# A Feature-Based Stochastic Permeability of Shale: Part 2–Predicting Field-Scale Permeability

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## Abstract

In a recent numerical study, it was demonstrated that characterizing reservoir permeability in terms of rock’s quality, as observed in lab and field, is the most important step before implementing an enhanced oil recovery operation or drilling a new well in a tight formation. In that study, it was shown that permeable features in shale-like organic matter (OM) and fractures were the only regions that allowed some reasonable movement of fluid, whereas inorganic matter (iOM) that occupies larger pore volume with significant saturation of hydrocarbons has extremely low permeability that did not allow any reasonable fluid movement to affect production. That study demonstrated the importance of characterizing reservoir heterogeneity in shale in order to economically exploit the shale resource. This study proposes a method to predict spatially heterogeneous field-scale permeability of shale in terms of natural fractures, and matrix (iOM and OM). The method developed in Part 1 is combined with a history-matching process that uses only readily available information from lab-scale and outcrop (information from geologists) to predict field-scale permeability. The method also ensures consistency between the underlying fracture distribution and optimally matched fracture lengths and their apertures, in addition to accounting for random distribution of fractures and their abundance. Optimized parameters of fracture distribution are used to generate multiple realizations of geological model, and the “best-fitting” (most-likely) permeability scenario is chosen by generating production response of each realization of the geological model and comparing them against the observed field production history. The novelty of the proposed to predict field-scale permeability is that it uses only readily available information while also ensuring consistency between the underlying fracture distribution and optimally matched fracture lengths and their apertures, in addition to accounting for random distribution of fractures and their abundance.

## Keywords

Fractures Tight oil Field scale Optimization Organic/inorganic## List of symbol

- \( k_{\text{core}} \)
Core-scale permeability obtained from lab (m

^{2})- \( k \)
Spatially varying permeability (m

^{2})- \( k_{\text{upscaled}} \)
Upscaled permeability (single value) (m

^{2})- \( \phi_{\text{eff}} \)
Effective porosity (single value)

- \( q_{\text{i}} \)
Field-scale ‘predicted’ cumulative production of well \( {\text{i}} \) (m

^{3}/s)- \( q_{{{\text{i}},{\text{obs}}}} \)
Field-scale ‘observed’ cumulative production of well \( {\text{i}} \) (m

^{3}/s)- \( L_{\text{x}} , L_{\text{y}} \)
Dimensions of reservoir along \( {\text{x}} \) and \( {\text{y}} \) direction (m)

- \( l_{\text{frac}} \left( {{\text{or}}\;l_{\text{f}} } \right) \)
Fracture length (m)

- \( {\text{aper}} \) (or \( a_{\text{f}} \))
Fracture aperture (m)

- \( N_{\text{w}} \)
Number of wells

- \( {\text{obj}} \)
Objective function to minimize in optimization

- \( \theta_{\text{f}} \)
Fraction orientation (dip angle) (°)

- \( \lambda \)
Dimensionless fracture intensity (between 0 and 1)

## Subscript/Superscript

- bf
“Best-fitting” scenario after optimization

- Min
Minimum

- Max
Maximum

- Obs
Observation

- Unc
Uncertainty represented by various realizations

- aper
Aperture

- OM
Organic matter

- iOM
Inorganic matter

- field
Field-scale

- core
Laboratory-scale core

## 1 Introduction

One of the challenges in optimal production from shale systems is characterization of reservoir heterogeneity. This was demonstrated by Singh and Cai (2018a), who numerically investigated the potential of enhanced oil recovery (EOR) from shales by comparing homogeneous and heterogeneous scenarios having equal upscaled porosity and permeability. Their study deduced that characterizing reservoir permeability in terms of rock’s quality as observed in the lab (Dong et al. 2017; Hawthorne et al. 2017; Tuero et al. 2017) and field (Witkowsky et al. 2012; Steiner et al. 2016; Kenomore et al. 2017) is the most important step before implementing an EOR operation in shales, as drilling a well around a non-permeable region may lead to uneconomical production. In other words, they showed that permeable features in shale (e.g., OM, and fractures) were the only regions that allowed some reasonable movement of fluid, whereas inorganic matter (iOM) that occupies larger pore volume with significant saturation of hydrocarbons has extremely low permeability that did not allow any reasonable fluid movement to affect production. Singh and Cai (2018a) showed that none of the fluids among water, gas, or surfactant could penetrate the tight iOM because of its low permeability even when capillary pressure was low. They concluded that the unswept oil around a well was a result of low permeability rocks and not because of capillary pressure. To date, the general tendency in EOR from shales is to prefer completions and expensive multi-well drilling (Ghaderi et al. 2017; Hustak et al. 2017) over characterization of reservoir heterogeneity that can be used to exploit the potential of shale in an economical way. Characterizing spatial variation in permeability at field-scale allows exploiting the potential of shale in an economical way by targeting permeable zones over an expensive option of drilling additional wells.

