 Solution

Let $S_g$ be the fraction of the pore space occupied by gas, and let $\varphi$ be the porosity. At fixed porosity and unchanged rock framework, the bulk density isReplacing water with much less dense gas therefore decreases the bulk density approximately linearly, as in Figure 1. The liquid and gas densities here are those at reservoir conditions.

A pore fluid has negligible static shear modulus, so the rock framework chiefly controls the effective shear modulus $\mu$. If $\mu$ changes little, $V_s=\sqrt{\mu /\rho }$ increases gently as the density falls. This does not imply that S waves travel through the pore gas itself: their restoring force comes from the connected solid framework.

The P wave speed depends on both bulk modulus and shear modulus:Gas is much more compressible than water. Even a small connected gas fraction can strongly reduce the effective fluid bulk modulus, approximately through the compliance average ${\kappa }_f^{-1}\simeq (1-S_g)/{\kappa }_w+S_g/{\kappa }_g$ for a pressure-equilibrated mixture. The saturated rock's effective bulk modulus consequently falls rapidly at first, overwhelming the smaller density reduction and causing the sharp initial fall in $V_p$. Once the fluid is already very compliant, further gas replacement changes the stiffness less; the continuing density decrease can then make $V_p$ rise again. Figure 1 assumes a particular saturation response: patchy fluids, changing effective pressure, and frequency-dependent fluid motion can alter it. The plotted $V_s$ is only nearly constant, so the plot should not be interpreted as an exact fixed-$\mu$ constitutive model.