Equivalent Modeling and Simulation of Fracture Propagation in Deep Coalbed Methane

Abstract

Deep coalbed methane (CBM) is challenging to develop due to considerable burial depth, high ground stress, and complex geological structures. However, modeling deep CBM in complex formations and setting reasonable simulation parameters to obtain reasonable results still needs exploration. This study presents a comprehensive equivalent finite element modeling method for deep CBM. The method is based on the cohesive element with pore pressure of the zero-thickness (CEPPZ) model to simulate hydraulic fracture propagation and characterize the effects of bedding interfaces and natural fractures. Taking Ordo’s deep CBM in China as an example, a comprehensive equivalent model for hydraulic fracturing was developed for the limestone layer–coal seam–mudstone layer. Then, the filtration parameters of the CEPPZ model and the permeability parameters of the deep CBM reservoir matrix were inverted and calibrated using on-site data from fracturing tests. Finally, the propagation path of hydraulic fractures was simulated under varying ground stress, construction parameters, and perforation positions. The results show that the hydraulic fractures are more likely to expand into layers with low minimum horizontal stress; the effect of a sizable fluid injection rate on the increase in hydraulic fracture length is noticeable; the improvement effect on fracture length and area gradually weakens with the increased fracturing fluid volume and viscosity; and when directional roof limestone/floor mudstone layer perforation is used, and the appropriate perforation location is selected, hydraulic fractures can communicate the coal seam to form a roof limestone/floor mudstone layer indirect fracturing. The results can guide the efficient development of deep CBM, improving the human society’s energy structure.

1. Introduction

Coalbed methane, shale oil and gas, and natural gas hydrates are important components of unconventional energy sources [1,2,3]. With the deepening exploration and development of fossil energy resources, shallow fossil energy reserves are insufficient to meet the demands of human industrial development [4]. Therefore, the exploitation of deep fossil energy resources is an inevitable trend [5,6]. Deep CBM (buried more than 2000 m) is a form of clean energy that helps reduce greenhouse gas emissions and optimize the energy structure [7]. China’s CBM resources at a buried depth greater than 2000 m amount to 10.01 × 10¹² m³, with 96% concentrates in the Junggar Basin, Ordos Basin, and Tuha-Santanghu Basin [8]. The early development of deep CBM in China primarily relies on large-scale fracturing technology, including “high injection rate + forced sand + dense cutting” used in shale gas extraction [9]. Due to the complex geological structures, high ground stress, bedding interfaces, and natural fractures in deep CBM reservoirs, as well as the high Poisson ratio and low elastic modulus of coal rock compared to sandstone and mudstone, traditional fossil energy development methods are not entirely suitable for deep CBM [10,11,12,13]. At present, how deep CBM reservoirs form complex fractures through hydraulic fracturing to obtain a more extensive reservoir reconstruction range remains to be studied. Therefore, developing a more realistic numerical model of deep CBM and studying the behavior of the hydraulic fracture propagation path is crucial for guiding the efficient development of deep CBM.

The mechanical properties of coal have been widely studied using various methods, including nanoindentation test, acoustic emission, digital image method, Brazilian splitting test, and triaxial compression test [14,15,16,17,18]. In order to clarify the law of hydraulic fracture expansion in thin coal seam groups, Haifeng Zhao et al. conducted a hydraulic fracturing test. They investigated the propagation of hydraulic fractures under different engineering factors and concluded that the hydraulic fracturing effect is mainly influenced by the perforation location [19]. Haifeng Zhao et al. conducted hydraulic fracturing tests on various coal macro lithotypes. They found that overall productivity increase in the coal seam relies on the morphology of horizontal large fractures and the proper proppant placement [20]. Wenzhuo Cao et al. conducted hydraulic fracturing tests on coal rocks containing natural fractures. They utilized acoustic emission and CT scanning to analyze the internal fracture geometry of the coal rock. They discovered that natural fractures in coal rocks had a significant impact on the behavior of hydraulic fractures [21]. Yuwei Li et al. conducted an experimental simulation to study the formation matrix of T-shaped hydraulic fractures in coal fracturing. They concluded that the occurrence of T-shaped hydraulic fractures is significantly influenced by the horizontal fracture angle [22]. Tingting Jiang et al. conducted a hydraulic fracturing experiment on coal rock with bedding development. They concluded that the bifurcation and turning of hydraulic fractures at the bedding interface were necessary to form a fracture network [23]. Haozhe Li et al. conducted a study on the propagation path of hydraulic fractures across the coal rock interface using a three-point bending test and a hydraulic fracturing simulation test. They suggested the use of variable injection rate fracturing construction to promote longitudinal penetration of hydraulic fractures at the initial stage, followed by a reduction in injecting rate to encourage lateral extension of fractures in the roof and coal seam [24]. These studies revealed that the propagation of hydraulic fractures in CBM is influenced by natural fractures, bedding interfaces, and construction parameters. However, these hydraulic fracturing experiments do not utilize deep coal rock, and the conclusion is the results of small-scale laboratory experiments. Therefore, they cannot fully depict the hydraulic fracture propagation path law under the conditions of a buried depth more fabulously than a 2000 m deep coal seam.

