saturated in brine of viscosit.9-3 Pas and densit 5 kg/m 3. NMR relaation measurements were performed using a MARAN bench top spectrometer (Resonance Instruments) at a temperature of 38K operating at MH. The Carr-Purcell-Meiboom-Gill (CPMG) sequence was used for transverse relaation measurements ( T -measurements) with an inter-echo spacing of 4µs. 74

89 A single data point was acquired at the centre of each echo, with 8, data points in total collected. The magnetiation deca curve was inverted into T distribution using Win DXP software (Resonance Instruments). Capillar pressure (KPa) Bentheimer Water saturation (S w ) Eperimental Network Simulation Berea Network Figure 8-7: Comparison of the eperimental capillar pressure data of the Bentheimer sandstone with simulation results from a tuned Berea network and the original Berea network Simulation parameters The diffusion coefficient and the bulk relaation time of brine used in the simulation were determined from the correlations given in equations (7.) and (7.). B substituting the temperature at which the eperiment was performed (38K) and the viscosit of the brine (.9-3 Pas), a diffusion coefficient of.9-9 m /s and bulk relaivit of.84 seconds were obtained and used in the simulations. The surface relaivit used in the simulation is 9.3µm/s which is tpical for Bentheimer sandstones (iaw et. al, 996) Simulation results The simulated magnetiation deca is consistent with eperimental values at earl times as shown in Figure 8-8a; however at late times the simulated magnetiation decas faster than the eperimental values. These differences could be as a result of the network not being able to full capture accuratel the topolog of this Bentheimer sandstone. 75

90 Normalied Amplitude. Bentheimer Eperimental Network Simulation. 3 Time (ms) (a).5. Bentheimer Eperimental Network Simulation Frequenc.5..5 T (ms) (b) Figure 8-8: (a) Comparison of magnetiation decas of the eperimental data of Bentheimer sandstone with simulation results of the tuned network. (b) The inverted T distributions from the magnetiation decas in (a). The inverted T distributions shown in Figure 8-8b are similar; their mean values are 677ms and 64ms for the eperiment and simulation respectivel. The computed permeabilit of the network is 95mD compared with eperimental values in the range,4md ± 8mD. 76

91 These results validate the algorithm used for the simulation of magnetiation deca in networks. Unlike in the analsis of the sand packs where surface relaivit was used to match eperimental data of the magnetiation deca, we used the tpical surface relaivit of Bentheimer sandstone and consistent results were obtained. The accurac of the simulation results using this network NMR algorithm could be improved considerabl if the actual surface relaivit of the porous medium is known and an appropriate network etracted from either a micro-ct image or a reconstructed microstructure. 77

92 9. Single-phase simulation results for carbonates The methods used in simulating NMR response in the previous sections are applied to carbonates. Carbonates often have bimodal pore sie distributions (micro-pores and macro-pores), with micro pores having sies in sub-micron range. Micro-CT scans with resolutions of a few microns cannot capture the features of these micro pores. In order to overcome this limitation, multiple point statistics have been used to reconstruct 3D images of carbonates from D thin section images (Okabe and Blunt, 4; Okabe and Blunt, 5). The challenge in pore scale modelling of carbonates lies in etracting a suitable network that is topologicall identical to the parent sample. However, predictions of transport properties such as permeabilit, formation factor, relative permeabilit and capillar pressures have been made on networks etracted from the reconstructed 3D images of carbonates using a maimal ball algorithm (Al- Kharusi and Blunt, 7). An alternative to generating networks without an underling microstructure is b tuning the properties of a known network to generate simulated capillar pressures that matches the measured data, (Valvatne, 3; Valvatne and Blunt, 4) as described in the previous section. 9. Heterogeneous carbonate C Carbonate C is a tpical calcrete or lithified paleosol, enclosing carbonate clastic detritus. Analsis of its thin-section images suggests it probabl developed as a weathering rind of limestones during eposure (uplift or sea-level drop). The porosit of its micro-ct image shown in Figure 9-a is 4.8% while its resolution is 5.345µm. This carbonate has highl heterogeneous pore structure with poorl connected pores as shown in Figure 9-. NMR simulations were performed on the micro-ct image and etracted network. 78

93 mm (a) (b) Figure 9-: (a) 3D micro-ct image of carbonate C (the dark areas are the pore spaces and the white represent grains). (b) Network etracted from the image in (a) using the maimal ball method. A central cubic section of voel sie 3 3 cropped from the micro-ct image shown in Figure 9-(b) was used for the simulations. The maimal ball algorithm was used to etract a network of 3,574 pores and 4,98 throats with an average coordination number of.3. This relativel low coordination number is as a result of a large number of poorl connected pores as shown in Figure 9.. Normalied Amplitude. Carbonate C Micro-CT Network Simulation. 3 4 Time (ms) Figure 9-a: Comparison of the magnetiation decas of the micro-ct image of carbonate C and the etracted network. 79