Numerical Solution of a Moving Boundary Problem of One-Dimensional Flow in Semi-Infinite Long Porous Media with Threshold Pressure Gradient

Jun Yao,Wenchao Liu,Zhangxin Chen

doi:10.1155/2013/384246

Abstract

A numerical method is presented for the solution of a moving boundary problem of one-dimensional flow in semi-infinite long porous media with threshold pressure gradient (TPG) for the case of a constant flow rate at the inner boundary. In order to overcome the difficulty in the space discretization of the transient flow region with a moving boundary in the process of numerical solution, the system of partial differential equations for the moving boundary problem is first transformed equivalently into a closed system of partial differential equations with fixed boundary conditions by a spatial coordinate transformation method. Then a stable, fully implicit finite difference method is adopted to obtain its numerical solution. Finally, numerical results of transient distance of the moving boundary, transient production pressure of wellbore, and formation pressure distribution are compared graphically with those from a published exact analytical solution under different values of dimensionless TPG as calculated from actual experimental data. Comparison analysis shows that numerical solutions are in good agreement with the exact analytical solutions, and there is a big difference of model solutions between Darcy's flow and the fluid flow in porous media with TPG, especially for the case of a large dimensionless TPG.

Highlights

With the increase of international oil price, unconventional reservoirs such as low-permeable reservoirs, heavy oil reservoirs, and shale gas reservoirs [1] have become new development targets in the field of petroleum engineering in modern times
The objective of this paper is to present a simple and novel method for numerical solution of the moving boundary problem of one-dimensional flow in semi-infinite long porous media with threshold pressure gradient (TPG) for the case of a constant flow rate at the inner boundary
The utility of the presented numerical method can be attributed to its simple approach of spatial coordinate transformation in numerically solving the moving boundary problem effectively by the stable, fully implicit finite difference method

Summary

Introduction

With the increase of international oil price, unconventional reservoirs such as low-permeable reservoirs, heavy oil reservoirs, and shale gas reservoirs [1] have become new development targets in the field of petroleum engineering in modern times. Due to the strong nonlinearity of the moving boundary problem, it is very difficult to obtain its exact analytical solution, whereas, recently, in the new published paper [29], exact analytical solutions for the non-Stefan moving boundary problems of one-dimensional flow in semi-infinite long porous media with TPG are presented through a similarity transformation method. It becomes very necessary to develop a verified numerical method by the published exact analytical solution [29] in order to solve more complicated moving boundary problems of fluid flow in porous media with TPG. The objective of this paper is to present a simple and novel method for numerical solution of the moving boundary problem of one-dimensional flow in semi-infinite long porous media with TPG for the case of a constant flow rate at the inner boundary.

Mathematical Model

Numerical Solution of the Problem

Verification of Numerical Solutions

Conclusions