With advancements in numerical simulation technology, it has become an efficient way to study hydraulic fracture propagation in CBM. At present, the most common numerical simulation methods include the finite element method (FEM), extended finite element method (XFEM), boundary element method (BEM), displacement discontinuity method (DDM), and discrete element method (DEM) [25]. During hydraulic fracturing, the FEM, combined with the cohesive element with the CEPPZ element, can accurately model the reservoir’s discontinuous interface. Yulong Liu et al. established a numerical model of hydraulic fracture propagation in layered coal seams using cohesive elements with pore pressure nodes. They investigated the initiation and propagation of hydraulic fractures in layered coal seams under different mechanical properties. The results indicated that the difference in elastic modulus and tensile strength was the primary factor influencing the vertical propagation of fractures [26]. Haifeng Zhao et al. studied the expansion behavior and influencing factors of H-shaped hydraulic fractures by the zero-thickness cohesive element with pore pressure nodes. The findings indicated that H-shaped hydraulic fractures are frequently observed in shallow coal seams [27]. Laisheng Huang et al. developed a finite element model to simulate hydraulic fracturing in a layered coal seam using cohesive elements. Their research focused on analyzing the propagation of hydraulic fractures under different ground stress, bedding inclination angle, and fluid injection rate. The study found that hydraulic fractures intersecting the bedding interface could cause shear failure at the interface, leading to the formation of secondary fractures [28]. Haozhe Li et al. established a two-dimensional finite element model of a coal seam with an interface by using the cohesive model. They concluded that ground stress and coal rock interface strength in geological factors are the main factors affecting whether the fractures can pass through the interface [29]. The numerical simulation results show that the expansion of hydraulic fracture in CBM is also affected by factors such as reservoir ground stress, rock mechanics parameters, and the strength of the bedding interface. Nevertheless, this model’s limitations include not accounting for geology conditions in deep CBM deeper than 2000 m. At the same time, they often rely on empirical values for some parameters that are challenging to obtain through experimental testing, such as cohesive element filtration and actual reservoir permeability parameters.

Countries such as China, North America, and Australia show potential for CBM extraction [30]. In recent years, China has been making progress in developing deep CBM. In the Ordos Basin, the maximum daily gas output from a single deep CBM horizontal well has reached 19.78 × 10⁴ m³, showing significant potential [31]. However, the different lithology reservoirs in horizontal well drilling present challenges in reservoir reconstruction [32]. The modeling of deep CBM in complex formations and setting simulation parameters to obtain reasonable results still needs exploration. This paper presents a comprehensive equivalent finite element modeling method for deep CBM, incorporating bedding interfaces and natural fractures using the CEPPZ model. Then, taking deep CBM buried more than 2000 m in Ordos Basin, China, as the research object, the model parameters were inverted using on-site data from fracturing tests. Finally, the hydraulic fracture propagation paths under different conditions were calculated. The research results can guide the efficient development of deep CBM and improve the human energy structure.

2. Mathematical Model

2.1. Fluid–Solid Coupling Equation of Deep Coalbed Methane

The deep CBM reservoir can be considered a fully saturated porous elastic medium. The expansion process of hydraulic fractures is quasi-static, and the failure process obeys small deformation.

The conservation momentum equation can be described as follows:

$$\mathsf{\nabla }·\mathsf{\sigma }+\rho \mathit{b} - \rho \stackrel{\cdot \cdot }{\mathsf{u}}=0,$$

(1)

where σ is the total stress acting on the porous media, ρ is the density of the reservoir rock, b is the body force, and $\ddot{\mathit{u}}$ denotes the acceleration vector of the solid.

According to Biot’s law, the relationship between effective stress and total stress can be expressed as follows [33]:

$$\mathsf{\sigma }=\overline{\mathsf{\sigma }}+\alpha p_w\mathit{I},$$

(2)

where σ represents the total stress acting on the porous medium; $\overline{\mathsf{\sigma }}$ is the effective stress; α is the Biot constant; P_w is the formation pore pressure; and I is the identity matrix.

Based on the principle of virtual work, the weak form of the stress balance equation can be expressed as follows [34]:

$${\displaystyle \underset{\Omega }{\int }\left(\stackrel{-}{\mathsf{\sigma }}+p_w\mathit{I}\right)}{\delta }_{\stackrel{.}{\epsilon }}d\Omega ={\displaystyle \underset{S}{\int }\mathit{t}{\delta }_V}dS+{\displaystyle \underset{\Omega }{\int }\mathit{f}{\delta }_V}d\Omega +{\displaystyle \underset{\Omega }{\int }\phi {\rho }_W\mathit{g}}{\delta }_{V}d\Omega ,$$

(3)

where ${\delta }_{\stackrel{.}{\epsilon }}$ is the virtual strain field; t is the outer surface force per integral area; δ_v is the node virtual velocity field; f is the volume force regardless of fluid gravity; ρ_w is the fluid density; S is the surface area; and Ω is the study area.

According to Darcy’s law, the fluid flow velocity v in the matrix can be expressed as follows [35]:

$$v=-{\displaystyle \frac{k}{\mu }}\cdot \mathsf{\nabla } p_w,$$

(4)

where k represents the permeability of the matrix; μ is the viscosity of the fracturing fluid; and p_w is the pore pressure of the matrix.

Based on the conservation of mass, the continuity equation for fluid flow in the matrix can be expressed as follows [36]:

$$\phi {\displaystyle \frac{\partial {\rho }_f}{\mathit{\partial }t}}+\mathsf{\nabla }\cdot \left({\rho }_{f}v\right)=0,$$

(5)

where φ represents the porosity of the matrix; ρ_f is the density of the fracturing fluid; v stands for the fluid flow velocity in the matrix; and t is the injection time of the fracturing construction.

