## Numerical Simulation Study on Seepage Theory of a Multi-Section Fractured Horizontal Well in Shale Gas Reservoirs Based on Multi-Scale Flow Mechanisms

### Abstract:

Aimed at the multi-scale fractures for stimulated reservoir volume (SRV)-fractured horizontal wells in shale gas reservoirs, a mathematical model of unsteady seepage is established, which considers the characteristics of a dual media of matrix and natural fractures as well as flow in the large-scale hydraulic fractures, based on a discrete-fracture model. Multi-scale flow mechanisms, such as gas desorption, the Klinkenberg effect, and gas diffusion are taken into consideration. A three-dimensional numerical model based on the finite volume method is established, which includes the construction of spatial discretization, calculation of average pressure gradient, and variable at interface, etc. Some related processing techniques, such as boundedness processing upstream and downstream of grid flow, was used to limit non-physical oscillation at large-scale hydraulic fracture interfaces.

#### Authors:

#### Chao Tang^{1}, Xiaofan Chen^{1}, Zhimin Du^{1}, Ping Yue^{1}, and Jiabao Wei^{2}

State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu 610500, Sichuan, China; [email protected] (C.T.); [email protected] (Z.D.); [email protected] (P.Y.)

Hekou Production Plant of Shengli Oilﬁeld, Dongying 257000, Shandong, China; [email protected] * Correspondence: [email protected]

Received: 3 August 2018; Accepted: 27 August 2018; Published: 4 September 2018

The sequential solution is performed to solve the pressure equations of matrix, natural, and large-scale hydraulic fractures. The production dynamics and pressure distribution of a multi-section fractured horizontal well in a shale gas reservoir are calculated. Results indicate that, with the increase of the Langmuir volume, the average formation pressure decreases at a slow rate. Simultaneously, the initial gas production and the contribution ratio of the desorbed gas increase. With the decrease of the pore size of the matrix, gas diffusion and the Klinkenberg effect have a greater impact on shale gas production.

By changing the fracture half-length and the number of fractured sections, we observe that the production process can not only pursue the long fractures or increase the number of fractured sections, but also should optimize the parameters such as the perforation position, cluster spacing, and fracturing sequence. The stimulated reservoir volume can effectively control the shale reservoir.

## Introduction

As an important part of unconventional oil and gas resources, shale gas resources have become a new hot spot in recent years. At present, numerical simulation models for shale gas mainly include dual media, multiple discrete media, and equivalent media, among which dual media models are widely used. Sawyer and Kucuk [1] first studied the pressure changes of shale gas reservoirs based on a dual-porosity continuous medium. Subsequently, Bumb and McKee [2] studied the effect of adsorption-desorption on transient behaviour by adding additional adsorption coefficients to the Langmuir isotherm equation. However, the above studies ignore the diffusion processes at the nano-microscales.

Carlson and Mercer [3] investigated the pressure changes in vertical wells of a shale gas reservoir by introducing diffusion and desorption terms into a dual-porosity media. The model predicts the productivity of shale gas accurately in a short term. However, the long-term prediction of productivity of gas wells is inaccurate, due to the failure to consider the slippage effect. Swami et al. [4] established a dual media model that considers the Knudsen diffusion, slippage, and sorption-desorption processes, and is validated by laboratory data.

Some researchers [5,6,7,8,9,10,11] have pointed out that although dual media models are widely used in commercial software, due to their inherent shortcomings, full or partial encryption still suffers from poor adaptability to multi-scale fracture network systems. In addition, due to the micro-pores in shale gas reservoirs, it is usually necessary to conduct fracturing to obtain a commercial gas production rate.

However, natural and artificial fractures have big differences in morphology and seepage ability. Kuuskraa et al. [12] propose to use multiple discrete media models to study the productivity of shale gas reservoirs.

Based on the concept of multiple media, Schepers [13] and Dehghanpour et al. [14] established a Darcy flow model that couples diffusion and desorption processes with matrix flow, respectively. Wu et al. [15] established a multi-discrete medium model of dense fractured reservoirs, considering the stress sensitivity and slippage effects of fractures. The fractures are divided into natural micro-fractures and artificial fractures. The slippage effect in the matrix is considered and the differences between the multi-discrete and dual-media models are compared. Aboaba and Cheng [16] used a linear flow model to study the typical productivity curve, which describes changes of fractured horizontal wells in shale gas reservoirs without regard to adsorption and diffusion.

In combination with the perturbation method and the point source function, a well test model for a horizontal well, considering diffusion and Darcy flow in a fracture, was proposed by Wang [17]. However, this model does not consider the influence of the reconstructed volume on the pressure change in a horizontal well. Fang et al. [18] considered the compressibility of tight reservoirs and the nonlinear seepage of matrix fluids, and established a multi-scale seepage discrete fracture model of two-dimensional volume fracturing.

In this paper, the authors summarize the law of flow in shale gas reservoirs and establish a three-dimensional (3D) composite model, which uses dual media to describe matrix-natural micro-fractures and utilizes discrete media to describe artificial fractures. The production of multi-section fractured horizontal wells in a rectangular shale gas reservoir is described, considering gas desorption, the Klinkenberg effect, and gas diffusion in the matrix. The stimulated volume is determined by parameter setting of the artificial fractures. The numerical solution is obtained by using the finite volume element method.

