2.8.1. Dam Breaching Parameters
In this study, breach parameters were applied to assess the characteristics of breach events, such as the dimension of the breach channel, breach outflow and failure time. According to previous studies, several breach parameters describe the characteristics of the breach channel, such as the depth of the breached channel (
Hb), and average width (
Bavg) and characteristics of the breach event, such as the height of the water surface from the bottom of the breached channel (
Hw), the total breach outflow volume (
Vw), the failure time of the breach (
Tf) and the peak outflow of the breach (
Qp) [
42]. Moreover, the relationships between parameters were examined by collecting actual dam breach cases [
43,
44]. As a result, several empirical relationships were found (see
Table 1 and
Table 2). In addition, previous studies reported that some breaching parameters are related to reservoir properties [
45], such as the dam height (
Hd), the dam width (
Wavg), and the reservoir storage (
V), as shown in
Figure 6. Thus, we compared the relationships found in this study with previous empirical relationships to clarify the similarities and differences of the spillway breaching event in the Swa Dam and other dam breach events.
The depth of the breached channel (
Hb) is mentioned as a breach height in many works in the literature. It is the vertical extent of the breach measured from the crest of the dam down to the bottom of the breach [
17,
18,
19,
20,
21,
22,
23,
24,
25,
26]. Previous studies advise that if there is no information for the breach height (
Hb), the height of the dam (
H) could be taken as the breach height. For the Swa Dam breach case,
Hb was taken as the height from the top of the spillway crest to the bottom of the final breach which was R.L 102.68 to 81 m (details are explained in
Section 2.8.3) From the spillway crest (R.L 102.68 m) to the final bottom elevation of the breach (R.L 81 m), the height of the breach was 21.68 m, and this height was applied as
Hb in the empirical calculation.
Meanwhile, the average breach width (
Bavg) is the average of the breaching width of the top and bottom width of the breached channel. For the Swa Dam case, the top width of the breach was about 148.73 m, the bottom width of the breach was about 71.93 m, and the average breach width was 110.33 m taken from the topographic survey data of the IWUMD (
Figure 7). This average width measured from the survey data was applied as the observed average breach width in the comparison of the resulting average breach widths (
Bavg) by empirical equations.
- 2.
Breach parameters for water discharge:
The height of the water surface from the breach inverted (
Hw) is the vertical extent of the water measured from the top of the water surface before dam failure to the bottom of the breach, whereas
Vw is the volume of water with respect to
Hw [
17,
18,
19,
20,
21,
22,
23,
24,
25,
26]. In previous studies, if there was no information for the breach water height (
Hw) and its corresponding water volume (
Vw), the dam height (
H) could be assumed as
Hw in some cases such as overtopping cases, and the full storage water level and full storage reservoir,
V, could be taken as parameters in other cases such as piping cases. For the Swa Dam case,
Hw was measured from the overflow water surface (R.L 103.2 m) before the breach day to the final breach bottom elevation (R.L 81 m), which was 22.2 m. This height was applied as
Hw in the empirical calculation.
The dam breach procedure should be divided into two stages (i.e., breach initiation and beach formation), so that the failure time could be separated into two parts (i.e., initiation time and formation time) corresponding to that in [
43]. The initiation time initiates with the first flow of water over or through the dam and ends when erosion reaches the upstream face of the dam and produces a rapid acceleration of breach outflow and an unstoppable failure of the dam. Meanwhile, the formation time begins when the initiation time ends and continues until the breach has reached its maximum size [
43]. However, previous studies stated that the formation time is not until the absolute draining end of the reservoir [
43]. Several previous studies also stated that the failure time of a breach (
Tf) is generally described as the time between the initiation of the breach and the development of its peak discharge (
Qp). For example, Froehlich (2008) [
24] stated that the failure time is from the initiation of a breach until it reaches its maximum size.
Two-time parameters were examined in the Swa Dam case for the failure time of a breach (
Tf) according to previous studies. In the Swa Dam, the water discharge rate increased from zero at 5:00 a.m. LT to the maximum breach discharge rate, which was about 7643 m
3/s at 7:00 a.m. LT on 29 August 2018 Therefore, the time taken to peak discharge was 2 h, which was from 5:00 a.m. to 7:00 a.m. LT. After that, the water discharge rate dramatically decreased to 3000 m
3/s at 9:00 a.m. After the drastic decrease, the water discharge decreased gradually with the decrease in the water level. Therefore, the breach development time or failure time for the Swa Dam breach was taken as 4 h. The detailed water discharge conditions are described in
Table A3.
2.8.2. Empirical Equations from Previous Studies
Empirical equations from previous studies were used to clarify the characteristics of the breach event that happened in the Swa Dam. The empirical equations were simplified equations derived from regression analyses of past dam failure cases [
42]. Several world dam failure datasets were compiled in previous studies and various empirical equations have been proposed on the basis of these datasets to qualify relationships between breach parameters such as peak outflow (
Qp), average width (
Bavg) and failure time of a breach (
Tf). From the empirical equations established between the 1980s and 2022, eight of them were selected to assess the Swa Dam breach. These empirical equations could be divided into two categories according to their distinguishing variables. Six equations were categorized as equations for any failure modes, levels of erodibility and dam types. They include those of the Bureau of Reclamation (USBR) (1982) [
17], MacDonald and Langridge-Monopolis (1984) [
18], Pierce et al. (2010) [
19], Soliman (2015) [
20], Sharharm (2013) [
21] and Tegos et al. (2022) [
22], as shown in
Table 1. Other equations include three equations by Froehlich established in 1995, 2008 and 2016 and that of Xu and Zhang (2009) [
26] (
Table 2).