The primary objective of this paper is to use method developed in Part 1 (Singh and Cai 2018b) of this two-part series to predict field-scale permeability through an optimization process that uses readily available information from lab-scale and outcrop (information from geologists) in a history-matching process. Although different approaches exist to predict field-scale permeability by history matching (Oliver and Chen 2011; Rwechungura et al. 2011), such as training image-based method (Caers 2003), ensemble-based methods (Jafarpour and McLaughlin 2008; Jung et al. 2018), etc., they usually require some reservoir scale information (e.g., reservoir geology, seismic, etc.) to constrain the optimization process. Additionally, most of these approaches were proposed to predict geological models without any fractures. Some methods exist to history-match fractured reservoirs (Gang and Kelkar 2006; Nejadi et al. 2014; Lu and Zhang 2015), but they work by optimizing the upscaled permeability of the fractured reservoir, which does not ensure consistency between the underlying fracture distribution and optimally matched fractured reservoir. Although Chai et al. (2018) proposed a method to history-match fractured shale reservoir that ensures the consistency between fracture lengths, their method does not ensure consistency for fracture aperture sizes in addition to ignoring random distribution of natural fractures and their abundance. The history-match process adopted in this study ensures consistency between the underlying fracture distribution and optimally matched fracture lengths and their apertures, in addition to accounting for random distribution of fractures and their abundance.

The proposed approach to predict field-scale permeability and detailed description of the optimization process is presented next in Sect. 2. Finally, field-scale permeability is predicted using the described method for two scenarios and the obtained predictions are compared against their reference “truths” (known synthetic field-scale permeability).

## 2 Method

In Part 1 (Singh and Cai 2018b) of this two-part series, a new method was proposed to model permeability of fractured tight rocks using a grid-based approach where permeability is estimated in each grid cell by identifying rock feature in each cell of the grid as matrix (organic and inorganic) or fracture, and using an appropriate analytical expression to estimate its permeability. To avoid duplication, the method presented in Part 1 is summarized here briefly. The proposed approach to estimate permeability involves the following two steps:

*Step 1* Develop a single discretized grid of the rock superimposed with relevant rock features at the given scale of rock. This step essentially involves developing separate grids for fractures, inorganic matter (iOM), and organic matter (OM), which are later superimposed on a single grid (as shown in Figs. 2 and 3 in Part 1).

*Step 2* Identify rock feature in each cell of the grid as matrix (organic and inorganic) or fracture and calculate permeability using appropriate expressions. Permeability of the rock matrix is calculated using the models proposed by others (Singh et al. 2014; Singh and Javadpour 2016), whereas the permeability of fracture is calculated using analytical expressions derived in Part 1 using steady-state Navier–Stokes equation for single and two-phase flow. In this Part 2 of the two-part series, we only consider a single-phase flow and provide a rationale for this assumption later in the study. The permeability expressions for single-phase flow in fractures are derived in Part 1.

### 2.1 Approach to Predict Field-Scale Permeability

### *Step 1* Create an initial permeability map

Create a permeability map (composed of iOM, OM and fractures) using the approach described in Part 1 of this two-part series. This permeability map is generated by guessing five unknowns that are fracture’s length and aperture distribution parameters \( \left( {l_{ \hbox{min} } , \;l_{ \hbox{max} } , \;\alpha_{\text{l}} , \;F_{\text{a}} ,\; \kappa_{\text{a}} } \right) \) described by List of symbols.

