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Transport in Porous Media

, Volume 126, Issue 3, pp 527–560 | Cite as

A Feature-Based Stochastic Permeability of Shale: Part 1—Validation and Two-Phase Permeability in a Utica Shale Sample

  • Harpreet SinghEmail author
  • Jianchao Cai
Article
  • 201 Downloads

Abstract

Estimate of permeability plays a crucial role in flow-based studies of fractured tight-rocks. It is well known that most of the flow through tight-rocks (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 tight-rocks do not account for such features present within the rock ranging from micro-scale to field-scale. Current permeability modeling approach assumes a single continuum without considering the presence of permeable features within the matrix (e.g., micro-fractures) or outside the matrix (e.g., natural fractures). Although the laboratory-measured permeability implicitly captures discrete features present in that sample (e.g., fractures, laminations, micro-fractures), 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 non-ideal 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 two-part 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 non-ideal nature of porous medium and two-phase 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 global-scale measurements on three fractured samples from laboratory. Finally, the method is used to predict two-phase flow permeability of supercritical CO2 displacing water within a fracture in a Utica shale sample. The proposed two-phase flow permeability equations can be used as a quick analytical tool to predict relative permeability estimates of two-phase flow in fractured shale samples. In Part 2, the proposed method is used to estimate field-scale permeability through an optimization process that uses field-scale production and other readily available information.

Keywords

Fractures Tight-rock Relative permeability Two phase Utica 

1 Introduction

Permeability is the most critical parameter in fluid flow analysis of tight-rocks like shale resources. Although several theoretical models have been proposed to predict shale permeability as a function of pressure (Darabi et al. 2012; Singh et al. 2014; Singh and Javadpour 2016), stress (Ma 2015; Cao et al. 2016; Wei et al. 2016), sorption (Kazemi and Takbiri-Borujeni 2015; Lopez and Aguilera 2015; Singh and Javadpour 2016; Wu et al. 2016), minerals (Naraghi and Javadpour 2015; Wang et al. 2015; Song et al. 2016; Naraghi et al. 2018), etc., they are based on the an approach that assumes only matrix as a continuum without considering the presence of permeable features within the matrix (e.g., micro-fractures, laminations, etc.) or outside the matrix (e.g., natural fractures). Although fracture permeability models have been proposed independently of matrix (Ma 2015), their primary purpose has been to evaluate the effect of stress on fractures (Cho et al. 2013) as these models cannot predict matrix permeability. Based on literature review of shale permeability models, it is apparent that a drawback of the current modeling approach is their inability to predict an upscaled permeability of a fractured rock with diverse features (e.g., matrix, micro-fractures, laminations, fractures) that can be significantly different from each other in terms of their permeabilities. This drawback in permeability modeling exists at all scales of a fractured tight-rock, which is composed of minerals with significant difference in their permeability (e.g., organic and inorganic), and other discrete features important at micro-scale (e.g., micro-fractures) to larger scales (e.g., fractures) that impact the permeability. This drawback in purely analytical approach can be visualized graphically in Fig. 1, which shows various features impacting shale permeability that are not possible to account reasonably by current permeability models. While some methods exist to predict permeability through rock discretization (Deutsch 1989; Øren and Bakke 2002; Okabe and Blunt 2004; Thompson 2004; Berre et al. 2007; Mostaghimi et al. 2013; Berg 2014; Guibert et al. 2015), they were proposed for application in conventional sandstone rocks with a single continuum.
Fig. 1

A schematic illustrating the ubiquitous presence of distinct permeable features in shale at different scales

The primary objective of this paper is to propose a new method to estimate permeability of fractured tight-rocks 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 two-phase flow permeability of scCO2 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 micro-scale (e.g., micro-fractures, 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

The procedure is to predict laboratory-scale permeability which is described in following steps and summarized in Fig. 2:
Fig. 2

A flow-chart summarizing the method to predict laboratory-scale permeability with fracture, and matrix (inorganic and organic) in shale rock

  • 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 micro-fracture.
    Fig. 3

    Schematic illustrating a grid superimposed with fracture, and matrix (inorganic and organic) of shale rock. The proposed method involves using appropriate analytical model to calculate permeability of each cell in the grid

  • 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 steady-state Navier–Stokes equation for single- and two-phase flow.

