An Approximate Solution for Fluid Infiltration near a Circular Opening in Unsaturated Media


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Abstract

A regular perturbation technique is employed to approximate the solution for fluid infiltration from a circular opening into an unsaturated medium. Introducing two empirical constitutive relations k=ks ea p and Θ=Θr+Θ(sr) eb p relating the permeabilityk and water content Θ with pore fluid pressure p, a nonlinear diffusion equation in terms of pore pressure is established. After rearranging the nonlinear diffusion equation, a parameter perturbation on γ=1-βb/α is performed and an approximate solution with an error of O(γ3) is obtained, which correlate to a condition in which α=β. This approximate solution is verified by a finite difference solution and compared also with a linear solution in which the diffusivity is constant. It is shown that the perturbation solution with terms up to and including first-order can give a reasonably accurate solution for the parameter range for αp0 selected in this paper. The solution procedure provided in this paper also avoids the numerical problem normally encountered for a small time solution. The solution may also be used to overcome difficulties arising in solution procedure by the similarity transformation (Boltzmann), commonly conducted on diffusion equation, which cannot be applied for a finite wellbore problem.

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