Patent Publication Number: US-9892321-B2

Title: Using maximal inscribed spheres for image-based rock property estimation

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     Priority is claimed from U.S. Provisional Application No. 62/185,567 filed on Jun. 27, 2015. 
    
    
     STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT 
     Not Applicable 
     NAMES OF THE PARTIES TO A JOINT RESEARCH AGREEMENT 
     Not Applicable. 
     BACKGROUND 
     This disclosure relates to the field of estimating physical properties of rock formations using images of samples of the formations. More specifically, the disclosure relates to the use of images of the rock mineral grain portion of the sample, rather than the void space, to estimate one or more physical properties of the rock formation. 
     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. See, for example, Fredrich, J. T., D. L. Lakshtanov, N. M. Lane, E. B. Liu, C. S. Natarajan, D. M. Ni, and J. J. Toms, 2014,  Digital Rocks: Developing an emerging technology through to a proven capability deployed in the business . Proceedings of the Society of Petroleum Engineers Annual Technical Conference and Exhibition, Amsterdam, The Netherlands, 27-29 October. 
     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 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 observed from a laboratory test perspective, an objective of an image-based technique would be the ability to predict an arbitrary stress path. Such prediction may use the pre-yield elastic behavior, the conditions for the onset of grain crushing and maximum rate of inelastic compaction. Such results could then be input into well-known geomechanical models for reservoir behavior forecasting as a result of depletion. 
     There continues to be a need for improved methods for interpreting images of rock formation samples to infer mechanical properties of the formations. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1A  shows a portion of a flow chart of a method according to one embodiments. 
         FIG. 1B  shows the remainder of the flow chart in  FIG. 1A . 
         FIG. 2  shows a computer system that may be used in some embodiments. 
     
    
    
     DETAILED DESCRIPTION 
     The experimental basis for the present disclosure is related to the stress conditions for failure under hydrostatic loading. Results from experiments on two porous sandstones, the Castlegate sandstone (porosity ˜26%) and the Boise sandstone (porosity ˜29%), reveal a very large difference (ratio of 3.7) between the pressures needed to initiate grain crushing and accelerated compaction, designated as P*. 
     As a means of accessing microstructural parameters that are thought to control mechanical behavior, the mineral framework of the samples was tested using a morphological analysis tool that has already been successfully applied to the simulation of mercury injection capillary pressure experiments in pore networks. Starting from micron-scale resolution X-ray CT images of both studied sandstone formations, the workflow consists of first performing a maximum 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 to provide a measure of a controlling intergranular contact radius. 
     In the following disclosure, after describing the mechanical data acquired during hydrostatic loading experiments, a description of the morphological analysis according to the present disclosure is provided 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 to refine 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 (the characteristic intergranular contact radius) is to render the ratio of intergranular contact radius to grain radius pressure and strain dependent, thus allowing for better description of failure envelopes as well as post-yield hardening behavior. Also, the proposed image analysis may offer an attractive alternative to digital grain separation, which is a computationally costly and difficult in natural aggregates. 
     The porosity and mineral composition for the Castlegate and Boise sandstones are provided in TABLE 1. The porosity difference between the two formations is moderate, although the Boise 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 Boise sandstone and some amount of carbonate in the Castlegate sandstone. 
     
       
         
           
               
               
               
               
             
               
                   
                 TABLE 1 
               
               
                   
                   
               
               
                   
                 Formation 
                 Castlegate 
                 Boise 
               
               
                   
                   
               
             
            
               
                   
                 Porosity (%) 
                 26 
                 29 
               
               
                   
                 Mineralogy (%) 
                 Quartz (85), 
                 Quartz (45), 
               
               
                   
                   
                 Calcite/ 
                 Feldspar (45), 
               
               
                   
                   
                 dolomite (10), 
                 Biotite (5), 
               
               
                   
                   
                 Clay (5) 
                 Clay (5) 
               
               
                   
                   
               
            
           
         
       
