I now present a very concise, but nevertheless complete,
derivation of Gassmann's famous results. For the sake of simplicity,
the analysis that follows is limited to isotropic systems, but it can be
generalized with little difficulty to anisotropic systems
(Gassmann, 1951; Brown and Korringa, 1975; Berryman, 1998).
Gassmann's (1951) equations relate the bulk and shear moduli
of a saturated isotropic porous,
monomineralic medium to the bulk and shear moduli of the same medium in the
drained case and shows furthermore that the shear modulus *must be*
mechanically independent of the presence of the fluid.
An important implicit assumption is that there is no chemical
interaction between porous rock and fluid that affects the moduli;
if such effects are present, we assume the medium is drained (rather
than dry) but otherwise neglect chemical effects for this argument.
Gassmann's paper is concerned with the quasistatic (low frequency)
analysis of the elastic moduli and that is what we emphasize here
also. Generalization to higher frequency effects and complications
arising in wave propagation due to frequency dispersion are well
beyond the scope of what we present.

In contrast to simple elasticity with stress tensor and
strain tensor *e*_{ij}, the presence of a saturating pore
fluid in porous media introduces the possibility of
an additional control field and an additional type of strain variable.
The pressure *p*_{f} in the fluid is the new field parameter that can be
controlled. Allowing sufficient time (equivalent to a low frequency
assumption) for global pressure equilibration will
permit us to consider *p*_{f} to be a constant throughout the
percolating (connected) pore fluid, while restricting the analysis
to quasistatic processes.
The change in the amount of fluid mass contained in the pores
is the new type of strain variable, measuring how much of the original
fluid in the pores is squeezed out during the compression of the
pore volume while including the effects of compression or expansion
of the pore fluid itself due to changes in *p*_{f}.
It is most convenient to write the resulting equations in terms of
compliances *s*_{ij} rather than stiffnesses *c*_{ij},
so for an isotropic porous medium (chosen only
for the sake of its simplicity)
the basic equation to be considered takes the form:

e_11 e_22 e_33 - =
s_11 & s_12 & s_12 & -
s_12 & s_11 & s_12 & -
s_12 & s_12 & s_11 & -
-& -& -&
_11 _22 _33 - p_f .
The constants and appearing in the matrix on the right hand side
will be defined later.
It is important to write the equations this way rather than using the
inverse relation in terms of the stiffnesses, because the compliances *s*_{ij}
appearing in (poro1) are simply and directly related
to the drained constants and
(the Lamé parameters for the isotropic porous medium in the drained
case) in the same way they are related in normal
elasticity (the matrix *s*_{ij} is just the inverse of the matrix
*c*_{ij}), whereas the individual stiffnesses *c ^{*}*

s_11 = 1E_dr = _dr+_dr_dr(3_dr+2_dr) = 19K_dr + 13_dr and

s_12
= - _drE_dr
= 19K_dr - 16_dr,
where the drained Young's modulus *E*_{dr} is defined
in terms of the drained bulk modulus *K*_{dr} and shear modulus by the second equality of (s11) and the drained Poisson's
ratio is determined by

Figure 1

The fundamental results of interest (Gassmann's equations) are found by considering the saturated (and undrained) case such that

0, which -- by making use of (poro1) -- implies that the pore pressure must respond to external applied stresses according to

p_f = - (_11 + _22 + _33).
Equation (buildup) is often called the ``pore-pressure buildup''
equation (Skempton, 1954).
Then, using this result to eliminate both and *p*_{f} from
(poro1), I obtain

e_11 e_22 e_33 =
s^*_11 & s^*_12 & s^*_12
s^*_12 & s^*_11 & s^*_12
s^*_12 & s^*_12 & s^*_11
_11 _22 _33
=
[s_11 & s_12 & s_12
s_12 & s_11 & s_12
s_12 & s_12 & s_11 -
^21 & 1 & 1
1 & 1 & 1
1 & 1 & 1 ]
_11 _22 _33 ,
where *s ^{*}*

s^*_11 = s_11 - ^2, and another for the off-diagonal compliance

s^*_12 = s_12 - ^2.
If *K ^{*}* and are respectively the undrained bulk and shear moduli,
then (s11) and (s12) together with
(s11*) and (s12*) imply that

19K^* + 13^* = 19K_dr + 13_dr - ^2, and

19K^* - 16^* = 19K_dr - 16_dr - ^2. Subtracting (s12imp) from (s11imp) shows immediately that or equivalently that

^* = _dr.
Thus, the first *result* of Gassmann is that, for purely mechanical
effects, the shear modulus
for the case with trapped fluid (undrained) is the same as that for
the case with no fluid (drained). Then, substituting
(shearresult) back into either (s11imp) or (s12imp)
gives one form of the result commonly known as Gassmann's equation
for the bulk modulus:

I want to emphasize that the analysis presented shows clearly that
(shearresult) is a definite *result* of this analysis,
*not an assumption*. In fact, we must have (shearresult)
in order for (Gassmanneqn) to hold, and furthermore, if
(Gassmanneqn) holds, then so must (shearresult).
Thus, monitoring any changes in shear modulus with changes of
fluid content (say through shear velocity measurements) provides a test
of both Gassmann's assumptions (homogeneous frame, no chemical
effects, & low frequencies) and results.

To obtain one of the more common forms of Gassmann's result for the bulk modulus, first note that

3= 1K_dr - 1K_g K_dr,
where *K*_{g} is the grain modulus of the solid constituent present
and is the Biot-Willis parameter (Biot and Willis, 1957).
Furthermore, the parameter is related through (buildup)
to Skempton's pore-pressure buildup coefficient *B*, so that

3 = B. Substituting these results into (Gassmanneqn) gives

K^* = K_dr1-B, which is another form (Carroll, 1980) of Gassmann's standard result for the bulk modulus.

10/25/1999