We consider the cases of a plane wave incident on the planar interfaces between dry sandstone and water-saturated rock, dry sandstone and oil-saturated rock, and oil-, and water- saturated rocks, respectively. In the following, we use several examples to test the derived formula numerically. Even though equations 12 and 13 have exactly the same form as in an acoustic case, the wavenumber, k ~ 1⁄4 ω ∕ V and the reflection coefficient are complex and dependent on frequency.

By substituting the impedance Z 1⁄4 ρ V ∕ cos θ, equation 12 can be rewritten as In an acoustic case, the wavenumber, ( is the velocity of acoustic wave), is real and the phase velocity neither is independent of frequency, nor is the reflection coefficient. Under the assumption that no fluid diffusion from one medium to another occurs on the boundary, the reflection coefficient R, can be obtained from the boundary conditions that requires the pressure and the particle velocity projection on the normal of the boundary to be continuous when waves propagate through the interface, following the derivation as Aki and Richards (2002) and Brekhovskikh (1980), with complex wavenumber and complex velocity, the result will be where 1 2, 1 2, and 1 2 are the complex wavenumbers, den- sities, and the complex velocity of media 1 and 2, respectively. The subscripts 1 and 2 denote the upper and lower media, respectively. The relationship between θ 1 and θ 2 is determined by the Snell ’ s law. Assume a plane wave being incident on a planar interface in a 2D model at z 1⁄4 0 that separates two diffusive-viscous media as shown in Figure 4, the wavenumbers can be formulated as where 1 is the incident angle and 2 is the angle of transmission.

Angle of reflection diagram