Simulation of Shale Gas Reservoir Production Considering the Effects of the Adsorbed Water Layer and Flow Differences

Abstract

Accurately describing the behavior of a gas-water two-phase flow in shale gas reservoirs is crucial for analyzing production dynamics in the field. Current research generally lacks consideration of the differences in physical properties and adsorption characteristics between the oleophilic organic matrix and the hydrophilic inorganic matrix. This study considers the organic matrix system as a single-phase gas flow, while the inorganic matrix and fracture systems involve a gas-water two-phase flow. Taking into account the impact of the adsorbed water layer on permeability at the surface of nanoscale pores in an inorganic matrix, the model comprehensively incorporates multiple mechanisms such as adsorption-desorption, the slippage effect, and Knudsen diffusion in the organic matrix and clay minerals. A multiscale gas-water two-phase comprehensive flow model for shale gas reservoirs has been established, and the results of the numerical model were validated against commercial software and actual field data. Simulation results over 1000 days indicate that early production from gas wells is primarily supplied by fractures, whereas free gas or desorbed gas from inorganic and organic matrices gradually contributes to the flow during the middle and later stages of production. As the Langmuir pressure and volume in the organic matrix and clay minerals increase, so does the corresponding gas production. The adsorbed water layer on the surface of inorganic nanopores reduces permeability, leading to a decrease in single-well cumulative gas production by 8.41%. The impact of the adsorbed water layer on gas production cannot be overlooked. The simulation method proposed in this study provides theoretical support for analyzing the gas-water two-phase flow behavior in shale gas reservoirs.

1. Introduction

Shale gas is not only a clean unconventional natural gas resource but also a strategically important asset for oil and gas exploration and development worldwide [1]. Shale gas reservoirs are characterized by multiscale porous media, which include nanopores, micropores, and microfractures [2]. Additionally, shale gas reservoirs also exhibit multiscale flow characteristics, encompassing processes such as desorption, diffusion, slippage, and seepage [3]. In shale formation, organic matter typically coexists with clay minerals [4]. The organic matter contains numerous nanopores with adsorption properties, while the inorganic matter not only contains hydrophilic nanopores but also possesses methane adsorption due to the unique crystalline structures of clay minerals [5]. A small portion of existing research roughly assumes that flow in the entire matrix is predominantly single-phase gas due to the low permeability of the matrix [6,7]. However, besides developing numerous nanopores, the clay and brittle minerals in the matrix also contain some micrometer-scale microfractures. When the formation undergoes gas-water two-phase flow, water tends to enter the hydrophilic inorganic matter and adsorb on the surface of clay minerals. Typically, water does not flow into the more oleophilic organic matter [8,9,10,11].

There is a paucity of studies about how the water phase in a shale inorganic matrix impacts its permeability. Research by Passey et al. [12] highlighted that the thickness of the adsorbed water layer on the nanopore wall in a shale inorganic matrix is comparable to the pore size itself. Shi et al. [13] observed that the effect of the adsorbed water layer on the permeability of the inorganic matrix varies with the level of water saturation. Li et al. [14] indicated the presence of an adsorbed water layer on the surface of clay minerals in shale, with the layer thickness predominantly determined by relative humidity and pore size. Jin and Firoozabadi [15] demonstrated through grand canonical Monte Carlo (GCMC) simulations that an adsorbed water layer forms on the surface of clay minerals, constricting the space available for gas flow. A molecular dynamics (MD) study by Liu et al. [16] revealed that the adsorption energy of water molecules on the surface of clay minerals surpasses that of methane molecules, resulting in a noticeable aggregation of water molecules in an adsorbed layer. Current research uniformly acknowledges the existence of an adsorbed water layer on the surface of inorganic clay minerals in shale, yet the specific effects on gas production remain unexplored. Moreover, precisely describing the multiscale porous structure and flow characteristics in shale gas reservoirs remains a significant challenge.

Shale gas wells undergo extensive multistage hydraulic fracturing to enhance production, involving substantial volumes of fracturing fluid injected into the reservoir. This results in the entire production lifecycle of shale gas wells being accompanied by the flowback of fracturing fluids. After fracturing, the reservoir exhibits hydraulic fracturing and gas-water two-phase flow [17], which complicates the accurate description of multiscale effects and dynamic changes in gas-water flow behavior in shale gas reservoirs. The flow model of shale gas reservoirs is key to describing the dynamic analysis of gas well production under multiscale effects, and many scholars have proposed different models for shale gas reservoir flow. For instance, some scholars have introduced matrix-fracture dual-porosity models for gas-water two-phase flow [18,19,20,21]. These models often consider the multiscale flow characteristics of shale gas reservoirs and use source function methods to obtain semi-analytical solutions or numerical methods like finite difference to achieve numerical solutions. However, they are not sufficient to precisely describe the multiscale porous media characteristics of shale formation. Additionally, fewer scholars have proposed gas-water two-phase triple-porosity models with comparative analyses presented in

Table 1. Comparison of triple-porosity models for gas-water flow in shale gas reservoirs.

