Research and Application of Deep Coalbed Gas Production Capacity Prediction Models

Abstract

The accurate prediction of single-well production performance necessitates considering the multiple factors influencing the dynamic changes in coal seam permeability during deep coalbed methane (CBM) extraction. This study focuses on Block D of the Ordos Basin. The Langmuir monolayer adsorption model was selected to describe gas adsorption behavior, and a productivity prediction model for deep CBM was developed by coupling multiple dynamic effects, including stress sensitivity, matrix shrinkage, gas slippage, and coal fines production and blockage. The results indicate that the stress sensitivity coefficients of artificial fracture networks and cleat fractures are key factors affecting the accuracy of CBM productivity predictions. Under accurate stress sensitivity coefficients, the predicted daily gas production rates of the productivity model for single wells showed errors ranging from 1.89% to 14.22%, with a mean error of 8.15%, while the predicted daily water production rates had errors between 0.35% and 17.66%, with a mean error of 8.68%. This demonstrates that the established productivity prediction model for deep CBM aligns with field observations. The findings can provide valuable references for production performance analysis and development planning for deep CBM wells.

1. Introduction

China’s coalbed methane (CBM) exploration and development is advancing into deeper formations where abundant resources exist. The estimated in-place CBM resources are at depths of 2000–3000 m total 40.71 × 10¹² m³, with recoverable resources of approximately 10.01 × 10¹² m³; among them, 18.47 × 10¹² m³ of these resources are located at depths of 2–3 km, primarily distributed in the Ordos Basin, Junggar Basin, Qinshui Basin, and Bohai Bay Basin [1,2,3,4].

The Ordos Basin, rich in deep CBM resources, has achieved significant milestones after two decades of exploration. The D gas field, for instance, boasts a proven natural gas recovery rate exceeding 75%. In order to accelerate three-dimensional exploration and find new successor layers, Well X-1 was put into operation in March 2023 as the first deep coal and rock gas risk exploration well, with stable and high production for more than 1 year, showing good exploration and development potential. However, exploration outcomes in the Ordos Basin and surrounding areas vary widely. While some deep CBM wells require large-scale hydraulic fracturing and prolonged dewatering to achieve economic yields, others exhibit early gas production with rapid post-fracturing output [5].

The post-fracturing production behavior in deep CBM reservoirs is governed by multiple factors, including stress sensitivity, matrix shrinkage, coal fines production, stimulation methods, gas–water two-phase flow, and production strategies [6,7]. Although the Langmuir equation is used for the characterization of coal-rock gas adsorption and desorption, its applicability needs to be combined with the actual mine due to the monolayer adsorption theory and the assumption of subcritical states. Alternative models, such as the BET multilayer adsorption formula, Freundlich isotherm, D-A equation, and DR equation, have been proposed to address these limitations [8,9,10,11,12].

Permeability changes in deep CBM reservoirs are influenced by adsorption characteristics, cleat fractures, and the stress–strain response of hydraulic fractures [13]. Various models have been developed to describe these dynamics, including the SD permeability model (Shi et al. [14]) and the PM dynamic permeability model (Palmer et al. [15]); however, these models primarily rely on single or partial mechanisms such as effective stress changes and matrix shrinkage, failing to comprehensively reflect the dynamic permeability response under multi-field coupling effects. In contrast to the PM and SD models, this study fully incorporates the interaction mechanisms of adsorption-induced strain and stress, breaking through the limitations of traditional models that focus solely on individual dominant factors. Consequently, it achieves a more comprehensive and accurate characterization of the dynamic evolution of permeability during drainage processes. Other researchers have attempted to establish dynamic permeability models by inverting production data, yet diverse geological and engineering challenges in actual operations make it nearly impossible to obtain production data that accurately represents reservoir permeability changes.

In CBM well productivity forecasting, King [16] derived a production prediction method based on the mass balance equation, while Clarkson and Qanbari [17] introduced the concept of dynamic boundaries to establish a yield prediction model. In addition, some scholars have established more targeted and adaptable mathematical models for different geological models and solved them numerically, such as the dual diffusion–gas–water seepage coupling model, the flow–solid–thermal coupling model, the stress matrix–seepage coupling model, and the thermal flow–solid damage coupling model. However, the accuracy of existing models for deep, low-permeability, and tectonically altered coal seams remains unverified. Moreover, these models take into account few influencing factors and cannot fully characterize the actual adsorption and diffusion of deep coalbed methane and the two-phase flow law in the cut fracture and artificial fracture, and the applicability of coalbed methane production capacity prediction at different depths needs to be verified. With the advancement of intelligent computing and big data, the use of machine learning and neural networks to build mathematical models for predicting gas well productivity has emerged as a new research direction [18,19]. However, these data-driven methods require extensive sample datasets to achieve high prediction accuracy, which limits their broad applicability. In this context, using the deep coal seams in Block D as a case study, this research develops a mathematical model for production prediction that incorporates multiple dynamic effects, including stress sensitivity, matrix shrinkage, gas slippage, and coal fines production and blockage. This physics-based model provides a mechanistic benchmark for data-driven machine learning methods, particularly suitable for scenarios with limited sample data availability. By calibrating the model with actual field production data, a productivity prediction model was established to guide the production of deep coalbed methane wells in practical applications.

