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International Journal of Civil, Structural, Environmental and Infrastructure Engineering Research and Development (IJCSEIERD) ISSN 2249-6866 Vol. 3, Issue 4, Oct 2013, 27-34 © TJPRC Pvt. Ltd.


Research Scholar, Department of Civil Engineering, IIT (BHU), Varanasi, Uttar Pradesh, India 2

Professor, Department of Civil Engineering, IIT (BHU), Varanasi, Uttar Pradesh, India


Associate Professor, Department of Civil Engineering, IIT (BHU), Varanasi, Uttar Pradesh, India

ABSTRACT Geographic Information System (GIS) has been used for SDUH model which is a time-area unit hydrograph technique to develop a cumulative travel time map of the watershed based on cell by cell estimates of overland and channel flow velocities. The model includes slope, land use, watershed position, channel characteristics, and rainfall excess intensity in determining flow velocities. The cumulative travel time map has been divided into isochrones which are used to generate a time-area curve and the resulting unit hydrograph. SDUH has been applied to estimate the flood hydrograph of Varuna River basin in India.

KEYWORDS: Unit Hydrographs, Spatial Distributions, Geographic Information Systems, Hydrologic Modelling INTRODUCTION The “spatially distributed unit hydrograph (SDUH)” proposed by Maidment (1993) is similar to the geomorphic instantaneous unit hydrograph (Rodriguez-Iturbe and Valdes, 1979), except that it uses a GIS to describe the connectivity of the links and the watershed flow network instead of probability arguments. Maidment (1993) calculated the flow distance from each cell to the watershed outlet. The travel time from each cell to the watershed outlet was calculated by dividing each flow length by a constant velocity. He developed a time-area curve based on the travel time from each grid cell. Maidment (1993) evaluated his methodology on a hypothetical watershed that consisted of 36 grid cells. His results were inconclusive; the time-area histogram did not result in the expected S-hydrograph shape. The poor results were blamed on the small size of the watershed. The SDUH is based on two assumptions: that the duration of excess runoff is a function only of the duration of the excess rainfall pulse and not of its magnitude and that the magnitude of the unit hydrograph ordinates is directly proportional to the volume of direct runoff contained within the hydrograph (Chow et al., 1988). In the SDUH the watershed is decomposed into subareas which are individual cells or zones of neighbouring cells. The unit hydrograph is found for each subarea and the response at the outlet to excess rainfall on each subarea is summed to produce the watershed runoff hydrograph. The cell to cell flow path to the watershed outlet is determined from a digital elevation model. A constant flow velocity is assigned to each cell and the time lag between subarea input and response at the watershed outlet is found by integrating the flow time along the path from the subarea to the outlet. The response function for a subarea is modelled as a lagged linear reservoir in which the flow time is equal to the sum of a time of translation and an average residence time in the reservoir. Ajward (1996) applied the spatially distributed unit hydrograph to two large watersheds in the Canadian Rockies, comparing the predictions from the spatially distributed unit hydrograph with the observed hydrograph for eight rainfall


Kailash Narayan, P. K. S. Dikshit & S. B. Dwivedi

events. He found that the model generally gave good predictions of the peak flow rate and the time to peak. He also found that the best predictions were produced when a unit hydrograph developed from a storm with a similar intensity to the test event was used. An “average” unit hydrograph always gave less accurate predictions of time to peak and peak flow rate than a unit hydrograph that was developed with an intensity that was close to the average intensity of the test event. Spatially distributed loss model and spatially distributed velocity field are two features added to the ModClark model (Peters and Easton, 1997). A new formula to calculate the spatially distributed velocity field was derived. The results were satisfactory (Bhattacharya at el. 2012). The main aim of the paper is to develop a SUDH model of Varuna river watershed by using GIS.

METHODOLOGY OF SDUH Mathematical representation of the unit hydrograph has a long history in hydrology. Clark (1945) formulated a unit hydrograph model by combining the time-area diagram of the watershed with a linear reservoir at the outlet. Nash (1957) proposed a cascade of linear reservoirs as a unit hydrograph model and Dooge (1959) presented a unit hydrograph theory combining linear channels and linear reservoirs. These approaches are based on the convolution integral for the watershed as a lumped system which defines the direct runoff hydrograph q(t) as a function of the excess rainfall hyetograph I   using the unit impulse response function I  t   as: t

q(t ) 

