Truncated MultiGaussian Fields and Effective Conductance of Binary Media

Truncated Gaussian fields provide a flexible model for defining binary media with dispersed (as opposed to layered) inclusions. General properties of excursion sets on these truncated fields are coupled with a distance-based upscaling algorithm and approximations of point process theory to develop an estimation approach for effective conductivity in two-dimensions. Estimation of effective conductivity is derived directly from knowledge of the kernel size used to create the multiGaussian field, defined as the full-width at half maximum (FWHM), the truncation threshold and conductance values of the two modes. Therefore, instantiation of the multiGaussian field is not necessary for estimation of the effective conductance. The critical component of the effective medium approximation developed here is the mean distance between high conductivity inclusions. This mean distance is characterized as a function of the FWHM, the truncation threshold and the ratio of the two modal conductivities. Sensitivity of the resulting effective conductivity to this mean distance is examined for two levels of contrast in the modal conductances and different FWHM sizes. Results demonstrate that the FWHM is a robust measure of mean travel distance in the background medium. The resulting effective conductivities are accurate when compared to numerical results and results obtained from effective media theory, distance-based upscaling and numerical simulation.

[1]  Touvia Miloh,et al.  On the effective conductivity of composites with ellipsoidal inhomogeneities and highly conducting interfaces , 1999, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[2]  Graeme W. Milton,et al.  Bounds on the Electromagnetic, Elastic, and Other Properties of Two-Component Composites , 1981 .

[3]  Jean-Raynald de Dreuzy,et al.  Hydraulic properties of two‐dimensional random fracture networks following a power law length distribution: 1. Effective connectivity , 2001 .

[4]  Salvatore Torquato,et al.  Flow in random porous media: mathematical formulation, variational principles, and rigorous bounds , 1989, Journal of Fluid Mechanics.

[5]  Arnold Verruijt,et al.  Flow and transport in porous media , 1981 .

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

[7]  Gedeon Dagan,et al.  Models of groundwater flow in statistically homogeneous porous formations , 1979 .

[8]  R. Ritzi,et al.  Hydrofacies distribution and correlation in the Miami Valley Aquifer System , 1995 .

[9]  G. Fogg,et al.  Transition probability-based indicator geostatistics , 1996 .

[10]  S. McKenna,et al.  Local Hydraulic Gradient Estimator Analysis of Long‐Term Monitoring Networks , 2006, Ground water.

[11]  R. Zimmerman.  Effective conductivity of a two-dimensional medium containing elliptical inhomogeneities , 1996, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[12]  Jesús Carrera,et al.  Binary upscaling—the role of connectivity and a new formula , 2006 .

[13]  Chin-Fu Tsang,et al.  A self‐consistent approach for calculating the effective hydraulic conductivity of a binary, heterogeneous medium , 2004 .

[14]  T. Harter,et al.  Effective conductivity of periodic media with cuboid inclusions , 2004 .

[15]  F. Alabert,et al.  Non-Gaussian data expansion in the Earth Sciences , 1989 .

[16]  J. Jaime Gómez-Hernández,et al.  ISIM3D: and ANSI-C three-dimensional multiple indicator conditional simulation program , 1990 .

[17]  F. Schwartz,et al.  An Analysis of the Influence of Fracture Geometry on Mass Transport in Fractured Media , 1984 .

[18]  On the conductivity of fiber reinforced materials , 1976 .

[19]  Jeffrey Marc Yarus,et al.  Stochastic Modeling and Geostatistics , 1994 .

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

[21]  Graham E. Fogg,et al.  Groundwater Flow and Sand Body Interconnectedness in a Thick, Multiple-Aquifer System , 1986 .

[22]  Karl J. Friston,et al.  Assessing the significance of focal activations using their spatial extent , 1994, Human brain mapping.

[23]  John L. Wilson,et al.  An approach to estimating hydraulic conductivity spatial correlation scales using geological characteristics , 1989 .

[24]  Muhammad Sahimi,et al.  Multiresolution wavelet coarsening and analysis of transport in heterogeneous media , 2002 .

[25]  S. McKenna,et al.  Fractured Continuum Approach to Stochastic Permeability Modeling , 2006 .

[26]  P. Fokker.  General Anisotropic Effective Medium Theory for the Effective Permeability of Heterogeneous Reservoirs , 2001 .

[27]  D. A. G. Bruggeman.  Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. III. Die elastischen Konstanten der quasiisotropen Mischkörper aus isotropen Substanzen , 1937 .

[28]  R. Adler.  On excursion sets, tube formulas and maxima of random fields , 2000 .

[29]  J. Kärger,et al.  Flow and Transport in Porous Media and Fractured Rock , 1996 .

[30]  A. W. Harbaugh.  MODFLOW-2005 : the U.S. Geological Survey modular ground-water model--the ground-water flow process , 2005 .

[31]  David J. Nott,et al.  Multi-phase image modelling with excursion sets , 2000, Signal Process..

[32]  G. Fogg,et al.  Modeling Spatial Variability with One and Multidimensional Continuous-Lag Markov Chains , 1997 .

[33]  Trevor Bailey,et al.  Statistical Analysis of Spatial Point Patterns. Second Edition. By PETER J. DIGGLE (London: Edward Arnold). [Pp. viii+159]. ISBN 0-340-74070-1. Price £40.00. Hardback , 2004, Int. J. Geogr. Inf. Sci..

[34]  T. Miloh,et al.  A generalized self‐consistent method for the effective conductivity of composites with ellipsoidal inclusions and cracked bodies , 1988 .

[35]  Peter J. Diggle,et al.  Statistical analysis of spatial point patterns , 1983 .

[36]  Graham E. Fogg,et al.  Geostatistical Modeling of Heterogeneity in Glaciofluvial, Buried-Valley Aquifers , 1994 .

[37]  Andrew R. Solow,et al.  Mapping by simple indicator kriging , 1986 .

[38]  Adriaan D. Poley.  Effective permeability and dispersion in locally heterogeneous aquifers , 1988 .

[39]  K. Mendelson,et al.  A theorem on the effective conductivity of a two‐dimensional heterogeneous medium , 1975 .

[40]  C. Langevin.  Stochastic Ground Water Flow Simulation with a Fracture Zone Continuum Model , 2003, Ground water.

[41]  J. Kong,et al.  Effective permittivity of dielectric mixtures , 1988 .

[42]  Christian Lantuéjoul,et al.  Geostatistical Simulation: Models and Algorithms , 2001 .

[43]  D. A. G. Bruggeman.  Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen , 1935 .

[44]  Alan C. Evans,et al.  Searching scale space for activation in PET images , 1996, Human brain mapping.

[45]  Donald M. Reeves,et al.  Transport of conservative solutes in simulated fracture networks: 1. Synthetic data generation , 2008 .

[46]  Robert J. Adler,et al.  Euler characteristics for Gaussian fields on manifolds , 2003 .

[47]  Muhammad Sahimi,et al.  Calculation of the effective permeabilities of field-scale porous media , 2000 .

[48]  S. Silliman,et al.  Monitoring Hydraulic Gradient Using Three-Point Estimator , 1998 .

[49]  L. Lake,et al.  A new approach to shale management in field scale simulation models , 1982 .

[50]  Clayton V. Deutsch,et al.  Geostatistical Software Library and User's Guide , 1998 .

[51]  S. Shtrikman,et al.  A Variational Approach to the Theory of the Effective Magnetic Permeability of Multiphase Materials , 1962 .

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