In the first set of equations, the USBR’s (1982) [
17] equation was based on 21 failed dams, whereas of the MacDonald and Langridge-Monopolis (1984) [
18] equation was from 42 dam failure cases, Pierce et al.’s (2010) [
19] equation was from 87 case studies, Soliman (2015) [
20] used 166 case studies and Sharharm (2013) [
21] collected data from 142 damaged dams. Each equation comprised three equations for three breach parameters: the average breach width (
Bavg), failure time (
Tf) and peak outflow (
Qp). Pierce (2010) [
19] only had an equation for peak outflow, whereas Soliman (2015) [
20] did not have a peak outflow equation and Sharharm (2013) [
21] had no equation for failure time, as described in
Table 1. Most of the equations depended on the water height from the breach bottom (
Hw) and the water volume at failure time (
Vw) (
Table 1). Some equations were governed by the height of a breach (
Hb) and the average dam width (
Wavg) (
Table 1). Soliman’s (2015) [
20] equation was based on dam characteristics such as the storage volume (
V), dam height (
H) and average dam width (
Wavg), as shown in
Table 1. Here, Tegos’s (2022) equation for peak breach outflow was also presented as it is the most updated empirical formula which was constructed on dam characteristics such as the storage volume (
V) and dam height (
H) using the database of 161 historical dam failures [
22].
The second category included Froehlich’s equations and Xu and Zhang’s (2009) [
26] equations, as shown in
Table 2. Froehlich proposed equations for breach parameters (
Bavg,
Tf and
Qp) using data from 111 past dam failure cases in 3 different papers (1995, 2008 and 2016) [
23,
24,
25]. Froehlich’s equations were mostly based on the height of a breach (
Hb), the water height from the breach bottom (
Hw) and the water volume at failure time (
Vw). Failure modes were distinguished by the coefficient
K0 in Froehlich’s average breach width equations proposed in 1995 and 2008 and by
Km in Froehlich’s 2016 average breach width equations. The values for the coefficient
K0 were 1.4 or 1.3 for the overtopping failure mode and 1 for piping or other failure modes. Similarly,
Km = 1.5 or 1.85 for the overtopping failure mode and 1 for piping or other failure modes. Froehlich’s 2016 equations for the average breach width and peak outflow had a factor for the breach height,
Kh, which was 1 for
Hb <
Hs and (
)
1/8 for
Hb >
Hs. A reference height
Hs = 6.1 m was used in Froehlich’s 2016 equations to separate large dams from small dams.
Xu and Zhang (2009) [
26] compiled 75 dam failure cases and proposed three equations for
Bavg,
Tf and
Qp as in
Table 2 [
45]. Meanwhile, Xu and Zhang (2009) [
26] distinguished the effects of the types of dams, failure modes and erodibility by inserting the coefficients
b3,
b4 and
b5, respectively, for each breach parameter, as described in
Table A2 (
Appendix A). The reference height
Hr = 15 m was used in Xu and Zhang’s equations to separate large dams from small dams according to the International Commission of Large Dams (1998) [
12], and the reference time
Tr = 1 h was applied in Xu and Zhang’s (2009) equation of failure time [
26]. Thus, the comparison between the calculated characteristics using previous empirical equations and the observed characteristics in the Swa Dam data is considered effective for clarifying the characteristics of the breach phenomenon in the Swa Dam compared with previous events. Further, as cases of spillway failure are very rare, this comparison may provide insight into the role of spillways on earthen embankment dams in reducing flood damage due to dam breaches.
2.8.3. Tested Scenarios for Failure Mechanisms
Most previous studies on dam failure have focused on two main mechanisms: overtopping erosion and piping failure of the dam body. Meanwhile, an ogee-type spillway failure happened in the Swa Dam case, indicating that the dam failure mechanism was not the same as those of previous events. However, two probable processes, overtopping erosion and piping erosion, were also considered in the spillway failure in the Swa Dam. Because the water level data indicated that the overflow occurred at the start of the breach, we cannot eliminate the possibility of overtopping erosion. Thus, we set two scenarios, Scenario 1 (overtopping erosion) and Scenario 2 (piping erosion) as described in
Figure 8, to apply the selected empirical equations from previous studies.
We set the same water height from the breach bottom (
Hw) and height of the breach (
Hb) for both scenarios based on the collected survey data for the Swa Dam’s collapse site (mentioned in
Section 2.8.1). The water surface level was about R.L 103.2 m at the beginning of the breach and the bottom level of the breach was R.L 81 m which was the final level of the breach opening (detailed description in
Section 2.8.1). The detailed setting for the two scenarios is described in
Table A4. In addition, the dam height (29.6 m), dam width (170.7 m) and storage volume (266,863,398 m
3) of the Swa Dam were applied as parameters related to reservoir properties such as the dam height (
H), dam width (
Wavg) and reservoir storage (
V). The specific values for the coefficients of the empirical equations of Xu and Zhang (2009) [
26] were set up for the Swa Dam breach according to
Table A2. The values for
b4 (failure mode) in Xu and Zhang (2009) [
26] were different for each scenario. Xu and Zhang (2009) [
26] showed three values for
b5 (erodibility), and we performed trial calculations using such three values as there was no direct evidence of such in the Swa Dam. Three values were also described for
b3 in Xu and Zhang’s equations, which were related to the types of dams, and two values were selected for the two dam types (concrete-faced dam and homogeneous earth-filled dam) for the Swa Dam according to the material of the dam and the failure that happened in the concrete spillway.