A FeatureBased Stochastic Permeability of Shale: Part 1—Validation and TwoPhase Permeability in a Utica Shale Sample
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Abstract
Estimate of permeability plays a crucial role in flowbased studies of fractured tightrocks. It is well known that most of the flow through tightrocks (e.g., shales) is controlled by permeable features (e.g., fractures, laminations, etc.), and there is negligible flow through the matrix. However, current approaches in the literature to model permeability of tightrocks do not account for such features present within the rock ranging from microscale to fieldscale. Current permeability modeling approach assumes a single continuum without considering the presence of permeable features within the matrix (e.g., microfractures) or outside the matrix (e.g., natural fractures). Although the laboratorymeasured permeability implicitly captures discrete features present in that sample (e.g., fractures, laminations, microfractures), most of the permeability models proposed for shale do not account for these features. Fracture permeability in the literature is typically modeled using an ideal slit assumption; however, this highly overestimates its permeability because fractures in real medium are nonideal in terms of their porosity and tortuosity, which affect their permeability. Additionally, the transition zone between fracture and matrix also affects the permeability of fracture. In this study, part of a twopart series, a new method to predict permeability of fractured shale by discretizing the medium into matrix (inorganic and organic) and fractures is presented. New analytical expressions of permeability are derived to account for nonideal nature of porous medium and twophase flow in fractures. Rock feature in each cell of the grid is identified as one of the three elements (organic matter, inorganic matter, or fracture), and permeability of that cell is estimated using a suitable analytical expression. This method allows estimating permeability at any scale of interest and more robustly than by a pure analytical approach. The proposed method is validated against local and globalscale measurements on three fractured samples from laboratory. Finally, the method is used to predict twophase flow permeability of supercritical CO_{2} displacing water within a fracture in a Utica shale sample. The proposed twophase flow permeability equations can be used as a quick analytical tool to predict relative permeability estimates of twophase flow in fractured shale samples. In Part 2, the proposed method is used to estimate fieldscale permeability through an optimization process that uses fieldscale production and other readily available information.
Keywords
Fractures Tightrock Relative permeability Two phase Utica1 Introduction
The primary objective of this paper is to propose a new method to estimate permeability of fractured tightrocks that can account for distinct features impacting the permeability at any given scale (Fig. 1). The approach and modeling for the proposed method, including derivation of new analytical expressions for fracture flow permeability, are discussed below in detail. Following its description, the proposed method is validated against three fractured samples from laboratory. Finally, the method is applied to predict twophase flow permeability of scCO_{2} displacing water through a fracture etched on a Utica shale sample.
2 Method
The proposed method to model permeability is motivated by the presence of permeable features at microscale (e.g., microfractures, imbibition from fractures to nearby matrix) to larger scales (e.g., laminations and fractures) that impact the permeability at any given scale as shown schematically in Fig. 1.
2.1 Approach
 Step 1: In order to accurately model the permeability of a fractured rock, including its direction dependence anisotropy, the fractured rock is first discretized into a grid by superimposing the elements of the fractures and the matrix on a single grid as shown schematically in Fig. 3. This step essentially involves developing separate grids for fractures and matrix (organic and inorganic) that are later superimposed on a single grid. The term fracture is used generically, which also includes microfracture.

Step 2: Identify rock feature in each cell of the grid as matrix (organic and inorganic) or fracture and calculate permeability by using appropriate expressions. Permeability of the rock matrix (divided into organic and inorganic part) is calculated using the model proposed by Singh et al. (2014), whereas the permeability of fracture is calculated using analytical expressions derived in this study from steadystate Navier–Stokes equation for single and twophase flow.