By substituting Equation (4) into Equation (5), the continuity equation of fluid flow in the matrix can be expressed as follows:

$$\phi {\displaystyle \frac{\mathit{\partial }{\rho }_f}{\mathit{\partial }t}}-\mathsf{\nabla }\cdot \left({\rho }_f{\displaystyle \frac{\mathit{k}}{\mu }}\cdot \mathsf{\nabla }{p}_w\right)=0,$$

(6)

By substituting the matrix fluid pressure test function δp_w into Equation (6), the weak form of matrix fluid seepage in the matrix pore structure can be expressed as follows:

$${\displaystyle {\int }_{\Omega }\delta \mathsf{\nabla } p_w\cdot \frac{\mathit{k}}{\mu }\cdot \mathsf{\nabla } p_w}d\Omega -{\displaystyle {\int }_{\Gamma }\delta p_{w}c\left(p_m-p_f\right)}d\Gamma =0,$$

(7)

2.2. CEPPZ Element of Fracture Propagation

The structure of the CEPPZ element is depicted in Figure 1. It consists of displacement nodes in the upper and lower layers, and fluid pressure nodes in the middle layer. In Figure 1b, the CEPPZ element is destroyed by the fluid pressure. Zone ① is the destroyed CEPPZ element, resulting in the formation of macroscopic hydraulic fractures, with the fluid flowing parallel to the hydraulic fracture surface (tangential flow). Simultaneously, filtration occurs along the upper and lower surfaces of the hydraulic fracture surface (normal flow). Zone ② is the fracture process region; there is no fluid in the region. Zone ③ is the CEPPZ element without failure.

The failure process of the CEPPZ element consists of the linear elastic stage prior to reaching peak load, followed by the damage evolution stage afterward.

The stress–strain relationship in the linear elastic stage is as follows [37]:

$$t=\left\{\begin{array}{c}{t}_n\\ t_s\\ t_t\end{array}\right\}=\left[\begin{array}{ccc}{E}_{nm}& E_{ns}& E_{nt}\\ E_{ns}& E_{ss}& E_{st}\\ E_{nt}& E_{st}& E_{tt}\end{array}\right]·\left\{\begin{array}{c}{\epsilon }_n\\ {\epsilon }_s\\ {\epsilon }_t\end{array}\right\},$$

(8)

where the symbols’ subscripts n, s, and t represent the normal direction and two tangential directions of the CEPPZ element. The symbols with subscripts indicate different physical quantities in different directions; t is the stress acting on the CEPPZ element; E is the stiffness of the cohesive element; and ε is the strain of the CEPPZ element.

When the load on the CEPPZ element reaches its peak strength, damage evolution begins. During the whole damage evolution stage, the normal stress t_n, tangential stress t_s, and t_t are determined by the damage variable, and the expression is as follows [37]:

$$t_n=\left\{\begin{array}{c}\left(1-D\right)\stackrel{-}{t_n}\hspace{1em}\stackrel{-}{t_n}\ge 0\\ \stackrel{-}{t_n}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\stackrel{-}{t_n}<0\end{array}\right.,$$

(9)

$$t_s=\left(1-D\right)\stackrel{-}{t_s},$$

(10)

$$t_t=\left(1-D\right)\stackrel{-}{t_t},$$

(11)

where $\overline{t_n}$, $\overline{t_s}$, and $\overline{t_t}$ are the normal peak stress Tmax and tangential peak stress τs-max and τt-max corresponding to the CEPPZ force element without damage. D is the damage variable.

Damage factor is utilized to depict the process of damage evolution, and the expression is as follows [38]:

$$D={\displaystyle \frac{{\delta }_m^f\left({\delta }_m^{\max }-{\delta }_m^o\right)}{{\delta }_m^{\max }\left({\delta }_m^f-{\delta }_m^o\right)}},$$

(12)

where D is the damage factor; ${\delta }_m^f$ is the displacement at the time of complete failure; ${\delta }_m^o$ is the displacement at initial damage; and ${\delta }_m^{\max }$ is the maximum displacement.

The calculation formula of effective displacement is as follows [37]:

$${\delta }_m=\sqrt{{〈{\delta }_n〉}^2+{\delta }_s^2+{\delta }_t^2},$$

(13)

where δ_n, δ_s, and δ_t are the normal, first, and second tangential damage displacement of the bedding plane.

The secondary stress criterion is used as the initiation criterion of hydraulic fractures. When the sum of the squares of the stress and strength ratio on the fracture surface is 1, the CEPPZ element begins to be damaged. The fracture initiation is expressed as follows [39]:

$${\left({\displaystyle \frac{〈t_n〉}{t_n^0}}\right)}^2+{\left({\displaystyle \frac{{\tau }_s}{{\tau }_s^0}}\right)}^2+{\left({\displaystyle \frac{{\tau }_t}{{\tau }_t^0}}\right)}^2=1,$$

(14)

where the symbols’ subscripts n, s, and t represent the normal direction and two tangential directions of the CEPPZ element. The symbols with subscripts indicate different physical quantities in different directions; t_n, τ_s, and τ_t are the normal stress and two tangential stresses on the CEPPZ element plane, respectively; and $t_n^0$, ${\tau }_s^0$, and ${\tau }_t^0$ are the tensile strength and shear strength of the CEPPZ element.

The BK criterion, as shown in Formula (8), is used as the hydraulic fracture propagation criterion. It takes into account the energy release rate of type I and type II in the fracture propagation process [40].

$$G_{IC}+\left(G_{IIC}-G_{IC}\right){\left({\displaystyle \frac{G_{II}}{G_I+G_{II}}}\right)}^{\eta }=G,$$

(15)

where $G_{IC}$ represents the type I critical energy release rate; $G_{IIC}$ represents the type II critical energy release rate; $G_I$ represents the type I energy release rate; $G_{II}$ represents the type II energy release rate; η is a constant related to the material properties and represents the contribution of the shear mode to the ratio of critical fracture energy (brittle materials, the value of η is closer to 2, while for plastic materials, it is closer to 3), and the value used in this paper is 2; G represents the compound energy release rate.

The damage factor changes monotonically from 0 to 1. When the damage factor is 0, the damage evolution begins and ends when the damage factor is 1. In Figure 1b, the damage factor of zone 1 is 1. The damage factor of zone 2 is between 0 and 1. The damage factor of zone 3 is 0.

