three-dimensional stochastic characterization of shale sem ...€¦ · transp porous med (2015)...

Transp Porous Med (2015) 110:521–531DOI 10.1007/s11242-015-0570-1

Three-Dimensional Stochastic Characterization of ShaleSEM Images

Pejman Tahmasebi1,2 · Farzam Javadpour1 ·Muhammad Sahimi2

Received: 26 March 2015 / Accepted: 27 August 2015 / Published online: 4 September 2015© Springer Science+Business Media Dordrecht 2015

Abstract Complexity in shale-gas reservoirs lies in the presence of multiscale networks ofpores that vary from nanometer to micrometer scale. Scanning electron microscope (SEM)and atomic force microscope imaging are promising tools for a better understanding ofsuch complex microstructures. Obtaining 3D shale images using focused ion beam-SEM foraccurate reservoir forecasting and petrophysical assessment is not, however, currently eco-nomically feasible. On the other hand, high-quality 2D shale images are widely available. Inthis paper, a new method based on higher-order statistics of a porous medium (as opposedto the traditional two-point statistics) is proposed in which a single 2D image of a shalesample is used to reconstruct stochastically equiprobable 3D models of the sample. Becausesome pores may remain undetected in the SEM images, data from other sources, such as thepore-size distribution obtained from nitrogen adsorption data, are integrated with the overallpore network using an object-based technique. The method benefits from a recent algorithm,the cross- correlation-based simulation, by which high-quality, unconditional/conditionalrealizations of a given sample porous medium are produced. To improve the ultimate 3Dmodel, a novel iterative algorithm is proposed that refines the quality of the realizations sig-nificantly. Furthermore, a new histogrammatching, which deals with multimodal continuousproperties in shale samples, is also proposed. Finally, quantitative comparison is made bycomputing various statistical and petrophysical properties for the original samples, as wellas the reconstructed model.

Keywords Shale-gas · Flow in nanoscale · Reconstruction · CCSIM

B Pejman [email protected]

1 Bureau of Economic Geology, Jackson School of Geosciences, The University of Texas at Austin,Austin, TX 78713, USA

2 Mork Family Department of Chemical Engineering and Materials Science, University of SouthernCalifornia, Los Angeles, CA 90089-1211, USA

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522 P. Tahmasebi et al.

1 Introduction

Thanks to recent improvements in fracking and horizontal-drilling technologies, gas and oilproduction from unconventional reservoirs, such as shale and tight gas, is becoming morefeasible (Hester and Harrison 2014). However, owing to the presence of the nanoscale porenetwork and the complexity of the governing equations for fluid flow in shale formations,characterization of such reservoirs is both tremendously difficult (Javadpour 2009; Darabiet al. 2012) and radically different from that of conventional reservoirs (Sahimi 2011). Shalereservoirs exhibit significant variability in their intrinsic properties, such as permeability,porosity, and mineralogy. Thus, traditional petrophysical methods, including those for bothwell and laboratory scales, cannot be used directly for characterization of such reservoirs.Flow in shale reservoirs is controlled mostly by pores in the organic matter (OM), inorganicmatter (iOM), and fractures (Naraghi and Javadpour 2015). Pores in these constituents giverise to complex multimodal pore networks in shale samples (see Fig. 1). Therefore, an accu-rate close-up study should be useful in improving the understanding of such reservoirs anddetermining their permeability and tortuosity.

Two-dimensional (2D) imagining studies (Javadpour et al. 2012; Loucks et al. 2012) arebecoming an essential part of shale characterization. The high-resolution focused ion beam-SEM (FIB-SEM)method is also gaining popularity for obtaining 3D images of pores in shalesamples. Yet, although the method reveals useful information about the pore network andpore connectivity in 3D, it is still expensive and time consuming. In addition, information onthe pores andmaterials of the milled layer could be lost during the milling process (Lemmenset al. 2011). Because of the high cost, computational difficulty, and the time-intensive effortrequired, providing 3D datasets for different samples has become a challenging problem.On the other hand, high-resolution 2D images can be obtained with ease and at a low cost(Tahmasebi and Sahimi 2012; Okabe and Blunt 2004).

