Flow–Solid Coupled Analysis of Shale Gas Production Influenced by Fracture Roughness Evolution in Supercritical CO₂–Slickwater Systems

Abstract

With the increasing global demand for energy, the development of unconventional resources has become a focal point of research. Among these, shale gas has drawn considerable attention due to its abundant reserves. However, its low permeability and complex fracture networks present substantial challenges. This study investigates the composite fracturing technology combining supercritical CO₂ and slickwater for shale gas extraction, elucidating the mechanisms by which it influences shale fracture roughness and conductivity through an integrated approach of theory, experiments, and numerical modeling. Experimental results demonstrate that the surface roughness of shale fractures increases markedly after supercritical CO₂–slickwater treatment. Moreover, the dynamic evolution of permeability and porosity is governed by roughness strain, adsorption expansion, and corrosion compression strain. Based on fluid–solid coupling theory, a mathematical model was developed and validated via numerical simulations. Sensitivity analysis reveals that fracture density and permeability have a pronounced impact on shale gas field productivity, whereas fracture dip angle exerts a comparatively minor effect. The findings provide a theoretical basis for optimizing composite fracturing technology, thereby enhancing shale gas extraction efficiency and promoting effective resource utilization.

1. Introduction

With the increasing global demand for energy, the development of unconventional resources has become a critical research focus in the energy sector [1]. Among these resources, shale gas reservoirs have garnered significant attention due to their abundant reserves and wide distribution [2,3]. However, the low permeability and complex fracture networks of shale gas reservoirs pose significant challenges to their development. Effectively improving the recovery rate of shale gas has become an urgent problem for researchers.

In recent years, the energy industry has been encouraged to recycle CO₂, particularly by applying supercritical CO₂ to shale gas development, to achieve dual benefits of resource utilization and environmental protection [4,5,6]. Under the high-temperature and high-pressure conditions of deep shale gas reservoirs, CO₂ readily reaches the supercritical state [7], exhibiting characteristics such as high density, high permeability, and low fracture initiation pressure [8,9,10]. These properties facilitate the formation of more complex fracture networks within the reservoir. Li et al. proposed a composite fracturing technology based on supercritical CO₂ and slickwater [11]. Field experiments conducted by Yanchang Petroleum in 2017 demonstrated that this composite fracturing technology significantly outperforms conventional large-scale hydraulic fracturing, achieving a reformed reservoir volume that is larger and more complex. This approach can increase production capacity by 3–5 times. The mechanism of supercritical CO₂–slickwater–shale interaction is one of the key scientific issues underlying this composite fracturing technology. Supercritical CO₂ and slickwater induce changes in the geometry of shale fractures and the shale matrix, which, in turn, affect the hydrodynamic behavior and pressure distribution within the fractures.

By exposing shale specimens to supercritical CO₂, Ozotta et al. [12] found that supercritical CO₂-induced changes can weaken shale by reducing its modulus of elasticity and increasing its Poisson’s ratio, ultimately leading to a loss of the shale’s mechanical properties. Lu et al. [13], using magnetic resonance imaging (MRI), unconfined compressive strength (UCS), and acoustic emission (AE) experiments on shale treated with supercritical CO₂, demonstrated reductions in the uniaxial compressive strength and Young’s modulus of the shale. Zhang et al. [14] showed that increasing the clay mineral content to a certain level significantly mitigates the adverse effects of hydration on the rock. Wang et al. [15] found that the action of slickwater leads to the softening of shale fractures and a significant reduction in the effective conductivity of the fractures, ultimately decreasing gas production. These studies separately investigated the effects of supercritical CO₂ and slickwater on shale, revealing how supercritical CO₂ significantly alters the mineral composition, pore structure, fracture characteristics, and mechanical behavior of shale, such as reducing the content of carbonate minerals and altering compressive strength and elastic modulus. Similarly, the hydration effect of slickwater significantly influences the mechanical and flow properties of shale. However, there is a lack of research simultaneously considering the interaction mechanisms between supercritical CO₂, slickwater, and shale.

In terms of fracture inflow, Liu et al. [16] found that in a supercritical CO₂ environment, the equivalent hydraulic aperture of vertical shear fracture is higher than that of parallel shear fracture at low effective stress; at this time, the vertical shear has stronger inflow capacity, and with the increase of effective stress, the parallel shear fracture has stronger inflow capacity. Wang et al. [17] found that supercritical CO₂ can cause significant changes in fracture aperture compared with primary core by injecting supercritical CO₂ into Niobrara shale core for fracturing and found that more natural fracture channels were formed by CT scanning experiments. It was found that supercritical CO₂ could cause significant changes in fracture apertures, and through CT scanning experiments, it was found that supercritical CO₂ opened more natural fracture channels, forming a fracture network with complexity compared to that of the primary core. In order to reveal the fracture inflow behavior even further, Figueiredo et al. [18], by numerically investigating the fractured rock body below 1000 m, found that the changes in the local stress and stress ratio distribution are anisotropic due to different fluid pore pressure levels, and that the equivalent permeability of the fractured rock body is the most sensitive to the normal stiffness of the fracture, the permeability in the tensile damage zone, and the power-law exponent of the change in permeability. Based on the Nierode-Kruk model, Lu et al. [19] considered factors such as closure stress, rock mechanical properties, and fracture surface roughness, and found that the morphology and roughness of the fractures were crucial for calculating the fracture conductivity of shale. The above studies mainly focus on the influencing factors of shale fracture inflow capacity, and studies on the mechanism of supercritical CO₂ and slickwater interaction with shale fracture are still limited; in particular, research on the coupling effect of flow and solid under the complex reservoir conditions needs to be further deepened.

This study employs a combination of “theory + experiment + numerical modeling” to conduct XRD, three-dimensional morphology scanning, and gas conduction experiments. By analyzing the geometric characteristics of shale fractures and the changing hydraulic apertures of fractures before and after the action of supercritical CO₂ and slickwater, a shale fracture roughness deformation model was established. Using numerical simulations, the multi-physics field was solved to investigate the changes in fracture inflow capacity and their impact on the yield of shale gas reservoirs under different fracture roughness conditions. This comprehensive approach reveals the effects of supercritical CO₂ and slickwater on the inflow characteristics of shale fractures, providing a theoretical basis for understanding the mechanisms behind composite fracturing technology.