2.2 Fracture Discretization
2.2.1 Fracture Attributes
Fracture attributes and their statistical distributions
Fracture attribute  Distribution  Expression  Parameters 

Number and location  Poisson point process  \( P\left[ {N\left( A \right) = n} \right] = \frac{{e^{  \lambda \left( A \right)} \cdot \left\{ {\lambda \left( A \right)} \right\}^{n} }}{n!}, \)n = 0, 1, 2,…  λ(A) 
Length  Power law  \( l_{\text{f}} = \left[ {l_{\hbox{min} }^{{1  \alpha_{\text{l}} }} + F_{\text{cdf,l}} \left( {l_{\hbox{min} }^{{1  \alpha_{\text{l}} }}  l_{\hbox{min} }^{{1  \alpha_{\text{l}} }} } \right)} \right]^{{\frac{1}{{1  \alpha_{\text{l}} }}}} \)  \( l_{\hbox{min} } , l_{\hbox{max} } ,\alpha_{\text{l}} \) 
Aperture  Power law  \( a_{\text{f}} = F_{\text{a}} \cdot \left( {l_{\text{f}} } \right)^{{\kappa_{\text{a}} }} \)  \( F_{\text{a}} , \kappa_{\text{a}} \) 
Orientation  Fisher  \( \theta_{\text{f}} = \cos^{  1} \left\{ {\frac{{\ln \left[ {e^{{K_{\text{fisher}} }}  F_{{{\text{cdf}},\theta }} \left( {e^{{K_{\text{fisher}} }}  e^{{  K_{\text{fisher}} }} } \right)} \right]}}{{K_{\text{fisher}} }}} \right\} \)  \( K_{\text{Fisher}} \) 
Here, cumulative distribution function in above expressions is given as \( F_{{{\text{c}}\,{\text{df}},x}} = \mathop \int \limits_{{x_{\hbox{min} } }}^{{x_{\hbox{max} } }} f_{{{\text{pdf}},x}} \cdot {\text{d}}x \).
2.3 Matrix Discretization
The rock matrix of tightrock is assumed to be composed of kerogen (OM) and inorganic matter (iOM). The matrix is divided into OM and iOM using the following method that spatially samples OM in a matrix based on values of bulk density and porosity, two rock quality parameters most widely available at both laboratoryscale and fieldscale. Although, some other methods exist to spatially sample the OM (Naraghi and Javadpour 2015; Tahmasebi et al. 2015; Yang et al. 2015), the advantage of this method is its computational simplicity while also honoring the physics of finding OM (small porosity and density).
2.3.1 Spatial Sampling of Organic Matter
 1.
Find probability map of physical occurrence of OM:
 2.
Locate OM positions on grid with a Bernoulli distribution:
 3.
Constrain the volumetric content of the OM:
Calculate the total volume of the cells occupied by OM as calculated in step 2. If the total volume of the cells is equal to the observed total organic carbon (TOC) for this rock, then save the map, otherwise go to step 2 and repeat the process until the observed volumetric TOC content is satisfied.
2.4 Analytical Permeability Expressions
Discretization of the rock into fractures and matrix (OM or iOM) helps in using appropriate permeability expression for a feature in each cell. Rock feature in each cell is identified as one of the three elements (fracture, OM or iOM), and permeability of that cell is calculated using an appropriate analytical expression. Before calculating permeability, appropriate properties are assigned to each cell, such as pore (or aperture) size, type of fluid, fluid property.
Although there has been a good progress in the proposed analytical expressions for permeability of matrix flow, the progress in deriving analytical expressions for permeability of fractures has been mostly focused to assess the impact of stress mechanics (Cho et al. 2013; Ma 2015) with relatively less effort (Liu et al. 2015; Chen et al. 2016; Zhang et al. 2017b) in studying impact of other parameters. We address this concern to some extent by (1) incorporating porous media effects (porosity, tortuosity, transition zone between fracture and matrix) for fracture flow, and (2) deriving new expressions for twophase flow permeability in fracture with capillary effect. Details of the proposed analytical permeability expressions in fracture are given below. Description of each parameter is given in Sect. 6 (Nomenclature).
2.4.1 Permeability Models for Fracture Flow
Fracture permeability in real porous medium cannot be defined using an ideal permeability of slit because of nonideal nature of fracture in porous medium, such as its length, its porosity, and its tortuosity, which affect the permeability. Additionally, the transition zone between fracture and matrix also imbibes some fluid from fracture walls into the matrix, which affects the permeability of fracture.
To account for nonideal nature of fractures, including single and twophase flows in fractures, new analytical expressions are derived. Permeability for single and twophase flow in fracture is derived by solving steadystate Navier–Stokes equation. Here we only present the final expressions, and complete derivation can be found in Appendix A. The final expression for fracture permeability in a porous medium is given a subscript pm to indicate porous media. Description of each parameter is given in Sect. 6 (Nomenclature).
TwoPhase Permeability of Gas and Water
The above twophase permeability model was derived for gas as the nonwetting phase and water as the wetting phase as depicted by the conceptual model in Fig. 22. The approach used in deriving these expressions does not differentiate between the type of fluid besides their wetting and nonwetting property. Therefore, these expressions are equally valid for oil and gas phases or for oil and water phases.
