Influence of Fracture Morphology on Gas Desorption Promotion in Coal by Hot Water Injection

Abstract

The influence of hydraulic fracture morphology on the flow of two-phase fluids and heat transfer characteristics during hot water injection into coal was studied by using a thermal-fluid–solid coupling model. Quantitative analysis was performed on the coal temperature distribution, fracture gas pressure, gas phase saturation dynamics, and total gas output variation. The results show that (1) with the change of fracture morphologies from single to complex combination, the average temperature of the coal body increases significantly. (2) The complexity of the fracture morphology increases the gas pressure of the fracture and expands the high-pressure area. (3) Fractures provide channels for hot water injection and gas migration, resulting in a gradient effect on gas saturation. (4) Gas output increases with the increase in fracture complexity. (5) The effective heat-affected area expands as the hot water injection duration extends and the fracture complexity rises. The results of this research have both vital practical worth and theoretical significance, enabling the execution of technology that integrates hydraulic fracturing with hot water injection to promote gas desorption within coal.

1. Introduction

In recent years, hot water injection technology has received significant attention as an effective method for improving coalbed methane recovery efficiency. , Cheng et al. found that the desorption and diffusion of CBM could be improved by hot water injection. Teng et al. established a thermo-hydro-mechanical model and found that injecting hot water could increase cumulative gas production by 70% over a 30 year period. Zhao et al. demonstrated that gas adsorption capacity and rate decreased at higher temperatures through constant-volume experiments. Some authors − have suggested that the adsorption amount of gas on the coal surface decreased as the temperature increases. Salmachi and Haghighi found that the recovery rate of CBM within 12 years after heat treatment was significantly improved, and the extraction rate was increased by nearly 7 times.

Hydraulic fracturing is an effective technique for enhancing the permeability of coal seams. Liang et al. effectively simulated the hydraulic fracturing process in fractured coal using a 3D coupled model, demonstrating that appropriately increasing the injection rate can enhance fracturing efficiency, while excessive pore pressure should be avoided. Huang et al. found that secondary fractures are more prevalent in high-strength coal seams, displaying a fishbone pattern, while being less developed in low-strength counterparts. Wang et al. demonstrated that high-rank coals with elevated mineral content exhibit more pronounced damage to pores and fractures after hydraulic fracturing. However, the complex fracture network in the coal can affect the fluid flow and heat transfer. Leiting et al. demonstrated that fracture density and angle exert a more pronounced influence on seepage capacity than fracture length and opening degree. Scholars at home and abroad have made many achievements in the study of seepage characteristics of porous media containing fractures. Tecklenburg et al. showed that the mass transfer behavior between the high-permeability fracture area and the low-permeability matrix greatly affects the flow behavior of the fluid. Cheng et al. revealed the complex seepage process of water injection into gas-bearing coal through experimental methods. Zhu et al. demonstrated that the effect of coal deformation on gas flow largely depends on the porosity and permeability of the coal mass. Zhang et al. studied the pore morphology and its impact on methane desorption in water-bearing coal, and explored the gas migration mechanism after water injection into coal seams. Graf and Therrien proposed a discretization method and a general conceptual model of seepage in fractured porous media, focusing on the influence of fractures on the seepage process. Meanwhile, fractures are also crucial in the heat transfer process of porous media, and the rational utilization of fractures can enhance the heat transfer efficiency. Yang et al. compared and analyzed the influence of different fracture types on the temperature change of rock mass and revealed the influence of five different fracture types on the heat transfer capacity. The above studies indicate that fractures are the dominant channels for hot water injection medium to flow in coal, and their morphological distribution has a significant influence on fluid flow and heat transfer. Therefore, the influence of fracture morphological distribution on coal temperature changes and gas desorption–transport characteristics during hot water injection still requires further investigation. This study aims to analyze the characteristics of coal temperature variation and the influence of fracture morphology on gas migration during the process of hot water injection in multifractured coal through numerical simulation.

This study addresses the fundamental scientific challenge of characterizing temperature field evolution in fractured coal during hot water injection and its regulatory mechanisms on gas desorption–migration dynamics. Through an integrated approach combining multiphysics numerical modeling and theoretical analysis, we systematically investigate: (1) by establishing a thermal-fluid–solid coupling numerical model of multifractured coal hot water injection and revealing the influence of different fracture morphologies on the temperature field distribution of coal; (2) the coupled thermo-physical mechanisms governing gas desorption kinetics and multiphase flow behavior under thermal stimulation; and (3) a comprehensive productivity analysis framework to evaluate fracture morphology-dependent gas recovery efficiency. The research results of this paper provide theoretical support for hot water injection technology to enhance coalbed methane production in low-permeability seams, with significant engineering value for improving recovery rates and reducing mining costs.

2. Theoretical Analysis

Based on the governing equations for gas–water two-phase migration fields, temperature fields, stress fields, and auxiliary equations established by previous researchers, and considering the actual conditions of hot water injection into fractured coal samples, the boundary and initial conditions for solving the equations are provided, and a thermo-hydromechanical model suitable for fractured coal samples under thermal effects is constructed.

