Current status and prospects of calculation methods for dam and dike failures (Ⅰ)：Theoretical models, parameter models, and one-dimensional mathematical models
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摘要:
国内外堤坝溃决事故屡有发生，堤坝安全对防范洪水灾害至关重要。溃口发展机理及数学计算模型的研究对于洪水预测与风险防范具有重要意义。以溃口数学计算模型的发展脉络为主线，对堤坝溃口特性及溃决过程进行了概括总结，并基于堤坝溃口类型与影响因素的分析，对其分类与演化规律进行了归纳梳理，对比已有模型试验结果与数学计算模型的关系。按照理论模型、参数模型和一维数学模型的演进路径，对堤坝溃决计算方法进行了全面综述，系统汇总了溃口数学模型的历史发展与研究现状，并对不同数学模型的特征和适用性进行了简要比较。以表格形式列举了常见溃口计算方法，便于学者查阅与模型对比，进一步把握溃口发展数学模型的研究方向。总体而言，理论模型与参数模型因计算简便，可快速应用于溃决应急与抢险，但难以描述溃口的动态演化；一维数学模型能够耦合复杂的水流运动、溃口几何变化与泥沙输移过程，更为精细地刻画堤坝溃决的动态特征，但仍存在物理过程简化与假设较多的问题。随着二维、三维数学模型的发展，以及机器学习与人工智能方法的引入，溃口计算模型正朝着物理过程刻画更为精细、计算效率更高的方向演进。
Abstract:Significance Dam and dike failures occur frequently worldwide, and embankment safety is crucial for flood prevention and disaster mitigation. Research on breach development mechanisms and mathematical models is of great significance for flood forecasting and risk prevention.
Progress Following the developmental context of breach mathematical models, this paper summarizes the characteristics and processes of dam breaches, analyzes breach types and influencing factors, summarizes their classification and evolutionary patterns, and compares the relationship between model test results and mathematical models. Theoretical models, parameter models, and one-dimensional mathematical models are systematically reviewed to summarize the historical evolution and current state of dam breach modeling. The distinct features and applicability of different modeling approaches are comparatively analyzed, and common breach calculation methods are summarized in a table to facilitate model reference and comparison, thereby providing a clearer understanding of research directions in mathematical models of breach development.
Conclusion and Prospect In general, theoretical and parameter models are computationally simple and can be quickly applied to emergency breach assessment and disaster response, yet they fail to capture the dynamic evolution of the breach process. One-dimensional mathematical models, by incorporating complex hydraulic flow, breach geometry variation, and sediment transport, provide more detailed representations of breach dynamics, though they still rely on simplifications and assumptions of physical processes. With the rapid advancement of two- and three-dimensional mathematical models, as well as the introduction of machine learning and artificial intelligence techniques, breach modeling is evolving toward more refined physical representation and higher computational efficiency.
