Estimating 3D elastic moduli of rock from 2D thin-section images usingdifferential effective medium theory

Sadegh Karimpouli1, Pejman Tahmasebi2, and Erik H. Saenger3

ABSTRACT

Standard digital rock physics (DRP) has been extensively usedto compute rock physical parameters such as permeability andelastic moduli. Digital images are captured using 3D microcom-puted tomography scanners that are not widely available andoften come with an excessive cost and expensive computation.Alternative DRPmethods, however, benefit from the highly avail-able low-cost 2D thin-section images and require a small amountof computer memory use and CPU. We have developed anotheralternative DRP method to compute 3D elastic parameters basedon differential effective medium (DEM) theory. Our investiga-tions indicate that the pore aspect ratio (PAR) is the most crucialfactor controlling the elastic moduli of rock. Based on digital rockmodeling in a dry calcite sample with 20% porosity, the bulkmodulus is reduced by 51%, 80.7%, and 96.8% for aspect ratios

of 1, 0.2, and 0.05, respectively. Similarly, the shear modulus isreduced by 52%, 73.8%, and 92.8% for the same PARs. Thesefindings confirm the importance of the PAR in wave propagationthrough porous media. Such an evaluation, however, can be veryexpensive for 3D images because one requires using several ofthem for drawing a reliable conclusion. Therefore, we aim to cap-ture the PAR distribution from 2D images. This distribution is,then, used to estimate 3D elastic moduli of sample by DEM equa-tions. Three orthogonal 2D images were used and results indi-cated that 2D PARs in orthogonal orientations could addresspore shapes more effectively. Moreover, a stochastic porous me-dia reconstruction method was also used to generate more scenar-ios of rock structure and those of which that are not seen in 2Dimages. Results from Berea sandstone and Grosmont carbonateindicated that using only 2D images our proposed method couldeffectively estimate 3D elastic moduli of rock samples.

INTRODUCTION

Digital rock physics (DRP) is established based on the computa-tion of numerical properties from rock images. High-resolution mi-crocomputed tomography (micro-CT) scanners have made it possibleto obtain a 3D image of the internal structure of rocks, including porespace, fractures, and minerals at a small scale. These images aresegmented and, subsequently, numerical algorithms are applied tosimulate physical processes (e.g., fluid flow, wave propagation, andelectrical current to obtain permeability, elastic moduli, and forma-tion factor) in the digital rock medium. Recently, DRP has been ex-tensively used to compute rock physical parameters, and the resultshave been compared with laboratory data (Arns et al., 2002; Keehmet al., 2004; Saenger et al., 2004, 2016; Andrä et al., 2013b; Karim-pouli and Tahmasebi, 2016; Tahmasebi et al., 2016a, 2017; Karim-pouli et al., 2017b).

Although the high-resolution data generated using the imagingtools provide detailed information about the rock structure, thereare still numerous limitations in practice. For example, micro-CTscanners are expensive and are therefore not available widely. More-over, 3D numerical simulations are time consuming and a hugeamount of memory as well as high-speed computers are required.To alleviate such limitations, some alternative DRP methods haverecently been developed that use the widely available 2D imagesand still predict the 3D properties reasonably well, e.g., Saxenaand Mavko (2016), Saxena et al. (2017), and Karimpouli and Tah-masebi (2016). The main purpose of these methods is to estimatethe 3D properties of rocks using just 2D images. For example,Karimpouli and Tahmasebi (2016) reconstruct a rock in three di-mensions with a crosscorrelation-based simulation method (CCSIM)(Tahmasebi et al., 2016a; Tahmasebi, 2017) and reproduce velocity-

Manuscript received by the Editor 1 August 2017; revised manuscript received 14 January 2018; published ahead of production 09 March 2018; publishedonline 31 May 2018.

1University of Zanjan, Mining Engineering Group, Faculty of Engineering, Zanjan, Iran. E-mail: s.karimpouli@znu.ac.ir.2University of Wyoming, Department of Petroleum Engineering, Laramie, Wyoming 82071, USA. E-mail: ptahmase@uwyo.edu.3University of Applied Sciences, International Geothermal Centre, Bochum, Germany and Ruhr-University, Bochum, Germany. E-mail: erik.saenger@rub.de.© 2018 Society of Exploration Geophysicists. All rights reserved.

