structural geology, propagation mechanics and - Stanford School of ...

periodic, f-scale block, denoted , is related to the c-scale pressure gradient by the
effective permeability tensor k * :
k G
* 1
u = - ⋅ . (6)
µ
To calculate k * , we solve the pressure equation (4) over the f-scale block and then
compute the average flux through it. The boundary conditions for this solution
assume periodicity of the system up to the c-scale, as well as the imposition of the
pressure gradient G. The boundary specifications require that flow across the domain
boundaries be periodic over the f-scale block, and that the pressure be periodic but with a
jump due to the large-scale gradient. Boundary conditions can be specified explicitly for
an f-scale block (Figure 6.8) by resolving G into its two components: G = G1i1 + G2i2,
where i1 and i2 are unit vectors in the coordinate directions f1 and f2, respectively.
Comparing pressures and velocities on opposing sides with unit normals n1 and n2 yields:
yields:
p ( f , f 0)
= p(
f , f = b)
− G × b
(7a)
1
2 = 1 2
2
( 1 , f 2 = 0)
⋅n
1 = −u(
f1,
f 2 = b)
n2
u f ⋅ . (7b)
Comparing pressures and velocities on opposing sides with unit normals n3 and n4
p ( f 0,
f ) = p(
f = a,
f ) − G × a
(7c)
1 = 2
1 2 1
( 1 = 0, f 2 ) ⋅n
3 = −u(
f1
= a,
f 2 ) n4
u f ⋅ . (7d)
To compute the complete effective permeability tensor k * , two problems must be
solved, one for which G1=0, G2≠0 and one for which G1≠0, G2=0. By specifying a
reference value for pressure at a single point in the domain, equation (4) can be solved
subject to equations (7). The flux through the f-scale block can then be determined by
integrating the velocities, and finally k * computed from equation (6). Durlofsky (1991)
presents this computational approach in full detail.
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