2.2 Fracture Discretization

The procedure to discretize the fractures is based on approximation of actual fractures on a finite-difference grid (Jenni et al. 2007) as shown schematically in Fig. 4. Each cell through which the approximate (discretized) fracture passes is assigned discrete fracture properties such as its length, aperture, and orientation.
Fig. 4

Schematic to illustrate fracture discretization on a grid

2.2.1 Fracture Attributes

The fracture properties are generally known at laboratory scale if we have sample images, which can be used to extract fracture attributes like lengths, apertures, and orientation. However, fracture attributes are unknown at field-scale except for some coarse-scale general information (like fracture orientation and density) that can help constrain the properties at field-scale. In absence of field data required to characterize the fractures, a common approach to characterize fractured media involves assigning fracture properties using statistical distributions that they are known to follow consistently at different scales (Bonnet et al. 2001; Neuman 2008; Liu et al. 2015; Cai et al. 2017). Theoretically, the approach of assigning fracture properties using statistical distributions can also be used at scales smaller, for example, in revealing micro-fractures invisible to the naked eye in laboratory-scale samples, it requires dynamic data to calibrate parameters of fracture distributions. The statistical distribution (Bonnet et al. 2001; Neuman 2008; Gutierrez and Youn 2015; Wei and Xia 2017) exhibited by each fracture attribute, its expression and the unknown parameter are shown in Table 1. These distributions will be used in Part 2 (Singh and Cai 2018b) to predict the unknown field-scale fracture distribution parameters.
Table 1

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 tight-rock 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 laboratory-scale and field-scale. 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. 1.

    Find probability map of physical occurrence of OM:

     
The approximate grain density of OM (kerogen) is 1.1 g/cc, and the porosity of OM is generally lower than the porosity of iOM (Singh and Cai 2018a). Therefore, the probability of finding an OM is inversely proportional to the product of bulk density and porosity \( \left( { \propto \frac{1}{{\rho_{\text{b}} \phi }}} \right) \). In other words, the presence of OM is sensitive to the combined change in bulk density and porosity. This probability can be quantified numerically as follows by scaling \( \frac{1}{{\rho_{\text{b}} \phi }} \) between 0 and 1:
$$ P_{\text{OM}} = \left[ {\frac{{\left( {\frac{1}{{\rho_{\text{b}} \phi }}} \right) - \left( {\frac{1}{{\rho_{\text{b}} \phi }}} \right)_{\rm{min} } }}{{\left( {\frac{1}{{\rho_{\text{b}} \phi }}} \right)_{\rm{max} } - \left( {\frac{1}{{\rho_{\text{b}} \phi }}} \right)_{\rm{min} } }}} \right] $$
(1)
Typically, bulk density and porosity of the laboratory samples can be obtained by using CT scanning, while the field-scale values of bulk density and porosity are obtained through well logs that can be used to determine full areal map by kriging. Kriging is an optimal linear interpolation to estimate values at unknown locations using weighted sum of surrounding data, where the accuracy of interpolation is primarily dependent on the weights that are obtained from a spatial correlation function (e.g., covariance, variogram) between the data, which is scale-dependent (Singh 2014; Singh and Srinivasan 2014) and must be accounted appropriately for reliable kriging estimates at field-scale. The spatial correlation function can be estimated using data from multiple wells (or cores) distributed across the reservoir, but in case of scarce data that may not be sufficient to accurately estimate a correlation function, it is recommended to assume a standard correlation function to approximate the underlying reservoir.
  1. 2.

    Locate OM positions on grid with a Bernoulli distribution:

     
Spatially locate cells on a grid occupied by OM using a Bernoulli distribution (Shapiro and Zahedi 1990) that takes probability map calculated in step 1 as an input.
  1. 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.

Figure 5 shows an example that illustrates our proposed method of sampling OM on a grid.
Fig. 5

Illustration of the algorithm to spatially sample OM. The red and blue legends in the binary image on bottom right represent the yOM and iOM, respectively

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 two-phase 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 non-ideal 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 non-ideal nature of fractures, including single and two-phase flows in fractures, new analytical expressions are derived. Permeability for single and two-phase flow in fracture is derived by solving steady-state 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).

Single-Phase Permeability
$$ k_{\text{f,pm}} = \left\{ {\begin{array}{*{20}l} {\left[ {k_{\text{f}} \cdot \exp \left\{ { - c_{\text{f}} \left( {p_{\text{c}} - \chi p_{\text{p}} } \right)} \right\}} \right] \cdot \frac{{\phi_{\text{f}} }}{{\tau_{\text{f}} }}\quad {\text{without}}\;{\text{damage}}} \hfill \\ {\left[ {k_{\text{f}} \cdot \left\{ {\exp \left( { - a\varepsilon_{1} } \right) + br_{\text{d}}^{3} } \right\}} \right] \cdot \frac{{\phi_{\text{f}} }}{{\tau_{\text{f}} }}\quad \quad {\text{with}}\;{\text{damage}}} \hfill \\ \end{array} } \right. $$
(2)
$$ k_{\text{f}} = k_{\text{m}} + k_{\text{slit}} + \sqrt {3k_{\text{f}} k_{\text{m}} } \cdot \tanh \left( 3 \right) $$
(3)
$$ k_{\text{slit}} \left( {i,j} \right) = \frac{{l_{{{\text{f}},i,j}} a_{\text{f}}^{3} }}{{12 \times \Delta x_{i,j} \times \Delta y_{i,j} }} $$
(4)
$$ \phi_{\text{f}} \left( {i,j} \right) = \frac{{l_{{{\text{f}},i,j}} a_{\text{f}} }}{{\Delta x_{i,j} \times \Delta y_{i,j} }} $$
(5)
$$ \tau_{\text{f}} \left( {i,j} \right) = \frac{{l_{{{\text{f}},i,j}} }}{{\hbox{min} \left( {\Delta x_{i,j} ,\Delta y_{i,j} } \right)}} $$
(6)

