A modified memory-based mathematical model describing fluid flow in porous media

This study presents a modified memory-based mathematical model describing fluid flow in porous media. For the first time such model is derived employing the Grünwald-Letnikov (G-L) definition of the Riemann-Liouville (R-L) time fractional operator via a generalized Darcy's equation. The proposed mathematical model is suitable to describe the anomalous diffusion behavior observed in a medium of fractal geometry, as well as in disordered and highly heterogeneous porous media. A numerical scheme based on existing discretization method is employed to handle the modified memory-based mathematical model. The accuracy of the numerical model is validated through the analytical solution for a simplified problem. In addition, numerical experiments are presented to demonstrate the effect of the anomalous diffusion exponent on the predicted reservoir pressure, wellbore pressure, and volumetric flux considering the flow of an under-saturated oil in a hydrocarbon reservoir as an example. Results show that decrease in magnitude of the anomalous diffusion exponent results in larger pore and wellbore pressure. This study would open a door to apply the G-L interpretation of the time fractional operator in numerical modeling of fluid flow through porous media.