Abstract of the paper ``Exact results for generalized Gassmann's equations
in composite porous media with two constituents'' with G. W. Milton
Wave propagation in fluid-filled porous media is governed by
Biot's equations of poroelasticity. Gassmann's relation
gives an exact formula for the poroelastic parameters
when the porous medium contains only
one type of solid constituent. The present paper generalizes
Gassmann's relation and derives
exact formulas for two elastic parameters needed to describe
wave propagation in a conglomerate of two porous phases. The
parameters were first introduced by Brown and Korringa
when they derived a generalized form of Gassmann's equation for
conglomerates. These elastic parameters
are the bulk modulus K_s associated with changes
in the overall volume of the conglomerate and the bulk modulus K_phi
associated with the pore volume when the fluid
pressure (p_f) and confining pressure (p_c) are increased, keeping the
differential pressure (p_d = p_c - p_f) fixed. These moduli are properties
of the composite solid frame (drained of fluid) and are shown
here to be completely
determined in terms of the bulk moduli associated with the
two solid constituents, the bulk moduli of the drained
conglomerate and the drained phases, and the porosities
in each phase. The pore structure of each phase is assumed uniform
and smaller than the grain size in the conglomerate.
The relations found are completely independent
of the pore microstructure and provide a means of analyzing
experimental data.
The key idea leading to the exact results is this: Whenever
two scalar fields (in our problem p_f and p_d) can be
independently varied in a linear composite containing only two
constituents, there exists a special value of the ratio of the increments
of these two fields corresponding to an overall expansion or contraction
of the medium with no change of relative shape.
This fact guarantees that a set of consistency relations exists
among the constituent moduli and the effective moduli, which then
determine all but one of the effective constants.
Thus, K_s and K_phi are determined in terms of the drained
frame modulus K and the constituents' moduli.
Because the composite is linear, the coefficients found for the special
value of the increment ratio are also the exact coefficients
for an arbitrary ratio.
Since modulus K is commonly measured while the other two are not, these exact
relations provide a significant advance in our ability to predict
the response of porous materials to pressure changes.
It is also shown that additional results (such as
rigorous bounds on the parameters) may be easily obtained
by exploiting an analogy between the equations of
thermoelasticity and those of poroelasticity.
The method used to derive these results may also be used to
find exact expressions for three component composite porous materials
when thermoelastic constants of the components and the
composite are known.
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