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6.1. Parallel
Effective permeability for any idealized parallel DB pattern can be calculated
analytically, as previously reported (Antonellini and Aydin, 1994; Taylor et al., 1999).
Our numerical modeling results matched these exact analytical solutions for parallel
patterns, so only the latter are described here for the purpose of building intuition. The
analytical methods are based on the general analysis of flow in layered media by Freeze
and Cherry, pages 30-34 (1979).
Band-parallel effective permeability, k , can be calculated as a weighted arithmetic
average of the band and matrix permeability values:
w k + * b b ( W − wb)km k = p
W
*
p
where kb is the DB permeability, km is the matrix permeability, wb is cumulative DB
thickness, and W is the width of the area studied. Note that using wb/W as a measure of
volume fraction assumes DB continuity in the third dimension, as suggested by field
observations in the Aztec. Band-normal effective permeability can be calculated as the
weighted harmonic average:
k
*
n
=
( ( w / k ) + ( W − w ) / k )
b
b
W
b
m
The band-parallel and band-normal effective permeabilities represent the maximum
and minimum (i.e. principal) values, respectively. An effective permeability tensor, k * ,
for the parallel system thus reads:
*
* ⎡k
⎤ p 0
k = ⎢ * ⎥
(10)
⎣ 0 kn
⎦
Band-normal and band-parallel effective permeability values were calculated for kb/km
ratios of both 10 -2 and 10 -3 and DB volume fractions (wb/W) from 0% to 100% (Figure
6.9). Band-normal effective permeability drops much more quickly at low wb/W than
does band-parallel effective permeability. For kb/km=10 -3 , band-normal effective
permeability is already reduced by an order of magnitude at wb/W=1% (1 mm bands
spaced at 10 cm) and 2 orders of magnitude at wb/W=10% (1cm bands spaced at 10 cm).
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