1. Introduction
The potential for the erosion of rock mass within an unlined spillway is determined by comparing the resistance of the rock mass with the erosive force of flowing water. That erosion in dam spillways poses a constant threat to the safety of both individuals and equipment. Instances of unexpected erosion have led to serious damage to numerous dam spillways, resulting in substantial maintenance expenses for several large dams. For example, the Oroville Dam in California experienced significant damage due to a large cavity in its concrete spillway, resulting from a substantial water discharge that caused over USD 2 million in damages and necessitated the evacuation of downstream residents [
1,
2]. Similarly, at the 113 m high Copeton embankment dam in Australia, significant water flow in the bedrock spillway created a 20 m deep gorge [
3]. To address this phenomenon, numerous studies involving laboratory tests using scaled-down physical models have been conducted. These physical models were designed to investigate the hydraulic characteristics of flows, assess erosion phenomena in granular materials or rock masses, and validate hypotheses used in the development of certain methods for predicting the erosion potential of rock mass within spillways. One subset of these models included the impact of hydraulic parameters on the erosion process, such as the slope of the flow channel, flow rate, flow velocity, and surface roughness of the flow channel [
4,
5,
6,
7,
8]. Moreover, some of these models focused on hydraulic parameters, aiming to simulate water flow or erosion downstream of weirs under specific conditions or to identify alternative solutions to reduce hydraulic power downstream of spillways [
9,
10,
11]. Other models sought to investigate the effects of specific geomechanical parameters of the rock mass on hydraulic erosion [
12,
13,
14,
15,
16,
17,
18].
Mostly based on the findings of these laboratory studies, several methods have been proposed to assess the potential risk of hydraulic erosion of a rock mass in dam spillways. These approaches are generally empirical, and all have limitations. Rock mass resistance is assessed through various indices of rock mass quality, whereas the erosive force of water is commonly represented by energy dissipation (
Table 1).
Rock mass resistance can be assessed by the Kirsten index (
K), the geological strength index for erodibility (
eGSI), and the rock mass erodibility index (
RMEIB), which all estimate rock mass quality. These indices use various geomechanical parameters related to the rock mass and the intact rock, such as the confined compressive strength of the intact rock (
Ms), the size (
Kb) or volume (
Vb in m
3) of the rock blocks, the shear strength of the rock joints (
Kd), and the relative structure of the blocks (either
Js or
Edoa); the latter considers the effect of the shape and orientation of the blocks with respect to the flow direction of water in the channel. The values of
K,
eGSI, and
RMEIB are determined using Equations (1), (2) and (3), respectively.
GSI (Equation (2)) is a rock mass classification index developed by Hoek et al. [
24]; it is also used by Pells as the basis for the
eGSI erodibility index.
RMEIB (Equation (3)) defines the resistance of the rock mass by weighting the various geomechanical parameters using factors of relative importance (
RF) and likelihood (
LF). The prefixes
P1 to
P5 in Equation (3) represent various sets of geomechanical parameters, including the viable mechanisms at the kinematic separation of the blocks, the nature of the potentially eroded surface, the nature of the joints contained in the rock mass, joint spacing, and block shape [
3].
Energy dissipation (
), which is favored in erosion-evaluation methods, is controlled by various parameters related to flow conditions. In the case of unlined spillway flow channels,
is determined using Equation (4) [
23].
where
is the density of water (kg∙m
−3),
is the gravitational acceleration (m∙s
−2),
is the flow rate per unit length of channel width
,
is the channel width (m),
is the water flow rate (m
3∙s
−1), and
is the energy loss during flow.
By combining the use of a rock mass resistance index with an index representing energy dissipation (
, graphic methods can assess the erosion potential (
Figure 1). Each point on these graphs represents a case of erosion that is categorized into damage classes on the basis of field observations, represented by using different symbols in various colors. These cases of erosion are also separated by theoretical damage classes represented by thresholds established through various methods, and these limits between the classes (and their number) can vary among authors. The Van Schalkwyk method includes three classes of erosion damage (
Figure 1a) [
20], Annandale’s has two damage classes (
Figure 1b) [
21], and Pells’ methods rely on five classes (
Figure 1c,d) [
3,
23].
A major inconsistency is apparent between the different damage classes (
Figure 1). A common inconsistency for all existing methods is that the erosion class from field observations differs from that of the erosion evaluation methods (threshold lines). For example, a large portion of the “negligible” erosion cases in
Figure 1a—according to field observations—are qualified as “moderate or serious” erosion levels when the Van Schalkwyk method is applied. In
Figure 1b, the Annandale method identifies some cases as scour when field observations find no scour. Finally, the Pells methods (
Figure 1c,d) show some overlap among the observed and theoretical erosion classes.