Researcher	Three Media	Characteristics of Model
Clarkson et al. [22] (2012)	Hydraulic fracture, inorganic matrix, organic matrix.	The desorption, diffusion, and slippage effect are considered. The matrix system is roughly regarded as single-phase gas flow.
Huang et al. [23] (2018)	Hydraulic fracture, natural fracture, matrix.	Both the matrix system and the natural fracture system are treated as a single porous medium. It failed to capture the distinct physical properties of the organic versus inorganic matrix.
Li et al. [24] (2019)	Hydraulic fracture, inorganic matrix, organic matrix.	The reservoir is divided into the unstimulated reservoir volume (USRV) region and the effectively stimulated reservoir volume (ESRV) region, without considering the slippage effect. It focuses on analyzing the gas-water flow behavior in different regions.
Cui et al. [25] (2020)	Natural fracture, inorganic matrix, organic matrix.	The model describes two-phase flow in fractures and single-phase gas flow in the matrix. It does not consider the influence of the water phase on gas production in the inorganic matrix.
Cheng et al. [26] (2020)	Fracture system (natural fracture and hydraulic fracture), inorganic matrix, organic matrix.	This model takes into account desorption, diffusion, and the slippage effect. It does not differentiate between the gas-water flow characteristics in the organic and inorganic matrices.
Xu et al. [27] (2021)	Hydraulic fracture, inorganic matrix, organic matrix.	The inorganic matrix solely considers the slippage effect, while both desorption and diffusion processes are overlooked.
Hu et al. [28] (2022)	Fracture system (natural fracture and hydraulic fracture), inorganic matrix, organic matrix.	The fracture system involves gas-water two-phase flow, while only single-phase gas flow is considered in the inorganic matrix. Both organic and inorganic matrices have ignored the slippage effect.
Wang et al. [29] (2023)	Hydraulic fracture, natural fracture, matrix.	The matrix system did not differentiate both the physical properties and gas-water flow differences of organic and inorganic matrices.
Wang et al. [30] (2024)	Hydraulic fracture, microfracture, matrix.

. It is evident that current research on gas-water two-phase flow in shale gas reservoirs is predominantly based on matrix-fracture dual-porosity models. Some scholars have further developed triple-porosity flow models based on the traditional dual-porosity models. These models provide a theoretical foundation for predicting the production dynamics of shale gas reservoirs. However, current research lacks a clear understanding of the gas-water two-phase flow mechanisms in the matrix, and the dynamic pressure characteristics of the organic matrix, inorganic matrix, and fractures during production are not sufficiently studied. It is necessary to consider the flow characteristics of organic and inorganic matrices separately in gas-water two-phase flow models. This is crucial for a more detailed description of the gas-water two-phase flow behavior under the multiscale porous media of shale gas reservoirs.

This study develops a comprehensive gas-water two-phase flow model for shale gas reservoirs. It includes independent physical systems for the organic matrix, inorganic matrix, and fractures. This model not only integrates dissolved gas in the organic matrix as a gas source into the matrix system but also considers multiscale flow phenomena in organic nanopores. It considers viscous flow driven by pressure differences, adsorption-desorption processes, the slippage effect, and Knudsen diffusion mechanisms in the organic matrix and clay minerals. Finally, we apply the model to multistage fractured horizontal wells in shale gas reservoirs and analyze the dynamic pressure changes, gas well production capacity, and the impact factors of the model. This has significant implications for the accurate simulation and prediction of production dynamics in shale gas wells.

2. Theory and Methods

2.1. Conceptual Model

A scanning electron microscope was utilized to observe the distribution of organic and inorganic matrices in the shale matrix, as depicted in Figure 1 [31,32,33]. Figure 1a reveals that the organic matrix is encircled by inorganic matters such as clay minerals, calcite, and pyrite [31]. In Figure 1b, “OM” signifies organic matrix, while “Cal-Clay” denotes a mixture of clay minerals and calcite [31]. Figure 1c illustrates the symbiotic coexistence of the organic matrix with clay minerals, quartz, and feldspar [32]. Figure 1d highlights the physical structural differences between organic and inorganic matrices [33]. It can be observed that a large number of organic pores develop in the shale organic matter, which coexists with clay minerals. The organic matter is either adsorbed on the surface of clay minerals or bound in an adhered state with them. Additionally, a small portion of the organic matter is associated with pyrite and calcite.

This study analyzes the coexistence of the organic matrix with clay minerals, pyrite, and calcite in the microscopic pore structures of shale. It models the organic matter as spherical aggregates based on the De Swaan spherical model [34]. A physical model is proposed where the organic matrix forms spherical aggregates symbiotically with the inorganic matrix. To enhance the mathematical description of distinct adsorption, desorption, and seepage behaviors of both organic and inorganic matters, the model also considers the impact of an adsorbed water layer on the permeability of inorganic nanopores. The shale organic matrix, inorganic matrix, and fractures are treated as three distinct systems. The physical model is shown in Figure 2, and the flow process is depicted in Figure 3. The model assumptions are as follows: (1) water is disregarded in the organic matter, with only single-phase gas flow considered; (2) the imbibition of fracturing fluid into the inorganic matter under capillary pressure is considered; (3) the fracture system exhibits gas-water two-phase flow; (4) capillary pressure is present only in the inorganic matrix and fracture systems, not in the organic matrix; (5) the effect of gravity is considered.