     
     Hydrostatic loading experiments were conducted at New England Research, Inc., White River Jct., VT on plugs of formation samples approximately 1 inch (25 mm) in diameter and two inches (51 mm) 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, increasing at a rate of 2.5 MPa (360 PSI) per minute up to a final pressure of 180 MPa (26100 PSI) for the Boise sandstone and 300 MPa (43500 PSI) for the Castlegate sandstone. Volumetric strain was obtained from the volume of expelled pore fluid. 
     X-ray CT imaging was performed on samples of the two above described formations by FEI Company, 16700 Park Row Drive, Houston, Tex. 77084 using a HeliScan micro CT scanner on 9 mm diameter plug samples of the two formations in the vicinity of the core samples used for the fluid pressure experiments. The voxel dimensions obtained in the images are 4.29 μm for the Boise sandstone and 3.47 μm for the Castlegate sandstone. From each of these images, a few regions of interest of 3003 and 4003 voxels 3  were selected for image analysis. Segmentation into pore and solid phases was conducted using default Otsu thresholding (segment at grey level that minimizes intra-phase variance) available in ImageJ [7]. The maximum inscribed sphere analysis was performed 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. 
     During testing it was determined that the Boise sandstone exhibits a much lower critical pressure P* (60 MPa or 8700 PSI) than the Castlegate (220 MPa or 31900 PSI), with a ratio of 3.7 between the two values. The Boise sandstone exhibited a fairly stable compaction regime in the Boise sandstone until enhanced hardening was 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. 
     From the CT images, subvolumes representing about 5 mm 3  of material were extracted for the maximum inscribed sphere analysis. The pore space in the CT images was segmented and MIS analysis was performed. MIS analysis reveals constriction that occurs at grain boundaries resulting in smaller maximal inscribed sphere or local thickness. 
     Cumulative distribution of the MIS results for all the subvolumes considered (4 subvolumes for Castlegate and 3 for Boise) together with 2 examples identified according to their local thickness indicate that for the two different formations the MIS distributions are clearly distinct and suggest for the Castlegate formation compared to the Boise formation: (1) a smaller mean grain size; (2) a narrower grain size distribution; and (3) a more homogeneous microstructure. The Boise formation subvolumes were selected from two zones that appeared clearly different on the full size CT image, whereas no notable difference from one zone to another could be observed in the Castlegate formation. 
     The foregoing experiments show that the MIS results already provide useful proxies for microstructural parameters. Next, the equivalent of a drainage simulation on the mineral framework was performed to test whether there exists a characteristic radius that controls the interconnectivity within the aggregate. 
     Simulated mercury injection curves exhibit 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 may be referred to as “characteristic intergranular contact radius” Rcc is a parameter that may be related to amplification of the stress effectively experienced by the grains when loading the aggregate. 
     To associate with Rcc, one may seek a measure of a grain size that corresponds to the state of the aggregate at the time of ‘breakthrough’ Rgc. To do that, the MIS data that belongs to the cluster invaded at breakthrough can be used. 
     The subset of the local thickness distribution can then be used to estimate Rgc. In the present case a simple average of the local thickness values was used. The values obtained in all subvolumes of Castlegate and Boise formations for Rcc and Rgc are provided in Table 2. 
     
       
         
           
               
               
             
               
                   
                 TABLE 2 
               
             
            
               
                   
                   
               
               
                   
                 Fm. 
               
            
           
           
               
               
               
            
               
                   
                 Castlegate 
                 Boise 
               
            
           
           
               
               
               
               
               
               
               
               
               
            
               
                   
                 Zone 
                 1 
                 2 
                 3 
                 4 
                 1 
                 2 
                 3 
               
               
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
            
               
                   
                 Rcc 
                 42 
                 47 
                 47 
                 47 
                 67 
                 53 
                 67 
               
               
                   
                 (+/−4 μm) 
               
               
                   
                 Rgc 
                 55 
                 60 
                 61 
                 61 
                 105 
                 75 
                 102 
               
               
                   
                 (+/−4 μm) 
               
               
                   
                   
               
            
           
         
       