2.1. Governing Equations

Several assumptions , are applicable to establish the governing equations: (a) coal is an isotropic elastic double-porosity medium; (b) gas and water are insoluble to each other, and their saturation sum is always 1; (c) the injected hot water moves through the fractures, following Darcy’s law, and the water adsorbed in the coal matrix is not considered during the two-phase flow of gas and water; (d) gas migrates through pores and fractures, with diffusion governed by Fick’s law and seepage by Darcy’s law; (e) the gas satisfies the ideal gas equation; (f) the desorption of the gas and the entry of water into the coal sample are instantaneous, and the desorption of the gas follows the Langmuir model.

2.1.1. Gas–Water Two-Phase Migration Field

Considering the capillary pressure factor, the governing equation of the gas–water two-phase migration field of coal is as follows.

2.1.1.1. Gas Diffusion Fields

The adsorption and desorption of gas as a function of pressure and temperature can be described using the Langmuir volume model: ,

$$V_{\mathrm{s}\mathrm{g}}=\frac{V_{\mathrm{L}}{p}_{\mathrm{m}}}{p_{\mathrm{m}}+P_{\mathrm{L}}}\exp \{-\frac{c_2(T-T_0)}{1+c_1{p}_{\mathrm{m}}}\}$$ 1

where V _sg is the adsorbed gas content in the unit coal, m³/kg; V _L is the Langmuir volume constant; P _L is the Langmuir pressure constant; p _m is the gas pressure of the coal matrix, Pa; c ₁ is the pressure constant of gas, 0.07 1/MPa; c ₂ is the temperature constant of gas, 0.02 1/K; T is the temperature after the actual change of coal, °C; and T ₀ is the initial temperature of coal, 30 °C.

Based on Fick’s diffusion law and the law of conservation of mass, the diffusion field equation of gas in a unit volume of the coal matrix is established:

$$\frac{\partial }{\partial t}[{\rho }_{\mathrm{a}}{\rho }_{\mathrm{c}}{V}_{\mathrm{s}\mathrm{g}}+{\varphi }_{\mathrm{m}}\frac{M{p}_{\mathrm{m}}}{RT}]=-\frac{M}{\tau RT}(p_{\mathrm{m}}-p_{\mathrm{f}\mathrm{g}})$$ 2

where ρ_c is the density of the coal seam; ρ_a is the standard-condition gas density, 0.717 kg/m³; ϕ_m is the porosity of the coal matrix; ρ_mg is the density of gas in the coal matrix, 0.717 kg/m³; and τ is time, 9000 s.

2.1.1.2. Gas Seepage Field

Gas seepage in the fracture of the coal body can be represented , as

$$\{\begin{array}{l}\frac{\partial ({\varphi }_{\mathrm{f}}{\rho }_{\mathrm{f}\mathrm{g}}{s}_{\mathrm{g}})}{\partial t}+\mathrm{\nabla }·\left({\rho }_{\mathrm{f}\mathrm{g}}\frac{-k\times k_{\mathrm{r}\mathrm{g}}}{{\mu }_{\mathrm{g}}}\mathrm{\nabla }{p}_{\mathrm{f}\mathrm{g}}\right)-Q_{\mathrm{f}\mathrm{g}}=(1-{\varphi }_{\mathrm{f}})\frac{M}{\tau RT}(p_{\mathrm{m}}-p_{\mathrm{f}\mathrm{g}})\\ {\rho }_{\mathrm{f}\mathrm{g}}=\frac{M\times p_{\mathrm{f}\mathrm{g}}}{R\times T}\\ \begin{array}{l}{k}_{\mathrm{r}\mathrm{g}}=k_{\mathrm{r}\mathrm{g}0}\times {[1-\left(\frac{s_{\mathrm{w}}-s_{\mathrm{w}\mathrm{r}}}{1-s_{\mathrm{w}\mathrm{r}}-s_{\mathrm{g}\mathrm{r}}}\right)]}^2\times [1-{\left(\frac{s_{\mathrm{w}}-s_{\mathrm{w}\mathrm{r}}}{1-s_{\mathrm{w}\mathrm{r}}}\right)}^2]\\ Q_{\mathrm{f}\mathrm{g}}={\rho }_{\mathrm{f}\mathrm{g}}h\frac{-k\times k_{\mathrm{r}\mathrm{g}}}{{\mu }_{\mathrm{g}}}\mathrm{\nabla }{p}_{\mathrm{f}\mathrm{g}}\end{array}\end{array}$$ 3

where ϕ_f is the fracture porosity; ρ_fg is the gas density in the fracture, 0.717 kg/m³; s _g represents the volume fraction of the gas in the fracture; k is the fracture permeability, m²; k _rg represents the relative permeability of gas in fractures, m²; μ_g is the gas dynamic viscosity coefficient; Q _fg is the gas output per unit volume of coal, kg/(m³·s); p _fg is the gas pressure in fractures, Pa; k _rg0 is the relative end-point permeability of gas in fractures, 0.756; s _w is the volume fraction of water in fractures of the coal sample; s _wr is the residual volume fraction of water in fractures of the coal sample, 0; s _gr is the residual volume fraction of gas in fractures of the coal sample, 0.1; and h is the thickness of coal body, m.

