A fast algorithm for the estimation of the equivalent hydraulic conductivity of heterogeneous media

Fast upscaling of hydraulic conductivity is a recurrent problem in modeling flow through heterogeneous porous media. We propose a new renormalization technique. It is based on the iterative application of the Cardwell and Parsons (1945) bounds on elementary groups of cells. The combination of the bounds with a heuristic formula allows anisotropy to be taken into account. The new technique is tested and compared with other fast techniques. Among the tested techniques the two most reliable ones are the tensorial renormalization and the new simplified renormalization. The numerical efficiency of the simplified renormalization leads us to recommend it when a diagonal tensor of equivalent conductivity is sufficient.

[1]  O. Wiener,et al.  Die theorie des Mischkörpers für das Feld der stationären Strömung , 1912 .

[2]  Michel Quintard,et al.  Ecoulement monophasique en milieu poreux: effet des hétérogénéités locales , 1987 .

[3]  Alex Hansen,et al.  A fast algorithm for estimating large-scale permeabilities of correlated anisotropic media , 1993 .

[4]  Benoît N∄tinger,et al.  The effective permeability of a heterogeneous porous medium , 1994 .

[5]  G. Marsily,et al.  Some current methods to represent the heterogeneity of natural media in hydrogeology , 1998 .

[6]  W. T. Cardwell,et al.  Average Permeabilities of Heterogeneous Oil Sands , 1945 .

[7]  S. Gorelick,et al.  Heterogeneity in Sedimentary Deposits: A Review of Structure‐Imitating, Process‐Imitating, and Descriptive Approaches , 1996 .

[8]  Yves Guéguen,et al.  flow in fractured media: A modified renormalization method , 1998 .

[9]  R. Ababou,et al.  Numerical simulation of three-dimensional saturated flow in randomly heterogeneous porous media , 1989 .

[10]  Benoit Noetinger,et al.  Calculation of Internodal Transmissivities in Finite Difference Models of Flow in Heterogeneous Porous Media , 1995 .

[11]  Benoit Nœtinger,et al.  Preferential Flow-Paths Detection for Heterogeneous Reservoirs Using a New Renormalization Technique , 1997 .

[12]  P. King.  The use of renormalization for calculating effective permeability , 1989 .

[13]  J. Feder,et al.  Real-Space Renormalization Estimates for Two-Phase Flow in Porous Media , 1997 .

[14]  G. Dagan.  Flow and transport in porous formations , 1989 .

[15]  X. Sanchez‐Vila,et al.  An evaluation of Jacob's Method for the interpretation of pumping tests in heterogeneous formations , 1998 .

[16]  H. Stanley,et al.  A Real-space Renormalization Group for Site and Bond Percolation? , 1977 .

[17]  H. Haldorsen,et al.  Stochastic Modeling (includes associated papers 21255 and 21299 ) , 1990 .

[18]  Ke Xu,et al.  Multiscale Structures to Describe Porous Media Part I: Theoretical Background and Invasion by Fluids , 1997 .

[19]  Benoit Noetinger,et al.  Influence of the Permeability Anisotropy Ratio on Large-Scale Properties of Heterogeneous Reservoirs , 1993 .

[20]  Uwe Jaekel,et al.  Renormalization group analysis of macrodispersion in a directed random flow , 1997 .

[21]  A. Saucier.  Scaling of the effective permeability in multifractal porous media , 1992 .

[22]  R. Romeu.  Ecoulement en milieux heterogenes : prise de moyenne de permeabilite en regimes permanent et transitoire , 1994 .

[23]  G. L. Loc’h.  Etude de la composition des perméabilités par des méthodes variationnelles , 1987 .

[24]  Julio M. Ottino,et al.  Effective transport properties of disordered, multi-phase composites: Application of real-space renormalization group theory , 1986 .

[25]  Modélisation des écoulements en milieu poreux hétérogènes-calcul des perméabilités équivalentes , 1996 .

[26]  G. Matheron.  Éléments pour une théorie des milieux poreux , 1967 .

[27]  A. De Wit,et al.  Correlation structure dependence of the effective permeability of heterogeneous porous media , 1995 .

