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<br />Formulation of Parametric Inftltration Models
<br />
<br />If the rainfall intensity is greater than the final
<br />limiting inf1Itration rate (or the saturated hydraulic
<br />conductivity), the rain infiltration process in general
<br />can be divided into two stages: One is before ponding
<br />and another after ponding. Therefore, a parametric
<br />infiltration model, if formulated, must consist of
<br />both stages. As mentioned previously, before ponding
<br />the inftltration rate is equal to the rainfall intensity
<br />and after ponding it is greater than or equal to the
<br />inftltration capacity, Mathematically the parametric
<br />infiltration model can be expressed as
<br />
<br />f . r(t)
<br />
<br />for05t~t
<br />p
<br />
<br />,(35)
<br />
<br />f - f(t)
<br />
<br />for t ~ t
<br />P
<br />
<br />,(36)
<br />
<br />where f = inftltration rate, r(t) = rainfall intensity
<br />which mayor may not vary with time, t = time, tp =
<br />time of ponding, f(t) = inf1Itration function which is
<br />an explicit function of time. lt should be noted that
<br />the f(t) expression in Eq. 36 can be any form of the
<br />algebraic inf1Itration equations except the Green-
<br />Ampt equation that is expressed implicitly as a
<br />function of f and t. Evidently, in addition to the
<br />parameters in f(t) that must be evaluated, the time of
<br />ponding, tp' must be determined before the para-
<br />metric inmtration model, Eqs, 35 and 36, can apply,
<br />Each of the available algebraic infiltration equations
<br />to be used in Eq, 36 is briefly discussed as follows.
<br />
<br />The Green.Ampt equation
<br />
<br />This is one of the algebraic infiltration equa.
<br />tions in which the parameters are made of physical
<br />properties of the soil-water system. The Green-Ampt
<br />equation, also called the "delta-function" solution by
<br />Philip (1969b), has been independently derived and
<br />studied by several other workers (Rode, 1965). The
<br />primary assumption imposed in the derivation of the
<br />Green-Ampt equation is that the soil surface is
<br />ponded by a pool of non.zero-depth water. Philip
<br />derived the average volumetric moisture content and
<br />hydraulic conductivity basing on an additional
<br />assumption that either similarity is preserved on the
<br />moisture proftles (Philip, 195Th) or the moisture
<br />diffusivity is the Dirac-delta function of the moisture
<br />content around the saturation point near the soil
<br />surface (Philip, 1954), Mein and Larson (1971 and
<br />1973) on the other hand evaluated the average
<br />suction at the wetting front from the soil suction-
<br />hydraulic conductivity relationship rather than inte-
<br />grating the suction over the soil depth, It is in fact the
<br />same as the Philip assumption that similarity on the
<br />moisture profiles, with Mein and Larson's assumed
<br />shape (Le" linear), is preserved, Talsma and Parlange
<br />(1972) however derived the delta-function solution
<br />from an integral method recently developed by
<br />
<br />Parlange (1971). Although the Green-Ampt equation
<br />and its solutions give no information about details of
<br />the moisture proftles, they do offer estimates of
<br />integral properties such as the infiltration rate, f(t),
<br />and the cumulative inftltration, F(t), A more
<br />generalized Green.Ampt equation can be formulated
<br />in the following way,
<br />
<br />The inftltration rate, f(t), can be mathe.
<br />matically expressed in two integro-differential forms,
<br />Eqs, 10 and II. Both expressions were used in the
<br />solution of the boundary-value problem of rain
<br />inftltration, Let L be the distance below' the soil
<br />surface (z = 0) at which non.zero depth of water is
<br />ponded, and be defined by
<br />
<br />('0
<br />J (6 - 60) dz ~ (as - eo) L(t)
<br />-r.,.
<br />
<br />for t. t '..' (37)
<br />p
<br />
<br />where Lw = soil deptll at the wetting front; 80 =
<br />initial volumetric moisture content; and Os =
<br />saturated volumetric moisture content, The L(t) value
<br />that varies with Lw(t) and 8(z,t), as defined in Eq,
<br />37, has such a physical meaning that the shaded areas
<br />in Figure 12a are equal, With this definition ofL, Eq,
<br />10 reduces to
<br />
<br />f _ (8 _ 0 ) dL + K
<br />5 0 dt 0
<br />
<br />for t ~ 0
<br />
<br />, ,(38)
<br />
<br />To express Eq. II in terms of L requires
<br />additional assumptions, Since the expression of the
<br />inftltration rate, as shown in Eq. II, is essentially that
<br />of Darcy's velocity, w(z,t), at the soil surface (z = 0),
<br />one may assume that
<br />
<br />~,o
<br />f ;; 1 [-w(z.t)] dz
<br />L
<br />-L
<br />w
<br />
<br />for t. t ,(39)
<br />p
<br />
<br />where
<br />w(z,t) _ -K(z,t) ,,,~~,t) - K(z,t) ,,(40)
<br />
<br />We cannot integrate Eq, 39 by simply substituting
<br />Eq, 40 into Eq, 39 unless K-distribution is known.
<br />For the K(z)-distribution having very sharp peak
<br />around the 8, [i.e" the tendency for soil diffusivity
<br />to have very large values around the 8,-Dirac-delta
<br />function distribution (Philip, 1969)], it may be
<br />justified to assume that
<br />
<br />_ 1 (0
<br />K(z~t) = L J K(z.t) dz'" Kg
<br />-L
<br />w
<br />
<br />, , , ,(41)
<br />
<br />K(z t) a~(z.t) = 1 Co K(z t)lli..~ dz
<br />, OZ L J ' az
<br />-L
<br />w
<br />
<br />_ a K(z,t) al)i(z.t) (42)
<br />'z ' ,
<br />
<br />32
<br />
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