2.1.1.3. Water Seepage Field

The flow of water in the fractures of the coal body can be expressed , as

$$\{\begin{array}{l}\frac{\partial ({\rho }_{\mathrm{w}}{\varphi }_{\mathrm{f}}{s}_{\mathrm{w}})}{\partial t}+\mathrm{\nabla }·\left({\rho }_{\mathrm{w}}\frac{-k\times k_{\mathrm{r}\mathrm{w}}}{{\mu }_{\mathrm{w}}}\mathrm{\nabla }{p}_{\mathrm{f}\mathrm{w}}\right)=N_{\mathrm{f}\mathrm{w}\text{\_}\mathrm{i}\mathrm{n}}-N_{\mathrm{f}\mathrm{w}\text{\_}\mathrm{o}\mathrm{u}\mathrm{t}}\\ k_{\mathrm{r}\mathrm{w}}=k_{\mathrm{r}\mathrm{w}0}\times {\left(\frac{s_{\mathrm{w}}-s_{\mathrm{w}\mathrm{r}}}{1-s_{\mathrm{w}\mathrm{r}}}\right)}^4\\ N_{\mathrm{f}\mathrm{w}\text{\_}\mathrm{i}\mathrm{n}}=\frac{Q_{\mathrm{f}\mathrm{w}\text{\_}\mathrm{i}\mathrm{n}}{\rho }_{\mathrm{w}}}{\pi r^2}\\ N_{\mathrm{f}\mathrm{w}\text{\_}\mathrm{o}\mathrm{u}\mathrm{t}}={\rho }_{\mathrm{w}}h\frac{-k\times k_{\mathrm{r}\mathrm{w}}}{{\mu }_{\mathrm{w}}}\mathrm{\nabla }{p}_{\mathrm{f}\mathrm{w}}\end{array}$$ 4

where ρ_w is the density of water, 1000 kg/m³; k _rw is the relative permeability of water in fractures, m²; μ_w is the water dynamic viscosity coefficient; p _fw is the water pressure in fractures, Pa; $N_{\mathrm{f}\mathrm{w}\text{\_}\mathrm{i}\mathrm{n}}$ is the amount of water injected per unit area per unit time at the water injection boundary of the coal body, kg/(m²·s); $N_{\mathrm{f}\mathrm{w}\text{\_}\mathrm{o}\mathrm{u}\mathrm{t}}$ is the output of water per unit area per unit time at the boundary of the gas–water outlet end, kg/(m²·s); $Q_{\mathrm{f}\mathrm{w}\text{\_}\mathrm{i}\mathrm{n}}$ is the water injection flow rate on the water injection boundary of the coal body, 3 mL/min; r is the radius of the coal sample, cm; and k _rw0 is the relative permeability of end of aqueous phase, 1.

2.1.2. Temperature Field

Considering the coupling relationship of temperature variations, heat transfer, and deformation of the coal body, the governing equation for the coal temperature field was established. This equation accounts for coal energy, heat convection, heat conduction, and coal skeleton thermal strain energy while excluding gas adsorption heat and adsorption-induced expansion effects. The expression − is as follows:

$$\{\begin{array}{l}[{\varphi }_{\mathrm{f}}({\rho }_{\mathrm{w}}{c}_{\mathrm{w}}{s}_{\mathrm{w}}+{\rho }_{\mathrm{f}\mathrm{g}}{c}_{\mathrm{g}}{s}_{\mathrm{g}})+{\varphi }_{\mathrm{m}}{\rho }_{\mathrm{m}\mathrm{g}}{c}_{\mathrm{g}}+(1-{\varphi }_{\mathrm{m}}-{\varphi }_{\mathrm{f}}){\rho }_{\mathrm{c}}{c}_{\mathrm{c}}]\frac{\partial T}{\partial t}-\mathrm{\nabla }·(k_{\mathrm{c}}(1-{\varphi }_{\mathrm{m}}-{\varphi }_{\mathrm{f}})\mathrm{\nabla }T)\\ +[{\rho }_{\mathrm{w}}{c}_{\mathrm{w}}\times \mathrm{\Delta }T\times \left(\frac{-k\times k_{\mathrm{r}\mathrm{w}}}{{\mu }_{\mathrm{w}}}\mathrm{\nabla }{p}_{\mathrm{f}\mathrm{w}}\right)+{\rho }_{\mathrm{f}\mathrm{g}}{c}_{\mathrm{g}}\times \mathrm{\Delta }T\times \left(\frac{-k\times k_{\mathrm{r}\mathrm{g}}}{{\mu }_{\mathrm{g}}}\mathrm{\nabla }{p}_{\mathrm{f}\mathrm{g}}\right)]+T{\alpha }_{\mathrm{s}}K\frac{\partial {\epsilon }_{\nu }}{\partial t}=W_{\mathrm{i}\mathrm{n}}\\ W_{\mathrm{i}\mathrm{n}}=N_{\mathrm{f}\mathrm{w}\text{\_}\mathrm{i}\mathrm{n}}{c}_{\mathrm{w}}(T_{\mathrm{i}\mathrm{n}}-T_{\mathrm{i}\mathrm{n}0})\end{array}$$ 5

where c _w, c _g, and c _c represent the specific heat capacity of water, gas, and coal, respectively; k _c is the thermal conductivity coefficient of coal; K is the bulk modulus of coal, GPa, K = D/3­(1 – 2ν); α_s is the thermal expansion coefficient of coal; ε_ν is the volumetric strain of coal; and W _in is the heat quantity of hot water injected on the water injection boundary, W/m.