     
     The microstructural parameter proxies that can be derived from this study may include porosity (through initial segmentation), grain size distribution (direct MIS result), characteristic intergranular contact radius Rcc and a characteristic grain radius Rgc which corresponds to the average radius of the connected grains at the time of maximum intrusion. 
     The data obtained by the MIS analysis 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. There may be some question 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 the present disclosure 
     The micromechanical model disclosed by 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 foregoing analysis, which invokes the mode I of 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 accordance with the micromechanical model of Zhang et al., (4) the value of P* is showed to be primarily controlled by a power of the product (ΦR) where Φ is the porosity and R is the mean rock grain radius. More specifically, the model predicts that P* may be proportional to (ΦR) −3/2 . Scatter, however, suggests that other parameters are likely to affect the results obtained. 
     Using the values we obtained in the morphological analysis, the expression of Zhang et al. predicts a P* ratio between the Castlegate and Boise 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 it is proposed that the information of the characteristic contact radius be incorporated into an expression for P* in the form an intensification factor γ=(Rcc/Rgc) 2 . 
     The dependency over R −3/2  in the expression of Zhang et al. or even simply over R −1  is appealing as it corresponds to 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 P* the way the porosity contributes to the strength so that finite boundaries are set, it is proposed to rewrite partially P* in the following fashion: 
     
       
         
           
             
               
                 
                   
                     P 
                     * 
                   
                   ∝ 
                   
                     
                       
                         1 
                         - 
                         Φ 
                       
                       
                         R 
                         gc 
                       
                     
                     ⁢ 
                     γ 
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     If it is accepted that there exists a scale dependency of the strength, a reference may be set that would balance Eq. (1) in terms of physical units and provide a value to use if the porosity tends towards zero. The foregoing may be expressed as: 
                     P   *     ∝       σ   ref   *     ⁢     R   ref     ⁢       1   -   Φ       R   gc       ⁢   γ             (   2   )               
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.
 
