Reservoir Foundation Boundary Condition

If there is no absorption or penetration of water into the reservoir bottom, the same dam-reservoir boundary condition that was obtained previously, can be used for reservoir-foundation boundary. In case of reservoir with sediment at the bottom, we can account it in a very simplified manner.

The boundary condition at the reservoir bottom relates the hydrodynamic pressure to the sum of the normal acceleration and acceleration due to interaction between impounded water and the reservoir bottom material. Here, we consider only interaction in the normal direction due to the assumption that the hydrodynamic pressure waves incident on the reservoir bottom only excite vertically propagating dilatational waves in the reservoir bottom materials. The hydrodynamic pressure p(n, t) in the water is governed by the one-dimensional wave equation:

Similarly, the interaction displacement x(n, t) in the layer of reservoir bottom materials is governed by:

Where Cr = A /—, Er is the Young's modulus of elasticity and pris the density

Y Pr of the reservoir bottom materials. At the reservoir bottom, the accelerative boundary condition states that the normal pressure gradient is proportional to the total acceleration:

Where k (0,t) is the acceleration of the reservoir bottom due to interaction between the impounded water and the reservoir bottom materials. Equilibrium at the surface of the reservoir bottom materials requires that:

the D'alembert solution to equation 2.30 is k = gr (n + Crt) where gr is the wavefront of the refracted wave propagating vertically downward in the reservoir bottom materials. An upward propagating wave does not exist because of the radiation condition for the assumed infinitely thick layer of reservoir bottom materials. Note that ^K- (0,t) = Cr g'r (Crt) and k (0,t) =

Crgr (Crt), where the prime indicates the derivative of gr with respect to argument (n + Crt). Differentiating equation 2.32 with respect to t gives:

dt C

The solution for k (0,t) from equation 2.33, when substituted into equation

Er Pr Cr dn dt

Where q = pC. For rigid reservoir bottom materials, Cr = to and q = 0, so the second term on the right hand side of the above equation is zero giving the boundary condition for a rigid reservoir bottom. The fundamental parameter that characterizes the effects of absorption of hydrodynamic pressure waves at the reservoir bottom is the admittance or damping coefficient q. The wave reflection coefficient k, 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 can be related to the damping coefficient.

The wave reflection coefficient k can be obtained by considering the reflection of a harmonic pressure wave in the impounded water impinging vertically on the reservoir bottom. The downward vertically propagating incident wave of unit amplitude is pi = exp [ic (n + Ct)~j, while the resulting upward propagating wave is pr = k exp [ic(-n + Ct)]. Both pi and pr satisfy the wave equation (equation 2.29) The substitution of the sum pi + pr of the two hydrodynamic pressure waves into the boundary condition at the reservoir bottom, equation 2.34, with no normal acceleration results in an equation which can be solved to results in:

The wave reflection coefficient is independent of excitation frequency u. In general, it depends on the angle of incident of the pressure wave at the reservoir bottom. The wave reflection coefficient may range within the limiting values of 1 and -1. For rigid reservoir bottom Cr = to and q = 0, resulting in k = 1. For very soft material, Cr approaches zero and q = to, resulting in k = -1. The material properties of the reservoir bottom medium are highly variable and depend upon many factors. It is believed the k values from 1 to 0 would cover the wide range of materials encountered at the bottom of actual reservoirs.

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