Two-Phase Permeability of Gas and Water

The approach adopted in deriving two-phase permeability model in fractures was proposed by Chima and Geiger (2012), which was later adopted to include the effect of reducible water saturation (Li et al. 2014) and capillary pressure (Lei et al. 2014). The two-phase permeability expressions derived in this study (given in Appendix A) accounts for the tortuosity of each fluid phase in fracture, fracture porosity, and the effect of stress mechanics, besides the effect of capillary pressure and irreducible water saturation.
$$ k_{{{\text{f}},i,{\text{pm}}}} = \left\{ {\begin{array}{*{20}l} {\left[ {k_{{{\text{f}},i}} \cdot \exp \left\{ { - c_{\text{f}} \left( {p_{\text{c}} - \chi p_{\text{p}} } \right)} \right\}} \right] \cdot \frac{{\phi_{\text{f}} }}{{\tau_{{{\text{f}},i}} }}\quad {\text{without}}\;{\text{damage}}} \hfill \\ {\left[ {k_{{{\text{f}},i}} \cdot \left\{ {\exp \left( { - a\varepsilon_{1} } \right) + br_{\text{d}}^{3} } \right\}} \right] \cdot \frac{{\phi_{\text{f}} }}{{\tau_{{{\text{f}},i}} }}\quad \quad {\text{with}}\;{\text{damage}}} \hfill \\ \end{array} } \right. $$
(7)
where i depicts the phase of fluid (\( = {\text{g}},\;{\text{w}} \)).
$$ k_{\text{f,w}} = \frac{{S_{\text{w}}^{2} \left( {S_{\text{w}} - S_{\text{wr}} } \right)}}{{1 - S_{\text{wr}} }}\left[ {2\left( {S_{\text{w}} - S_{\text{wr}} } \right) + 3S_{\text{g}} } \right] \cdot \left( {\frac{{H^{2} }}{24}} \right) $$
(8)
$$ k_{\text{f,g}} = \frac{{S_{\text{g}}^{2} }}{{1 - S_{\text{wr}} }}\left[ {2S_{\text{g}}^{2} + 3\alpha \left( {S_{\text{w}} - S_{\text{wr}} } \right)^{2} + 6\alpha \left( {S_{\text{w}} - S_{\text{wr}} } \right)S_{\text{g}} } \right] \cdot \left( {\frac{{H^{2} }}{24}} \right) $$
(9)
$$ \tau \left( {S_{\text{w}} } \right) = \frac{1}{{0.74S_{\text{w}}^{\text{norm}} + 0.26}} $$
(10)
$$ \tau \left( {S_{\text{g}} } \right) = \frac{1}{{0.43S_{\text{g}}^{2} + 0.38S_{\text{g}} + 0.19}} $$
(11)
$$ S_{\text{w}}^{\text{norm}} = \frac{{S_{\text{w}} - S_{\text{wr}} }}{{1 - S_{\text{wr}} }} $$
(12)

The above two-phase permeability model was derived for gas as the non-wetting 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 non-wetting 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

