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.

scholarly journals Multi-Scale Insights on the Threshold Pressure Gradient in Low-Permeability Porous Media

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 364 ◽  
Author(s):  
Huimin Wang ◽  
Jianguo Wang ◽  
Xiaolin Wang ◽  
Andrew Chan
Keyword(s):  
Porous Media ◽  
Pressure Gradient ◽  
Water Saturation ◽  
Flow Behavior ◽  
Low Permeability ◽  
Threshold Pressure ◽  
Threshold Pressure Gradient ◽  
Linear Flow ◽  
Multi Scale ◽  
Non Linear

Low-permeability porous medium usually has asymmetric distributions of pore sizes and pore-throat tortuosity, thus has a non-linear flow behavior with an initial pressure gradient observed in experiments. A threshold pressure gradient (TPG) has been proposed as a crucial parameter to describe this non-linear flow behavior. However, the determination of this TPG is still unclear. This study provides multi-scale insights on the TPG in low-permeability porous media. First, a semi-empirical formula of TPG was proposed based on a macroscopic relationship with permeability, water saturation, and pore pressure, and verified by three sets of experimental data. Second, a fractal model of capillary tubes was developed to link this TPG formula with structural parameters of porous media (pore-size distribution fractal dimension and tortuosity fractal dimension), residual water saturation, and capillary pressure. The effect of pore structure complexity on the TPG is explicitly derived. It is found that the effects of water saturation and pore pressure on the TPG follow an exponential function and the TPG is a linear function of yield stress. These effects are also spatially asymmetric. Complex pore structures significantly affect the TPG only in the range of low porosity, but water saturation and yield stress have effects on a wider range of porosity. These results are meaningful to the understanding of non-linear flow mechanism in low-permeability reservoirs.

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