Abstract
The macroscopic equations that govern the processes of one- and two-phase flow through heterogeneous porous media are derived by using the method of multiple scales. The resulting equations are mathematically similar to the point equations, with the fundamental difference that the local permeabilities are replaced by effective parameters. The method allows the determination of these parameters from a knowledge of the geometrical structure of the medium and its heterogeneities. The technique is applied to determine the effective parameters for one- and two-phase flows through heterogeneous porous media made up of two homogeneous porous media.
Similar content being viewed by others
Abbreviations
- C f :
-
Compressibility coefficient
- f :
-
Vector function, defined in Equation (80)
- g :
-
Gravity acceleration vector
- h :
-
Vector function, defined in Equation (24)
- n :
-
Normal unit vector
- k :
-
Relative permeability
- k eff :
-
Effective relative permeability
- K :
-
Absolute permeability tensor
- K eff :
-
Effective permeability tensor
- p :
-
Pressure
- p c :
-
Capillary pressure
- P :
-
Modified pressure
- P c :
-
Modified capillary pressure
- r :
-
Position vector
- S :
-
Without subscript, saturation of the α-phase With subscript i, saturation of the i-phase
- t :
-
Time
- T :
-
Temperature Characteristic time
- v :
-
Superficial velocity vector
- x :
-
Position vector of the macroscopic scale
- y :
-
Position vector of the microscopic scale
- z :
-
Position vector equivalent to r, constrained to the unit cell
- δ :
-
Parameter defined by Equation (91)
- ε :
-
Perturbation parameter
- κ :
-
Absolute permeability ratio
- μ :
-
Viscosity
- ξ :
-
Volumetric fraction of porous medium 1
- ϱ :
-
Density
- τ :
-
Tortuosity tensor
- φ :
-
Porosity
- ψ :
-
Gravitational potential
- Ψ:
-
Scalar function, defined in Equation (23)
- 0:
-
Denotes reference value
- 1:
-
Referring to porous medium 1
- 2:
-
Referring to porous medium 2
- (i):
-
Denotes ith-order term in an asymptotic expansion
- α :
-
Relative to the α-phase
- β :
-
Relative to the β-phase
References
Aifantis, E. C., 1980, On Barenblatt's problem, Int. J. Engng. Sci. 18, 857–867.
Anderson, T. B. and Jackson, R., 1967, A fluid mechanical description of fluidized beds, Ind. Eng. Chem. Fund. 6, 527–533.
Bachmat, Y. and Bear, J., 1986, Macroscopic modelling of transport phenomena in porous media. 1: The continuum approach, Transport in Porous Media 1, 213–240.
Barenblatt, G. I., Zheltov, I. P., and Kochina, I. N., 1960, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks (strata), J. Appl. Math. U.S.S.R. 24, 852–864.
Barenblatt, G. I., 1964, On the motion of a gas-liquid mixture in porous fissured media, Izv. Akad. Nauk. S.S.S.R., Mekh. Machionst 3, 47–50.
Bear, J. and Bachmat, J., 1986, Macroscopic modelling of transport phenomena in porous media. 2: Applications to mass, momentum and energy transport, Transport in Porous Media 1, 241–270.
Bensoussan, A., Lions, J. L., and Papanicolau, G., 1978, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam.
Bourgeat, A., 1984, Homogenized behavior of two-phase flows in naturally fractured reservoirs with uniform fractures distribution, Comp. Meth. Applied Mech. Eng. 47, 205–216.
Cala, M. A. and Greenkorn, R. A., 1986, Velocity effects on dispersion in porous media with a single heterogeneity, Water Resour. Res. 22, 919–926.
Carbonell, R. G. and Whitaker, S., 1984, Heat and mass transport in porous media, in J. Bear and Y. Corapcioglu (eds.), Fundamentals of Transport in Porous Media, Martinus Nijhoff, Dordrecht.
Chang, H.-C., 1982, Multiscale analysis of effective transport in periodic heterogeneous media, Chem. Engng. Commun. 15, 83–91.