Matrix permeability is modeled based on the study by Singh et al. (2014), which accounts for the convective flux, Knudsen diffusion, and the effect of sorption (surface diffusion) on pore size for a gas permeability in nanopores. The effect of sorption on permeability is only considered in the case of OM in matrix. The final expression of permeability in a porous medium is given as:
$$ k_{\text{m,pm}} = \left[ {\left( {k_{\text{iom,pm}} } \right)^{{1 - x_{\text{toc}} }} \times \left( {k_{\text{om,pm}} } \right)^{{x_{\text{toc}} }} } \right] $$
(13)
where xtoc denotes the fraction of OM pores in the matrix, and subscript pm indicates permeability that accounts for the effect of porous media, i.e., tortuosity (τ) and porosity (ϕ). The expressions for kiom,pm and kom,pm in above equation will depend on the type of fluid, i.e., gas or oil, stored in matrix and they are given as follows (details in Appendix A.3.):
Gas in Matrix
$$ k_{\text{iom,pm}} = \frac{{2\mu_{\text{g}} d_{\text{iom}} }}{\pi }\left( {\frac{{\pi d_{\text{iom}} Z}}{{64\mu_{\text{g}} }} + \frac{1}{{3p_{\text{p}} M}}\sqrt {2\pi MRT} } \right) \times \frac{{\phi_{\text{iom}} }}{{\tau_{\text{iom}} }} $$
(14)
$$ k_{\text{om,pm}} = \frac{{2\mu_{\text{g}} d_{\text{om}} }}{\pi }\left( {\frac{{\pi d_{\text{om}} Z}}{{64\mu_{\text{g}} }} + \frac{1}{{3p_{\text{p}} M}}\sqrt {2\pi MRT} } \right) \times \frac{{\phi_{\text{om}} }}{{\tau_{\text{om}} }} $$
(15)
Oil in Matrix
$$ k_{\text{iom,pm}} = \frac{{d_{\text{iom}}^{2} }}{32} \times \frac{{\phi_{\text{iom}} }}{{\tau_{\text{iom}} }} $$
(16)
$$ k_{\text{om,pm}} = \frac{{d_{\text{om}}^{2} }}{32} \times \frac{{\phi_{\text{om}} }}{{\tau_{\text{om}} }} $$
(17)

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 non-uniform cell sizes.

2.5 Upscaling Grid-Based Permeability to a Single Value

The discretized model that describes porosity and permeability of the rock in each grid cell depicts the rock heterogeneity. However, at laboratory scale we generally measure a single value of porosity and permeability that represents the estimate of the entire rock sample. Therefore, in order to validate the permeability estimated by the proposed method we must be able to compare our results with the laboratory-measured value. The discretized porosity and permeability are converted to corresponding single value (ϕeff and kupscaled) for the entire sample by taking the weighted average and geometric average, respectively, of all grid values.
$$ \phi_{\text{eff}} = \phi_{\text{f}} \cdot \left( {\frac{{A_{\text{f}} }}{{A_{\text{b}} }}} \right) + \phi_{\text{m}} \cdot \left( {\frac{{A_{\text{b}} - A_{\text{f}} }}{{A_{\text{b}} }}} \right) $$
(18)
$$ k_{\text{upscaled}} = \left( {\mathop \prod \limits_{i = 1}^{{N_{X} }} \mathop \prod \limits_{j = 1}^{{N_{Y} }} k_{ij} } \right)^{{\frac{1}{{N_{X} N_{Y} }}}} $$
(19)

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 laboratory-scale experimental studies, each with a distinct fractured rock sample. Following the validation, the method is applied to predict two-phase flow permeability of scCO2 displacing water through a fracture etched on a Utica shale sample.

3.1 Laboratory-Scale Validation

Three different laboratory-scale studies are used in validating proposed model, each with a distinct fractured rock sample. The three samples are a clay-rich sample from a European shale gas interval, a carbonate-rich sample with some organic matter from a basin in Germany, and a synthetic sample created in laboratory using Portland cement. Each study provides, at minimum, porosity and permeability of the bulk samples, which we reproduce using our model. Besides these two common measurements, each study provides additional data specific to their case, such as permeability of gas/water, local permeability of fracture, and a pressure pulse-decay response, which we also reproduce to validate our model. The primary fitting parameters in the validation process are a coupling parameter (cf) that reflects the sensitivity of stress on fracture permeability and coefficient of pore fluid pressure (χ). The compressibility coefficient, or coupling parameter (cf) that reflects the sensitivity of stress on fracture permeability, is generally obtained by matching the model to the data (Fink et al. 2017; Cui et al. 2018). The approach to validate the proposed method is summarized in Fig. 6.
Fig. 6

A flow-chart summarizing the approach used in validating the proposed method

3.1.1 Fractured Clay-Rich Shale Sample

The sample shown in Fig. 7 comes from a depth of ~ 3700 m within a prospective shale gas interval in Europe. This clay-rich shale sample, almost free of any black shale or organic matter, was used by Backeberg et al. (2017) to measure porosity and permeability. The parameters used to measure permeability are shown in Fig. 7, which shows two values of permeability that were measured using water and argon.
Fig. 7

Image of the sample used in permeability measurement by Backeberg et al. (2017)

Validation of Sample Porosity, and Permeability of Gas and Water

Using the information about sample size, fracture orientation and its length from Fig. 7, we develop a discretized model for the sample and estimate fracture aperture, sample porosity, and upscaled permeability to gas and water. The values of cf and χ used to obtain the match are 1.55 M Pa−1 and 0.8, respectively. The effective porosity of the sample is estimated to be ϕeff = 3.6%, which is close to the measured value of 3.5%. Estimation of permeability to gas and water that matches the measured data required using a residual saturation value of 0.28, which gives permeability to water and gas as \( k_{\text{h-water}} = 6.28 \times 10^{ - 20} \;{\text{m}}^{2} \) and \( k_{\text{h-argon}} = 2.0 \times 10^{ - 20} \;{\text{m}}^{2} \), respectively. These results are presented in Fig. 8. Although Backeberg et al. (2017) does not differentiate between the sample’s effective permeability and that of fracture, it can be seen in Fig. 8 (bottom right plot) that fracture permeability is ~ 0.0023 mD or 2.3 μD.
Fig. 8

