Abstract
Heterogeneity, as it occurs in porous media, is characterized in terms of a scaling exponent, or fractal dimension. A feature of primary interest for two-phase flow is the mixing length. This paper determines the relation between the scaling exponent for the heterogeneity and the scaling exponent which governs the mixing length. The analysis assumes a linear transport equation and uses random fields first in the characterization of the heterogeneity and second in the solution of the flow problem, in order to determine the mixing exponents. The scaling behavior changes from long-length-scale dominated to short-length-scale dominated at a critical value of the scaling exponent of the rock heterogeneity. The long-length-scale-dominated diffusion is anomalous.
This is a preview of subscription content, log in via an institution to check access.
References
Y. Amirat, K. Hamdache, and A. Ziani, Homogénisation d'équations hyperboliques du première ordre et application aux écoulements miscibles en milieu poreux,Ann. Inst. H. Poincaré 6:397–417 (1989).
J.-P. Bouchaud, A. Georges, J. Kopik, A. Provata, and S. Redner, Superdiffusion in random velocity fields, Levich Institute Preprint.
J. Cannon, Continuous sample paths in quantum field theory,Commun. Math. Phys. 35:215–234 (1974).
P. Collela and O. Lanford, inConstructive Quantum Field Theory, G. Velo and A. Wightman, eds. (Springer-Verlag, New York, 1973), pp. 44–70.
F. J. Dyson, Existence of a phase transition in a one-dimensional Ising ferromagnet,Commun. Math. Phys. 12:212–215 (1969).
J. Glimm and A. Jaffe,Quantum Physics: A Functional Integral Point of View, 2nd ed. (Springer-Verlag, New York, 1987).
N. Goldenfield, O. Martin, Y. Oono, and F. Liu, Anomalous dimensions and the renormalization group in a nonlinear diffusion process,Phys. Rev. Lett. 64:1361–1364 (1990).
T. A. Hewett, Fractal distributions of reservoir heterogeneity and their influence on fluid transport, SPE 15386 (1986).
T. A. Hewett and R. A. Behrens, Conditional simulation of reservoir heterogeneity with fractals, SPE 18326 (1988).
L. Lake and H. Carroll, eds.,Reservoir Characterization (Academic Press, 1986).
A. Lallemand-Barres and P. Peaudecerf, Recherche des relations entre le valeur de la dispersivité macroscopic, ses autres caractéristiques et les conditions de mesure,Bull BRGM 2e Ser. Sec. III 1978(4).
J. F. Pickens and G. E. Grisak,Water Resources Res. 17:1191–1121 (1981).
S. W. Wheatcroft and S. W. Tyler,Water Resources Res. 24:566–578 (1988).
W. Zielke, Frequency-dependent friction in transient pipeline flow,J. Basic Eng., Trans. ASME (1968).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Glimm, J., Sharp, D.H. A random field model for anomalous diffusion in heterogeneous porous media. J Stat Phys 62, 415–424 (1991). https://doi.org/10.1007/BF01020877
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01020877