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y identical blocks all subject to the same regional pressure gradient that drives flow.
This modeling approach derives from an application of homogenization theory as
presented in detail by Durlofsky (1991; 1992).
5.1. Governing equations
Single-phase, steady state, incompressible flow through a heterogeneous porous
media in the absence of sources or sinks can be described by Darcy’s law and the
requirements of continuity as:
1
u = − k ⋅∇p
µ
∇⋅u = 0 (2)
where u is the local fluid velocity vector, ∇ p is the local fluid pressure gradient
vector, µ is the fluid viscosity, and k is the tensor representing directional permeability.
The units of k are length squared, as can be demonstrated by dimensional analysis of
Equation (1). In a 2-D x-y coordinate system, k can be represented as:
⎡kxx
kxy⎤
k = ⎢ ⎥ . (3)
⎣kyx
kyy⎦
where kxy is the permeability component relating fluid velocity in the x-direction to
pressure gradient in the y-direction. Permeability is commonly measured in darcys (1
darcy = 9.87 x 10 -9 cm 2 ), and the native permeability of uniform sandstone ranges from
about 10 -2 to 10 2 darcys in any given direction (Freeze and Cherry, 1979).
The homogenization approach considers k to vary over two spatial scales—a fine
scale, f, and a coarse scale, c, with f