Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 275057, 12 pages

http://dx.doi.org/10.1155/2015/275057

## Numerical Investigations of the Effect of Nonlinear Quadratic Pressure Gradient Term on a Moving Boundary Problem of Radial Flow in Low-Permeable Reservoirs with Threshold Pressure Gradient

^{1}School of Petroleum Engineering, China University of Petroleum (Huadong), Qingdao 266580, China^{2}Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, China

Received 11 May 2014; Revised 18 July 2014; Accepted 18 July 2014

Academic Editor: Jun Liu

Copyright © 2015 Wenchao Liu and Jun Yao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The existence of a TPG can generate a relatively high pressure gradient in the process of fluid flow in porous media in low-permeable reservoirs, and neglecting the QPGTs in the governing equations, by assuming a small pressure gradient for such a problem, can cause a significant error in predicting the formation pressure. Based on these concerns, in consideration of the QPGT, a moving boundary model of radial flow in low-permeable reservoirs with the TPG for the case of a constant flow rate at the inner boundary is constructed. Due to strong nonlinearity of the mathematical model, a numerical method is presented: 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; and then a stable, fully implicit finite difference method is used to obtain its numerical solution. Numerical result analysis shows that the mathematical models of radial flow in low-permeable reservoirs with TPG must take the QPGT into account in their governing equations, which is more important than those of Darcy’s flow; the sensitive effects of the QPGT for the radial flow model do not change with an increase of the dimensionless TPG.

#### 1. Introduction

Due to a continuously decreasing crude oil output from conventional reservoirs and a high international gasoline price in recent years, unconventional reservoirs such as low-permeable reservoirs and shale oil and gas reservoirs have become urgent development resources in the petroleum industry. Consequently, considerable attention has been paid to the relevant research on the kinematic principles for the fluid flow behavior in these unconventional reservoirs [1–5] at present. Abundant experimental and theoretical analyses [6–13] have demonstrated that the fluid flow, in low-permeable porous media, does not obey the classical Darcy’s law: the seepage velocity is not proportional to the formation pressure gradient, and there exists a threshold pressure gradient (TPG) *λ*; the fluid flow happens only if the formation pressure gradient is larger than TPG.

Much research on these relevant moving boundary models has been conducted [14–22]. The computed formation pressure distributions corresponding to these moving boundary problems of the fluid flow in the porous media with TPG show big difference from the ones based on Darcy’s law (see Figure 1): the formation pressure gradient is much steeper, it decreases until up to zero at a certain value of a dimensionless distance from a well, that is, the position of a moving boundary, and the pressure distribution curve shows a property of compact support [21]; whereas, for Darcy’s flow problem, the formation pressure drop can propagate to any infinite distance transiently according to the exact analytical solution [20], and the formation pressure gradient is much more smooth. Actually, the pressure distribution difference between Darcy’s flow and fluid flow in the porous media with TPG can be explained through the angle between the dimensionless formation pressure curve and the dimensionless distance at the place of moving boundary, as shown in Figure 1: for the case of the existence of TPG, the tangent of is equal to the derivative of pressure with respect to distance at the place of moving boundary, that is, the TPG; and, as decreases gradually and is equal to zero, it becomes a limit case, which corresponds to Darcy’s flow; that is, TPG is equal to zero. And if a constant value of TPG is given, the tangent of will not change transiently. Therefore, it can be concluded that the formation pressure gradient is relatively much higher for the fluid flow in porous media with large TPG. The main physical reason is that the presence of TPG can make the formation pressure drop propagate more slowly, which causes a larger pressure gradient in a relatively shorter pressure disturbed distance from a production well.