2.1.3. Stress Field

Considering the fluid pressure in the coal body, the thermal stress generated by hot water injection, and the expansion and contraction deformation of the coal matrix caused by gas adsorption and desorption, the governing equation of the stress field ,− is as follows:

$$G{u}_{i,jj}+\frac{G}{1-2\upsilon }{u}_{j,ji}-{\alpha }_{\mathrm{m}}\mathrm{\nabla }{p}_{\mathrm{m}}-{\alpha }_{\mathrm{f}}\mathrm{\nabla }{p}_{\mathrm{f}}-{\alpha }_{\mathrm{s}}K\mathrm{\nabla }(T-T_0)-K\frac{{\alpha }_{\mathrm{s}\mathrm{g}}{V}_{\mathrm{L}}{P}_{\mathrm{L}}}{{(p_{\mathrm{m}}+P_{\mathrm{L}})}^2}\mathrm{\nabla }{p}_{\mathrm{m}}+\mathrm{\nabla }f=0$$ 6

where G = D/2­(1 + ν) is the shear modulus of coal, Pa; D = 1/[1/E _s + 1/(aK _n)] is the equivalent elastic modulus of the coal matrix, GPa; u is the displacement of the coal body, m; υ is Poisson’s ratio of coal, 1; α_m and α_f are the Biot effective stress coefficients corresponding to pores and fractures, respectively, 1; p _m and p _f are the gas pressure of the coal matrix and the pressure of the gas–water mixed fluid in the fracture, respectively, Pa; α_sg is the adsorption strain coefficient, 0.0156 kg/m³; and f is an external force, N.

2.2. Cross-Coupling Terms

Barenblatt et al. proposed a dual-porosity model and established a porosity equation that accounts for multiple factors, including stress, fluid pressure, coal matrix shrinkage due to gas desorption, and thermal stress. ,, According to the cubic law obtained by Liu et al., the relationship between the fracture permeability and the porosity of the coal is expressed. The expressions for the porosity of the coal matrix, fracture porosity, and fracture permeability of the coal sample are as follows:

$$\{\begin{array}{l}{\varphi }_{\mathrm{m}}=\frac{1}{1+S}[{{\varphi }_{\mathrm{m}}}_0(1+S_0)+{\alpha }_{\mathrm{m}}(S-S_0)]\\ {\varphi }_{\mathrm{f}}={\varphi }_{\mathrm{f}0}-\frac{3{\varphi }_{\mathrm{f}0}}{{\varphi }_{\mathrm{f}0}+\frac{3K_{\mathrm{f}}}{K}}[\frac{{\alpha }_{\mathrm{s}\mathrm{g}}{V}_{\mathrm{L}}{p}_{\mathrm{m}}}{P_{\mathrm{L}}+p_{\mathrm{m}}}\exp (-\frac{c_2(T-T_0)}{1+c_1{p}_{\mathrm{m}}})+{\alpha }_{\mathrm{s}}(T-T_0)-\mathrm{\Delta }{\epsilon }_{\nu }]\\ k=k_0\times {\left(\frac{{\varphi }_{\mathrm{f}}}{{\varphi }_{\mathrm{f}0}}\right)}^3\\ S={\epsilon }_{\nu }+p_{\mathrm{m}}/K_{\mathrm{s}}-{\alpha }_{\mathrm{T}}T-\frac{{\alpha }_{\mathrm{s}\mathrm{g}}{V}_{\mathrm{L}}{p}_{\mathrm{m}}}{P_{\mathrm{L}}+p_{\mathrm{m}}}\exp (-\frac{c_2(T-T_0)}{1+c_1{p}_{\mathrm{m}}})\\ S_0={\epsilon }_{\nu 0}+p_0/K_{\mathrm{s}}-{\alpha }_{\mathrm{T}}{T}_0-\frac{{\alpha }_{\mathrm{s}\mathrm{g}}{V}_{\mathrm{L}}{p}_0}{P_{\mathrm{L}}+p_0}\exp (-\frac{c_2(T_0-T_0)}{1+c_1{p}_0})\\ p_{\mathrm{f}}=s_{\mathrm{g}}{p}_{\mathrm{f}\mathrm{g}}+s_{\mathrm{w}}{p}_{\mathrm{f}\mathrm{w}}\\ p_{\mathrm{f}\mathrm{w}}=p_{\mathrm{f}\mathrm{g}}-p_{\mathrm{e}}\end{array}$$ 7

where ϕ_m0 is the initial porosity of matrix; ϕ_f0 is the initial porosity of fracture; K _f is the improved fracture stiffness, 4.5 GPa; K _s is the bulk modulus of the coal skeleton, 8.1 GPa; k ₀ is the initial permeability of fracture; ε_ν0 is the initial volumetric strain of coal; p ₀ is the initial gas pressure, 0.5 MPa; and p _e is the capillary inlet pressure, 0.02 Pa.

The above seven equations constitute a thermo-hydromechanical (THM) coupling mathematical model for fractured coal samples under thermal stimulation (Figure ), with the corresponding solution conditions established to address specific problems under the thermal action.

1.

THM coupling diagram of fractured coal under thermal stimulation.