[28]  H. S. Price,et al.  Flow in Heterogeneous Porous Media , 1961 .

[29]  A. J. Desbarats,et al.  Spatial averaging of hydraulic conductivity in three-dimensional heterogeneous porous media , 1992 .

[30]  W. G. Price,et al.  Renormalization calculations of immiscible flow , 1993 .

[31]  Peter K. Kitanidis,et al.  Determination of the effective hydraulic conductivity for heterogeneous porous media using a numerical spectral approach: 1. Method , 1992 .

[32]  Clayton V. Deutsch,et al.  Calculating effective absolute permeability in sandstone/shale sequences , 1989 .

[33]  J. F. McCarthy,et al.  Comparison of fast algorithms for estimating large-scale permeabilities of heterogeneous media , 1995 .

[34]  E. M. Lifshitz,et al.  Electrodynamics of continuous media , 1961 .

[35]  Peter R. King,et al.  The use of field theoretic methods for the study of flow in a heterogeneous porous medium , 1987 .

[36]  D. Li,et al.  A New Efficient Averaging Technique for Scaleup of Multimillion-Cell Geologic Models , 1999 .

[37]  A. Desbarats,et al.  Numerical estimation of effective permeability in sand-shale formations , 1987 .

[38]  Jesús Carrera,et al.  A Synthesis of Approaches to Upscaling of Hydraulic Conductivities , 1995 .

[39]  K. Wilson.  Renormalization Group and Critical Phenomena. I. Renormalization Group and the Kadanoff Scaling Picture , 1971 .

[40]  H. H. Hardy,et al.  Fractals in Reservoir Engineering , 1994 .

[41]  C. Axness,et al.  Three‐dimensional stochastic analysis of macrodispersion in aquifers , 1983 .

[42]  Dominique Guerillot,et al.  Conditional Simulation of the Geometry of Fluvio-Deltaic Reservoirs , 1987 .

[43]  J. Feder,et al.  Effective renormalization group algorithm for transport in oil reservoirs , 1991 .

[44]  Gedeon Dagan,et al.  Higher-order correction of effective permeability of heterogeneous isotropic formations of lognormal conductivity distribution , 1993 .

[45]  Dominique Guerillot,et al.  Conditional simulation of the geometry of fluvio-deltaic reservoirs. SPE 16753. , 1987 .

[46]  J. R. Macmillan,et al.  Stochastic analysis of spatial variability in subsurface flows: 2. Evaluation and application , 1978 .

[47]  S. Bachu,et al.  Effects of core‐scale heterogeneity on steady state and transient fluid flow in porous media: Numerical analysis , 1990 .

[48]  E. Teodorovich.  Calculation of the effective permeability of a randomly inhomogeneous porous medium , 1997 .

[49]  S. P. Neuman,et al.  Prediction of steady state flow in nonuniform geologic media by conditional moments: Exact nonlocal , 1993 .

[50]  J. Gómez-Hernández,et al.  Upscaling hydraulic conductivities in heterogeneous media: An overview , 1996 .

[51]  P. Indelman,et al.  Effective permittivity of log-normal isotropic random media , 1995 .

[52]  S. M. Kozlov.  A CENTRAL LIMIT THEOREM FOR MULTISCALED PERMEABILITY , 1993 .

[53]  K. Wilson.  The renormalization group: Critical phenomena and the Kondo problem , 1975 .

[54]  Gudmundur S. Bodvarsson,et al.  Effective transmissivity of two-dimensional fracture networks , 1995 .

[55]  J. Bernasconi.  Real-space renormalization of bond-disordered conductance lattices , 1978 .

[56]  P. Renard,et al.  Calculating equivalent permeability: a review , 1997 .

[57]  Chin-Fu Tsang,et al.  A semianalytical approach to spatial averaging of hydraulic conductivity in heterogeneous aquifers , 1997 .

[58]  Emmanuel Ledoux,et al.  Two Dimensional Stochastic Modelling of Flow in Non-Uniform Confined Aquifers. Correction of the Systematic Bias Introduced by Numerical Models when they are used Stochastically , 1990 .