In this paper, a new iterative method for reconstruction of 3D shale samples from a single2D SEM image is presented. A successful 3D reconstruction that is based on a single 2Dimage would considerably reduce both the cost and the time required. As described in thispaper, such a goal is achieved by using a new stochastic modeling method. Since somepores could remain undetected in the SEM images, data from other sources such as the pore-

Fig. 1 SEM image of Eagle Ford shale sample that shows pore structures (>100nm)

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Three-Dimensional Stochastic Characterization of Shale SEM. . . 523

size distribution obtained from nitrogen adsorption or capillary pressure data can also beintegrated with the pore network using an object-based technique (see below). Integration ofsuch data with the SEM images present a more realistic evaluation of shale samples. Thus,using the method proposed in this paper, one can even assimilate quantitative data with thequalitative shale images.

The rest of this paper is organized as follows. In Sect. 2, the algorithm that we havedeveloped is described, and an iterative method and a histogram-matching technique arealso introduced for more accurate reproduction of the bimodal porosity distribution in shalesamples. Section 3 presents the results of several sets of simulations, including a reproductionof the porosity in the OM and iOM. Section 4 summarizes the paper.

2 Methods

The proposed method in this paper is based on the cross-correlation-based simulation(CCSIM) (Tahmasebi et al. 2012; Tahmasebi and Sahimi 2012, 2013), which we have refinedby adding new concepts from optimization, in order to create a technique that improves theinitial model iteratively. The main idea of the CCSIM algorithm is based on the use of adigital image (DI) and the assumption that it represents the heterogeneities in a shale reser-voir (or any other reservoir for that matter). The DI must be selected carefully and shouldbe large enough to include most of the expected variations in and heterogeneity of a porousmedium. It should be noted that the one can consider several DIs for reservoirs with complexmorphological and spatial variations. Therefore, a set of the DIs can be viewed as a “library”of patterns than exist for any given shale reservoir. This allows a better coverage of the asso-ciated uncertainty in the structure of shales. Next, generation of an ensemble of realizationsof the shale sample within a stochastic framework is attempted after identifying the candidateDI. The method, in its original form, does not use any computationally intensive techniqueand provides a fast and efficient way of simulating complex media.

The basic CCSIM algorithm is summarized as follows. Let G represent the simulationgrid that is used to reconstruct a model of a shale sample. G is partitioned into smaller partscalled templates (grid blocks), denoted by T, while the data event—the temporary data thatmay change during the iteration—at position u inT is denoted byDT (u). The algorithm usesa 1D raster path along which G is constructed template by template. To maintain continuitybetween the patterns in the templates, an overlap (OL) region is defined. Then, using across-correlation function (CCF), the DI is sampled and one of its patterns is selected andsuperimposed on the current DT . The CCF quantifies the similarity between DI and DT andis given by,

C (i, j) =�x−1∑

x=0

�y−1∑

y=0

DI (x + i, y + j)DT (x, y) , (1)

where 0 ≤ i < Lx+�x−1 and 0 ≤ j < Ly+�y−1. DI (x, y) represents the location at point(x, y) of a target system of size Lx ×Ly , with x ∈ {0, . . . , Lx − 1} and y ∈ {

0, . . . , Ly − 1}.

The OL region DT of size �x × �y is used to match the pattern in the DI.The stochastically sampled data in theOL region are used for computing the CCF between

the OL and the DI; hence, a computed cross-correlation is ascribed to each selected pattern.Then, the patterns are sorted according to the numerical value of their CCFs, and the firstsorted patterns that indicate a high similarity with OL region (high CCF) are selected as thecandidate patterns. Eventually, a randomly selected pattern among the candidates is selected

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and assigned to theG. In this study, we used the first 10 candidates to select the final pattern.More information about the method is found in Tahmasebi et al. (2012, 2014).

The algorithm is used as the foundation for reconstructing 3D models from a single 2DDI and is refined further to adapt to the new situation, namely a shale sample having complexmorphology. To this end, the CCSIM algorithm in its conditional mode is utilized. In theconditional mode, hard (quantitative) data are given, and the reconstructed model is made tohonor them exactly. Generally speaking, the hard data are those that assume to be reproducedin the reconstructed model and act as constraints. In this context, the hard data are used tobetter control the generated model. Using the hard data, one can control the porous mediumand its structure (see Tahmasebi et al. 2012, 2014). Thus, the conditional CCSIM algorithmis used to generate the frames that maintain the exterior continuity of the porous medium.First, the original DI is set as the first layer (plane) at the bottom of 3D gridG. Then, the otherfour frames (i.e., front, left, back, and right) are generated using the CCSIM. Next, optimallocations of conditional (i.e., hard) data for the second layer are determined according tothe Shannon entropy, which is computed on the basis of pixel values in the image for eachtemplate. If the entropy is too high (implying that the template contains strong heterogeneity),the template is split into four equal, smaller templates, and the entropy is computed for eachsmaller template. The computation continues until the right template size (with low-enoughShannon entropy) is determined. We used a threshold of 0.5 for the normalized entropy inthis study. An example of template splitting is provided in Fig. 2. In the next step, a fraction ofthe data in the templates of the correct size is selected as the hard data and used to reconstructthe next layer. This method makes it possible to pick the point hard data efficiently in orderto carry the spatial variability to the next layer. Note that to preserve continuity betweenneighboring templates, each new layer is conditioned at the edges and the previous layer.Then, the next layer is generated using the extracted hard data. The procedure continues untilall the (2D) layers have been reconstructed. Eventually, all the generated layers are stackedtogether to create the reconstructed 3D model.