2. Effect of Supercritical CO₂ and Slickwater on Shale Fracture Aperture

2.1. Experimental Equipment and Methods

The shale samples utilized in this study were collected from the Lower Silurian Longmaxi Formation in the southeastern Sichuan Basin, China. The sides and ends of the test specimens were wrapped with tape, leaving only the fracture surfaces exposed to ensure full immersion. The test samples were first immersed in slickwater at a constant temperature of 40 °C for 5 days. Subsequently, the supercritical CO₂ injection pressure was set at 15 MPa, with the water bath temperature maintained at 60 °C, and the immersion time was also 5 days. Due to the presence of high and low roughness bumps on the shale fracture surfaces, direct analysis using a fixed method was not feasible. Therefore, the fracture surfaces of each specimen were scanned for 3D morphology [20], and the joint roughness coefficient (JRC) was used to quantitatively characterize the surface roughness of the fractures (

Table 1. 3D morphology scanning results.

Sample	Treatment Condition	A1	A2	ΔA	Change Rate	ΔH	Change Rate
S	Before immersing	3662.53	3821.96	1.044	26.59%	5.102	21.54%
	After immersing		4838.16	1.321		6.201

). A₁ is the projected area in mm², A₂ is the actual area in mm², ΔA is the area ratio, ΔH is the difference in height of the projection in mm, as follows:

$$e_h={\displaystyle \frac{e_m^2}{JR{C}^{2.5}}}$$

(1)

where e_m and e_h are the fracture mechanical tension and fracture equivalent hydraulic aperture in m, respectively.

Subsequently, the gas inflow experiment was conducted using a fracture inflow capacity testing device. This device can apply a maximum axial pressure of 2000 kN, a maximum peripheral pressure of 50 MPa, and a flow rate ranging from 0 to 20 mL/min. To ensure the accuracy of the experimental data, nitrogen was selected as the flow-conducting medium because it does not react with shale and is not adsorbed onto the pore space surface. During the experiment, the gas pressure was gradually increased from 0 to 0.2 MPa, and the total effective stress values for the experiment were set as follows: 1 MPa, 2 MPa, 4 MPa, 6 MPa, 8 MPa, 12 MPa, 16 MPa, 20 MPa, and 25 MPa.

2.2. Shale Fracture Aperture Variation

It has been confirmed that there is a significant correlation between the fracture hydraulic aperture and the roughness. To further illustrate the effect of the nodal roughness coefficient (JRC) [21] on the equivalent hydraulic apertures, Equation (2) and the experimental results obtained in this study were used for fitting (

Table 2. Fitting results.

Treatment Condition	σ	eh	JRC/eh	Treatment Condition	σ	eh	JRC/eh
Before treatment	1	56.917	0.24038	After treatment	1	76.495	0.36171
	2	46.76239	0.29258		2	61.44677	0.4503
	4	36.60779	0.37374		4	46.39855	0.59634
	6	30.66772	0.44613		6	37.5959	0.73596
	8	26.45318	0.5172		8	31.35032	0.88258
	10	23.18413	0.59013		10	26.50588	1.04389
	15	17.24406	0.79341		15	17.70323	1.56295
	20	13.02952	1.05005		20	11.45765	2.41491
	25	9.76047	1.40174		25	6.61321	4.18393

). As can be observed from Table 2, the ratio of the nodal roughness coefficient (JRC) to the equivalent hydraulic aperture increases progressively with rising effective stress in shale fractures treated with supercritical CO₂ and slickwater, as follows:

$$e_h=e_i\times JRC-{\displaystyle \frac{c_i{\sigma }_n}{d_i+{\sigma }_n}}+f_i$$

(2)

where c_i, d_i, e_i, and f_i are the fitting parameters.

3. Numerical Simulation Methods

3.1. Theoretical Model

3.1.1. Model Hypothesis

In this work, the modeling assumptions are as follows: (1) The gases contained in the shale pores are ideal, with constant viscosity, and the shale matrix is saturated with shale gas [22]. The adsorption of gases by the shale follows the Langmuir equation, and both shale gas and CO₂ are considered ideal gases. (2) Strain occurring in shale is non-negligible, and the shale is modeled as a transversely isotropic elastoplastic porous medium [23]. (3) Shale reservoirs are fractured reservoirs consisting of a matrix–system.

3.1.2. Non-Linear Medium Models

According to the nonlinear medium flow theory [24], the gas percolation equation in the matrix is as follows:

$${\displaystyle \frac{\partial }{\partial t}}\left({\rho }_g\phi +\left(1-\phi \right){\rho }_s{\rho }_{gsc}{\displaystyle \frac{p{V}_L}{p+P_L}}\right)+\nabla \cdot \left(-{\rho }_g{\displaystyle \frac{2k_0}{{\mu }_g\cdot \left(1+\sqrt{1+4{\rho }_g\beta {\left(k_0/{\mu }_g\right)}^2\left|\nabla p\right|}\right)}}\left(1-{\displaystyle \frac{c_1}{\nabla p_q-c_2}}\right)\nabla p\right)={\rho }_g{q}_w$$

(3)

The gas seepage equations in microfractures and in hydraulic fractures are as follows:

$${\displaystyle \frac{\partial }{\partial t}}\left({\rho }_g\phi \right)+\nabla \cdot \left(-{\rho }_g{\displaystyle \frac{2k_0}{{\mu }_g\cdot \left(1+\sqrt{1+4{\rho }_g\beta {\left(k_0/{\mu }_g\right)}^2\left|\nabla p\right|}\right)}}\left(1-{\displaystyle \frac{c_1}{\nabla p_q-c_2}}\right)\nabla p\right)={\rho }_g{q}_w$$