MR211

GEOPHYSICS, VOL. 83, NO. 4 (JULY-AUGUST 2018); P. MR211–MR219, 11 FIGS., 1 TABLE.10.1190/GEO2017-0504.1

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porosity and permeability-porosity trends. Besides being easier, lessexpensive, and computationally faster, they show that the results arecomparable with the standard DRP. Another alternative method wasintroduced by Saxena and Mavko (2016). They develop a power-lawrelation between the 2D and 3D elastic parameters of rock based on a2D plane strain computation through a thin-section image and em-pirical relations. They also apply this approach and introduce 2D-to-3D transformations for estimating the permeability using Kozeny-Carman and flow path division approaches (Saxena et al., 2017).In a recent work, Karimpouli et al. (2017a) apply such methodson a real case study and find promising results even in comparisonwith borehole data for clean samples. Basically, these alternativemethods use the available 2D thin-section images that are computa-tionally faster and require much less memory. Finding a representa-tive trend for rock physical parameters such as velocity-porosity and/or permeability-porosity trend is one of the main purposes of usingsuch methods (Dvorkin et al., 2011). These trends, indeed, are oftencontrolled by pore space heterogeneity such as pore size, type, andspatial distribution in all scales. In other words, trends are believed tobe scale invariant from microscales (DRP) to macroscales (laboratoryor well data) (Dvorkin et al., 2011).In this study, a new approach is introduced to estimate the elastic

parameters of rock using 2D images. Theoretically speaking, esti-mation of the effective elastic moduli of rock depends on (Mavkoet al., 2009) (1) the volume fractions of the individual components,(2) the elastic properties of each component, and (3) the geometricdetails of the components such as shapes, size, and spatial distribu-tions. Among all, pore type, which is usually expressed by the poreaspect ratio (PAR) — the ratio of smallest to largest radius of porespace, strongly affects seismic-wave velocities, especially in car-bonate rocks (Eberli et al., 2004). For example, for a given porosity,mineral composition and fluid type, velocity variation is due tochanges in pore shape (i.e., PAR) (Anselmetti and Eberli, 1999; As-sefa et al., 2003; Karimpouli and Malehmir, 2015; Karimpouli andTahmasebi, 2017). Sun (2004) finds that the variation of velocitycaused by different PARs could be up to 2.5 km∕s or even more,which highlights the importance of the PAR for interpretation andevaluation of rock seismic responses.To consider the PAR effect theoretically, several methods based

on effective medium theory have been developed. For example,Kuster and Toksöz (1974) introduce a formulation based on along-wavelength first-order scattering theory that evaluates the ef-fective elastic parameters of a medium containing different poretypes. The self-consistent approximation (O’Connell and Budian-sky, 1974) uses mathematical solutions for elastic deformation ofa single inclusion with a specific shape and approximates the inter-action of inclusions by replacing the background medium with theas-yet-unknown effective medium (Mavko et al., 2009). In differ-ential effective medium (DEM) theory (Berryman, 1992), modelsare built by incrementally adding inclusions with varying PARsto a matrix.On the experimental side, some studies have been carried out to

determine the effect of PAR on wave velocities in rocks basedon constructing real rock models with known pore structure andstudying their effect using laboratory measurements. For example,Ass’ad et al. (1992) embed penny-shaped rubber disks into an ep-oxy resin matrix. Rathore et al. (1995), and similarly Tillotson et al.(2012), entrench disc and penny-shaped aluminum foils into a sandand silica cement matrix, respectively, and leach them out by acids

