Coupled thermo-hydro-mechanical cohesive phase-field model for hydraulic fracturing in deep coal seams

doi:10.1007/s10483-025-3236-7

Abstract

Abstract:

A coupled thermal-hydro-mechanical cohesive phase-field model for hydraulic fracturing in deep coal seams is presented. Heat exchange between the cold fluid and the hot rock is considered, and the thermal contribution terms between the cold fluid and the hot rock are derived. Heat transfer obeys Fourier’s law, and porosity is used to relate the thermodynamic parameters of the fracture and matrix domains. The net pressure difference between the fracture and the matrix is neglected, and thus the fluid flow is modeled by the unified fluid-governing equations. The evolution equations of porosity and Biot’s coefficient during hydraulic fracturing are derived from their definitions. The effect of coal cleats is considered and modeled by Voronoi polygons, and this approach is shown to have high accuracy. The accuracy of the proposed model is verified by two sets of fracturing experiments in multilayer coal seams. Subsequently, the differences in fracture morphology, fluid pressure response, and fluid pressure distribution between direct fracturing of coal seams and indirect fracturing of shale interlayers are explored, and the effects of the cluster number and cluster spacing on fracture morphology for multi-cluster fracturing are also examined. The numerical results show that the proposed model is expected to be a powerful tool for the fracturing design and optimization of deep coalbed methane.

Key words: phase-field method, thermo-hydro-mechanical coupling, indirect fracturing, cohesive zone model, deep coal seam

2010 MSC Number:

O347

Jianping LIU, Zhaozhong YANG, Liangping YI, Duo YI, Xiaogang LI. Coupled thermo-hydro-mechanical cohesive phase-field model for hydraulic fracturing in deep coal seams. Applied Mathematics and Mechanics (English Edition), 2025, 46(4): 663-682.

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Figures/Tables 17

Fig. 1

Fig. 2

Table 1

Model parameters for hydraulic fracturing problem in multilayer reservoir"

Parameter	Symbol	Value		Unit
Parameter	Symbol	Coal rock	Shale	Unit
Young’s modulus of the intact matrix	$E 0$	6	15	GPa
Permeability of the intact matrix	$k 0$	0.01	0.05	mD
Porosity of the intact matrix	$φ 0$	0.05	0.05	–
Young’s modulus of the skeleton grains	$E s$	8	20	GPa
Poisson’s ratio	$ν$	0.3	0.21	–
Fracturing fluid viscosity	$μ$	5		mPa·s
Critical energy release rate	$G c$	500	1 250	N/m
Breaking strength	$f t$	5	12	MPa
Density of porous media	$ρ$	1 500	2 550	kg/m³
Fluid density	$ρ f$	1 000		kg/m³
Density of skeleton grains	$ρ s$	1 700	2 800	kg/m³
Compressibility of the fluid	$c f$	$4 × 10 − 4$		1/MPa

Table 1

Table 2

Fracturing experimental program"

Specimen number	Vertical stress/MPa	Maximum horizontal geostress/MPa	Minimum horizontal geostress/MPa	Displacement/ $(mL ⋅ min − 1)$
1	20		Shale interlayer: 13
		15	Coal seam: 8	100
2	16		Shale interlayer: 13

Table 2

Fig. 3

Fig. 4

Fig. 5

Table 3

Thermodynamic parameters of the heterogeneous non-isothermal coal seams"

Parameter	Symbol	Value		Unit
Parameter	Symbol	Coal rock	Shale	Unit
Thermal expansion coefficient	$α T$	$5.5 × 10 − 6$	$1 × 10 − 5$	1/℃
Thermal conductivity of fluid	$λ f$	0.65		W/(m ·℃)
Thermal conductivity of skeleton grains	$λ s$	1	1.5	W/(m ·℃)
Specific heat capacity of the fluid	$(c p) f$	4 000		J/(kg ·℃)
Specific heat capacity of the skeleton grains	$(c p) s$	1 000		J/(kg ·℃)

Table 3

Table 4

Model parameters of the coal cleats"

Case number	$ϕ 0$	$k 0$ /mD	$G c$ / $(N ⋅ m − 1$ )	$E 0$ /GPa	$f t$ /MPa
1	0.2	20	25	1	0.75
2	0.15	5	75	1.25	1
3	0.1	1	125	1.5	1.25

Table 4

Fig. 6

Fig. 7

Fig. 8

Fig. 9

Fig. 10

Fig. 11

Fig. 12

Fig. 13

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