Using maximal inscribed spheres for image-based compaction forecasting

ARMA 16-47 Using maximal inscribed spheres for image-based compaction forecasting Louis, L., and Boitnott, G. New England Research, White River Junction, Vermont, USA Bhattad, P., and Knackstedt, M. FEI, Houston, Texas, USA Copyright 2016 ARMA, American Rock Mechanics Association This paper was prepared for presentation at the 50 th US Rock Mechanics / Geomechanics Symposium held in Houston, Texas, USA, 26-29 June 2016. This paper was selected for presentation at the symposium by an ARMA Technical Program Committee based on a technical and critical review of the paper by a minimum of two technical reviewers. The material, as presented, does not necessarily reflect any position of ARMA, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of ARMA is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 200 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgement of where and by whom the paper was presented. ABSTRACT: The present paper concerns itself with the use of morphological information from the 3D image of a rock microstructure to extract parameters needed to model the compaction of porous sandstones. We propose to test on the mineral framework a tool that is already employed in the simulation of mercury injection capillary pressure experiments (MICP) (see [1-2] for implementations and [3] for background on MICP) and, in particular, we investigate the existence of a characteristic grain contact radius. As a starting point, results from hydrostatic loading of two porous sandstones of similar porosities, the sandstone (~26%) and the sandstone (~29%), are presented. The mechanical data reveals a factor of almost 4 between the values measured for the critical grain crushing pressure P*. As a way to connect microstructural parameters to the observed strength contrast, we test the use of the morphological analysis on high resolution X-ray CT images of both rocks. In comparing our findings with the already existing model of Zhang and Wong [4-5], we propose that the intergranular contact radius information extracted from the image analysis be explicitly incorporated into the modeling of the strength of porous sandstones. INTRODUCTION Mechanical response to depletion, which comprises irrecoverable volumetric strain as well as elastic deformation, strongly depends on in situ conditions and on the nature of the corresponding perturbation in terms of stress path, strain rate, fluid substitution, etc. The ability to forecast this behavior, whether for pressure support or subsidence risk assessment, hinges on our understanding of deformation mechanisms at the scale of the aggregate, their interplay with preexisting heterogeneities and their manifestation at the scale of the reservoir. Modeling of mechanical properties traditionally relies on microstructural parameters such as porosity, mineralogy, coordination number, cemented contact area, grain size and shape, which are combined to account for trends obtained in laboratory measurements. The now widespread availability of 3D pore scale imaging techniques allows one to access the intimate make-up of a rock, offering in principle a means to fully quantify and validate the parameters used for pore-scale modeling. It also provides an opportunity to identify which of these parameters control resulting behavior, whether redundancies exist from a physical point of view, and whether they can even be measured in a meaningful way. The option of performing direct numerical simulations based on pore scale images is being increasingly utilized to complement costly laboratory measurements [6]. However, an understanding of the key controls of the observed behavior remains essential for generalizations to be made. Several approaches to compaction modeling exist depending on the application. In basin modeling and pore pressure prediction, empirical relations of exponential and power law types are often used as porosity predictors. In soil mechanics, the Cam-clay model relates the logarithm of the applied pressure to the void ratio to describe both elastic and permanent deformation. For cohesive siliciclastic aggregates, yet other approaches have been proposed that use fracture mechanics to establish the conditions for grain crushing and/or pore collapse to occur. In the case of a porous sandstone as seen from a laboratory test perspective, the objective of an image-based technique would be the ability to predict for an arbitrary stress path: The pre-yield elastic behavior The conditions for the onset of grain crushing A maximum rate of inelastic compaction

Such results could then be plugged into geomechanical models for reservoir behavior forecasting as a result of depletion. The experimental basis for the present work is restricted to the stress conditions for failure under hydrostatic loading. Results from experiments on two porous sandstones, the sandstone (~26%) and the sandstone (~29%), reveal a very large difference (ratio of 3.7) between the pressures needed to initiate grain crushing and accelerated compaction P*. As a means of accessing microstructural parameters that are thought to control mechanical behavior, we test on the mineral framework a morphological analysis tool that has already been successfully applied to the simulation of mercury injection capillary pressure experiments in pore networks. Starting from micronscale resolution X-ray CT images of both studied sandstones, the workflow consists of first