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Analytical and Numerical Solutions to Level Pool Routing Equations for Simplified Shapes of Inflow Hydrographs

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A Correction to this article was published on 15 February 2022

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Abstract

Reservoir routing is a technique to predict an outflow hydrograph expected from a dam outlet due to a specific inflow hydrograph. This study develops a series of analytical solutions for determining outflow hydrographs with detailed calculation procedures when a triangular, abrupt wave, flood pulse, broad peak, and double-peak inflow hydrographs occur upstream of a typical reservoir with an orifice outlet. The proposed analytical solutions are used as benchmarks to evaluate 16 schemes of a conventional numerical method so-called Runge–Kutta method, and the main equations for sensitivity analyses. The results indicate that the best scheme among those 16 Runge–Kutta schemes is the Kutta–Merson scheme for all types of inflow hydrographs except the double-peak inflow hydrograph in which the Cash–Karp method outperforms others, and the peak of the inflow hydrograph and the water area supplied by the reservoir are the two most sensitive parameters influencing outflow hydrographs results.

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Abbreviations

\(S\) :

The storage volume of water (\(m^{3}\))

\(t\) :

Time (\(s\))

\(I\) :

Inflow (\({{m^{3} } \mathord{\left/ {\vphantom {{m^{3} } s}} \right. \kern-\nulldelimiterspace} s}\))

\(Q\) :

Discharge released from the reservoir outlet (\({{m^{3} } \mathord{\left/ {\vphantom {{m^{3} } s}} \right. \kern-\nulldelimiterspace} s}\))

\(A\) :

Area (\(m^{2}\))

\(h\) :

Water head (\(m\))

\(\lambda\) :

The orifice correction factor

\(C\) :

The orifice coefficient

\(a\) :

Total cross-sectional area of the orifice (\(m^{2}\))

\(g\) :

Gravity acceleration (\({m \mathord{\left/ {\vphantom {m {s^{2} }}} \right. \kern-\nulldelimiterspace} {s^{2} }}\))

\(I_{p}\) :

The peak of inflow hydrograph (\({{m^{3} } \mathord{\left/ {\vphantom {{m^{3} } s}} \right. \kern-\nulldelimiterspace} s}\))

\(I_{p1}\) :

The first peak of double-peak inflow hydrograph (\({{m^{3} } \mathord{\left/ {\vphantom {{m^{3} } s}} \right. \kern-\nulldelimiterspace} s}\))

\(I_{p2}\) :

The second peak of double-peak inflow hydrograph (\({{m^{3} } \mathord{\left/ {\vphantom {{m^{3} } s}} \right. \kern-\nulldelimiterspace} s}\))

\(I_{p3}\) :

The third peak of double-peak inflow hydrograph (\({{m^{3} } \mathord{\left/ {\vphantom {{m^{3} } s}} \right. \kern-\nulldelimiterspace} s}\))

\(\delta t_{d}\) :

The corresponding time to \(I_{p}\) as the peak of inflow hydrograph (\(s\))

\(\delta_{1} t_{d}\) :

The corresponding time to \(I_{p1}\) as the first peak of double-peak inflow hydrograph (\(s\))

\(\delta_{2} t_{d}\) :

The corresponding time to \(I_{p2}\) as the second peak of double-peak inflow hydrograph (\(s\))

\(\delta_{3} t_{d}\) :

The corresponding time to \(I_{p3}\) as the third peak of double-peak inflow hydrograph (\(s\))

\(t_{d}\) :

The finishing time of inflow hydrograph (\(s\))

\(t_{1}\) :

The starting time associated with the peak of inflow hydrograph (\(s\))

\(t_{2}\) :

The finishing time associated with the peak of inflow hydrograph (\(s\))

\(g\) :

Gravity acceleration (\({m \mathord{\left/ {\vphantom {m {s^{2} }}} \right. \kern-\nulldelimiterspace} {s^{2} }}\))

\(f(t)\;{\text{and}}\;g(t)\) :

General functions of time

\(B\) :

The constant value which is \(\delta t_{d}\) for triangular inflow hydrograph, \(t_{1}\) for broad peak inflow hydrograph, and \(\delta_{1} t_{d}\) for double-peak inflow hydrograph

J :