(4)

$$d_f{\displaystyle \frac{\partial \left({\rho }_g{\phi }_f\right)}{\partial t}}-\nabla \cdot \left(d_f{\rho }_g{\displaystyle \frac{2k_{f0}}{{\mu }_g\cdot \left(1+\sqrt{1+4{\rho }_g\beta {\left(k_0/{\mu }_g\right)}^2\left|\nabla p\right|}\right)}}\left(1-{\displaystyle \frac{c_1}{\nabla p_q-c_2}}\right)p\right)=d_f{\rho }_g{q}_{wf}$$

(5)

where ρ_g is the gas density in kg/m³; φ is the porosity; ρ_s is the rock density in kg/m³; ρ_gsc is the mass density of natural gas under standard conditions in kg/m³; p is the pore pressure in Pa; V_L is the Langmuir volume in m³/kg; P_L is the Langmuir pressure in Pa; k₀ is the initial permeability in m²; μ_g is the gas viscosity in Pa·s; β is the Darcy–Forchheimer coefficient; c₁ and c₂ are the nonlinear fitting parameters; ∇p_q is the initiating pressure gradient in MPa/m; q_w is the source or sink of gas in kg/(m³·s); q_wf is the source or sink of fracture gas in kg/(m³·s); d_f is the fracture aperture in m; and φ_f is the fracture porosity.

3.1.3. Shale Fracture Roughness Model

Based on the equivalent hydraulic apertures model and the porosity definition [25], p/k_s and JRC were fitted with e_h to establish a statistical relationship. Through analysis of experimental data, we derived the supercritical CO₂–slickwater-induced deformation model for shale fractures with different roughness under varying JRC conditions as follows:

$${\epsilon }_{JRC}=-0.017286\times \ln \left({\displaystyle \frac{p}{k_s}}\right)-0.13932+0.0017882\cdot JRC$$

(6)

where ε_JRC is the roughness strain and k_s is the shale skeleton modulus in MPa. Roughness strain is a physical quantity derived from the fitting of experimental data.

The roughness deformation model can be written as follows:

$${\epsilon }_{JRC}=a_J\cdot \ln \left({\displaystyle \frac{p}{k_s}}\right)+b_J+c_J\cdot JRC$$

(7)

where a_J = −0.017286, b_J = −0.13932, c_J = 0.0017882, is the roughness coefficient.

Combined with the definition of relative roughness, the supercritical CO₂–slickwater modified deformation model for different shale fracture roughness under different relative roughness is obtained according to the geometric equation:

$$\Delta JRC=e^{\left(-4.4133+0.38664\cdot p\right)}$$

(8)

Combined with Equation (2), the roughness deformation model can be written as:

$${\epsilon }_{JRC}=a_J\cdot \ln \left({\displaystyle \frac{p}{k_s}}\right)+b_J+c_J\cdot JRC\cdot e^{\left(d_J+e_J\cdot {\displaystyle \frac{p}{k_s}}\right)}$$

(9)

where d_J = −4.4133 and e_J = 0.38664 is the corrected roughness coefficient after the effect of supercritical CO₂–slickwater.

3.1.4. Controlling Equations for Shale Matrix Deformation

Porosity and permeability are critical physical parameters for describing shale deformation and shale gas flow [26]. They are dynamically related to shale deformation and shale gas pressure [27]. When shale is subjected to disturbances, it undergoes deformation and damage, leading to changes in porosity and permeability. In engineering practice, the primary deformations in shale result from pressure-induced stress, adsorptive expansion stress [27], and corrosion pressure [28]. However, in the supercritical CO₂–slickwater system, shale roughness deformation varies significantly under different pressures [21]. The nonlinear flow characteristics within fractures are essential for accurately understanding and simulating fluid behavior in this complex multiphase system. These characteristics directly influence the fluid pressure distribution and flow paths, which, in turn, affect effective stresses in the reservoir and the dynamics of the fractures. Therefore, roughness-induced deformations must be considered. The deformation process is assumed to occur under isothermal conditions. Accordingly, the change in the amount of shale solid particles is expressed as [29]:

$$n={\displaystyle \frac{V_p}{V_b}}={\displaystyle \frac{V_{p0}+\Delta V_p}{V_{b0}+\Delta V_b}}=1-{\displaystyle \frac{V_{s0}+\Delta V_s}{V_{b0}+\Delta V_b}}=1-{\displaystyle \frac{1-n_0}{1+{\epsilon }_V}}\left(1+{\displaystyle \frac{\Delta V_s}{V_{s0}}}\right)$$

(10)

where V_p is the shale pore volume, V_b is the total volume of shale, V_p0 is the initial pore volume of shale, V_b0 is the initial total volume of shale, n₀ is the initial porosity, V is the shale volumetric strain, and $V_s$ is the volume of a solid skeleton in a porous medium.

Pressure-induced shale deformation is as follows:

$${\epsilon }_p={\displaystyle \frac{\Delta p}{k_s}}$$

(11)

where ε_p is the strain caused by CO₂ pressure in %, Δp is the CO₂ change value in MPa, and k_s is the solid skeleton modulus of shale in MPa.

Shale deformation due to adsorption expansion is [30] as follows:

$${\epsilon }_s={\displaystyle \frac{2a_{s}RT}{3V_m{k}_s}}\ln \left(1+b_{s}p\right)$$

(12)

where ε_s is the adsorption expansion strain in %, a_s is the limit adsorption capacity in m³/t, b_s is the adsorption constant in MPa⁻¹, R is the gas molar constant, T is the shale temperature in K.

Corrosion-induced shale deformation is [28] as follows:

$${\epsilon }_r=a_r\ln \left({\displaystyle \frac{p}{k_s}}\right)+b_r$$

(13)

where ε_r is the corrosion compression strain in %, a_r and b_r are the corrosion constants, p is pore pressure in MPa.