2. Geological Setting of the Study Area

The D gas field is located in the northern part of the Yishan slope in the Ordos Basin of China, and the main coal seam is a deep coal seam with a well depth of more than 2000 m, which is generally a gentle monoclinic with a high northeast and low southwest, with an average slope drop of 6~9 m/km and a formation dip angle of 0.3~0.6°. There are 5~10 sets of coal developed longitudinally, of which 5# and 8# coal are the main source rocks and marker layers in the area, which are widely distributed in the basin [20]. The Paleozoic strata within the gas field have a greater burial depth, gentle layers, weak tectonic deformation, and high coal evolution degree. The geological conditions differ to some extent from those of coalbed methane in the eastern margin of the Ordos Basin. The latest geophysical data show that several multi-stage active strike-slip faults have developed in this area, and low-amplitude nasal structures have developed on the gentle slope of the SW trend. The burial depth of the top surface of the coal seam gradually increases from NE to SW, and the buried depth range of the 8# coal seam of the Taiyuan Formation is 2450~2950 m. The porosity of the 8# coal seam is 0.97~8.97%, the average value is 4.80%, the permeability is (0.0001~0.76) × 10⁻³ μm², and the average value is 0.2400 × 10⁻³ μm², which belongs to the ultra-low-porosity–ultra-low-permeability reservoir. The comprehensive logging curve of the study interval is shown in Figure 1.

3. Deep Coalbed Methane Capacity Evaluation Model

3.1. Physical Model and Assumptions

After fracturing, the deep coalbed methane reservoir is a three-stage continuous flow system consisting of a coal matrix, natural cleat fracture network, and artificial fracture network. In the original state, the adsorbed methane gas mainly exists in the coal matrix block rich in micropores, while the formation water is mainly saturated in the natural cleat fractures surrounding the matrix. During development, the reduced reservoir pressure causes methane gas to desorb from the matrix surface and migrate to the adjacent cleat fracture system through diffusion. In the cleat fracture, the desorbed gas and the existing formation water form a gas–water two-phase flow, and percolation occurs on a two-dimensional plane, according to Darcy’s law. Subsequently, these gas–water mixtures from the cleaved fractures are fed into a network of highly conductive artificial fractures (including primary fractures and secondary networks) formed by hydraulic fracturing. In the artificial fracture network, the gas–water two-phase mainly flows in a one-dimensional linear flow along the fracture extension direction pointing to the horizontal wellbore and is finally recovered. The whole flow process is cascade coupling, the outflow (sink) of the upstream region is the inflow (source) of the downstream area, the gas of the matrix desorption and diffusion is the source of the cleat fracture, and the gas–water mixture of the seepage of the cleat fracture is the source of the artificial fracture network. The desorption diffusion and gas–water two-phase flow models of coalbed methane are shown in Figure 2.

Consider the gas–water flow characteristics in the deep coalbed methane reservoir after fracturing and make the following assumptions: (1) there is a two-phase percolation of gas and water in the hydraulic fracture network, which follows Darcy’s law and ignores gravity and the capillary force. In the matrix, there is a vapor phase considering single-phase desorption and diffusion. (2) The cleat fracture and artificial fracture are rigid and incompressible, and the law of conservation of flow between the media is followed. (3) According to the classical equivalent medium theory, cleats and fractures are treated as an equivalent continuous medium, which is homogeneous and isotropic. The matrix blocks and cleat fractures are handled as a dual-porosity, single-permeability system. (4) The thickness of the coal reservoir is uniform, with a one-dimensional flow in the hydraulic fracture and the two-dimensional flow in the cleat fracture. (5) The productivity evaluation model considers the steady-state flow for a certain instantaneous state of the coalbed methane reservoir, which is independent of time.

The steady-state model offers significant practical value in the engineering evaluation, scheme design, and dynamic analysis during the pseudo-steady-state phase due to its simple form and minimal parameter requirements. However, it has limitations in characterizing core mechanisms such as dynamic permeability changes and gas desorption kinetics. Therefore, when constructing the mathematical model, incorporating dynamic permeability variations and gas adsorption–desorption behavior—while combining steady-state with unsteady-state numerical simulation methods—not only accounts for the strong transient characteristics of coalbed methane wells but also simplifies the parameter input and model solution. This integrated approach systematically enhances the comprehensiveness and accuracy of productivity evaluation.

3.2. Mathematical Model Establishment

3.2.1. Mathematical Model of Matrix Capacity Evaluation

The gas supply capacity of deep coal matrices is intrinsically controlled by methane adsorption–desorption behavior, where adsorption isotherm models define the maximum gas storage potential under in situ pressures and temperatures, while desorption kinetics govern the release rate of adsorbed gas into fracture networks during pressure depletion. Elevated confining stress, temperature, and pore pressure in deep reservoirs critically modulate this process: stress compaction reduces matrix porosity and impedes diffusion, temperature suppresses the equilibrium adsorption capacity yet accelerates desorption kinetics, and a high initial pore pressure necessitates significant drawdown to initiate gas release despite providing substantial adsorbed-phase resources. The accurate quantification of these pressure-constrained adsorption–desorption mechanisms is thus critical for predicting coalbed methane productivity.

• Gas adsorption desorption model

In this study, D-8# deep coal samples were selected to carry out isothermal adsorption experiments, and the experimental data were fitted and analyzed by six classical adsorption models, including Langmuir, Freundlich, Langmuir–Freundlich, BET, DA, and DR [8,9,10,11,12], and the applicability of each model to the adsorption/desorption characteristics of deep coalbed methane was systematically evaluated.

According to the deformation fitting equation of different adsorption and desorption models, different parameter combinations are selected to form the horizontal and vertical coordinates, and the experimental isothermal adsorption/desorption data are then plotted and fitted with a linear regression. A model is considered suitable for characterizing the adsorption/desorption characteristics of coal and rock matrices in the target formation when it demonstrates both a (1) strong linear correlation (R² > 0.95) and (2) good fitting performance.

2.

Optimization of adsorption and desorption model of deep coal matrix

Based on the isothermal adsorption experimental data of 8# coal and rock samples in the D block, the applicability of the six adsorption models was evaluated, and the appropriate deep coalbed methane adsorption and desorption models were selected according to the linear fitting effect. Taking samples XH1-5 and XH1-15 as examples, the experimentally measured isothermal adsorption data under the conditions of 85 °C and a pressure range of 0–30 MPa are shown in

Table 1. Isothermal adsorption experimental data for sample XH1-5.

