Acoustic Wave

Key Takeaways

• The acoustic wave (P-wave) velocity in a material is given by Vp = sqrt((K + 4G/3) / rho), where K is the bulk modulus (resistance to volumetric compression, in pascals), G is the shear modulus (resistance to shape change, in pascals), and rho is the bulk density (in kg/m³). The 4G/3 term arises because a compressional wave also involves some shear deformation as the wave passes, so the shear modulus contributes even in a compressional wave. In a fluid where G = 0, the formula simplifies to Vp = sqrt(K/rho): the P-wave velocity is entirely controlled by the bulk modulus and density. This is why gas dramatically slows acoustic wave velocity (gas has a very low bulk modulus, roughly 100,000 to 200,000 Pa, compared to 2.2 GPa for water or 36 GPa for quartz), and why the P-wave can travel through water and air but loses the shear modulus contribution that would otherwise increase its velocity if the medium were solid.
• Acoustic wave velocities span a wide range depending on the stiffness and density of the medium. In air, the acoustic wave travels at approximately 330 to 340 m/s. In fresh water it travels at approximately 1,480 m/s and in seawater at 1,480 to 1,530 m/s depending on temperature and salinity. Unconsolidated sediments at shallow depth have velocities close to water (1,500 to 2,000 m/s) because the frame stiffness of loosely packed grains is very low. Well-cemented sandstones range from 2,500 to 5,000 m/s. Shales range widely from 2,000 to 4,500 m/s depending on compaction, clay content, and anisotropy. Limestones range from 4,000 to 6,500 m/s and dolomites from 5,000 to 7,500 m/s reflecting their high mineral stiffness. Anhydrite is among the fastest common sedimentary minerals at 5,500 to 6,500 m/s. Halite (rock salt) has a relatively low velocity for an evaporite at 4,420 to 4,560 m/s despite its density of 2.16 g/cc. These velocity contrasts produce acoustic impedance differences that generate seismic reflections and allow different rock types to be identified on seismic sections and sonic logs.
• Gas saturation has a strong and non-linear effect on acoustic wave velocity because gas has a bulk modulus that is orders of magnitude smaller than brine. Even a small amount of gas in the pore space (10 to 20 percent gas saturation) reduces the bulk modulus of the pore fluid mixture to nearly the value of 100 percent gas, because gas is so compressible that it dominates the mixture compressibility at any concentration. This means that the reduction in Vp caused by gas is nearly as large at 10 percent gas saturation as at 100 percent saturation. The practical consequence for seismic exploration is that Vp is a poor quantitative indicator of gas saturation level: you can tell that gas is present from a low Vp, but you cannot easily determine how much gas is there from Vp alone. The ratio Vp/Vs (or equivalently Poisson's ratio) is much more diagnostic because the shear wave velocity Vs is relatively insensitive to fluid saturation, so a low Vp/Vs ratio (typically below 1.6) in a porous formation indicates gas saturation, while a higher ratio (above 1.8 to 2.0) indicates brine or oil saturation.
• At every interface where acoustic wave velocity changes, incoming acoustic energy is split: part is reflected back toward the source and part is transmitted onward into the next layer, with refraction (bending) of the transmitted wave according to Snell's law: sin(θ₁)/V₁ = sin(θ₂)/V₂, where θ₁ and θ₂ are the angles of the incident and refracted rays measured from the interface normal, and V₁ and V₂ are the acoustic velocities in the two layers. When V₂ is greater than V₁ and the angle of incidence exceeds the critical angle (arcsin(V₁/V₂)), the refracted wave travels along the interface as a head wave. Head waves return to the surface at the critical angle and arrive at surface receivers before the direct wave over sufficiently long source-receiver distances; mapping these first-arrival head-wave travel times as a function of offset allows the velocity and depth of the refracting layer to be computed by the time-distance slope. This refraction principle is used in refraction seismic surveys for near-surface velocity characterisation and in borehole sonic logging, where the formation P-wave velocity is faster than the borehole mud velocity, so the refracted formation arrival reaches the receiver array before the direct fluid wave and is picked as the first arrival.
• Time-lapse (4D) seismic monitoring uses repeated measurements of acoustic wave velocity and reflectivity to track changes in reservoir fluid saturation and pressure over the producing life of a field. When oil is replaced by injected water during a waterflood, the acoustic wave velocity in the swept zone increases (water has higher bulk modulus than oil), increasing the acoustic impedance and strengthening the reflection at the base of the reservoir. When reservoir pressure declines, effective stress on the grain framework increases, stiffening the rock skeleton and also increasing the acoustic wave velocity. By comparing 3D seismic surveys shot at different times during production (typically 3 to 10 years apart), production geoscientists can map which parts of the reservoir have been swept by water, identify bypassed pay zones that the waterflood has not reached, and calibrate reservoir simulation models against the observed velocity changes. Time-lapse seismic is used in major oilfield developments in the North Sea, the Gulf of Mexico, offshore Brazil, and offshore Newfoundland to optimise recovery and inform infill well placement decisions.