The spatial variations in porous media (aquifers and petroleum reservoirs) occur at all length scales (from the pore to the reservoir scale) and are incorporated into the governing equations for multiphase flow problems on the basis of random fields (geostatistical models). As a consequence, the velocity field is a random function of space. The randomness of the velocity field gives rise to a mixing region between fluids, which can be characterized by a mixing length = (t). Here we focus on the scale-up problem for tracer flows. Under very general conditions, in the limit of small heterogeneity strengths it has been derived by perturbation theories that the scaling behavior of the mixing region is related to the scaling properties of the self-similar (or fractal) geological heterogeneity through the scaling law (t) ∼ t γ , where γ = max{1/2, 1 − β/2}; β is the scaling exponent that controls the relative importance of short vs. large scales in the geology. The goals of this work are the following: (i) The derivation of a new, mathematically rigorous scaling analysis for the tracer flow problem subject to self-similar heterogeneities. This theoretical development relates the large strength to the small strength heterogeneity regime by a simple scaling of solutions. It follows from this analysis that the scaling law derived by perturbation theory is valid for any strength of the underlying geology, thereby extending the current available results. To the best of the knowledge of the authors this is the only rigorous result available in the literature for the large strength heterogeneity regime. (ii) The presentation of a Monte Carlo study of highly resolved simulations, which are in excellent agreement with the predictions of our new theory. This indicates that our Monte Carlo results are accurate and can be applied to other models for stochastic geology.
SUMMARYA new multiscale procedure is proposed to compute flow in compressible heterogeneous porous media with geology characterized by power-law covariance structure. At the fine scale, the deformable medium is modeled by the partially coupled formulation of poroelasticity with Young's modulus and permeability treated as stationary random fields represented by their Karhunen-Loève decompositions. The framework underlying the multiscale procedure is based on mapping these random parameters to an auxiliary domain and constructing a family of equivalent stochastic processes at different length scales characterized by the same ensemble mean and covariance function. The poromechanical variables inherit a space-time version of the scaling relations of the random input parameters which allows for constructing a set of multiscale solutions of the same governing equations posed at different space and time scales. A notable feature of the multiscale method proposed herein is the feasibility of solving both the poroelastic model and the Fredholm integral equation for the eigenpairs of the Karhunen-Loève expansion in an auxiliary domain with much lower computational effort and then derive the long term behavior at a coarser scale from a straightforward rescaling of the auxiliary solution. Within the framework of the finite element approximation, in conjunction with the Monte Carlo algorithm, numerical simulations of fluid withdrawal and injection problems in a heterogeneous poroelastic reservoir are performed to illustrate the potential of the method in drastically reducing the computational burden in the computation of the statistical moments of the poromechanical unknowns in large-scale simulations.
SUMMARYIn this paper, we develop a new Godunov‐type semi‐discrete central scheme for a scalar conservation law on the basis of a generalization of the Kurganov and Tadmor scheme, which allows for spatial variability of the storage coefficient (e.g. porosity in multiphase flow in porous media) approximated by piecewise constant interpolation. We construct a generalized numerical flux at element edges on the basis of a nonstaggered inhomogeneous dual mesh, which reproduces the one postulated by Kurganov and Tadmor under the assumption of homogeneous storage coefficient. Numerical simulations of two‐phase flow in strongly heterogeneous porous media illustrate the performance of the proposed scheme and highlight the important rule of the permeability–porosity correlation on finger growth and breakthrough curves. Copyright © 2013 John Wiley & Sons, Ltd.
SUMMARYThis paper introduces a new methodology to generate numerically multi-scale random fields. The generation of numerical samples of random fields plays an important role in Monte Carlo simulation methods. We are concerned with porous media flow studies where random field generators are used widely as a tool to model rock heterogeneities for applications in hydrocarbon recovery and groundwater flow.Reservoir rock properties such as permeability and porosity vary in space and may be characterized by their distributions. As the true distribution of these properties is unknown, we assume that they can be approximated by Gaussian (or transforms of Gaussian) random fields. In particular, we focus on correlated random fields with a power-law covariance structure.Our new method for the numerical generation of random fields uses a hierarchy of independent Gaussian variables defined on multiple length scales and is based on a theoretical construction of random fields described in Glimm and Sharp (J. Stat. Phys. 1991; 62(1-2):415-424). Numerical results are presented to show that the proposed method is accurate and numerically efficient.
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