 I   u  t    d 0


Then, the total discharge at the outlet t

Q(t )  A I   u  t    d 0


Where A is the total drainage area of the Watershed. The watershed area (Figure 3(a)) is subdivided into J subareas (Figure 3(b)) of area  A j where, j= 1,2,. . . , J , that the excess rainfall rate varies by subarea as denoted by I j  t  and that a subarea unit impulse response function I j  t    can be calculated for each subarea independent of all other subareas. The response at the outlet can then be

found by summing the responses from each subarea as J


j 1


Q(t )   Aj  I j   u j  t    d


The concept of a spatially distributed unit hydrograph, proposed by Maidment (1993), is based on the fact that the unit hydrograph ordinate at time t is given by the slope of the watershed time-area diagram over the interval  t  t , t  . The time-area diagram is a graph of cumulative drainage area contributing to discharge at the watershed outlet within a specified time of travel. The validity of the above can be proved by considering the S-hydrograph method. An Shydrograph, defined as the runoff at the outlet of a watershed resulting from a continuous excess rainfall occurring at rate i, over the watershed, is given by: Qs  t   ie A  t 



Spatially Distributed Unit Hydrograph for Varuna River Basin of India

Where A(t) is the watershed area contributing to flow Qs(t) at the outlet at time t. The direct runoff hydrograph discharge at time t, resulting from a pulse of excess rainfall Pe  ie t is equal to the difference between the S-hydrograph value at time t and its value lagged by time t , i.e. QD  t   ie A  t   ie A  t  t 


The unit hydrograph ordinates are U  t   QD  t  / Pe U t  

A  t   A  t  t  t



It is appropriate to determine the circumstances under which the subarea unit impulse response function can be calculated independently of all other subareas. These conditions may be stated as follows: 

The watershed is subdivided into a finite number of non-overlapping subareas which collectively span the whole drainage area and each subarea is connected to the watershed outlet by a single, continuous flow path.

The impulse response function at the watershed outlet to excess rainfall on the subarea does not depend on the magnitude of the excess rainfall.

In the event that the flow paths from two subareas share some path segments in common, the impulse response function for flow from one subarea is not affected by the presence in these segments of flow from the other subarea. Condition-1 is satisfied provided that a path exists from each cell to the outlet. This is accomplished by defining a

flow direction grid as shown in Figure 3(b) using the eight-direction pour-point algorithm (Figure 2) which specifies for each cell a single neighboring cell along the line of steepest descent which receives its drainage along a flow path between cell centres. Provided there are no pits in the landscape the linkage created between the cell centres, creates a flow network over the landscape, which is effectively a spanning tree drawn from the watershed outlet with branches reaching every interior cell. The construction of the flow direction grid is a standard GIS function, called Flow direction in Arc/Info, which requires pre-processing of the digital elevation model of the land surface terrain by adjusting the elevation of cells which would cause artificial pits in the landscape to ensure a continuous flow path from each cell to the watershed outlet. Condition-2 is the standard condition for a linear system response which ensures that the principles of proportionality and superposition will apply to convolution of the excess rainfall inputs with the unit hydrograph from the subarea. Condition-3 is accomplished by assigning a representative velocity to each cell, as shown in Figure 3(b) and assuming that flow through this cell is transmitted with this velocity independent of the magnitude of the discharge. As the size of the cells is fixed, assuming a representative velocity is equivalent to assuming a representative residence time of water in the cell. Residence time can also be defined as the ratio of the storage in the cell to the discharge passing through it, so a linear reservoir with a fixed reservoir constant k is one model which satisfies condition-3. A linear channel model with a fixed lag time is another model which also satisfies this condition. A basic assumption made here with regard to the flow velocity field is that it depends on local constant variables


Kailash Narayan, P. K. S. Dikshit & S. B. Dwivedi

such as bottom roughness, land slope, drainage area, etc. and not on time-varying variables such as flow or storage. Accordingly, the flow velocity field is time and discharge-invariant. This requirement for constant residence times in the elements of the system making up the unit hydrograph was recognized by Clark (1945). Pilgrim (1976) conducted a field study in which he released labeled dye into the drainage waters at various points in a small watershed and traced its travel time to the outlet. He found that at low to medium flows there is a significant variation of the velocity with the discharge, but ‘at medium to high flows the travel times and average velocities become almost constant, indicating that linearity is approximated in this range of flows’. Application of SDUH Model: Study Area (Varuna River Basin in India) India is divided into seven hydro-meteorological zones and is further divided into twenty six hydrometeorological sub-zones [i.e. sub-zones 1(a) to 1(g), 2(a) to 2(c), 3(a) to 3(i), 4(a) to 4(c), 5(a), 5(b), 6 and 7] (Jain et al., 2007). The River Varuna is a minor tributary of the Ganges. The Varuna rises at 25°36′ N 82°7′ E, flows east-to-southeast for some 110 km from Phulpur Tahsil of Allahabad, and joins the Ganges at 25°19′46″N 83°02′40″E.It drains approximately 3315 km2 area of the central-east part of the Uttar Pradesh. The extracted drainage network of the basin using the ArcGIS is shown in Figure 1. The shape of the basin is nearly oblong in nature. Topography is nearly flat with low terrains at upstream end of the basin. The maximum and minimum elevation of the basin is 95 m and 65 m above MSL, respectively. The climate of the basin is ranging from semi arid to sub humid tropical with average annual rainfall at different locations is 850-1100 mm. Approximately, 75 percent of total rainfall is due to the occurrence of North-West Monsoon (Mid June to Mid October). The mean minimum and maximum temperature over the basin is 1° to 48° C with daily mean sunshine of 8 hours. The relative humidity varies between 10-90 percent. The potential evapotranspiration experienced in the basin is nearly 1200 mm (Narayan et al., 2012).

