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#### Hydrological Theory

##### Calculating Effective Rainfall

The Horton Equation

From the MIDUSS Version 2 Reference Manual - Chapter 7
(c) Copyright Alan A. Smith Inc.

One of the first attempts to describe the process of infiltration was made by Horton in 1933.  He observed that the infiltration capacity reduced in an exponential fashion from an initial, maximum rate f0 to a final constant rate fc.

The Horton equation for infiltration capacity  fcapac  is given by equation [7-21] which shows the variation of the maximum infiltration capacity with time t. where

fcapac  =              maximum infiltration capacity of the soil

f0            =              initial infiltration capacity

fc            =              final (constant) infiltration capacity

t               =              elapsed time from start of rainfall

K             =              decay time constant

At any point in time during the storm, the actual infiltration rate must be equal to the smaller of the rainfall intensity i(t) and the infiltration capacity fcapac.  Thus the Horton model for abstractions is given by equations [7-22] and [7-23]. for  i > fcapac for   i <= fcapac

where

f              =              actual infiltration rate (mm/hr or inches/hr)

i               =              rainfall intensity (mm/hr or inches/hr).

Figure 7-8 below shows a typical problem in which the average rainfall intensity in each time step is shown as a stepped function.  It is clear that if the total volume of rain in time step 1 (say) is less than the total infiltration volume in that time step it is more reasonable to assume that the reduction in f is dependent on the infiltrated volume rather than on the elapsed time.  It is therefore usual to use a 'moving curve' technique in which the ft  curve is shifted by an elapsed time which would produce an infiltrated volume equal to the volume of rainfall. Figure 7-6:  Representation of the moving curve Horton equation

Figure 7-6 shows a dashed infiltration curve shifted by a time which is defined as follows.

Let If then and If then is defined implicitly by  the equation and Solving for involves the implicit solution of equation [7-26] Application of equations [7-24] ‑  [7-26] to every time step of the storm results in a hyetograph of effective rainfall intensity on either the impervious or pervious fraction.  If the surface has zero surface depression storage, this is the net rainfall that will generate the overland flow.  However, if the depression storage is finite, this is assumed to be a first demand on the effective rainfall and the depth must be filled before runoff can occur.

You are prompted to enter a total of five parameters comprising Manning's 'n', the initial and final infiltration rates f0  and fc  (mm/h or inch/h), the decay time constant K (in hours, not 1/hrs) and the depression surface storage depth (millimetres or inches).  For the impervious fraction you can enter either very small or zero values for all the parameters except the Manning roughness coefficient 'n'.

(c) Copyright 1984-2019 Alan A. Smith Inc.
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