2.4.2 Permeability Model for Matrix
Although the permeability of OM changes with desorption (function of pressure), the permeability considered here is only at a given pressure. Another other difference between the permeability of OM and iOM is because of their different pore sizes (and porosities). In general, permeability of OM is higher than permeability of iOM because of their larger pores, which can be about four times larger than the pore size in iOM (Ma et al. 2016). It is possible that OM and iOM may occur at different scales at the scale of interest under study, which can be accounted by considering a grid with nonuniform cell sizes.
2.5 Upscaling GridBased Permeability to a Single Value
There are other complex methods of permeability upscaling (Masihi et al. 2016) such as effective medium theory (Dagan 1979; Ghanbarian and Javadpour 2017) and critical path analysis (Hunt and Idriss 2009); however, the advantage of geometric average is that, despite being a simple approach, it has been found to match well with the observed permeability values.
3 Results
We validate our proposed method by reproducing the measured values using three different laboratoryscale experimental studies, each with a distinct fractured rock sample. Following the validation, the method is applied to predict twophase flow permeability of scCO_{2} displacing water through a fracture etched on a Utica shale sample.
3.1 LaboratoryScale Validation
3.1.1 Fractured ClayRich Shale Sample
Validation of Sample Porosity, and Permeability of Gas and Water
3.1.2 Fractured Carbonate Sample
Validation of Sample Porosity and Permeability
3.1.3 Fractured Cement Core
Validation of Sample Porosity, Permeability, and Pulse Decay
3.2 TwoPhase Flow Permeability in a Fractured Shale Sample
In above expressions, saturations of scCO_{2} and water at any given time are extracted from the images shown in Fig. 15 as the fraction of each fluid in fracture space. The other required parameters in above expressions are given as p_{c} = 10.34 MPa and p_{p} = 8.3 MPa, \( \alpha = \frac{{\mu_{\text{g}} }}{{\mu_{\text{w}} }} = 0.053 \). The other parameters that were not provided in Porter et al. (2015), but are required for our twophase permeability model are c_{f} and χ, which we assume as ~ 1.5 MPa^{−1} and 0.8, respectively, based on range of values for these two parameters from 9 different shale formations (Fink et al. 2017). Additionally, a constant value of matrix porosity was assumed as 4%.
3.2.1 Relative Permeability of Water and scCO_{2} in a Fractured Utica Shale Sample
To estimate the twophase permeability, we assumed S_{wr} = 0 because of lack of data. It should be noted that although the relative permeability estimates for the three saturations (S_{w} ~ 0.66, 0.69, and 0.75) were computed with an assumption of S_{wr} = 0 because of lack of data, model estimates (Eqs. 20 through 25) at other saturations (shown with dashed lines) predict that “effective” (approaching k_{ri,f} ≈ 0) residual water saturation is S_{wr} ∼ 0.20. Another interesting point to note from Fig. 18 is that, although the proposed twophase model does not assume any residual saturation for the gasphase, “effective” (approaching k_{ri,f} ≈ 0) residual gas saturation based on k_{rg},_{f} ≈ 0 in this sample can be considered as S_{gr} = 0.20.
3.3 Impact of Grid Resolution on Predictions
It must be noted that permeability prediction is a function of grid resolution just like other gridbased prediction methods whose global values can be constrained to some extent, but changing a grid resolution would affect their regional values. This is the hypothesis behind upscaling (Singh et al. 2014) where a global value of a property is intended to be constrained based on variation in regional values that depend on grid resolution.
This final equation shows that porosity is a function of grid resolution.
3.3.1 Sensitivity of Permeability Estimate to Grid Resolution
4 Discussions
The proposed method was validated against measured permeability of three different fractured rocks. Besides the permeability, the porosity value estimated by our method was also validated simultaneously against its measured value in each sample. The method was also validated against additional data specific to each experiment, such as permeability of gas/water, local permeability of a fracture, and a pressure pulsedecay response, respectively. Although, the three experimental samples were widely different in their geological properties, the fragmented modeling approach of the proposed method allows enough flexibility to incorporate different types of data, including mineralogy that can be accounted by the porescale properties in the analytical permeability model for the matrix.
