Effective Permeabilities in a Porous Medium with Log-Stable Statistics

Abstract

Equations for the effective coefficients of random permeability fields for fluid flow through a porous medium with log-stable distributions are derived using the Wilson renormalization group approach. Results of the theoretical modeling are compared with data of numerical modeling.Introduction. Full-scale and laboratory observations have shown that the fluid permeability field is nonuniform and that the spread of the nonuniformity scale increases with an increase in the number of measurements in a bounded interval. This has led to the development of fractal permeability models with log-stable permeability distributions [1]. Shvidler [2] and Dagan [3] studied statistical models with lognormal permeability distributions. Kuz'min and Soboleva [4,5], using the Wilson renormalization group (RG) approach [6], derived subgrid formulas for the effective permeability coefficients and studied the diffusion of the interface between the fluids for their joint flow in a multiscale porous medium for lognormal permeability and porosity distributions. Teodorovich [7] derived the Landau-Lifshits formula for the effective permeability within the framework of a rigorous field renormalization group and analyzed previous studies of the field RG. In particular, mention is made of the arguments of [8], according to which renormalization group methods partially allow for the highest orders of perturbation theory and should improve the accuracy of the formulas derived. The same arguments are also applicable to subgrid modeling. If the medium is assumed to satisfy the improved Kolomogorov similarity hypothesis [9,4], the effective coefficients take especially simple form. In the present paper, the ideas of the Wilson RG method are employed to find subgrid modeling formulas for solving filtration problems for fractal porous media with a lognormal permeability distribution. Differential equations for obtaining effective constants are also derived for media that do not satisfy the improved similarity hypothesis. The derived formulas are verified by direct numerical modeling.Formulation of the Problem. At small Reynolds numbers, the filtration velocity v and the pressure p are related by the Darcy law v = −ε(x)∇p, where ε(x) is the permeability (a random function of the coordinates). For an incompressible fluid, we have the equation

Resistivity logging in a multiscale isotropic porous medium with log-normal distributed conductivity

“…In the opposite case (the second correlation function does not exist), the integral over dl/l tends to aˇeld described by a stable distribution. The case of a stable distribution is considered in [4]. In our work, it is assumed that at l 0 < l < L ϕ(x, l) has normal distribution.…”

Subgrid modeling of flow processes in an anisotropic fractal medium

“…The Kolmogorov similarity hypothesis does not assume logarithmic normality and uncorrelatedness for the logarithm of the scale. Therefore, another obvious possibility of extending the above approach is to study stable distributions [12,13] and to consider the case of conformal symmetry of conductivity. The procedure proposed in the present work allows the estimation of the effect of small-scale pulsations on the large-scale component of various fields, which is especially important if the inhomogeneity scales are in a wide range.…”