Roughness-induced shale deformation is as follows:

$${\epsilon }_{JRC}=a_J\cdot \ln \left({\displaystyle \frac{p}{k_s}}\right)+b_J+c_J\cdot JRC\cdot e^{\left(d_J+e_J\cdot {\displaystyle \frac{p}{k_s}}\right)}$$

(14)

Combined with Equations (9)–(14), the porosity expression is as follows:

$$n=1-{\displaystyle \frac{1-n_0}{1+{\epsilon }_V}}\left(1-{\epsilon }_p+{\epsilon }_{sw}-{\epsilon }_r-{\epsilon }_{JRC}\right)$$

(15)

The Kozeny–Carman equation combined with the porosity expression leads to the shale solid permeability expression of [31,32] as follows:

$$k={\displaystyle \frac{k_0}{1+{\epsilon }_V}}{\left[1+{\displaystyle \frac{{\epsilon }_V+\left({\epsilon }_p-{\epsilon }_s+{\epsilon }_r+{\epsilon }_{JRC}\right)\left(1-n_0\right)}{n_0}}\right]}^3$$

(16)

where k₀ is the initial permeability of shale containing supercritical CO₂ and slickwater.

Assuming that shale behaves as an elastomer with uniform mechanical properties [33], its pore pressure is equal in all directions. Consequently, the shale experiences uniform stress in all directions. Furthermore, its relationship with force follows Hooke’s law [34]. Based on these assumptions, the adsorption expansion stress [35], corrosion compression stress, and roughness deformation stress of shale can be calculated using the following formula:

$${\sigma }_s=E\cdot {\epsilon }_s={\displaystyle \frac{2E{a}_{s}RT}{3V_m{k}_s}}\ln \left(1+b_{s}p\right)$$

(17)

$${\sigma }_r=E\cdot {\epsilon }_r=E\left[a_r\ln \left({\displaystyle \frac{p}{k_s}}\right)+b_r\right]$$

(18)

$${\sigma }_{JRC}=E\cdot {\epsilon }_{JRC}=E\left[a_J\ln \left({\displaystyle \frac{p}{k_s}}\right)+b_J+c_J\cdot JRC\cdot e^{\left(d_J+e_J\cdot {\displaystyle \frac{p}{k_s}}\right)}\right]$$

(19)

where E is the elastic modulus of shale in MPa.

According to the effective stress principle, and considering the stresses induced by adsorption, corrosion, and roughness deformation, the effective stress equation for supercritical CO₂–slickwater-bearing shale can be expressed as follows [32]:

$${\sigma }_{ij}^{\prime }={\sigma }_{ij}-\alpha p{\delta }_{ij}-{\sigma }_s{\delta }_{ij}-{\sigma }_r{\delta }_{ij}-{\sigma }_{JRC}{\delta }_{ij}$$

(20)

where ${\sigma }_{ij}^{\prime }$ is the effective stress of shale containing supercritical CO₂ and slickwater in MPa; ${\sigma }_{ij}$ is the overall stress of shale containing supercritical CO₂ and slickwater in MPa; and α is the Biot coefficient.

Assuming that the supercritical CO₂–slickwater-bearing shale is an isotropic linear elastic medium, the variation of the stress field is given by the following equation:

$${\sigma }_{ij,j}+F_i=0$$

(21)

where F_i is the bulk stress in N/m³.

Substituting Equations (16)–(20) into Equation (21) yields:

$${\sigma }_{ij,j}^{\prime }+{\left(\alpha p{\delta }_{ij}\right)}_{,j}+{\sigma }_s{\sigma }_{ij,j}+{\sigma }_r{\sigma }_{ij,j}+{\sigma }_{JRC}{\sigma }_{ij,j}+F_i=0$$

(22)

Geometric equation as follows:

$${\epsilon }_{ij}={\displaystyle \frac{1}{2}}\left(u_{i,j}+u_{j,i}\right)$$

(23)

Stress–strain relationship as follows:

According to the basic assumptions of the model, shale follows the generalized Hooke’s law during elastic deformation as follows:

$${\sigma }_{ij}^{\prime }=\lambda {\delta }_{ij}{\epsilon }_V+2G{\epsilon }_{ij}$$

(24)

where λ is the Lame constant, G is the modulus of shear, and δ_ij is the Kroeker number.

The main factors contributing to shale deformation include geopathic stress, supercritical CO₂–slickwater system pressure, adsorption expansion stress, corrosion compression stress, and roughness deformation stress. Accordingly, shale strain can be expressed as follows:

$$\begin{array}{l}{\displaystyle \frac{G}{1-2\nu }}{u}_{j,ji}+G{u}_{i,jj}+\left(\alpha +{\displaystyle \frac{2G}{3k_s}}\right)p_{,i}+\left\{\left[{\displaystyle \frac{2E}{3}}-{\displaystyle \frac{4G}{9}}\right]{\displaystyle \frac{a_s{b}_s{\rho }_{s}RT}{V_m{k}_s\left(1+b_{s}p\right)}}\right\}{p}_{,i}\\ +\left\{\left[E+{\displaystyle \frac{2G}{3}}\right]\left[{\displaystyle \frac{a_r+a_J}{p}}+c_J\cdot JRC\cdot e_J\cdot e^{e_J\cdot {\displaystyle \frac{p}{k_s}}}\right]\right\}{p}_{,i}+F_i=0\end{array}$$

(25)

where ν is the Poisson’s ratio.

3.1.5. Coupled Solution Methods

This work proposes and implements a coupled solver approach that combines the advanced capabilities of COMSOL Multiphysics 6.1 for stress field modeling with the efficiency of the Matlab Reservoir Simulation Toolbox (MRST v1.2; SINTEF Digital, Trondheim, Norway) for seepage field analysis (Figure 1). The primary foundational models include the embedded discrete fracture model (EDFM), the nonlinear dual-medium model, and the roughness deformation model. The core methodology involves coupling the stress-field-enabled EDFM grid constructed in MRST with the flow field in COMSOL, followed by validation against field data and the flow field in MRST. If the results align, a C-R model is successfully established; otherwise, model parameters are iteratively adjusted.