performing a maximal inscribed sphere (MIS) analysis on the mineral framework to obtain some information about grain size and grain size distribution, then using the MIS result to simulate an injection that has the potential of yielding a measure of a controlling intergranular contact radius. In the following, after reporting on the mechanical data acquired during hydrostatic loading experiments, we perform the proposed morphological analysis to yield proxies for average grain size and characteristic intergranular contact radius. The information of a characteristic intergranular contact radius provides an additional parameter that has the potential of refining the expression of P* as a function of microstructural parameters as proposed in the model by Zhang and Wong [4-5]. The advantage of adding this new independent parameter is to render the ratio of intergranular contact radius to grain radius pressure and strain dependent, hence allowing for better description of failure envelopes as well as of the post-yield hardening behavior. Also, the proposed image analysis offers an attractive alternative to digital grain separation, which is a computationally costly and difficult task in natural aggregates. 1. MATERIALS AND METHODS The porosity and mineral composition for the and sandstones are provided in Table 1. The porosity difference between the two rocks is moderate, although the sandstone exhibits a comparatively high plug scale heterogeneity (both in porosity and grain size). Concerning the mineral composition, the main difference between the two rocks is the presence of a substantial amount of Feldspar in the sandstone, and some amount of carbonate in the sandstone. Table 1. Porosity and composition of the sandstones studied Porosity (%) 26 29 Mineralogy (%) Quartz (85), Calcite/dolomite (10), Clay (5) Quartz (45), Feldspar (45), Biotite (5), Clay (5) Hydrostatic loading experiments were conducted at New England Research on plugs of approximately 1 inch in diameter and two inches in length. The samples were saturated with tap water and the pore pressure was kept constant at 5 MPa (725 PSI) throughout the experiments. Hydrostatic pressure was then applied at a rate of 2.5 MPa (360 PSI) per minute up to 180 MPa (26100 PSI) for the sandstone and 300 MPa (43500 PSI) for the sandstone. Volumetric strain was obtained from the volume of expelled pore fluid. X-ray CT imaging was performed by FEI in Houston using a HeliScan micro CT scanner on 9mm diameter plugs cored in the vicinity of the ones used for the experiments. The voxel dimensions obtained in the images are 4.29 m for the sandstone and 3.47 m for the sandstone. From each of these images, a few regions of interest of 300 3 and 400 3 voxels 3 were picked for image analysis. Segmentation into pore and solid phases was conducted using the default Otsu thresholding (segment at grey level that minimizes intraphase variance) available in ImageJ [7]. The maximal inscribed sphere analysis was done using the implementation of Dougherty and Kunzelmann [8] available as a plugin in ImageJ under the name Local Thickness. The principle of this analysis is to find and tag each voxel with the diameter of the largest sphere containing that voxel that can be fit inside the analyzed 3D object. The injection simulation into the mineral framework was implemented using ImageJ Macro language and consists of sequentially capturing clusters connected to the injection faces (6 for isotropic injection and 2 for directional injection) in step by step segmentation starting from the largest local thickness. 2. RESULTS AND DISCUSSION 2.1. Mechanical data Figure 1 presents the stress strain data obtained in these experiments. The sandstone exhibits a much lower critical pressure P* (60 MPa or 8700 PSI) compared to the (220 MPa or 31900 PSI), with a ratio of 3.7 between the two values. Note the fairly stable compaction regime in the sandstone until enhanced hardening can be observed. If continued further, it is expected that these stress vs. strain curves

would start exhibiting a progressively more elastic behavior, reducing plastic compaction to a transient phenomenon, as suggested in the probabilistic damage model of Zhu et al. [9] (though this was applied to permeability modeling during shear-enhanced compaction). A fully predictive compaction model should be able to provide as a function of the stress path (i) an elastic loading modulus, (ii) the stress conditions for the onset of grain crushing or substantial irrecoverable deformation and (iii) the rate at which this compaction occurs. 300 Net hydrostatic stress (MPa) Normalized cumulative frequency scan. In Figure 3b, the pore space was segmented and the MIS analysis was run. Figure 3c shows the result of the analysis on the mineral framework instead of the pore space. Note the constriction that occurs at grain boundaries resulting in colder colors (smaller maximal inscribed sphere or local thickness). (a) (b) (c) 250 200 150 100 50 0 P * CAS=220 MPa P * BOI=60 MPa 0 2 4 6 8 10 12 14 Volumetric Strain (%) Fig. 1. Stress-strain curves for hydrostatically loaded samples of and sandstones. 2.2. Imaging and image analysis The and sandstones were CT scanned at a resolution of 3.47 m and 4.29 m, respectively. Visualizations of 300 3 voxels 3 volumes are showed in Figure 2. (a) Fig. 2. 