MODEL OF DEEP BED FILTRATION IN A POROUS MEDIUM WITH HETEROGENEOUS POROSITY
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Abstract
Modeling the transport and sedimentation of small particles of suspensions and colloids in porous rocks is an important problem in subsurface hydromechanics. Particles entrained in fluid are transported and retained in the rock pores. The filtration process is determined by the number and size of pores and is characterized by porosity—the ratio of the void volume to the total soil volume. In homogeneous materials, porosity can be considered constant. However, in practical problems describing filtration of suspended particles in multilayered and heterogeneous soils, the rock porosity is variable, and a constant-porosity model is inapplicable. A one-dimensional model of suspension and colloid filtration in a porous medium with nonuniform porosity is considered. The problem includes a mass balance equation accounting for variable porosity and a kinetic equation for sediment growth. The model describes the injection of a suspension or colloid of constant concentration into a porous medium containing pure water without suspended or sedimented particles. Unlike the case of uniform porosity, the transport velocity of suspended particles in pores is variable, and the boundary between the suspension and pure water, called the concentration front, is curved. Previously, such problems were solved only numerically. In this article, a system of filtration equations in a medium with variable porosity is solved analytically using the method of characteristics. An explicit formula is obtained for the curved front of suspended and sedimented particle concentrations, and exact analytical closed-form solutions are constructed ahead of and behind the front. An explicit solution is found for a model with a linear filtration function.
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