to form cracks with known geometry. Recently, Wang et al. (2015)use soft penny-shaped silicone disks with physical properties sim-ilar to those of water and expandable polystyrene balls into cleancrystalline grains of carbonate cuttings as the matrix to build sam-ples with different PARs and sizes. Although the experimental stud-ies are realistic, controlling some parameters such as porosity value,embedding a range of PARs, and even constructing similar samplesis not straightforward. Alternatively, in this study, we use digitalrock modeling (DRM) (Garboczi and Day, 1995; Roberts and Gar-boczi, 2000) to explore the effect of pore geometry and shape onbulk and shear moduli of digital samples. In this method, syntheticrocks are considered as a digital image with distinct phases forporosity and minerals. This flexible method allows one to constructa synthetic model containing porosity and/or minerals with any ar-bitrary shape, size, orientation, and distribution. This model can besolved with a finite-element algorithm to compute the linear elasticproperties (Garboczi and Day, 1995; Andrä et al., 2013b). Robertsand Garboczi (2000) describe how DRM could analyze and help inour understanding of the actual behavior of porous materials.In this paper, we use the DRM to first show that PAR is the main

geometric factor affecting elastic properties. Second, we use PARdistributions, extracted from the original and reconstructed 2D im-ages, and DEM equations to approximate the aforementioned prop-erties. Many studies showed that DEM equations are appropriate, inpractice, for a wide range of pore shapes from cracks and fracturesto ball shape porosities in sandstones (Mukerji et al., 1995; Arnset al., 2002; Saenger et al., 2004) and carbonate (Xu and Payne,2009; Neto et al., 2015; Wang et al., 2015).

MODELING OF EFFECTIVE ELASTIC MODULI

DEM theory

In DEM theory, two-phase composites are modeled by incremen-tally adding inclusions of one phase (e.g., porosity) to the otherphase (matrix). When the porosity is zero, the matrix is consideredas phase 1. Porosity is iteratively added as phase 2 with a knowngeometry until it reaches the desired value. In each iteration, effec-tive bulk and shear moduli are computed and, then, used as a back-ground module for the next iteration (Mavko et al., 2009). Thecoupled system of ordinary differential equations for the effectivebulk and shear moduli, namely, K� and μ�, are (Berryman, 1992)

ð1 − yÞ ddy

½K�ðyÞ� ¼ ðK2 − K�ÞPð�2ÞðyÞ; (1)

ð1 − yÞ ddy

½μ�ðyÞ� ¼ ðμ2 − μ�ÞQð�2ÞðyÞ; (2)

with initial conditions K�ð0Þ ¼ K1 and μ�ð0Þ ¼ μ1, where K1 andμ1 are the bulk and shear moduli of the initial matrix (phase 1), K2and μ2 are the bulk and shear moduli of the incrementally addedporosity (phase 2), and y is the concentration of phase 2. For fluidinclusions and voids, y is equal to porosity ϕ. The terms P and Qare geometric factors that are described by Mavko et al. (2009), andthe superscript �2 on P and Q indicates that these terms are for theinclusion of phase 2 (porosity) in the background medium witheffective moduli of K� and μ�.

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Digital rock modeling

A digital rock often refers to a 2D or 3D numerical matrix witharrays labeled as pore or minerals. Relevant values are allocated toeach phase representing their physical properties. Physical simula-tions such as flow transport, electrical current flow, and elastic de-formation can be numerically computed through this model, whichresult in permeability, resistivity, and elastic-moduli/elastic-wavevelocity, respectively (Andrä et al., 2013b). A real digital rock isnormally produced using direct imaging. Some image processingalgorithms are applied for segmentation to usually obtain a binaryimage with two phases of pore and mineral (Andrä et al., 2013a).However, the synthetic digital sample can be generated numericallyin a predefined pattern of pore and mineral phases. By using DRM,one can generate various synthetic 2D or 3D samples with any poreand/or mineral shape, size, orientation, and distribution. The over-lap between different phases can also be controlled under the dryand saturated conditions. However, discretization still remains aprominent challenge. In other words, the model size must be largeenough, with respect to the largest feature (e.g., pore or mineralphase), to detract the discretization effect on the containing shape.For example, having a pore with a small PAR is not possible unlessan appropriately large model is used. Some examples of such syn-thetic rocks with known pore-size distributions are shown inFigure 1.