The tangential flow of fluid in the hydraulic fracture is expressed as follows [41]:

$${\mathit{q}}_f=-{\displaystyle \frac{w^3}{12{\mu }_w}}\mathsf{\nabla }{p}_f,$$

(16)

where p_f is the fluid pressure in the hydraulic fracture, µ_w is the fluid viscosity, w is the hydraulic fracture width, and q_f is the fluid flux of tangential flow in the hydraulic fracture.

The normal flow in the hydraulic fracture is expressed as follows [42]:

$$q_t=c_t\left(p_f-p_t\right); q_b=c_b\left(p_f-p_b\right),$$

(17)

where q_t and q_b represent the fluid filtration along the upper and lower surfaces of the hydraulic fracture, respectively; c_t and c_b are the fluid filtration coefficients on upper and lower surfaces of the hydraulic fracture, respectively (c = c_t = c_b in this paper); p_f is the fluid pressure in the hydraulic fracture; and p_t and p_b are the pore pressure of deep CBM at upper and lower surfaces of the hydraulic fracture, respectively.

The continuity equation of fluid flow in the hydraulic fracture can be expressed as follows [43]:

$${\displaystyle \frac{\mathit{\partial }w}{\mathit{\partial }t}}+\mathsf{\nabla }\cdot q_f+q_t+q_b=Q\left(t\right)\delta \left(x\right),$$

(18)

where Q(t) is the injection rate; t is the injection time; and δ(x) is a Dirac function.

By bringing Equations (3) and (4) into Equation (5), the continuity equation considering fluid flow and filtration can be obtained as follows:

$${\displaystyle \frac{\mathit{\partial }w}{\mathit{\partial }t}}-{\displaystyle \frac{w^3}{12{\mu }_w}}{p}_f+c_t\left(p_f-p_t\right)+c_b\left(p_f-p_b\right)=Q\left(t\right)\delta \left(x\right),$$

(19)

The fluid pressure test function δpf is put into Equation (6), and the weak form of the fluid flow continuity equation in the hydraulic fracture is obtained as follows:

$${\displaystyle {\int }_{\Omega f}}\delta p_f{\displaystyle \frac{\mathit{\partial }w}{\mathit{\partial }t}}d\Omega +{\displaystyle {\int }_{\Omega f}}{\displaystyle \frac{w^3}{12{\mu }_w}}\delta \mathsf{\nabla }{p}_f\cdot \mathsf{\nabla }{p}_{f}d\Omega -2{\displaystyle {\int }_{\Gamma }}\delta p_{f}c\left(p_f-p_t\right)d\Gamma -\delta p_{f}Q\left(t\right){|}_{x = x_0}= 0,$$

(20)

where Γ is the fracture surface.

2.3. Finite Element Modeling and Solution

Displacement (u), fluid pressure in hydraulic fracture (p_f), pore pressure (p_w), and hydraulic fracture width (w) function are as follows:

$$u\left(x,t\right)={\displaystyle \sum N_I^u}u\left(t\right),$$

(21)

$$p_f\left(x,t\right)={\displaystyle \sum N_I^f}{p}_f\left(t\right),$$

(22)

$$p_w\left(x,t\right)={\displaystyle \sum N_I^w}{p}_w\left(t\right),$$

(23)

By defining fracture criteria and setting boundary conditions, Formulas (7), (12), and (14) are approximated by form functions and solved.

3. Comprehensive Equivalent Modeling and Parameters Invert

3.1. Reservoir Characteristics Overview

The target deep CBM is situated in the Ordos Basin. According to the logging curve of the pilot well, the coal seam is positioned at a depth of 2873–2885 m and has a thickness of 12 m. There is a 10 m thick layer of limestone on top of the coal seam and a 12 m thick layer of mudstone at the bottom. Based on the results of the reservoir rocks triaxial compression test, the elastic modulus of coal rock ranges from 6.41 to 8.28 GPa, the elastic modulus of limestone ranges from 18.64 to 22.36 GPa, and the elastic modulus of mudstone ranges from 10.76 to 16.89 GPa. The Poisson ratio is 0.186 to 0.246 for coal rock, 0.165 to 0.235 for limestone, and 0.17 to 0.242 for mudstone. The reservoir pore pressure is 28.9 MPa. The vertical ground stress is 64.28 MPa.

Table 1. Ground stress of reservoir.

Reservoir Type	Minimum Horizontal Ground Stress (MPa)	Maximum Horizontal Ground Stress (MPa)
Limestone	60.04	70.11
Coal seam	53.66	61.07
Mudstone	57.16	67.01

illustrates the horizontal ground stress of the reservoir, and it indicates that the minimum ground stress of the coal seam is lower than that of limestone and mudstone layers.

CT scans and triaxial compression experiments were carried out on the coal and rock in the block. Meanwhile, combined with the existing research results, the parameters, as shown in

Table 2. Initial condition setting of the model.

Reservoir Type	Parameter	Unit	Value	Value Source
Limestone	Elastic modulus	GPa	22.36	Triaxial compression results
	Poisson’s ratio	Dimensionless	0.165	Triaxial compression results
	Porosity	Dimensionless	0.0409	Document [13]
	Matrix tensile strength	MPa	5	Document [27,44]
Coal	Elastic modulus	GPa	8.28	Triaxial compression results
	Poisson’s ratio	Dimensionless	0.186	Triaxial compression results
	Porosity	Dimensionless	0.0499	CT scan results
	Tensile strength of matrix	MPa	1	Document [27,44]
	Tensile strength of interface	MPa	0.25	Document [27,44]
Mudstone	Elastic modulus	GPa	16.89	Triaxial compression results
	Poisson’s ratio	Dimensionless	0.17	Triaxial compression results
	Porosity	Dimensionless	0.0138	Document [13]
	Tensile strength of matrix	MPa	4.5	Document [27,44]

, were determined.

3.2. Equivalent Modeling and Parameters Invert

It is challenging to obtain the fluid filtration coefficient of the CEPPZ elements and the permeability coefficient of the deep CBM reservoir through experimentation. These coefficients are determined through hydraulic fracturing tests. The detailed steps are as Figure 2.