Discretized sample with predicted properties (porosity, and permeability) used in validation. The measured value of sample porosity is 3.5%, whereas measured permeability with water and gas are \( k_{\text{h-water}} = 6.28 \times 10^{ - 20} \;{\text{m}}^{2} \) and \( k_{\text{h-argon}} = 2.0 \times 10^{ - 20} \;{\text{m}}^{2} \), respectively

Figure 9 shows permeability variation within the matrix estimated using Singh et al. (2014) based on typical log-normal pore size distribution (Ghanbarian and Javadpour 2017). Figure 9 shows that although there is variation in permeability magnitude within the matrix, the magnitude of the permeability is much smaller compared to the fracture that it appears almost as a constant value in Fig. 8.
Fig. 9

Discretized sample with predicted matrix permeability used in validation. The two fracture locations, shown in white, are not included with this permeability. Matrix permeability is estimated using Singh et al. (2014) based on typical log-normal pore size distribution (Ghanbarian and Javadpour 2017) with its magnitude varying between 1.338126 × 10−20 and 1.338134 × 10−20 m2

3.1.2 Fractured Carbonate Sample

The sample shown in Fig. 10 comes from a carbonate hydrocarbon reservoir of the NW German basin, which was originally composed of calcitic ooides before dolomitization from diagenesis resulted in nearly horizontal millimeter fine-scale layering. The sample, containing some organic matter, was used in a study (Dürrast and Siegesmund 1999) to measure permeability of the sample and the large single crack/fracture. The permeability was measured using argon, and the measured parameters are shown in Fig. 10. The data in this study (Dürrast and Siegesmund 1999) also include a measure of permeability for the major crack/fracture besides the effective permeability of the sample.
Fig. 10

Image of the sample used in permeability measurement by Dürrast and Siegesmund (1999)

Validation of Sample Porosity and Permeability

Using the information about sample size, fracture orientation and its length from Fig. 10, we develop a discretized model for the sample and estimate fracture aperture, sample porosity, and upscaled permeability to argon gas. The values of cf and χ used to obtain the match are 0.04 M Pa−1 and 0.1, respectively. The effective porosity of the sample is estimated to be ϕeff = 9%, which is close to the measured value of 9.6%. The upscaled permeability of the sample with argon gas, and the permeability of the fracture come out to be \( k_{\text{h - argon}} = 0.188\; {\text{mD}} \) and \( k_{\text{f,pm}} = 4.0 \;{\text{mD}} \), respectively, which are close to the measured values. These results are presented in Fig. 11.
Fig. 11

Discretized sample with predicted properties (porosity, and permeability) used in validation. The measured value of sample porosity is 9.6%, whereas measured permeability of sample (with gas) and fracture are \( k_{\text{h-argon}} = 0.188\;{\text{mD}} \), and \( k_{\text{f,pm}} \sim 5.86\;{\text{mD}} \)

3.1.3 Fractured Cement Core

The core sample shown in Fig. 12 is composed of class H Portland cement without additives in which a planar fracture was created along the center using the Brazilian Disk method. The sample, containing no organic matter, was used by Cronin (2014) to measure permeability and porosity of the matrix before fracturing the sample as well as after inducing a planar fracture. In both scenarios, permeability was measured through a pulse-decay experiment using argon gas, and the parameters used in the measurement are shown in Fig. 12. Cronin (2014) also provide a measure of the time it takes for the pressure pulse to reach equilibrium after starting the pulse-decay experiment (~ 4 s) for the fractured sample.
Fig. 12

Image of the sample used in permeability measurement by Cronin (2014). The permeability of the un-fractured matrix in above image is indicated as km

Validation of Sample Porosity, Permeability, and Pulse Decay

Using the information about sample size, fracture orientation and its length from Fig. 12, we develop a discretized model for the sample and estimate fracture aperture, sample porosity, and upscaled permeability to argon gas. The values of cf and χ used to obtain the match are 5 × 10−8 M Pa−1 and 0.1, respectively. The effective porosity of the sample is estimated to be ϕeff = 14.1%, which is close to the measured value of 14.3%. The upscaled permeability of the fractured sample using argon gas comes out to be \( k_{\text{h - eff}} = 0.318\; {\text{mD}} \). These results are presented in Fig. 13.
Fig. 13

