工程科学与技术   2019, Vol. 51 Issue (5): 137-142

Study on Stage–discharge Relationship Curve in Mountain Rivers
LUO Ming, DING Rui, HUANG Er, FAN Niannian
State Key Lab. of Hydraulics and Mountain River Eng., Sichuan Univ., Chengdu 610065, China
Abstract: The relationship between stage and discharge caused by flood fluctuation shows a clear loop curve and most of the current researches are based on plain rivers. Due to larger river-bed slope, bigger size of sediment the difficult flow characteristics affected by many factors, some assumptions used are inconsistent so that the result is not very accurate. Therefore, this paper, considering the differences of river bed morphology and sediment particle size between mountain rivers and flat rivers, introduces new parameters of equations about river grade and sediment size and reintroduces the new stage–discharge relationship based on Saint–Venant equations. The results of examples application show this method can better fit the actual data of stage–discharge of the cross-sections in mountain rivers and plain rivers. When adapting the line, this method only need to make some corrections to the parameters according to the actual situation, which is more convenient to modify and has good practicality, high precision and strong adaptability.
Key words: mountain rivers    stage–discharge relationship    loop-rating curve    diffusion wave    mathematical model

1 水位流量数学模型

 $Q=AV=AC\sqrt {R{S_{\rm f}}}$ (1)

 $\frac{{\partial Q}}{{\partial x}} + \frac{{\partial A}}{{\partial t}} = 0$ (2)
 $\frac{1}{g}\frac{{\partial V}}{{\partial t}} + \frac{V}{g}\frac{{\partial V}}{{\partial x}} + \frac{{\partial h}}{{\partial x}} + {S\!_{\rm f}} - {S_0} = 0$ (3)

 ${S_{\rm f}}={{{S}}_{\rm{0}}} = \frac{{{V^2}}}{{{C^2}R}} = \frac{{{Q^2}}}{{{{(CA)}^2}R}}$ (4)

 $C = \frac{1}{n}{R^{\tfrac{1}{6}}} = \kappa \frac{{{g^{0.43}}{Q^{0.14}}}}{{d_{90}^{0.52}S_0^{0.26}}}{R^{\tfrac{1}{6}}}$ (5)

 $\frac{B}{h} = \frac{1}{{f(G)}}=\alpha$ (6)
 $R = \frac{A}{\chi } = \frac{{Bh}}{{B + 2h}} = \frac{\alpha }{{\alpha + 2}}h=\beta h$ (7)

 ${S_{\rm f}}=\frac{{\partial h}}{{\partial x}} - {S_0}$ (8)

 $Q=AC\sqrt R \sqrt {{S_0} - \frac{{\partial h}}{{\partial x}}} ={Q_0}\sqrt {1 - \frac{1}{{{S\!_0}}}\frac{{\partial h}}{{\partial x}}}$ (9)

 ${Q_0}{\rm{ = }}\frac{1}{n}A \cdot {\beta ^{\tfrac{2}{3}}}{h^{\tfrac{2}{3}}}S_0^{\!\tfrac{1}{2}}{\rm{ = }}\frac{{{\alpha ^{\tfrac{5}{3}}}}}{{{{(\alpha {\rm{ + }}2)}^{\tfrac{2}{3}}}}} \cdot \frac{1}{n}{h^{\tfrac{8}{3}}}S_0^{\!\tfrac{1}{2}}$ (10)

 $Q{\rm{ = }}\frac{{{\alpha ^{\tfrac{5}{3}}}}}{{{{(\alpha {\rm{ + }}2)}^{\tfrac{2}{3}}}}} \cdot \frac{1}{n}{h^{\tfrac{8}{3}}}S_0^{\!\tfrac{1}{2}}\sqrt {1 - \frac{1}{{{S_0}}}\frac{{\partial h}}{{\partial x}}}$ (11)

 $Q = \left(M \cdot N \cdot {h^{\tfrac{8}{3}}}S_0^{0.24}\sqrt {1 - \frac{1}{{{S_0}}}\frac{{\partial h}}{{\partial x}}} \right)^{1.16}$ (12)

 $h = {\left[\frac{{{Q^{0.86}}}}{{M \cdot N \cdot S_0^{0.24}}} \cdot {\left(1 - \frac{1}{{{S_0}}}\frac{{\partial h}}{{\partial x}}\right)^{ - \tfrac{1}{2}}}\right]^{\tfrac{3}{8}}}$ (13)

 $H \!=\! {H_0}\! + \!h \!= \!{H_0} \!+\! {\left[\dfrac{{{Q^{0.86}}}}{{M \cdot N \cdot S_0^{0.24}}} \cdot {\left(1 \!-\! \dfrac{1}{{{S_0}}}\dfrac{{\partial h}}{{\partial x}}\right)^{-\tfrac{1}{2}}}\right]^{\tfrac{3}{8}}}$ (14)

 图1 扩散波的水位–流量关系 Fig. 1 Stage–discharge relation in diffusion wave

2 实例验证

 图2 龙河流域 Fig. 2 Longhe river area

 图3 $h/ B$ – $G$ 拟合曲线 Fig. 3 Fitting curve of $h/ B$ – $G$

 图4 石柱站水位流量关系曲线 Fig. 4 Stage–discharge relation of Shizhu station

 图5 石柱站单场洪水绳套关系曲线 Fig. 5 Flood loop-rating curves of Shizhu station

 $H = {H_0} + {\left[\frac{{nQ}}{a} \cdot {\left({S_0} - \frac{{\partial h}}{{\partial x}}\right)^{ - \tfrac{1}{2}}}\right]^{\tfrac{3}{8}}}$ (15)

 图6 小浪底站水位流量关系 Fig. 6 Stage–discharge relation of Xiaolangdi station

3 结　论

1）引入了河道级别和泥沙粒径区别山区河流与平原河流，修正了在山区河流中用 $h$ 来代替R的办法，更加符合实际情况。同时，在推广至平原河流时，利用该公式也是完全可行的，精度在中高水位时更高，更有利于水位流量关系曲线的高水位延长。

2）利用公式对水位流量曲线进行配线时，只需根据变化情况对参数作一些更正，修改起来比较方便，对于受多种作用影响、复杂多变的水位流量绳套关系，本文方法具有实用性好、精度高及适应性强等特点，同时绳套曲线的大小与 ${{\partial h} / {\partial x}}$ $S_0$ 有关，一般它们的比值越大，绳套越大。

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