Step 1: Establish the geometric model of deep CBM.

The hydraulic fracturing geometry model of a deep CBM reservoir is established based on logging reservoir interpretation, well trajectory, and fracturing design.

Step 2: Insert the CEPPZ elements for equivalent characterization.

Due to the complexity of natural fractures and bedding interfaces in the deep CBM reservoir, it is challenging to create a model that includes both the bedding interfaces and natural fractures in a three-dimensional model. To address this, the tensile strength of the CEPPZ elements is used to characterize the impact of bedding interfaces and natural fractures on the strength of the deep CBM reservoir matrix. Additionally, the influence of bedding interfaces and natural fractures on fluid flow in the deep CBM reservoir matrix is characterized based on the fluid filtration parameters of the CEPPZ elements.

Step 3: Set the model parameters.

The model boundary conditions and injection conditions are set based on the field fracturing test conditions. Model iteration and solution parameters are adjusted, and model convergence is tested until the model converges.

Step 4: Inversion and calibration of model parameters.

Considering the fluid injection process before the pump stops, the bottom-hole pressure is calculated by adjusting the fluid filtration coefficients of CEPPZ elements and the permeability coefficients of the deep CBM reservoir matrix during the fluid injection process. The simulation pressure is then compared with the actual bottom-hole pressure. If the two pressures are consistent, the model parameters are deemed reasonable, and the model is considered correct.

According to the Pearson correlation coefficient (r), as shown in Formula (21) [45], whether the two are consistent is judged. The Pearson correlation coefficient ranges from −1 to +1. Positive values indicate positive linear correlation, and negative values indicate negative linear correlation. The closer the value is to +1 or −1, the stronger the linear correlation. In this paper, the condition for compliance is r > 0.95.

$$r=\left[{\displaystyle \sum _{i=1}^n\left(x_i-\stackrel{-}{x}\right)\left(y_i-\stackrel{-}{y}\right)}/\sqrt{{\displaystyle \sum _{i=1}^n{\left(x_i-\stackrel{-}{x}\right)}^2}}·\sqrt{{\displaystyle \sum _{i=1}^n{\left(y_i-\stackrel{-}{y}\right)}^2}}\right],$$

(24)

where x_i is the simulation pressure at the time of t_i; and y_i is the actual bottom-hole pressure at the time of t_i.

Based on the results of logging interpretation, considering the symmetry of hydraulic fracture propagation, Figure 3 depicts the establishment of a 1/2 geometric model based on the geological structure of the deep CBM. The model measures 250 m in length, 60 m in width, and 54 m in height. It includes a 12 m thick coal seam, 10 m thick limestone layer, 12 m thick mudstone layer, and 10 m thick upper and lower cap layers. The CEPPZ elements (solid red line) were placed in the middle of the model to simulate the propagation path of the main hydraulic fracture. Additionally, the insertion of 10 layers of CEPPZ elements (red dotted line) at medium intervals in the coal seam effectively represents the influence of natural fractures and bedding surfaces. The minimum horizontal ground stress, maximum horizontal ground stress, and vertical ground stress correspond to the x, y, and z directions, respectively. The displacement boundary condition of the model is 0, and the pore pressure boundary condition is 28.9 MPa.

The process of calculating actual bottom-hole pressure based on the surface-wellhead pressure is as follows.

The relationship between actual bottom-hole pressure and surface-wellhead pressure can be expressed as Formula (25).

$$P_b=P_s+{\rho }_{\mathrm{w}}gh-P_f-P_{perf},$$

(25)

where P_b and P_s are the actual bottom-hole and surface-wellhead pressure, respectively; ρ_w is the fluid density; g is the acceleration of gravity; h is vertical depth; P_f is the friction pressure loss; and P_perf is the frictional pressure drop at the perforation.

$$p_f=2{\displaystyle \frac{f{L}_s{\rho }_L{v}_L^2}{D}},$$

(26)

where L_s is the total length of casing; vL is the average flow rate of fracturing fluid; and D is the inner diameter of casing.

The friction coefficient f is calculated by Formula (27) [46].

$$f={\displaystyle \frac{a}{N_{\mathrm{Re}}^b}},$$

(27)

where for laminar flow, a = 16, b = 1; for turbulent flow, a = 0.0719, b = 0.25.

The Reynolds number N_Re can be calculated by Formula (28) [46].

$$N_{\mathrm{Re}}={\displaystyle \frac{v_{L}D{\rho }_L}{{\mu }_{\mathrm{w}}}},$$

(28)

where μ_w is the fluid viscosity.

The Bernoulli equation is used to describe the frictional pressure drop at the perforation, and the relationship can be expressed as Formula (29) [47].

$$p_{perf}=0.807249\times {\displaystyle \frac{{\rho }_L}{N^2{D}_p^4{C}^2}}{Q}^2$$

(29)

where Q is the flow rate; D_p is the perforation hole diameter; C is the empirical coefficient, which is 0.56 when no perforation wear occurs, and 0.89 when perforation wear occurs (0.56 in this paper); and N is the number of perforations.

According to the hydraulic fracturing test, the viscosity of the fracturing fluid is 1 mPa·s. The initial vertical ground stress condition is 64.28 MPa. The initial horizontal ground stress conditions are detailed in Table 1. The hydraulic fracturing test includes increasing the injection rate, decreasing the injection rate, and the pressure drop after stopping the pump. Figure 4 illustrates the construction curve of the hydraulic fracturing test. The peak injection rate reaches 20 m³/min at 2370 s, and the pump stops at 2972 s.