2.2. Mathematical Models

2.2.1. Permeability Correction Model

In the shale matrix, nanopore sizes typically range from a few nanometers to several hundred nanometers, resulting in the flow of shale gas no longer being linear Darcy flow. Therefore, it is necessary to correct permeability. Beskok et al. [35] developed a corrected permeability model that accounts for Knudsen diffusion and slip effect, applicable across the entire range of Knudsen numbers, including continuum flow, slip flow, transition flow, and free molecular flow. The corrected form of Darcy’s law can be expressed as [36,37]:

$$v_g=-{\displaystyle \frac{k_a}{{\mu }_g}}(1+aKn)\left(1+{\displaystyle \frac{4Kn}{1-bKn}}\right){\displaystyle \frac{\mathrm{d}p}{\mathrm{d}x}}$$

(1)

The Knudsen number can be calculated using the following equation:

$$Kn={\displaystyle \frac{K_{B}T}{\sqrt{2}\pi {\delta }^{2}p{r}_n}}$$

(2)

The rarefaction coefficient can be calculated using the following equation [38]:

$$a={\displaystyle \frac{128}{15{\pi }^2}}{\tan }^{-1}\left(4K{n}^{0.4}\right)$$

(3)

Therefore, the apparent permeability of the shale matrix considering Knudsen diffusion and slip effect can be expressed as follows:

$$k=k_a\xi ,$$

(4)

$$\xi =(1+aKn)\left(1+{\displaystyle \frac{4Kn}{1-bKn}}\right)$$

(5)

2.2.2. Single-Phase Flow in Organic Matrix System

When the shale formation exhibits gas-water two-phase flow, water typically does not flow into the oleophilic organic matrix. Therefore, it is unnecessary to consider the impact of the adsorbed water layer on permeability in the organic matrix. Attention should be focused solely on Knudsen diffusion and slip effect. Additionally, as development progresses and formation pressure decreases to the desorption pressure of shale gas, the adsorbed gas on the surface of clay minerals or the organic matrix begins to desorb. This process is commonly represented by the Langmuir isotherm adsorption model. The gas-phase flow differential equation in the organic matrix system can be derived from the law of mass conservation as follows:

$$\nabla \left[{\displaystyle \frac{k_{\mathrm{o}}{\xi }_{\mathrm{o}}}{B_{\mathrm{g}}{\mu }_{\mathrm{g}}}}\left(\nabla p_{\mathrm{og}}-{\rho }_{\mathrm{og}}g\nabla D\right)\right]-q_{\mathrm{o}-\mathrm{c},\mathrm{g}}={\displaystyle \frac{\partial }{\partial t}}\left({\displaystyle \frac{{\varphi }_{\mathrm{o}}{s}_{\mathrm{og}}}{B_{\mathrm{g}}}}+{\displaystyle \frac{{\rho }_{\mathrm{bi}}{V}_{\mathrm{Lo}}{p}_{\mathrm{og}}}{p_{\mathrm{Lo}}+p_{\mathrm{og}}}}\right)$$

(6)

The gas exchange term from the organic matrix to the inorganic matrix can be expressed as follows:

$$q_{\mathrm{o}-\mathrm{c},\mathrm{g}}={\displaystyle \frac{\alpha k_{\mathrm{o}}{\xi }_{\mathrm{o}}}{B_{\mathrm{g}}{\mu }_{\mathrm{g}}}}\left(p_{\mathrm{og}}-p_{\mathrm{cg}}\right)$$

(7)

2.2.3. Two-Phase Flow in Inorganic Matrix System

When shale reservoirs experience gas-water two-phase flow, water enters the inorganic matter under the influence of capillary pressure. Water also adsorbs onto the surface of inorganic nanopores. Therefore, it is necessary to consider the impact of the adsorbed water layer on the gas-water two-phase flow in inorganic matrix pores [12]. Based on the research by Shi et al. [13], corrections are made to the permeability of the inorganic matrix to account for the adsorbed water layer. The model of inorganic nanopores with the adsorbed water layer is shown in Figure 4.

The equation for the water saturation in the inorganic matrix with a pore diameter of $D_{\mathrm{c}}$ is:

$$s_{\mathrm{w}}={\displaystyle \frac{\pi D_{\mathrm{c},0}^{2}L/4-\pi D_{\mathrm{c}}^{2}L/4}{\pi D_{\mathrm{c},0}^{2}L/4}}=1-{\displaystyle \frac{D_{\mathrm{c}}^2}{D_{\mathrm{c},0}^2}}$$

(8)

Assuming $\beta$ is the influence factor of the adsorbed water layer, Equation (8) can be further simplified to:

$$\beta ={\displaystyle \frac{D_{\mathrm{c}}}{D_{\mathrm{c},0}}}={\left(1-s_{\mathrm{w}}\right)}^{0.5}$$

(9)

By combining the permeability correction Equations (4) and (5) and considering the effect of the adsorbed water layer, Knudsen diffusion, and slippage, the equation for the corrected permeability in the inorganic matrix is as follows:

$$k=k_{\mathrm{c}}{\xi }_{\mathrm{c}},$$

(10)

$${\xi }_{\mathrm{c}}=\left[1+{\displaystyle \frac{128}{15{\pi }^2}}{\tan }^{-1}\left[4{(Kn{\beta }^{-1})}^{0.4}\right]Kn{\beta }^{-1}\right]\left(1+{\displaystyle \frac{4Kn{\beta }^{-1}}{1-bKn{\beta }^{-1}}}\right)$$

(11)