3. Model Description and Model Validation

3.1. Model Description

During the hydraulic fracturing process, the injected pressurized water splits and penetrates the fractures within the coal body, resulting in a series of changes, including fracture expansion and widening within the coal structure as well as reorganization of the channel network. This leads to a continuous expansion of the seepage space in the coal body, which is manifested in the constant change of the permeability coefficient of the coal body. Based on this, the high permeability of hydraulic fractures and main fluid seepage channel characteristics were simulated by quantifying the fracture complexity with the fractal dimension. It is equivalent to the actual hydraulic fracturing in terms of hot water injection stimulation effect and can be used as a simplified alternative method for laboratory research on the stimulation mechanism of hydraulic fracturing.

Figure shows the process of the coal sample processing and prefabricated fractures. The coal samples were obtained from Coal Seam #3 at Zhangcun Coal Mine, Shanxi Province, located at a depth interval of 400–500 m below the ground surface. The study area in the Zhangcun Mine contains anthracite, with fundamental characterization parameters of the coal matrix presented in Table . The coal samples were initially cored parallel to the bedding plane using a rock coring machine to obtain cylindrical samples of ϕ 50 mm × H 100 mm, which were subsequently processed into three dimensions through cutting and polishing: ϕ 50 mm × H 25 mm, ϕ 50 mm × H 50 mm, and ϕ 50 mm × H 100 mm. Following fracture generation using the Brazilian splitting method, five standardized cylindrical coal samples were fabricated by assembling coal samples of varying heights (Figure e) and bonding them with electrical insulation tape (Figure f). The experiments were conducted using a custom-developed true triaxial seepage system under constant axial pressure (6 MPa) and confining pressure (4 MPa), involving initial gas desorption and seepage tests at 30 °C followed by hot water injection (70 °C at 6 mL/min), with continuous monitoring of temperature and fluid production throughout the process.

2.

Coal sample processing and prefabrication fracture process.

1. Basic Parameters of Coal Samples.

		proximate analyses (%)			mineral contents (%)
coal rank	R o,max (%)	M ad	A ad	V ad	total	clay	carbonate
anthracite	2.34	2.15	10.26	12.27	96.34	2.65	1.13

The laboratory testing conditions are limited, so it is impossible to monitor the temperature change and gas–water flow in the coal after hot water injection. Since the directions of fluid flow and heat transfer by hot water are mainly along the single-fracture face prefabricated by Brazilian splitting and the vertical fracture face formed during coal sample combination, perpendicular to the flow direction, the laboratory test coal sample was simplified to a two-dimensional plane model for numerical calculation feasibility and effectiveness. , The laboratory standard cylindrical specimens were simplified into two-dimensional rectangular coal samples, and the three-dimensional fractures were simplified into two-dimensional rectangular fractures. The two red points in Figure indicate measurement positions for both laboratory tests and numerical calculations. Figure shows the numerical calculation models of four coal samples with fractures. The model dimensions are 100 mm (width) × 51 mm (height), where the dark central area represents the high-permeability area and the light area represents the low-permeability area.

3.

Comparison diagram between the laboratory test coal sample and the numerical model.

4.

Physical model of coal samples with different combined fracture morphologies.

3.2. Simulation Plan and Parameter Settings

The modeling scheme for investigating the impact of hot water injection on gas–water seepage in coal is given in Table . Under the condition of a single fracture, the displacement effect of the coal body injected with hot water was compared by changing the number of vertical fractures to construct different fracture distribution patterns.

2. Numerical Simulation Scheme of Hot Water Injection.

number of single fractures/strip	number of vertical fractures/strip	water injection temperature/°C
1	0	70
1	1	70
1	2	70
1	3	70

All of the parameters of coal in the numerical simulation are obtained from the experimental results of the same samples of the 3# coal seam in Zhangcun Coal Mine in Shanxi Province. Other parameters required for numerical simulations are derived from other scholars’ works in the same research field. , The simulation parameters are listed in Table .

3. Physical Parameters of Numerical Simulation of Hot Water Injection.

symbol	parameter	value	unit
ρc	density of coal	1300	m3/kg
E s	elastic modulus of the coal skeleton	3740	MPa
ϕm0	initial porosity	0.05
k 0	initial permeability	6 × 10–11	m2
c w	specific heat capacity of water	4200	J/(kg·K)
M	molar mass of CH4	16	g/mol
V L	Langmuir volume constant of CH4	0.02778	m3/kg
P L	Langmuir pressure constant of CH4	1.116	MPa
μg	dynamic viscosity coefficient of CH4	1.5 × 10–5	Pa·s
R	ideal gas constant	8.31441	J/(mol·K)
c g	specific heat capacity of gas	2160	J/(kg·K)
k c	the thermal conductivity coefficient of coal	0.478	W/(m·K)
αs	thermal expansion coefficient of the coal skeleton	1 × 10–7	1/K
c c	specific heat capacity of coal	1350	J/(kg·K)

3.3. Boundary Setting and Initial Conditions

Figure shows the initial conditions and boundary conditions.

5.

Initial conditions and boundary conditions.