EFFECTS OF GEOMETRIC DETAILS OFINCLUSIONS ON ELASTIC MODULI

As mentioned before, we aim to explore the effect of pore geom-etry on the elastic properties using synthetic models. Geometricparameters are pore shape, size, orientation, and spatial distribution.Among them, pore size, orientation, and spatial distribution areassumed to be random in our generated models, which is acceptablein most cases of the real samples. However, pore orientation couldbe discussed in anisotropic media. We, here, just explore the effectsof pore shape (or PAR) and, thus, effects of other parameters areneglected by computing average properties of several generatedsamples.We used the PAR as a known criterion for generating different

pore shapes in a model. Accordingly, the PAR values of 1, 0.1,and 0.01 are considered as ball, discs, and penny shape or crackporosity. We developed an in-house 2D and 3D code to producedigital models with desired pore geometry (see Figure 1). Compu-tation time is one of the main limiting factors for studying 3D mod-els. The runtime for 3D models is much higher than 2D ones.Therefore, because a sensitivity analysis is an aim of this study, the2D models are studied and the results are extended to three dimen-sions using an alternative DRP method introduced by Saxena andMavko (2016). Then, the effect of pore structure is studied in threedimensions.Three different models, such as, A, B, and C with PAR of 1, 0.2,

and 0.05, respectively, were generated with porosities of 5%, 10%,15%, and 20%. Table 1 summarizes characteristics of these models.For having porosities with the same size, but different PAR, largeradii of pores were fixed and, subsequently, small radii of pores wereaccordingly computed using the PAR values. Figure 2 illustratessome of these synthetic models. Pores were considered to have a ran-dom size, distribution, and orientation, which is realistic and commonin a real rock. To extend the generality of our proposed method, three

models were generated in each case (i.e., for a specific PAR andporosity value), and an average value was considered as the result.In each model, calcite elastic properties (K ¼ 68.3, μ ¼ 28.4 GPa)

were allocated to the mineral phase. For the pore phase, air properties(K ¼ μ ¼ 0) were assigned to mimic the dry rock condition. To havea fluid-saturated condition, fluid properties can be used instead of air.Then, using the method proposed by Garboczi and Day (1995),effective elastic parameters of each model were numerically calcu-lated by a finite-element linear elastic algorithm. Figure 3 illustratesthe results of the bulk and shear moduli of synthetic rock samples byDRM (solid lines). As a validation, the elastic properties of thesemodels were also computed using DEM equations (the dashed lines).Figure 3 shows that the results by these two methods are highly com-parable. However, the power of DRM is to compute physical proper-ties of models with a mixture of arbitrary pores and mineral shapes,which is not the case in this study.According to these results, in each pore type (or PAR), elastic

moduli generally decrease nonlinearly with increasing the porosity.For example, in a dry rock with 20% porosity, the bulk and shearmoduli are reduced to 45% and 38% for the PAR of 1, 68% and 50%for the PAR of 0.2, and 93% and 82% for the PAR of 0.05, com-pared with a sample with zero porosity. This implies that the reduc-tion rate of elastic moduli also increases with declining PAR.

Figure 1. Some examples of (a) 2D and (b and c) 3D synthetic digitalrock model with a specific pore size and geometry distribution.

Table 1. Characteristics of the generated models to study thePAR effect.

Model PAR Pore shapeMajor and minorpore radii (pixel)

Porosityvalue (%)

A 1 Ball 20, 20 5, 10, 15, and 20

B 0.2 Disc 20, 4 5, 10, 15, and 20

C 0.05 Penny 20, 1 5, 10, 15, and 20

Figure 2. Examples of the generated models (a) A, (b) B, and (c) Cwith PARs of 1, 0.2, and 0.05, respectively (see Table 1).

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Moving from a ball to a disc shape, and also to the penny-shapepore type, makes the rock softer, which gives rise to a reductionin the wave velocity of the sample.