The proposed method was used to predict twophase permeability for water and scCO_{2} through a fracture etched on a rock sample from Utica shale (Porter et al. 2015). The proposed method predicted the permeability of the sample and the fracture as ~ 0.016, and \( 30\;{\text{mD}} \), respectively. Twophase permeability of each water and scCO_{2} phase is significantly lower than the absolute permeability of the fracture. The presence of nonwetting phase (scCO_{2}) significantly reduces the permeability of the wetting phase (water) from 11.28 to 7.29 mD as the saturation of scCO_{2} increases from 0.25 to 0.34. Although permeability of the wetting phase is expected to decrease with an increasing saturation of nonwetting phase, estimating the quantitative values generally requires performing experiments or computationally expensive porescale simulations. The proposed method allows estimating quantitative values of twophase permeability that is much simpler compared to other methods (e.g., experiments or porescale simulations). In addition to estimation of twophase permeability within fractures using the proposed method, it is also possible to leverage this method to estimate relative permeability in a nonempirical way, and/or perform sensitivity analysis either at laboratory or fieldscale. Model estimates of relative permeability (using Eqs. 20 through 25) at various saturations predict that “effective” (practically relevant) residual water saturation in the Utica shale sample as S_{wr} ∼ 0.20. Similarly, “effective” (practically relevant) residual gas saturation based on k_{rg,f} ≈ 0 in this sample was estimated as S_{gr} = 0.20. These residual saturations are estimated based on the results where relative permeability for a particular phase goes to 0; we obtain these nonzero residual saturation estimates despite the fact that we assumed S_{wr} = 0 due to lack of data and the mathematical formulation of the model does not require any residual gas saturation. The results presented in Fig. 18 demonstrate how the proposed twophase permeability model can be used as a quick analytical tool to estimate relative permeability estimates of twophase flow in fractured shale samples. The proposed twophase permeability model can also predict “effective” (approaching k_{ri,f} ≈ 0) estimates of residual saturations when there is no data on residual saturation by assuming S_{wr} = 0. Although laboratory measurement of relative permeability data in ultratight fractured rocks is a challenging task as evident by the presence of only few relative permeability datasets (Liu et al. 2010; Honarpour et al. 2012; Yassin et al. 2016; Guo et al. 2017; Ojha et al. 2017), laboratory estimate can be used with the proposed twophase permeability in an inversemodeling approach to predict important physical parameters in fractured shale (e.g., fracture aperture, fracture porosity).
Although the sensitivity of grid resolution on permeability estimation is significant, it is lower than the uncertainty experienced in measurement of a tightrock permeability that can vary by as much as 200% (Fisher et al. 2016) between steadystate and pulsedecay methods and more than 300% (Zhang et al. 2013) between MICP and pulsedecay methods of measurement. Even within pulsedecay method, the uncertainty in permeability measurement can be up to 50% (Finsterle and Persoff 1997; Song et al. 2013).
5 Summary and Conclusions
This study proposed a novel stochastic method to predict the permeability of a fractured shale system. The proposed method requires discretizing shale into a grid by superimposing the elements of the fractures and the matrix on a single grid. Rock feature in each cell of the grid is identified as one of the three elements (organic matter, inorganic matter, or fracture), and permeability of that cell is calculated using an appropriate analytical expression. The proposed method was validated against local and globalscale measurements on three fractured samples from laboratory. Finally, we used the method to predict twophase permeability for water and scCO_{2}, including their relative permeability estimates, through a fractured rock sample from Utica shale. The proposed method provides a quick and easy way to quantitatively estimate permeability of fractured tightrocks, including its regional and upscaled values, compared to other methods (e.g., experiments or porescale simulations). The twophase permeability expressions proposed in this study can be used as a quick analytical tool to predict relative permeability estimates of twophase flow in fractured shale samples. The proposed twophase permeability model can also be used to predict important physical parameters in shale (e.g., fracture aperture, fracture porosity) through inverse modeling.
In a recent numerical study (Singh and Cai 2018a), it was demonstrated that characterizing reservoir heterogeneity in terms of permeability is the most important step before an enhanced oil recovery operation or drilling a new well in shales. In Part 2 (Singh and Cai 2018b), the proposed method is used to predict fieldscale permeability through an optimization process that uses fieldscale production and other readily available information.
