## Abstract

Unconventional shale gas production is often characterized by a short period of high production followed by a rapid decline in the production rate. Given the high costs of hydraulic fracturing and horizontal drilling, it is critical to identify the mechanisms behind the production loss. The existing shale gas production models often assume constant matrix permeability. However, laboratory observations show that matrix permeability can decrease significantly with increasing effective stress, which highlights the necessity of considering the stress-dependent properties of shale matrix in production analysis.

#### Authors:

##### Huiying Tang^{1}, Yuan Di^{1}, Yongbin Zhang^{2} and Hangyu Li^{3}

^{1}College of Engineering, Peking University, Beijing 100871, China. ^{2}Tarim Oil Company, PetroChina, Korla 841000, Xinjiang, China. ^{3}Department of Energy Resources Engineering, Stanford University, Stanford, CA 94305, USA

Received: 17 June 2017; Accepted: 12 July 2017; Published: 14 July 2017

© 2017 by the authors. Licensee MDPI, Basel, Switzerland.

Moreover, the compaction of pore space will also increase the matrix permeability by enhancing the gas-slippage effect. In this paper, a matrix permeability model which couples the effect of pore volume compaction and non-Darcy slip flow is derived. Numerical simulations are conducted to understand the role of matrix permeability evolution during production. Changes of fractures’ permeability and contact area during depletion process are also taken into account. The results indicate that the loss of fracture permeability has a greater impact at the early stage of the depletion process, while matrix permeability evolution is more important for the long-term production.

## Introduction

A significant amount of natural gas in the United States is produced from shale gas reservoirs. Although the initial production rates might be high for shale gas wells, the rates decline steeply [1,2], which significantly affects the economics of many shale gas projects. Therefore, it is of great importance to identify the mechanisms responsible for the rapid production loss for shale gas reservoirs.

In shales, the transport processes take place on scales covering several orders of magnitude. These processes range from the slip flow which occurs on the nanometer scale, to fracture flow which takes place on the centimeter scale. The slip flow, also known as Klinkenberg effect [3,4], occurs when the mean free path length of a gas molecule is on the same order of magnitude as the pore size. In this situation, the gas molecules might be accelerated along the flow path which results in an increase in the apparent permeability. The slip flow and the compaction of pores have been studied independently in literature.

However, because the strength of gas slippage is related to the size of the pore-throat, the pore size reduction caused by compaction will also influence the slippage process. It has been proved by coal experiments that a strong coupling effect exists between the pore compressibility and the gas slippage [5]. Thus, it is necessary to build a coupled permeability model to combine the gas-slippage effect and pore size compaction for unconventional reservoirs.

In addition to the matrix gas flow, the fracture flow is also critically important in unconventional reservoirs. The conductivities of fractures were found to be very sensitive to the applied effective stress based on the experiment results [6]. The impact of stress-dependent fracture conductivities on gas production has been studied using numerical simulations [7,8]. Besides, the contact area between matrix and fractures will affect gas transport between matrix and fractures. For example, Huo and Benson [9] reported a loss of contact area of opened fractures by scanning shale samples layer by layer. Later, Pyrak-Nolte and Nolte [10] conducted numerical simulations on fluid flow through porous media and the contact surface area was found to decrease with increasing effective stress as well. However, a mathematical model that represents the impact of contact surface area loss on shale gas production has not been investigated yet.

In this study, we develop a matrix permeability model which couples the poro-elastic and the gas-slippage effects. The impacts of stress-dependent fracture and matrix permeabilities on shale gas production are investigated to understand the relative importance of these effects. The mechanism of fracture-matrix contact area loss is also discussed and integrated into the stress-dependent model.

## Model Description

### Stress-Dependent Matrix Permeability

As the effective stress increases, both the matrix permeability and porosity will be reduced due to the compaction of pores (Figure 1). The dependence of matrix permeability on effective stress can be described as a poro-elastic effect which has an exponential form [11,12]:

where k_{d} is the matrix permeability at low Knudsen number (<0.01, Heller et al. [13]) when only Darcy flow occurs, subscript d indicates Darcy flow, c_{m} is the poro-elastic coefficient, x is the Biot’s constant,σ_{c }is the confining stress, p_{f} is the fluid pressure, and k_{d},0 denotes the matrix permeability under zero effective stress.

**Figure 1.** Schematic illustrating the effect of pore space compaction.

We assume that the pore space in shale matrix consists of a series of parallel tubes with varying radii as shown in Figure 2. The intrinsic permeability, k_{int}, of a cylinder tube with the radius of r is given by [14]:

**Figure 2.** Schematic showing the pore structures in shale matrix.

Based on Equation (2), the apparent permeability (permeability measured by total flux) of the shale matrix system shown in Figure 2 can then be expressed by:

where r_{i} is the radius of tube i (i ranges from 1 to N), N is the number of tubes in the matrix, and A is the total area of the cross-section perpendicular to the tubes. The porosity can be written as a function of tube radius as well:

By combining Equations (1) and (3), with further assumption that the compressibility, the change of pore radius with per unit increase of effective stress, of each tube is the same, the radius of tube i under the effective stress can be written as:

where r_{i},_{0} is the radius of tube i under zero effective stress, and all the other variables are as defined previously. The compressibility of the pores can then be written as:

The mean free path *λ* of an ideal gas is given by:

where *µ*_{g} is the gas viscosity, Z_{g} is the gas compressibility factor, R is the universal gas constant which equals to 8.314 J/mol/K, and M and T are the molecule weight and temperature of the ideal gas, respectively. Knudsen number is defined as the ratio of the molecule mean free path to the flow path radius (e.g., radius of the tube):

By substituting Equation (5) into Equation (8), the Knudsen number can be expressed with the fluid pressure as:

With the decrease in fluid pressure during the production, the Knudsen number will increase not only due to the pressure decline but also due to the shrinkage of the tubes.

In the nanometer-scale pores in shale matrix, the mean free path is comparable with the tube radius. In this circumstance, the gas molecules may slip along the tube which results in an increase in the apparent matrix permeability. A variety of models to correct the permeability with gas-slippage effect have been proposed in the literature [15,16]. Here, the formulation developed by Zhang et al. [17] is adopted and is given by:

where k_{int,t} is the instrinsic permeability with consideration of gas slippage, c is a constant derived based on the kinetic theory of gases and the value of c is taken as 6 in most situations [17]. By substituting Equations (2) and (9) into Equation (10), for the tube with radius r_{i} , the intrinsic permeability with gas-slippage effect can then be expressed as:

By calculating the permeability of each cube with Equation (11), the volumetric flow rate across the cross-section in Figure 2 can then be calculated as:

where k_{m} is the apparent matrix permeability and can be computed as:

If we assume all tubes have the same radius, the above equation can be simplified as:

One assumption made in Equation (14) is that the slippage effect can be neglected at the initial pore pressure because the pressure before production is considered high. Only with this assumption, Only with this assumption, we can use k_{d,0 }to replace (πN/A)(r^{4}_{0}/8).