|
<br />which indicates the decrease in the total moisture
<br />content remaining in the soil, If the initial condition
<br />is specified at the static equilibrium where Ko. 0, this
<br />of course happens only when r<O in,/hr (I.e"
<br />equivalent to evaporation).
<br />
<br />It is noted that before ponding, the 1/1(8) and
<br />K(8) relationships of a given soil have no bearing on
<br />the infiltration rate, f(t), as long as
<br />
<br />f(t) . r(t)
<br />
<br />, . . ' , , . . , ' , (9)
<br />
<br />which is apparent from Eq, 7 because f(t) is evaluated
<br />by
<br />
<br />J.o ae
<br />fCe) = - dz + K
<br />~lEIat 0
<br />
<br />. , ' , , , , ,(10)
<br />
<br />Equation 10 can readily be obtained from Eq. I in a
<br />similar fashion deriving Eq. 7 with the help of the
<br />following definition of the infiltration rate on the soil
<br />surface:
<br />
<br />f(t) = [K(6)~ + K(e) ] I
<br />az z~o
<br />
<br />, ,(11)
<br />
<br />As soon as the soil surface starts ponding, Eq, 9
<br />is no longer valid, The >I^- 8) and K(O) relationships of
<br />soil properties, initial soil moisture content, 80, soil
<br />moisture content at saturation, f} s' rainfall intensity I
<br />r, and ponding depth, h, all come into play with the
<br />infiltration rate, f, which must be computed by either
<br />Eq, 10 or 11. Use of Eq, 10 has a slight advantage
<br />over that of Eq, II because the evaluation of the 8,
<br />distribution seems to be more accurate than that of
<br />the soil capillary potential gradient, a1/J(8)/az, at the
<br />soil surface (z = 0) in terms of known values at grid
<br />points.
<br />
<br />Some investigators (Smith and Woolhiser, 1971;
<br />Smith, 1971), assuming initial water movement to be
<br />negiigible, ignored the Ko term in Eq, 10 in their
<br />evaluation of f(t) after ponding, This may result in a
<br />big error if Ko. Ks, Without the Ko term in Eq. 10,
<br />it can readily be shown from Eq, I that as t
<br />approaches infinity, f(t) cannot be asymptotic to Ks '
<br />In other words, integration of Eq, I with respect to z
<br />as t approaches infinity gives
<br />
<br />11m J~ 0 as
<br />- dz -)- K - K
<br />t~ at S 0
<br />-~
<br />
<br />, , , , ,(12)
<br />
<br />Thus, incorporating Eq, 12 into Eq, 10 yields
<br />
<br />11m
<br />t~ f(t) -+ Ks
<br />
<br />, , , , , , , , , ,(13)
<br />
<br />which does not seem to vary with any of the factors
<br />mentioned previously,
<br />
<br />After ponding, the soil profile becomes fully
<br />saturated near the soil surface with the saturated zone
<br />overlying the unsaturated zone, as shown in Figure 1.
<br />As described before, the saturation front advances
<br />downward, starting at the soil surface, The flow
<br />equation (Richards equation, Eq, I) used in the
<br />unsaturated zone can also apply in the saturated
<br />zone; however, because 8 = Os and Ks = constant, it
<br />can be simplified to the Laplace equation in terms-of
<br />the hydraulic head, h = '" + z, or '" as
<br />
<br />a'",
<br />"-L = 0
<br />az'
<br />
<br />. . , , ,(14)
<br />
<br />Note that Eq. 14 is equivalent to the Darcy law
<br />having a constant vertical velocity component, f(t),
<br />Integration of Eq, 14 with respect to z with the help
<br />of Eqs. 4 and II at the soil surface yields
<br />
<br />(f(t) - K )
<br />~ = Kg S z + h(t)
<br />
<br />, . , , ,(IS)
<br />
<br />because in the saturated zone the vertical velocity,
<br />though it does not vary with z, varies with t and
<br />hence is not constant.
<br />
<br />At z = 0, Eq. 15 is identical to Eq. 4, the soil
<br />surface condition after ponding, On the other hand,
<br />at the saturation front (z = -Lf), '" is zero and hence
<br />f(t) can be expressed from Eq, IS as
<br />
<br />h(t) + Lf(t)
<br />f(t) . Ks Lf(t)
<br />
<br />In application, use of Eq, 16 in the problem of the
<br />infiltration rate computation after ponding requires a
<br />knowledge of Le(t) which is, of course, unknown.
<br />The following simple method was developed to
<br />determine the Le (t), Equating Eq, 16 to Eq, 10 yields
<br />
<br />, , , ' . ,(16)
<br />
<br />K h(t)
<br />Lf(t) . s
<br />
<br />10 ~d +K
<br />J at Z 0
<br />-~
<br />
<br />. ' ,(17)
<br />
<br />- K
<br />.
<br />
<br />Therefore, by knowing the total rate of change of the
<br />soil moisture content in the unsaturated zone (the
<br />first term in the denominator on the right side of Eq,
<br />17, includes the total rate of change of 8 in the
<br />saturated zone, but the total rate of change of 8 in
<br />the saturated zone is assumed to be zero by implica'
<br />tion), the L e( t) value can readily be computed. Use
<br />of Eqs, 16 and 17 at some critical points of time
<br />requires that a few comments be made here. At the
<br />time of ponding, because h(tp) and Le(tp) are all
<br />zero, Eq, 16 becomes indeterminate, In other words,
<br />Eq, 16 is not valid at t = tp whereas Eq, 17 reduces to
<br />Le(tp) = 0, On the other hand, as t .., one obtains
<br />
<br />16
<br />
|