The constant value that can be calculated by applying the initial condition of zero at t = td for triangular hydrograph, zero at t = 0 and specific water depth at for abrupt wave hydrograph, specific depths at t = td and for flood pulse hydrograph, and zero at for broad peak hydrograph

J′:

The constant value that can be calculated by applying the initial condition

J″:

A constant parameter that can be calculated by applying the initial conditions

D :

The constant value, that is \(\frac{{I_{p} }}{{1 - \delta }}\) for triangular and abrupt wave hydrographs, \(\frac{{\left( {I_{p} } \right)t_{d} }}{{t_{d} - t_{2} }}\) for broad peak hydrograph, \(I_{{p1}} - \frac{{\delta _{1} (I_{{p2}} - I_{{p1}} )}}{{(\delta _{2} - \delta _{1} )}}\), \(I_{{p2}} - \frac{{\delta _{2} (I_{{p3}} - I_{{p2}} )}}{{(\delta _{3} - \delta _{2} )}}\), and \(\frac{{I_{{p3}} }}{{1 - \delta _{3} }}\) for \(\delta _{1} t_{d} \le t < \delta _{2} t_{d}\), \(\delta _{2} t_{d} \le t < \delta _{3} t_{d}\), and \(\delta _{3} t_{d} \le t < t_{d}\) sections of the double-peak hydrograph, respectively

F :

The time coefficient, which is \(\frac{{I_{p} }}{{\left( {\delta - 1} \right)t_{d} }}\) for triangular and abrupt wave hydrographs, \(\frac{{I_{p} }}{{t_{d} - t_{2} }}\) for broad peak hydrograph, \(\frac{{(I_{{p2}} - I_{{p1}} )}}{{(\delta _{2} - \delta _{1} )t_{d} }},\;\frac{{(I_{{p3}} - I_{{p2}} )}}{{(\delta _{3} - \delta _{2} )t_{d} }}\;{\text{and}}\,\frac{{I_{{p3}} }}{{\left( {\delta _{3} - 1} \right)t_{d} }}\;{\text{for}}\;\delta _{1} t_{d} \le t < \delta _{2} t_{d} ,\;\delta _{2} t_{d} \le t < \delta _{3} t_{d} \;{\text{and}}\;\delta _{3} t_{d} \le t < t_{d}\) sections of the double-peak hydrograph, respectively

Cons.:

The constant value related to the Eq. 45 solution

n :

The number of iteration for the numerical scheme

\(\Delta t\) :

The time step (s)

f :

The general function which is defined as the right side of an ODE equation

k :

The Runge-Kutta schemes' coefficients

a, b and c :

The equation's coefficients for Runge-Kutta methods

s :

The total number of equations for Runge-Kutta methods

i :

The equation's number

m :

The iteration number for each Runge-Kutta scheme

N :

The total number of iterations for each numerical scheme

\({Q}_{n}^{{\text{analytical}}}\) :

The outflow calculated from the analytical solution for the time step corresponding to the iteration number of n (m/s3)

\({Q}_{n}^{{\text{numerical}}}\) :

The outflow calculated from the numerical solution for the time step corresponding to the iteration number of n (m/s3)

\(\overline{{Q_{n}^{{\text{analytical}}} }}\) :

The average of outflow calculated from the best analytical scheme solution for the time step corresponding to the iteration number of n (m/s3)

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Authors and Affiliations

  1. Department of Civil and Environmental Engineering, Shiraz University, Shiraz, Iran

    Banafsheh Nematollahi & Majid Niazkar

  2. Department of Civil and Environmental Engineering, Head of Environmental Research and Sustainable Development Center, Shiraz University, Shiraz, Iran

    Nasser Talebbeydokhti

Authors
  1. Banafsheh Nematollahi
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  2. Majid Niazkar
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  3. Nasser Talebbeydokhti
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Correspondence to Majid Niazkar.

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The original Online version of this article was revised : The co-author's affiliation and the email address has been incorrectly published

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Nematollahi, B., Niazkar, M. & Talebbeydokhti, N. Analytical and Numerical Solutions to Level Pool Routing Equations for Simplified Shapes of Inflow Hydrographs. Iran J Sci Technol Trans Civ Eng 46, 3147–3161 (2022). https://doi.org/10.1007/s40996-021-00757-x

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