Using the Biot theory, we have performed calculations to show that viscous fluid flow affects seismic wave amplitudes and reflection coefficients at a gas-water boundary in a porous sand reservoir. Our formulation of the boundary value problem for fluid-filled porous rocks is applicable at all angles and follows parallel to the classical reflection and transmission problem solved by Knott, Zöeppritz, and others for two elastic media in perfect contact. It is pointed out that there are two major differences between the two: (1) in the present case, we deal with the propagation of inhomogeneous waves, and (2) there are interference fluxes of energy between various types of waves. The latter exists only at an oblique incidence.
In our model, loss of seismic energy is mainly due to mode converted type II waves of Biot at the gas-water contact which propagate away from the boundary in the manner of a diffusion process at low frequencies. Both seismic energy loss and reflection coefficients are studied over a wide range of frequencies and angles of incidence. We find that for a porous unconsolidated sandstone [Vp (brine-filled) = 7234 ft/sec, Vp (gas-filled) = 4920 ft/sec, Vs = 3044 ft/sec, Φ = 0.3, permeability = 1 darcy], the energy loss associated with a single gas-water boundary is approximately 2.5 percent at 100 Hz for P-waves incident at 30 degrees and that the loss increases with frequency as f½. Further, at a given frequency, the loss of energy is minimal at normal incidence (~1.2 percent at 100 Hz) and maximal (~6 percent at 100 Hz) in the neighborhood of the critical angle (~42 degrees). We find that the inelastic losses are significantly higher at the frequencies at which seismic logging tools operate (50 percent at 10 kHz near the critical angle). As far as the amplitude reflection coefficients are concerned, we find that the effect of the fluid flow across a single gas-water boundary is to reduce the reflection coefficient with increasing frequency (when the wave is incident from the gas saturated rock) at all angles of incidence. In the exploration frequency band, the reduction is about 1.5–3 percent, whereas the reduction is approximately 40–50 percent in the logging frequency band.
For high frequencies (10 kHz and higher) classical interpretation breaks down, and hence it appears that for frequencies at which the seismic logging tool operates corrections should be made to the classical theory.