3.2. Geometrical Modelling and Boundary Conditions

The model dimensions are 1200 × 300 × 90 m, containing a total of 28 hydraulic fractures. The boundary conditions for the supercritical CO₂–slickwater–shale reservoir are defined as follows: (1) uniaxial strain, (2) constant overburden stress at the upper boundary, (3) wellbore pressure applied at the wellbore boundary, and (4) no-flow conditions at the other boundaries. The basic reservoir parameters are listed in

Table 3. Basic parameters of reservoir.

Parameter	Value	Units	Parameter	Value	Units
Reservoir pressure	20.34	MPa	Fracture compressibility	1.0 × 10−8	1/Pa
Temperature	352	K	Matrix permeability	2 × 10−19	m2
Rock density	2500	kg/m3	Fracture permeability	2 × 10−13	m2
Langmuir pressure	4.47	MPa	Fracture width	0.003	m
Langmuir volume	0.00272	m3/kg	Fracture spacing	30.5	m
Matrix porosity	0.03		Fracture half-length	47.2	m
Matrix compressibility	1.5 × 10−10	1/Pa	Well BHP	3.69	MPa
Gas Viscosity	0.02	mPa·s	Gas density	0.6571	kg/m3

, and four characteristic monitoring points were selected, as summarized in

Table 4. Feature location.

Name	Position (m, m)	Name	Position (m, m)
Point A	173.08, 149.53	Point C	203.58, 243.93
Point B	203.58, 149.53	Point D	173.08, 243.93

. To ensure numerical robustness and reproducibility, we expanded the description of the C–R algorithm in the revised manuscript. The coupled solver proceeds iteratively by first updating the stress field using COMSOL, followed by updating the flow field using MRST, and finally performing a coupling correction step. This sequence is repeated until convergence is reached at each time step. Convergence is achieved when the relative residuals of both pressure and displacement fall below 10⁻⁵; in practice, 5–10 iterations per time step were typically sufficient. In addition, a grid sensitivity analysis was performed with coarse (Δx = 10 m), medium (Δx = 5 m), and fine (Δx = 2 m) meshes. The resulting difference in cumulative production was within 3%, confirming that the simulation results are stable and not significantly affected by grid resolution. These details collectively strengthen the transparency and reliability of the numerical methodology.

3.3. Validation of the Model

The analyses in Figure 2 demonstrate that the modeled production captures the overall trend of the measured field data obtained from a shale gas well in the Lower Silurian Longmaxi Formation (Sichuan Basin, China). For the first 200 days, the simulated results agree closely with the observed production, while during 200–1600 days the model slightly overestimates production. Nevertheless, the overall trend remains consistent with field measurements. To quantitatively validate the model, the root mean square errors (RMSE) were calculated, yielding values of 80.5%, which confirm a high level of agreement. The slight overprediction in the mid-to-late stage may be attributed to the presence of water in the shale matrix, which acts as a temporary barrier to gas flow. Flowback water suppresses production during the initial stage, leading to lower measured values compared to the simulation. As production continues and the residual water is gradually removed, shale gas dominates the reservoir, resulting in slightly higher simulated values during later periods. Moreover, the pressure drop behavior at monitoring points A–D provides additional evidence of model validity. Point B exhibits the fastest pressure decline, followed by points A, C, and D. This reflects the strong conductivity of the fracture network: point B lies between two hydraulic fractures and is influenced by both, while point A is affected by only one. In the early production stage (first 50 days), the pressure at point B decreased only slightly, indicating that the stimulated reservoir volume (SRV) region between two fractures was the most effective in sustaining fluid mobility. In contrast, point A, with one side bounded by undisturbed matrix, showed a different depletion trend. The total volumetric strain (εV), incorporating adsorption, pressure-induced deformation, corrosion, and roughness deformation, further explains variations in the rate of pressure decline.

4. Results

4.1. Reservoir Pressure Evolution

During the early stages of extraction, the high permeability of the hydraulically fractured region, driven by differential pressure, creates a significant pressure gradient in the adjacent simulated region. As fluid within the hydraulic fracture continues to discharge, the pressure in this region decreases until it reaches equilibrium with the bottomhole pressure. This prompts the surrounding fluid to gradually migrate toward the fracture region, triggering a pressure reduction in the areas adjacent to the hydraulic fracture. As the extraction process progresses, the pressure across the entire reservoir exhibits a decreasing trend. Fluids in the shale matrix system, located at the far end of the hydraulic fracture, begin to flow toward the fracture area. With the continued drop in reservoir pressure, gas originally adsorbed on the matrix surface desorbs and migrates into pores and microfractures. Additionally, the fracture and its controlled stimulated reservoir volume (SRV) region show a faster rate of pressure reduction, with pressures in this region significantly lower than those in the non-SRV region. This phenomenon likely results from the higher permeability of the microfractures compared to the matrix, causing the reduction in gas pressure within the matrix to lag behind the microfracture system. When the pressure in the area around the hydraulic fracture decreases significantly, the pressure in the SRV region under its control also drops rapidly (Figure 3).

4.2. Multiple Strain Evolution

The simulation results indicate that all types of strain decrease over time during the gas production process. However, the adsorbed strain at characteristic point B is lower than that at characteristic point A due to the influence of the hydraulic fracture in the stimulated reservoir volume (SRV) region. This suggests that locations farther from the fracture region and undisturbed by fractures are more influenced by adsorbed gases in the primary pore matrix, which impacts extraction efficiency. Additionally, as shale gas continues to be extracted, the effective stress in the shale matrix increases, leading to further compaction of the natural pores and fractures within the matrix (Figure 4).

4.3. Permeability Evolution

Figure 5 shows that the permeability decreases faster from 0 to 500 days, which is mainly due to the high permeability of the fracture region, resulting in a large amount of fluid being rapidly extracted, while the matrix around the hydraulic fracture desorbs gas in the microfracture and seeps into the fracture, leading to a decrease in gas pressure, which in turn leads to a pressure strain, and is accompanied by the dual effects of the corrosive effect of supercritical CO₂ on the shale and the alteration of the fracture roughness after its action, which further weakened the permeability. From 500 to 1500 days, the decreasing trend of permeability slowed down, and at 3000 days, the final permeability was 95.5% of the initial permeability, although the degree of influence of permeability was small, this weak change had a significant effect on the yield. When the geomechanical effect is not considered, the final cumulative production decreases from 7.29 × 10⁷ m³ to 7.06 × 10⁷ m³ and 6.53 × 10⁷ m³. The difference in yield becomes more and more obvious at different stages of production, especially from the early to the middle and late stages. This is mainly due to the fact that in the early stage of production, gas production is mainly affected by the high conductivity of hydraulic fractures, and as gas production enters the middle and late stages, the migration of gas from the matrix and microfractures to the fractures is intensified, and the complementary effect of gas in the matrix on the production begins to be significant, the supercritical CO₂–slickwater modifies the roughness, and the influence of geomechanical factors is also enhanced.

