# Gassmann equation

The Gassmann equation is the most common way of performing a fluid substitution model from one known parameter.

## Procedure

These formulations are from Avseth et al (2006)[1].

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, and $K_\mathrm{fluid}^{(1)}$ and $K_\mathrm{fluid}^{(2)}$ are the bulk moduli of the fluids themselves.

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

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

## Assumptions

### Saturated shear modulus does not change with different saturating fluid

Gassmann's equations assume that the shear modulus of the saturated rock is the same as the shear modulus of the dry rock, $\mu_\mathrm{sat}= \mu_\mathrm{dry}$, for all saturating fluids. This assumes that the chemical interactions between the fluids and the solids are always the same (or have the same consequences for rigidity). For example, certain clay minerals may swell only when exposed to water causing the rigidity of the rock to increase or decrease depending on the geometry of the clay grains.

### 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. 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 (10-100Hz), the error in using Gassmann's equation may be negligible. However, when constraining the necessary parameters with sonic measurements at logging frequencies (~20kHz), 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.

## References

1. Avseth, P, T Mukerji & G Mavko (2006), Quantitative seismic interpretation, Cambridge University Press, 2006.