Pore Fluid Effects on Shear Modulus in a Model of Heterogeneous Rocks, Reservoirs, and Granular Media Page: 4 of 17
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elastic and poroelastic coefficients are almost all scale in-
variant physical properties so the basic physical picture
presented works at all scales. The only exception is for
results concerning fluid permeability, which is definitely
not scale invariant, but therefore also excluded from con-
sideration here.) Then, the overall behavior of this sys-
tem can be determined/estimated using another method
from the theory of composites: the well-known Hashin-
Shtrikman bounds [Hashin and Shtrikman, 1962b]. In
this case the bounds of interest for the types of crystal
symmetry that arise are those first obtained by Pesel-
nick and Meister [1965] and later refined by Watt and
Peselnick [1980]. These bounds have been refined fur-
ther recently by the author [Berryman, 2004b; 2005]. In
particular, these latest refinements provide sufficient in-
sight into the resulting equations that self-consistent esti-
mates (lying between the rigorous bounds) of the elastic
constants can be formulated and very easily computed.
We find that the Peselnick-Meister-Watt upper and lower
bounds are already quite close together for this model
material, so the resulting self-consistent estimates are
very well constrained. The bounds then serve as error
bars on the self-consistent model estimates.
Although the model being proposed has several of the
key features of real rocks, reservoirs, and/or granular me-
dia such as soils, it is neither a claim of this presenta-
tion nor an intent of this work that the model should
be viewed as anything more than a caricature of these
complicated media. In this regard, the model follows in
the long-established tradition of effective medium theo-
ries [Maxwell, 1873; Einstein, 1906; Bruggeman, 1937]
see review by Berryman [1995] none of which when
initially proposed was viewed as anything more than such
a caricature; but nevertheless these models permitted
inclusion of desired features and enabled calculation of
useful estimates. It was only many years after the intro-
duction of these effective medium theories that bounding
methods were developed [Hill, 1952; Hashin and Shtrik-
man, 1962a,b], and then still later when it was shown
that there was a realizable microstructure implicit in
some of these effective medium theories so that their
predictions would necessarily always lie inside the rig-
orous bounds [Milton, 1985; Norris, 1985; Avellaneda,
1987]. The alternative approach we are developing here
uses instead a new prototype for effective medium theo-
ries: We start with a realizable microstructure; then we
compute the rigorous bounds; finally, we construct an
effective medium estimate that is consistent with those
bounds. This direct approach seems likely to be very ad-
vantageous in many situations both for scientific and for
engineering design purposes.
The method being introduced can be applied to a wide
variety of difficult technical issues concerning poroelas-
ticity of rocks and/or geomechanical coefficients of reser-
voirs and soils. The one issue that will be addressed at
length here is the question of how shear moduli in fully
saturated, partially saturated, and/or patchy saturated
porous rock or soil may or may not depend on mechanicalproperties of the pore fluids. The well-known fluid substi-
tution formulas of Gassmann [1951] (also see Berryman
[1999]) show that for isotropic, microhomogeneous
(single solid constituent) porous media the undrained
bulk modulus depends strongly on a pore-liquid's bulk
modulus, but in sharp contrast the undrained shear
modulus is not at all affected by changes in the pore-
liquid modulus. Since the system we are considering vi-
olates Gassmann's microhomogeneity constraint as well
as the the isotropy constraint in the vicinity of layer in-
terfaces, we expect that the shear modulus will in fact
depend on the fluid properties in this model [Mavko and
Jizba, 1991; Berryman and Wang, 2001; Berryman et al.,
2002b]. The semi-analytical model presented here allows
us to explore this issue in some detail, to show that over-
all shear modulus does depend on pore-fluid mechanical
properties, and to quantify these effects.
The next section introduces the basic tools used later
in the layer analysis. The third section reviews the
Peselnick-Meister-Watt bounds and presents the new for-
mulation of them. The fourth section summarizes the
results needed from poroelastic analysis. The fifth sec-
tion presents the main new results of the paper, including
four distinct scenarios that help to elucidate the behav-
ior of the overall shear modulus and contrast it to that
of the bulk modulus. The final section summarizes our
conclusions. Appendix A provides a brief proof of one
of the results used in the text concerning the behavior
of the effective stress coefficient for patchy saturation.
Appendix B shows that Hill's equation [Hill, 1963; 1964]
should be used cautiously in analysis of heterogeneous
rocks, reservoirs, and soils.
II. ELASTICITY OF LAYERED MATERIALS
We assume that a typical building block of the random
system is a small grain of laminate material whose elas-
tic response for such a transversely isotropic (hexagonal)
system can be described locally by:/11 e/c
022 C12
033 C13
0'23
0'31
012 \C12
C1
C13C13
C13
C332c44
ell
e22
33 (1)
C23'
e31
2c66 \E12/where a2j are the usual stress components for i, j =1 -3
in Cartesian coordinates, with 3 (or z) being the axis
of symmetry (the lamination direction for such a lay-
ered material). Displacement u2 is then related to strain
component e2j by e2j (a2/xt + &uk/&x2)/2. This
choice of definition introduces some convenient factors of
two into the 44, 55, 66 components of the stiffness matrix
shown in (1).
For definiteness we also assume that this stiffness ma-
trix in (1) arises from the lamination of N isotropic con-
stituents having bulk and shear moduli K, pm, in the2C44
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Berger, E. L. Pore Fluid Effects on Shear Modulus in a Model of Heterogeneous Rocks, Reservoirs, and Granular Media, article, March 23, 2005; Livermore, California. (https://digital.library.unt.edu/ark:/67531/metadc793705/m1/4/: accessed April 25, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.