A semi-analytical mathematical model for transient pressure behavior of multiple fractured vertical well in coal reservoirs incorporating with diffusion, adsorption, and stress-sensitivity

• Micro-seismic imaging strongly showed multiple fractured vertical wells (MFVW) do exist.
• Little work about pressure performance of MFVW in coal reservoirs was reported.
• Mathematic model of MFVW was established with complex mechanisms.
• Model validation and sensitivity analysis were conducted.

Vertically fractured wells have been widely used in coal gas reservoirs. Investigating transient pressure behaviors (TPB) of these wells is a considerably significant task for fracturing evaluation as well as productivity prediction. Although some studies have been done in this area, most of them focus on two-wing fractured wells. Both micro-seismic fracture imaging and samples testing strongly showed that multiple fractures can develop during hydraulic fracturing in naturally fractured reservoirs like coal seam. However, little work has been done to study the TPB of multiple fractured vertical wells (MFVW) in coal reservoirs.In this paper, to study this problem thoroughly, a well testing model of MFVW at a constant production rate was established with consideration of multiple mechanisms including diffusion, adsorption, and stress-sensitivity effect. Applying line source function, superposition principle, Pedrosa's substation, Gauss elimination, and Stehfest numerical inversion, the transient pressure solution of the mathematical model was solved. Based on this solution, model validation was conducted by selecting a case in literature. In addition, the influences of some critical parameters on TPB were studied, containing fracture length, fracture conductivity, fracture number, fracture length asymmetry, fracture conductivity asymmetry, and fracture angle asymmetry.This work provides important references for some reservoir engineers in performance forecast and completion design of MFVW in coal gas reservoirs.