Numerical modeling of a memory-based diffusivity equation and determination of its fractional order value

Abstract

Conventional diffusion equations for fluid flow through porous media do not consider the effects of the history of rock, fluid, and flow. This limitation can be overcome by the incorporation of “memory” in the model, using fractional-order derivatives. Inclusion of fractional-order derivatives in the diffusion equation, however, adds complexity to both the equation and its numerical approximation. Of particular importance is the choice of temporal mesh, which can dramatically affect the convergence of the scheme. In this article, we consider a memory-based radial diffusivity equation, discretized on either uniform or graded meshes. Numerical solutions obtained from these models are compared against analytical solutions, and it is found that the simulation using properly chosen graded meshes gives substantially smaller errors compared to that using uniform meshes. Experimental data from one-dimensional flow through a porous layer with constant pressure gradient are collected from the literature and used to fit the fractional order in the diffusivity equation considered here. A reasonable value of the fractional order is found to be 0.05; this is further validated by performing numerical simulations to match these experiments, demonstrating substantial improvement over the classical Darcy’s model.