Derivation of fractional-derivative models of multiphase fluid flows in porous media

Abstract

This paper is devoted to deriving several fractional-order models for multiphase flows in porous media, focusing on some special cases of the two-phase flow. We derive the mass and momentum conservation laws of multiphase flow in porous media. The mass conservation-law has been developed based on the flux variation using Taylor series approximation. The fractional Taylor series's advantage is that it can represent the non-linear flux with more accuracy than the first-order linear Taylor series. The divergence term in the mass conservation equation becomes of a fractional type. The model has been developed for the general compressible flow, and the incompressible case is highlighted as a particular case. As a verification, the model can easily collapse to the traditional mass conservation equation once we select the integer-order. To complete the flow model, we present Darcy’s law (momentum conservation law in porous media) with time/space fractional memory. The modified Darcy’s law with time memory has also been considered. This version of Darcy’s law assumes that the permeability diminishes with time, which has a delay effect on the flow; therefore, the flow seems to have a time memory. The fractional Darcy’s law with space memory based on Caputo's fractional derivative is also considered to represent the nonlinear momentum flux. Then, we focus on some cases of fractional time memory of two-phase flows with countercurrent-imbibition mechanisms. Five cases are considered, namely, traditional mass equation and fractional Darcy’s law with time memory; fractional mass equation with conventional Darcy’s law; fractional mass equation and fractional Darcy’s law with space memory; fractional mass equation and fractional Darcy’s law with time memory; and traditional mass equation and fractional Darcy’s law with spatial memory.