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 eij, 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 pf 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 pf 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 pf. It is most convenient to write the resulting equations in terms of compliances sij rather than stiffnesses cij, 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 sij 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 sij is just the inverse of the matrix cij), whereas the individual stiffnesses c*ij (the * superscript indicates the constants for the saturated case) obtained by inverting the equation in (poro1) must contain coupling terms through the parameters and that depend on the porous medium and fluid compliances. Using the standard relations for the isotropic moduli, I find that
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 Edr is defined in terms of the drained bulk modulus Kdr and shear modulus by the second equality of (s11) and the drained Poisson's ratio is determined by
_dr = _dr2(_dr+_dr).
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 pf 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*ij is the desired compliance including the effects of the trapped fluid, while sij is the compliance in the absence of the fluid. Since for elastic isotropy there are only two independent coefficients (s11 and s12), I find that (poro2) reduces to one expression for the diagonal compliance
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:
1K^* = 1K_dr - 9^2.
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 Kg 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.