6 Nomenclature
Parameter  Description  SI unit 

k _{m}  Matrix permeability  m^{2} 
k _{f}  Fracture permeability with imbibition  m^{2} 
k _{slit}  Ideal slit permeability  m^{2} 
δ _{trans}  Thickness of transition zone  m 
α  Ratio of gas to water viscosity  – 
k _{f,pm}  Fracture permeability in porous media  m^{2} 
k _{f, i,pm}  Fracture permeability of phase \( i ( = {\text{g}},{\text{w)}} \) in porous media  m^{2} 
\( k_{\text{m, pm}} \)  Permeability of matrix in porous media  m^{2} 
k _{(i)om,pm}  Permeability of (in)organic matter in porous media  m^{2} 
c _{f}  Stressdependent parameter for fracture permeability  Pa^{−1} 
p _{p}  Pore pressure  Pa 
p _{c}  Confining pressure  Pa 
χ  Coefficient of pore fluid pressure  – 
H  Thickness of fracture (aperture)  m 
S _{ i }  Saturation of phase \( i ( = {\text{g}},{\text{w}}) \) in fracture  – 
S _{wr}  Residual water saturation in fracture  – 
μ _{ i }  Viscosity of phase \( i ( = {\text{g}},{\text{w}}) \) in fracture  
ϕ  Porosity  – 
τ  Tortuosity  – 
k _{upscaled}  Upscaled permeability (single value)  m^{2} 
A _{f}  Fracture area in bulk rock  m^{2} 
A _{b}  Rock bulk area  m^{2} 
a, b  Rock materialspecific parameters  – 
ɛ _{1}  Compressible deformation  – 
r _{d}  Ratio representing weight of dilatant deformation  – 
d _{(i)om}  Pore diameter of (in)organic matter  m 
l  Fracture length  m 
P  Probability  – 
ρ _{b}  Bulk density of shale rock  kg/m^{3} 
β_{b}, β_{s}, β  Compressibility of the bulk sample, matrix, and the fluid  Pa^{−1} 
\( V_{\text{u}} , V_{\text{d}} \)  Volumes of upstream and downstream reservoirs  m^{3} 
T  Shear stress  Pa 
β  Correction factor for pore shape  – 
b _{f}  Fracture width  m 
Subscript  Description  Superscript  Description 

f  Fracture  T  Top 
m  Matrix  B  Bottom 
pm  Porous media  
avg  Average  
OM  Organic matter  
iOM  Inorganic matter  
t  Transition zone  
min  Minimum  
max  Maximum  
aper  Aperture 
Notes
Acknowledgements
This research was supported in part by an appointment to the National Energy Technology Laboratory Research Participation Program, sponsored by the US Department of Energy and administered by the Oak Ridge Institute for Science and Education. This research was also supported in part by the National Natural Science Foundation of China (No. 41722403).
References
 Backeberg, N.R., Iacoviello, F., Rittner, M., Mitchell, T.M., Jones, A.P., Day, R., Wheeler, J., Shearing, P.R., Vermeesch, P., Striolo, A.: Quantifying the anisotropy and tortuosity of permeable pathways in clayrich mudstones using models based on Xray tomography. Sci. Rep. 7(1), 14838 (2017)Google Scholar
 Berg, C.F.: Permeability description by characteristic length, tortuosity, constriction and porosity. Transp. Porous Media 103(3), 381–400 (2014)Google Scholar
 Berre, I., Lien, M., Mannseth, T.: A levelset corrector to an adaptive multiscale permeability prediction. Comput. Geosci. 11(1), 27–42 (2007)Google Scholar
 Bonnet, E., Bour, O., Odling, N.E., Davy, P., Main, I., Cowie, P., Berkowitz, B.: Scaling of fracture systems in geological media. Rev. Geophys. 39(3), 347–383 (2001)Google Scholar
 Brace, W.F., Walsh, J.B., Frangos, W.T.: Permeability of granite under high pressure. J. Geophys. Res. 73(6), 2225–2236 (1968)Google Scholar
 Cai, J., Perfect, E., Cheng, C.L., Hu, X.: Generalized modeling of spontaneous imbibition based on Hagen–Poiseuille flow in tortuous capillaries with variably shaped apertures. Langmuir 30(18), 5142–5151 (2014)Google Scholar
 Cai, J., Wei, W., Hu, X., Liu, R., Wang, J.: Fractal characterization of dynamic fracture network extension in porous media. Fractals 25(02), 1750023 (2017)Google Scholar
 Cao, P., Liu, J., Leong, Y.K.: A fully coupled multiscale shale deformationgas transport model for the evaluation of shale gas extraction. Fuel 178, 103–117 (2016)Google Scholar
