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EXACT SOLUTION OF NONLINEAR DIFFUSION EQUATION FOR FLUID FLOW IN FRACTAL RESERVOIRS

Volume 1, Issue 3, 2010, pp. 193-204
DOI: 10.1615/SpecialTopicsRevPorousMedia.v1.i3.10
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ABSTRACT

The fundamental theory of fluidflow in porous media should be consistent with material balance. However, the quadratic gradient term in the nonlinear diffusion equations is usually neglected, according to the slightly compressible fluid assumption, during the process of linearization, which will lead to errors for large time values. In this paper, fractal geometry theory is used to combine with seepage flow mechanics to establish the nonlinear diffusion equation of fluids flow in fractal reservoirs, including the quadratic gradient term. A method is used to scale the fractal properties of a fractal reservoir by double parameters (df, ds) and to describe the generalized flow characteristics of the nonlinear diffusion equation by four parameters, α, ds, df, and cD. A nonlinear flow model for fractal medium is presented, and all terms in the nonlinear diffusion equation are retained. A methodology to solve the diffusion equation with the quadratic gradient term is proposed. The solution technique, which is based on Laplace transform and Weber transform, is well suited for solving the flow model of fractal mediums. After analyzing the typical curves, we find that the relative error caused by the effects of the quadratic gradient term on pressure may amount to several to tens of percent in fractal reservoir flow. Especially for live oil and low-permeability reservoirs, linearization by neglecting the quadratic gradient term may generate inaccurate values for large time values.

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