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 [721] 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 [722]
and [723].
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 78 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 76:
Representation of the moving curve Horton equation
Figure 76 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 [726]
Application of
equations [724] ‑ [726] 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'.