The differential flow equation for the gas phase in the inorganic matrix system can be expressed as:

$$\nabla \cdot \left[{\displaystyle \frac{k_{\mathrm{c}}{\xi }_{\mathrm{c}}{k}_{\mathrm{crg}}}{B_{\mathrm{g}}{\mu }_{\mathrm{g}}}}\left(\nabla p_{\mathrm{cg}}-{\rho }_{\mathrm{cg}}g\nabla D\right)\right]-q_{\mathrm{c}-\mathrm{mf},\mathrm{g}}+q_{\mathrm{o}-\mathrm{c},\mathrm{g}}={\displaystyle \frac{\partial }{\partial t}}\left({\displaystyle \frac{{\varphi }_{\mathrm{c}}{s}_{\mathrm{cg}}}{B_{\mathrm{g}}}}+{\displaystyle \frac{{\rho }_{\mathrm{bi}}{V}_{\mathrm{Lc}}{p}_{\mathrm{cg}}}{p_{\mathrm{Lc}}+p_{\mathrm{cg}}}}\right)$$

(12)

The gas exchange term from the inorganic matrix to microfractures can be expressed as:

$$q_{\mathrm{c}-\mathrm{mf},\mathrm{g}}={\displaystyle \frac{\alpha k_{\mathrm{c}}{\xi }_{\mathrm{c}}{k}_{\mathrm{crg}}}{B_{\mathrm{g}}{\mu }_{\mathrm{g}}}}\left(p_{\mathrm{cg}}-p_{\mathrm{mfg}}\right)$$

(13)

According to the law of mass conservation, the differential equation for the water-phase flow in the inorganic system can be derived as follows:

$$\nabla \left[{\displaystyle \frac{k_{\mathrm{c}}{k}_{\mathrm{crw}}}{B_{\mathrm{w}}{\mu }_{\mathrm{w}}}}\left(\nabla p_{\mathrm{cw}}-{\rho }_{\mathrm{cw}}g\nabla D\right)\right]-q_{\mathrm{c}-\mathrm{mf},\mathrm{w}}={\displaystyle \frac{\partial }{\partial t}}\left({\displaystyle \frac{{\varphi }_{\mathrm{c}}{s}_{\mathrm{cw}}}{B_{\mathrm{w}}}}\right)$$

(14)

The water exchange term from the inorganic matrix to microfractures can be expressed as:

$$q_{\mathrm{c}-\mathrm{mf},\mathrm{w}}={\displaystyle \frac{\alpha k_{\mathrm{c}}{k}_{\mathrm{crw}}}{B_{\mathrm{w}}{\mu }_{\mathrm{w}}}}\left(p_{\mathrm{cw}}-p_{\mathrm{mfw}}\right)$$

(15)

The saturation of the gas phase and the water phase in the inorganic matrix satisfy the following equation:

$$s_{\mathrm{cg}}+s_{\mathrm{cw}}=1$$

(16)

Capillary pressure exists in the inorganic matrix, and it satisfies the following equation:

$$p_{\mathrm{cc}}\left(s_{\mathrm{cw}}\right)=p_{\mathrm{cg}}-p_{\mathrm{cw}}$$

(17)

2.2.4. Two-Phase Flow in Fracture System

The shale gas reservoir exhibits gas-water two-phase flow after hydraulic fracturing, with the fracture system serving as the primary conduit for this flow. Additionally, with continuous development of the reservoir, the decrease in formation pressure leads to the gradual closure of fractures, which reduces porosity and permeability. The stress sensitivity of porosity and permeability in microfractures can be respectively represented as follows [39,40]:

$${\varphi }_{\mathrm{s},\mathrm{mf}}={\varphi }_{\mathrm{mf}0}\exp \left(-C(p_{\mathrm{mf}0}-p_{\mathrm{mf}})\right),$$

(18)

$$k_{\mathrm{s},\mathrm{mf}}=k_{\mathrm{mf}0}\exp \left(-2C(p_{\mathrm{mf}0}-p_{\mathrm{mf}})\right)$$

(19)

The equation for gas-phase flow in microfractures is as follows:

$$\nabla \cdot \left[{\displaystyle \frac{k_{\mathrm{s},\mathrm{mf}}{k}_{\mathrm{mfrg}}}{B_{\mathrm{g}}{\mu }_{\mathrm{g}}}}\left(\nabla p_{\mathrm{mfg}}-{\rho }_{\mathrm{mfg}}g\nabla D\right)\right]-q_{\mathrm{mf}-\mathrm{hf},\mathrm{g}}+q_{\mathrm{c}-\mathrm{mf},\mathrm{g}}={\displaystyle \frac{\partial }{\partial t}}\left({\displaystyle \frac{{\varphi }_{\mathrm{s},\mathrm{mf}}{s}_{\mathrm{mfg}}}{B_{\mathrm{g}}}}\right)$$

(20)

The gas exchange term from microfractures to hydraulic fractures can be expressed as:

$$q_{\mathrm{mf}-\mathrm{hf},\mathrm{g}}=\alpha {\displaystyle \frac{k_{\mathrm{s},\mathrm{mf}}{k}_{\mathrm{mfrg}}}{B_{\mathrm{g}}{\mu }_{\mathrm{g}}}}\left(p_{\mathrm{mfg}}-p_{\mathrm{hfg}}\right)$$

(21)