3.3.1. Transport Field

The initial time is filled with gas to achieve an adsorption equilibrium. The initial pore pressure is 0.5 MPa, and the initial water saturation is 0. The mass flux boundary for hot water injection is set at the left boundary of the physical model, and hot water is injected into the physical model at a flow rate of 6 mL/min. The right boundary of the physical model is defined as the fluid outlet boundary of the outflow of gas and water, and the outlet boundary pressure is set to 13 kPa. The other boundaries are no-flux boundaries. The initial and boundary conditions of the gas diffusion field are essentially the same as those of the seepage field. To ensure the continuity of the high- and low-permeability zones at the contact boundary, the variable values on their inner common boundary are set equal. In the Darcy seepage field, the boundary pressure of the high-permeability area is set to be equal to that of the adjacent low-permeability area.

3.3.2. Temperature Field

The initial temperature is 30 °C. The left boundary of the physical model is set as a temperature boundary with a value of 70 °C, which is the temperature of the injected hot water. The right boundary of the physical model is set to the outflow boundary. The other boundaries are set as thermally insulated boundaries. Since the high-permeability area is adjacent to the low-permeability area, it is necessary to set the temperature values of the two calculation regions at the contact boundary of the two to be equal. The same temperature value was set at the contact boundary between the high-permeability area and the low-permeability area.

3.3.3. Stress Field

The initial displacement of the physical model is zero. A 6 MPa axial pressure is applied to the model’s left/right boundaries, with 4 MPa confining pressure on the top/bottom boundaries.

3.4. Model Validation against Experimental Data

Figure demonstrates that the numerical calculation results of fractured coal sample model F1 agree with the experimental results at the same monitoring point. The temperature at the monitoring point gradually rises with the increase in hot water injection time.

6.

Variation of the temperature of coal sample monitoring points with time under the two methods.

Moreover, the maximum temperature difference between the two research methods at the monitoring point is 2.7 °C, and the maximum error between the two data sets is less than 10%. This suggests that the mathematical model is reasonable and can be applied to subsequent calculations and analyses.

4. Results and Analysis

4.1. Analysis of the Displacement Effect in the Process of Hot Water Injection

The displacement effect of hot water injection in the fractured coal sample model F2 can be analyzed from the temperature distribution, gas pressure distribution, and gas phase saturation.

As shown in Figure , the high-temperature area gradually spreads from the left boundary to the right over time. The injected hot water gradually penetrates into the coal body, and the heat it carries transfers from the hot water injection boundary to the surrounding coal. At the beginning of hot water injection, the influence distance of temperature changes significantly, but with the increase of time, the growth rate of the influence distance gradually decreases. At 9000 s, the thermal boundary front of the temperature reached 70 mm. As hot water injection continues, the coal temperature becomes uniformly distributed, demonstrating that hot water effectively elevates temperatures throughout the coal mass.

7.

Temperature distribution of coal during the process of hot water injection.

As shown in Figure , gas pressure in the fractures fluctuates during hot water injection. The pressure wave propagates through coal fractures and gradually spreads rightward. At 300 s, the gas pressure front in the coal fractures reached 60 mm. At 1800 s, the area affected by gas pressure almost covered the entire coal body. When the hot water injection duration is below 5400 s, the gas pressure in the fracture changes significantly. With continued hot water injection, the gas pressure variation in the fractures slows. Especially after 6000 s, the gas pressure rises due to the heat energy from the hot water. Because the hot water injection pressure exceeds the initial coal gas pressure, the pressure peak migrates toward the gas–water outflow boundary. With time, hot water injection leads to the gas pressure being evenly distributed.

8.

Gas pressure distribution in coal fractures during the flooding process of hot water injection.

Figure demonstrates significant changes in gas saturation distribution within coal during hot water injection. In the initial stage, gas saturation is high. As hot water injection continues, gas saturation gradually diffuses to the right, indicating that the gas in the coal is gradually displaced by hot water, causing the gas phase saturation to decrease. Finally, gas saturation in the coal tends to be uniformly distributed, but the overall saturation decreases, suggesting that gas saturation in the coal changes dynamically during hot water injection.

9.

Distribution of gas saturation in coal during hot water injection.

4.2. Temperature Variation Characteristics with Different Fracture Patterns

The hot water injection effect in coal samples with different fracture distribution patterns was analyzed based on the temperature distribution under identical hot water injection pressures.

As shown in Figure , the temperature distributions of coal with different fracture morphologies present different characteristics. The fractured high-permeability zone serves as the dominant flow channel during hot water injection. Heat exchange between hot water and coal occurs during the hot water flow within this zone, causing the temperature of the coal near the high-permeability fracture to rise continuously. The high-temperature zones of each coal sample are mainly concentrated near the hot water injection boundary on the left and the high-permeability zone of fractures. Hot water injection causes the temperature distribution range to expand into the coal sample along the hot-water seepage direction in fractures and the directions on both sides of the fractures. Consequently, the temperature of the coal sample rises and the heat-affected area expands.

10.

Coal temperature distribution at 9000 s under different fracture distribution patterns: (a) F1, (b) F2, (c) F3, (d) F4.

Under different fracture morphologies, the distribution and diffusion ranges of coal temperatures vary significantly. As shown in Figure , with a constant number of individual fractures, increasing vertical fractures enlarge both the thermal influence range and temperature rise in coal samples during hot water injection. This demonstrates that fracture networks effectively expand the heat-affected area, elevating average coal temperatures and enhancing adsorbed gas desorption within this region.

