Rock Scour Model for Unlined Plunge Pools and Stilling Basins

Abstract

This paper presents a new scour model (SM) that is able to evaluate the rock scour development, shape, ultimate depth and probability of occurrence. The model takes into account two main scour mechanisms: (1) the maximum rock crack extension that mainly depends on the geology, rock characteristics/mechanics and hydraulic fracturing analysis; and (2) the isolated rock block stability that depends on the flow hydraulics and rock block dimensions. The former defines the ultimate extent of the fractured area in a rock matrix where the isolated rock blocks can be detected; only in this area, the rock block stability analysis can be performed afterwards. The isolated rock block stability has been studied in the literature; on the contrary, the information related to the rock crack propagation in dissipaters is limited, which is investigated in this paper. Here, the expected maximum rock crack extension is defined by a threshold on the stress intensity factor. The rock stress intensity factor is investigated via stochastic approaches, compatible with the stochastic nature of the turbulent bottom pressure field, with the support of physical hydraulic modelling results. The proposed SM is a physically based model that couples the hydraulics and geological aspects of scour phenomena, allowing a realistic evaluation of the rock scour. This is drawn out by the comparison between the maximum scour depth computed via the rock block stability analysis and the maximum depth of rock matrix fractured area where the isolated rock blocks can be formed by the incident flow; the smallest depth gives the expected scour. The ultimate scour could be significantly smaller than the one computed via the approaches in the literature, especially for compact rocks suitable for unlined dissipaters. This highlights the relevance of this study that takes into account the high-velocity flow characteristics such as aeration and turbulence effects as well as the main geomechanical characteristics of the whole rock mass obtained by geological surveys. Furthermore, the SM stochastic approach makes it suitable in the risk-based design of dams, hydropower outlet works and other hydraulic structures as well as dam stability assessment. The model is validated using well-known real-life scour cases in the literature such as Wivenhoe and Cabora Bassa dams. A design example is included in the paper demonstrating the evaluation of the rock scour downstream of a large dam.

Highlights

• The paper presents a novel probabilistic rock scour model applicable to hydraulic structures.

• The model is able to evaluate the rock scour development, ultimate depth and probability of occurrence in unlined dissipaters.

• This physically-based model couples the hydraulics and geological aspects of scour phenomena allowing a realistic evaluation of the rock scour.

• The model is suitable for the risk-based design of dams, hydropower outlet works and other hydraulic structures as well as dam stability assessment.

• The model is validated using knowing real-life scour cases in the literature such as Wivenhoe and Cabora Bassa dams.

Appendix A: Cabora Bassa Dam: Detailed Calculations

The bottom turbulent pressure fluctuations in the case of a plunge pool depend on: (1) the kinetic energy of the jet impact on the water surface; (2) the submerged jet ratio H/y_e; and (3) the jet break-up length ratio, L_j/L_b, where L_j is the jet length in the atmosphere and L_b is the jet break-up length.

The bottom pressure fluctuations are defined by the pressure fluctuations coefficient, ${c}_{{p} }^{^{\prime}}=\frac{{\sigma }_{{p}}(0)}{0.5 \rho {V}^{2}}$. The values of ${c}_{{p} }^{^{\prime}}$ in a plunge pool are computed according to the experimental formulas of Castillo et al. (2015), as a function of H/y_e, and L_j/L_b that takes into account the air effects on the phenomena.

For 1 < L_j/L_b < 1.3, the ${c}_{{p} }^{^{\prime}}$ coefficient is given by the following formula for H/y_e ≤ 14:

$${c}_{{p} }^{^{\prime}}=a{\left(\frac{H}{{y}_{{e}}}\right)}^{3}+b{\left(\frac{H}{{y}_{{e}}}\right)}^{2}+c\left(\frac{H}{{y}_{{e}}}\right)+d,$$

(16)

with a = − 0.000005, b = − 0.0022, c = 0.016, and d = 0.35, while for H/y_e > 14 it is:

$${c}_{{p} }^{^{\prime}}=a{e}^{-b\left(H/{y}_{{e}}\right)},$$

(17)

with a = 1 and b = 0.15.

This formula allows to compute the standard deviation of pressure fluctuations:

$${\sigma }_{{p}}\left(0\right)=0.5 \rho {V}^{2}{c}_{{p} }^{\mathrm{^{\prime}}}$$

(18)

from the jet kinetic energy at the impact on the plunge pool water surface with water velocity, V.