Figure 5 shows that permeability decreases rapidly during the first 0 to 500 days. This is primarily due to the high permeability of the fracture region, which results in a large amount of fluid being rapidly extracted. Simultaneously, the matrix around the hydraulic fracture desorbs gas from microfractures, which seeps into the fracture, leading to a decrease in gas pressure. This drop in pressure induces pressure strain and is accompanied by the dual effects of supercritical CO₂ corrosion on the shale and alterations in fracture roughness, further weakening permeability.

From 500 to 1500 days, the rate of permeability decline slows, and by 3000 days, the final permeability stabilizes at 95.5% of the initial permeability. Although the impact on permeability is relatively minor at this stage, even this slight change significantly affects yield. When geomechanical effects are not considered, the final cumulative production decreases from 7.29 × 10⁷ m³ to 7.06 × 10⁷ m³ and 6.53 × 10⁷ m³. The difference in yield becomes increasingly pronounced at different production stages, particularly from the early to the middle and late stages. In the early stages, gas production is primarily driven by the high conductivity of hydraulic fractures. However, as production progresses to the middle and late stages, the migration of gas from the matrix and microfractures to the fractures intensifies. At this point, the complementary effect of gas from the matrix on production becomes significant. Additionally, supercritical CO₂–slickwater modifies the roughness, and the influence of geomechanical factors is enhanced.

Based on the shown changes in L₂ intercepts, it can be observed that within the SRV zone, which is more influenced by hydraulic fracturing, the permeability shows an overall decreasing trend over time, but in different spatial domains, the permeability shows an evolutionary trend of growth. This phenomenon is attributed to the fact that as the pore pressure decreases over a given period of time, the areas away from the SRV are more influenced by the low porosity and low permeability matrix, and thus the pressure gradient is smaller, resulting in a slower permeability decline that diminishes over time. In contrast, the permeability evolution on the L₁ cut-off shows an increasing trend in different spatial domains, but the permeability evolution in the temporal domain does not consistently show a decrease. As seen in the figure, the order of permeability evolution with time is 100 d, 500 d, 1000 d, 1500 d, 2000 d, 2500 d, and 3000 d. This suggests that at 1500 days, the effect of decreasing permeability begins to slow down. It is inferred that at this time, the area where the L₁ cutoff is located has gradually changed from a low-pressure area to an area close to the bottomhole pressure, while microfractures in the distant matrix continue to deliver gas to the area to replenish the pressure, and thus the permeability gradually increases at this time period with the advancement of time (Figure 6).

4.4. Roughness Effects

Figure 7 highlights a significant difference in the final cumulative reservoir production between scenarios that account for the influence of the JRC and those that do not. Both scenarios indicate that, in the early stages of extraction, fracture roughness is modified by the effects of supercritical CO₂–slickwater, which creates additional seepage channels and enhances production. Under conditions of high fracture aperture (w_f = 0.002 m), a larger volume of supercritical CO₂–slickwater can enter the fracture channels, reducing seepage obstructions and facilitating greater extraction of shale gas from the formation. Conversely, under conditions of low fracture aperture (w_f = 0.0005 m), a significant portion of the supercritical CO₂–slickwater becomes trapped in the formation without entering the fractures. This retention further blocks the already low-permeability channels, impeding fluid flow and negatively impacting production.

5. Discussion

In the actual process of shale gas reservoir extraction and production, the diversity of reservoir conditions results in significant variations in the roughness of hydraulic fractures formed after fracturing. These variations are influenced by both geomechanics and the fracturing process. In real reservoirs, the fracture network comprises numerous natural and hydraulic fractures, which directly affect the seepage characteristics of the reservoir. Specifically, the total number of natural fractures per unit area in the shale gas reservoir is used to characterize fracture density, serving as the basis for simulation. To enhance computational efficiency, a model size reduced to 1/10 of the original was selected for the simulation.

5.1. Effect of Fracture Characteristics on Yield

5.1.1. Fracture Permeability and Fracture Density

In this study, two key parameters in shale gas reservoirs were investigated: natural fracture permeability and density. The permeability was set to vary between 1 and 100 × 10⁻¹⁵ m², while the density ranged from 0 to 0.24 m⁻². According to Figure 8, the following observations can be made: An increase in fracture density leads to higher cumulative production, indicating that fracture density is directly related to the number of gas release channels, effectively improving the development efficiency of the gas reservoir [36]. The permeability of natural fractures plays a critical role in modulating the impact of fracture density on production. Under low-permeability conditions, an increase in fracture density has only a limited positive effect on production. Conversely, under high-permeability conditions, an increase in natural fracture density significantly enhances production. These findings suggest that high-permeability fractures effectively promote gas flow, where even small increases in density can result in a substantial rise in yield.

5.1.2. Fracture Density and Fracture Inclination

The model conditions were adjusted to calculate the effect of different combinations of fracture density and tilt angle on the cumulative production of a gas reservoir with a natural fracture permeability of 1 × 10⁻¹⁵ m². The fracture density gradient was set at 0.04 m⁻², ranging from 0 to 0.24 m⁻² across six groups. Similarly, the tilt angle gradient was set at 15°, ranging from 0° to 90° across seven groups. According to Figure 9, the cumulative production of the reservoir decreased as the tilt angle increased. Conversely, as fracture density increased, the cumulative production of the shale gas reservoir also increased, further confirming the positive correlation between fracture density and production. The 3D surface map highlights the greater sensitivity of production to changes in tilt angle, indicating that tilt angle has a more pronounced effect on cumulative production. However, as fracture density increased, the impact of tilt angle on production gradually diminished, suggesting that tilt angle has a relatively minor influence in high-density fracture scenarios.

