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The main results used here can be derived using
uniform expansion, or selfsimilar, methods
analogous to ideas used in thermoelasticity
by Cribb (1968) and in singleporosity poroelasticity
by Berryman and Milton (1991). Cribb's method provided a simpler and more intuitive derivation of Levin's
earlier results on thermoelastic expansion coefficients (Levin, 1967).
Our results also provide a simpler
derivation of results obtained by Berryman and Pride (2002)
for the doubleporosity coefficients.
Related methods in other applications to micromechanics are called ``the theory of uniform fields'' by some authors
(Dvorak and Benveniste, 1997).
First assume two distinct phases at the macroscopic level:
a porous matrix phase with the effective properties K_{d}^{(1)}, G_{d}^{(1)},
K_{m}^{(1)}, (which are drained bulk and shear moduli,
grain or mineral bulk modulus, and porosity of phase 1 with analogous
definitions for phase 2), occupying volume fraction V^{(1)}/V = v^{(1)}
of the total volume and a macroscopic crack or joint phase occupying the
remaining fraction of the volume V^{(2)}/V = v^{(2)} = 1  v^{(1)}.
The key feature distinguishing the two phases  and therefore
requiring this analysis  is the very high fluid permeability
of the crack or joint phase and the relatively lower permeability
(but higher fluid volume content)
of the matrix phase.
doublepor
Figure 2
Blowup showing a detail that illustrates how each one of the grains
is composed of two very different types of porous materials: one being
a storage material (high porosity and low permeability) and one
a transport material (low porosity and high permeability).

 
In the doubleporosity model,
there are three distinct pressures: confining pressure ,porefluid pressure [for the storage porosity],
and jointfluid pressure [for the transport porosity].
(See Figures 1 and 2.)
Treating and
as the independent variables in our double
porosity theory, I define the dependent
variables ,,and
,which are respectively the total volume dilatation, the
increment of fluid content in the matrix phase, and the
increment of fluid content in the joints.
The fluid in the matrix is the same as that
in the cracks or joints, but the two fluid regions may
be in different states of average stress and, therefore, need
to be distinguished by their respective superscripts.
Linear relations among strain, fluid content, and pressure take the
symmetric form
 
(1) 
following Berryman and Wang (1995) and Lewallen and Wang (1998). It is easy to check that a_{11} = 1/K_{d}^{*}, where K_{d}^{*} is the overall
drained bulk modulus of the system.
I now find analytical expressions for the remaining five constants
for a binary composite system.
The components of the
system are themselves porous materials 1 and 2, but each is assumed to
be what I call a ``Gassmann material'' satisfying
 
(2) 
for material 1 and a similar expression for material 2.
The new constants appearing on the right are the drained bulk modulus
K_{d}^{(1)} of material 1, the corresponding BiotWillis
(Biot and Willis, 1957) coefficient , and the Skempton (1954)
coefficient B^{(1)}.
The volume fraction v^{(1)} appears here in order to correct for the
difference between a global fluid content and the corresponding
local variable for material 1.
The main special characteristic of a Gassmann (1951)
porous material
is that it is composed of only one type of solid constituent,
so it is ``microhomogeneous'' in its solid component, and
in addition the porosity
is randomly, but fairly uniformly, distributed so there is a
welldefined constant porosity associated with material 1, etc.
To proceed further, I ask this question: Is it possible to
find combinations of ,, and so that the expansion or
contraction of the system is spatially uniform or selfsimilar?
Or equivalently, can I find uniform
confining pressure , and porefluid pressures
and , so that all
these scalar conditions can be met simultaneously?
If so, then results for system constants
can be obtained purely algebraically without ever having to solve
equilibrium equations of the mechanics. I initially set , as this
condition of uniform confining pressure is clearly a requirement for the
selfsimilar thought experiment to be
a valid solution of stress equilibrium equations.
So, the first condition to be considered is the equality of
the strains of the two constituents:
 
(3) 
If this condition is satisfied, then the two constituents are
expanding or contracting at the same rate and it is clear that
selfsimilarity prevails, since
 
(4) 
If I imagine that and
are fixed, then I need an appropriate value
of , so that (3) is
satisfied. This requires that
 
(5) 
showing that, for undrained conditions,
can almost always be chosen so the
uniform expansion takes place. Using (5), I now eliminate from
the remaining equality so
 
(6) 
where is given by (5).
Making the substitution and then noting that and
were chosen independently and arbitrarily,
I find the resulting coefficients must each vanish. The
two equations I obtain are
 
(7) 
and
 
(8) 
Since a_{11} is assumed to be known, (7)
can be solved directly,
giving
 
(9) 
Similarly, with a_{13} known, substituting into (8)
gives
 
(10) 
So, formulas for three of the six coefficients are now known.
[Also, note the similarity of the formulas (9) and
(10), i.e., interchanging indices 1 and 2 on the
right hand sides takes us from one expression to the other.]
To evaluate the remaining coefficients, I consider what
happens to fluid increments during the selfsimilar expansion.
I treat only material 1, but the
equations for material 2 are completely analogous.
From the preceding equations,
 
(11) 
Again substituting for from
(5) and noting that the resulting equation contains
arbitrary values of and , the
coefficients of these terms must vanish separately. Resulting
equations are
 
(12) 
and
 
(13) 
Solving these equations, I obtain
 
(14) 
and
 
(15) 
Performing the corresponding calculation for produces
formulas for a_{32} and a_{33}. Since (14) is already
symmetric in component indices, the formula for a_{32} provides
nothing new. The formula for a_{33} is easily
seen to be identical in form to a_{22}, but indices 1 and 2
are interchanged.
Formulas for all five of the nontrivial coefficients of
double porosity have now been determined.
These results also show how the constituent properties
K_{d}, , B upscale at the macrolevel for a twoconstituent
composite (Berryman and Wang, 1995; Berryman and Pride, 2002).
I find
 
(16) 
and
 
(17) 
Note that all the important formulas [(8),(9),(11)(14)] depend on the
overall drained bulk modulus K_{d}^{*} of the system. So far this quantity
is unknown and therefore must still be determined independently
either by experiment or by another analytical method.
It should also be clear that some parts (but not all) of the preceding
analysis generalize to the multiporosity problem (i.e., more
than two porosity types). A discussion of the issues surrounding
solvability of the multiporosity problem has been presented elsewhere
(Berryman, 2002).
Next: UPSCALING MODEL FOR GEOMECHANICS
Up: Berryman: Geomechanical analysis with
Previous: INTRODUCTION
Stanford Exploration Project
10/31/2005