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Key words:
- dam and dike /
- breach development /
- theoretical model /
- parameter model /
- mathematical model
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图 1 河道堤防溃口发展过程示意图
Figure 1. Schematic diagram of breach development in a river embankment
图 3 瞬时溃决Ritter解示意图
x. 距离溃口断面的距离，向下游为正；t. 时间；g. 重力加速度；h0. 初始上游水深；h. 溃口当前水深；下同
Figure 3. Schematic diagram of Ritter solution for instantaneous breach
图 4 任意梯形断面溃口形状示意图
m1，m2. 溃口两侧坡度；h'0. 等效梯形断面水深与h的差值；a. 梯形断面的斜底高差；b. 溃口最大可裁剪梯形断面底部宽度；b0. 等效梯形断面底部宽度；下同
Figure 4. Schematic diagram of breach shape with arbitrary trapezoidal cross-section
图 5 湿床河道溃坝波示意图[38]
1～4. 溃坝波段编号；M，N，R. 溃坝波段交界位置；O为原点位置；t0. 激波与稀疏波的时间差；u. 2段处流速；ξ. 激波的波前速度；ξt. t时刻激波速度；ht. 2段处水深；hc. 3段处水深；h1. 下游段水深；uc. 3段处流速；xM. 1段和2段交界位置M处坐标；xN. 2段和3段交界位置N处坐标；xR. 3段和4段交界位置R处坐标；下同
Figure 5. Schematic diagram of dam-break wave in a wet-bed channel
图 6 典型梯形溃口示意图
hw. 溃口底部以上水深; hb. 溃口深度; hd. 坝高; m. 溃口平均坡度; Bave. 溃口最终平均宽度; 下同
Figure 6. Schematic diagram of a typical trapezoidal breach
图 8 溃口横向扩展示意图[109]
z0. 溃口底部初始高度；β为梯形斜面与水平面的夹角；βc. 割线夹角；βr. 切线夹角
Figure 8. Schematic diagram of lateral breach expansion
表 1 3个现场溃决试验对比
Table 1. Comparison of three field dam breach experiments
现场试验名称 堤坝材料类型 坝高/m 相关计算模型 欧洲IMPACT
计划项目黏性土、非黏性
土、卵砾石4.5～6 HR BREACH、NWS
BREACH、DEICH、
SIMBA、Sobek、Firebird美国农业部
试验粉砂、黏性土 1.3～2.3 SIMBA、WINDAM 南京水利科学
研究院试验黏性土 9.7 DB-NHRI 
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表 2 国内外常见峰值流量参数模型汇总
Table 2. Summary of commonly used peak discharge parameter models worldwide
参数模型 峰值流量表达式 案例量 参数量 KIRKPATRICK[39] $ {Q}_{\text{p}}=1.268{({{h}_{\text{w}}}+0.3)}^{2.5} $ 19 1 Soil Conservation Seerive[41] $ {Q}_{\text{p}}=16.6{({{h}_{\text{w}}})}^{1.85} $ 13 1 HAGEN[42] $ {Q}_{\text{p}}=0.54{({{h}_{\text{d}}}S)}^{0.5} $ 6 2 SINGH等[43] $ {Q}_{\text{p}}=13.4{({{h}_{\text{d}}})}^{1.89},{Q}_{\text{p}}=1.776{({{V}_{\text{w}}})}^{0.47} $ 28 1 MACDONALD等[44] $ {Q}_{\text{p}}=1.154{({{V}_{\text{w}}}{{h}_{\text{w}}})}^{0.412} $ 23 2 COSTA[45] $ {Q}_{\text{p}}=0.981{({{h}_{\text{d}}}{{V}_{\text{w}}})}^{0.42} $ 31 2 EVANS[46] $ {Q}_{\text{p}}=0.72{({{V}_{\text{w}}})}^{0.53} $ 29 1 U. S. Bureau of Reclamation[47] $ {Q}_{\text{p}}=19.1{({{h}_{\text{w}}})}^{1.85} $ 21 1 FROCHLICH[48] $ {Q}_{\text{p}}=0.607{({{V}_{\text{w}}})}^{0.295}{({{h}_{\text{w}}})}^{1.24} $ 22 2 WALDER等[49] $ {Q}_{\text{p}}=0.031{(g)}^{0.5}{({{V}_{\text{w}}})}^{0.47}{({{h}_{\text{w}}})}^{0.15}{({{h}_{\text{b}}})}^{0.94} $ 18 3 XU等[50] $ {Q}_{\text{p}}=0.175{g}^{0.5}V_{\text{w}}^{5/6}{({{h}_{\text{d}}}/{{h}_{\text{r}}})}^{0.199}{({V_{\text{w}}^{1/3}}/{{h}_{\text{w}}})}^{-1.274}{{\mathrm{e}}}^{{{B}_{4}}} $ 