Number	Pressure (MPa)	Gas Density (g/mL)	Adsorption Capacity (m3/t)
0	0	0	0
1	0.5	0.00269	1.38
2	2.0	0.01091	3.87
3	4.0	0.02213	5.61
4	6.0	0.03358	6.79
5	8.0	0.04534	7.55
6	10.0	0.05725	8.00
7	12.0	0.06917	8.28
8	14.0	0.08103	8.46
9	16.0	0.09274	8.57
10	18.0	0.10407	8.62
11	20.0	0.11514	8.67
12	22.0	0.12579	8.60
13	24.0	0.13592	8.62
14	26.0	0.14558	8.61
15	28.0	0.15474	8.54
16	30.0	0.16348	8.53

and

Table 2. Isothermal adsorption experimental data for sample XH1-15.

Number	Pressure (MPa)	Gas Density (g/mL)	Adsorption Capacity (m3/t)
0	0	0	0
1	0.5	0.00271	2.27
2	2.0	0.01095	6.02
3	4.0	0.02215	8.44
4	6.0	0.03360	9.84
5	8.0	0.04536	10.74
6	10.0	0.05725	11.33
7	12.0	0.06917	11.73
8	14.0	0.08108	11.98
9	16.0	0.09272	12.11
10	18.0	0.10412	12.20
11	20.0	0.11522	12.23
12	22.0	0.12590	12.24
13	24.0	0.13598	12.19
14	26.0	0.14564	12.19
15	28.0	0.15481	12.12
16	30.0	0.16341	11.99

. A comparison of the fitting performance of different adsorption models is illustrated in Figure 3, Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8.

Comparative analysis of isothermal adsorption experimental data from samples XH1-5 and XH1-15 demonstrated that the Langmuir adsorption model exhibited a significantly superior fitting performance (R² > 0.98) compared to other models. Consequently, the Langmuir model was selected as the optimal characterization method for subsequent analysis. This validated model was then applied to analyze 22 additional samples from the No. 8 coal seam, with the derived adsorption parameters (Langmuir volume VL and Langmuir pressure PL) systematically presented in

Table 3. Langmuir adsorption model statistical table.

Sample Number	Parameter
	Slope	Intercept	VL (m3/t)	PL (MPa)	R2
XH1-5	0.0811	0.3474	12.39	4.38	0.9998
XH1-7	0.0905	0.1429	14.03	3.27	0.9992
XH1-9	0.8101	5.7542	1.31	8.7	0.9998
XH1-10	0.0749	0.3242	13.41	4.43	0.9998
XH1-14	0.2106	0.8385	4.79	4.18	0.9993
XH1-15	0.0625	0.2031	16.07	3.34	0.9998
XH1-17	0.0852	0.2276	11.82	2.81	0.9995
XH1-18	0.0676	0.2758	14.81	4.11	0.9999
XH1-19	0.1214	0.338	8.29	2.92	0.9996
XH1-22	0.7011	3.2503	1.44	4.86	0.9988
XH1-24	0.3699	1.3588	2.73	3.87	0.9993
XH1-26	0.0609	0.1967	16.47	3.28	0.9999
XH1-27	0.1112	0.3882	9.05	3.61	0.9998
XH1-28	0.0673	0.2284	14.91	3.48	0.9998
XH1-29	0.0669	0.2394	15.02	3.67	0.9999
XH1-30	0.1043	0.4306	9.64	4.25	0.9998
XH1-31	0.0607	0.2295	16.53	3.86	0.9999
XH1-32	0.1066	0.6657	9.5	6.56	0.9991
XH1-33	0.1238	0.6186	8.13	5.16	0.9996
XH1-34	0.1401	0.5385	7.17	3.94	0.9997
XH1-35	0.0671	0.2769	14.96	4.21	0.9998
XH1-36	0.1556	0.7457	6.52	5.11	0.9985

.

3.

Matrix block gas supply capacity equation

The matrix is the storage medium of coalbed methane, and the gas content of ther matrix determines the final gas production. Coalbed methane is transported from the matrix to the cleat fracture by desorption diffusion, the matrix blocks and cleavages of coal seams are considered as a dual-pore and single-permeability system. Each matrix block supplies equal amounts of gas to the cleavages, and the matrix block supplies the same amount of gas as follows:

$$q_{\mathrm{scfg}}={\displaystyle \frac{5184k_{\mathrm{m}}{a}_{\mathrm{m}}\left(p_{\mathrm{m}}-p_{\mathrm{f}}\right)}{B_{gm}{\mu }_{\mathrm{gm}}}}$$

(1)

The pressure at the junction between the matrix and the cleat fracture is continuous, therefore the internal boundary condition is as follows:

$$p_{\mathrm{m}}{|}_{\xi \to 0}=p_{\mathrm{fg}}$$

(2)

The matrix pressure is the average pressure of the reservoir; thus, the outer boundary condition is as follows:

$$p_{\mathrm{m}}{|}_{\xi \to \infty }=\overline{p}$$

(3)

where q_scfg is the reservoir flow rate of gas supplied by each coal matrix block to the cleat fracture, m³/d; k_m is the permeability of the matrix, 10⁻³ µm²; a_m is the size of the matrix block, m; p_m is the average pressure of the matrix, MPa; p_f is the pressure of the cleat fracture, MPa; B_gm is gas volume coefficient, sm³/m³; and μ_gm is the average viscosity of the gas, mPa·s.