### *Step 2* Simulate field production

Using the permeability map created in step 1, simulate oil production response of the field. This field production data will be used in step 3 to optimize unknown parameters.

### *Step 3* Obtain unknown parameters by optimization

### *Step 4* Obtain most-likely field-scale permeability

The optimized parameters predicted in step 3 are used to generate various realizations of permeability that represent the remaining, but secondary, uncertainty after optimizing the primary unknown parameters of fracture distributions. This secondary uncertainty is reduced by simulating production response for each realization of permeability (generated post optimization) and comparing it against the observed field production. The realization that gives the best production match with the observed production history is chosen as the “best-fitting” permeability scenario.

### 2.2 Modeling Geologic Fractures and Organic Matter

The procedure to discretize the fractures is described in Part 1. The fracture discretization method is based on the approximation of actual fractures on a finite-difference grid where each cell through which the approximate (discretized) fracture passes is assigned discrete fracture properties such as its length, aperture, and orientation. In the absence of field data required to characterize the fractures, a common approach to characterize fractured medium 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). 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 given by List of symbols.

The procedure to spatially sample the OM involves quantifying the probability of finding an OM, which is inversely proportional to the product of bulk density and porosity, or in other words the presence of OM is sensitive to the combined change in bulk density and porosity. Density and porosity from multiple well logs can be used to develop a full reservoir-scale areal map of the density and porosity through kriging, which is used to obtain values at unknown locations per the process described in Part 1.

### 2.3 Optimization

#### 2.3.1 Approach

#### 2.3.2 Method of Optimization

The study seeks to optimize the natural fracture distribution in a shale reservoir, which is assumed stochastic in nature. Traditional optimization methods, based on gradients (or derivatives) of the objective functions, require the objective function to be smooth (twice continuously differentiable), while the stochastic objective functions, such as in this study, are non-smooth. Therefore, derivative estimates of stochastic objective functions using finite difference can provide an unreliable minimum because of randomness. For this reason, traditional optimization methods based on gradients (derivatives) are not suitable for this study. For stochastic optimization problems, a generally accepted minimization technique is based on “response surface models” (also referred as “surrogate modeling approach” or “pattern search”), which use regression techniques to fit the stochastic objective function with a “response surface” and then minimize over it using only the objective function values without any derivatives. The minimized solution can be improved by iteratively refining the model around the current minimum. However, optimization of a stochastic objective function by the “response surface” technique requires more function evaluation to find the approximate global minimum than derivative-based optimization methods, which makes them computationally expensive than traditional derivative-based optimization methods.

The optimization method to predict field-scale permeability of a shale reservoir includes a five-dimensional stochastic objective function and each evaluation of that function requires generating a static geological model (permeability and porosity), and running fluid flow simulations on that geological model. The optimization technique used in this study is based on a surrogate modeling approach (Queipo et al. 2005; Eriksson et al. 2015), which approximates the underlying objective function with a response surface, then optimizes by searching for an improved solution within the surrogate model (Eriksson et al. 2015). As already mentioned, the “response surface” optimization technique requires several evaluations of the objective function to find a robust approximate minimum; therefore, optimization was performed in parallel using 35 cores.

### 2.4 Numerical Flow Simulations

*Bakken*(Singh and Cai 2018a) with an initial reservoir pressure of 8000

*ψ*, and a fluid as compressible light oil whose viscosity and density vary with pressure. We only consider first year of production history; therefore, it can be safely assumed that the reservoir pressure does not drop below bubble point pressure (~ 1500

*ψ*in

*Bakken*; Singh and Cai 2018a) and no free gas exists in the reservoir. The oil is produced from two horizontal wells operating at a constant bottom-hole pressure of 2500

*ψ*as shown in Fig. 3. The grid size used for field-scale simulation is 100 × 100, which is sufficient to reliably represent a 3000 × 3000 m

^{2}field. Further details about numerical simulation of field production are given in “Appendix A”.

Well operations in a field sometimes require a well to be shut-in for a short period, which may lead to fluctuations in production history of a well that are not physical in nature and can lead to erroneous results if the data is not handled well. In order to remove such sources of errors, the time instantaneous well rates are converted to cumulative sum of oil production with time, which is depicted as \( q_{\text{obs}} \) in objective function of optimization.