The equation for water-phase flow in microfractures is as follows:

$$\nabla \cdot \left[{\displaystyle \frac{k_{\mathrm{s},\mathrm{mf}}{k}_{\mathrm{mfrw}}}{B_{\mathrm{w}}{\mu }_{\mathrm{w}}}}\left(\nabla p_{\mathrm{mfw}}-{\rho }_{\mathrm{mfw}}g\nabla D\right)\right]-q_{\mathrm{mf}-\mathrm{hf},\mathrm{w}}+q_{\mathrm{c}-\mathrm{mf},\mathrm{w}}={\displaystyle \frac{\partial }{\partial t}}\left({\displaystyle \frac{{\varphi }_{\mathrm{s},\mathrm{mf}}{s}_{\mathrm{mfw}}}{B_{\mathrm{w}}}}\right)$$

(22)

The water exchange term from microfractures to hydraulic fractures can be expressed as:

$$q_{\mathrm{mf}-\mathrm{hf},\mathrm{w}}=\alpha {\displaystyle \frac{k_{\mathrm{s},\mathrm{mf}}{k}_{\mathrm{mfrw}}}{B_w{\mu }_w}}\left(p_{\mathrm{mfw}}-p_{\mathrm{hfw}}\right)$$

(23)

The saturation of the gas phase and the water phase in microfractures satisfy the following equation:

$$s_{\mathrm{mfg}}+s_{\mathrm{mfw}}=1$$

(24)

Capillary pressure exists in microfractures, satisfying the following equation:

$$p_{\mathrm{mfc}}\left(s_{\mathrm{mfw}}\right)=p_{\mathrm{mfg}}-p_{\mathrm{mfw}}$$

(25)

After hydraulic fracturing of the shale gas well, the gas-water two-phase flow in the hydraulic fracture system must be considered. The differential equation for the gas-phase flow in hydraulic fractures can be derived as follows:

$$\nabla \cdot \left[{\displaystyle \frac{k_{\mathrm{s},\mathrm{hf}}{k}_{\mathrm{hfrg}}}{B_{\mathrm{g}}{\mu }_{\mathrm{g}}}}\left(\nabla p_{\mathrm{hfg}}-{\rho }_{\mathrm{hfg}}g\nabla D\right)\right]-q_{\mathrm{g}}+q_{\mathrm{mf}-\mathrm{hf},\mathrm{g}}={\displaystyle \frac{\partial }{\partial t}}\left({\displaystyle \frac{{\varphi }_{\mathrm{s},\mathrm{hf}}{s}_{\mathrm{hfg}}}{B_{\mathrm{g}}}}\right)$$

(26)

The differential equation for the water-phase flow in hydraulic fractures is:

$$\nabla \cdot \left[{\displaystyle \frac{k_{\mathrm{s},\mathrm{hf}}{k}_{\mathrm{hfrw}}}{B_{\mathrm{w}}{\mu }_{\mathrm{w}}}}\left(\nabla p_{\mathrm{hfw}}-{\rho }_{\mathrm{hfw}}g\nabla D\right)\right]-q_{\mathrm{w}}+q_{\mathrm{mf}-\mathrm{hf},\mathrm{w}}={\displaystyle \frac{\partial }{\partial t}}\left({\displaystyle \frac{{\varphi }_{\mathrm{s},\mathrm{hf}}{s}_{\mathrm{hfw}}}{B_{\mathrm{w}}}}\right)$$

(27)

The gas-phase saturation and water-phase saturation in hydraulic fractures satisfy the following equation:

$$s_{\mathrm{hfg}}+s_{\mathrm{hfw}}=1$$

(28)

Capillary pressure exists in hydraulic fractures, satisfying the following equation:

$$p_{\mathrm{hfc}}\left(s_{\mathrm{hfw}}\right)=p_{\mathrm{hfg}}-p_{\mathrm{hfw}}$$

(29)

2.3. Solution-Determining Conditions

2.3.1. Initial Conditions

The gas-water two-phase flow model of shale gas reservoirs developed in this study satisfies the following initial conditions:

$$\left\{\begin{array}{l}{\left.p_{\mathrm{hfg}}(x,y,z,0)\right|}_{t=0}=p_{\mathrm{hfg}}^0(x,y,z)\\ {\left.p_{\mathrm{mfg}}(x,y,z,0)\right|}_{t=0}=p_{\mathrm{mfg}}^0(x,y,z)\\ {\left.p_{\mathrm{cg}}(x,y,z,0)\right|}_{t=0}=p_{\mathrm{cg}}^0(x,y,z)\\ {\left.p_{\mathrm{og}}(x,y,z,0)\right|}_{t=0}=p_{\mathrm{og}}^0(x,y,z)\\ {\left.s_{\mathrm{hfw}}(x,y,z,0)\right|}_{t=0}=s_{\mathrm{hfw}}^0(x,y,z)\\ {\left.s_{\mathrm{mfw}}(x,y,z,0)\right|}_{t=0}=s_{\mathrm{mfw}}^0(x,y,z)\\ {\left.s_{\mathrm{cw}}(x,y,z,0)\right|}_{t=0}=s_{\mathrm{cw}}^0(x,y,z)\end{array}\right.$$

(30)