 Chen, C.Y., Horne, R.N.: Twophase flow in roughwalled fractures: experiments and a flow structure model. Water Resour. Res. 42(3), W03430 (2006)Google Scholar
 Chen, D., Pan, Z., Ye, Z., Hou, B., Wang, D., Yuan, L.: A unified permeability and effective stress relationship for porous and fractured reservoir rocks. J. Nat. Gas Sci. Eng. 29(Supplement C), 401–412 (2016)Google Scholar
 Chima, A., Geiger, S.: An analytical equation to predict gas/water relative permeability curves in fractures. In: SPE Latin America and Caribbean Petroleum Engineering Conference, Society of Petroleum Engineers (2012)Google Scholar
 Cho, Y., Ozkan, E., Apaydin, O.G.: Pressuredependent naturalfracture permeability in shale and its effect on shalegas well production. SPE Reserv. Eval. Eng. 16(02), 216–228 (2013)Google Scholar
 Cronin, M.B.: Corescale heterogeneity and dualpermeability pore structure in the Barnett shale. Thesis (2014)Google Scholar
 Cui, J., Sang, Q., Li, Y., Yin, C., Li, Y., Dong, M.: Liquid permeability of organic nanopores in shale: Calculation and analysis. Fuel 202(Supplement C), 426–434 (2017)Google Scholar
 Cui, G., Liu, J., Wei, M., Shi, R., Elsworth, D.: Why shale permeability changes under variable effective stresses: new insights. Fuel 213, 55–71 (2018)Google Scholar
 Dagan, G.: Models of groundwater flow in statistically homogeneous porous formations. Water Resour. Res. 15(1), 47–63 (1979)Google Scholar
 Darabi, H., Ettehad, A., Javadpour, F., Sepehrnoori, K.: Gas flow in ultratight shale strata. J. Fluid Mech. 710, 641–658 (2012)Google Scholar
 Deutsch, C.: Calculating effective absolute permeability in sandstone/shale sequences. SPE Form. Eval. 4(3), 343–348 (1989)Google Scholar
 Dürrast, H., Siegesmund, S.: Correlation between rock fabrics and physical properties of carbonate reservoir rocks. Int. J. Earth Sci. 88(3), 392–408 (1999)Google Scholar
 Feng, R., Liu, J., Harpalani, S.: Optimized pressure pulsedecay method for laboratory estimation of gas permeability of sorptive reservoirs: part 1—background and numerical analysis. Fuel 191(Supplement C), 555–564 (2017)Google Scholar
 Fenton, L.: The sum of lognormal probability distributions in scatter transmission systems. IRE Trans. Commun. Syst. 8(1), 57–67 (1960)Google Scholar
 Fink, R., Krooss, B.M., Gensterblum, Y., AmannHildenbrand, A.: Apparent permeability of gas shales—superposition of fluiddynamic and poroelastic effects. Fuel 199(2), 532–550 (2017)Google Scholar
 Finsterle, S., Persoff, P.: Determining permeability of tight rock samples using inverse modeling. Water Resour. Res. 33(8), 1803–1811 (1997)Google Scholar
 Fisher, Q.J., Grattoni, C., Rybalcenko, K., Lorinczi, P., Leeftink, T.: Laboratory measurements of porosity and permeability of shale. In: Fifth EAGE Shale Workshop. https://doi.org/10.3997/22144609.201600389 (2016)
 Franken, A.C.M., Nolten, J.A.M., Mulder, M.H.V., Bargeman, D., Smolders, C.A.: Wetting criteria for the applicability of membrane distillation. J. Membr. Sci. 33(3), 315–328 (1987)Google Scholar
 Ghanbarian, B., Javadpour, F.: Upscaling pore pressuredependent gas permeability in shales. J. Geophys. Rese. Solid Earth 122(4), 2016JB013846 (2017)Google Scholar
 Guibert, R., Nazarova, M., Horgue, P., Hamon, G., Creux, P., Debenest, G.: Computational permeability determination from porescale imaging: sample size, mesh and method sensitivities. Transp. Porous Media 107(3), 641–656 (2015)Google Scholar
 Guo, P., Zhang, H., Du, J., Wang, Z., Zhang, W., Ren, H.: Study on gas–liquid relative permeability experiments of fracturedporous reservoirs. Petroleum 3(3), 348–354 (2017)Google Scholar
 Gutierrez, M., Youn, D.J.: Effects of fracture distribution and length scale on the equivalent continuum elastic compliance of fractured rock masses. J. Rock Mech. Geotech. Eng. 7(6), 626–637 (2015)Google Scholar