75 3 PIERCE等[51] $ {Q}_{\text{p}}=0.017\;6{({{V}_{\text{w}}}{{h}_{\text{w}}})}^{0.606}，{Q}_{\text{p}}=0.038{({{V}_{\text{w}}})}^{0.475}{({{h}_{\text{w}}})}^{1.09} $ 87 2 THORNTON等[52] $ {Q}_{\text{p}}=0.120\;2{L}^{1.785\;6}\text{,}{Q}_{\text{p}}=0.863{V}^{0.335}h_{\text{d}}^{1.833}W_{\text{ave}}^{-0.663}\text{,}{Q}_{\text{p}}=0.012{V}^{0.493}h_{\text{d}}^{1.205}L_{}^{0.226} $ 38 3 PENG等[53] $ \dfrac{{Q}_{\text{p}}}{{g}^{0.5}h_{\text{d}}^{2.5}}={\left(\dfrac{{h}_{\text{d}}}{{H}_{\text{r}}}\right)}^{-1.417}{\left(\dfrac{{h}_{\text{d}}}{{W}_{\text{d}}}\right)}^{-0.265}{\left(\dfrac{V_{\text{d}}^{1/3}}{{h}_{\text{d}}}\right)}^{-0.471}{\left(\dfrac{V_{\text{l}}^{1/3}}{{h}_{\text{d}}}\right)}^{1.569}{{\mathrm{e}}}^{\alpha } $ 45 4 DE LOORENZO等[54] $ \begin{aligned}{Q}_{\text{p}}&=0.321{g}^{0.258}(0.07{V_{\text{w}}^{}})^{0.485}h_{\text{b}}^{0.802}（漫顶）\\{Q}_{\text{p}}&=0.347{g}^{0.263}(0.07{V_{\text{w}}^{}})^{0.474}h_{\text{b}}^{-2.151}h_{\text{w}}^{2.992}（渗透破坏）\end{aligned} $ 14 3 HOOSHYARIPOR等[55] $ {Q}_{\text{p}}=0.021\;2{({{V}_{\text{w}}})}^{0.542\;9}{({{h}_{\text{w}}})}^{0.871\;3},{Q}_{\text{p}}=0.045\;4{({{V}_{\text{w}}})}^{0.448}{({{h}_{\text{w}}})}^{1.156} $ 93 2 AZIMI等[56] $ {Q}_{\text{p}}=0.016\;6{(gS)}^{0.5}h $ 70 2 FROEHLICH[40] $ {Q}_{\text{p}}=0.017\;5{k}_{\text{M}}{k}_{\text{H}}{\text{(}g{{V}_{\text{w}}}{{h}_{\text{w}}}{h_{\text{b}}^{\text{2}}}/{{W}_{\text{ave}}}\text{)}}^{0.5} $ 41 4 黄委会科研所[57] $ {Q}_{\text{p}}=0.296(\sqrt{g}){({B_{\mathrm{R}}}/{B})}^{0.4}{B}{H}^{1.5} $ − 3 戴荣尧等[58] $ {Q}_{\text{p}}=0.27(\sqrt{g}){({{L}_{0}}/{B_{\mathrm{R}}})}^{1/10}{({B_{\mathrm{R}}}/B)}^{1/3}B{(H-K{\textit{z}})}^{1.5} $ − 3 国家防汛抗旱总指挥部等[59] $ \begin{aligned}{Q}_{\text{b}}&={c}_{1}\sigma B\sqrt{2g}h_{\text{w}}^{\text{1.5}}（堤防溃口出流）\\{Q}_{\text{b}}&={c}_{\text{v}}{k}_{\text{s}}{c}_{\text{d}}\sqrt{2\mathrm{g}}\left[\dfrac{2}{3}{b}_{\text{s}}{(H-{\textit{z}})}^{1.5}+\dfrac{8}{15m}{(H-{\textit{z}})}^{2.5}\right]（土石坝漫顶溃决）\end{aligned} $ − 2 邓刚等[60] $ {Q}_{\text{p}}={B}_{\text{b}}{c}_{\text{d}}\sqrt{2g}{(H-{\textit{z}})}^{3/2}+\dfrac{4{c}_{\text{d}}m\sqrt{2g}}{5}\left\{{(H-{\textit{z}})}^{5/2}[1-{(1-k)}^{5/2}]\right\} $ − 2 石振明等[61] $ {Q}_{\text{p}}=3.130h_{\text{d}}^{0.120}W_{\text{d}}^{0.302}V_{\text{d}}^{\text{-0.106}}V_{\text{l}}^{\text{0.453}}{{\mathrm{e}}}^{\alpha } $ 26 4 梅世昂等[62] $ \dfrac{{Q}_{\text{p}}}{{V}_{\text{w}}{g}^{0.5}h_{\text{w}}^{-0.5}}=\begin{cases} {\left(\dfrac{V_{\text{w}}^{1/3}}{{h}_{\text{w}}}\right)}^{-1.58}{\left(\dfrac{{h}_{\text{w}}}{{h}_{\text{b}}}\right)}^{-0.76}{\left(\dfrac{{h}_{\text{d}}}{{h}_{\text{r}}}\right)}^{0.10}{\mathrm{e}}^{-4.55}（均质坝）\\{\left(\dfrac{V_{\text{w}}^{1/3}}{{h}_{\text{w}}}\right)}^{-1.51}{\left(\dfrac{{h}_{\text{w}}}{{h}_{\text{b}}}\right)}^{-1.09}{\left(\dfrac{{h}_{\text{d}}}{{h}_{\text{r}}}\right)}^{-0.12}{\mathrm{e}}^{-3.61}（心墙坝）\end{cases} $ 154 3 齐子杰等[63] $ \dfrac{{Q}_{\text{p}}}{{g}^{0.5}h_{\text{d}}^{2.5}}=0.828{\left(\dfrac{{h}_{\text{d}}}{{H}_{\text{r}}}\right)}^{-0.128}{\left(\dfrac{{h}_{\text{d}}}{{W}_{\text{d}}}\right)}^{-0.432}{\left(\dfrac{V_{\text{d}}^{1/3}}{{h}_{\text{d}}}\right)}^{-0.394}{\left(\dfrac{V_{\text{l}}^{\text{1/3}}}{{h}_{\text{d}}}\right)}^{1.151} $ 65 4 单熠博等[64] $ {Q}_{\text{p}}={\left({h}_{\text{d}}\right)}^{-0.229}{\left({W}_{\text{d}}\right)}^{-0.04}{\left({V}_{\text{l}}\right)}^{0.558}{\mathrm{e}}^{{{C}_{\text{m}}}} $ 44 3 FROEHLICH[65] $ {Q}_{\text{p}}={K}_{Q}{\left(\dfrac{V_{\text{l}}^{1/3}}{{h}_{\text{d}}}\right)}^{2}\sqrt{gh_{\text{d}}^{5}} $ 42 2 焦煦等[66] $ \begin{aligned}{Q}_{\text{p}}&=10.03h_{\text{d}}^{2.816}V_{\text{l}}^{\text{1.557}}h_{\text{b}}^{\text{-3.661}}({h}_{\text{d}} > 70),\;{Q}_{\text{p}}=h_{\text{d}}^{\text{1.629}}V_{\text{l}}^{\text{0.826}}h_{\text{b}}^{\text{-0.053}}(30\leq {h}_{\text{d}}\leq 70)\\{Q}_{\text{p}}&=194.7h_{\text{d}}^{1.163}V_{\text{l}}^{\text{-0.084}}h_{\text{b}}^{-0.636}({h}_{\text{d}} < 30)\end{aligned} $ 75 3 GUAN等[67] $ {Q}_{\text{p}}=910h_{\text{d}}^{-0.25}V_{\text{l}}^{0.56}{{\mathrm{e}}}^{{{\alpha }_{2}}} $ 46 2 冯震宇等[68] $ {Q}_{\text{p}}=0.043{\left(\dfrac{{h}_{\text{d}}}{{H}_{\text{r}}}\right)}^{1.717}{\left(\dfrac{{W}_{\text{d}}}{{h}_{\text{d}}}\right)}^{1.725}{\left(\dfrac{V_{\text{d}}^{1/3}}{{h}_{\text{d}}}\right)}^{-2.03}{\left(\dfrac{V_{\text{l}}^{\text{1/3}}}{{h}_{\text{d}}}\right)}^{1.865}{{\mathrm{e}}}^{{{\alpha }_{1}}}{{\mathrm{e}}}^{{{\beta }_{1}}} $ 48 4 注：Qp为峰值流量；Qb为溃口实时流量；hw为溃口底部以上水深；S为库容；Vw为溃口底部以上水库库容；hr为参考坝高15 m；L为坝体长度；Wave为坝体平均宽度；Hr为参考坝高1 m；Wd为坝体宽度；Vd为坝体体积；Vl为溃坝时库容；L0为库长；BR为库宽；B为溃口平均宽度；H为坝前水深；z为坝体残留高度；K为系数，详见文献[58]；B4为考虑坝型、溃决类型与坝高的拟合指数，见文献[50]；kM、kH为系数，具体见文献[40]；c1为自由溢流流量系数，σ为淹没系数，cv为行进流速改正系数，cd为流量系数，ks为流量修正系数，bs为溃口实时底宽，具体见文献[59]；Bb为溃口最终底宽；k为水头跌落系数，见文献[60]；α为侵蚀度，见文献[61]；Cm为考虑颗粒组成的冲蚀因子，具体见文献[64]；KQ为坝体可蚀系数，具体取值见文献[65]；α2为可蚀系数，具体取值见文献[67]；α1为坝体材料系数，β1为坝体诱因系数，具体取值见文献[68]；下同
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表 3 国内外常见溃口平均宽度参数模型汇总
Table 3. Summary of commonly used average breach width parameter models worldwide
