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To obtain the a_{ij} coefficients in the squirt model, we first note that
these coefficients are defined under conditions where
(no fluid passing between the porous grains and the principal porespace).
Under these conditions,
the rate of
fluid depletion of a sample
(rate of fluid volume being extruded
from the principal pore space via
the exterior sample surface as normalized by the sample volume)
is due to the difference between the rate of dilatation of the principal
porespace (denoted here as )
and the rate at which
fluid in the pores is dilating . If we
also perform a volume average of equation (3) over the porous grain space
and use the notation that
we obtain
the following three equations
 
(81) 
 (82) 
 (83) 
The macroscopic dilatation of interest is .
In order to obtain the macroscopic
compressibility laws for porousgrain/principalporespace composite, we
introduce linear response laws of the form
 
(84) 
 (85) 
where the a_{i} and b_{i} must be found.
We note immediately that from the definition
one has
 
(86) 
which must hold true for any variation of the independent pressure variables so
that a_{1}=1/v_{2}, a_{2} =  v_{1}/v_{2}, a_{3} = 0.
To obtain the b_{i}, we now
combine the above into the macroscopic laws
 

 (87) 
 (88) 
 (89) 
and use the fact that the coefficients of the matrix must be symmetric
(a_{ij} = a_{ji}). With a_{11} = 1/K corresponding to the overall
drained frame modulus of the composite (to be independently specified),
we obtain v_{1} b_{1} = 1/K  1/K_{2}^{d}, v_{1} b_{2} = 1/K + (1+v_{1})/K_{2}^{d},
and . The final a_{ij} coefficients are
exactly
 
(90) 
 (91) 
 (92) 
 (93) 
 (94) 
 (95) 
Reasonable models for K and K^{d}_{2} will be discussed shortly.
Next: Squirt Transport
Up: SQUIRTFLOW MODEL
Previous: SQUIRTFLOW MODEL
Stanford Exploration Project
10/14/2003