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The effective homogeneous behavior of heterogeneous porous media

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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.

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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

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Authors and Affiliations

  1. Departmento de Termodinámica y Fenómenos de Transferencia, Universidad Simón Bolívar, P.O. Box 89000, 1086A, Caracas, Venezuela

    A. E. Sáez

  2. Tecnología de Yacimientos, INTEVEP, S.A., P.O. Box 76343, 1070-A, Caracas, Venezuela

    Carlos J. Otero & Isak Rusinek

Authors
  1. A. E. Sáez
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  2. Carlos J. Otero
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  3. Isak Rusinek
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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

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  • DOI: https://doi.org/10.1007/BF00138037

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