Anomalous Filtration of Liquid in a Plane-Radial Homogeneous Porous Medium

Abstract

The paper considers the radial problem of anomalous filtration of a homogeneous liquid in a plane-radial formulation. The filtration process was modeled by a differential equation with fractional derivatives with respect to time and pressure. This differential equation was derived on the basis of Darcy’s fractional relaxation law, which takes into account relaxation effects both in the filtration velocity and in the pressure gradient. Fractional derivatives are defined in the Caputo sense, which is preferable to other definitions of fractional derivatives, such as Riemann-Liouville, Grundwald-Letnikov, etc. The problem is solved numerically by the finite difference method. The integral representation of the fractional Caputo derivative is discretized using known quadrature formulas. The pressure profiles are determined for different values of the order of the fractional derivative with respect to time both with respect to pressure and filtration velocity, and the effect of process anomality on the filtration characteristics is estimated. The effect of changing the orders of fractional derivatives on the pressure distribution at different moments in time is established. The effect of relaxation times for the pressure gradient and filtration velocity on the pressure distribution in the medium at different moments in time is also estimated. A comparative analysis of the effect of the orders of fractional derivatives and relaxation times in Darcy’s law on the pressure distribution in the medium is carried out.