Analytical solution for one-dimensional radial flow caused by line source in porous medium with threshold pressure gradient

Abstract

• A special free boundary problem in the cylindrical coordinate is discovered. • An analytical solution of the free boundary problem is established. • Application of the analytical solution in an inverse problem is presented. One-dimensional radial flow caused by a line source in a porous medium with a threshold pressure gradient is investigated. The problem is a free boundary problem in the cylindrical coordinate, and it is different from traditional free boundary problems owing to the presence of a space-dependent internal source and an implicit condition at the free boundary. To ensure the existence of a similarity solution, a line source with an intensity varying in proportion to the square root of time is considered, and an analytical solution is subsequently established using the similarity transformation technique and the theory of the Kummer functions. As an application, the analytical solution is incorporated into an inverse problem. A sensitivity analysis is initially conducted, and the results indicate that the time-varying pressure is sensitive to the variations in the permeability but insensitive to the variations in the threshold pressure gradient. Therefore, the inverse problem is designed to estimate the permeability of the porous medium from the measured data of time-varying pressure, and the Levenberg–Marquardt method is applied to minimize the objective function. Computational examples of the inverse problem with different error levels are also presented, and the effectiveness of the inverse analysis is confirmed.