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The fluid pressure and confining pressure
in the grains can again be developed as asymptotic series
in [as in equations (60)(61)].
The zeroorder response corresponds to the static limit in
which the fluid pressure is everywhere the same and given by
with
B_{o} = (a_{12}+a_{13})/(a_{22} + 2 a_{23} + a_{33}) and with the
a_{ij} as given by equations (90)(95).
The detailed result for B_{o} can be expressed
 

 (98) 
which reduces to the standard Gassmann expression given in the
appendix (with a total porosity given by ) when
B_{2} and are themselves given by the Gassmann expressions.
In this same zeroorder limit, the undrained bulk modulus
is defined as 1/K_{o}^{u} = a_{11} + (a_{12} + a_{13}) B_{o} which also reduces to
the standard Gassmann expression when B_{2} and are
themselves given by Gassmann expressions.
The leadingorder in correction to uniform fluid pressure
is thus governed by the problem
 
(99) 
 (100) 
 (101) 
Here, p_{c2}^{(0)} is the local confining pressure in the grain
space in the static limit that can be written
.
The average static confining pressure throughout the grains is
determined from equation (84) with and
to yield
 
(102) 
The deviations thus volume integrate to zero
and
are formally defined
 
(103) 
The local perturbations are thus highly sensitive to the detailed
nature of the grain packing and grain geometry. Fortunately, these perturbations
do not play an important role in the theory.
The fluid pressure in the grains is now written in the scaled
form
 
(104) 
where the potential is independent of and
is a solution of the elliptic problem
 
(105) 
 (106) 
 (107) 
To leadingorder in , an average of equation (104) gives
 
(108) 
 (109) 
where the squared length L_{2}^{2} is defined
 
(110) 
with overlines denoting volume averages over the grain space and with
the potential defined as the solution of
 
(111) 
 (112) 
 (113) 
Although it is not generally true that
for all grain geometries, we nonetheless expect this integral to be small
because is a smooth function and .
The local perturbations in the static confining pressure
require a solution of the static displacements throughout
the entire grain spacea daunting numerical task. Whenever
the length L_{2} needs to be estimated, such as in the numerical
results that follow, our approach is to simply
use
the reasonable approximation that
.
Last, from the definition of the internal transfer
we have that to leading order in
 
(114) 
 (115) 
 (116) 
The normal in equation (114) is outward to phase 1
which accounts for the sign change in equation(115). Note as
well that equation (115) is a volume average of equation (99)
while equation (116) follows from equations (102) and
(109).
The desired limit is thus
.
Next: Highfrequency limit of
Up: Squirt Transport
Previous: Squirt Transport
Stanford Exploration Project
10/14/2003