## 3 Results

We predict a field-scale permeability map of a synthetic tight-oil reservoir using only a year of production history from two wells. Field-scale permeability (composed of iOM, OM, and fractures) is predicted using the method summarized in Fig. 1. The method involves optimizing five parameters with constant values, which are required to characterize fractures, by minimizing the following objective function shown in Eq. (1).

### 3.1 Knowns and Unknowns in Optimization

#### 3.1.1 Knowns

In the above equation, \( q_{\text{obs}} \left( t \right) \) is the well production history from a field where permeability is to be estimated. It is assumed that the core-scale permeability (\( k_{\text{core}} \)) measured in the laboratory is representative of the upscaled permeability of the field. The upscaled permeability (given by Eq. 19 in Part 1) is an averaged estimate of the shale stratum comprising matrix and fractures.

##### 3.1.1.1 Rock Matrix

Although we have assumed the matrix permeability as known to avoid unnecessary complexity, it is possible to assume the matrix as unknown and optimize them along with fracture distribution parameters. However, typically matrix can be well characterized at laboratory scale unlike fracture networks; therefore, this is one of the reasons to assume matrix as known. The above assumed values of matrix permeability varying between 1 to 20 μD indicate a tight matrix. These values for iOM and OM, assumed per laboratory and field observations (Singh and Cai 2018a), suggest that iOM occupy larger pore volume with significant saturation of hydrocarbons, but have extremely low permeability in comparison to the OM.

##### 3.1.1.2 Fracture Orientation and Abundance

Unlike the matrix, fracture properties are unknown at field-scale, except for some information on fracture orientation and their abundance that are generally evaluated from outcrops or well logs. Distribution of fracture orientation and its abundance in the reservoir are assumed to mimic their outcrop distributions. Although the method proposed in this study can allow for these two parameters to be considered as unknowns, we chose to follow a generally accepted practice of treating them as known values observed from outcrop. It is assumed that the majority of fractures are dipping along two different directions of 0, and − 50°, respectively, relative to a horizontal plane.

Fracture abundance in the subsurface is commonly estimated as fracture spacing (Narr 1996), density, or intensity (Dershowitz and Herda 1992) from an analogous outcrop through scanline-based methods or selection method (Manda and Mabee 2010). Although fracture abundance estimated through these common methods is a function of fracture lengths, these methods ignore the wide spectrum of fracture lengths while estimating fracture abundance. This study uses a normalized measure of fracture intensity (Ortega et al. 2006) that accounts for the distribution of fracture lengths under study by estimating cumulative-frequency fracture-size distribution using a scanline data set. This normalized fracture intensity is used to calculate a dimensionless number that we refer as *dimensionless fracture intensity* (\( \lambda \)), which is required to spatially sample the fracture seed through Bernoulli distribution (Shapiro and Zahedi 1990). This *dimensionless fracture intensity* (\( \lambda \)) is estimated by dividing the normalized fracture intensity of each fracture length with the largest value of normalized fracture intensity. This *dimensionless fracture intensity* is used as a probability in a Bernoulli distribution to spatially sample the fracture seeds.

#### 3.1.2 Unknowns

The other information required to characterize a fracture network, such as its length and aperture, is generally unknown. The distribution of fracture lengths and apertures can be characterized by five parameters of two power-law distributions (given by List of symbols), which are \( l_{ \hbox{min} } ,\; l_{ \hbox{max} } , \;\alpha_{\text{l}} , \;F_{\text{a}} ,\; \kappa_{\text{a}} \).

