MODELING THE STORM RUNOFF FROM URBAN WATERSHEDS

Address for correspondence: Parthenonos 2, 67100 Xanthi, Greece

e-mail: pangelid@xan.forthnet.gr and kotsovin@demo.cc.duth.gr  Tel:003054126923

Abstract: This work represents a simulation model that provides an analysis of the quantity of runoff from urban watersheds. A mathematical - numerical model has been developed, which simulates the precipitation – runoff process and computes flow hydrographs at various places. The program accepts as input the real morphology of the urban watershed using a digitizer interfaced to a computer. Then it divides automatically the hydrological basin into very small subbasins. The model substitutes each small subbasin with an idealized V-shaped overland flow element of equivalent area, which consists of two equal orthogonal planes connected at one side. Successful application of this approach begins with the description of unsteady, uniform flow over the idealized overland flow elements. The overland flow of each element is collecting by a corresponding collector channel of the drainage system. A dendritic channel system – the drainage system – is generated automatically. The overland flow from each element is input as uniformly distributed lateral inflow into the corresponding collector channel, which routes both this lateral inflow and the inflow from upstream channel too. The kinematic wave approximation has been proven to be an accurate and efficient method of simulating storm water runoff for both overland flow and stream channel routing. A finite difference method is used for approximating the governing partial differential equations with simple difference equations.

Keywords: runoff simulation, overland flow, kinematic wave approximation

1  INTRODUCTION

The basic objective of this work is to present a simulation model for the precipitation – runoff process on urban watersheds. Various investigators have developed similar models, see for example Beasley et al (1977), Berkowitz (1979), Donigian et al (1977), Hydrologic Engineering Center (1973 and 1979), Henderson and Wooding (1964). During the past 20 years more than 100 mathematical models have been developed, see for review Bonazountas et al (1997). The simplest of all hydrologic model for rough estimates of runoff is the Rational Formula (flow rate equals precipitation times watershed area times the runoff coefficient; a fraction between 0 and 1). The hydrology routines of most operational watershed models are derived from classical methods of hydrograph separation. One of the earliest and most comprehensive numerical models based on this approach is the Stanford Watershed Model (SWM). SWMM, the Stormwater Management Model, is designed to simulate nonpoint source runoff, primarily from urban areas. USDAHL demands the specification of many more input parameters than SWMM, with the resulting advantage that it relies less on calibration versus observed hydrographs. CREAMS sediment algorithms represent a detailed description of processes known to affect sediment detachment and transport. AGRUN is a nonpoint source model applicable to agricultural watersheds. MRI includes a nonpoint source loading function. STORM, the Storage, Treatment, Overflow, Runoff Model, is a continuous simulation model that provides an analysis of the quantity and quality of runoff from urban or nonurban watersheds. ANSWERS is designed to simulate the areal, nonpoint source watershed environment response.

2  DESCRIPTION OF THE MODEL

The mathematical-computer model, which has been developed in this work, accepts as input the real morphology (map) of the urban watershed using a digitizer interfaced to a computer. The program divides automatically the hydrological basin into very small finite elements (small subbasins) as shown in Figure 1. The model substitutes each finite element with a V-shaped overland flow element of equivalent area, which consists of two equal orthogonal planes connected at one side of length L. It is assumed in the model that water initially travels over these surfaces as sheet flow, see Figure 2. This (overland) flow is collecting by the corresponding collector channel. The collector channel as shown in Figure 3, has as lateral inflow the overland flow from the two orthogonal planes, uniformly distributed along the length L. Then the model automatically generates and draws a dendritic channel system – the storm water drainage system (see Figure 1). So each channel carries both inflows from upstream channel as well as lateral flows supplied by the corresponding overland flow element.  The model calculates a runoff hydrograph at any point of the urban watershed.

Fig. 1  Overland flow elements and stream collectors

3  OVERLAND FLOW – CHANNEL FLOW

Numerical techniques used to simulate overland and channel flows can only approximate the actual response of real systems because of the complex nature of natural drainage basins and because simplifications must be made to the mathematics to make the model efficient and economical to execute. The kinematic wave approximation has been proven to be an accurate and efficient method of simulating stormwater runoff for both overland flow and stream channel routing (Lighthill and Witham 1955, Overton and Meadows 1976). This approximation has been chosen in our model to simulate both the thin sheet – flow (overland flow), which occurs on the overland flow elements and the stream flows too.

