A mathematical model for a hydraulically fractured well in a coal seam reservoir by considering desorption, viscous flow, and diffusion*

This study is an extension of previous works, which develops a mathematical model to simulate the transient performance of a multi-wing fractured well (MWFW) in a coal seam reservoir. Including the characteristics of initial pressure and pore structure of coal seam, the proposed model simultaneously considers the Langmuir isothermal adsorption, Knudson diffusion, and Darcy seepage in coal seam matrix as well as the viscous flow in fracture system. Then, by coupling the seepage flow differential equation in matrix system to that in fracture system, the continuous line-source solution is derived. Finally, the methods of superposition principle, Gauss elimination, and Stehfest numerical inversion are applied to obtain the transient pressure response and production dynamics.

Sensitivity analysis reveals that transient performance is mainly affected by properties of hydraulic fractures, Knudsen diffusion coefficient, and Langmuir volume. The findings of this study can improve our understanding of the well test interpretation and production performance of MWFWs in coal seam reservoirs.