Anomalous diffusion performance of multiple fractured horizontal wells in shale gas reservoirs

• A model for MFHWs in shale gas reservoirs is presented based on fractional Darcy's law.
• The effects of desorption and anomalous diffusion are considered.
• The resulting seepage equation involves fractional calculus.
• Semi-analytical solution is obtained and pressure behaviors of MFHWs are analyzed.

Shale gas reservoirs are naturally fractured reservoirs which mainly consist of organic matrix and natural fracture system. The disordered distribution of natural fractures results in anomalous diffusion taking place in shale gas reservoirs, and thus the classical Darcy's law is no longer suitable for simulating gas flow under this condition. This paper introduces fractional Darcy's law to establish a novel model for multiple fractured horizontal wells (MFHWs) in shale gas reservoirs with consideration of desorption and anomalous diffusion. As a result the resulting seepage equations are differential equations with fractional calculus. Laplace transform, point source function, numerical discrete method, and Gaussian elimination method are applied to obtain the semi-analytical solution in Laplace space. Duhamel's principle is then applied to consider the effects of the wellbore storage and skin on the pressure responses. Stehfest numerical inversion method is finally used to invert the pressure responses from Laplace space to real space. Type curves of the pressure responses are plotted, and a detailed analysis of the pressure characteristics is presented. The model presented here can be applied to accurately interpret the pressure data of an MFHW in a shale gas reservoir.