The basic assumptions made in development of the basic differential equations of one dimensional unsteady flow are that: a) the slope of the plane is small (less than 1:10), b) streamlines are essentially straight, c) the pressure distribution is approximately hydrostatic, d) resistance to flow may be described by empirical resistance equations such as the Manning equation, and e) momentum carried to the fluid from lateral inflow is negligible. Under the above mentioned conditions, the mechanics of unsteady open channel flow may be expressed mathematically in terms of the St. Venant equations (for channels of unit width):

Fig. 2  Idealized overland flow element

Fig. 3  Collector channel

Continuity

                                                          (1)

Momentum

                                                   (2)

where

x     = distance measured in downstream flow direction

y     = mean depth

t     = time

Q    = discharge per unit width of channel

q_L   = total lateral inflow per unit length of channel

i     = rainfall intensity

f     = infiltration rate

u     = x-component of mean velocity

g     = acceleration of gravity

S_o   = average bottom slope

S_f   = friction slope defined by the Manning equation

v  = y-component of velocity for lateral inflow

An order of magnitude analysis shows that inertia and pressure terms are not important in the momentum equation (2) which is then reduced to the well known relation for steady, uniform flow in a “channel”:

S₀ = S_f                                                                       (3)

i.e. the bed slope is approximately equal to the friction slope. We calculate the volume flux Q at any point in the "channel" from Manning’s formula

                                                       (4)

where R_h is hydraulic radius, A is crossectional area, n is Manning's resistance factor, and  a and m are parameters related to flow geometry and surface roughness. For a wide rectangular channel (overland flow) R_h ~ y so that

                                            (5)

i.e. a=(S₀)^1/2/n  and  m=5/3  

Similarly the kinematic wave equations for channel flow routing are:

                                                   (6)

                                              (7)

where

A_C = cross sectional area of channel flow

Q_c   = discharge of channel

q₀    = lateral inflow per unit length from overland flow

a_c, m_c      = kinematic wave parameters for a particular cross sectional shape, slope and roughness.

Fig.4  Comparison of hydraulic solution of overland flow depth and
of our computer simulation (slope So = 0.04, Manning
coefficient n = 0.1, rainfall intensity = 4 cm/hr, duration 3 hours).

We can find for circular section of diameter D_c for the pipes of the drainage system:

    and  m_c =1.25                                       (8)

where the subscript c denotes channel flow. Combining the above equations (1) and (4) we find for overland flow:

                                               (9)

and combining equations (6) and (7) we find for channel flow:

                                  (10)

We use finite difference method for approximating the governing partial differential equations with simple difference equations for an array of stationary grid points located in the space - time (x-t) plane. We compare the analytical solution of overland flow in a plane surface for the steady varied flow region and for the equilibrium depth profile (see Morgali, 1970) with our computer simulation at Figure 4. As we see the two solutions are very closed.

4  COMPUTATION PROCEDURE

The computer program initially accepts as input the real morphology (contour lines) of the hydrological basin using a digitizer interfaced to a computer. Then it requires hydrological data i.e. a precipitation hydrograph i(x,y,t), where i the rainfall intensity (cm/hr) at any point with horizontal Cartesian coordinates x,y at a time t, and soil characteristics regarding infiltration and detention losses.  It is pointed out that in this program the rainfall and infiltration are not necessarily uniform over the subbasin. The area where rainfall falls does not coincides in general with the subbasin.

Because of the complexities in dealing with detailed interception losses, it is assumed in our model that the amount intercepted is a constant per cent of the rainfall, which must be input by the user. The program uses the Horton’s equation to calculate the infiltration loss. The infiltration rate f is given by the equation:

f = fc + (fo – fc) e ^–kt                                         (11)

where f is the infiltration rate at time t in centimetres per hour, fo and  fc are the initial and final infiltration rates in centimetres per hour, and k is the infiltration constant, which is allegedly a function of soil and vegetation.  These parameter values vary widely with soil type and initial moisture conditions.

The program divides automatically the hydrological basin into very small subbasins, it creates the idealized overland flow elements, it generates a dendritic channel system and it calculates areas for overland flow and lengths and slopes for channels. Next at each time step Δt the program determinates the overland flow on each overland flow element. This flow is input as uniformly distributed lateral inflow into the corresponding collector channel, which routes both this lateral inflow and the inflow from upstream channel too. The output is hydrographs at selected places.

This model was applied to study the storm runoff from the urban area of city of Xanthi in Northern Greece. At Figure 1 is given the map (contour lines) of this urban area and the drainage system, which has been generated automatically. For demonstration reasons the total area was divided into two zones: zone 1 is marked as shaded and zone 2 is not shaded. Over each of them it was supposed different rain (see Figure 5). The hydrographs at the exit of each zone and the total hydrograph are shown at the same figure too. As we see, the effect of the different rain to the resulted hydrographs is obvious.

Fig.5  Computed hydrographs for the city of Xanthi in Northern Greece

5  CONCLUSIONS

The mathematical – numerical model presented in this paper simulates the precipitation – runoff process and computes flow hydrographs at any desired place of urban area. The advantages of the model are: a) it accepts as input the real morphology of the urban watershed with a simple, quick and accurate way, using a digitizer interfaced to a computer, b) it divides automatically the hydrological basin into very small subbasins, c) it generates automatically the dendritic channel system. Regarding the complex nature of natural drainage basins this model approximates with reasonable accuracy the actual response of real systems.

References

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Berkowitz, S.T.: 1979, Modification and evaluation on the computer program LANDBRUN for modeling runoff and sediment yield from small agricultural watersheds, Report, Department of Civil Engineering, University of Wisconsin, Modison, WI.

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