This algorithm is next integrated with an iterative scheme for further improvement. Theiterative process minimizes the discontinuities and improves the reconstruction quality of theglobal structure. Consequently, the similarity between the existing features in the input 3Dmodel obtained by the CCSIM and the DIs increases iteratively. For the increase to occur,

Fig. 2 An example of template splitting using the Shannon entropy for the shown image in (a). The optimaltemplate size is shown in (b)

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Fig. 3 Iterative 3D modeling: extraction of three data events in output 3D model (qx , qy , qz), with the bestthree candidates (px , py , pz) being found in the DIs with the size of 150 × 150 pixels (5 × 5 µm2) andproposed histogram matching being used for selection of one pixel to modify the output 3D

the algorithm must take the previously generated 3D model and proceed, voxel-by-voxel,according to the following method. For each voxel in the reconstructed model generated bythe CCSIM, three different patterns (qx , qy, qz) in three orthogonal planes passing throughthe voxel (i.e., the visiting point) are selected (Fig. 3). We used one-tenth of the DI as thesize of the qx , qy , and qz . Then, the CCF between the planes and the entire 2D DI imageare computed, thus generating three new 2D images in which every point or voxel is thecomputed CCF between the planes and the original 2D image. The minimum voxel in eachimage is identified, and the patterns (px , py, pz) corresponding to it in the original 2D imageare located. An average of the selected patterns may simply be assigned to the simulationvoxel, but the averagingmay produce artifacts and/or a too smooth of a 3Dmodel (see below).The above steps are repeated until all of the voxels in the 3D mode are visited.

To alleviate the artifacts, a new histogram-matching algorithm is proposed so that oneof the candidate voxels can be selected more efficiently. After each selected pattern px , py ,and pz is inserted into the 3D model, the histogram of the 3D model is constructed. Then,the distance dJS (p,q) between the resulting histograms of the three patterns (pi ) within thereconstructed 3Dmodel with the DI (qi ) is calculated. The distance is defined by the Jensen–Shannon divergence formula (Cover and Thomas 1991; Endres and Schindelin 2003), whichis the average of two Kullback–Leibler divergences:

dJS (p,q) = 1

2

∑ipi log

(piqi

)+ 1

2

∑iqi log

(qipi

). (2)

Next, the one that has the minimum dJS (p,q) is selected as the final value of the voxel inthe 3D model. As shown later, by conditioning the histogram of the 3D model to the DIs,both the global features and statistical properties of the sample are reproduced. Moreover,reproducing the histogram exactly is crucial because shale reservoirs exhibit a multimodaldistribution of pore sizes. Using the histogram of the DI as a constraint allows both nano-and micropores to appear in the final 3D model. Such a direct use of the DI patterns in thefinal model eliminates any possible artificial smoothness and artifacts. Indeed, histogrammatching helps in reproducing the global statistics, while simultaneously preserving localfeatures.

Implementing the computations in the frequency domain can reduce the CPU time sig-nificantly. The algorithm can, however, be integrated with a multiscale approach (Tahmasebiet al. 2014) that can be helpful by (1) accelerating computational modeling and (2) allowingthe algorithm to be applicable to larger templates, hence resulting in more accurate repro-duction of the structural and large-scale connectivity of the sample. Increasing the templatesize in the multiscale framework imposes no computational cost as well.

