GASSMANN'S EQUATIONS AND FRACTURED MEDIA

It is also important to make a connection between the fracture results quoted above and Gassmann's results (Gassmann, 1951; Berryman, 1999) for fluid saturated and undrained porous media. Even very flat cracks could harbor some fluid at various times and the question is how this fluid affects the response of the cracks.

Recall first that Gassmann's result on the effect of fluids in a porous medium in anisotropic media can be expressed by using a compliance correction matrix of the form:  

$$\Delta S_{ij} = -\gamma^{-1}\left(\begin{array} {cccccc} \be... ... & 0 & & \cr & & & & 0 & \cr & & & & & 0 \cr\end{array}\right),$$ (10)

where the fluid bulk modulus appears only in the factor $\gamma$,and the coefficients $\beta_1$,$\beta_2$,$\beta_3$ satisfy a sumrule of the form $\sum_{i=1}^3 \beta_i = \alpha/K_d = 1/K_d - 1/K_m$,and $\alpha = 1 - K_d/K_m$ is the Biot-Willis coefficient (Biot and Willis, 1957). The bulk moduli K_d and K_m are, respectively, the drained (porous) bulk modulus of the overall system and the mineral modulus whenever there is only one mineral present in the system (as I will assume here for the present). [The scalar drained modulus K_d for an anisotropic system is identical to the Reuss average for the bulk modulus of the compliance matrix.] When the system is responding anisotropically as in the case of a set of cracks having vertical symmetry axis in the example (3), we can easily make (10) compatible with the structure of (3) by first ignoring the coefficient $\eta_1$ (which is known to be quite small anyway), and then setting $\beta_1 = \beta_2 = 0$.No other possibiities are available. This means that Gassmann's results are introduced into the anisotropic problem by making a fluid-dependent perturbation to $\eta_2\rho$ and ignoring $\eta_1\rho$, since its value is two orders of magnitude smaller.

Gassmann's formula for an isotropic medium can be written in the form:

$$K_u = K_d + \frac{\alpha^2}{(\alpha-\phi)/K_m + \phi/K_f},$$ (11)

where K_u is the undrained bulk modulus, K_d is the drained bulk modulus, K_m is the mineral or solid modulus, K_f is the bulk modulus of the pore fluid, $\phi$ is the porosity and $\alpha = 1 - K_d/K_m$.This result can be rearranged in order to express it in terms of compliances, instead of stiffnesses, as  

$$\frac{1}{K_u} - \frac{1}{K_d} = -\frac{\alpha}{K_d}\left[1 +... ... \phi}{K_f \alpha}\left(1-\frac{K_f}{K_m}\right)\right]^{-1}.$$ (12)

Now for fractured media having no other porosity $\phi$ except the fractures themselves, I have  

$$\phi = \frac{4\pi}{3} b^3\left(\frac{a}{b}\right)\frac{N}{V} = \frac{4\pi}{3}\left(\frac{a}{b}\right)\rho,$$ (13)

where the crack density is $\rho = Nb^3/V$,N/V is the number of cracks per unit volume, b is the radius of the (assumed) penny-shaped crack, and a/b is its aspect ratio. Substituting (13) into (12) shows that crack density $\rho$ is always multiplied by the factor (1 - K_f/K_m) in Gassmann's formula, and this result thereby provides a convenient means of introducing the fluid effects into the formulas for compliance in the presence of distributions of cracks.

The preceding review of Gassmann's original derivation shows that it is not appropriate to replace all occurrences of $\rho$ in (1) by $\rho(1 - K_f/K_m)$. Only those terms that determine the strain response to the principle stresses need to be considered. Furthermore, the analysis of the symmetry conditions has shown that only those terms involving $\eta_2$ need to be modified. If we neglect $\eta_1$ for the remainder of this argument (it is small anyway -- a 1% effect -- as stated previously), then we find that, for an isotropic system,  

$$\Delta S_{ij} \simeq \frac{2\eta_2\rho}{3}\left(\begin{array... ... 2 & & \cr & & & & 2 & \cr & & & & & 2 \cr\end{array}\right).$$ (14)

Changes in the pertinent compliances are therefore given by

$$\Delta\left(\frac{1}{K_u}\right) = 2\eta_2\rho(1-K_f/K_m)$$ (15)

for the undrained bulk modulus K_u, and

$$\Delta\left(\frac{1}{G_{eff}^r}\right) = 4\eta_2\rho(1-K_f/K_m)/3,$$ (16)

which is the change in the uniaxial shear modulus due the presence of cracks containing fluids. Then, I recover the result of Mavko and Jizba (1991) easily by noting first that the change in the undrained shear modulus for the isotropic system is

$$\Delta\left(\frac{1}{G_u}\right) = \frac{2}{5}\Delta\left(\frac{1}{G_{eff}^r}\right) + \frac{3}{5}\times0,$$ (17)

since there are two equal shear contributions from the upper left hand $3\times3$ submatrix of (14), while the three remaining contributions to shear compliance exhibit no fluid dependence and so do not contribute here. Then, finally, I have  

$$\Delta\left(\frac{1}{G_u}\right) = \frac{4}{15}\Delta\left(\frac{1}{K_u}\right),$$ (18)

in agreement with previous results of this type (Berryman et al., 2002b). So this formalism provides an efficient means of correctly deriving both old and new results.

Berryman et al. (2002b) show that the factor $\frac{4}{15}$in (18) holds strictly only for very flat cracks, and that the appropriate factor in other situations can be either higher or lower than $\frac{4}{15}$, depending on details. The neglected terms depending on the Sayers and Kachanov parameter $\eta_1$ provide very small corrections to the drained moduli, but actually have no dependence on fluid saturation, and so have no influence on the relationship between undrained (fluid saturated) moduli shown in (18).