Productivity equation of fractured well in CBM reservoirs

Due to complex process of CBM transport associated with desorption and diffusion, there has not been an explicit and accurate prediction formula of CBM production for fractured wells. This article presented a productivity equation of fractured well in CBM reservoir regarding desorption and diffusion. Elliptical flow pattern exists around the hydraulically fractured well, and the flow field was divided into two regions. One is high-velocity non-linear flow in artificial fracture, and the other is Darcy flow in elliptical region controlled by artificial fracture. Mathematical models for the elliptical gas flow were established based on conservation of mass and momentum equations, in which Langmuir equation, Fick pseudo-steady state law, and function of pseudo pressure were combined to consider effect of desorption and diffusion. The productivity equation of hydraulically fractured well was presented by coupling the analytical solutions to elliptical gas flow. Effect on gas rate of reservoir properties and production parameters such as desorption rate, diffusion coefficient, drawdown pressure, half-length of artificial fracture, and flow conductivity were clarified based on the productivity equation. It is seen that there exist optimal drawdown pressure and optimal half-length of hydraulic fracture dependent on other parameters. In the excess of optimal values, gas rate will reach a plateau and respond with little increment along with the increase of drawdown pressure or half-length. Consequently, the research provides direct insight of the effect of various parameters on gas rate and theoretical foundation for optimization design of CBM development.