3.2.2. Mathematical Model for Productivity Evaluation of Cleat Fractures

In the process of coalbed methane development, the permeability of coal reservoirs is affected by many aspects, including effective stress, the matrix shrinkage effect, gas slip effect, and pulverized coal production and blockage, which leads to the dynamic change in coal reservoir permeability [15,21,22,23]. To accurately characterize these complex interactions, we employ a dynamic permeability model that couples with multiple media flow equations. This integrated approach enables a more reliable productivity evaluation for deep coal reservoirs, particularly in accounting for the stress sensitivity and multi-physics coupling effects unique to these formations.

• Coalbed methane stress sensitivity model

The stress sensitivity of coalbed methane conforms to the attenuation law of power exponential function [24], satisfying the following formula:

$$k=k_i\exp (-cp)$$

(4)

where k is the permeability under known stress conditions, 10⁻³ µm²; k_i is the permeability when the initial effective stress is 0, 10⁻³ µm²; p is the effective stress, MPa; and c is the attenuation coefficient, MPa⁻¹.

Considering the stress sensitivity of the coal seam, as the formation pressure decreases, the change formula of coal seam porosity is as follows:

$$\varphi ={\varphi }_i{e}^{c_p\left(p-p_i\right)}$$

(5)

where $\varphi$ is the porosity of the coal reservoir, decimal; $\varphi$_i is the porosity of the original coal reservoir, decimal; c_p is the porosity compression coefficient of the coal seam, MPa⁻¹; p is the current coal reservoir pressure, MPa; and p_i is the original coal reservoir pressure MPa.

The permeability model under the influence of stress sensitivity is as follows:

$$k=k_i{e}^{c_k(p-p_i)}$$

(6)

where k is the coalbed rock permeability, 10⁻³ µm²; k_i is the original coalbed rock permeability, 10⁻³ µm²; c_k is the coal seam stress sensitivity index, MPa⁻¹; p is the current coal reservoir pressure, MPa; and p_i is the original coal reservoir pressure, MPa.

2.

Coalbed shrinkage effect model

The matrix shrinkage effect refers to the process where the coal matrix shrinks with gas desorbing from the coal seam surface, resulting in the enlargement of the coal fracture space and the corresponding increase in coal reservoir permeability.

The formula for the change in porosity when only the shrinkage effect of the matrix is considered is as follows [25]:

$${\displaystyle \frac{\varphi }{{\varphi }_i}}=1+c_a\left({\displaystyle \frac{p_d}{p_d+p_L}}-{\displaystyle \frac{p_f}{p_f+p_L}}\right)$$

(7)

Considering the effect of matrix shrinkage on porosity, the permeability change formula is as follows:

$${\displaystyle \frac{k}{k_i}}={\left({\displaystyle \frac{\varphi }{{\varphi }_i}}\right)}^3={\left[1+c_a\left({\displaystyle \frac{p_d}{p_d+p_L}}-{\displaystyle \frac{p_f}{p_f+p_L}}\right)\right]}^3$$

(8)

where $\varphi$ is the porosity of the coal seam, decimal; $\varphi$_i is the porosity of the original coal reservoir, decimal; k is the permeability of the coal seam, 10⁻³ µm²; k_i is the permeability of the original coal reservoir, 10⁻³ µm²; c_a is the shrinkage coefficient of the coal seam matrix; p_d is the critical desorbed pressure, MPa; p_L is the Langmuir pressure, MPa; and p is the current coal reservoir pressure, MPa.

3.

Gas slippage effect model

The permeability of cleavages in coal reservoirs is affected by the gas slippage effect. The permeability change formula is as follows [26]:

$$k=k_i(1+{\displaystyle \frac{b}{p}})$$

(9)

where k is the coal seam permeability, 10⁻³ µm²; k_i is the original coal reservoir permeability, 10⁻³ µm²; b is the gas slip coefficient, MPa; and p is the current coal reservoir pressure, MPa.

4.

Coal fines production and migration-induced blockage model

The coal texture is soft and weak, and it is easy to produce coal fines particles under a variety of actions. The equation for the change in permeability under the influence of coal fines production and migration-induced blockage effect is as follows [23]:

$$k_v=k_{\max }\left[1-{\displaystyle \frac{D_{v,\max }{\left(v-v_{cr2}\right)}^n}{v_{0.5}^n+{\left(v-v_{cr2}\right)}^n}}\right]$$

(10)

where k_v is the permeability of the coal corresponding to the flow velocity v in the experimental process, 10⁻³ µm²; k_max is the maximum permeability during the experiment, 10⁻³ µm²; D_v,max is the theoretical maximum permeability damage rate, dimensionless; v is the fluid flow velocity, m/d; v_cr2 is the minimum critical flow velocity when the permeability begins to decrease, m/d; v − v_cr2 is the relative flow velocity of the fluid, m/d; v_0.5 is the relative flow velocity corresponding to 0.5D_v,max, m/d; and n is the permeability damage rate index, dimensionless.

5.

Comprehensive dynamic permeability model of cleat fractures

Considering the comprehensive effects of the stress-sensitive effect, coal matrix shrinkage effect, gas slip effect, and coal fines production/migration-induced blockage effect on deep coal and rock reservoirs, a coupled dynamic permeability model of coal reservoirs under the influence of multiple factors is established as follows:

$$k_{\mathrm{f}}=R_{\mathrm{k}}{k}_{\mathrm{fi}}{\left[{\mathrm{e}}^{\left(-{\displaystyle \frac{C_{\mathrm{k}}}{3}}\left(p_{\mathrm{i}}-p_{\mathrm{f}}\right)\right)}+C_{\mathrm{a}}\left({\displaystyle \frac{p_{\mathrm{d}}}{p_{\mathrm{d}}+p_{\mathrm{L}}}}-{\displaystyle \frac{p_{\mathrm{f}}}{p_{\mathrm{f}}+p_{\mathrm{L}}}}\right)\right]}^3\left(1+{\displaystyle \frac{b}{p_{\mathrm{f}}}}\right)$$

(11)

$$R_{\mathrm{k}}=\left\{\begin{array}{l}1,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}v\le v_{\mathrm{cr}2}\\ 1-{\displaystyle \frac{D_{\mathrm{v},\max }{\left(v-v_{\mathrm{cr}2}\right)}^n}{v_{0.5}^n+{\left(v-v_{\mathrm{cr}2}\right)}^n}},\hspace{1em}\hspace{1em}v>v_{\mathrm{cr}2}\end{array}\right.$$

(12)

6.