The red curve in Figure 4 represents the bottom-hole pressure curve obtained through numerical simulation. The low simulation pressure is considered to be the cause of near-wellbore friction. The Pearson correlation coefficient (r) between the simulation pressure and the actual bottom-hole pressure is 0.97, which is greater than 0.95. This indicates that the model parameters are reasonable. The finite element model of the deep CBM is accurate, and the model parameters obtained are detailed in

Table 3. The model parameters are obtained.

Parameter	Unit	Value
Bedding interfaces fluid filtration coefficient of coal reservoir	m3·s−1·Pa−1	5 × 10−11
Permeability coefficient of the coal matrix reservoir	m·s−1	3.5 × 10−7
Hydraulic fracture fluid filtration coefficient of coal reservoir	m3·s−1·Pa−1	5 × 10−11
Permeability coefficient of the limestone matrix reservoir	m·s−1	5.5 × 10−8
Hydraulic fracture fluid filtration coefficient of limestone reservoir	m3·s−1·Pa−1	5.5 × 10−12
Permeability coefficient of the mudstone matrix reservoir	m·s−1	5 × 10−8
Hydraulic fracture fluid filtration coefficient of mudstone reservoir	m3·s−1·Pa−1	5 × 10−12

.

According to Figure 4, the bottom-hole pressure calculated by the numerical simulation is lower than the actual bottom-hole pressure. The difference is relatively small in the early stage of fluid injection, but gradually increases in the later stage. However, the overall difference is relatively close. During the actual hydraulic fracture process, the filtrate loss of the fracturing fluid in the hydraulic fracture gradually decreases due to the addition of materials such as proppants. However, this was not considered in the numerical simulation calculation process.

Table 1, Table 2 and Table 3 show the basic data. When calculating the hydraulic fracture propagation paths under different conditions, only the corresponding parameter conditions are changed.

4. Numerical Simulation Result

Based on the model parameters obtained from the hydraulic fracturing test, the propagation path of hydraulic fracturing in the main fracturing section is studied under different conditions. According to the main fracturing section construction parameters, the injection rate is 20 m³/min, with a total liquid volume of 3600 m³.

4.1. Minimum Horizontal Ground Stress

Keep the minimum horizontal ground stress of the coal seam (σ_hc) while adjusting the minimum horizontal ground stress of the mudstone (σ_hm), and limestone (σ_hl) layers individually. Based on the ground stress data in Table 1, under default conditions, the minimum horizontal ground stress difference between the coal seam and limestone layer is −6.38 MPa, and the minimum horizontal ground stress difference between the coal seam and mudstone layer is −3.5 MPa. Figure 5 illustrates the morphology of hydraulic fractures, while Figure 6 illustrates the hydraulic fracture parameters under these conditions.

Only change the minimum horizontal ground stress of the limestone layer. When the minimum horizontal stress difference between the coal seam and the limestone layer (σ_hc − σ_hl) increases from −6.38 MPa to 7 MPa, the half-length of hydraulic fracture increases from 83.33 m to 183.33 m, and the maximum hydraulic fracture height increases from 9 m to 21 m, and the area of the hydraulic fracture increases from 961.28 m² to 2895.13 m². In the case of σ_hc − σ_hl = 7 MPa, the hydraulic fracture extends from the coal seam into the limestone layer, with the half-length of the hydraulic fracture being 55.56 m in the coal seam and 183.33 m in the limestone layer.

Only change the minimum horizontal ground stress of the mudstone layer. When the minimum horizontal ground stress difference between the coal seam and mudstone layer (σ_hc − σ_hm) increases from −3.5 MPa to 7 MPa, the half-length of hydraulic fracture increases from 83.33 m to 150 m, the maximum hydraulic fracture height increases from 9 m to 23 m, and the hydraulic fracture area increases from 961.28 m² to 2968.02 m². In the case of σ_hc − σ_hm = 3.5 MPa, the far end of the hydraulic fracture passes through the interface between the coal seam and the mudstone layer, with only a small portion of the hydraulic fracture expanding in the coal seam. In the case of σ_hc − σ_hm = 7 MPa, the hydraulic fracture extends through the interface between the coal seam and mudstone layer into the mudstone layer. The half-length of the hydraulic fracture is 66.67 m in the coal seam and 150 m in the limestone layer.

When the minimum horizontal ground stress of the limestone/mudstone layer changes, the minimum horizontal stress difference between the coal and limestone/mudstone layer is set to be the seam. When σ_hc − σ_hl/σ_hm = 3.5 MPa, the hydraulic fracture extends through the interface between the coal seam and limestone layer into the limestone layer. The half-length of the hydraulic fracture reaches a maximum value of 133.33 m, and the hydraulic fracture area reaches a maximum of 2778.71 m². When σ_hc − σ_hl/σ_hm = 7 MPa, the hydraulic fracture also enters the limestone layer after cracking through the interface between the coal seam and the limestone layer. The half-length of the hydraulic fracture in the coal seam is 61.11 m, the half-length of the hydraulic fracture in the limestone layer is 94.44 m, and the hydraulic fracture area is 2028.58 m².

Based on the analysis above, hydraulic fractures are more likely to expand into layers with low minimum horizontal stress. Increasing the minimum horizontal stress difference between the coal seam and the limestone/mudstone layer contributes to the formation of a larger hydraulic fracture length and fracture area. However, the hydraulic fracture in the coal, limestone, and mudstone layers may spread forward asynchronously.

4.2. Fracturing Fluid Injection Rate

The injection rate is set at 5 m³/min, 10 m³/min, 15 m³/min, and 20 m³/min, respectively. Figure 7 illustrates the morphology of hydraulic fractures, while Figure 8 illustrates the hydraulic fracture parameters under these conditions. As the injection rate increases from 5 m³/min to 20 m³/min, the half-length of the hydraulic fracture increases from 12.53 m to 83.33 m, and the hydraulic fracture area increases from 427.03 m² to 961.28 m². There is a linear relationship between the half-length of hydraulic fracture, fracture area, and the injection rate. Hydraulic fractures propagate completely in the coal seam. The hydraulic fracture height is 8 m when the injection rate is 5 m³/min, and 9 m under other conditions.