参数模型 溃口宽度表达式 案例量 参数量 U. S. Bureau of Reclamation[47] $ {B}_{\text{ave}}=3{h}_{\text{w}} $ 63 1 VONTHUN等[69] $ {B}_{\text{ave}}=2.5{h}_{\text{w}}+{C}_{\text{b}} $ 63 1 FROCHLICH[48] $ {B}_{\text{ave}}=0.180\;3{K}_{0}{({{V}_{\text{w}}})}^{0.32}{({{h}_{\text{b}}})}^{0.19} $ 63 2 XU等[50] $ \dfrac{{B}_{\text{ave}}}{{h}_{\text{b}}}=0.787{\left(\dfrac{{h}_{\text{d}}}{{h}_{\text{r}}}\right)}^{0.133}{\left(\dfrac{V_{\text{w}}^{1/3}}{{h}_{\text{w}}}\right)}^{0.652}{\mathrm{e}}^{{{B}_{3}}} $ 47 2 PENG等[53] $ \begin{aligned}\dfrac{{B}_{\text{f}}}{{H}_{\text{r}}}&={\left(\dfrac{{h}_{\text{d}}}{{H}_{\text{r}}}\right)}^{0.752}{\left(\dfrac{{h}_{\text{d}}}{{W}_{\text{d}}}\right)}^{0.315}{\left(\dfrac{V_{\text{d}}^{\text{1/3}}}{{h}_{\text{d}}}\right)}^{-0.243}{\left(\dfrac{V_{\text{l}}^{\text{1/3}}}{{h}_{\text{d}}}\right)}^{0.682}{\mathrm{e}}^{\alpha }（溃口顶宽）\\\dfrac{{B}_{\text{b}}}{{h}_{\text{d}}}&=0.004\left(\dfrac{{h}_{\text{d}}}{{H}_{\text{r}}}\right)+0.050\left(\dfrac{{h}_{\text{d}}}{{W}_{\text{d}}}\right)-0.044\left(\dfrac{V_{\text{d}}^{\text{1/3}}}{{h}_{\text{d}}}\right)+0.088\left(\dfrac{V_{\text{l}}^{\text{1/3}}}{{h}_{\text{d}}}\right)+\alpha （溃口底宽）\end{aligned} $ 10 4 FROEHLICH[40] $ {B}_{\text{ave}}=0.27{k}_{\text{M}}{({{V}_{\text{w}}})}^{1/3} $ 63 1 黄委会科研所[57] $ {B}_{\text{ave}}=k{S}^{0.25}{B_{\mathrm{R}}}^{0.25}{H}^{0.5} $ - 3 戴荣尧等[58] $ {B}_{\text{ave}}=k{S}^{0.25}{B_{\mathrm{R}}}^{1/7}{H}^{0.5} $ - 3 国家防汛抗旱总指挥部等[59] $ \begin{aligned}{B}_{\text{ave}}&=4.5({{\lg }}{{B}_{\text{r}}})^{3.5}+50（汇流点）\\{B}_{\text{ave}}&=1.9({{\lg }}{{B}_{\text{r}}})^{4.8}+20（其他）\end{aligned} $ - 1 刘建康等[70] $ {B}_{\text{ave}}=0.367{\left(\dfrac{S}{{V}_{\text{s}}}\right)}^{0.195}{\left(\dfrac{{B}_{\text{e}}}{\tan \varphi }\right)}^{0.337}{H}^{0.5} $ 31 3 石振明等[61] $ \begin{aligned}{B}_{\text{f}}&=1.593{h}_{\text{d}}+85.249\dfrac{{h}_{\text{d}}}{{W}_{\text{d}}}-3.438\dfrac{V_{\text{d}}^{1/3}}{{h}_{\text{d}}}+15.963\dfrac{V_{\text{l}}^{\text{1/3}}}{{h}_{\text{d}}}+\alpha （溃口顶宽）\\{B}_{\text{b}}&=-0.006h_{\text{d}}^{\text{2}}-0.047\dfrac{h_{\text{d}}^{\text{2}}}{{W}_{\text{d}}}+0.017V_{\text{d}}^{\text{1/3}}+0.047V_{\text{l}}^{\text{1/3}}+\alpha {h}_{\text{d}}（溃口底宽）\end{aligned} $ 16 4 梅世昂等[62] $ \dfrac{{B}_{\text{ave}}}{{h}_{\text{b}}}=\begin{cases} {\left(\dfrac{V_{\text{w}}^{1/3}}{{h}_{\text{w}}}\right)}^{0.84}{\left(\dfrac{{h}_{\text{w}}}{{h}_{\text{b}}}\right)}^{2.30}{\left(\dfrac{{h}_{\text{d}}}{{h}_{\text{r}}}\right)}^{0.06}{\mathrm{e}}^{-0.90}（均质坝）\\{\left(\dfrac{V_{\text{w}}^{1/3}}{{h}_{\text{w}}}\right)}^{0.55}{\left(\dfrac{{h}_{\text{w}}}{{h}_{\text{b}}}\right)}^{1.97}{\left(\dfrac{{h}_{\text{d}}}{{h}_{\text{r}}}\right)}^{-0.07}{\mathrm{e}}^{-0.09}（心墙坝）\end{cases} $ 63 3 齐子杰[63] $ \begin{aligned}\dfrac{{B}_{\text{f}}}{{H}_{\text{r}}}&=1.162{\left(\dfrac{{h}_{\text{d}}}{{H}_{\text{r}}}\right)}^{1.016}{\left(\dfrac{{h}_{\text{d}}}{{W}_{\text{d}}}\right)}^{0.429}{\left(\dfrac{V_{\text{d}}^{1/3}}{{h}_{\text{d}}}\right)}^{0.444}{\left(\dfrac{V_{\text{l}}^{\text{1/3}}}{{h}_{\text{d}}}\right)}^{0.381}（溃口顶宽）\\\dfrac{{B}_{\text{b}}}{{H}_{\text{r}}}&=-12.55+0.208\dfrac{{h}_{\text{d}}}{{H}_{\text{r}}}+17.38\dfrac{{h}_{\text{d}}}{{W}_{\text{d}}}-1.941\dfrac{V_{\text{d}}^{\text{1/3}}}{{h}_{\text{d}}}+7.371\dfrac{V_{\text{l}}^{\text{1/3}}}{{h}_{\text{d}}}（溃口底宽）\end{aligned} $ 12 4 冯震宇等[68] $ \begin{aligned}{B}_{\text{f}}&=0.462{\left(\dfrac{{h}_{\text{d}}}{{H}_{\text{r}}}\right)}^{1.77}{\left(\dfrac{{W}_{\text{d}}}{{h}_{\text{d}}}\right)}^{0.081}{\left(\dfrac{V_{\text{d}}^{1/3}}{{h}_{\text{d}}}\right)}^{1.419}{\left(\dfrac{V_{\text{l}}^{\text{1/3}}}{{h}_{\text{d}}}\right)}^{-0.457}{\mathrm{e}}^{{{\alpha }_{1}}}{\mathrm{e}}^{{{\beta }_{1}}}（溃口顶宽）\\{B}_{\text{b}}&=7.83\times {10}^{-9}{\left(\dfrac{{h}_{\text{d}}}{{H}_{\text{r}}}\right)}^{6.634}{\left(\dfrac{{W}_{\text{d}}}{{h}_{\text{d}}}\right)}^{1.874}{\left(\dfrac{V_{\text{d}}^{1/3}}{{h}_{\text{d}}}\right)}^{-1.07}{\left(\dfrac{V_{\text{l}}^{\text{1/3}}}{{h}_{\text{d}}}\right)}^{-1.027}{\mathrm{e}}^{{{\alpha }_{1}}}{\mathrm{e}}^{{{\beta }_{1}}}（溃口底宽）\end{aligned} $ 11 4 注：Bf为溃口最终顶宽；Br为河道宽度；Vs为坝体单宽体积；Be为有效坝长；φ为内摩擦角；Cb为经验拟合系数，取值见文献[69]；B3为系数，详见文献[32]；k为系数，见文献[57]；对于漫顶溃坝，K0=1.4，对于渗透破坏溃坝，K0=1.0；对于漫顶溃坝，kM=1.3，对于渗透破坏溃坝，kM=1.0；下同
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表 4 国内外常见溃决历时参数模型汇总
Table 4. Summary of commonly used breach duration parameter models worldwide
参数模型 溃坝历时表达式 案例量 参数量 MACDONALD等[44] $ {T}_{\text{f}}=0.017\;9{(0.026\;1{{({{V}_{\text{w}}}{{h}_{\text{w}}})}^{0.769}})}^{0.364} $ 39 2 U.S. Bureau of Reclamation[47] $ {T}_{\text{f}}=0.011{B}_{\text{ave}} $ 39 1 FROCHLICH[48] $ {T}_{\text{f}}=0.002\;54{({{V}_{\text{w}}})}^{0.53}{({{h}_{\text{b}}})}^{-0.9} $ 39 2 XU等[50] $ \dfrac{{T}_{\text{f}}}{{T}_{\text{r}}}=0.304{\left(\dfrac{{h}_{\text{d}}}{{H}_{\text{r}}}\right)}^{0.707}{\left(\dfrac{V_{\text{w}}^{\text{1/3}}}{{h}_{\text{w}}}\right)}^{1.228}{\mathrm{e}}^{{{B}_{5}}} $ 34 2 国家防汛抗旱总指挥部等[59] $ {T}_{\text{f}}=1.527({B}_{\text{ave}}-10) $ − 1 FROEHLICH[40] $ {T}_{\text{f}}=63.2{({{V}_{\text{w}}}/(g{{h}_{\text{b}}}{{}^{2}}))}^{0.5}/3\;600 $ 39 2 PENG等[53] $ \dfrac{{T}_{\text{f}}}{{T}_{\text{r}}}={\left(\dfrac{{h}_{\text{d}}}{{H}_{\text{r}}}\right)}^{0.262}{\left(\dfrac{{h}_{\text{d}}}{{W}_{\text{d}}}\right)}^{-0.024}{\left(\dfrac{V_{\text{d}}^{\text{1/3}}}{{h}_{\text{d}}}\right)}^{-0.103}{\left(\dfrac{V_{\text{l}}^{\text{1/3}}}{{h}_{\text{d}}}\right)}^{0.705}{\mathrm{e}}^{{{a}_{9}}} $ 