Discretized sample with predicted properties (porosity, and permeability) used in validation. The measured value of sample porosity is 14.3%, whereas measured permeability of sample (with gas) is \( k_{\text{h-argon}} = 0.31\;{\text{mD}} \)

Cronin (2014) also provided the time that the pressure pulse took to reach equilibrium after starting the pulse-decay experiment, which is ~ 4 s. This time is used in further validation by estimating the time it takes for a pressure pulse to reach equilibrium in our discretized model of the rock with a fracture and matrix, as shown in Fig. 13. Our results are shown in Fig. 14, which shows that it takes ~ 4 s to reach equilibrium using our proposed method. Details of pressure pulse-decay numerical model are given in Appendix B.
Fig. 14

Pulse-decay result used in validation with the discretized model

3.2 Two-Phase Flow Permeability in a Fractured Shale Sample

In this section we predict two-phase flow permeability through a fracture using the expressions (Eqs. 7 through 12) for two-phase permeability derived in Appendix A.2. Two-phase permeability is predicted for scCO2 displacing water while passing through a fracture etched on a rock sample from Utica shale as shown in Fig. 15 for three different time snapshots that were captured in an experimental study by Porter et al. (2015). The primary objective of Porter et al. (2015) study was to demonstrate the experimental system of capturing such images using real shale rocks in micromodels, although, they did not measure any rock property, including porosity or permeability. We will use the three images shown in Fig. 15 to predict permeability of water and scCO2 for each time. It is assumed that the fracture is 100% water saturated at t = 0.
Fig. 15

Time-lapse images of scCO2 displacing water while passing through a fracture etched on a shale rock.

Adapted from Porter et al. (2015)

The permeability of water and scCO2 in fracture is predicted using the following expressions that are derived in Appendix A.2:
$$ k_{{{\text{f}},i,{\text{pm}}}} = \left[ {k_{{{\text{f}},i}} \cdot \exp \left\{ { - c_{\text{f}} \left( {p_{\text{c}} - \chi p_{\text{p}} } \right)} \right\}} \right] \cdot \frac{{\phi_{\text{f}} }}{{\tau_{{{\text{f,}}i}} }} $$
(20)
where i depicts the phase of fluid (\( = {\text{g}}, {\text{w}} \)).
$$ k_{\text{f,w}} = \frac{{S_{\text{w}}^{2} \left( {S_{\text{w}} - S_{\text{wr}} } \right)}}{{1 - S_{\text{wr}} }}\left[ {2\left( {S_{\text{w}} - S_{\text{wr}} } \right) + 3S_{\text{g}} } \right] \cdot \left( {\frac{{H^{2} }}{24}} \right) $$
(21)
$$ k_{\text{f,g}} = \frac{{S_{\text{g}}^{2} }}{{1 - S_{\text{wr}} }}\left[ {2S_{\text{g}}^{2} + 3\alpha \left( {S_{\text{w}} - S_{\text{wr}} } \right)^{2} + 6\alpha \left( {S_{\text{w}} - S_{\text{wr}} } \right)S_{\text{g}} } \right] \cdot \left( {\frac{{H^{2} }}{24}} \right) $$
(22)
$$ \tau \left( {S_{\text{w}} } \right) = \frac{1}{{0.74S_{\text{w}}^{\text{norm}} + 0.26}} $$
(23)
$$ \tau \left( {S_{\text{g}} } \right) = \frac{1}{{0.43S_{\text{g}}^{2} + 0.38S_{\text{g}} + 0.19}} $$
(24)
$$ S_{\text{w}}^{\text{norm}} = \frac{{S_{\text{w}} - S_{\text{wr}} }}{{1 - S_{\text{wr}} }} $$
(25)

In above expressions, saturations of scCO2 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 pc = 10.34 MPa and pp = 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 two-phase permeability model are cf 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%.

Before predicting the two-phase permeability with scCO2 and water, we discretize the fractured rock shown in Fig. 15 and estimate its rock properties using Eqs. 2 through 6. The predicted value of sample porosity and permeability is 20.9%, and \( 0.016\;{\text{mD}} \), respectively, where permeability of fracture is ~ 30 mD as shown by Fig. 16. The permeability of matrix estimated from the Singh et al. (2014) and using typical log-normal pore size distribution (Ghanbarian and Javadpour 2017), is 3.2 μD.
Fig. 16

Discretized sample with predicted properties (porosity, and permeability) used the proposed method. The predicted value of sample porosity and permeability are 20.9%, and 0.016 mD, respectively. The permeability of fracture is ~ 30 mD

The two-phase permeability for water and scCO2 at three times of 0.2, 0.4, and 1 s, respectively, are shown in Fig. 17. Permeability of scCO2 increases from ~ 0.08 to ~ 0.25 mD when Sg increases from 0.25 to 0.34, whereas the corresponding permeability of water decreases from ~ 11.28 to 7.29 mD. These estimates are based on the assumption that Swr = 0.
Fig. 17