The source of these inconsistencies could stem from either the rock mass resistance index or energy dissipation index, the latter representing the erosive force of water. The primary source of this inconsistency arises from incomplete observations of the erosion process. Indeed, since these data were obtained from laboratory tests using scaled-down physical models, they do not permit a comprehensive evaluation of the erosion process in rock masses [
25,
26]. Moreover, Boumaiza et al. evaluated the representativeness of various geomechanical parameters used in rock mass resistance indices [
27,
28]. They concluded that the various indices rely on some geomechanical parameters that are not relevant to hydraulic erosion when defining rock mass resistance. However, the observed inconsistency (e.g.,
Figure 1) could also stem from the erroneous assessment of the water erosive force, which is the focus of this paper.
Apart from the use of energy dissipation to represent the erosive force of water in many erosion-assessment methods, the average flow velocity (
in m∙s
−1) and shear stress applied at the bottom of the flow channel (
in kPa) were identified as relevant for representing the water erosive force. These indices can be estimated using Equations (5) and (6). However, much criticism in regard to
and the complexity of estimating
have discouraged their use in erosion-assessment methods.
where
is the Manning resistance coefficient,
is the hydraulic radius (m),
is the slope of the channel equal to
,
is the distance along the channel (m),
is the elevation above a datum (m),
is the angle of inclination of the channel (°),
is the Darcy [
3,
29] flow resistance coefficient,
is the Chézy [
3,
29] resistance coefficient, and
is the total energy gradient.
Moreover, the erosive force of water varies depending on the hydraulic conditions, including the water flow rate, the flow velocity, and the configuration of the spillway flow channel (
Figure 2); thus, estimates of the erosive force can be affected by the problem of non-uniqueness. In
Figure 2, four hydraulic conditions, A, B, C, and D, are presented. In conditions A and B,
is set at 7 m·s
−1, although the hydraulic conditions (flow rate and turbulence) in condition A are five times greater than those in B. Likewise, it is also apparent in
Figure 2 that a single measure of the average shear stress or energy dissipation is not associated with a single particular condition of flow rate and hydraulic head. For example, the erosive force in conditions C and D may not be equivalent, even though the energy dissipation is constant [
3].
In
Figure 2, we note that
can be affected by the problem of non-uniqueness, which can produce an under- or overestimate of the erosive force of the water. The average flow velocity represents a characteristic parameter of flow in the channel. This parameter is related to the channel slope and the hydrostatic force of the water in the dam reservoir. The average shear stress of the flow channel results from the normal stress applied to the bottom of the flow channel and represents an important component of the hydraulic erosion process. In terms of energy dissipation, this parameter correlates with the turbulence intensity of the flow [
21,
22]. Otherwise, few laboratory studies show a correlation between
and the magnitude of pressure fluctuations in a flow channel. Thus, these different parameters have the same utility for representing the erosive force of water. The question arises of why there is the use of
rather than
to represent the erosive force. According to the literature, the following apply:
- -
The non-representativeness of
is linked to its sensitivity to the problem of non-uniqueness in the erosion process [
3].
- -
The non-representativeness of
is linked to its complexity for estimating all probable mechanisms of erosion, including erosion by the dynamic expulsion of rock blocks and the erosion by the fragile failure of the rock mass into smaller pieces because of turbulent flow, a basic physical mechanism of erosion [
30].
However,
which is used by various erosion-assessment methods, is not always reliable because it does not integrate all the complexities of erosion; it is used to represent water erosive force because of its simplicity, not because of its representativeness. Thus, the selection of
to represent the erosive force is based on a general qualitative analysis. This led Pells [
3] to state that the recommendation for using
is pragmatic and concessional but not optimal.
Therefore, the determination of the best parameters to represent the erosive force of water is not based on sound analysis, and the justifications for rejecting some parameters, such as and are qualitative. Hence, there is a need to verify the applicability of these different parameters to actual observed erosion data. In this context, we present a method to assess the applicability of the various and presumably relevant hydraulic parameters to represent the erosive force of water; these parameters include , and the pressure head (). We first describe the steps of our approach. We then apply our methodology to 100 observed cases of erosion within dam spillways. Finally, we identify the most relevant hydraulic parameters for quantifying the water erosive force within dam spillways.