14 4 石振明等[61] $ {T}_{\text{f}}=h_{\text{d}}^{0.275}W_{\text{d}}^{-1.224}V_{\text{d}}^{\text{0.439}}V_{\text{l}}^{\text{0.232}}{\mathrm{e}}^{\alpha } $ 12 4 梅世昂等[62] $ \dfrac{{T}_{\text{f}}}{{T}_{0}}=\begin{cases} {\left(\dfrac{V_{\text{w}}^{1/3}}{{h}_{\text{w}}}\right)}^{0.56}{\left(\dfrac{{h}_{\text{w}}}{{h}_{\text{b}}}\right)}^{-0.85}{\left(\dfrac{{h}_{\text{d}}}{{h}_{\text{r}}}\right)}^{-0.32}{\mathrm{e}}^{-0.20}（均质坝）\\{\left(\dfrac{V_{\text{w}}^{1/3}}{{h}_{\text{w}}}\right)}^{1.52}{\left(\dfrac{{h}_{\text{w}}}{{h}_{\text{b}}}\right)}^{-11.36}{\left(\dfrac{{h}_{\text{d}}}{{h}_{\text{r}}}\right)}^{-0.43}{\mathrm{e}}^{-1.57}（心墙坝）\end{cases} $ 39 3 齐子杰等[63] $ \dfrac{{T}_{\text{f}}}{{T}_{\text{r}}}=0.05{\left(\dfrac{{h}_{\text{d}}}{{H}_{\text{r}}}\right)}^{1.237}{\left(\dfrac{{h}_{\text{d}}}{{W}_{\text{d}}}\right)}^{-1.371}{\left(\dfrac{V_{\text{d}}^{1/3}}{{h}_{\text{d}}}\right)}^{-0.905}{\left(\dfrac{V_{\text{l}}^{\text{1/3}}}{{h}_{\text{d}}}\right)}^{0.736} $ 16 4 冯震宇等[68] $ {T}_{\text{f}}=2.352{\left(\dfrac{{h}_{\text{d}}}{{H}_{\text{r}}}\right)}^{1.4324}{\left(\dfrac{{W}_{\text{d}}}{{h}_{\text{d}}}\right)}^{-0.58}{\left(\dfrac{V_{\text{d}}^{1/3}}{{h}_{\text{d}}}\right)}^{0.623}{\left(\dfrac{V_{\text{l}}^{\text{1/3}}}{{h}_{\text{d}}}\right)}^{1.625}{\mathrm{e}}^{{{\alpha }_{1}}}{\mathrm{e}}^{{{\beta }_{1}}} $ 18 4 注：$T_{\mathrm{f}} $为溃决历时；$T_{\mathrm{r}} $为单位时间1 h；$a_9 $为经验拟合系数，见参考文献[53]；B5为系数，见文献[50]；下同
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表 5 国内外常见溃口深度参数模型汇总
Table 5. Summary of commonly used breach depth parameter models worldwide
参数模型 溃口深度表达式 案例量 参数量 PENG等[53] $ \dfrac{{h}_{\text{b}}}{{H}_{\text{r}}}={\left(\dfrac{{h}_{\text{d}}}{{H}_{\text{r}}}\right)}^{0.882}{\left(\dfrac{{h}_{\text{d}}}{{W}_{\text{d}}}\right)}^{-0.041}{\left(\dfrac{V_{\text{d}}^{\text{1/3}}}{{h}_{\text{d}}}\right)}^{-0.099}{\left(\dfrac{V_{\text{l}}^{\text{1/3}}}{{h}_{\text{d}}}\right)}^{0.139}{\mathrm{e}}^{\alpha } $ 21 4 石振明等[61] $ {h}_{\text{b}}=h_{\text{d}}^{0.840}W_{\text{d}}^{-0.169}V_{\text{d}}^{0.089}V_{\text{l}}^{\text{0.040}}{\mathrm{e}}^{\alpha } $ 26 4 齐子杰等[63] $ \dfrac{{h}_{\text{b}}}{{H}_{\text{r}}}=2.266+0.403\dfrac{{h}_{\text{d}}}{{H}_{\text{r}}}-14.668\dfrac{{h}_{\text{d}}}{{W}_{\text{d}}}-0.409\dfrac{V_{\text{d}}^{\text{1/3}}}{{h}_{\text{d}}}+1.181\dfrac{V_{\text{l}}^{\text{1/3}}}{{h}_{\text{d}}} $ 10 4 焦煦等[66] $ {h}_{\text{b}}=\alpha (1.636h_{\text{d}}^{0.791}+0.003V_{\text{l}}^{\text{1.183}}) $ 23 2 冯震宇等[68] $ {h}_{\text{b}}=0.102{\left(\dfrac{{h}_{\text{d}}}{{H}_{\text{r}}}\right)}^{0.68}{\left(\dfrac{{W}_{\text{d}}}{{h}_{\text{d}}}\right)}^{0.299}{\left(\dfrac{V_{\text{d}}^{1/3}}{{h}_{\text{d}}}\right)}^{-0.434}{\left(\dfrac{V_{\text{l}}^{\text{1/3}}}{{h}_{\text{d}}}\right)}^{0.174}{\mathrm{e}}^{{{\alpha }_{1}}}{\mathrm{e}}^{{{\beta }_{1}}} $ 19 4
