= Solution
Let $S_g$ be the fraction of the <pore space> occupied by <gas>, and let $\phi$ be the <porosity>. At fixed <porosity> and unchanged <rock> framework, the bulk <density> is
$$
\rho=(1-\phi)\rho_s+\phi\bigl[(1-S_g)\rho_w+S_g\rho_g\bigr].
$$
Replacing <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 \b[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>:
$$
V_p=\sqrt{\frac{\kappa+4\mu/3}{\rho}}.
$$
<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.
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