*2.1. Coal Deformation Equation*

Coal deformation mainly refers to the effective stress acting on the coal skeleton to deform the coal skeleton. In this paper, the double medium model is cited [24]. The stress effect is also affected by gas adsorption and desorption. The effective stress expression is:

$$
\sigma\_{\vec{\text{ij}}}^{\text{eff}} = \sigma\_{\vec{\text{ij}}} - (\alpha\_{\text{m}} \mathbf{P}\_{\text{m}} + \alpha\_{\text{f}} \mathbf{P}\_{\text{f}}) \delta\_{\vec{\text{ij}}} - \alpha\_{\text{s}} \varepsilon\_{\text{s}(\vec{\text{ij}})} \tag{1}
$$

In the formula, σeff ij is the effective stress, MPa; σij is the total stress, MPa; *α*<sup>m</sup> and *α*<sup>f</sup> are the effective stress coefficients of coal pore and fracture, respectively; Pm and Pf are gas pressure in the coal matrix and fracture, respectively, MPa; δij is a Kronecker tensor; *α*<sup>s</sup> is the volumetric strain coefficient caused by gas adsorption and desorption in coal, kg/m3 ; and εs(ij) is adsorption–desorption strain of coal under isothermal condition. Among them, *α*<sup>m</sup> = <sup>K</sup> Km <sup>−</sup> <sup>K</sup> Ks ; *<sup>α</sup>*f<sup>=</sup> <sup>1</sup><sup>−</sup> <sup>K</sup> Km ; <sup>ε</sup>s(ij) <sup>=</sup> aPm Pm+<sup>b</sup> . K is the bulk modulus of coal, MPa; Km, and Ks are the bulk modulus of the coal matrix and skeleton, respectively, MPa.

Equilibrium equation, geometric equation, and constitutive equation constitute the volumetric deformation equation of coal containing gas.

The force balance of each surface of the unit body is analyzed. According to the conservation of momentum, the equilibrium equation is:

$$
\sigma\_{\vec{\text{ij}},\vec{\text{j}}} + \mathbf{F}\_{\vec{\text{i}}} = \mathbf{0} \tag{2}
$$

The coal skeleton undergoes slight deformation, and the geometric equation is:

$$
\varepsilon\_{\vec{\imath}\vec{\jmath}} = \frac{1}{2} \left( \mu\_{\vec{\imath},\vec{\jmath}} + \mu\_{\vec{\jmath},\vec{\imath}} \right) \tag{3}
$$

Assuming that the coal is a linear elastic body, according to Hooke's law, the constitutive equation is:

$$\sigma\_{\vec{\text{ij}}} = 2\mathbf{G} \frac{\mathbf{v}}{1 - 2\mathbf{v}} \varepsilon\_{\text{V}} \delta\_{\vec{\text{ij}}} + 2\mathbf{G} \varepsilon\_{\vec{\text{ij}}} + (\alpha\_{\text{m}} \mathbf{P}\_{\text{m}} + \alpha\_{\text{f}} \mathbf{P}\_{\text{f}}) \delta\_{\vec{\text{ij}}} + \alpha\_{\text{s}} \varepsilon\_{\text{s}(\vec{\text{ij}})} \tag{4}$$

In the formula, G is the shear modulus, MPa; v is Poisson's ratio; and ε<sup>v</sup> is the volume strain. Among them, G = <sup>E</sup> <sup>2</sup> (1+v), E is elastic modulus of coal, MPa.

The change in gas pressure will induce the deformation of the coal matrix and coal skeleton. Gas adsorption and desorption in the coal matrix will also affect the deformation of the coal matrix. Combining the three Equations (1)–(3), based on the dual pore medium model, the governing equation of coal deformation can be obtained:

$$\mathbf{G}\mathbf{u}\_{\mathrm{i,ij}} + \frac{\mathbf{G}}{1 - 2\boldsymbol{\nu}}\mathbf{u}\_{\mathrm{j,ji}} - \boldsymbol{\kappa}\_{\mathrm{m}}\mathbf{P}\_{\mathrm{m},\mathrm{i}} - \boldsymbol{\kappa}\_{\mathrm{f}}\mathbf{P}\_{\mathrm{f},\mathrm{i}} - \boldsymbol{\kappa}\_{\mathrm{s}}\boldsymbol{\varepsilon}\_{\mathrm{s},\mathrm{i}} + \mathbf{F}\_{\mathrm{i}} = \mathbf{0} \tag{5}$$
