Effect of quadratic pressure gradient term on a one-dimensional moving boundary problem based on modified Darcy’s law

2015 ◽  
Vol 32 (1) ◽  
pp. 38-53 ◽  
Author(s):  
Wenchao Liu ◽  
Jun Yao ◽  
Zhangxin Chen ◽  
Yuewu Liu
Keyword(s):  
Pressure Gradient ◽  
Boundary Problem ◽  
Moving Boundary ◽  
Darcy’S Law ◽  
Darcy's Law ◽  
Moving Boundary Problem ◽  
Gradient Term ◽  
One Dimensional ◽  
Pressure Gradient Term ◽  
Modified Darcy’S Law

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

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Jun Yao ◽  
Wenchao Liu ◽  
Zhangxin Chen
Keyword(s):  
Porous Media ◽  
Partial Differential Equations ◽  
Pressure Gradient ◽  
Numerical Solution ◽  
Boundary Problem ◽  
Moving Boundary ◽  
Moving Boundary Problem ◽  
Threshold Pressure ◽  
Threshold Pressure Gradient ◽  
One Dimensional

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.

scholarly journals 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

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Wenchao Liu ◽  
Jun Yao
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Pressure Gradient ◽  
Boundary Problem ◽  
Moving Boundary ◽  
Radial Flow ◽  
Moving Boundary Problem ◽  
Strong Nonlinearity ◽  
Governing Equations ◽  
Partial Differential

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.

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