Estimation of the Time of Setting Steady-State Conditions for Newtonian Fluid Pressure Flow through a Flat Porous Layer

Abstract

Unsteady elastic filtration conditions for a viscous incompressible fluid isothermally pumped through an isotropic porous layer served as a basis for formulating initial-boundary value problems for a one-dimensional pressure field, assuming that the filtration mode obeys Darcy's law. The problems describe the pumping on/off modes, as well as the steady-state conditions for the flow of a Newtonian fluid supplied from the upper surface of the porous medium and exiting through its lower bounding surface, with infinitely large boundary permeability. The equations of the formulated initial-boundary value problems are partial differential equations with Dirichlet and/or von Neumann boundary conditions. In the case of pulsed input, the boundary condition is represented by a structure consisting of the difference of Heaviside functions, one of which has a shift equal to the pulse duration. Due to their linear nature, analytical solutions can be obtained using a one-sided integral Laplace transform in the form of explicit relationships for dimensionless pressure and velocity profiles along the height of a porous layer. As these solutions are represented by rapidly converging Fourier series with a multiplicative exponent, the concept of a “regular” mode is used (only the first term is considered in the expansion). Approximate relationships are derived to determine the duration of the setting modes during the start and stop of pumping the liquid medium through a porous layer. The obtained solutions are generalized using an impulse boundary condition. Computational experiments have shown that it takes approximately 0,537 dimensionless time for the pressure profile to approach linearity during both the on and off pumping modes, with a relative accuracy of 0,01, depending on the velocity of the fluid at the outlet of the porous layer. An example is given to demonstrate that the setting time can reach a significant value, which should be taken into account in the operation of various technical devices that contain porous layers.