What Moves and What Doesn't

Part 4, Part 4: Gassmann and Fluid Substitution

Learning objectives

Sort the six elastic quantities into what the fluid moves directly, what it leaves fixed, and what follows
Explain why Vp feels both the bulk modulus and the density while Vs feels only the density
Read the brine-to-gas numbers: Vp down about 10 percent, impedance down about 21 percent, Vs up
Recognize the Vp-down, Vs-up pattern and the falling Vp/Vs as the seismic gas signature

Six Quantities, Three Behaviors

A fluid swap can move six numbers: the bulk modulus $K$ , the shear modulus $\mu$ , the density $\rho$ , and then the three that follow from those, $V_P$ P, $V_S$ S, and the impedance $I_P$ P. The whole of fluid detection lives in the fact that they do not all move the same way. They sort into three behaviors, and knowing which quantity does which is what lets an interpreter look at an amplitude and reason about a fluid.

What the Fluid Touches

Start with the three that the fluid sets directly. The bulk modulus rises when you put in a stiffer fluid and falls when you put in a softer one; that is the Gassmann term of Part 4.1 doing its work. The shear modulus does not move at all, Gassmann's second statement, because a fluid has no shear stiffness to lend. The density rises with a heavier fluid and falls with a lighter one, a plain volume average of grain and fluid (Part 3.5). So of the three inputs a velocity is built from, the fluid moves two of them, $K$ and $\rho$ , and leaves the third, $\mu$ , exactly where it was.

Now the velocities inherit that split. The P-wave, $V_P = \sqrt{\dfrac{K + \tfrac{4}{3}\mu}{\rho}}$ P=sqrtdfracK+tfrac43murho, feels both the bulk modulus and the density, and in a soft rock it moves a great deal. The S-wave, $V_S = \sqrt{\dfrac{\mu}{\rho}}$ S=sqrtdfracmurho, feels only the shear modulus and the density; since $\mu$ is fixed, $V_S$ S can change only through the density, and it moves the opposite way to it, a lighter fluid lifting $V_S$ S because the denominator has shrunk.

The Numbers, and the Signature

Put the brine-to-gas swap on the soft sand through it. The bulk modulus falls from 12.35 to 6.12 GPa, the shear modulus holds at 7, and the density falls from 2.20 to 1.94 g/cc. The P-wave drops from 3.142 to 2.826 km/s, a fall of about 10 percent, and the impedance drops from 6.90 to 5.47, a fall of about 21 percent, because both $V_P$ P and the density fell. The S-wave does the surprising thing: it rises, from 1.785 to 1.902 km/s, because the only quantity it felt was the density dropping away beneath a fixed shear modulus.

That pattern, $V_P$ P down while $V_S$ S up, is the classic gas signature and one of the most exploited facts in quantitative interpretation. Two readouts sharpen it. The ratio $V_P/V_S$ falls hard, from about 1.76 to about 1.49, since its numerator dropped and its denominator rose at once, and a low $V_P/V_S$ is a direct hydrocarbon indicator. And that same divergence is what lets amplitude-versus-offset separate a gas sand from a brine sand that happens to share its impedance, because the two rocks still split on how $V_P$ P and $V_S$ S part company with angle. A model that predicts the full set is worth more than any one number, and every number in that set came from four inputs (the dry frame, the porosity, the mineral, and the fluid) that are never known exactly. How much the answer moves when those inputs are wrong, and which of them matters most, is the honest question of the next section.

References

Mavko, G., Mukerji, T., & Dvorkin, J. (2009). The Rock Physics Handbook (2nd ed.). Cambridge University Press.
Smith, T. M., Sondergeld, C. H., & Rai, C. S. (2003). Gassmann fluid substitutions: A tutorial. Geophysics, 68(2), 430-440.