Nonlinear Pressure Diffusion in Flow of Compressible Liquids Through Porous Media

Abstract

In the flow of liquids through porous media, nonlinear effects arise from the dependence of the fluid density, porosity, and permeability on pore pressure, which are commonly approximated by simple exponential functions. The resulting flow equation contains a squared gradient term and an exponential dependence of the hydraulic diffusivity on pressure. In the limiting case where the porosity and permeability moduli are comparable, the diffusivity is constant, and the squared gradient term can be removed by introducing a new variable y, depending exponentially on pressure. The published transformations that have been used for this purpose are shown to be special cases of the Cole-Hopf transformation, differing in the choice of integration constants. Application of Laplace transformation to the linear diffusion equation satisfied by y is considered, with particular reference to the effects of the transformation on the boundary conditions. The minimum fluid compressibilities at which nonlinear effects become significant are determined for steady flow between parallel planes and cylinders at constant pressure. Calculations show that the liquid densities obtained from the simple compressibility equation of state agree to within 1% with those obtained from the highly accurate Wagner-Pruß equation of state at pressures to 20 MPa and temperatures approaching 600 K, suggesting possible applications to some geothermal systems.

Fluid dynamics in naturally fractured tectonic reservoirs

“…For homogeneous media, the flow law used is Darcy's law (Pedrosa 1986;Chin and Raghavan 2000;Marshall 2009), satisfying the Reynolds number Re \ 1. Figure 6 shows a tectonic fracture represented as two parallel surfaces.…”

“…Also, it has been used to solve a nonlinear diffusion problem for the flow of compressible liquids through homogeneous porous media (Marshall 2009). The Cole-Hopf transformation is a mathematical technique, through which a nonlinear partial differential equation may be reduced to linear partial differential equation.…”

“…It was observed that the transformation y = F(p f ) applied in Burgers equation generated a linear partial equation of type qy/qt = Dr 2 y, and this concept was utilized to solve the nonlinear diffusivity equation (Ames 1972;Burgers 1974;Marshall 2009):…”

Parametric analysis of diffusivity equation in oil reservoirs

“…Some useful approaches have been applied to solve the mentioned equation such as Laplace transform, Boltzmann transform, dimensionless form and Ei function (Loucks and Guerrero 1961;Odeh and Babu 1988;Marshall 2009). Also, Cole-Hopf transform is used to simplify pressure diffusivity equation included pressure dependence permeability, porosity and density which cause the flow equation non-linear (Marshall 2009). The Ei function approach is functional for the current case of study of analytical solution.…”

Nonlinear flow model for well production in an underground formation

“…When porosity, permeability and fluid density depend exponentially on pressure, the diffusivity equation reduces to a diffusion equation containing a squared gradient term. Many published articles have described analytical solutions for this equation through variable modifications (Chakrabarty et al, 1993a, b;Jelmert and Vik, 1996;Odeh and Babu, 1998;Wang and Dusseault, 1991), which are special cases of the Hopf-Cole transformation (Marshall, 2009). Applications in dual-porosity and fractal (Tong and Wang, 2005) media have also been described.…”