Effect of quadratic pressure gradient term on a on

Authors: Wenchao Liu Jun Yao Zhangxin Chen Yuewu Liu

Publish Date: 2015/10/29

Volume: 32, Issue: 1, Pages: 38-53

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Abstract

A relatively high formation pressure gradient can exist in seepage flow in lowpermeable porous media with a threshold pressure gradient and a significant error may then be caused in the model computation by neglecting the quadratic pressure gradient term in the governing equations Based on these concerns in consideration of the quadratic pressure gradient term a basic moving boundary model is constructed for a onedimensional seepage flow problem with a threshold pressure gradient Owing to a strong nonlinearity and the existing moving boundary in the mathematical model a corresponding numerical solution method is presented First a spatial coordinate transformation method is adopted in order to transform the system of partial differential equations with moving boundary conditions into a closed system with fixed boundary conditions then the solution can be stably numerically obtained by a fully implicit finitedifference method The validity of the numerical method is verified by a published exact analytical solution Furthermore to compare with Darcy’s flow problem the exact analytical solution for the case of Darcy’s flow considering the quadratic pressure gradient term is also derived by an inverse Laplace transform A comparison of these model solutions leads to the conclusion that such moving boundary problems must incorporate the quadratic pressure gradient term in their governing equations the sensitive effects of the quadratic pressure gradient term tend to diminish with the dimensionless threshold pressure gradient increasing for the onedimensional problemThe authors would like to acknowledge the funding by the project Grant 51404232 sponsored by the National Natural Science Foundation of China the National Science and Technology Major Project Grant 2011ZX05038003 and the China Postdoctoral Science Foundation project Grant 2014M561074 In particular Wenchao Liu would also like to express his deepest gratitude to the China Scholarship Council for its generous financial support of the research Special thanks go to Dr Yongfei Yang Dr Lili Xue and Dr Lei Zhang for their tremendous help in improving the writing and wording of the paper

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