The ratio H/y_e allows to compute the rock scour, because it is:

$$d_{{{s}}} + h_{{{t}}} = H\sin (\alpha_{i} ).$$

(19)

From Figs. 3 and 4, the ratio $\frac{{\sigma }_{{KI}}}{{\sigma }_{{p}}(0)\sqrt{L}}$ = β can be estimated. Having ${\sigma }_{{p}}(0)$, the standard deviation of the stress intensity factor is:

$${\sigma }_{\mathrm{KI}} = {\beta \sigma }_{{p}}(0)\sqrt{L.}$$

(20)

In Table 2, the computation procedure is reported; that is the ratio H/y_e (column no. 1); the ${c}_{{p} }^{^{\prime}}$ coefficient (column no. 2); ${\sigma }_{{p}}(0)$ (column no. 3); ${\sigma }_{{KI}}$ (column no. 4); and the probability of occurrence of ${K}_{{I}}>{K}_{th}$ and ${K}_{{I}}>{K}_{Ic}$, according to the Gaussian probability distribution, are reported in columns no. 5 and no. 6, respectively. These probabilities are computed using Microsoft Excel software from the ratios ${K}_{th}$ /${\sigma }_{{KI}}$ and ${K}_{Ic}/$ ${\sigma }_{{KI}}$, respectively. In column no. 7, the scour depth d_s + h_t is shown.

The computation is performed for V = 42 m/s, y_e = 6.5 m, α_i ≈ 43°, L = 3 m and β = 2.73 according to the Cabora Bassa dam via Eqs. (16)–(20) with ${K}_{th}$ = 0.9 MPa m^0.5 and ${K}_{Ic}$ ≈ 2.7 MPa m^0.5.

Table 2 Computation procedure for crack propagation

From Table 2, one can observe that the stress intensity factor is larger than ${K}_{th}$ (i.e. ${K}_{{I}}>{K}_{th}$) with a probability level larger than 0.01 for d_s + h_t close to ~ 71 m (i.e. 70 m), while assuming a probability level of 0.001, the stress intensity factor is larger than ${K}_{th}$ for d_s + h_t close to ~ 80 m (i.e. 78 m). The role of the brittle fracture becomes more important for d_s + h_t smaller than 58 m.

The stability analysis of the isolated rock block can be done according to Maleki and Fiorotto (2019b). It defines the maximum hydrodynamic force acting on a prismatic block, with square base L_x = L_y:

$$F = 6\sigma_{{{F}}} ,$$

(21)

The coefficient F/σ_F was experimentally verified, and it is equal to the ratio (P′_min + P′_max)/(2 σ_p(0)) where P′_max and P′_min are the maximum and minimum pressure fluctuations (absolute value) measured in experiments with long duration (e.g. Maleki and Fiorotto 2019b; Castillo et al. 2015; Fiorotto and Rinaldo 1992a).

The standard deviation of the uplift force is:

$$\sigma_{{{F}}} = \Omega \;L_{x}^{2} {\sigma }_{{p}}(0),$$

(22)

where Ω = uplift coefficient, L_x = length of the rock block in the stream wise direction, and L_y = length of the rock block in the cross-stream wise direction; in this case L_y = L_x. The coefficient Ω is computed according to the following experimental formula (Maleki and Fiorotto 2019b):

$$\Omega = 0.35(1 - \exp ( - L_{x} /b\;0.89))).$$

(23)

that depends on the ratio L_x/b in 0 < L_x/b < 4. Here, b = distance from the jet axis to the point where the mean velocity halves its maximum value, that is:

$$b = 0.228\;y_{{e}} + 0.0913 \times 0.65H.$$

(24)

It depends linearly on the jet length from the bottom, H = (d_s + h_t)/sin(α_i), where h_t = tailwater depth, d_s = scour depth, and α_i = impact angle of the jet on the water surface (Fig. 1).

The stability force is equal to:

$$F_{s} = \left( {\gamma_{s} - \gamma } \right)S \, L_{x} \, L_{y} ,$$

(25)

where ${\gamma }_{s}$ = rock block specific weight, $\gamma$ = water specific weight, and $S$ = rock block thickness.

Equating the maximum hydrodynamic force, F, and the stability force, ${F}_{s}$, one obtains the ultimate scour depth d_s + h_t. The computation, for S = L_x = L_y = 2 m, is reported in Table 3, which includes H/y_e (column no. 1); the ${c}_{{p} }^{^{\prime}}$ value (column no. 2); ${\sigma }_{{p}}(0)$ (column no. 3); b (column no. 4); L_x/b (column no. 5); Ω (column no. 6); σ_F (column no. 6); uplift force, F (column no. 7); and the scour depth, d_s + h_t, in column no. 8.

Table 3 Computation procedure for rock block stability

For a rock block with S = 2 m, the stability force is ${F}_{s}=\left({\gamma }_{s}- \gamma \right)S$ L_x L_y = 16,500 × 2 × 2 × 2 = 132,000 N. From Table 3, one can note that the stability force is larger than the uplift force for d_s + h_t larger than 80 m.