Gassmann's equation

Gassmann's equations are a set of two equations describing the isotropic elastic constants of an ensemble consisting of an isotropic, homogeneous porous medium with a fully connected pore space, saturated by a compressible fluid at pressure equilibrium.

First published in German by Fritz Gassmann, the original work was only later translated in English long after the adoption of the equations in standard geophysical practice.

Gassmann's equations remain the most common way of performing fluid substitution—predicting the elastic behaviour of a porous medium under a different saturant to the one measured.

Procedure
These formulations are from Avseth et al. (2006).

Given an initial set of velocities and densities, $$V_{P}^{(1)}$$, $$V_{S}^{(1)}$$, and $$\rho ^{(1)}$$ corresponding to a rock with an initial set of fluids, you can compute the velocities and densities of the rock with another set of fluid. Often these velocities are measured from well logs, but might also come from a theoretical model.

Step 1: Extract the dynamic bulk and shear moduli from $$V_\mathrm{P}^{(1)}$$, $$V_\mathrm{S}^{(1)}$$, and $$\rho ^{(1)}$$:


 * $$K_\mathrm{sat}^{(1)} = \rho \left ((V_\mathrm{P}^{(1)})^{2}-\frac{4}{3}(V_\mathrm{S}^{(1)})^{2} \right) $$


 * $$\mu_\mathrm{sat}^{(1)} = \rho (V_\mathrm{S}^{(1)})^{2}$$

Step 2: Apply Gassmann's relation, of the following form, to transform the saturated bulk modulus:


 * $$\frac{K_\mathrm{sat}^{(2)}}{K_\mathrm{mineral}-K_\mathrm{sat}^{(2)}}-\frac{K_\mathrm{fluid}^{(2)}}{\phi (K_\mathrm{mineral}-K_\mathrm{fluid}^{(2)})}=\frac{K_\mathrm{sat}^{(1)}}{K_\mathrm{mineral}-K_\mathrm{sat}^{(1)}}-\frac{K_\mathrm{fluid}^{(1)}}{\phi (K_\mathrm{mineral}-K_\mathrm{fluid}^{(1)})}$$

where $$K_\mathrm{sat}^{(1)}$$ and $$K_\mathrm{sat}^{(2)}$$ are the rock bulk moduli saturated with fluid 1 and fluid 2, $$K_\mathrm{fluid}^{(1)}$$ and $$K_\mathrm{fluid}^{(2)}$$ are the bulk moduli of the fluids themselves, and $$\phi$$ is the rock's porosity.

Step 3: Leave the shear modulus unchanged (rigidity is independent of fluid type):


 * $$\mu_\mathrm{sat}^{(2)}=\mu_\mathrm{sat}^{(1)}$$

Step 4: Correct the bulk density for the change in fluid:


 * $$ \rho^{(2)}= \rho^{(1)}+\phi (\rho_\mathrm{fluid}^{(2)} -\rho_\mathrm{fluid}^{(1)})$$

Step 5: recompute the fluid substituted velocities


 * $$V_\mathrm{P}^{(2)}=\sqrt \frac{K_\mathrm{sat}^{(2)}+\frac{4}{3} \mu_\mathrm{sat}^{(2)}}{\rho^{(2)}} $$


 * $$V_\mathrm{S}^{(2)} = \sqrt \frac{\mu_\mathrm{sat}^{(2)}}{\rho^{(2)}}$$

Rearranging for Ksat
Given


 * $$\frac{K_\mathrm{sat}^{(2)}}{K_\mathrm{mineral}-K_\mathrm{sat}^{(2)}}-\frac{K_\mathrm{fluid}^{(2)}}{\phi (K_\mathrm{mineral}-K_\mathrm{fluid}^{(2)})}=\frac{K_\mathrm{sat}^{(1)}}{K_\mathrm{mineral}-K_\mathrm{sat}^{(1)}}-\frac{K_\mathrm{fluid}^{(1)}}{\phi (K_\mathrm{mineral}-K_\mathrm{fluid}^{(1)})}$$

Let


 * $$ S = \frac{K_\mathrm{sat}^{(1)}}{K_\mathrm{mineral}-K_\mathrm{sat}^{(1)}} $$

and


 * $$ F_1 = \frac{K_\mathrm{fluid}^{(1)}}{\phi (K_\mathrm{mineral}-K_\mathrm{fluid}^{(1)})}\ \ \ \ F_2 = \frac{K_\mathrm{fluid}^{(2)}}{\phi (K_\mathrm{mineral}-K_\mathrm{fluid}^{(2)})} $$

then


 * $$ K_\mathrm{sat}^{(2)} = \frac{K_\mathrm{mineral}}{\frac{1}{S-F_1+F_2} + 1} $$

Or, expanded


 * $$ K_\mathrm{sat}^{(2)} = \frac{K_\mathrm{mineral}}{ \left[ {\frac{K_\mathrm{sat}^{(1)}}{K_\mathrm{mineral}-K_\mathrm{sat}^{(1)}}-\frac{K_\mathrm{fluid}^{(1)}}{\phi (K_\mathrm{mineral}-K_\mathrm{fluid}^{(1)})}+\frac{K_\mathrm{fluid}^{(2)}}{\phi (K_\mathrm{mineral}-K_\mathrm{fluid}^{(2)})}} \right]^{-1} + 1} $$

Load induced pore pressure is homogeneous and identical in all pores
This assumption imply that shear modulus of the saturated rock is the same as the shear modulus of the dry rock, $$\mu_\mathrm{sat}= \mu_\mathrm{dry}$$.

Porosity does not change with different saturating fluids
Gassmann fluid substitution requires that the porosity remain constant. The assumption being that, all other things being equal, different saturating fluids should not affect the porosity of the rock. This does not take into account diagenetic processes, such as cementation or dissolution, that vary with changing geochemical conditions in the pores. For example, quartz cement is more likely to precipitate in water-filled pores than it is in hydrocarbon-filled ones (Worden and Morad, 2000). So the same rock may have different porosity in different locations due to the local water saturation.

Frequency effects are negligible in the measurements
Gassmann's equations are essentially the lower frequency limit of Biot's more general equations of motion for poroelastic materials. At seismic frequencies (10–100 Hz), the error in using Gassmann's equation may be negligible. However, when constraining the necessary parameters with sonic measurements at logging frequencies (~20 kHz), this assumption may be violated. A better option, yet more computationally intense, would be to use Biot's frequency-dependent equation to calculate the fluid substitution effects. If the output from this process will be integrated with seismic data, the obtained elastic parameters must also be corrected for dispersion effects.

Rock frame is not altered by the saturating fluid
Gassmann's equations assumes no chemical interactions between the fluids and the solids.