## Mathematical Model

### Assumptions

Figure 1a shows a multi-section fractured horizontal well in a shale gas reservoir. The x–y plane represents the horizontal plane, and the z-axis represents the vertical direction. Artificial fractures are represented by two-dimensional elemental bodies. Segments of the horizontal well are represented by one-dimensional, line-element entities. In order to simplify the model, we propose the following assumptions:

**Figure 1.** Diagram of a mathematical model. (a) Multi-section fractured horizontal well grid section diagram; (b) artificial fracture diagram; (c) grid of natural micro-fractures and matrix.

1- The gas reservoirs are rectangular, and the flow is an isothermal flow. The gas reservoirs are divided into artificial fractures, natural micro-fractures, and matrix;

2- Flows in artificial fractures and natural micro-fractures are described by Darcy’s law. The gas desorption in a matrix pore is described by the Langmuir isotherm equation;

3- Horizontal wells produce at constant pressure. There is only a single-phase gas in gas reservoirs;

4- The fractures are perpendicular to the horizontal wellbore and symmetrical about the wellbore;

5- Permeability anisotropy and gravity effects are ignored, and natural gas can only flow into the horizontal wellbore through artificial fractures;

6- Shale gas consists of methane, and does not consider the effect of competitive adsorption on the adsorption-desorption process;

7- Gas diffusion process in shale gas matrix is a non-equilibrium, quasi-steady-state process, which obeys Fick’s first law.

### Governing Equation

#### Flow Equation

According to the real gas state equation, the shale gas density can be defined as

Where i=m or f represents the matrix and the fractures, respectively; ρ_{g} is gas density (kg m^{−3}); Mg is the molecular mass (kg mol^{−1}); Z is the gas deviation factor (dimensionless); p is pressure (Pa); R is the universal gas constant (J mol^{−1} k^{−1}); and T is temperature (K).

Based on the above assumptions, the governing ﬂow equation of shale gas in the matrix can be obtained from the law of conservation of mass, as follows:

Where φ_{m} is the shale matrix porosity (value), v_{m} is the apparent gas velocity (m s^{−1}), q_{m} is the matrix desorption rate (m^{3} s^{−1}), and q_{m-f} is the crow-flow rate from the matrix to the fracture (m^{3} s^{−1}).

The first term in the formula represents the change of fluid mass in the unit volume element of the matrix. The second term is the flux flowing through the surface of the element, which must be modified by introducing the shale-gas-transport mechanisms in nanopores. In this paper, we consider gas molecular diffusion, slippage, and desorption. The third term is the desorption capacity of the matrix. The fourth item is the cross-flow from the matrix to the fracture.

Generally speaking, water is not able to enter the micro-pores in the matrix of shale gas reservoirs. Therefore, it is reasonable to consider that there is only gas phase in the matrix. In other words, the gas in the micro-pores can be divided into adsorption gas adsorbing on the surface of the matrix and the free gas flowing in the micro-pores.

According to the study by Javadpour [19], the Knudsen number is in the transition zone of the viscous flow and Knudsen diffusion under shale gas formation conditions. At this time, the mass exchange of gas in the matrix is affected by viscous flow, Knudsen diffusion, and desorption. Therefore, corrections should be made to the mass flow in the matrix:

Where V^{v}_{m} is the corrected gas velocity considering the Klinkenberg effect, and v^{k}_{m} is the corrected gas velocity considering the diffusive transport. The measure of pores in the matrix is usually tiny compared to other reservoir types.

The additional contribution of the Klinkenberg effect to gas transport may be due to frequent collisions of gas molecules with the wall of the pores, causing the gas viscosity in the Knudsen layer to gradually deviate from the traditional gas viscosity. According to the study by Karniadakis et al., the gas effective viscosity can be expressed as

where µ_{eff} is the gas effective viscosity (mPa·s), µ_{g} is gas viscosity (mPa·s), α is the rarefaction coefficient (dimensionless), and Kn is the Knudsen number (dimensionless).

Combing Equation (4) and Darcy’s law, the corrected gas velocity with the Klinkenberg effect can be expressed as follows:

There are two fundamental modes: the advection and diffusion of fluid transport. The flow governing equation usually neglects the diffusive contribution, which is reasonable for most reservoirs—having a medium-high permeability, it may be unreasonable for shale gas.

According to the mechanism of fluid dynamics [20], the gas diffusive velocity then can be expressed as

where D_{g} is the Knudsen molecule diffusivity (m^{2} s^{−1}), c_{g} is the gas compression factor (Pa^{−1}; δ^{m}), and τ^{m} is the constrictivity and tortuosity of the shale matrix, respectively. The value of δ^{m}/ τ^{m} is always less than one. Therefore, Equation (3) can be rewritten as

Another contribution to gas production from shale reservoirs comes from the desorption of the gas (mostly to the kerogen) absorbed in shale, which is quantified via the change in the gas adsorption amount.

The amount of gas adsorption per unit matrix volume at any pressure can be described by the Langmuir isotherm; then, the matrix desorption rate can be expressed as follows

where Vm is the adsorption capacity of per unit volume matrix (m^{3}), V_{L} is the Langmuir volume (m^{3} kg^{−1}), p L is the Langmuir pressure (Pa), and V std is the mole volume of gas at temperature (273.15 K) and pressure (101,325 Pa).