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Fig. 4 Application of the multiscale approach to a large DI (left). The algorithm first identifies a matchingpattern in the coarsest version of DI (S = 2), and then, its location (white dot) is projected onto the next finerDI (S = 1). Next, a cropped DI (white window) is considered the new search domain, and a matching patternis selected in the new area. Finally, location of the identified pattern along a new search window is projectedonto the original DI (S = 0), and the ultimate pattern is selected

A multiscale approach captures the larger-scale structures of the OMs and, therefore,generates a more accurate pore network. The approach in this paper is similar to that ofTahmasebi et al. (2014) in which the original DI is downscaled into a few successive imagesusing an interpolation method (see Fig. 4). In this paper, because small-scale pores in theOM must be dealt with, the nearest neighborhood is selected as the interpolator. After theDIs have been prepared at different scales, the algorithm proceeds as follows. The locationof the matching pattern is selected in the coarsest DI, which is S = 2 in Fig. 3. Then, theidentified location is projected onto the next finer DI, S = 1, and a search window is placedaround the transferred coordinate. This step not only reduces the CPU time, but also increasespattern reproducibility because (1) a lower resolution DI provides the same information asthat of a large 2D DI, and (2) the search space is limited to a cropped area in a finer DI. Thus,incorporating this approach significantly alleviates the CPU burden of the iterative method.

Note that orthogonal template patterns can be selected from different DIs. The ability toselect from various DIs is useful when dealing with a highly heterogeneous porous mediumin which patterns in each direction are different, i.e., there is significant anisotropy. It isalso worth mentioning that the described methodology can be adopted with non-stationarysystems and the cases in which various information need to be integrated (Tahmasebi andSahimi 2015a, b).

3 Results and Discussion

In this paper, an SEM image of a shale sample, shown in Fig. 5a, was used. The imageindicates a spatially stationary distribution of the OMs along their associated nanopores. AsingleDIwas therefore used for different directions, and 50 realizations using the given imageand the proposed method were generated. Three reconstructed (realizations) shale models of600 × 600 × 600 voxels are shown in Fig. 5b. As they manifest, the global structure in the2D DI was reproduced, which is also well connected. Thus, even a visual inspection revealsthat the proposed method is capable of preserving the pore structure, both vertically andhorizontally, which is necessary for accurate permeability prediction. If larger templates Tare used, however,Tmay decrease the variability between the realizations and, consequently,the uncertainty space will be underestimated.

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Fig. 5 a 2D SEM image of shale sample showing pore space within the OM. In this image, black spots poresin the OM, dark-gray the OM, and light-gray the iOM. b Three reconstructed models in which dark pointsnanopores in the OM. The size of these images is 9 × 9 × 9 (µm3)

The initial DI in Fig. 5a contains no pore space in the iOM. However, nitrogen adsorptionand helium porosity measurement of the sample suggest additional pores in the iOM partof the sample. Thus, to improve the model further, nitrogen adsorption data were used.Figure 6 presents the data. Then, the pore-size distribution of the iOM region was extractedusing data presented by Hall et al. (1986) and Naraghi and Javadpour (2015). The statisticswere then used to construct random pores whose sizes followed the statistics, and weredistributed randomly in the image of Fig. 5a. The result is shown in Fig. 7a. The resultingimage was then used to reconstruct a new 3D model, shown in Fig. 7b, which indicates thatthe vertical and horizontal connectivities are well reproduced and the new 3Dmodel containsa pore space consisting of both the OM and iOM. Finally, the reconstructed models shownin Fig. 7b were used to compute the statistics of the porosity distribution and tortuosity τ ,as well as the effective permeability. The effective permeability was calculated numericallyusing the Stokes’ equation, assuming that the fluid is incompressible and Newtonian, andthe flow is at steady state and in the laminar regime. We used the Aviso software (Version9, FEI Company). A pressure difference of 3 × 104 (Pa) [upstream = 13 × 104 (Pa), anddownstream = 10 × 104 (Pa)] was imposed on the system in one direction and used tocalculate the permeability. The fluid viscosity was assumed to be 0.001 (Pa s). The resultsare presented in Table 1.

The probability density of colors and the multiple connectivity probability (Krishnan andJournel 2003), i.e., the probability that r points are connected in a given direction, for boththe DI and the generated 3D realizations, are compared in Fig. 8 (for r = 100). According tothe results, the uncertainty space is well covered, and the reconstructed samples can forecastthe petrophysical properties. For example, there is good agreement between the probability

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0

0.2

0.4

0.6

0.8

1

1.2

1.4

1 5 25 125

PDF

of p

ore

size

dist

ribut

ion

Pore diameter (nm)

Fig. 6 Bimodal pore-size distribution of the DI sample based on nitrogen adsorption experiment (Naraghiand Javadpour 2015)

Fig. 7 a Integrated iOM and OM pores for representing the total porosity. b The result of the newly recon-structed, integrated model. Black spots and white ellipses show the porosity in the OM and iOM, respectively.c A cross-sectional view of b. The sizes of the images are the same as Fig. 5

density of the DI and the generated models, which indicates that all the voxels are reproducedand that they also follow the same distribution as in DI. Therefore, by using the describedhistogram matching in this algorithm, similar distributions is produced in stochastic models.