ESTIMATING ELASTIC MODULI OF ROCK FROM2D IMAGES

Based on the results shown in the previous section, one can ob-serve that the elastic parameters of rock are highly sensitive to PARdistribution. Therefore, the main idea in this study is to reproducethe distribution of PAR values from 2D images and, finally, com-pute the elastic parameters of the 3D medium using DEM equations.Here, the question is: How comparable are 2D and 3D PARs? Inaddition, How do we construct the distribution of PAR values from2D images? In fact, instead of using one value for describing thePAR of a 3D pore, three PAR values obtained from 2D orthogonalsections of pores can be used, which in turn is more informative toexplain an arbitrary pore shape. According to Figure 4, in the 3Dcase, the PAR is defined as C∕A, which does not contain any in-formation along the y-axis (B). On the other hand, by using threePAR values for 2D cases (B∕A;C∕A;C∕B), the pore shape can bedescribed more effectively.In this study, three orthogonal 2D images are used in any case

(see Figure 5a). However, one may discuss that in Figure 4, 2DPARs come from one pore, whereas, in reality they are not acces-sible for all pores in just three orthogonal 2D images. This may be atrue statement, but due to the implemented statistical basis in this

approach, other PAR values will likely be considered using simu-lated 2D images and frequency histogram of PAR. To add morevariance to PAR values, the CCSIM is used to reproduce those prob-able pores that are not seen in these 2D images, but may exist in thereal rock sample (Figure 5b). In each 2D image, individual porescan be distinguished, and the corresponding PARs are computed.Using the PAR values obtained from three 2D images and severalsimulated images, a distribution of PARs are reproduced (see Fig-ure 5c). The DEM equations are then solved by applying PARs toestimate the 3D elastic parameters of a rock sample (Figure 5e). Asimple graphical flowchart is shown in Figure 5.

CCSIM

The conditional reconstruction method used in this paper is basedon the CCSIM algorithm (Tahmasebi and Sahimi, 2012, 2016; Tah-masebi et al., 2016b). In this algorithm, a 1D raster path startingfrom one corner of the simulation grid and ending in the other cor-ner is implemented. First, a random pattern is selected from the ini-tial digital image and inserted in the simulation grid. To keep thesimilarity of the current pattern with the next one, an overlap regionis used for finding a similar candidate pattern. At the end, a cross-correlation function is applied. The more candidates that exist, themore variable the realization will be. Due to the complexity of theused images, in this study, we used five candidates.

PAR computation

For using the PAR equations, an ellipse is fitted to the pore space.PAR is defined as the ratio of the smallest to the largest radius ofthe ellipse. As illustrated in Figure 6, for a digital rock image withan arbitrarily shaped porosity, the porosity (or here we call it mass)and mineral phases are considered to be one and zero, respectively.The center of porosity (or mass of the image) (ic, jc) (Figure 7) canbe obtained using the moments of the image along each axis dividedby the total mass of the image (Corke, 2017):

ic ¼M10M00

jc ¼M01M00

; (3)

where M10 and M01 are the moment of the image around the iand j axes (Figure 7) and M00 is the total mass of the image.

Figure 3. Crossplot of (a) bulk and (b) shear moduli of differentmodels A (circle), B (square), and C (diamond) with different PARsin dry (solid line) and water-saturated (dashed line) conditions.

Figure 4. There are one and three PARs in the 3D and 2D cases,respectively. It is clear that three PARs obtained from orthogonalsections can characterize the pore shape more realistically.

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Subsequently, the inertia matrix of the image can be further definedas (Corke, 2017)

IM ¼�τ02 τ11τ11 τ20

�; (4)

where τ02 and τ20 are the central second-order moment (moments ofinertia) and τ11 is the axial second-order moment (product of iner-tia). The equivalent ellipse (Figure 7) is the ellipse that has the sameinertia matrix as the porosity. The eigenvalues and eigenvectorsof IM are related to the radii of the ellipse and the orientationof its major and minor axes. The radii of the equivalent ellipseare (Figure 7)

A ¼ 2ffiffiffiffiffiffiffiffiffiγ1M00

r; B ¼ 2

ffiffiffiffiffiffiffiffiffiγ2M00

r; (5)

where γ1 and γ2 are the eigenvalues of IM. PAR of the equivalentellipse (or porosity) is calculated using