5.1.3. Fracture Permeability and Fracture Inclination Angle

The model conditions were changed to use the natural fracture density of 0.08 mm⁻² to calculate the effects of different combinations of tilt angles on the cumulative production of gas reservoirs, in which the tilt angle was 15° as the gradient. According to Figure 10, it can be observed that: the rate of change in the direction of natural fracture permeability was significantly lower than that in the direction of tilt angle, indicating that the tilt angle is more sensitive to the effect of shale gas reservoir production; the cumulative production of shale gas reservoirs showed a decreasing trend with the increase of the tilt angle. As a larger tilt angle may reduce the efficiency of hydraulic fracturing, it affects the effective release and flow of gas; the increase of tilt angle makes the docking of hydraulic fracture and natural fracture more difficult, which affects the flow path of gas; under the condition of high permeability fracture, the adverse effect of tilt angle is more significant; at the same time, when the tilt angle increases, the positive effect of permeability is weakened.

5.1.4. Analysis of the Weighting of Factors

To quantitatively evaluate the impact of factors on shale yield, their contributions can be assessed using analysis of variance (ANOVA). The variance contribution rate and p-value are important indicators for evaluating the influence of factors. The variance contribution rate reflects the relative explanatory power of each factor on the variance of the experimental results; a higher value indicates a more significant effect of the factor on the results. The p-value measures the probability of observing the current results under the null hypothesis (i.e., there is no significant difference in the means of the factors). When the p-value is <0.05 and the variance contribution rate is high, it indicates that the factor has a significant and substantial effect on the results.

Table 5. Analysis table of permeability, density, and inclination angle.

Factor	Square Sum	Freedom	Mean Square	F	p	Variance Contribution Rate
Permeability	41,069,351.93	4	10,267,337.98	9.04	2.16 × 10−6	23.76
Density	51,822,826.89	6	8,637,137.82	8.14	2.50 × 10−7	29.98
Angle	22,870,810.14	6	3,811,801.70	2.90	0.071	13.23
Residual	57,069,557.71	104	548,745.75			33.02

shows that fracture density had the most significant effect on cumulative yield. It had a variance contribution of 29.98% and a p-value of 2.5 × 10⁻⁷ (much less than 0.05), which indicates a statistically significant effect on yield. Higher fracture density may have increased gas or liquid flow paths through the fracture, thus contributing to yield. This result suggests that an appropriate increase in fracture density can effectively increase cumulative yield and is a key factor of interest. Secondly, the variance contribution ratio of fracture permeability was 23.76% with a p-value of 2.16 × 10⁻⁶, which was also much less than 0.05, indicating that the effect of fracture permeability on yield was also more significant. Higher fracture permeability implies less resistance to flow of the medium, which can further facilitate the flow and output of resources. Therefore, increasing fracture permeability is one of the effective means to enhance cumulative production. Although it is less influential than fracture density, it still has a significant effect on production. In contrast, the influence of fracture inclination on cumulative production is small. The variance contribution ratio is only 13.23% with a p-value of 0.71, which is higher than 0.05, indicating that it does not statistically significantly affect the production. Although fracture inclination may affect the flow path under certain geological conditions, its effect is limited under the conditions of this simulation. Therefore, more attention to fracture density and permeability can be considered when optimizing gas production schemes in the future, while fracture inclination can be used as a secondary factor.

5.2. Extraction Methods’ Impact on Yield

5.2.1. Effect of Hydraulic Fracture Geometry

Shale gas reservoirs consist of nanopore and microfracture systems, which serve as critical spaces for gas storage. Due to the extremely low porosity and permeability of shale, a significant number of high-permeability fractures created through hydraulic fracturing become essential channels for substantially increasing shale gas production. These include both primary and secondary fractures.

Shale gas production relies heavily on hydraulic fracturing, and different construction parameters result in varying fracture geometries, thereby affecting the productivity of shale gas reservoirs. This study compared and analyzed the developmental effects of six main fracture geometries under conditions of fixed total fracture length, number of fractures, and fracture spacing (Figure 11). By comparing Geometries 1, 3, and 4, it is evident that as the angle between the fracture and the horizontal well decreased, the pressure propagation range also diminished. When the fracture was perpendicular to the horizontal well, the stimulated reservoir volume (SRV) control area formed a rectangular shape. In contrast, when the fracture formed an angle with the horizontal well, the SRV control area became a parallelogram, which was smaller than the rectangular type. Additionally, as the angle between the hydraulic fracture and the horizontal well decreased, the SRV control region reduced further, which is unfavorable for shale gas extraction. Comparing of the pressure distributions for Geometries 5 and 6 reveals that their SRV control regions were essentially identical. However, in the regions at the top or bottom of the fractures, the spacing between the fractures was larger, resulting in a slower pressure drop, which was unfavorable for pressure propagation. Overall, the vertical orientation of hydraulic fractures relative to horizontal wells was advantageous for the development of shale gas reservoirs.

As observed in Geometry 2, this geometry was more favorable for the propagation of gas pressure compared to a symmetrical distribution, as it had a wider control area. Although each fracture was the same length, the longer end of the fracture in Geometry 2 extended connectivity to a more distant area of the wellbore. In contrast, the shorter end compensated for the lack of reservoir connectivity caused by the larger distance between the two longer ends. To verify the size of the controlled SRV region, a pressure value of 10.498 MPa at 1500 days was selected as the node, and contour plots were drawn for different moments with various geometries. These plots were converted to binary images using MATLAB R2017a (The MathWorks, Natick, MA, USA) to estimate the SRV region sizes analyzed in this paper. It was observed that the SRV area sizes ranked in descending order as follows: 2 > 1 > 4 > 5 > 6 > 3. Additionally, Figure 12 confirms the results of the above analysis. The control area of Geometry 1 was slightly smaller than that of Geometry 2, resulting in a slightly lower yield. Geometry 4 cannot effectively control the area due to excessive hydraulic fracture spacing, leading to a lower cumulative yield compared to Geometries 1 and 2. Meanwhile, the cumulative yields of Geometries 5 and 6 were similar, at 7.02 × 10⁷ m³ and 6.99 × 10⁷ m³, respectively. Geometry 3 had the smallest cumulative yield, at 6.67 × 10⁷ m³. These results highlight that different fracture distribution patterns significantly impact shale gas development. In particular, the comparison between Geometries 3 and 4 illustrates that inappropriate inclination does not result in a significant increase in production.