2.3.2. Boundary Conditions

The model in this study assumes a closed outer boundary, and the pressure at the outer boundary satisfies the following relationship:

$$\left\{\begin{array}{l}{\left.{\displaystyle \frac{\partial p_{\mathrm{hfg}}}{\partial n}}\right|}_{\tau }=0\\ {\left.{\displaystyle \frac{\partial p_{\mathrm{mfg}}}{\partial n}}\right|}_{\tau }=0\\ {\left.{\displaystyle \frac{\partial p_{\mathrm{cg}}}{\partial n}}\right|}_{\tau }=0\\ {\left.{\displaystyle \frac{\partial p_{\mathrm{og}}}{\partial n}}\right|}_{\tau }=0\end{array}\right.$$

(31)

The model assumes a constant bottomhole flowing pressure at the inner boundary. According to the Peaceman model [41], the production can be expressed as:

$$q_{\mathrm{vg}}={\displaystyle \frac{2\pi h{k}_{\mathrm{hf}}{k}_{\mathrm{hfrg}}}{B_{\mathrm{g}}{\mu }_{\mathrm{g}}\left(\ln r_{\mathrm{e}}/r_{\mathrm{w}}+S\right)}}\left(p_{\mathrm{hfg}i,j,k}-p_{\mathrm{wf}}\right)\delta (i,j,k)$$

(32)

$$q_{\mathrm{vw}}={\displaystyle \frac{2\pi h{k}_{\mathrm{hf}}{k}_{\mathrm{hfrw}}}{B_{\mathrm{w}}{\mu }_{\mathrm{w}}\left(\ln r_{\mathrm{e}}/r_{\mathrm{w}}+S\right)}}\left(p_{\mathrm{hfw}i,j,k}-p_{\mathrm{wf}}\right)\delta (i,j,k)$$

(33)

For anisotropic formation, the shale permeability and the equivalent radius of the grid block can be expressed as:

$$k_{\mathrm{hf}}=\sqrt{k_l{k}_m},$$

(34)

$$r_{\mathrm{e}}=0.28{\displaystyle \frac{{\left[{\left(k_l/k_m\right)}^{1/2}\Delta m^2+{\left(k_m/k_l\right)}^{1/2}\Delta l^2\right]}^{1/2}}{{\left(k_l/k_m\right)}^{1/4}+{\left(k_m/k_l\right)}^{1/4}}},$$

(35)

where the coordinate transformation related to permeability is referenced in

Table 2. Coordinate transformation table.

Coordinates	Horizontal Well		Vertical Well
	Horizontal Section is Parallel to the X-Axis	Horizontal Section is Parallel to the y-Axis
L	Y	X	X
M	Z	Z	Y
N	X	Y	Z

.

2.4. Mathematical Transformation and Production Method for the Well

2.4.1. Mathematical Transformation for the Well

If a well exists in a grid of shale formation, it is typically treated as a point source or sink in mathematical models. According to numerical simulation theory, the production from a well in formation grids can be expressed as:

$$q_l={\displaystyle \frac{2\pi h{k}_{\mathrm{hf}}{k}_{\mathrm{hfr}l}}{B_l{\mu }_l}}{\displaystyle \frac{1}{\ln r_{\mathrm{e}}/r_{\mathrm{w}}+S}}\left(p_{\mathrm{hfl}i,j,k}-p_{\mathrm{wf}}\right)$$

(36)

The productivity index of the well can be defined as:

$$PID={\displaystyle \frac{2\pi h{k}_{\mathrm{hf}}}{\ln r_{\mathrm{e}}/r_{\mathrm{w}}+S}}$$

(37)

2.4.2. Constant Bottomhole Flowing Pressure

When the well produces at a constant bottomhole flowing pressure, the gas production from the kth completed section can be expressed as:

$$Q_{\mathrm{g}k}={\left(PID\cdot {\displaystyle \frac{{\lambda }_{\mathrm{g}}}{B_{\mathrm{g}}}}\right)}_k^n{\left(p^n-p_{\mathrm{wf}}\right)}_k$$

(38)

The water production from the kth completed section is:

$$Q_{\mathrm{w}k}={\left(PID\cdot {\displaystyle \frac{{\lambda }_{\mathrm{w}}}{B_{\mathrm{w}}}}\right)}_k^n{\left(p^n-p_{\mathrm{wf}}\right)}_k$$

(39)

3. Model Validation

Capillary pressure is required in numerical simulation calculations. For the capillary pressure present in the inorganic matrix and microfractures, this study uses the model proposed by Brooks et al. [42] for calculation:

$$s_{\mathrm{ew}}={\left({\displaystyle \frac{p_{\mathrm{e}}}{p_{\mathrm{c}}}}\right)}^{\lambda }$$

(40)

The normalized water saturation can be expressed as:

$$s_{\mathrm{ew}}={\displaystyle \frac{s_{\mathrm{w}}-s_{\mathrm{wc}}}{1-s_{\mathrm{wc}}}}$$

(41)

Relative permeability curves are also needed in numerical simulation calculations. This study uses the method proposed by Li et al. [43] to calculate the relative permeability of the matrix and microfractures. The relative permeability of the water phase can be calculated by the following equation:

$$k_{\mathrm{rw}}={\left({\displaystyle \frac{s_{\mathrm{w}}-s_{\mathrm{wc}}}{1-s_{\mathrm{wc}}}}\right)}^2{\left({\displaystyle \frac{s_{\mathrm{w}}-s_{\mathrm{wc}}}{1-s_{\mathrm{wc}}}}\right)}^{{\displaystyle \frac{2+\lambda }{\lambda }}}$$