 Honarpour, M.M., Nagarajan, N.R., Orangi, A., Arasteh, F., Yao, Z.: Characterization of critical fluid PVT, rock, and rockfluid properties—impact on reservoir performance of liquid rich shales. In: SPE Annual Technical Conference and Exhibition, Society of Petroleum Engineers (2012)Google Scholar
 Hunt, A.G., Idriss, B.: Percolationbased effective conductivity calculations for bimodal distributions of local conductances. Philos. Mag. 89(22–24), 1989–2007 (2009)Google Scholar
 Jenni, S., Hu, L.Y., Basquet, R., de Marsily, G., Bourbiaux, B.: History matching of a stochastic model of fieldscale fractures: methodology and case study. Oil Gas Sci. Technol. Revue de l’IFP 62(2), 265–276 (2007)Google Scholar
 Kazemi, M., TakbiriBorujeni, A.: An analytical model for shale gas permeability. Int. J. Coal Geol. 146, 188–197 (2015)Google Scholar
 Lei, G., Dong, P., Yang, S., Li, Y., Mo, S., Gai, S., Wu, Z.: A new analytical equation to predict gas–water twophase relative permeability curves in fractures. In: International Petroleum Technology Conference (2014)Google Scholar
 Li, Y., Li, X., Teng, S., Xu, D.: Improved models to predict gas–water relative permeability in fractures and porous media. J. Nat. Gas Sci. Eng. 19(Supplement C), 190–201 (2014)Google Scholar
 Liu, R., Liu, H., Li, X., Wang, J., Pang, C.: Calculation of oil and water relative permeability for extra low permeability reservoir. In: International Oil and Gas Conference and Exhibition in China, Society of Petroleum Engineers (2010)Google Scholar
 Liu, R., Jiang, Y., Li, B., Wang, X.: A fractal model for characterizing fluid flow in fractured rock masses based on randomly distributed rock fracture networks. Comput. Geotech. 65(Supplement C), 45–55 (2015)Google Scholar
 Lopez, B., Aguilera, R.: Sorptiondependent permeability of shales. In: SPE/CSUR Unconventional Resources Conference, Society of Petroleum Engineers (2015)Google Scholar
 Ma, J.: Review of permeability evolution model for fractured porous media. J. Rock Mech. Geotech. Eng. 7(3), 351–357 (2015)Google Scholar
 Ma, L., Taylor, K.G., Lee, P.D., Dobson, K.J., Dowey, P.J., Courtois, L.: Novel 3D centimetreto nanoscale quantification of an organicrich mudstone: the Carboniferous Bowland Shale, Northern England. Mar. Pet. Geol. 72, 193–205 (2016)Google Scholar
 Masihi, M., Gago, P.A., King, P.R.: Estimation of the effective permeability of heterogeneous porous media by using percolation concepts. Transp. Porous Media 114(1), 169–199 (2016)Google Scholar
 Mostaghimi, P., Blunt, M.J., Bijeljic, B.: Computations of absolute permeability on microCT images. Math. Geosci. 45(1), 103–125 (2013)Google Scholar
 Naraghi, M.E., Javadpour, F.: A stochastic permeability model for the shalegas systems. Int. J. Coal Geol. 140, 111–124 (2015)Google Scholar
 Naraghi, M.E., Javadpour, F., Ko, L.T.: An objectbased shale permeability model: nonDarcy gas flow, sorption, and surface diffusion effects. Transp. Porous Media. (2018). https://doi.org/10.1007/s112420170992z
 Neuman, S.P.: Multiscale relationships between fracture length, aperture, density and permeability. Geophys. Res. Lett. 35(22), L22402 (2008)Google Scholar
 Ojha, S.P., Misra, S., Tinni, A., Sondergeld, C., Rai, C.: Relative permeability estimates for Wolfcamp and Eagle Ford shale samples from oil, gas and condensate windows using adsorption–desorption measurements. Fuel 208, 52–64 (2017)Google Scholar
 Okabe, H., Blunt, M.J.: Prediction of permeability for porous media reconstructed using multiplepoint statistics. Phys. Rev. E 70(6), 066135 (2004)Google Scholar
 Øren, P.E., Bakke, S.: Process based reconstruction of sandstones and prediction of transport properties. Transp. Porous Media 46(2–3), 311–343 (2002)Google Scholar
 Porter, M.L., JiménezMartínez, J., Martinez, R., McCulloch, Q., Carey, J.W., Viswanathan, H.S.: Geomaterial microfluidics at reservoir conditions for subsurface energy resource applications. Lab Chip 15(20), 4044–4053 (2015)Google Scholar