This shows that the high injection rate can increase the length of the hydraulic fracture and form a larger area of hydraulic fracture. When the injection rate is 20 m³/min, the half-length of the hydraulic fracture and fracture area increase by 565% and 125%, respectively, compared to 5 m³/min.

4.3. Fracturing Fluid Volume

The fracturing fluid volume is set at 250 m³, 500 m³, 1500 m³, and 3600 m³, respectively. Figure 9 illustrates the morphology of hydraulic fractures, while Figure 10 illustrates the hydraulic fracture parameters under these conditions. As the fracturing fluid volume increased from 250 m³ to 3600 m³, the half-length of the hydraulic fracture increased from 54.28 m to 83.33 m, and the hydraulic fracture area increased from 1238.08 m² to 1611.33 m². A clear power function relationship exists between the half-length of hydraulic fracture, fracture area, and fracturing fluid volume. Hydraulic fractures propagate completely in the coal. The hydraulic fracture height is 8 m when the liquid injection displacement is 5 m³/min, and 9 m under other conditions.

This shows that increasing the fracturing fluid volume helps the hydraulic fracture expand further. However, with the increase in the hydraulic fracture length, the effect of the fracturing fluid volume on the hydraulic fracture length is gradually weakened. Compared to 250 m³, the half-length and fracture area at 500 m³ increased by 33.05% and 13.59%, respectively. Compared to 500 m³, the half-length and fracture area at 1500 m³ increased by 14.04% and 9.43%, respectively. Compared to 1500 m³, the half-length and fracture area at 3600 m³ increased by only 1.18% and 4.7%, respectively.

According to the Table 3, since the fluid filtration coefficient and the permeability coefficient of the coal seam are one order of magnitude larger than those of limestone and mudstone layer, the larger the length and area of hydraulic fractures are, the faster the filtration rate of the fracturing fluid is, and the friction loss during fluid flow in hydraulic fractures increases. As a result, the improvement effect on fracture length and area gradually weakens with the increased fracturing fluid volume.

4.4. Fracturing Fluid Viscosity

The fracturing fluid viscosity is set at 1 mPa·s, 50 mPa·s, 75 mPa·s, and 100 mPa·s. Figure 11 illustrates the morphology of hydraulic fractures, while Figure 12 illustrates the hydraulic fracture parameters under these conditions. As the fracturing fluid viscosity increases from 1 mPa·s to 100 mPa·s, the half-length and hydraulic fracture area increase initially and then decrease. The maximum half-length of hydraulic fracture is 105.56 m when the fracturing fluid viscosity is 50 mPa·s and 75 mPa·s. The maximum hydraulic fracture area is 2326.39 m² when the fracturing fluid viscosity is 50 mPa·s. When the viscosity of the fracturing fluid is +1 mPa·s, the hydraulic fracture is limited to propagation in the coal seam, with a fracture height of 9 m. However, when the viscosity is 50 mPa·s, 75 mPa·s, and 100 mPa·s, the hydraulic fractures extend through the interface between the coal seam and mudstone layer, entering the mudstone layer to propagate. The hydraulic fracture height increases to 23 m.

This shows that a large hydraulic fracture area can be achieved by increasing the fracturing fluid viscosity to 50–75 mPa·s. However, reaching a viscosity of 100 mPa·s does not result in the maximum hydraulic fracture length and area.

A lower viscosity will lead to a significant loss of the fracturing fluid, with less frictional resistance in the fractures; a higher viscosity will reduce the fluid loss of the fracturing fluid, but it will also increase the frictional resistance in the fractures. Therefore, when the viscosity is lower than 50 mPa·s, due to the lower viscosity of the fracturing fluid, the fluid is more likely to filter into the formation, resulting in a smaller length and area of the hydraulic fracture. When the viscosity is greater than 75 mPa·s, due to the higher viscosity of the fracturing fluid, the frictional resistance in the fractures is also higher, making it difficult for the hydraulic fracture to expand.

4.5. Perforation Position

According to the horizontal wellbore trajectory, perforation holes are present in coal, limestone, and mudstone. The hydraulic fractures under different perforation positions are calculated. Figure 13 illustrates the morphology of hydraulic fractures, while Figure 14 illustrates the hydraulic fracture parameters under these conditions.

4.5.1. Perforation Position in Coal Seam

The No.① perforating position is located in the upper part of the coal seam, 3 m away from the limestone layer. The No.② perforating position is located in the middle of the coal seam, 6 m away from the limestone and mudstone layer, and the No.③ perforating position is located at the bottom of the coal seam, 9 m away from the limestone layer.

When perforating the upper part of the coal seam, the half-length of the hydraulic fracture is 105.56 m, the hydraulic fracture height is 9 m, and the hydraulic fracture area is 1354.87 m². When perforating in the middle of the coal seam, the half-length of the hydraulic fracture is 83.33 m, the hydraulic fracture height is 9 m, and the hydraulic fracture area is 961.28 m². When perforating at the bottom of the coal seam, the half-length of the hydraulic fracture is 83.33 m, the hydraulic fracture height is 10 m, and the hydraulic fracture area is 1139.22 m².

4.5.2. Perforation Position in Limestone

The No.① perforating position is located in the upper part of the limestone layer, 7.5 m away from the coal seam. The No.② perforating position is located in the middle of the limestone layer, 5 m away from the coal seam, and the No.③ perforating position is located at the bottom of the limestone layer, 2.5 m away from the coal seam.

