The Use of Implicit Evolution Equations with Generalized Mittag-Leffler Kernels to Elucidate a Connection Between Distributed Microstructure Models in Porous Media and Fractional Calculus
Ian Turner, Patrick Perré · Multiscale Modeling and Simulation · 2025
Abstract. Multiscale flow in heterogeneous porous media finds application across numerous fields of science and engineering. In this work, we first review the well-known averaged-parameter and distributed microstructure modeling frameworks for simulating diffusive transport phenomena in a porous (binary) medium comprised of a connected phase surrounding disconnected inclusions. We show that while the averaged-parameter model performs well when the diffusivity ratio of the inclusion to that of the surrounding phase is large, it fails to describe the diffusive transport phenomena for small diffusivity ratios. The distributed microstructure model, on the other hand, produces a more realistic representation of the transport behavior over all diffusivity ratios. However, this class of models is computationally demanding because of the need to compute over the macroscopic and microscopic scales concurrently. By representing the integral exchange term appearing in the macroscopic averaged equation as a convolution integral with a kernel describing the fading memory effects, it is well known that the fully coupled dual-scale model reduces to a single implicit evolution equation. We explore a connection between the dual-scale modeling framework and fractional calculus through the choice of two specific kernel functions. The first is a power-law function used with good success in the modeling of subsurface hydrology, and the second is the ubiquitous Mittag-Leffler function. Both choices produce implicit evolution equations involving time-fractional operators; however, it is only the generalized Mittag-Leffler function kernel that is capable of capturing the evolution of the average concentration over the inclusion across a wide range of diffusivities for the associated microcell problem. We measure the accuracy of this new fractional model against a semianalytical solution and make comparisons with the full dual-scale model to highlight not only its predictive capability but also its superior computational performance.