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Table 1 Statistics of petrophysical properties of the reconstructed samples

Volume Property

Avg. porediameter (µm)

STD (µm) Porosity (%) Keff (µD) T

Porosity in OM 0.029 0.006 2.5 1.61 5.59

Porosity in OM and iOM 0.107 0.003 9.53 4.18 3.6

Porosity in iOM 0.095 0.002 7.03 3.69 3.38

Top line shows the statistics based on the data of Fig. 3a. Second line consists of the statistics calculated on thebasis of Fig. 4a. The bottom line presents computed results for the iOM region on the basis of the reconstructedmodel of Fig. 4b

(a) (b)

Fig. 8 Comparison of a the density and b the connectivity probability. Black and gray curves are for the DIand the generated realizations, respectively. The density plot shows the relative likelihood of the 3D modelsto take a given value in a bin. The connectivity probability, p(r), represents the probability that r points areconnected. For example, the probability of seeing 10 connected points in the DI is 0.04

In order to evaluate the effect of the various parameters, some of the important factors,such as the size of the input DI and the effective permeability in distinct directions werestudied using various regions of interest. The results for the variability of the porosity and thesample size are shown in Fig. 9. Note that, because an optimal size of the input sample forthis study is around 11 µm2, a 2D image should be used with a size equal to and/or greaterthan the identified optimal size.

Next, the ability of the reconstruction method to generate the permeability anisotropywithin the porous medium was investigated by computing the effective permeabilities inthree directions (see Fig. 10). The predicted permeabilities are comparable with the resultspresented by Naraghi and Javadpour (2015). Roughly speaking, the median of the effectivepermeability in the Z (vertical) direction is smaller than those in the X and Y directions.

4 Summary and Outlook

A new method based on a higher-order statistical method—the CCSIM—is proposed, whichcan take a 2D shale image and reconstruct its 3D model accurately. This method first uses aninput 2DDI and reconstructs the second layer, which is then used to reconstruct another layer.

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4

5

6

7

8

9

10

11

12

1 3 5 7 9 11 13 15 17 19

Poro

sity

(%)

Sample size (µm2)

Fig. 9 Porosity variation versus sample size. Mean porosity in a sample size of 11 µm2 reaches a plateau

Fig. 10 Boxplot of the directional effective permeabilities versus different regions of interest, e.g., a size of27 = 3 × 3 × 3. For example, the dark blue boxes show the distribution of the effective permeabilities ofdifferent sizes in the Z (vertical) direction. Each box shows the data distribution from the first quartile to thethird. In each box, the median and mean are shown by a horizontal black line and black asterisk, respectively.The smallest and largest non-outliers are shown as two thin lines (after the dash lines) at the bottom and upperpart of box, respectively. The outliers are plotted as black points

The sequential reconstruction proceeds until the entire 3D porous medium is reconstructed.An adaptive template splitting method and hard-data sampling were used. Furthermore,a new iterative algorithm based on the histogram was introduced. The algorithm reducesartifacts and simultaneously increases pattern reproducibility. Moreover, due to exhibiting amultimodal distribution of the pore sizes, the histogram reproduction is crucial. Therefore,a new histogram matching using the Jensen–Shannon divergence was implemented, whichguarantees reproduction of a similar histogram in the reconstructed 3D models. Altogether,the new technique is able to accurately reconstruct the complex features in shale samples.Properties of the reconstructed models agree well with the original DI.

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The results also indicate that dual pore networks consisting of the pores in the OM andiOM are dominant in shale reservoirs, which must be simulated separately as two small- andlarge-scale problems. Our currently ongoing work is focused on bridging the gap between thephysics of shale reservoirs, consisting of both small and large pores, and 3D reconstructionmethods using a multiscale approach. The results will be reported in future papers.

Acknowledgments Work at USCwas supported by RPSEA, and at The University of Texas at Austin by theNanoGeosciences Lab. Dr. R. Reed provided the SEM images. We would like to thank FEI Company for useof Aviso software. We also thank Lana Dieterich for her help. Publication authorized by the Director, Bureauof Economic Geology.

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