     Having explained the theoretical foundation and test results for a method according to the present disclosure, an example implementation will now be explained with reference to  FIGS. 1A and 1B . Referring first to  FIG. 1A , at  10 , a 3D image of the rock sample is obtained. At  12 , pre-processing to enhance image quality may be performed, for example and without limitation, noise reduction. At  14  image segmentation may be performed to isolate the solid fraction from the pore space using a technique such as Otsu thresholding. At  16 , rock sample porosity Φ may be determined from the volume occupied by the pore space in the image. 
     At  18 , maximal inscribed sphere type of analysis may be performed to obtain an image of the local thickness of the solid fraction of the formation. The objective of such analysis consists of tagging each volume element (voxel) of the domain analyzed with the diameter (hence the concept of local thickness) of the largest sphere inscribed in the domain and containing that voxel. Examples of implementation can be found in references [2] and [8]. At  20 , proxies may be derived for grain size distribution, mean grain size and grain size standard deviation from the values of local thickness used either as is (e.g., by considering that the distribution of the local thicknesses directly yields a relevant proxy for grain size distribution as applies to the physical property of interest) or processed to yield, for example, spheres or disks as equivalent object counts. If the solid fraction is idealized as an aggregate of substantially spherical particles, then the particle count can be obtained by dividing the population of each bin in the distribution by the volume of the sphere corresponding to the diameter or thickness associated with that bin (e.g., 4/3π(d/2) 3 ). If the solid fraction is idealized as a set of two-dimensional disks (tube-like geometry), then the particle count can be obtained by dividing the population of each bin in the distribution by the cross sectional area corresponding to the diameter or thickness associated with that bin (e.g., π(d/2) 2 ). 
     At  22 A, isotropic morphological analysis of the solid fraction may be performed. This signifies that the injection simulation will be conducted from all sides of the probed volume simultaneously. At  24 A a variable x is given to the highest local thickness value measured. At  26 A, the number of voxels bearing a local thickness value equal to x is returned. At  28 A, a decision element is made based on whether returned voxels individually participate in a cluster that includes all previously retained voxels (except for the first stage where no voxels have been retained yet) and that is connected to one of the faces of the original volume. If yes, i.e., for those voxels that passed the foregoing test, at  30 A, those voxels are included in the count for that local thickness category. If no, i.e., for those voxels that failed the test, at  31 A, those voxels are ignored. At  32 A, the number of voxels retained bearing a local thickness value of x are returned to a table. At  34 A, the value of x is decreased by a predefined decrement ensuring that no voxels will be ignored by the analysis, and a check is performed to determine if a lowest value is reached. If the lowest value is not reached, the process returns to  26 A and the analysis in blocks  28 A,  30 A,  31 A,  32 A sand  34 A is repeated until at  34 A the lowest value is reached. At  36 , the resulting table bears a distribution of volumes that constitute the sequential increments that are invaded when simulating the acquisition of a ‘mercury injection capillary pressure’ (MICP) curve. The increments in that distribution can be sequentially added to create a cumulative distribution starting from the volumes corresponding to the largest local thickness. 
     At  22 B through  34 B, the same process as in  22 A through  34 A is started with the difference between processes being that the process beginning at  22 B will be conducted by checking for connectivity to either of one pair of volume faces at a time along a direction of interest, hence the denomination of directional morphological analysis. This process ends at  36  as for the isotropic process of  22 A through  34 A with a similar result table. 
     At  40 , the local thickness value is extracted that corresponds to the largest volume increment in the cumulative distribution using either the data as is or after performing some smoothing. At  42 , the extracted local thickness value is named the “characteristic contact diameter” and after division by 2, the “characteristic contact radius.” At  44 , the directional analysis of  22 B may be sequentially performed in more than one direction and an anisotropy of the characteristic contact radius value can be computed. 
     At  45 , the object that is formed by the solid cluster at the time of largest volume increment, which corresponds to the characteristic contact diameter, can be used to derive an alternate proxy for grain size distribution, mean grain size and grain size standard deviation following the same procedure as in step  20  and as 
     At  46 , a ratio of the characteristic contact dimension (squared or not) with respect to the mean grain dimension (cubed or not) obtained previously defines a stress intensification coefficient γ. 
     At  48 A, the calculated proxies may be used in a rock physics effective medium model to predict elastic properties such as static bulk modulus, Young&#39;s modulus and shear modulus, as well as dynamic P-wave and S-wave moduli and their anisotropy. At  48 B, The calculated proxies may be used in a rock mechanics model to predict mechanical strength in both hydrostatic and triaxial stress loading conditions, and the anisotropy thereof. At  48 C, the calculated proxies may be used in a rock mechanics model to predict behavior during compaction associated with pore collapse and grain crushing (e.g., as may be caused by fluid pressure depletion in a porous subsurface formation). 
       FIG. 2  shows an example computing system  100  in accordance with some embodiments. The computing system  100  may be an individual computer system  101 A or an arrangement of distributed computer systems. The individual computer system  101 A may include one or more analysis modules  102  that may be configured to perform various tasks according to some embodiments, such as the tasks explained with reference to  FIGS. 1A and 1B . To perform these various tasks, the analysis module  102  may operate independently or in coordination with one or more processors  104 , which may be connected to one or more storage media  106 . A display device  105  such as a graphic user interface of any known type may be in signal communication with the processor  104  to enable user entry of commands and/or data and to display results of execution of a set of instructions according to the present disclosure. 
     The processor(s)  104  may also be connected to a network interface  108  to allow the individual computer system  101 A to communicate over a data network  110  with one or more additional individual computer systems and/or computing systems, such as  101 B,  101 C, and/or  101 D (note that computer systems  101 B,  101 C and/or  101 D may or may not share the same architecture as computer system  101 A, and may be located in different physical locations, for example, computer systems  101 A and  101 B may be at a well drilling location, while in communication with one or more computer systems such as  101 C and/or  101 D that may be located in one or more data centers on shore, aboard ships, and/or located in varying countries on different continents). 
     A processor may include, without limitation, a microprocessor, microcontroller, processor module or subsystem, programmable integrated circuit, programmable gate array, or another control or computing device. 
     The storage media  106  may be implemented as one or more computer-readable or machine-readable storage media. Note that while in the example embodiment of FIG. the storage media  106  are shown as being disposed within the individual computer system  101 A, in some embodiments, the storage media  106  may be distributed within and/or across multiple internal and/or external enclosures of the individual computing system  101 A and/or additional computing systems, e.g.,  101 B,  101 C,  101 D. Storage media  106  may include, without limitation, one or more different forms of memory including semiconductor memory devices such as dynamic or static random access memories (DRAMs or SRAMs), erasable and programmable read-only memories (EPROMs), electrically erasable and programmable read-only memories (EEPROMs) and flash memories; magnetic disks such as fixed, floppy and removable disks; other magnetic media including tape; optical media such as compact disks (CDs) or digital video disks (DVDs); or other types of storage devices. Note that computer instructions to cause any individual computer system or a computing system to perform the tasks described above may be provided on one computer-readable or machine-readable storage medium, or may be provided on multiple computer-readable or machine-readable storage media distributed in a multiple component computing system having one or more nodes. Such computer-readable or machine-readable storage medium or media may be considered to be part of an article (or article of manufacture). An article or article of manufacture can refer to any manufactured single component or multiple components. The storage medium or media can be located either in the machine running the machine-readable instructions, or located at a remote site from which machine-readable instructions can be downloaded over a network for execution. 
     It should be appreciated that computing system  100  is only one example of a computing system, and that any other embodiment of a computing system may have more or fewer components than shown, may combine additional components not shown in the example embodiment of  FIG. 2 , and/or the computing system  100  may have a different configuration or arrangement of the components shown in  FIG. 2 . The various components shown in FIG. may be implemented in hardware, software, or a combination of both hardware and software, including one or more signal processing and/or application specific integrated circuits. 
     Further, the acts of the processing methods described above may be implemented by running one or more functional modules in information processing apparatus such as general purpose processors or application specific chips, such as ASICs, FPGAs, PLDs, or other appropriate devices. These modules, combinations of these modules, and/or their combination with general hardware are all included within the scope of the present disclosure. 
     The methods disclosed herein suggest methods for 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 subsurface fluid reservoir depletion 
     REFERENCES CITED IN THE PRESENT DISCLOSURE INCLUDE THE FOLLOWING 
     