Chavent, G., 1976, A new formulation of diphasic incompressible flows in porous media, Lecture Notes in Mathematics 503, 258–270.
Dagan, G., 1986, Statistical theory of groundwater flow and transport: Pore to laboratory, laboratory to formation, and formation to regional scale, Water Resour. Res. 22, 120S-134S.
Gelhar, L. W., 1986, Stochastic subsurface hydrology. From theory to applications, Water Resour. Res. 22, 135S-145S.
Gilman, J. R. and Kazemi, H., 1982, Improvements in the simulation of naturally fractured reservoirs, paper SPE 10511, presented at the 6th. SPE Symposium on Reservoir Simulation, New Orleans.
Haldorsen, H. H. and Lake, L. W., 1984, A new approach to shale management in field scale problems, Soc. Pet. Eng. J. (August) 447–457.
Keller, J. B., 1977, Effective behavior of heterogeneous media, in U. Landman (ed.), Statistical Mechanics and Statistical Methods in Theory and Application, Plenum, New York.
Keller, J. B., 1980, Darcy's law for flow in porous media and two-space method, in R. L. Sternberg, A. J. Kalinowsky, and J. S. Papadakis (eds.), Nonlinear Partial Differential Equations in Engineering and Applied Science, Marcel Dekker, New York.
Mantoglou, A. and L. W. Gelhar, 1987a, Stochastic modeling of large-scale transient unsaturated flow systems, Water Resour. Res. 23, 37–46.
Mantoglou, A. and Gelhar, L. W., 1987b, Capillary tension head variance, mean soil moisture content, and effective specific soil moisture capacity of transient unsaturated flow in stratified soils, Water Resour. Res. 23, 47–56.
Mantoglou, A. and Gelhar, L. W., 1987c, Effective hydraulic conductivity of transient unsaturated flow in stratified soils, Water Resour. Res. 23, 57–67.
Marle, C. M., 1967, Écoulements monophasiques en milieux poreux, Rev. Inst. Français du Pétrole 22, 1471–1509.
Pironti, F. and Whitaker, S., 1986, Aplicación del Método de Escalas Múltiples al Proceso de Difusión con Reacción en Catalizadores Bimodales, Acta Cientifica Venezolana, in press.
Ryan, D., Carbonell, R. G., and Whitaker, S., 1980, Effective diffusivities under reactive conditions, Chem. Engng. Sci. 35, 10–15.
Slattery, J. C., 1967, Flow of viscoelastic fluids through porous media, AIChE J. 13, 1066–1071.
Suddicky, E. A. and Frind, E. O., 1982, Contaminant transport in fractured porous media. Analytical solutions for a system of parallel fractures, Water Resour. Res. 18, 1634–1642.
Thomas, E. P., 1986, Understanding fractured oil reservoirs, Oil & Gas J. (July 7), 75–79.
Van Golf-Racht, T. D., 1981, Fundamentals of Fractured Reservoir Engineering, Elsevier, Amsterdam.
Warren, J. E. and Root, P. J., 1963, The behavior of naturally fractured reservoirs, Soc. Pet. Eng. (September), 245–255.
Whitaker, S., 1967, Diffusion and dispersion in porous media, AIChE J. 13, 420–427.
Whitaker, S., 1986a, Transport processes with heterogeneous chemical reaction, in A. E. Cassano and S. Whitaker (eds.), Chemical Reactor Analysis and Design, Gordon and Breach, New York.
Whitaker, S., 1986b, Flow in porous media I: A theoretical derivation of Darcy's law, Transport in Porous Media 1, 3–26.
Whitaker, S., 1986c, Flow in porous media II: The governing equations for immiscible, two-phase flow, Transport in Porous Media 1, 105–126.
Whitaker, S., 1986d, Flow in porous media III: Deformable media, Transport in Porous Media 1, 127–154.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Sáez, A.E., Otero, C.J. & Rusinek, I. The effective homogeneous behavior of heterogeneous porous media. Transp Porous Med 4, 213–238 (1989). https://doi.org/10.1007/BF00138037
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00138037