Equation of gas–water two-phase flow in cleat fractures

The cleat fracture is the main flow path for the gas and water. The two-dimensional steady-state seepage of the two phases of gas and water is considered in the cleat fractures. Therefore, when the steady-state flow equation is established, the inflow and outflow phases consist of two parts: the x direction and the y direction.

The mathematical equation for the steady-state flow of the gas phase in the cleat fracture is as follows:

$${\displaystyle \frac{\partial }{\partial x}}[{\displaystyle \frac{k_{\mathrm{f}}\left(p_{\mathrm{f}}\right)k_{\mathrm{rg}}\left(s_{\mathrm{fg}}\right)}{{\mu }_{\mathrm{g}}{B}_{\mathrm{g}}}}{\displaystyle \frac{\partial p_{\mathrm{f}}}{\partial x}}]+{\displaystyle \frac{\partial }{\partial y}}[{\displaystyle \frac{k_{\mathrm{f}}\left(p_{\mathrm{f}}\right)k_{\mathrm{rg}}\left(s_{\mathrm{fg}}\right)}{{\mu }_{\mathrm{g}}{B}_{\mathrm{g}}}}{\displaystyle \frac{\partial p_{\mathrm{f}}}{\partial y}}]+{\tilde{q}}_{\mathrm{scfg}}+{\tilde{q}}_{\mathrm{scFg}}=0$$

(13)

The mathematical equation for the steady-state flow of the water phase in the cleat fracture is as follows:

$${\displaystyle \frac{\partial }{\partial x}}[{\displaystyle \frac{k_{\mathrm{f}}\left(p_{\mathrm{f}}\right)k_{\mathrm{rw}}\left(s_{\mathrm{fg}}\right)}{{\mu }_{\mathrm{w}}{B}_{\mathrm{w}}}}{\displaystyle \frac{\partial p_{\mathrm{f}}}{\partial x}}]+{\displaystyle \frac{\partial }{\partial y}}[{\displaystyle \frac{k_{\mathrm{f}}\left(p_{\mathrm{f}}\right)k_{\mathrm{rw}}\left(s_{\mathrm{fg}}\right)}{{\mu }_{\mathrm{w}}{B}_{\mathrm{w}}}}{\displaystyle \frac{\partial p_{\mathrm{f}}}{\partial y}}]+{\tilde{q}}_{\mathrm{scFw}}=0$$

(14)

The gas reservoir is the supply boundary; the outer boundary condition of the cleat fracture is as follows:

$$p_{\mathrm{f}}{|}_{\sqrt{{\mathrm{x}}^2+{\mathrm{y}}^2}=\mathrm{R}}=\overline{p}$$

(15)

The cleat fracture is connected with hydraulic fractures; the inner boundary condition of cleat fractures is as follows:

$$p_{\mathrm{f}}{|}_{(\mathrm{x},\mathrm{y})=\mathrm{Link}}=p_{\mathrm{F}}$$

(16)

where p_f is the pressure in the cleavage, MPa; s_fg is the average gas saturation, dimensionless; μ_g and μ_w are the viscosity of gas and water, respectively, mPa⋅s; B_g and B_w are the formation volume coefficients of gas phase and water phase, respectively, sm³/m³, respectively; k_f is the permeability of cleat fractures, 10⁻³ μm²; $\tilde{q}$_scfg is the gas unit volume flow rate from the matrix to the cleat fracture, d⁻¹; $\tilde{q}$_scFg and $\tilde{q}$_scFw are the unit volume flow rates of gas and water from the cleat fracture to the hydraulic fracture, d⁻¹; t is the time, d; and R represents the radius of the supply boundary of the gas reservoir, m.

3.2.3. Mathematical Model of Evaluation of Artificial Fractures Capacity

• Dynamic conductivity model of artificial fractures in deep coal and rock

During reservoir depletion, decreasing pore pressure leads to increased effective stress, resulting in the stress-sensitive behavior of hydraulic fractures. This phenomenon dynamically alters fracture conductivity, with the permeability–pressure relationship expressed as follows [26]:

$$k{w}_{\mathrm{F}}=k{w}_{\mathrm{Fi}}\cdot e^{-C_{\mathrm{F}}\left(p_{\mathrm{i}}-p_{\mathrm{F}}\right)}$$

(17)

where kw_F is the conductivity of the hydraulic fractures under the current pressure p_F, mD·m; kw_Fi is the conductivity of hydraulic fractures under the pressure p_i of the original coal reservoir, mD·m; C_F is the stress sensitivity index of hydraulic fractures, MPa⁻¹; p_i is the original coal reservoir pressure, MPa; and p_F is the current hydraulic fractures pressure, MPa.

2.