Discretized sample with two-phase permeability for water and scCO2 at three times of 0.2, 0.4, and 1 s, respectively. Blue color within the fracture represents water and the green color represents carbon dioxide. The permeability of fracture with single-phase flow (t = 0) is ~ 30 mD

3.2.1 Relative Permeability of Water and scCO2 in a Fractured Utica Shale Sample

Figure 18 shows relative permeability of two-phase flow for water and scCO2 within the fracture of the Utica shale sample. The relative permeability is calculated as the ratio of the two-phase flow permeability and the absolute fracture permeability as follows:
$$ k_{{{\text{r}}i,{\text{f}}}} = \frac{{k_{{{\text{f}},i}} }}{{k_{{{\text{f}},{\text{pm}}}} }},\;i = {\text{w}},{\text{g}} $$
(26)
where kri,f is relative permeability of fluid i (= w, g) within a fracture, which is given by Eqs. 20 through 25.
Fig. 18

Two-phase relative permeability for water and scCO2 within the fracture at three saturation values (Sw ~ 0.66, 0.69, and 0.75; shown by solid lines) captured from Fig. 15 and model estimates at other saturations shown with dashed lines. The permeability of fracture with single-phase flow (t = 0) is ~ 30 mD

To estimate the two-phase permeability, we assumed Swr = 0 because of lack of data. It should be noted that although the relative permeability estimates for the three saturations (Sw ~ 0.66, 0.69, and 0.75) were computed with an assumption of Swr = 0 because of lack of data, model estimates (Eqs. 20 through 25) at other saturations (shown with dashed lines) predict that “effective” (approaching kri,f ≈ 0) residual water saturation is Swr ∼ 0.20. Another interesting point to note from Fig. 18 is that, although the proposed two-phase model does not assume any residual saturation for the gas-phase, “effective” (approaching kri,f ≈ 0) residual gas saturation based on krg,f ≈ 0 in this sample can be considered as Sgr = 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 grid-based 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.

Below, it is shown mathematically how two properties (e.g., porosity, permeability) are functions of grid resolution. Porosity is a function of grid resolution because local values of porosity are important, unlike length and apertures that must be looked at their entirety, i.e., overall length and complete thickness of the aperture; these two parameters are not important if defined for a local section within a fracture. We can look at this mathematically as follows:
$$ \phi_{\text{f}} \left( {i,j} \right) = \frac{{l_{{{\text{f}},i,j}} a_{\text{f}} }}{{\Delta x_{i,j} \times \Delta y_{i,j} }} $$
(27)
\( \because \;a_{\text{f}} = F_{\text{a}} \cdot \left( {l_{\text{f}} } \right)^{{\kappa_{\text{a}} }} \), and lf (total fracture length) is independent of \( \Delta x \left( {{\text{or}}\;\Delta y} \right) \), we have:
$$ \phi_{\text{f}} \left( {i,j} \right) \propto \frac{{l_{{{\text{f}},i,j}} \left( {l_{\text{f}} } \right)^{{\kappa_{\text{a}} }} }}{{\Delta x_{i,j} \times \Delta y_{i,j} }} \propto \frac{{l_{{{\text{f}},i,j}} }}{{\Delta x_{i,j} \times \Delta y_{i,j} }} $$
(28)
If \( \Delta y_{i,j} = \Delta x_{i,j} \), then we have
$$ \Rightarrow \phi_{\text{f}} \left( {i,j} \right) \propto \frac{{l_{{{\text{f}},i,j}} }}{{\Delta x_{i,j} \times \Delta y_{i,j} }} \propto \frac{{\Delta x_{i,j} }}{{\Delta x_{i,j} \times \Delta x_{i,j} }} \propto \frac{1}{{\Delta x_{i,j} }} $$
(29)

This final equation shows that porosity is a function of grid resolution.

Similarly, permeability would also be dependent on grid resolution as shown below:
$$ k_{\text{slit}} \left( {i,j} \right) = \frac{{l_{{{\text{f}},i,j}} a_{\text{f}}^{3} }}{{12 \times \Delta x_{i,j} \times \Delta y_{i,j} }} \propto \frac{{l_{{{\text{f}},i,j}} \left( {l_{\text{f}} } \right)^{{3\kappa_{\text{a}} }} }}{{\Delta x_{i,j} \times \Delta y_{i,j} }} $$
(30)
If \( \Delta y = \Delta x \), then
$$ k_{\text{slit}} \left( {i,j} \right) \propto \frac{{\Delta x_{i,j} }}{{\Delta x_{i,j} \times \Delta x_{i,j} }} \propto \frac{1}{{\Delta x_{i,j} }} $$
(31)

