HORTON'S
INFILTRATION MODEL
The infiltration process was
thoroughly studied by Horton in the early 1930s [4]. An outgrowth of his work,
shown graphically in Fig. 7.1, was the following relation for determining
infiltration capacity:
where
= the infiltration capacity (depth/time)
at some time t
k = a constant representing the
rate of decrease in / capacity
It indicates
that if the rainfall supply exceeds the infiltration capacity, infiltration
tends to decrease in an exponential manner. Although simple in form,
difficulties in determining useful values for
and k restrict the use of this
equation. The area under the curve for any time interval represents the depth
of water infiltrated during that interval. The infiltration rate is usually
given in inches per hour and the time t in minutes, although other time
increments are used and the coefficient k is determined accordingly.
By observing the
variation of infiltration with time and developing plots of f versus t as shown in Fig. 7.5, we can estimate
and k. Two sets of f and t are selected from the curve and entered in Eq. 7.1.
Two equations having two unknowns are thus obtained; they can be solved by successive
approximations for
and k.
Typical
infiltration rates at the end of 1 hr (
)
are shown in Table 7.1. A typical relation between
and the infiltration rate throughout a
rainfall period is shown graphically in Fig. 7.6a; Fig.7.6b shows an
infiltration capacity curve for normal antecedent conditions on turf. The data
given in Table 7.1 are for a turf area and must be multiplied by a suitable
cover factor for other types of cover complexes. A range of cover factors is
listed in Table 7.2.
Total volumes of
infiltration and other abstractions from a given recorded rainfall are
obtainable from a discharge hydro graph (plot of the stream flow rate versus time)
if one is available. Separation of the base flow (dry weather flow) from the
discharge hydrograph results in a direct runoff hydrograph (DRH), which
accounts for the direct surface runoff, that is, rainfall less abstractions.
Direct surface runoff or precipitation excess in inches uniformly distributed
over a watershed can readily be calculated by picking values of DRH discharge
at equal time increments through
FIGURE7.6
(a) Typical infiltration curve.
(b) Infiltration capacity and mass curves for normal antecedent conditions of
turf areas.
hydrograph and applying the
formula [5]:
(7.2)
Where Pe = precipitation excess (in.)
A = drainage area (mi2)
For most cases
the difference between the original rainfall and the direct runoff can be
considered as infiltrated water. Exceptions may occur in areas of excessive
subsurface drainage or tracts of intensive interception potential. The
calculated value of infiltration can then be assumed as distributed
according to an equation of the form of Eq. 7.1 or it may be
uniformly spread over the storm period. Choice of the method employed
depends on the accuracy requirements and size of the watershed.
To circumvent
some of the problems associated with the use of Horton's
infiltration model, some adjustments can be made [6]. Consider Fig. 7.5. Note
that where the infiltration capacity curve is above the hyetograph, the actual
rate of infiltration is equal to that of the rainfall intensity,
adjusted for interception, evaporation, and other losses. Consequently, the
actual infiltration is given by:
f(t) = min [fp(t), i(t)]
(7.3)
where f(t) is the actual
infiltration into the soil and i(t) is the rainfall intensity. Thus the
infiltration rate at any time is equal to the lesser of the infiltration
capacity fp(t) or the rainfall intensity.
Commonly, the typical values of
and
are greater than the prevailing rainfall
intensities during a storm. Thus, when Eq. 7.1 is solved for fp as a
function of time alone, it shows a decrease in infiltration capacity
even when rainfall intensities are much less than
.
Accordingly,
a reduction in infiltration capacity is made regardless of the amount of
water that enters the soil.
To adjust for this deficiency,
the integrated form of Horton's equation may be ~
used: .
where F is the cumulative
infiltration at time tp as shown in Fig. 7.7. In the figure, it is
assumed that the actual infiltration has been equal to
. As previously
noted, this is not usually the case, and the true cumulative infiltration must
be determined. This can be done using:
where f(t) is determined
using Eq. 7.3.
Equations 7.4
and 7.5 may be used jointly to calculate the time tp, that is, the equivalent
time for the actual infiltrated volume to equal the volume under the
infiltration capacity curve (Fig. 7.7). The actual accumulated infiltration
given by Eq. 7.5 is equated to the area under the Horton curve, Eq. 7.4, and
the resulting expression is solved for tp. This equation:
cannot be solved explicitly for tp
but an iterative solution can be obtained. It should be understood that the
time tp is less than or equal to the actual elapsed time t. Thus
the available infiltration capacity as shown in Fig. 7.7 is equal to or exceeds
that given by
FIGURE7.7
Cumulative
infiltration
Eq. 7.1. By making the
adjustments described, fp becomes a function of the actual amount of
water infiltrated and not just a variable with time as is assumed in the
original Horton equation.
In selecting a
model for use in infiltration calculations, it is important to know its
limitations. In some cases a model can be adjusted to accommodate shortcomings;
in other cases, if its assumptions are not realistic for the nature of the use
proposed, the model should be discarded in favor of another that better fits
the situation.
The first eight
chapters of this book deal with the principal components of the hydrologic
cyc1~.In later chapters; the emphasis is on putting these components together
in various hydrologic modeling processes. When these models are designed for
continuous simulation, the approach is to calculate the appropriate components
of the hydrologic equation, Eq. 1.4, continuously over time. A discussion of
how infiltration could be incorporated into a simulation model follows. It
exemplifies the use of Horton's equation in a storm water management model
(SWMM) [6].
First, an
initial value of tp is determined. Then, considering that the value of fp
depends on the actual amount of infiltration that has occurred up to that
time, a value of the average infiltration capacity, ~, available over
the next time step is calculated using:
Equation 4.3 is then used to find
the average rate of infiltration,
:
where i is the average
rainfall intensity over the time step.
Following this,
infiltration is incremented using the expression:
where
F
=
t
is
the added cumulative infiltration (Fig. 7.7).
The next step is
to find a new value of
. This is done
using Eq. 7.6. If
Example 7.1
Given an initial infiltration
capacity
of 2.9 in./hr and a time constant k of
0.28 in/hr derive an infiltration capacity versus time curve if the ultimate infiltration
capacity is 0.50 in./hr. For the first 8 hours, estimate the total volume of
water infiltrated in inches over the watershed.
Solution:
1. Using Horton's
equation (Eq. 7.1), values of infiltration can be computed for various
times. The equation is:
2. Substituting the
appropriate values into the equation yields:
3.
For
the times shown in Table 7.3,values of f are computed and entered into the table.
Using a spreadsheet graphics package, the curve of Fig. 7.8 is derived.
To
find the volume of water infiltrated during the first 8 hours, Eq. 7.1 can be
integrated over the range of 0-8:
The
volume over the watershed is thus 11.84 in.
What is E ?
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