When perforating the upper part of the limestone layer, the hydraulic fracture passes through the interface between the limestone layer and the coal seam after initiation. The half-length of hydraulic fracture in the limestone layer is 16.67 m, the half-length of hydraulic fracture in the coal same is 88.89 m, the hydraulic fracture height is 19 m, and the hydraulic fracture area is 1254.48 m². When perforating the middle of the limestone layer, the hydraulic fracture also passes through the interface between the limestone layer and the coal same after initiation. The half-length of the hydraulic fracture in the limestone layer is 44.44 m, the half-length of the hydraulic fracture in the coal same is 83.33 m, the hydraulic fracture height is 19 m, and the hydraulic fracture area is 1438.52 m². When perforating the bottom of the limestone layer, the hydraulic fracture does not propagate forward after initiation, and the hydraulic fracture area is only 422.22 m².

4.5.3. Perforation Position in Mudstone

The No.① perforating position is located in the upper part of the mudstone layer, 2.4 m away from the coal seam. The No.② perforating position is located in the middle part of the mudstone layer, 4.8 m away from the coal seam. The No.③ perforating position is located at the bottom of the mudstone layer, 7.2 m away from the coal seam.

When perforating in the upper part of the mudstone layer, the hydraulic fracture does not propagate forward after initiation, resulting in a fracture area of only 503.19 m². When perforating in the middle part of the mudstone layer, the hydraulic fracture passes through the interface between the mudstone layer and the coal seam after initiation. The half-length of the hydraulic fracture is 105.56 m, with the hydraulic fracture height of 23 m, and a resulting fracture area of 2399.55 m². When perforating the bottom of the mudstone layer, the hydraulic fracture also passes through the interface between the mudstone layer and the coal seam after initiation. The half-length of the hydraulic fracture is 111.11 m, with the hydraulic fracture height of 24 m, and a resulting fracture area of 2453.03 m².

Based on the parameters in Table 1 and Table 2, the minimum horizontal ground stress in limestone, coal rock, and mudstone is 60.04 MPa, 53.66 MPa, and 57.16 MPa, respectively. The matrix tensile strengths are 5 MPa, 1 MPa, and 4.5 MPa, respectively. Since the hydraulic fractures are vertical fractures, during the expansion process of the fractures, the rock matrix’s tensile strength in the reservoir and the minimum horizontal ground stress need to be overcome. Therefore, when perforating in the coal seam, the hydraulic fractures are restricted to expand within the coal seam; when perforating in limestone, the hydraulic fractures are more inclined to expand into the coal seam; when perforating in mudstone, the hydraulic fractures are also more inclined to expand into the coal seam.

4.5.4. Damage of the Bedding Interface

Figure 15 shows the hydraulic fractures morphology with the bedding interface under the perforation position in the limestone bottom part and the mudstone upper part. After hydraulic fractures initiate, the bedding interface in the coal seam becomes damaged, resulting in the formation of “+” and “T” shaped hydraulic fractures, leading to shorter hydraulic fracture length and smaller hydraulic fracture area.

When perforating in a coal seam, different perforation locations have little effect on the hydraulic fracture morphology and hydraulic fracture parameters. However, in limestone and mudstone layers, the position of perforations significantly affects the hydraulic fracture morphology and parameters. Perforating close to the coal seam (bottom of limestone layer and upper of mudstone layer) makes hydraulic fractures susceptible to being confined by the bedding interface, restricting their propagation. In the process of deep CBM construction in the past, it was usually considered that the physical properties of the roof limestone/floor mudstone layer were poor, which may cause the difficulty of hydraulic fracture initiation and propagation. The simulation results show that the deep CBM reservoir’s roof limestone/floor mudstone layer is not entirely unsuitable for perforation fracturing. When directional roof limestone/floor mudstone layer perforation is used, and the appropriate perforation location is selected, hydraulic fractures can communicate the coal seam to form a roof limestone/floor mudstone layer indirect fracturing.

5. Conclusions

This paper conducts an equivalent modeling of the deep coalbed methane in the Ordos Basin. A finite element model for hydraulic fracture propagation in the complex limestone–coal–mudstone reservoir was established based on the CEPPZ elements. The model parameters were inverted based on field fracturing tests. Finally, the propagation paths of hydraulic fractures were calculated, leading to the following conclusions.

(1)

Hydraulic fractures are more likely to expand into layers with low minimum horizontal stress. Increasing the minimum horizontal stress difference between the coal seam and the limestone/mudstone layer forms a more considerable hydraulic fracture length and fracture area. However, the hydraulic fracture in the coal seam, limestone, and mudstone layers may propagate asynchronously.

(2)

The high injection rate can increase the length of the hydraulic fracture and form a larger area of hydraulic fracture. Increasing the fracturing fluid volume helps the hydraulic fracture expand further. However, the improvement effect on fracture length and area gradually weakens with the increased fracturing fluid volume.

(3)

As the fracturing fluid viscosity increases, the half-length of the hydraulic fracture and hydraulic fracture area increase initially and then decrease. The maximum fracture half-length and area can be obtained when the fracturing fluid viscosity is 50–75 mPa·s.

(4)

Different perforation locations have little effect on the hydraulic fracture morphology and hydraulic fracture parameters when perforating in the coal seam. When perforating in limestone and mudstone layers, perforating close to the coal seam (bottom of limestone layer and upper of mudstone layer) makes hydraulic fractures susceptible to being confined by the bedding interface, restricting their expansion.

(5)

When directional roof limestone/floor mudstone layer perforation is used, and the appropriate perforation location is selected, hydraulic fractures can communicate the coal seam to form a roof limestone/floor mudstone layer indirect fracturing.

(6)

The hydraulic fracture propagation model for deep coal seam gas reservoirs established in this paper can conduct numerical simulations of hydraulic fracture propagation, but it still has certain limitations. These include not considering the changes in reservoir pore pressure caused by the adsorption–desorption behavior of coalbed methane, the interference between clusters during the expansion process of multiple clusters of hydraulic fractures, and the influence of coal seam plasticity. Subsequent research needs to address these issues.