PAR ¼ BA: (6)

Inserting the PAR distribution in DEM equations

Each pore type is expressed by a value of PAR. Then, the PARdistribution represents the frequency of each pore type with a spe-cific PAR. Therefore, by dividing the PAR range (i.e., [0, 1]) by “n”

parts (bin number of histogram), the fractional porosity of each binvalue (PAR1 : : : PARn) relative to total porosity (ϕ) can be com-puted by the PAR frequencies (f1 : : : fn). Using these values, themodel starts with phase 1 (or a host mineral) and the proportionalporosity of f1 × ϕ is incrementally added with PAR of PAR1 for thefirst part (or pore type). The term PAR1 is used for calculating thegeometric factors (P� and Q� in equations 1 and 2). Then, the ef-fective elastic properties of the medium (K�1, μ

�1) are computed using

DEM equations (equations 1 and 2). In the second step, these valuesare considered as background properties (or the host phase) and theproportional porosity of f2 × ϕ is incrementally added with PAR ofPAR2. This procedure is continued until all parts are added and thefinal effective elastic properties (K�3D; μ

�3D) are computed.

DIGITAL ROCK SAMPLES

Andrä et al. (2013a) introduce a set of benchmark digital rocksamples such as Berea and Fontainebleau sandstone and Grosmontcarbonate. In this study, we used the segmented samples of Bereasandstone and Grosmont carbonate to evaluate the proposedworkflow.

Berea sandstone

Berea sandstone is mostly composed of quartz grains with a mi-nor content of clay, K-feldspar, ankerite, and zircon. Petrographystudies, as well as microprobe results, have demonstrated an iso-tropic solid matrix in this sample (Madonna et al., 2012). The con-nected laboratory porosity of the used sample is approximately 20%with permeability between 200 and 500 mD. Andrä et al. (2013b)

Figure 5. Flowchart of the proposed method.

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show that the average computed values of the bulk and shear moduliof a 700 × 700 × 1024 digital sample are 19.7 and 21.5 GPa, re-spectively. In this study, a 400 × 400 × 400 sample is extracted fromthe original digital sample (10243) (see Figure 8a) (Andrä et al.,2013b). The porosity of this sample is 20%, and its computed bulkand shear moduli are 19.1 and 20.1 GPa, respectively.

Grosmont carbonate

Grosmont carbonate is composed of dolomite and karst breccia.The laboratory results show that its porosity is approximately 21%with a permeability between 150 and 470 mD. Simulation resultsfrom Andrä et al. (2013b) indicate that, for a 10243 digital sample,the porosity is 24.7%, and the computed bulk and shear moduli are23.2 and 11.5 GPa, respectively. In this study, a sample with the sizeof 200 × 200 × 200 voxels is extracted from the original segmentedsample (10243) (see Figure 8b) (Andrä et al., 2013a). The porosityof this sample is 25.4%, and its computed bulk and shear moduli are23.1 and 13.4 GPa, respectively.

RESULTS

According to the proposed method, four sets of 2D orthogonalimages were arbitrarily extracted from each 3D digital rock sam-ples. Each set contains three 2D images in the x-, y-, and z-direc-tions, as shown in Figure 5a. Requirements of the CCSIMalgorithm, such as template size, overlap size, the number of can-didates, and other parameters (Tahmasebi, 2017), were adjusted ineach set to generate the most reliable reconstructed 2D images. Foreach 2D image, 10 realizations are reconstructed, which results in33 (3þ 3 × 10) images for each set.To find the PAR values in each 2D image, after implementing a

median filter, individual pores are detected and separated using awatershed algorithm (Beucher and Meyer, 1992). Then, the eigenval-ues of the inertia matrix of each pore are calculated, and the corre-sponding radii of the equivalent ellipses are obtained, which are thenused for calculating the PAR. After calculating the PARs of all 2Dimages, the corresponding histogram frequency of PARs is computed.Then, the PAR values and their corresponding frequencies are used inthe DEM equations and, subsequently, the elastic parameters of the3D medium of the rock are computed. The average porosity of thethree actual images is used as the porosity of the final 3D model.Figure 9 illustrates the results for Berea sandstone and Grosmont