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表 6 国内外常见一维计算模型汇总
Table 6. Summary of commonly used one-dimensional computational models worldwide
来源 溃口形状 冲蚀计算公式 流量计算公式 备 注 CRISTOFANO[75] 恒定底宽的梯形 — 宽顶堰公式 不考虑溃口的横向冲蚀，时间步内只考虑溃口的竖向冲蚀量 H-W[76] 抛物线形 Schoklitsch推移质公式 宽顶堰公式 溃口形状为抛物线，溃口顶宽为溃口深度3.5倍；溃口坡度为坝体材料的内摩擦角并保持不变 BRDAM[77] 抛物线形 Schoklitsch推移质公式 宽顶堰公式 在Harris-Wagner模型基础上改进，溃口侧壁坡度恒定为45° LOU[78] 根据剪应力求解 Duboy推移质模型&Einstein悬移质公式 Saint-Venant方程 根据剪应力来分析溃口的形状 P-T[79] 根据水流速度计算溃口顶宽 Meyer-Peter&Muller推移质公式 Saint-Venant方程 根据河床稳定理论确定溃口宽度和水流速度的关系，流量达到峰值后溃口宽度不再变化 DAMBRK[74] 由溃口底宽、高度和侧壁坡度决定 经验公式 宽顶堰公式 根据Froelich冲蚀关系经验方程计算溃口侵蚀过程 NOGUEIRA[80] 断面水深为与溃口纵断面距离的余弦函数 Meyer-Peter & Muller推移质公式 Saint-Venant方程 在Lou模型的基础上进行改进，考虑了溃口侧壁滑塌对溃口扩展的影响 BEED[81] 由溃口底宽、高度和侧壁坡度决定 Einstein – Brown推移质公式 宽顶堰公式 根据楔形体稳定分析方法确定溃口侧壁形状，考虑渗透力、扬压力分析楔形体稳定性 BREACH[82] 矩形和梯形 Smart改进的Meyer-Peter & Muller推移质公式 宽顶堰公式 溃口稳定性根据材料性质判断，同时考虑了水压力过大造成的溃口坍塌影响 NCP BREACH[89] 抛物线形 经验公式 经验公式 根据室内试验数据得到溃口流量参数、溃口形状和冲蚀计算公式参数 ED BREACH[90] 梯形 Meyer-Peter & Muller推移质公式 宽顶堰公式 对BREACH模型的改进模型 BRES[91] 梯形 Bagnold-Visser、Engelund-Hansen和Van Rijn公式 宽顶堰公式 分为5个阶段研究溃坝过程，各阶段分别采用不同的冲蚀公式进行计算 HR BREACH[92] Mohammed法 Yang公式、Visser公式和Chen-Anderson 公式 宽顶堰公式 在BRES模型冲蚀计算方法的基础上，引入有效剪应力计算溃口形状 SIMBA[93] 矩形或梯形 引入坝料冲蚀系数模拟坝体冲蚀 宽顶堰公式 模拟陡坎侵蚀，将溃口过程分为4个阶段，采用基于能量法或剪应力法的陡坎移动公式模拟溯源冲刷 WINDAM[94] 矩形或梯形 引入坝料冲蚀系数模拟坝体冲蚀 宽顶堰公式 在SIMBA模型的基础上，考虑了坡面植被的影响 陈生水等[95]
DB-NHRI矩形或梯形 修正的Meyer-Peter & Muller 推移质公式 宽顶堰公式 采用溃口边坡稳定性分析来计算边坡楔形体的稳定性，从而判断溃口扩展，通过下游坝体冲槽和坝顶溃口流量平衡建立二者发展过程的相互影响 傅旭东等[96] 梯形 泥沙输移公式 Saint-Venant方程 基于Exner方程，考虑床沙活动层和泥沙的侧向补给，得到河床变形高度，考虑并采用Osman和Thorne[97]提出的河堤稳定性分析方法 CHANG等[98] 梯形 经验公式 宽顶堰公式 冲蚀速率经验公式由原位试验得到，并在计算过程中考虑了溃口深度对冲蚀的影响 DHI Water Environment[99]
MIKE11DB梯形 Engelund-Hansen推移质公式，也可自定义冲蚀系 宽顶堰公式 可采用两种冲蚀计算防范，自定义输入参数，作为溃坝计算边界条件 WU[100] 矩形、梯形 直线型公式 宽顶堰公式 使用Osman和Thorne提出的河堤稳定性分析方法，并考虑了溯源冲刷过程 CHEN等[88]
DB-IWHR圆弧形 经验双曲线型公式 宽顶堰公式 以流速为步长进行计算，在溃口深度采用冲蚀计算的基础上，溃口横向扩展采用土坡稳定分析中的圆弧滑动法进行 BRUNNER[101]
HEC-RAS多边形 泥沙输移公式 Saint-Venant方程 溃决过程采用BREACH模型，洪水下游演进，采用四点隐式有限差分格式对非线性方程进行离散求解 ZHAO[102] 梯形 基于力矩平衡的侵蚀率公式 宽顶堰公式 将溃口发展过程分为表面侵蚀、切头侵蚀、横向侵蚀3个阶段，切头侵蚀与横向侵蚀过程以黏土块的形式进行 沈鸿杰等[103]
改进DB-IWHR梯形 经验双曲线型公式 宽顶堰公式 采用修正的Shields曲线改进DB-IWHR模型中临界剪应力的计算 巨江等[104]
DB-D梯形 溃口下切扩散方程 宽顶堰公式 采用溃坝冲刷过程扩散模型，并给出溃口纵剖面侵蚀下切与横断面展宽的解析解及冲刷系数计算公式 吕佳豪等[105] 梯形条分 — Saint-Venant方程 基于Godunov有限体积格式的一维溃坝洪水演进模型，采用条分法处理复杂河道断面
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