 Shapiro, S.S., Zahedi, H.: Bernoulli trials and discrete distributions. J. Qual. Technol. 22(3), 193–205 (1990)Google Scholar
 Shi, L., Zeng, Z., Bai, B., Li, X.: Effect of the intermediate principal stress on the evolution of mudstone permeability under true triaxial compression. Greenh. Gases Sci. Technol. 8(1), 37–50 (2018)Google Scholar
 Singh, H.: Scaleup of reactive processes in heterogeneous media. Dissertation, The University of Texas at Austin (2014)Google Scholar
 Singh, H., Srinivasan, S.: Some perspectives on scaleup of flow and transport in heterogeneous media. Bull. Can. Pet. Geol. (2014)Google Scholar
 Singh, H., Javadpour, F.: Langmuir slipLangmuir sorption permeability model of shale. Fuel 164, 28–37 (2016)Google Scholar
 Singh, H., Cai, J.: Screening improved recovery methods in tightoil formations by injecting and producing through fractures. Int. J. Heat Mass Transf. 116(Supplement C), 977–993 (2018a)Google Scholar
 Singh, H., Cai, J.: A featurebased stochastic permeability of shale: Part 2–Predicting fieldscale permeability. Transp. Porous Med. (2018b). https://doi.org/10.1007/s1124201810764 Google Scholar
 Singh, H., Javadpour, F., Ettehadtavakkol, A., Darabi, H.: Nonempirical apparent permeability of shale. SPE Reserv. Eval. Eng. 17(03), 414–424 (2014)Google Scholar
 Song, I., Rathbun, A.P., Saffer, D.M.: Uncertainty analysis for the determination of permeability and specific storage from the pulsetransient technique. Int. J. Rock Mech. Min. Sci. Complete 64, 105–111 (2013)Google Scholar
 Song, W., Yao, J., Li, Y., Sun, H., Zhang, L., Yang, Y., Zhao, J., Sui, H.: Apparent gas permeability in an organicrich shale reservoir. Fuel 181(Supplement C), 973–984 (2016)Google Scholar
 Tahmasebi, P., Javadpour, F., Sahimi, M.: Threedimensional stochastic characterization of shale SEM images. Transp. Porous Media 110(3), 521–531 (2015)Google Scholar
 Thompson, Karsten E.: Porescale modeling of fluid transport in disordered fibrous materials. AIChE J. 48(7), 1369–1389 (2004)Google Scholar
 Wang, J., Liu, H., Wang, L., Zhang, H., Luo, H., Gao, Y.: Apparent permeability for gas transport in nanopores of organic shale reservoirs including multiple effects. Int. J. Coal Geol. 152(Part B), 50–62 (2015)Google Scholar
 Wei, W., Xia, Y.: Geometrical, fractal and hydraulic properties of fractured reservoirs: a minireview. Adv. Geo Energy 1(1), 31–38 (2017)Google Scholar
 Wei, M., Liu, J., Feng, X., Wang, C., Zhou, F.: Evolution of shale apparent permeability from stresscontrolled to displacementcontrolled conditions. J. Nat. Gas Sci. Eng. 34(Supplement C), 1453–1460 (2016)Google Scholar
 Wu, K., Li, X., Guo, C., Wang, C., Chen, Z.: A unified model for gas transfer in nanopores of shalegas reservoirs: coupling pore diffusion and surface diffusion. SPE J. 21(05), 1–583 (2016)Google Scholar
 Yang, Y., Yao, J., Wang, C., Gao, Y., Zhang, Q., An, S., Song, W.: New pore space characterization method of shale matrix formation by considering organic and inorganic pores. J. Nat. Gas Sci. Eng. 27(Part 2), 496–503 (2015)Google Scholar
 Yassin, M.R., Dehghanpour, H., Wood, J., Lan, Q.: A theory for relative permeability of unconventional rocks with dualwettability pore network. SPE J. 21(06), 1–970 (2016)Google Scholar
 Zhang, X., Spiers, C.J., Peach, C.J., Hebing, A., Geoconsultants, P.: Tight rock permeability measurement by pressure pulse decay and modeling. In: Proceedings of the International Symposium of the Society of Core Analysts, Napa Valley, California, USA (2013)Google Scholar
 Zhang, Q., Su, Y., Wang, W., Lu, M., Sheng, G.: Apparent permeability for liquid transport in nanopores of shale reservoirs: coupling flow enhancement and near wall flow. Int. J. Heat Mass Transf. 115(Part B), 224–234 (2017a)Google Scholar
 Zhang, T., Li, Z., Adenutsi, C., Lai, F.: A new model for calculating permeability of natural fractures in dualporosity reservoir. Adv. Geo Energy Res. 1(2), 86–92 (2017b)Google Scholar