structural geology, propagation mechanics and - Stanford School of ...
structural geology, propagation mechanics and - Stanford School of ...
structural geology, propagation mechanics and - Stanford School of ...
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6.1. Parallel<br />
Effective permeability for any idealized parallel DB pattern can be calculated<br />
analytically, as previously reported (Antonellini <strong>and</strong> Aydin, 1994; Taylor et al., 1999).<br />
Our numerical modeling results matched these exact analytical solutions for parallel<br />
patterns, so only the latter are described here for the purpose <strong>of</strong> building intuition. The<br />
analytical methods are based on the general analysis <strong>of</strong> flow in layered media by Freeze<br />
<strong>and</strong> Cherry, pages 30-34 (1979).<br />
B<strong>and</strong>-parallel effective permeability, k , can be calculated as a weighted arithmetic<br />
average <strong>of</strong> the b<strong>and</strong> <strong>and</strong> matrix permeability values:<br />
w k + * b b ( W − wb)km k = p<br />
W<br />
*<br />
p<br />
where kb is the DB permeability, km is the matrix permeability, wb is cumulative DB<br />
thickness, <strong>and</strong> W is the width <strong>of</strong> the area studied. Note that using wb/W as a measure <strong>of</strong><br />
volume fraction assumes DB continuity in the third dimension, as suggested by field<br />
observations in the Aztec. B<strong>and</strong>-normal effective permeability can be calculated as the<br />
weighted harmonic average:<br />
k<br />
*<br />
n<br />
=<br />
( ( w / k ) + ( W − w ) / k )<br />
b<br />
b<br />
W<br />
b<br />
m<br />
The b<strong>and</strong>-parallel <strong>and</strong> b<strong>and</strong>-normal effective permeabilities represent the maximum<br />
<strong>and</strong> minimum (i.e. principal) values, respectively. An effective permeability tensor, k * ,<br />
for the parallel system thus reads:<br />
*<br />
* ⎡k<br />
⎤ p 0<br />
k = ⎢ * ⎥<br />
(10)<br />
⎣ 0 kn<br />
⎦<br />
B<strong>and</strong>-normal <strong>and</strong> b<strong>and</strong>-parallel effective permeability values were calculated for kb/km<br />
ratios <strong>of</strong> both 10 -2 <strong>and</strong> 10 -3 <strong>and</strong> DB volume fractions (wb/W) from 0% to 100% (Figure<br />
6.9). B<strong>and</strong>-normal effective permeability drops much more quickly at low wb/W than<br />
does b<strong>and</strong>-parallel effective permeability. For kb/km=10 -3 , b<strong>and</strong>-normal effective<br />
permeability is already reduced by an order <strong>of</strong> magnitude at wb/W=1% (1 mm b<strong>and</strong>s<br />
spaced at 10 cm) <strong>and</strong> 2 orders <strong>of</strong> magnitude at wb/W=10% (1cm b<strong>and</strong>s spaced at 10 cm).<br />
158<br />
(8)<br />
(9)