Arunoff models orrainfall-runoff model describes howrainfall is converted intorunoff in adrainage basin (catchment area or watershed). More precisely, it produces asurface runoffhydrograph in response to a rainfall event, represented by and input as ahyetograph. Rainfall-runoff models need to becalibrated before they can be used.
A well known runoff model is thelinear reservoir, but in practice it has limited applicability.The runoff model with anon-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 usingflood routing techniques.
The hydrology of a linear reservoir (figure 1) is governed by two equations.[1]
where:
Q is therunoff or discharge
R is theeffective rainfall orrainfall excess orrecharge
A is the constantreaction factor orresponse 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 adifferential equation, whose solution is:
This is therunoff equation ordischarge 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, thetotal hydrograph can be obtained using a successive number of time steps and computing, with therunoff 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 theinstantaneous unit hydrograph (IUH) because the Q herein equals Q2 of the foregoing runoff equation usingR = 0, and taking S asunity which makes Q1 equal to A according to equation (1).
The availability of the foregoingrunoff equation eliminates the necessity of calculating thetotal hydrograph by the summation of partial hydrographs using theIUH as is done with the more complicatedconvolution method.[2]
Determining the response factor A
When theresponse factor A can be determined from the characteristics of the watershed (catchment area), the reservoir can be used as adeterministic model oranalytical model, seehydrological modelling.
Otherwise, the factor A can be determined from a data record of rainfall and runoff using the method explained below undernon-linear reservoir. With this method the reservoir can be used as ablack box model.
Conversions
1 mm/day corresponds to 10 m3/day per ha of the watershed
1 L/s per ha corresponds to 8.64 mm/day or 86.4 m3/day per ha
Contrary to the linear reservoir, the non linear reservoir has a reaction factor A that is not a constant,[3] 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 hasno usableunit hydrograph.
During periods without rainfall or recharge, i.e. whenR = 0, the runoff equation reduces to
or, using aunit time step (T2 − T1 = 1) and solving for Aq:
Hence, the reaction or response factor Aq can be determined from runoff or discharge measurements usingunit time steps during dry spells, employing anumerical method.
Figure 3 shows the relation between Aq (Alpha) and Q for a small valley (Rogbom) in Sierra Leone.
Figure 4 shows observed andsimulated orreconstructed dischargehydrograph of the watercourse at the downstream end of the same valley.[4][5]
The recharge, also calledeffective rainfall orrainfall excess, can be modeled by apre-reservoir (figure 6) giving the recharge asoverflow. The pre-reservoir knows the following elements:
The recharge during a unit time step (T2−T1=1) can be found fromR = Rain − Sd
The actual storage at the end of aunit time step is found as Sa2 = Sa1 + Rain −R − Ea, where Sa1 is the actual storage at the start of the time step.
TheCurve Number method (CN method) gives another way to calculate the recharge. Theinitial abstraction herein compares with Sm − Si, where Si is the initial value of Sa.
The Nash model[7] uses a series (cascade) of linear reservoirs in which each reservoir empties into the next until the runoff is obtained. Forcalibration, the model requires considerable research.
Figures 3 and 4 were made with the RainOff program,[8] designed to analyse rainfall and runoff using the non-linear reservoir model with a pre-reservoir. 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.[9]
Raven is a robust and flexible hydrological modelling framework, designed for application to challenging hydrological problems in academia and practice. This fully object-oriented code provides complete flexibility in spatial discretization, interpolation, process representation, and forcing function generation. Models built with Raven can be as simple as a single watershed lumped model with only a handful of state variables to a full semi-distributed system model with physically-based infiltration, snowmelt, and routing. This flexibility encourages stepwise modelling while enabling investigation into critical research issues regarding discretization, numerical implementation, and ensemble simulation of surface water hydrological models. Raven is open source, covered under the Artistic License 2.0.
The SMART hydrological model[10] includes agricultural subsurface drainage flow, in addition to soil and groundwater reservoirs, to simulate the flow path contributions to streamflow.
Vflo is another software program for modeling runoff.Vflo usesradar rainfall andGIS data to generate physics-based, distributed runoff simulation.
TheWEAP (Water Evaluation And Planning) software platform models runoff and percolation from climate and land use data, using a choice of linear and non-linear reservoir models.
TheRS MINERVE software platform simulates the formation of free surface run-off flow and its propagation in rivers or channels. The software is based on object-oriented programming and allows hydrologic and hydraulic modeling according to a semi-distributed conceptual scheme with different rainfall-runoff model such as HBV,[11] GR4J, SAC-SMA or SOCONT.
TheIHACRES is a catchment-scale rainfall-streamflow modelling methodology. Its purpose is to assist the hydrologist or water resources engineer to characterise the dynamic relationship between basin rainfall and streamflow.[12]