Analytical solution for one-dimensional non-Darcy flow with bilinear relation in porous medium caused by line source

Abstract

One-dimensional non-Darcy flow in a porous medium caused by a line source is investigated. Owing to several reasons, a bilinear relation between the fluid velocity and the pressure gradient is considered, and the problem can be seen as an extension of a special one-phase free boundary problem reported in literature to a two-phase situation. An analytical solution for the problem is established based on the similarity type general solution of the governing equation. The analytical solution contains an unknown coefficient, which describes the free boundary movement and needs to be determined by a nonlinear equation, and the existence and uniqueness of this coefficient is proven. A numerical solution is also developed using the finite volume method, and special attention is spent on the two control volumes near the free boundary location so as to track the movement of the free boundary accurately. Computational examples are presented. The application of the analytical solution as a benchmark is introduced, and the accuracy of the numerical solution is verified. The non-Darcy flow for boreholes with different radii is studied, and the error caused by neglecting the size of borehole is discussed. The non-Darcy flow with a bilinear relation is compared with that with a threshold pressure gradient, and the error caused by neglecting the permeability under low pressure gradients is analyzed.