carbonate. In each sample, the actual trend of the rock was producedusing subsamples extracted from the original sample (Karimpouliand Tahmasebi, 2017). The size of subsamples was half of theoriginal sample (i.e., 2003 and 1003 for Berea and Grosmont, respec-tively). Thus, eight subsamples are extracted in each case. We appliedthe finite-element method of Garboczi and Day (1995) to computethe static elastic moduli of the 3D sample and subsamples. Thismethod solves the basic Hooke’s law equations of linear elasticity.

Figure 8. The 3D view and an orthogonal section of (a) Berea sand-stone (4003) and (b) Grosmont carbonate (2003) used in this study.

Figure 7. A schematic example for computing PAR from digitalrock image. The white and dark cells are supposed to be the mineraland pore with zero and one values, respectively.

Figure 6. Sensitivity analysis of bin number (n) of the PAR histo-gram on bulk modulus computation for (a) Berea sandstone and(b) Grosmont carbonate. Original values were obtained from origi-nal trends. In the linear division, the PAR range of [0, 1] is linearlydivided to n bins, whereas in the 100 nonlinear case, each range of[0, 0.1] and [0.1, 1] is divided into 100 bins individually. The 100nonlinear division is similar to the 1000 linear division.

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Due to multiple simulation steps, in each set of 2D images, differ-ent values are obtained for the elastic moduli in each run of thealgorithm. As mentioned, the reconstructed images contain variouspore structures, which induces more variability in the ultimate re-sults. This variation changes in each set depends on the pattern ofthe pore structure of the actual 2D images. According to Figure 9,the bulk modulus is reasonably well-estimated in both cases. How-ever, the shear modulus is slightly higher than the real trend but isstill acceptable. The main reason for obtaining such promising re-sults is that the method is based on the distribution of PARs, the keyparameter for the evaluation of elastic moduli. A comparison withresults computed by Andrä et al. (2013b) also shows how reliablethis method is.

DISCUSSIONS

There are some critical points during the computation process,which are discussed here. The first point is tiny linear pores (Fig-ure 10). Because the smallest radius of the equivalent ellipse is com-puted as zero for these pores, their PAR is also zero. These are, infact, penny-shaped pores or microcracks with PARs of 0.01, whichstrongly reduce the elastic properties of the rock. Our investigationsshowed that an empirical value in the range of0.01–0.02 is reasonable to replace the PARs ofthese pores. In our cases, values of 0.018 and0.01 were allocated for Berea sandstone andGrosmont carbonate, respectively (Figure 9).The other point is the bin number of the histo-

gram. To compute the histogram plot, the PARrange is divided by n number of bins and thefrequencies are counted in each bin. The centervalue of each bin is assumed as PAR, whichmeans the larger the bin, the more averaged isthe value for the PAR is used. It is inevitable toincrease the bin number to have a more accuratecomputation. For the investigation of this binnumber effect, we selected those results, whichare very close to original values (obtained fromoriginal trend), with a trial and error method inboth samples. Then, the bin number was changedalong a wide range from 5 to 1000 linearly alongPAR values (i.e., [0, 1]). The results displayed inFigure 6 show that with increasing bin number,the results converge to the original value. Itshould be noted that not all values of PAR needto be divided linearly because their effect is notsimilar. According to Figure 3, it is implied thatthe effect of pores from ball to disc shape (PARof 1 to 0.1) is comparable with pores from disc topenny-shaped pores (PAR of 0.1 to 0.01). Therefore, we dividedeach of [0, 0.1] and [0.1, 1] into 100 bins and called it nonlineardivision. Our results showed that 100 nonlinear divisions canproduce the same result as 1000 linearly division but in a shortertime.The final point is the effect of importing large to small and small

to large PAR pores to DEM equations. For multiple inclusionshapes, the effective moduli depend not only on the final volumefractions of the constituents but also on the order in which the in-cremental additions are made (Mavko et al., 2009). To explore thiseffect, three 2D orthogonal images from each Berea sandstone and