(42)

Similarly, the relative permeability of the gas phase can be calculated by:

$$k_{\mathrm{rg}}={\displaystyle \frac{s_{\mathrm{g}}}{{\left(1-s_{\mathrm{wc}}\right)}^2}}\left[{\displaystyle \frac{2{\mu }_{\mathrm{g}}\left(s_{\mathrm{w}}-s_{\mathrm{wc}}\right)+{\mu }_{\mathrm{w}}{s}_{\mathrm{g}}}{{\mu }_{\mathrm{w}}}}\right]\left[1-{\left({\displaystyle \frac{s_{\mathrm{w}}-s_{\mathrm{wc}}}{1-s_{\mathrm{wc}}}}\right)}^{{\displaystyle \frac{2+\lambda }{\lambda }}}\right]$$

(43)

The relative permeability curves for microfractures and matrices during gas-water two-phase flow in shale gas reservoirs were obtained through calculation, as depicted in Figure 5. However, due to the high conductivity of hydraulic fractures, multiphase flows do not significantly affect each other’s flow, and there is an approximate linear relationship between relative permeability and saturation [44].

The simplified flowchart of the solver simulator developed in this study is shown in Figure 6. In order to validate the accuracy of the numerical solution method for the model in this study, we compared our model with the commercial software (Eclipse 2018.1). Since the commercial software lacks a module for dividing shale gas formation into organic matrix, inorganic matrix, and fractures, we used a simplified validation method. Based on the parameter settings simulated for single-well production by Huang et al. [23], detailed parameter settings are shown in

Table 3. Parameter settings for comparison validation [23].

Parameter	Value	Parameter	Value
x-direction dimension (dimensionless)	10	Initial pressure (MPa)	28
y-direction dimension (dimensionless)	10	Formation temperature (K)	373.15
z-direction dimension (dimensionless)	3	Porosity (dimensionless)	0.1
x-direction grid step (m)	40	Gas compressibility factor (MPa−1)	1.29 × 10−2
y-direction grid step (m)	40	Water compressibility factor (MPa−1)	4.35 × 10−4
z-direction grid step (m)	10	Rock compressibility factor (MPa−1)	1.0 × 10−4
x-direction permeability (m2)	1 × 10−15	Water density (kg/m3)	1000
y-direction permeability (m2)	1 × 10−15	Gas specific gravity (dimensionless)	0.648
z-direction permeability (m2)	1 × 10−16	Water viscosity (MPa·s)	1 × 10−9
Gas saturation (dimensionless)	0.5	Wellbore radius (m)	0.08
Reservoir depth (m)	2180	Bottomhole flowing pressure (MPa)	16

. By setting the same parameters in both the simplified numerical model and the Eclipse software for simulating production, the comparison results are depicted in Figure 7. By comparing the simulated production from the model in this study with that of the Eclipse software, the average absolute deviation is 4.8%. This demonstrates that the numerical solution method of the model in this study is accurate and reliable.

The use of multistage fracturing technology has been successful in developing shale gas reservoirs, significantly increasing the production of shale gas wells. To further validate that the model developed in this study is suitable for the production process of multistage fractured horizontal wells in shale gas reservoirs, we simulated the gas production for 1000 days after fracturing. We based this simulation on field production data and related simulation parameters reported by Grieser et al. [45]. The comparison of simulation results with field data appears in Figure 8. The daily gas production calculated by the model established in this study shows a high degree of consistency with the field data. This consistency confirms the correctness and effectiveness of the model in this study.

4. Results and Discussion

In order to simulate production dynamics and perform parameter sensitivity analysis, a simulation model was constructed with dimensions of 600 m in length, 220 m in width, and 30 m in height. Since the porosity and permeability data of organic and inorganic matrices in actual shale gas reservoirs cannot be distinguished and obtained separately, the parameters for the numerical simulation in this study were based on actual shale gas formation parameters and relevant literature [46,47,48]. The detailed parameter settings are shown in

Table 4. Numerical simulation parameters.

Parameter	Value	Parameter	Value
Reservoir grid size (dimensionless)	60 × 22 × 3	Water compressibility factor (MPa−1)	4.5 × 10−4
Grid side length (m)	10	Inorganic matrix Langmuir volume (m3/kg)	2 × 10−3
Organic matrix porosity (%)	5	Inorganic matrix Langmuir pressure (MPa)	5
Organic matrix permeability (m2)	1 × 10−20	Initial water saturation in inorganic matrix	0.3
Inorganic matrix porosity (%)	4	Organic matrix Langmuir pressure (MPa)	10
Inorganic matrix permeability (m2)	1 × 10−19	Organic matrix Langmuir volume (m3/kg)	4 × 10−3
Microfracture porosity (%)	2	Organic matrix nanopore throat radius (m)	2 × 10−9
Microfracture permeability (m2)	7.5 × 10−17	Inorganic matrix nanopore throat radius (m)	1 × 10−9
Gas-phase critical pressure (MPa)	4.6	Initial water saturation in fractures (dimensionless)	0.5
Gas-phase critical temperature (K)	181	Fracture compressibility factor (MPa−1)	2 × 10−3
Slip factor (dimensionless)	−1	Hydraulic fracture porosity (%)	1.6
Initial formation pressure (MPa)	30	Number of hydraulic fractures (dimensionless)	4
Reservoir temperature (K)	362.15	Hydraulic fracture spacing (m)	100
Shale rock density (kg/m3)	2500	Hydraulic fracture length (m)	140
Rock compressibility factor (MPa−1)	1.5 × 10−3	Hydraulic fracture width (m)	3 × 10−3
Fick diffusion coefficient (m2/s)	5.86 × 10−8	Hydraulic fracture permeability (m2)	1.5 × 10−14
Wellbore radius (m)	0.06	Formation water density (kg/m3)	1100
Reservoir depth (m)	2500	Formation water viscosity (MPa·s)	1 × 10−9
Total days of simulation (day)	1000	Bottomhole flowing pressure (MPa)	16