Figure 1: Drainage Network of Varuna River Derivation of the time-area diagram requires the knowledge of the distribution of flow directions and velocities over the watershed. To obtain this information a simple digital elevation model of the watershed (approx. A = 3315 km2) was constructed by digitizing the watershed into 3 x 3 km grid cells, and by assigning an average elevation to each of these cells. Water on a grid cell is permitted to flow to one of its eight nearest neighbor cells. A grid of flow directions can then be created by choosing for each cell the direction of steepest descent among the eight permitted choices. This grid is shown in Figure 3(b) as a set of arrows. Connecting cell centers along the arrows creates an equivalent channel network as shown

Spatially Distributed Unit Hydrograph for Varuna River Basin of India


in Figure 3(a), which compares well with the actual channel network shown on a topography map. The generated equivalent channel network is used to compute flow velocities through channel reaches, and subsequently the times of travel to the watershed outlet from various locations within the watershed.

Figure 2: Eight Direction Flow

Figure 3(a): Equivalent Channel Network

Figure 3(b): Grid of Flow Direction of Varuna River

RESULTS AND DISCUSSIONS The average channel velocity of Varuna River is from 2.25 m/s (during lean period) to 3.75 m/s (during peak discharge time). Thus, in order to analyses the SDUH model, average channel velocity of 3.0 m/s is taken by considering the same design storm pattern (Narayan et al., 2012). The time-area diagram resulting from rainfall having an intensity of 2.6 mm/h, uniformly distributed over the watershed, is shown in Figure 4. It should be noted that the excess rainfall generated from this total rainfall input is not spatially uniform, because of varying runoff curve numbers in the grid cells. In general, it is possible to use a spatially distributed total rainfall as input, digitized into the grid cells, if such information is available. Thus, it is possible to derive a time-area curve for each individual rainfall event, and obtain the corresponding response function (the direct runoff hydrograph) by numerical differentiation. This approach leads to distributed modelling of the storm hydrograph by superposition of individual response functions to rainfall blocks in the storm's hyetograph.


Kailash Narayan, P. K. S. Dikshit & S. B. Dwivedi

Figure 4: The Time-Area Diagram for 2.6 mm/h Uniform Rainfall A SDUH is derived from the time-area curve shown in Figure 4 by applying the standard S-hydrograph method (Chow et al., 1988); i.e. the time-area curve is lagged by one hour and subtracted from the original curve. The SDUH onehour unit hydrograph was used to simulate an observed hydrograph produced by a 24-hour storm which was recorded at two adjoining rain gauge stations. The storm hyetograph, digitized into one-hour incremental rainfalls, was applied uniformly over the watershed's 370 grid cells. Each cell generated an excess rainfall hyetograph according to its assigned runoff curve number. The incremental one-hour excess rainfalls were spatially averaged to obtain a representative excess rainfall hyetograph for the watershed. This average hyetograph was convoluted with the one-hour distributed unit hydrograph and resulted in the predicted hydrograph shown in Figure 5.

Figure 5: GIUH (Black-Circle) and SDUH (Red-Star) Direct Runoff Hydrographs The agreement between SDUH and observed GIUH hydrograph (Figure 5) is very good considering that no parameter optimization was performed. The distributed unit hydrograph used in this simulation was derived from a uniform total rainfall input of 2.6 mm/h, which is the average rainfall intensity for the storm.

CONCLUSIONS The SDUH can actually be derived for ungauged watersheds without observed rainfall and runoff data, because the time-area curve is computed on the basis of watershed hydraulics, for which data may be obtained by field survey. with the same data, it is possible to derive a family of SDUH for a watershed from inputs of various magnitudes, thus emulating the geomorphic climatic unit hydrograph concept in which the shape of the unit hydrograph depends on rainfall intensity.

Spatially Distributed Unit Hydrograph for Varuna River Basin of India


The most crucial part of the SDUH model is the computer algorithm for the computation of the time-area diagram. In the present paper the travel times were computed, for the sake of simplicity, from equilibrium flow conditions and later adjusted empirically for neglected effects of storage and unsteady flow. The SDUH model of a watershed can be efficiently implemented by the use of a GIS. The GIS enables capture and to utilization a number of watershed and rainfall distributed parameters in the model, which is not possible when using the classical unit hydrograph approach. So far, experience with the SDUH modelling indicates that it is a very promising new tool for the prediction of flood hydrographs.


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