alt="images"/> is the compression wave speed, Er is the modulus of elasticity, and ρr is the density of the reservoir bottom materials. The equilibrium condition at the surface of the layer of reservoir bottom materials (y′ = 0) is that the pressure in the fluid equals the normal stress; thus
The solution of Eq. (A2.5) subject to the equilibrium condition of Eq. (A2.6) and the radiation condition in the negative y′‐direction gives
(A2.7)
By definition,
The compliance function
The substitution of Eqs. (A2.4) and (A2.8) into Eq. (A2.3) gives the boundary condition at the absorptive reservoir bottom:
where the damping coefficient ξ = ρ/ρrCr. This boundary condition for time‐harmonic motion takes the following form for transient motion:
which is identical to Eq. (2.3.4a).
We next relate the wave reflection coefficient, α, which is the ratio of the amplitude of the reflected hydrodynamic pressure wave to the amplitude of a vertically propagating pressure wave incident on the reservoir bottom, to the damping coefficient ξ. Consider a downward traveling wave in the fluid domain that strikes the fluid‐sediment boundary. Hydrodynamic pressures are governed by the one‐dimensional version of Eq. (2.3.8):
(A2.11)
The general solution of this equation is
where A(ω) is the amplitude of the hydrodynamic pressure wave incident to the reservoir bottom, and B(ω) is the amplitude of the reflected wave.
An equation for the ratio B(ω)/A(ω), termed the reflection coefficient, α, can be obtained by substituting Eq. (A2.12) into the boundary condition of Eq. (A2.9) with
which is independent of the excitation frequency.
The wave reflection coefficient α is a more physically meaningful description than is the damping coefficient ξ of the behavior of hydrodynamic pressure waves at the reservoir bottom. Although the wave reflection coefficient depends on a pressure wave's angle of incidence at the reservoir bottom, the value α for vertically incident waves, as given in Eq. (A2.13), is used here as a metric to characterize the absorptiveness of the reservoir bottom materials. The wave reflection coefficient, α, may range within the limiting values of 1 and −1. For rigid reservoir bottom materials, Cr = ∞ and ξ = 0, resulting in α = 1, i.e. full reflection or no absorption of hydrodynamic pressures waves. For very soft reservoir bottom materials, Cr approaches zero and ξ tends to ∞, resulting in α = − 1. It is believed that α values from 1 to 0 would cover the wide range of materials encountered at the bottom of actual reservoirs (Ghanaat and Redpath 1995); α = 0 indicates no reflection of hydrodynamic pressure waves.
A2.2 Application to Flexible Foundation
The preceding theory leading up to Eqs. (A2.10) and (A2.13) is also applicable to modeling of wave refraction in the underlying foundation. Such a derivation was presented even earlier by Hall and Chopra (1982).
The wave‐reflection coefficient, α, at the water–foundation boundary can be computed from Eq. (A2.13), wherein ξ = ρ/ρrCr and
A2.3 Comments on the Absorbing Boundary
Wave absorption – or, alternatively, wave refraction – at the reservoir bottom is represented only approximately by the boundary condition of Eq. (A2.10). A hydrodynamic pressure wave impinging on the reservoir bottom results in a reflected hydrodynamic pressure wave in the water and two refracted waves, dilatational and rotational, in the sediments or underlying rock. The angle of reflection is equal to the angle of incidence and the angles of refraction of the two refracted waves are given by Snell's law. Although the boundary condition given by Eq. (A2.10) allows for proper reflection of hydrodynamic pressure waves for any angle of incidence, the only refracted waves allowed in the foundation or in the sediments are downward, vertically propagating waves.
NOTES
1 † “Reservoir” is the place of