### 3.2 Reference “Truth” Model

#### 3.2.1 Observed Production History

### 3.3 Results of Optimization

#### 3.3.1 Optimized and Reference Truth Parameters

#### 3.3.2 Uncertainty Post-Optimization and Most-Likely Scenario

### 3.4 An Ultra-Tight Field

## 4 Discussions

The method proposed in Part 1 of the two-part series was used in this study to predict field-scale permeability using a history-matching process to find five unknown parameters that characterize the fracture distribution in the field. The history-match process adopted in this study ensures consistency between the underlying fracture distribution and optimally matched fracture lengths and their apertures, in addition to accounting for random distribution of fractures and their abundance. The optimization procedure requires some readily available data from lab-scale and outcrop (information from geologists) as well as field production history. The predicted field-scale porosity and permeability were compared against the (i) reference truth that was used to generate the observed field history, and (ii) the lab-scale value using upscaled value of the predicted permeability. The effective porosity and upscaled permeability value from the best-fitting realization (Fig. 8) are \( \phi_{\text{eff}} = 2.1\% \) and \( k_{\text{upscaled}} = 3.68 \;\upmu{\text{D}} \), which compare well with their corresponding values from reference truth (\( \phi_{\text{eff}} = 2.5\% \) and \( k_{\text{upscaled}} = 2.90 \;\upmu{\text{D}} \)). To verify the robustness of field-scale permeability prediction method, we considered another case with a lower upscaled permeability (\( k_{\text{core}} = 0.05 \;\upmu{\text{D}} \)), while other known information is unchanged. The effective porosity and upscaled permeability value from the best-fitting realization (Fig. 8) of this case are \( \phi_{\text{eff}} = 2.3\% \) and \( k_{\text{upscaled}} = 0.05 \;\upmu{\text{D}} \), which match closely with their corresponding values from reference truth (\( \phi_{\text{eff}} = 2.5\% \) and \( k_{\text{upscaled}} = 0.05 \;\upmu{\text{D}} \)). In case of shales with similar upscaled (average) permeability values, spatial variability of permeability plays a crucial role in the production potential of the shale resource, especially the variability and magnitude of permeability around a well (Singh and Cai 2018a). This regional heterogeneity is naturally accounted by the proposed method through different rock features and by appropriate permeability expression assigned to each cell on the basis of its rock feature. This heterogeneity around the well along with the magnitude of permeability affects the production and this is the reason there is a large band of production uncertainty for an ultra-tight case (Fig. 12) than in case of the field with relatively higher permeability (Fig. 7). This difference in uncertainty between the two cases is primarily the end result of the well location that is not able to sample the permeability variation along with entire reservoir; however, with more number of wells this uncertainty tends to decrease as observed in a study by Singh and Cai (2018a).

Although, the predicted permeability field and its upscaled value compare well against the reference truth, there will be some local differences between the reference truth and prediction; the primary reason for this difference is because of the stochastic (random) nature of the permeability at field scale, while the secondary reason being that the optimization solution for a stochastic problem is non-unique, i.e., there are multiple solutions that can minimize the objective function. The reason fractures are treated as stochastic in nature which is because of the uncertainty in fracture network inside a reservoir as they are not visible and, hence, cannot be characterized deterministically. Although this uncertainty (after optimization) is not desirable in general, it can be reduced further when more data are available in future by comparing the production response of all realizations like in Fig. 7 (or Fig. 12).

## 5 Summary and Conclusions

This study used the method proposed in Part 1 of this two-part series to predict field-scale permeability of a naturally fractured shale system through a history-matching optimization process that uses only readily available information. The method proposed in Part 1 requires discretizing shale into a grid by superimposing the elements of the fractures and the matrix (iOM and OM) on a single grid. Fractures at field scale are represented stochastically using statistical distributions mimicking their random nature in terms of lengths, aperture sizes, abundance and spatial distribution. The predicted field-scale permeability for the two cases (tight and ultra-tight permeability, respectively) compared well against their respective reference truth. The “best-fitting” (most-likely) solution for field-scale permeability was chosen from an ensemble of stochastic realizations; multiple realizations obtained after optimization also allow assessing the performance of each realization with other customized approaches with additional well history in future, or by using other forms of available field data (e.g. bottomhole pressure).

Characterization of heterogeneous permeability of shale in terms of its most common rock features (iOM, OM, fractures) would allow exploiting the shale reservoir in an economical way by screening out the areas that might otherwise prove to be an uneconomical prospect. To our knowledge, this is a first study that uses only readily available information to predict spatially heterogeneous shale permeability (with iOM, OM, and fractures) while also ensuring consistency between the underlying fracture distribution and optimally matched fracture lengths and their apertures, in addition to accounting for random distribution of fractures and their abundance.

## Notes

### Acknowledgement

This research was supported in part by an appointment to the National Energy Technology Laboratory Research Participation Program, sponsored by the U.S. 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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