Equation of gas–water two-phase flow in artificial fractures

The gas and water in the hydraulic fractures show a one-dimensional steady state seepage, and the relative permeability needs to be considered in the presence of gas–water two-phase seepage. The mathematical model of the steady-state flow of the gas phase is as follows:

$${\displaystyle \frac{\partial }{\partial \epsilon }}[{\displaystyle \frac{k_{\mathrm{F}}\left(p_{\mathrm{F}}\right)k_{\mathrm{rg}}\left(s_{\mathrm{Fg}}\right)}{{\mu }_{\mathrm{g}}{B}_{\mathrm{g}}}}{\displaystyle \frac{\partial p_{\mathrm{F}}}{\partial \epsilon }}]+{\tilde{q}}_{\mathrm{scFg}}+{\tilde{q}}_{\mathrm{scWg}}=0$$

(18)

The mathematical model of steady-state flow of the water phase is as follows:

$${\displaystyle \frac{\partial }{\partial \epsilon }}[{\displaystyle \frac{k_{\mathrm{F}}\left(p_{\mathrm{F}}\right)k_{\mathrm{rw}}\left(s_{\mathrm{Fg}}\right)}{{\mu }_{\mathrm{w}}{B}_{\mathrm{w}}}}{\displaystyle \frac{\partial p_{\mathrm{F}}}{\partial \epsilon }}]+{\tilde{q}}_{\mathrm{scFw}}+{\tilde{q}}_{\mathrm{scWw}}=0$$

(19)

The end of the artificial seam is closed, and the outer boundary conditions of the artificial seam are as follows:

$${\left.{\displaystyle \frac{\partial p_{\mathrm{F}}}{\partial \epsilon }}\right|}_{\epsilon =\mathrm{Tips}}=0$$

(20)

Considering that the horizontal well is a constant pressure in the entire production process, the inner boundary conditions of the hydraulic fracture are as follows:

$$p_{\mathrm{F}}{|}_{\epsilon =\mathrm{wellbore}}=p_{\mathrm{wf}}$$

(21)

where p_F is the pressure in the artificial seam, MPa; S_Fg is the average gas saturation in the artificial seam, dimensionless; μ_g and μ_w are the viscosity of gas and water, mPa⋅s, respectively; B_g and B_w are the formation volume coefficients of gas and water, sm³/m³, respectively; k_F is the permeability of artificial seams, 10⁻³ μm²; $\tilde{q}$_scFg and $\tilde{q}$_scFw are the unit volume flow rates of gas and water from the c to the artificial fracture, d⁻¹, respectively; $\tilde{q}$_scWg and $\tilde{q}$_scWw are the gas–water unit volume production of horizontal wells, d⁻¹, respectively; Tips is indicate the end of the crack; and p_wf represents the bottom-hole flow pressure, MPa.

According to the structure of the deep coal-rock reservoir and the cascading flow mode of coalbed methane from the coal-rock matrix to the horizontal wellbore, the gas–water two-phase flow equation of the coal seam matrix, cleat fracture, and artificial fracture network was established to establish a productivity evaluation model of deep coal-rock gas. The equation includes the flow term, the source-sink term, and the internal and external boundary condition equations; the time-related compression term is not considered under the steady-state condition, and the capacity evaluation model can be coupled and solved based on the principle of flow conservation.

3.3. Mathematical Model Solution

This study employs a dual-porosity, dual-permeability model where the matrix and cleat systems share the same set of three-dimensional structured grids. Hydraulic fractures are characterized using the Embedded Discrete Fracture Model (EDFM). This method discretizes each fracture into two-dimensional units and couples them with the dual-medium grids through Non-Neighboring Connections (NNC), thereby accurately capturing the fluid exchange between fractures, cleats, and the matrix. The total number of grids in the model is approximately 850,000.

Given the prolonged production period of coalbed methane desorption and the significant matrix–cleat interporosity flow effect, achieving simulation convergence is particularly challenging. Therefore, this study adopts strict convergence criteria: pressure residuals must be below 10⁻⁶, and the gas and water component residuals must be below 10⁻⁵. Additionally, the daily gas production rate at the wellhead must exhibit less than 2% variation over 200 consecutive iteration steps.

Employing the finite volume method, an implicit numerical matrix for gas–water two-phase flow (incorporating boundary condition handling) is constructed. Convergence of the implicit iterative algorithm is contingent upon well-defined initial pressure fields and permeability parameters. The analytical solution is derived by using the source function and the principle of potential superposition, the matrix gas supply is regarded as the surface source, the artificial fracture microelement is regarded as the line source, and the potential contribution of the two to the cleat fracture is superimposed and the pressure equation is constructed. The matrix flow directly adopts the analytical solution, and the channeling relationship is established based on the assumption of uniform channeling. Finally, through the principle of flow conservation (matrix → cleat fracture → the steady-state flow rate of the artificial fracture → the wellbore is equal), all equations are combined to solve the artificial fracture pressure p_F, the cleat fracture pressure p_f, the matrix channeling flow rate q_scfg, and the fracture-to-artificial fracture flow $\tilde{q}$_scFg, forming a complete coupling matrix which provides the key parameters for productivity prediction.

4. Results and Discussion

Through the established capacity prediction model, the production history of typical Well J1 and Well J2 are fitted, and the stress sensitivity index of the target block is corrected by combining the physical property parameters of the reservoir and the fracturing fracture parameters of the well. Through the production verification of typical wells, it is believed that the capacity prediction model constructed in this paper is suitable for deep coal rocks, and on this basis, the production capacity of the other four fracturing horizontal wells in the block is predicted, and the predicted production and bottom-hole inflow dynamic curves of a single well are obtained. At the same time, we conduct a sensitivity analysis on different parameters, and the results can be used as a reference for the fracturing design of new well pressing in the block.

4.1. Typical Well Production Verification

According to the low permeability characteristics of deep coal and rock gas reservoirs, the fracture parameters of the horizontal Well J1 were inverted, the morphological diagram of the artificial fracture network was drawn according to the inversion results, and it was considered that the artificial fracture network of the typical well did not intersect. The formation pressure is 28.82 MPa, the reservoir thickness is 12.4 m, the coal matrix permeability is 0.04 mD, the permeability of the cleat fracture is 0.8 mD, the matrix block size a_m is taken as 0.005 m, and the average gas saturation is 0.4. The fracture parameters of each section are shown in

Table 4. Well J1 inversion artificial fracture parameters.