Geosmont carbonate were selected, and the corresponding elasticproperties were computed using the proposed method (Figure 5)for 10 iterations. In each iteration, PAR values were arranged toimport from (1) small to large and (2) large to a small value duringcomputation. Figure 11 shows these results. According to thisfigure, when PARs are introduced to DEM equations from largeto small values, an approximately 2% and 3% softer sample is ob-tained relative to the condition when small to large PARs are im-ported for Berea sandstone and Grosmont carbonate, respectively.Because the effect direction is not significant, one could either usethe average of both directions or just use a single direction andneglect the other direction.The presented results in Figure 4 indicated that among all geo-

metric parameters, the pore shape (or PAR) is the most importantparameter that highly affects the elastic moduli. Thus, the core ofthis method is, indeed, the DEM equations that are solved withPARs obtained from the real and reconstructed media of rocks.Although these PARs are 2D, orthogonal cross sections along threemain axes compensate for the lack of PAR information in three di-mensions. They could even address the shape of a 3D pore betterthan one 3D PAR value does. The implemented method in thispaper is fast and computationally effective, and it requires a small

Figure 9. Bulk and shear modulus of (a) Berea sandstone and (b) Grosmont carbonateobtained from the proposed alternative DRP method. Results by Andrä et al. (2013b) arefrom a cropped sample with 4003 pixels from original 10243 sample. The real trend ineach plot is obtained using 3D subsamples extracted form a real image of the rock.

Figure 10. Penny-shaped pores or microcracks. The white and darkcells are the mineral and pore, respectively. The PAR of these kindsof pores should be an empirical value in the range of 0.01–0.02.

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amount of memory. Two reasons explain the computationalefficiency of our proposed method: (1) a fast 2D reconstructionand (2) an efficient analytical computation. Going from 2D samplesto 3D samples is not only computationally expensive, but it alsorequires extensive knowledge and assumptions. Therefore, in thispaper, the 2D samples are used directly. Furthermore, performingthe computations on a 3D sample can take several hours. In contrast,the proposed method only uses the 2D samples and conducts thecomputations on the same samples, which only takes around afew seconds (approximately 30 and 20 s for the Berea and Gros-mont samples, respectively). Input data are 2D thin-section images,meaning that the proposed method can be used along with or insteadof the standard DRP, especially when high-quality 3D images arenot available. A notable application, which is expected for thismethod, is estimating elastic parameters using cuttings from drill-ing. We are working on this problem, and the results will be re-ported in a future study.

CONCLUSION

Alternative DRP methods, working with simple ubiquitous 2Dthin-section images, have recently been developed due to their fastcomputation, their required data, and a small amount of memoryusage. In this paper, an alternative method is introduced to computethe elastic moduli of rock samples. Our investigation also showedthat PAR is an important parameter controlling the elastic moduli ina determined porosity. The proposed method solves the equations ofthe DEM equations, and geometric coefficients are obtained usingPAR histograms extracted from thin-section images of rock samplesin pore scale. Because these images are in two dimensions, threeorthogonal images were proposed to be used and, therefore, three

PARs were used instead of one 3D PAR. Our results showed thatthree PARs could effectively explain the effect of the pore shape. Toreproduce a representative PAR distribution, the CCSIM recon-struction method was used and several realizations were recon-structed for each orthogonal 2D image. This increases the varianceof the PAR values and reconstructs those pore shapes that cannot beobserved in the actual 2D images but may exist in the real sample.The obtained results in this paper revealed that the proposed

method could efficiently estimate the elastic moduli of Berea sand-stone and Grosmont carbonate. The values of the bulk modulus ofrocks, predicted by this method, are close to the real trend of therock samples obtained from 3D images, whereas the shear moduluswas slightly overestimated.

ACKNOWLEDGMENT

Authors would like to thank C. Werner for the very useful hints toimprove the manuscript.

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