= Solution
For an approximately <isotropic> elastic solid,
$$
V_P=\sqrt{\frac{K+4\mu/3}{\rho}},\qquad V_S=\sqrt{\frac\mu\rho},
$$
where $K$ is <bulk modulus>, $\mu$ is <shear modulus> and $\rho$ is <mass density>. Five relevant controls are:
* \b[Mineral composition and texture:] different elastic moduli and densities give different <seismic velocities>; compliant clay-rich material generally differs from crystalline igneous framework rock.
* \b[<Porosity>, cracks and fractures:] compliant voids reduce the effective moduli and usually reduce both speeds; aligned fractures produce <seismic anisotropy>.
* \b[Effective confinement:] increasing <effective stress> closes compliant cracks and tends to raise speeds. Higher <pore fluid pressure> at fixed total confinement can reopen or preserve cracks and lower the effective stiffness.
* \b[Fluid saturation and phase:] replacing gas by liquid commonly increases the saturated <bulk modulus> and $V_P$; fluids have no static shear rigidity, so $V_S$ changes through the frame, density and frequency-dependent interactions rather than a direct fluid <shear modulus>.
* \b[<Temperature> and alteration:] heating commonly reduces elastic moduli and may create thermal cracks; <hydrothermal alteration> can replace strong minerals by clay or other weaker phases. The sign depends on alteration products and crack sealing, so reservoir-specific measurements are needed.
These controls interact: a dense rock is not automatically faster if its moduli are also different.
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