Well Section	Section Long/m	Main Seam Height/m	Main Seam Half-Length/m	Stitch Density/m	Flow Diversion Capacity/D·cm
J1-1	127	27	145	0.65	24.2
J1-2	95	28	120	0.89	30.5
J1-3	138	32	114	0.31	42.2
J1-4	111	35	158	0.79	38.2
J1-5	133	35	119	0.4	34.2
J1-6	96	35	175	1.05	39.2
J1-7	90	34	112	0.78	28.5
J1-8	96	38	117	0.26	36.6
J1-9	112	32	135	0.54	26.8
J1-10	87	32	141	0.75	35.4
J1-11	94	31	124	0.55	28.5

, and the fracture pattern of the fracturing well is shown in Figure 9.

According to the actual condition of Well J1, the well is divided into 11 sections and 25 clusters.

Gas–water two-phase seepage generally exists in cleavages and hydraulic fractures. The different characterization of the relative permeability model of this well, the conventional relative permeability curve, which is used for hydraulic fractures, and the gas–water relative permeability characteristic described by Corey’s theoretical model are used for cleavage systems, as shown in Figure 10.

Based on the established productivity evaluation model, the model is applied to a typical fractured horizontal well in J1, and its production and pressure field diagrams are obtained. The analysis of the pressure field shows that there are certain differences in the pressure difference between different media, such as the pressure difference between the matrix and the cleat fracture, which is small, while the pressure difference between the cleat fracture and the artificial fracture is large, which is caused by the low permeability of the cleat fracture and the high permeability of the artificial fracture. The pressure at the artificial fracture network is relatively low, close to the flow pressure at the bottom of the well, and the pressure from the cleat fracture to the matrix block gradually increases. The morphology of the pressure drop funnel can also be seen near the horizontal well, as shown in Figure 11.

Selecting production data on days 10, 50, 90, and 140, the model’s predicted daily gas and water production were verified. The model’s predicted production compared to the actual production is shown in Figure 12, and the prediction error values are presented in

Table 5. Well J1 error table between forecasted and actual production.

Production Time (Day)	10	50	90	140
Actual daily gas production (104 m3/d)	2.61	3.57	3.84	3.73
Forecast daily gas production (104 m3/d)	2.80	3.94	4.17	3.98
Error	7.53%	10.75%	8.65%	6.69%
Actual daily water production (m3/d)	144.71	62.03	43.25	29.08
Forecast daily water production (m3/d)	133.46	67.36	43.68	24.78
Error	7.77%	8.6%	1.00%	14.77%

.

According to the verification results, the error range of the daily gas production prediction is 6.69–10.75%, and the error range of the daily water production prediction is 1.00–14.77%. The initial production deviation was significant according to the fitted curve analyzing the productivity prediction results, and it can be found that the fractures near the well are filled with fracturing fluid in the early stage of production, resulting in a high initial water production, and the initial daily water production error predicted by the model is large. In the middle and late stages of production, the flowback and discharge of the fracturing fluid gradually decreased, and most of the fractures near the well were formation water, coal and rock gas, and a small amount of residual fracturing fluid, which met the assumptions of the model, and the prediction results of the model were within the error range.

Similarly, the model’s production prediction performance was validated using another typical well in the block, J2. Fracture parameters for Well J2 were inverted based on actual fracturing data: Well J2 was fractured in 11 stages, generating 24 fractures in total, with detailed results provided in

Table 6. Well J2 inversion artificial fracture parameters.

Well Section	Section Long/m	Main Seam Height/m	Main Seam Half-Length/m	Stitch Density/m	Flow Diversion Capacity/D·cm
J2-1	89	32	143	1.01	27.6
J2-2	90	35	165	1.56	37.5
J2-3	84	30	135	0.97	39.3
J2-4	86	33	149	1.10	35.3
J2-5	94	38	111	0.96	37.5
J2-6	86	34	122	0.98	34.2
J2-7	105	32	135	1.04	32.5
J2-8	91	39	86	0.90	29.7
J2-9	99	38	145	0.40	38.8
J2-10	85	38	137	0.94	29.8
J2-11	94	25	149	0.63	32.2

.

According to the geological data of Well J2, the formation pressure is 28.02 MPa, and the reservoir thickness is 10.6 m. As both wells are situated in the same reservoir formation, identical geological parameters and relative permeability curves were applied, including a coal matrix permeability of 0.04 mD, a cleat permeability of 0.8 mD, the matrix block size a_m is taken as 0.005 m, and an average gas saturation of 0.4. The productivity prediction model was used to forecast the production performance of Well J2, and the results were compared with actual production data. A comparison of daily production rates is shown in Figure 13, and the error analysis between the predicted and actual production is presented in

Table 7. Well J2 error table between forecasted and actual production.

Production Time (Day)	10	50	90	140
Actual daily gas production (104 m3/d)	1.83	2.25	2.20	2.79
Forecast daily gas production (104 m3/d)	1.60	2.29	2.54	2.72
Error	14.22%	1.89%	13.15%	2.33%
Actual daily water production (m3/d)	105.32	52.62	35.5	19.72
Forecast daily water production (m3/d)	92.19	50.06	35.3	23.94
Error	14.24%	5.11%	0.35%	17.66%

.

According to the verification results, the error range of daily gas production prediction is 1.89–14.22%, and the error range of daily water production prediction is 0.35–17.66%.

Based on the error analysis between the predicted and actual production data from both wells, Well J2 exhibits larger errors than Well J1. Nevertheless, both maintain a high prediction accuracy, with average errors of 8.15% for daily gas production and 8.86% for daily water production. These results demonstrate that the deep coalbed methane productivity prediction model established in this study aligns well with field production performance.