5.2.2. Effect of Complex Fracture Geometry

During the composite fracturing process, not only were primary fractures formed, but also multi-stage secondary fractures were generated along with the CO₂ phase change releasing energy extension effect. These secondary fractures continued to extend and contact and interact with existing natural fractures in the reservoir, forming a complex fracture network. This fracture network was critical to gas flow and pressure propagation, affecting the efficiency of gas extraction and ultimate yield. And the shape of the fracture network was irregular with multiple directions, which increased the complexity of the fluid flow path.

Table 6. Schematic diagram of different complex fracture combinations.

Sample	Illustration
Different fracture intersection quantities	a = 2	b = 4	c = 6
Different fracture densities	d = 0.08 m/m2	e = 0.12 m/m2	f = 0.16 m/m2

further illustrates the dynamic impact of these fracture networks on gas reservoir yield by analyzing the number of crossings between different secondary fractures and hydraulic fractures and the density of different secondary fractures. Using the two models, Geometry 1 and Geometry 2, as a basis, the pressure distributions (under conditions c and f in Table 6) were calculated at different points in time when secondary fractures were considered. Through Figure 13, it can be visualized that the model containing a complex network of secondary fractures has a significant advantage in pressure propagation compared to the model with only primary hydraulic fractures.

The modeling of Geometry 1 and Geometry 2, presented in Figure 14 and Figure 15, reflects the dynamic changes in pressure distribution under different cross-fracture configurations during hydraulic fracturing. The effect of secondary fractures on pressure propagation is revealed as follows: the larger the area of the SRV region controlled by secondary fractures at different time points, the greater the connectivity between secondary fractures and the matrix, which helps to expand the pressure transmission range. Since fluid flow in fractures is primarily controlled by permeability, primary fractures serve as the preferred path for pressure propagation due to their higher permeability. When secondary fractures intersect primary fractures, they significantly enhance pressure propagation compared to secondary fractures that do not intersect primary fractures. This facilitates faster pressure transfer from the primary fractures to the surrounding matrix.

In addition, the density of secondary fractures is a critical factor. A high-density secondary fracture network implies more fracture intersections, which not only increases the interaction among secondary fractures but also raises the likelihood of their interaction with primary fractures. Such a network structure contributes to the formation of a wider pressure propagation region, promoting the spread of pressure over a larger area. In cases of higher secondary fracture density, secondary fractures more effectively propagate the pressure drop generated by primary fractures, further enhancing the connectivity of the entire fracture network.

The comparative analysis presented in Figure 16 reveals the impact of a complex fracture network on the cumulative production of a gas reservoir. The results emphasize the importance of secondary fractures in the hydraulic fracturing process, especially how they affect the effective development of gas reservoirs in the long term. Firstly, the area of SRV region controlled by secondary fractures is a key factor affecting the cumulative production. A larger SRV area indicates a wider contact area between the fracture network and the matrix, which contributes to more gas flow from the shale matrix into the fracture network, thus increasing the production. Secondly, the number of intersections between secondary and primary fractures also has a significant effect on cumulative yield. A higher number of intersections means that more channels are available to facilitate the propagation of pressure and fluids from the primary to the secondary fractures, which in turn reaches a wider area of the matrix and enhances gas recoverability. In addition, an increase in secondary fracture density increases the probability that a secondary fracture will intersect the primary fracture, thereby increasing the migration of gas from the matrix to the secondary fracture and then to the primary fracture, further enhancing production. When the density of secondary fractures is certain, the number of their intersections with the primary fractures becomes a key factor affecting the cumulative yield. Additionally, how secondary fractures of different shapes and extension directions increase the complexity of gas flow and pressure propagation further emphasizes the importance of considering these complexities when modelling and predicting yield. Considering only the primary fractures may underestimate the long-term yield potential, as this approach ignores the additional flow paths provided by secondary fractures.

6. Conclusions

In this work, the deformation model of supercritical CO₂–slickwater affecting shale fracture roughness under different pressures was fitted using experimental data. A mathematical model incorporating supercritical CO₂–slickwater–shale fluid–solid coupling was established, and a C-R solver was constructed and validated by leveraging the coupling advantages of two solvers. Finally, the full-coupling model was solved using the C-R solver. The main results of this work are as follows:

The experimental results confirm that the surface of shale fractures becomes rougher after supercritical CO₂–slickwater treatment. A relationship between the equivalent hydraulic aperture of shale fractures and fracture roughness was established. Based on the fundamental definitions of permeability and porosity, a dynamic mathematical model of porosity and permeability was developed. Additionally, a fluid–solid coupling mathematical model incorporating supercritical CO₂–slickwater–shale interactions was constructed. This model considers the effects of adsorption expansion strain, corrosion compression strain, and roughness strain on permeability evolution, thereby enhancing the accuracy of the shale permeability model.

The error of the C-R solver was within an acceptable range, and the overall model simulation yield matched field-measured data. Under roughness conditions, the permeability evolution varied across different spatial and temporal domains, propagating outward with the fracture as the center. Sensitivity analysis was performed on fracture permeability, density, and inclination. Among these factors, the variance contribution rates of fracture density and permeability were higher, with p-values much less than 0.05. This indicates that fracture density and permeability should receive greater attention when optimizing gas production schemes in the future. Conversely, the yield is affected differently by variations in fracture inclination angle, suggesting that fracture inclination angle can be considered a secondary factor.