Investigation on nonlinear multi-scale effects of unsteady flow in hydraulic fractured horizontal shale gas wells

A unified mathematical model is established to simulate the nonlinear unsteady percolation of shale gas with the consideration of the nonlinear multi-scale effects such as slippage, diffusion, and desorption. The continuous inhomogeneous models of equivalent porosity and permeability are proposed for the whole shale gas reservoir including the hydraulic fracture, the micro-fracture, and the matrix regions. The corresponding semi-analytical method is developed by transforming the nonlinear partial differential governing equation into the integral equation and the numerical discretization. The nonlinear multi-scale effects of slippage and diffusion and the pressure dependent effect of desorption on the shale gas production are investigated.