         
         1. Hilpert, M. and C. T. Miller. 2001. Pore-morphology 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 spheres.  Physica A: Statistical and Theoretical Physics  371(2): 336-360. 
         3. Pittman, E. D. 1992. Relationship of porosity and permeability to various parameters derived from mercury injection—capillary pressure curves for sandstone.  American Association of Petroleum Geologists Bull.  76(2): 191-198. 
         4. Zhang, J., T.-f. Wong, and D. M. Davis. 1990. Micromechanics of pressure-induced grain crushing in porous rocks.  J. Geophys. Res.  95: 341-352. 
         5. Wong, T.-f, C. David, and W. Zhu. 1997. The transition from brittle faulting to cataclastic flow in porous sandstones: Mechanical deformation.  J. Geophys. Res.  102: 3009-3025. 
         6. Fredrich, J. T., D. L. Lakshtanov, N. M. Lane, E. B. Liu, C. S. Natarajan, D. M. Ni, and J. J. Toms. 2014. Digital Rocks: Developing an emerging technology through to a proven capability deployed in the business. In  Proceedings of the Society of Petroleum Engineers Annual Technical Conference and Exhibition , Amsterdam, The Netherlands, 27-29 Oct. 2014. 
         7. Rasband, W. S. 1997-2015. ImageJ, U. S. National Institutes of Health, Bethesda, Md., USA, http://imagej.nih.gov/ij/. 
         8. Dougherty, R. and K. Kunzelmann. 2007. Computing local thickness of 3D structures with ImageJ.  Microsc. Microanal.  13: 1678-1679. 
         9. Zhu, W., L. Montesi, and T.-f Wong. 2007. A probabilistic damage model of stress-induced permeability anisotropy during cataclastic flow.  J. Geophys. Res.  112: B10207 
         10. Brzesowsky, R. H., C. J. Spiers, C. J. Peach, and S. J. T. Hangx. 2014. Time-independent compaction behavior of quartz sands.  J. Geophys. Res. Solid Earth  119: 936-956. 
       
    
     While the invention has been described with respect to a limited number of embodiments, those skilled in the art, having benefit of this disclosure, will appreciate that other embodiments can be devised which do not depart from the scope of the invention as disclosed herein. Accordingly, the scope of the invention should be limited only by the attached claims.