3D views of the sandstones imaged. (a). (b). Each cube is 300 voxels along each edge. From the CT images, subvolumes representing about 5 mm 3 of material were extracted for the maximal inscribed sphere analysis. In Figure 3, we show an example output of the MIS analysis in a clean well cemented sandstone. Figure 3a shows the original CT (b) Fig. 3. Result of maximum inscribed sphere (MIS) analysis. (a) Original image. (b) MIS in the pore space. (c) MIS on the mineral framework. Figure 4 shows the cumulative distribution of the MIS results for all the subvolumes considered (4 subvolumes for and 3 for ) together with 2 examples colored according to their local thickness (cold colors are for low values and hot colors for high values). The distributions are clearly distinct and suggest for the compared to the (1) a smaller mean grain size, (2) a narrower grain size distribution and (3) a more homogeneous microstructure. The subvolumes were picked from two zones that appeared clearly different on the full size tomogram, whereas no notable difference from a zone to another could be observed in the. 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 4 40 400 1/2 Local thickness (m) Fig. 4. MIS results on image subvolumes. The MIS results already provide us with useful proxies for microstructural parameters. Next, we perform the equivalent of a drainage simulation on the mineral framework to test whether there exists a characteristic radius that controls the interconnectivity within the aggregate. An example result is given in Figure 5. Here, the cumulative plot represents progressive injection,

which is illustrated by the two visualizations. The yellow curve is for a simulation where injection is allowed from all sides, and the green curve is the average of simulations conducted along the three reference axes, X, Y and Z. Normalized cumulative frequency 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 Drain ISO XYZ Mean XYZ R cc 0 20 Radius (microns) 200 Fig. 5. Simulated injection performed on the digitized mineral framework. Yellow curve: Injection is performed from all sides. Grey curves: injection along one reference axis (X, Y or Z). Green curve: average result of the directional injections. These curves present the same characteristics as MICP data in the sense that a conformance effect can be observed in the early stages of the injection (artifact associated with asperities directly in contact with the surface), but more importantly that there exists a radius for which a maximum incremental intrusion is observed, before the mineral framework becomes fully saturated through ever smaller increments. This radius which we will call characteristic intergranular contact radius R cc is a parameter that may participate in the amplification of the stress effectively experienced by the grains when loading the aggregate. (a) Fig. 6. grain volume occupied at breakthrough (b) compared with the full volume in (a) colored by local thickness. To associate with R cc, one may seek a measure of a grain size that corresponds to the state of the aggregate at the time of breakthrough R gc. To do that, the MIS data that belongs to the cluster invaded at breakthrough can be (b) used. Figure 6 shows a volume of sandstone (Figure 6a) and the fraction effectively invaded at the time of breakthrough (Figure 6b) colored by local thickness. The subset of the local thickness distribution can then be used to estimate R gc. In the present case we use a simple average of the local thickness values. The values obtained in all subvolumes of and for R cc and R gc are provided in Table 2. Table 2. Results of simulated injection on mineral framework Zone 1 2 3 4 1 2 3 R cc (+/- 4m) 42 47 47 47 67 53 67 R gc (+/- 4m) 55 60 61 61 105 75 102 2.3. Parameters use in compaction forecasting The microstructural parameters proxies that can be derived from this study are a porosity (through initial segmentation), a grain size distribution (direct MIS result), a characteristic intergranular contact radius R cc and a characteristic grain radius R gc which corresponds to the average radius of the connected grains at the time of maximum intrusion. It is worth mentioning here that the data obtained by the MIS is not strictly a grain size distribution in the sense of a count but rather a distribution of volumes occupied by grains of given diameters or radii. One may propose to normalize these statistics by the square of the radius (assuming a pipe geometry) or by the cube of the radius (assuming a spherical geometry) to approach quantities that would be closer to actual counts. Though it may be debated whether counts should actually be used since volume fractions may be considered more appropriate for an effective medium type of approach. The MIS distribution was used as is in this paper. The micromechanical model of Zhang et al., 1990 [4] which is based on Hertz theory, proposes to predict the pressure P* necessary for the onset of inelastic compaction induced by grain fracturing and pore collapse. The analysis, which invokes the mode I opening of a microcrack at the grain-to-grain contact as a means for initiation of grain crushing, was reused more recently to model shear-enhanced compaction by Brzesowsky et al., 2014 [10]. As recast in Wong et al., 1997 [5] and in Figure 7, and in accordance with the micromechanical model of Zhang et al., the value of P* is showed to be primarily controlled by a power of the product (R) where is the porosity and R the mean grain radius. More specifically, the model predicts that P* be proportional to (R) -3/2. Scatter, however, suggests that other parameters are likely to be playing a role.

