Modeling and discretization of flow in porous media with thin, full-tensor permeability inclusions

When modeling fluid flow in fractured reservoirs, it is common to represent the fractures as lower-dimensional inclusions embedded in the host medium. Existing discretizations of flow in porous media with thin inclusions assume that the principal directions of the inclusion permeability tensor are aligned with the inclusion orientation. While this modeling assumption works well with tensile fractures, it may fail in the context of faults, where the damage zone surrounding the main slip surface may introduce anisotropy that is not aligned with the main fault orientation. In this article, we introduce a generalized dimensional reduced model which preserves full-tensor permeability effects also in the out-of-plane direction of the inclusion. The governing equations of flow for the lower-dimensional objects are obtained through vertical averaging. We present a framework for discretization of the resulting mixed-dimensional problem, aimed at easy adaptation of existing simulation tools. We give numerical examples that show the failure of existing formulations when applied to anisotropic faulted porous media, and go on to show the convergence of our method in both two-dimensional and three-dimensional.publishedVersio

Estimation of the Effective Permeability of Heterogeneous Porous Media by Using Percolation Concepts

In this paper we present new methods to estimate the effective permeability (k_eff) of heterogeneous porous media with a wide distribution of permeabilities and various underlying structures, using percolation concepts. We first set a threshold permeability (k_th) on the permeability density function (pdf) and use standard algorithms from percolation theory to check whether the high permeable grid blocks (i.e. those with permeability higher than k_th) with occupied fraction of “p” first forms a cluster connecting two opposite sides of the system in the direction of the flow (high permeability flow pathway). Then we estimate the effective permeability of the heterogeneous porous media in different ways: a power law (k_eff=k_th p^m), a weighted power average (k_eff=[p.k_th^m+(1-p).k_g^m ]^(1/m) with k_g the geometric average of the permeability distribution) and a characteristic shape factor multiplied by the permeability threshold value. We found that the characteristic parameters (i.e. the exponent “m”) can be inferred either from the statistics and properties of percolation sub-networks at the threshold point (i.e. high and low permeable regions corresponding to those permeabilities above and below the threshold permeability value) or by comparing the system properties with an uncorrelated random field having the same permeability distribution. These physically based approaches do not need fitting to the experimental data of effective permeability measurements to estimate the model parameter (i.e. exponent m) as is usually necessary in empirical methods. We examine the order of accuracy of these methods on different layers of 10th SPE model and found very good estimates as compared to the values determined from the commercial flow simulators