Fluid transport in Forchheimer porous medium with spatially varying porosity and permeability - Siraev, Ramil’ Rifgatovich (Computational Continuum Mechanics, 2019)
Filtration of incompressible fluid in a saturated heterogeneous porous medium is theoretically studied. A mathematical model is based on the Forchheimer equation with an additional term that takes into account the heterogeneity of the medium. The effect of spatially varying permeability and porosity on the filtration in a flat channel is numerically investigated. For simplicity, we make an assumption about the lack of correlation between porosity and permeability and take them as independent parameters of the medium. The results show that the spatial variations of permeability and porosity affect transport in porous media in different ways. In the first case, the fluid flowing around the areas with low permeability obeys the Forzheimer law. The structure of the hydrodynamic fields and the flow rate do not vary when the flow direction changes to the opposite. In the media with varying porosity, the flow asymmetry is observed: filtration rate in the direction of increasing porosity is greater than in the opposite direction. This difference can cause a secondary flow, which is illustrated by the problem studying filtration in a channel with a periodic flow rate. We consider the case when the external periodic action has a high frequency as compared to hydrodynamic times, which provides an application of the averaging procedure to the system. Equations that describe the averaged flow arising on the background of an oscillating motion are obtained. Secondary flow arises on the background of the oscillating motion under the action of the force which is represented as a term with porosity gradient in the equations of the averaged flow and can evolve even in the absence of a constant pressure drop. For one-dimensional flow, an analytical solution is obtained. The intensity of the averaged flow is determined by the permeability and porosity gradient of the medium, as well as by the amplitude and frequency of periodic loads.
Keywords: heterogeneous porous medium; mathematical model of fluid filtration; flow asymmetry; oscillating fluid motion; secondary averaged flow