.

4.1. Analysis of Dynamic Pressure

Reservoir pressure is a crucial parameter during the production of shale gas reservoirs, and it directly affects the output and production efficiency of shale gas wells. The model established in this study can calculate changes in the gas-phase pressure, as illustrated in Figure 9. By the 100th day of simulation, it was observed that fracture pressure decreased the fastest, followed by a slower decrease in inorganic matrix pressure, and the organic matrix pressure decreased the slowest. By the 500th day, the pressure drop had propagated to the grid boundary. The fracture pressure continued to decrease, and the inorganic matrix contributed to the energy supply, resulting in a pressure drop. The organic matrix pressure decreased at a slower rate compared to the inorganic matrix. By the 1000th day of production, the pressure drop had further propagated to the boundary. All grid fracture pressure continued to decrease, with a noticeable drop in the inorganic matrix pressure at the boundary, while the organic matrix pressure decreased slowly.

Simulation results from the 100th day to the 1000th day reveal that natural gas in the fracture system was developed first during the early stages of production. As the fracture system served as the main conduit for output, the overall production process experienced a rapid decrease in fracture pressure. As production continued, free gas in the inorganic matrix began to flow towards the fracture system, and adsorbed gas further desorbed to participate in the flow as free gas, leading to a more noticeable pressure decrease in the inorganic matrix in the middle and later stages. Additionally, a decrease in the organic matrix pressure was observed in the later stage, and it indicates that free and adsorbed gases in the organic matrix also participated in the flow. We can observe that the dynamic changes in pressure during the production process align with the actual development process of shale gas reservoirs. This further validates the reasonableness and accuracy of the model presented in this study.

4.2. Analysis of Langmuir Adsorption Constants

The simulation calculates the impact of Langmuir pressure and Langmuir volume of organic and inorganic matrices on gas production, as illustrated in Figure 10 and Figure 11. It is observed that an increase in the adsorption constant leads to higher gas production, and this confirms the contribution of adsorbed gas on the surface of organic matrix and clay minerals to the gas output.

4.3. Analysis of Hydraulic Fracturing Parameters

The effects of the number and length of hydraulic fractures on gas production are shown in Figure 12. The results indicate a positive correlation between gas production and both the number and length of hydraulic fractures. More and longer fractures enhance the reservoir’s gas production. Therefore, the number and length of hydraulic fractures greatly influence production, and optimal fracturing parameters should be designed based on the actual conditions of the shale gas reservoir.

4.4. Analysis of Adsorbed Water Layer

The adsorbed water layer on the surface of inorganic nanopores decreases the diameter of pore throats and affects permeability. The influence of the adsorbed water layer on the inorganic nanopore surface on gas production in shale gas reservoirs is illustrated in Figure 13. After 1000 days of production, the cumulative gas production per well considering the adsorbed water layer effect is 930.63 × 10⁴ m³, whereas the cumulative gas production is 1008.88 × 10⁴ m³, which excludes the adsorbed water layer. The adsorbed water layer reduces the cumulative gas production of a single well by 78.25 × 10⁴ m³, which is equivalent to a reduction rate of 8.41%. This decrease is due to the reduction in the pore throat diameter and permeability caused by the adsorbed water layer, which significantly impacts gas output.

5. Conclusions

This study develops a gas-water two-phase comprehensive flow model for shale gas reservoirs and validates the model with actual production parameters from shale gas wells. Through validation with commercial software and field production data, the mathematical model and its numerical solution method established in this paper have been proven to be accurate and reliable. Simulating production for 1000 days leads to the following conclusions:

(1) Analysis of dynamic pressure changes reveals that as production progresses, the fastest decrease in reservoir pressure occurs in the fractures, followed by the inorganic matrix, and then the organic matrix. Early in production, gas output is primarily from the fractures. As production continues, the pressure drop gradually spreads to the boundary, and gas from the inorganic and organic matrices contributes to the flow.

(2) Results of the model’s influential factor analysis indicate that increasing the Langmuir pressure and Langmuir volume for organic and inorganic matrices assists in enhancing gas well production. This is because higher Langmuir constants enhance the storage capacity of the shale matrix, thereby increasing gas output during production. As the number and length of hydraulic fractures increase, the gas well production increases.

(3) The adsorbed water layer decreases the diameter of pore throats in the inorganic matrix, thereby reducing its permeability and leading to a reduction in the gas production of a single well by 78.25 × 10⁴ m³. Although this work considers the impact of the adsorbed water layer on permeability, future work should further analyze the effect of shale hydration on production.