# Mathematical Model of Coalbed Gas Flow with Klinkenberg Effects in Multi-Physical Fields and its Analytic Solution

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## Abstract

The deep-mining coal seam impacted by high in situ stress, where Klinkenberg effects for gas flow were very obvious due to low gas permeability, could be regarded as a porous and tight gas-bearing media. Moreover, the Klinkenberg effects had a significant effect on gas flow behavior of deep-mining coal seam. Based on the gas flow properties of deep-mining coal seams affected by in situ stress field, geothermal temperature field and geo-electric field, a new mathematical model of coalbed gas flow, which reflected the impact of Klinkenberg effects on coalbed gas flow properties in multi-physical fields, was developed by establishing the flow equation, state equation, and continuity equation and content equation of coalbed gas. The analytic solution was derived for the model of one-dimensional steady coalbed gas flow with Klinkenberg effects affected by in situ stress field and geothermal temperature field, and a sensitivity analysis of its physical parameters was carried out by comparing available analytic solutions and the measured values. The results show that the analytic solutions of this model of coalbed gas flow with Klinkenberg effects are closer to the measured values compared to those without Klinkenberg effects, and this model can reflect more accurately gas flow of deep-mining coal seams. Moreover, the analytic solution of this model is more sensitive to the change of Klinkenberg factor *b* and temperature grad *G* than depth *h*.

## Keywords

Porous media Klinkenberg effects Multi-physical fields Mathematical model of coalbed gas flow Analytic solution## Nomenclature

*A*Ash of coal

*a*Langmuir volume parameter, m

^{3}kg^{−1}*B*Moisture of coal

*b*Klinkenberg factor, Pa

*c*Langmuir pressure parameter, Pa

^{−1}*C*_{1},*C*_{2}Integral constant

*D*_{0}Gas diffusion coefficient when the pore pressure equals

*p*_{0}*d*Average coal micro-grain size of coal seam, m

- div
Divergence of a vector

*E*Electric intensity

- grad
Gradient of a scalar

*h*Depth, m

*K*_{g}Gas permeability of coal, m

^{2}*K*_{∞}Absolute gas permeability under very high pressure, m

^{2}*k*_{0}^{σ}Value of

*K*_{ σ }(0) when effective stress*σ*equals zero*k*_{0}^{T}Value of

*K*_{ T }(0) when the temperature of coal theoretically equals zero*L*Oblique length of coal seam,

*L*=*h*/sin*θ*, m*m*Methane content reserved in the unit volume coal, m

^{3}m^{−3}*m*_{0}Gas content in coal seam at initial time

*t*_{0}, m^{3}m^{−3}*m*_{a}Absorbed gas content in coal seam, m

^{3}m^{−3}*m*_{d}Desorbable average gas content in coal seam, m

^{3}m^{−3}*m*_{f}Free gas content, m

^{3}m^{−3}**n**Outward unit normal vector on the domain boundary

*p*Gas pressure,

*p*= (*p*_{1}+*p*_{2})/2, MPa*p*_{1},*p*_{2}Gas pressure of top and bottom of coal sample respectively, Pa

*p*_{h}Gas pressure calculated with Klinkenberg effects, MPa

*p*´_{h}Gas pressure calculated without Klinkenberg effects, MPa

*p*_{H}Gas pressure measured, MPa

*p*_{0}Initial gas pressure, MPa

- Δ
*p* Gas pressure difference, MPa

*R*Universal gas constant, J kmol

^{−1}K^{−1}*T*Absolute temperature of coalbed gas, °C

*T*_{0}Temperature of air, °C

*T*_{a}Temperature of coal, °C

- Δ
*T* Geothermal temperature increment

*t*Time, s

*v*Gas flow velocity vector, m s

^{−1}*α*_{1}Klinkenberg effects coefficient

*α*_{2},*α*_{3}Experimental constant

*α*_{4}Experimental constant

*β*Compressibility factor of gas

*β*_{0}Compressibility factor of gas under standard atmosphere, generally equals one

*β*_{a}Compressibility factor when the temperature of coal equals

*T*_{a}*ø*Porosity of coal

*ø*_{0}Porosity of coal without the impact of loads under a norm temperature

*λ*_{1}Bulk compressibility factor of coal

*λ*_{2}Heat swelling factor of coal

*μ*_{g}Gas dynamic viscosity, Pas

*ρ*Gas density, kg m

^{−3}*ρ*_{c}Density of coal, kg m

^{−3}*σ*Effective stress, MPa

*σ*_{1}Axial pressure of three-axis loading, Pa

*σ*_{3}Confining pressure of three-axis loading, Pa

- Γ
In situ stress, MPa

- Δ
*σ* Effective stress increment

*θ*Obliquity of coal seam, °

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