4.2. Typical Well Production Forecasts

A coupled productivity evaluation model integrating dynamic geological parameter variations and permeability evolution was developed to predict the gas–water two-phase production performance for four representative horizontal wells (D1, D2, Y1, and Y2) in the target block. The numerical simulation results show that when the stress sensitivity coefficient C_f of hydraulic fractures is 0.2 MPa⁻¹ and the stress sensitivity coefficient of cleat fractures C_k is 0.05 MPa⁻¹, the predicted gas production and water production of Well D1 are 53,121.8 m³/d and 22.9 m³/d, respectively. Well D2 obtains a gas production of 45,527 m³/d and a water production of 19.04 m³/d. Well Y1 produced 41,799.5 m³/d of gas and had a water production of 25 m³/d. Well Y2 produced 28,168.7 m³/d of gas and had a water production of 28 m³/d. The predicted IPR curves of the four wells are shown in Figure 14, Figure 15, Figure 16 and Figure 17.

4.3. Sensitivity Analysis

In actual production, the analysis of IPR curve characteristics can intuitively understand the production status of gas wells, which is the key basis for determining the reasonable downhole flow pressure. By adjusting the bottom-hole flow pressure, the corresponding gas well production can be obtained so as to establish a model for the relationship between production and pressure. When production conditions change, the IPR curve characteristics change accordingly. Numerical simulations were conducted with the following sensitivity parameters.

• Stress-sensitive effects of artificial seams

Figure 18 illustrates the relationship between the flowing bottom-hole pressure and daily gas production rate under different stress sensitivity indices (Cf) of the artificial fracture network. Overall, as daily gas production increases, the flowing bottom-hole pressure corresponding to each Cf demonstrates a declining trend. Specifically, a smaller Cf results in a slower decline rate of the flowing bottom-hole pressure, and a higher daily gas production rate is required to approach a flowing bottom-hole pressure of 0 MPa. When Cf = 0.3, the flowing bottom-hole pressure drops sharply to nearly 0 MPa once the daily gas production exceeds approximately 4 × 10⁴ m³/d. In contrast, when Cf = 0.05, the daily gas production must reach nearly 16 × 10⁴ m³/d for the flowing bottom-hole pressure to approach 0 MPa. This indicates that a lower dimensionless fracture conductivity (Cf) is more conducive to maintaining flowing bottom-hole pressure under high gas production conditions. In other words, for the same flowing bottom-hole pressure requirement, a well with lower stress sensitivity indices of the artificial fracture network can achieve a higher daily gas production rate.

2.

Stress sensitivity effect of cleat fractures

Figure 19 demonstrates the relationship between the flowing bottom-hole pressure and daily gas production rate under different stress sensitivity indices (Ck) of cleat fractures. Overall, as daily gas production increases, the flowing bottom-hole pressure corresponding to each Ck exhibits a declining trend. Specifically, a smaller Ck results in a slower decline rate of the flowing bottom-hole pressure and leads to a higher flowing bottom-hole pressure at the same daily gas production level. When daily gas production reaches approximately 7 × 10⁴ m³/d, the flowing bottom-hole pressure for Ck = 0.2 approaches 0 MPa, whereas for Ck = 0.02, it remains at a relatively high level. This indicates that a lower stress sensitivity index of cleat fractures is beneficial for maintaining bottom-hole pressure during gas production, which contributes positively to the long-term stable production of gas wells.

3.

Average pressure influence

Figure 20 shows the relationship between bottom-hole flowing pressure and daily gas production under different reservoir pressures (P). As production increases, the pressure first decreases and then rises. Lower reservoir pressure (P = 12 MPa) causes a faster pressure decline and earlier rebound (near 0.5 × 10⁴ m³/d). In contrast, higher pressure (P = 28 MPa) results in a more gradual decline, sustained higher pressure levels, and a later rebound (near 7 × 10⁴ m³/d). These results indicate that higher reservoir pressure helps maintain bottom-hole pressure during production.

4.

Effect of average gas saturation

Figure 21 shows the relationship between the bottom-hole pressure and gas production rate under different gas saturations (Sg). As production increases, pressure decreases across all Sg values. Lower Sg (0.2) leads to a faster pressure drop, reaching near 0 MPa at about 1 × 10⁴ m³/d. Higher Sg (0.5) results in a slower decline, maintaining higher pressure until production reaches approximately 12 × 10⁴ m³/d. Higher gas saturation better maintains bottom-hole pressure during production.

5. Conclusions

(1) Based on experimental data from 22 core samples obtained from the 8# coal measure strata (provided by D), we systematically evaluated the applicability of six adsorption models for characterizing methane adsorption behavior in deep coal reservoirs. Among the tested models (including Langmuir, Freundlich, and others), the Langmuir adsorption model demonstrated a superior performance in fitting the experimental data. Consequently, it was identified as the most suitable model for describing methane adsorption characteristics in the target formation and was selected as the preferred model for subsequent analyses.

(2) A comprehensive multi-field coupling dynamic permeability model has been developed for coal reservoir fracture systems, integrating key physical mechanisms to achieve dynamic coupling among the stress–strain field, adsorption-induced swelling field, and seepage field. Validation based on the actual production data from two wells shows that, while Well J2 exhibits slightly larger errors than Well J1, both maintain a high prediction accuracy—with average errors of 8.15% in daily gas production and 8.86% in daily water production—demonstrating strong alignment between the model and field performance.

(3) Using the productivity evaluation model, the stress sensitivity coefficients of cleat fractures and hydraulic fractures were fitted, and the production of three typical wells was predicted. The simulation results show that the constructed deep coalbed methane productivity evaluation model is suitable for steady-state flow, which can predict the instantaneous state of coalbed methane reservoirs and evaluate the production of deep coalbed methane wells.