Runoff model (reservoir)

Runoff model (reservoir)

A runoff model is a mathematical model describing the rainfall-runoff relations of a rainfall catchment area or watershed. More precisely, it produces the surface runoff hydrograph as a response to a rainfall hydrograph as input. In other words, the model calculates the conversion of rainfall into runoff.
A well known runoff model is the "linear reservoir", but in practice it has limited applicability.
The runoff model with a "non-linear reservoir" is more universally applicable, but still it holds only for catchments whose surface area is limited by the condition that the rainfall can be considered more or less uniformly distributed over the area. The maximum size of the watershed then depends on the rainfall characteristics of the region. When the study area is too large, it can be divided into sub-catchments and the various runoff hydrographs may be combined using flood routing techniques.
Rainfall-runoff models need to be calibrated before they can be used.

Linear reservoir

The hydrology of a linear reservoir (figure 1) is governed by two equations [ J.W. de Zeeuw, 1973. Hydrograph analysis for areas with mainly groundwater runoff. In: Drainage Principle and Applications, Vol. II, Chapter 16, Theories of field drainage and watershed runoff. p 321-358. Publication 16, International Institute for Land Reclamation and Improvement, Wageningen, The Netherlands.] .
#flow equation: Q = A.S , with units [L/T] , where L is length (e.g. mm) and T is time (e.g. hr, day)
#continuity or water balance equation: R = Q + dS/dT , with units [L/T] where:
Q is the "runoff" or" discharge"
R is the "effective rainfall" or "rainfall excess" or "recharge"
A is the constant "reaction factor" or "response factor" with unit [1/T]
S is the water storage with unit [L]
dS is a differential or small increment of S
dT is a differential or small increment of T

Runoff equation
A combination of the two previous equations results in a differential equation, whose solution is:
* Q2 = Q1 exp { −A (T2 − T1) } + R [ 1 − exp { −A (T2 − T1) } ] This is the "runoff equation" or "discharge equation", where Q1 and Q2 are the values of Q at time T1 and T2 respectively while T2−T1 is a small time step during which the recharge can be assumed constant.

Computing the total hydrograph
Provided the value of A is known, the "total hydrograph" can be obtained using a successive number of time steps and computing, with the "runoff equation", the runoff at the end of each time step from the runoff at the end of the previous time step.

Unit hydrograph
The discharge may also be expressed as: Q = − dS/dT . Substituting herein the expression of Q in equation (1) gives the differential equation dS/dT = A.S , of which the solution is: S = exp(− A.t) . Replacing herein S by Q/A according to equation (1) , it is obtained that: Q = A exp(− A.t) . This is called the instantaneous unit hydrograph (IUH) because the Q herein equals Q2 of the foregoing runoff equation using "R" = 0 , and taking S as "unity" which makes Q1 equal to A according to equation (1).
The availability of the foregoing "runoff equation" eliminates the necessity of calculating the "total hydrograph" by the summation of partial hydrographs using the "IUH" as is done with the more complicated convolution method [ D.A. Kraijenhoff van de Leur, 1973. Rainfall-runoff relations and computational models. In: Drainage Principle and Applications, Vol. II, Chapter 16, Theories of field drainage and watershed runoff. p 245-320. Publication 16, International Institute for Land Reclamation and Improvement, Wageningen, The Netherlands.] .

Determining the response factor A
When the "response factor" A can be determined from the characteristics of the watershed (catchment area), the reservoir can be used as a "deterministic model" or "analytical model" , see hydrological modelling.
Otherwise, the factor A can be determined from a data record of rainfall and runoff using the method explained below under "non-linear reservoir". With this method the reservoir can be used as a black box model.

1 mm/day corresponds to 10 m3/day per ha of the watershed
1 l/sec per ha corresponds to 8.64 mm/day or 86.4 m3/day per ha

Non linear reservoir

Contrary to the linear reservoir, the non linear reservoir has a reaction factor A that is not a constant [R.J.Oosterbaan, 1994. Land drainage and soil salinity: some Mexican experiences. In: ILRI Annual Report 1995, p. 44-53. International Institute for Land Reclamation and Improvement, Wageningen, The Netherlands] , but it is a function of S or Q (figure 2, 3). Normally A increases with Q and S because the higher the water level is the higher the discharge capacity becomes. The factor is therefore called Aq instead of A.
The non-linear reservoir has "no" usable unit hydrograph.

During periods without rainfall or recharge, i.e. when "R" = 0, the runoff equation reduces to
* Q2 = Q1 exp { − Aq (T2 − T1) }, or:or, using a "unit time step" (T2 − T1 = 1) and solving for Aq:
* Aq = − ln (Q2/Q1)Hence, the reaction or response factor Aq can be determined from runoff or discharge measurements using "unit time steps" during dry spells, employing a numerical method.

Figure 3 shows the relation between Aq (Alpha) and Q for a small valley (Rogbom) in Sierra Leone.
Figure 4 shows observed and "simulated" or "reconstructed" discharge hydrograph of the watercourse at the downstream end of the same valley.
The data were obtained from Huizing [A.Huizing, 1988. Rainfall-Runoff relations in a small cultivated valley in Sierra Leone. Wetland Utilization Research Project. International Institute for Land Reclamation and Improvement, Wageningen, The Netherlands] .


The recharge, also called "effective rainfall" or "rainfall excess", can be modeled by a "pre-reservoir" (figure 5) giving the recharge as "overflow". The pre-reservoir knows the following elements:
*a maximum storage (Sm) with unit length [L]
*an actual storage (Sa) with unit [L]
*a relative storage: Sr = Sa/Sm
*a maxmimum escape rate (Em) with units length/time [L/T] . It corresponds to the maximum rate of evaporation plus "percolation" and groundwater recharge, which will not take part in the runoff process (figure 6)
*an actual escape rate: Ea = Sr.Em
*a storage deficiency: Sd = Sm + Ea − Sa

The recharge during a unit time step (T2−T1=1) can be found from "R" = Rain − Sd
The actual storage at the end of a "unit time step" is found as Sa2 = Sa1 + Rain − "R" − Ea, where Sa1 is the actual storage at the start of the time step.

The Curve Number method (CN method) gives another way to calculate the recharge. The "initial abstraction" herein compares with Sm − Si, where Si is the initial value of Sa.


ee also

*Surface runoff
*Hydrological transport model
*Hydrological modelling


The "RainOff" program was designed to analyse rainfall and runoff using the non-linear reservoir model with a pre-reservoir. The figures 3 and 4 were made with it.
"Free download" from : [] .
The program also contains an example of the hydrograph of an agricultural subsurface drainage system for which the value of A can be obtained from the system's characteristics.

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