Using the values we obtained in the morphological analysis, the expression of Zhang et al. predicts a P* ratio between the and sandstones in the range 1.6 to 3.1 with an average of 2.3, what is somewhat lower to the ratio of 3.7 observed in the hydrostatic experiment. It is clear that many other factors, some of which will be listed below, are likely to contribute to this matching exercise. As a first order improvement we propose that the information of the characteristic contact radius be incorporated into an expression for P* in the form an intensification factor = (R cc / R gc) 2. (1) in terms of physical units and provide a value to go by if the porosity tends towards zero. This would write: (1 ) P* * ref Rref (2) R Where a reference mineral strength is added together with the length scale at which that reference is valid. The reference mineral strength also provides a means to introduce an effective mineralogical contribution which is thought to exert substantial influence on inelastic compaction behavior. gc Fig. 7. Critical effective pressure P* for the onset of grain crushing in sandstone as a function of initial porosity and grain radius (from Wong et al., 1997 [5]). The dependency over R -3/2 in the expression of Zhang et al. or even simply over R -1 is appealing as it echoes the observation that strength tends to decrease with sample size increase, which can be intuitively understood as the effect of an increasing probability of finding defects as the dimensions increase or as the fact that once division of a particle has occurred due to the presence of a weakness, the resulting two particles are in essence devoid of that weakness. Taking into account that dependency of strength over length scale, and rewriting the way the porosity contributes to the strength so that finite boundaries are set, we propose to write partially P* in the following fashion: (1 ) P* (1) R gc If it is admitted that there exists a scale dependency of the strength, a reference should be set that would balance 3. CONCLUSION The paper presented here is an exploration of the use of an existing morphological tool to describe the mineral framework of an aggregate, a direct application of which being the ability to predict the onset of grain crushing in a rock subjected to an increasing effective compressive stress such as in the case of reservoir depletion. There are many uncertainties involved in this analysis, the major one being directly associated with image resolution. Imaging the rocks studied here at a better resolution, which would be in fact very straightforward to do, would certainly modify substantially the values measured for the critical intergranular contact radius. Also, in comparing the values of P*, we have not considered the effect of the mineralogy, which would typically be factored into equation (2) and might be impacting the measurements done on and sandstones since the contains a large fraction of Feldspar in place of Quartz. In the short term, it would be valuable to first extend this study to more sandstones in order to observe whether the trend that was established by Zhang and Wong can be noticeably improved. Finally, as the analysis that is done here explicitly considers the role of the intergranular contact radius, and granted that its value can be confirmed in hydrostatic tests, it could be extended to be used as a predictor of conditions for failure along arbitrary stress paths (yield envelope), as well as for the prediction of hardening behavior past the onset of grain crushing. REFERENCES 1. Hilpert, M. and C.T. Miller. 2001. Poremorphology based simulation of drainage in totally wetting porous media. Advances in Water Resources 24: 243 255. 2. Silin, D.B. and T.W. Patzek. 2006. Pore space morphology analysis using maximal inscribed

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