3.3.1 Sensitivity of Permeability Estimate to Grid Resolution

An effective approach to obtain accurate predictions from a grid-based method is by making use of some prior information, preferably including regional data, which can be used to calibrate the results. All the results under validation section were calibrated on a 50 × 50 grid dimension. Figure 19 shows results obtained for a sample from Sect. 3.1.1., but with a different grid dimension (60 × 60) than used in Fig. 8. Results shown in Fig. 8 were calibrated using measured values on a grid dimension of 50 × 50, while results shown in Fig. 19 use a grid dimension of 60 × 60 (~ 44% more grids than 50 × 50) to show the effect of grid size on permeability. The global (upscaled) results obtained in Fig. 19 are similar in the case of porosity (ϕeff), but there’s an error of ~ 22 and ~ 25% in the case of upscaled permeability to water and gas, respectively. The errors in regional (local) values of porosity and permeability (single phase) are ~ 27 and ~ 42%, respectively.
Fig. 19

A sample from a validation case (3.1.1.) discretized here on a grid dimension of 60 × 60. The measured value of sample porosity is 3.5%, whereas measured permeability with water and gas are \( k_{\text{h-water}} = 6.28 \times 10^{ - 20} \;{\text{m}}^{2} \) and \( k_{\text{h-argon}} = 2.0 \times 10^{ - 20} \;{\text{m}}^{2} \), respectively

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 pulse-decay 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 pore-scale properties in the analytical permeability model for the matrix.

The proposed method was used to predict two-phase permeability for water and scCO2 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. Two-phase permeability of each water and scCO2 phase is significantly lower than the absolute permeability of the fracture. The presence of non-wetting phase (scCO2) significantly reduces the permeability of the wetting phase (water) from 11.28 to 7.29 mD as the saturation of scCO2 increases from 0.25 to 0.34. Although permeability of the wetting phase is expected to decrease with an increasing saturation of non-wetting phase, estimating the quantitative values generally requires performing experiments or computationally expensive pore-scale simulations. The proposed method allows estimating quantitative values of two-phase permeability that is much simpler compared to other methods (e.g., experiments or pore-scale simulations). In addition to estimation of two-phase permeability within fractures using the proposed method, it is also possible to leverage this method to estimate relative permeability in a non-empirical way, and/or perform sensitivity analysis either at laboratory or field-scale. 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 Swr ∼ 0.20. Similarly, “effective” (practically relevant) residual gas saturation based on krg,f ≈ 0 in this sample was estimated as Sgr = 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 Swr = 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 two-phase permeability model can be used as a quick analytical tool to estimate relative permeability estimates of two-phase flow in fractured shale samples. The proposed two-phase permeability model can also predict “effective” (approaching kri,f ≈ 0) estimates of residual saturations when there is no data on residual saturation by assuming Swr = 0. Although laboratory measurement of relative permeability data in ultra-tight 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 two-phase permeability in an inverse-modeling 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 tight-rock permeability that can vary by as much as 200% (Fisher et al. 2016) between steady-state and pulse-decay methods and more than 300% (Zhang et al. 2013) between MICP and pulse-decay methods of measurement. Even within pulse-decay 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 global-scale measurements on three fractured samples from laboratory. Finally, we used the method to predict two-phase permeability for water and scCO2, 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 tight-rocks, including its regional and upscaled values, compared to other methods (e.g., experiments or pore-scale simulations). The two-phase permeability expressions proposed in this study can be used as a quick analytical tool to predict relative permeability estimates of two-phase flow in fractured shale samples. The proposed two-phase 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 field-scale permeability through an optimization process that uses field-scale production and other readily available information.

6 Nomenclature

Parameter

Description

SI unit

k m

Matrix permeability

m2

k f

Fracture permeability with imbibition

m2

k slit

Ideal slit permeability

m2

δ trans

Thickness of transition zone

m

α

Ratio of gas to water viscosity

k f,pm

Fracture permeability in porous media

m2

k f, i,pm

Fracture permeability of phase \( i ( = {\text{g}},{\text{w)}} \) in porous media

m2

\( k_{\text{m, pm}} \)

Permeability of matrix in porous media

m2

k (i)om,pm

Permeability of (in)organic matter in porous media

m2

c f

Stress-dependent 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)

m2

A f

Fracture area in bulk rock

m2

A b

Rock bulk area

m2

a, b

Rock material-specific 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/m3

βb, βs, β

Compressibility of the bulk sample, matrix, and the fluid

Pa−1

\( V_{\text{u}} , V_{\text{d}} \)

Volumes of upstream and downstream reservoirs

m3

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).

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.National Energy Technology LaboratoryMorgantownUSA
  2. 2.Hubei Subsurface Multi-Scale Imaging Key Laboratory, Institute of Geophysics and GeomaticsChina University of GeosciencesWuhanPeople’s Republic of China

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