Analysis of diffusion process in fractured reservoirs using fractional derivative approach

•Analytical solution for the pressure response of naturally fractured reservoirs (NFR) during transient period.•Analytical solution for the pressure response of NFR during boundary-dominated flow periods, in the light of FC.•Investigating the response of fractured systems on the basis of fractional derivative.•Discussing of the analytical solutions by several synthetic examples.•Developing some standard type curves.

The fractal geometry is used to model of a naturally fractured reservoir and the concept of fractional derivative is applied to the diffusion equation to incorporate the history of fluid flow in naturally fractured reservoirs. The resulting fractally fractional diffusion (FFD) equation is solved analytically in the Laplace space for three outer boundary conditions. The analytical solutions are used to analyze the response of a naturally fractured reservoir considering the anomalous behavior of oil production. Several synthetic examples are provided to illustrate the methodology proposed in this work and to explain the diffusion process in fractally fractured systems.