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<br />~ - , -
<br />
<br />a~1 2
<br />o
<br />2Ks (h . ~,)
<br />
<br />, , .(55)
<br />
<br />13 "" --1..-..__ [I + L (1
<br />(It - .~o) . I 11
<br />
<br />- :~O)J
<br />
<br />, , ,(56)
<br />
<br />and
<br />
<br />1
<br />a - (h _ , )
<br />"0
<br />
<br />[,2
<br />h + (1 _ n) b
<br />
<br />(->I> + b) "0 - ---"--- ] ,(57)
<br />o (l - n)
<br />
<br />respectively. Especially for h = 0, Eqs, 55, 56, and 57
<br />reduce to
<br />
<br />B = t (1 + ::)
<br />
<br />, , , ' , , , . ,(58)
<br />
<br />~_-'- (.a(,o_,) ",.,.,. ,(59)
<br />a.o
<br />
<br />and
<br />
<br />1 [ b
<br />B = ljIo 1'=fl
<br />
<br />(-.0 + b)1'0J ' ,(60)
<br />
<br />bn
<br />- ..,.-:-n
<br />
<br />respectively. For soils having spacial variation in soil
<br />properties over an areal extent, one may use the
<br />average hydraulic conductivity_ ~s a function of
<br />average soil capillary potential, K(1/i), in the foregoing
<br />analysis,
<br />
<br />As pointed out previously, the Green-Ampt
<br />approach does not give information about details of
<br />the moisture profiles, However, the role of the i3
<br />factor playing in the determination of L, Lp, and tp,
<br />respectively, from Eqs, 47, 49, and 50 should not be
<br />ignored, The i3 factor, as defined in Eq, 42, can be
<br />regarded as a gross measure of the effect of moisture
<br />profiles on the infiltration process of a Green-Ampt
<br />type. Consequently, to assume i3 = I and Ko = 0 in
<br />Eq, 46 by some investigators may result in an
<br />erroneous solution of L from Eq, 47 and hence of f
<br />from Eq, 44,
<br />
<br />The Kostiakov equation
<br />
<br />This is strictly an empirical formula, which was
<br />developed independently by Lewis (1937), Kostiakov
<br />(1932) expressed the infiltration capacity, f, as a
<br />negative power function of time, t:
<br />
<br />f = At-a (0 < ex < t) for t ~ tp . .(61)
<br />
<br />where A and a are parameters. Despite simplicity in
<br />its form, the applicable range of time for Eq, '61 is
<br />rather limited, as pointed out by Philip (1957b). In
<br />
<br />other words, in order to fit the whole range of t, the
<br />value of a and hence of A must vary with t, which in
<br />essence detracts from its usefulness. Conversely, if the
<br />values of a and A are kept constant, Eq, 61 provides
<br />an infinite initial f, but asserts footo approach zero as
<br />t increases, rather than a constant non-zero f (= K s)'
<br />However, this awkwardness in the form of Eq. 61 can
<br />be remedied by assuming the form
<br />
<br />-.
<br />f = fo> + A(t - to)
<br />
<br />for t ~ t ,,(62)
<br />p
<br />
<br />as generalized by Smith (1970) and Smith and Chery
<br />(1973) (henceforth called the modified Kostiakov
<br />equation), In Eq. 62, to is another parameter, in
<br />addition to A and a, needs to be determined
<br />from soil data, The form of Eq. 62 is simple, but the
<br />values of A, n, and to cannot be predicted in advance.
<br />Furthermore, there is no provision or criterion for
<br />predicting when ponding occurs under rainfall (I.e.,
<br />the time of ponding, lp)' Smith (1970) has attempted
<br />to express tp as a negative power function of the.
<br />rainfall intensity, r, using the numerical solutions
<br />obtained from the boundary-value problem of rain
<br />inf1ltration for six soils, His strictly empirical
<br />formulation of tp' thuugh the values of A, a, and to
<br />may already be given or determined from experi-
<br />ments, hardiy makes Eq, 62 useful under conditions
<br />other than those tested. The usefulness of an
<br />algebraic infiltration equation must lie in the validity
<br />and applicability of its simple expression over a wide
<br />range of conditions imposed or given, Whether and
<br />how Eq, 62 can be applied to the computation of tp
<br />is investigated herein.
<br />
<br />A review of Smith's (1970) results reveals that
<br />the values of A and a for soil under various rainfall
<br />intensities tested are fairly constant, This finding
<br />suggests the possibility of applying Eq, 62 to a soil
<br />under the same initial moisture content, Bo. and the
<br />same soil surface condition, h, but under various
<br />rainfall intensities, say I1, r2, and so forth, as shown
<br />in Figure 13. Therefore, with the same 00 and h
<br />values, one may have
<br />
<br />f = foo + A(t
<br />
<br />".
<br />t01)
<br />
<br />for t ;:?: tp1
<br />
<br />,(63)
<br />.(64)
<br />
<br />f = fo> + A(t
<br />
<br />-.
<br />toZ)
<br />
<br />for t;:?: tp2
<br />
<br />where to 1 and .02 are parameters corresponding to r 1
<br />and r2, respectively; and tp1 and tp2 are the times of
<br />ponding corresponding to II and r 2' respectively.
<br />Because the same soil having the identical initial and
<br />boundary conditions is subjected to two different
<br />application rates, rl and r2' it may be assumed that
<br />the total cumulative inf1ltration, F(oo), for the soil
<br />with the same water.storage potential, though under
<br />the different rainfall intensities, must be equal.
<br />Physical!x,. this assumption implies that the shaded
<br />aIeas, l!J and Q), in Figure 13, are equal, or
<br />mathematically it can be expressed as
<br />
<br />35
<br />
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