4.3. Gas Pressure Characteristics in Fractures of Different Fracture Patterns

As shown in Figure , the fracture distribution pattern in coal samples becomes increasingly complex from left to right with significant variations in gas pressure distribution within the fractures.

11.

Gas pressure distribution in fractures at 9000 s under different fracture distribution patterns: (a) F1, (b) F2, (c) F3, (d) F4.

In Figure a, the fracture distribution is relatively simple with only a single fracture and the gas pressure within it is low. With the complexity of fracture distribution and morphology, the high-pressure area gradually appears and expands. In particular, in Figure c,d, the high-pressure areas (red and yellow areas) are significantly increased and more widely distributed, indicating that the gas pressure in the fractures increases with an increase in the number of vertical fractures.

The injection of hot water raises the local temperature, promoting the desorption of adsorbed gases in the coal. Due to the local increase in gas pressure caused by hot water injection, a localized high-pressure zone begins to appear in Figure b. As shown in Figure c,d, as the fracture distribution becomes further complicated, the local high-pressure area gradually moves away from the hot-water injection boundary. This indicates that the more complex the fracture morphology distribution, the greater the impact on fracture gas pressure over a larger area.

The distribution of the fracture morphology directly affects the formation and distribution of high-pressure gas zones in fractures. In coal with a complex fracture morphology, the gas pressure in the fracture is more likely to be concentrated, forming a high-pressure area. Enhanced fracture connectivity facilitates the flow and accumulation of gas in the fracture, resulting in the formation of high-pressure zones in specific areas. Fracture morphology distribution determines the gas pressure distribution in fractures. The more complex the fracture morphology, the more obvious the formation and distribution of high-pressure zones.

4.4. Characteristics of Gas Saturation with Different Fracture Patterns

As shown in Figure , fractures provide channels for hot water injection and gas migration, causing a gradient effect in the spatial distribution of gas saturation. Due to the thermal desorption effect of hot water injection, the adsorbed gas desorbs and migrates from the coal. The gas saturation at the hot water injection boundary and its nearby area decreased at 9000 s. In areas where there are no fractures and away from the hot water injection boundary, the gas saturation tends to remain high and uniform, indicating that gas migration is unrestricted. However, in the fracture, the gas saturation distribution is nonuniform. The fracture provides a path for gas migration from the high-pressure region to the low-pressure region, causing a decrease in gas saturation in the fracture.

12.

Distribution of gas saturation at 9000 s under different fracture distribution patterns: (a) F1, (b) F2, (c) F3, (d) F4.

The complex morphological distribution of fractures can enhance the propagation and connectivity of fractures, boost the migration capacity of the desorbed gas, and lead to the redistribution of gas saturation over a wider range. As shown in Figure , the gas saturation near the fracture region is low, which may be due to gas migrating along the fracture to other regions.

4.5. Variation of Total Gas Output and Average Temperature with Time under Different Fracture Distribution Patterns

Fracture morphologies significantly affect the temperature changes of coal. As shown in Figure , with the fracture morphology of the coal sample becoming more complex (from F1 to F4), the average temperature rose gradually from the initial 30 °C to 63.240 °C, 64.960 °C, 66.128 °C, and 67.124 °C at 9000 s, showing a gradual upward trend. Among these, the fractured coal sample model F4 shows the largest average temperature increase. This indicates that compared with a single-fractured coal sample, the fracture morphology of the combined-fractured coal sample is more complex. This can effectively enlarge the heat exchange area of the coal body, greatly enhance the heat transfer effect after hot water injection, transfer more heat to the coal body, and then improve the efficiency of influencing the temperature field so that the coal temperature can be greatly increased.

13.

Average temperature of coal samples with different fracture patterns at 9000 s.

Figure shows the variation in the total gas output over time for coal samples with different fracture geometries under thermal stimulation. As shown in Figure , model F4 exhibits the highest gas output among the fractured coal samples, approximately 0.079 cm³, and reaches a steady state rapidly after about 1000 s. These results indicate that the temperature and pressure conditions in model F4’s fracture network most effectively promote gas desorption and diffusion. In contrast, fractured coal sample F3 has the second highest gas output, stabilizing at about 0.069 cm³ eventually. It also reached a steady state after approximately 1000 s but slightly lower than that of the model F4. The gas output of the F2 and F1 models is relatively low, stable at about 0.059 cm³ and 0.058 cm³, respectively, and both reach a steady state after about 1000 s, with F2 slightly higher than F1.

14.

Total gas output of coal samples with different fracture patterns under thermal action.

These four models exhibit a common trend: within the first 1000 s, the gas production increases rapidly and then stabilizes gradually. This suggests that the gas desorption and diffusion rates are relatively fast at the beginning of the experiment and that the production rate gradually decreased over time until reaching a steady state. From the perspective of total gas output, it can be seen that the order of fracture morphology effectiveness in the four coal sample models is F4 > F3 > F2 > F1. The model F4 has the highest gas production efficiency, while model F1 has the lowest. The total gas output of F4 is 13.62% of that of F1.

The temperature variations in coal samples with different fracture morphologies demonstrate that hot water flow through more complex fracture networks enhances heat transfer to the coal matrix, increasing its temperature and promoting adsorbed gas desorption through enhanced heat absorption.

