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However, an accurate simulation of the time to ponding and infiltration capacity of the soil
may require very small depth and time increments to resolve the rapid changes in hydraulic
gradients in time and space that control the change in infiltration capacity. Computer run time
therefore becomes an issue, particularly where distributed predictions of infiltration capacity
may be required simulating the generation of surface runoff in a heterogeneous catchment.
Thus, there remains a need for simpler analytical solutions of infiltration into the soil surface.
There are many such equations reported in the literature. A selection of the most widely used
are discussed in the following sections.
B5.2.1 The Horton Infiltration Equation
Horton (1933, 1940) described this type of curve by an empirical function of the form
= f o
f c exp
kt +
f ( t )
f c
(B5.2.1)
where f o is an initial infiltration capacity, f c is a final infiltration capacity and k is an empirical co-
efficient. The three parameters, f o , f c and k , are a function of soil type but may also depend on the
antecedent state of the soil. The final infiltration capacity, f c , will be close to the hydraulic con-
ductivity of the soil at field saturation. Although this form of equation was based on empirical
evidence, Eagleson (1970) has shown that it is an approximate solution of the Richards equation
under certain simplifying assumptions. As noted in Chapter 1, however, Horton did not think
of the decline of infiltration as being controlled by the soil moisture profile. He observed that
the surface of the soil could saturate without the profile being saturated and concluded that the
infiltration capacity was controlled by processes at the surface, for example the redistribution
of fine particles by rainsplash that would block the larger pores. He therefore assigned the de-
cline in capacity to what he called “extinction phenomena” (Beven, 2004b). Others have also
suggested that the Richards equation may not be the most appropriate description of infiltration
into soils (e.g. Beven and Germann, 1981, 1982; Germann, 1985; Germann et al. , 2007).
B5.2.2 The Green-Ampt Infiltration Equation
There are other, more direct, ways of deriving an infiltration equation from the Richards equa-
tion. Soil physical theory suggests that infiltration can be described (at least in the absence of
major macropores) by the Richards equation which is based on the nonlinear form of Darcy's
law for partially saturated flow (see Box 5.1). While there are no general analytical solutions
to the Richards equation, a number of different solutions are available for infiltration at the
soil surface based on different simplifying assumptions. Green and Ampt (1911), for example,
assumed that the infiltrating wetting front forms a sharp jump from a constant initial moisture
content ahead of the front to saturation at the front. This allows a simple form of Darcy's law
to be used to represent the infiltration such that infiltration rate f is calculated as
K s h o +
1
f
f ( t )
=
+
(B5.2.2)
z f
where K s is the hydraulic conductivity of the soil at field saturation, h o is the depth of ponded
water on the soil surface, f is a parameter related to the difference in capillary potential
across the wetting front and z f is the depth of penetration of the wetting front. The h o + f
z f term is
due to the capillary potential gradient, here estimated from an effective difference in capillary
potential across the wetting front averaged over the depth of penetration. The magnitude of this
term reduces as the wetting front goes deeper leading to the decline in infiltration capacity of
the soil. This term is added to the gravitational term (1). It stays at unity regardless of the depth
of the wetting front and, when multiplied by the effective saturated hydraulic conductivity,
gives the final infiltration capacity.
 
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