5. Discussions

Referring to the laws established by predecessors and integrating previous experimental results, the area where the temperature along the hot water injection boundary exceeds 50 °C after hot water injection can be approximated as the effective heat-affected area. In this area, due to the desorption of a large amount of adsorbed gas after it absorbs heat, the permeability of the coal increases and the gas production also rises. Table shows the effective heat-affected area of the coal sample model with different fracture distribution morphologies during the hot water injection process. In this study, the influence of hot water injection on gas desorption in fractured coal is characterized by the effective heat-affected area S, and the number of vertical fractures N was used to characterize the complexity of fracture morphology.

4. Effective Heat-Affected Area by Injecting Hot Water under Different Fracture Morphologies.

	the effective heat affected area/cm2
time/s	F1	F2	F3	F4
1000	2.39	2.55	2.56	6.56
3000	15.09	20.90	23.00	28.56
5000	34.62	39.96	46.25	51.00
7000	51.00	51.00	51.00	51.00
9000	51.00	51.00	51.00	51.00

As shown in Figure , under the same hot water injection conditions, the effective heat-affected area exhibits an exponential relationship with injection time t. The effective heat-affected area has an exponential function relationship S = at ^b with injection time t. The growth rate of the effective heat-affected area varies across models, with model F4 showing the fastest rate. As shown in Figure , coefficient a follows an exponential relationship with the number of vertical fractures, while coefficient b shows a linear relationship with the number of vertical fractures, i.e., S = (A e^–N/B + C)t ^{1.558–0.112N} , 0 ≤ N ≤ 3. Analyzing the relationship among the effective heat-affected area, the time of hot water injection, and the number of vertical fractures can guide the development and implementation of hot water injection technology in hydraulic fracturing.

15.

Variation curves of the effective heat-affected area of coal samples with different fractures.

16.

Relationship between coefficients and the number of vertical fractures.

Although the simplified 2D rectangular fracture model effectively captures the dominant patterns of temperature distribution and gas desorption, it is imperative to acknowledge its inherent limitations: (1) the actual three-dimensional hydraulic fractures exhibit preferential flow channels and terminal branching, which significantly influence fluid seepage and heat transfer in coal matrices, whereas the idealized vertical fracture morphology overlooks both the tortuosity and anisotropy of hydraulic fracture networks, potentially leading to overestimated thermal transport efficiency along fracture pathways. (2) The two-dimensional model fails to account for fracture aperture variation and three-dimensional fluid migration effects, which are critical for accurately assessing the thermal influence zone of hot water injection. The model in this study primarily characterizes fracture complexity, with future work employing three-dimensional fracture networks to enhance model fidelity and enable comprehensive analysis.

6. Conclusion

The results demonstrate that complex fracture networks enhance coal-bed methane production during thermal stimulation. These findings are derived from homogeneous coal samples containing idealized fracture morphologies and differ from the actual thermal performance in hydraulically fractured coal seams. The main conclusions are as follows:

• 1.As fracture morphology changed from single to complex combinations, the average coal temperature increased significantly. From F1 to F4, the average temperature of coal samples increased from 30 °C to 63.240 °C, 64.960 °C, 66.128 °C, and 67.124 °C. The complex fracture morphology of the coal sample with combined fractures can greatly increase the heat exchange area of the coal, improve the heat transfer effect, and greatly increase the temperature of the coal.
• 2.With the complexity of fracture distribution and morphology, the gas pressure in the fracture increases and a high-pressure area gradually appeared and expanded. The fracture gas pressure in a single-fracture coal sample was low, whereas that in a multifractured coal sample had a significantly increased high-pressure area with wide distribution. The more complex the fracture morphology, the farther the local high-pressure area was from the boundary of hot water injection, and the more likely it was to form a high-pressure area.
• 3.The fracture provided a channel for hot water injection and gas migration, causing the gas saturation to present a gradient effect. At the hot water injection boundary and the adjacent area, the gas saturation decreased. In the fracture region, the distribution of gas saturation was nonuniform and low. The complexity of the fracture morphology enhanced the migration ability of the gas, leading to the redistribution of gas saturation in a larger range.
• 4.The total gas output increased significantly with the increase of fracture morphological complexity. At 9000 s, the fractured coal sample model F4 had the highest gas output, about 0.079 cm³, which was 13.62 times that of the model F1. Generally, gas output increased rapidly initially and then stabilized. Moreover, the more complex the fracture morphology, the higher the gas production efficiency.
• 5.The effective heat-affected area increases significantly with the growth of hot water injection time and fracture complexity. The area of the effective heat-affected area has an exponential relationship with the time of hot water injection, and the more complex the fracture morphology, the faster the growth rate of the effective heat-affected area. The effective heat-affected area of the fractured coal model F4 increased the fastest, indicating that the complex fracture morphology can more effectively transfer heat to the coal by enhancing the penetration and heat transfer of hot water, expanding the thermal influence range, and improving the gas desorption efficiency.

Data will be made available on request.

Yan Li: Writingoriginal draft. Weiji Sun: Conceptualization. Bing Liang: Supervision. Xiaoyang Zhang: Visualization. Xintao Chen: Validation.

The authors declare no competing financial interest.