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<br />of the implicit scheme, Noteworthy is the con-
<br />vergence problem of the implicit scheme (Smith and
<br />Woolhiser, 1971), The fact that an implicit-difference
<br />equation is unconditionally stable does not neces-
<br />sarily guarantee a convergent solution. Convergence
<br />of the numerical scheme depends on the form of the
<br />equation, and on some parameters, which are func-
<br />tions of the coefficients in the differential equation
<br />and the mesh size in both time and space,
<br />
<br />Mathematical Statement of the Problem
<br />
<br />In the formulation of the mathematical model,
<br />the following assumptions are made:
<br />
<br />(1) A water system in the soil will be regarded
<br />as a continuous medium.
<br />
<br />(2) Soil will be treated as a semi.infinite,
<br />homogeneous, isotropic porous body of stable struc-
<br />ture.
<br />
<br />(3) The flow of the water system is assumed to
<br />be uoi-directional (Le., in the vertical or gravitational
<br />direction only) and to obey Darcy's law,
<br />
<br />(4) The physical properties of soil, such as
<br />capillary tension, hydraulic conductivity, and mois-
<br />ture diffusivity are unique, single-valued, continuous
<br />functions of soil moisture content. ]n other words,
<br />there is no hysteresis as long as only the wetting parts
<br />of the relationships are considered.
<br />
<br />(5) For simplicity, the initial moisture content
<br />will be assumed uniform, Note that in reality the
<br />initial moisture content is rarely uniformly dis-
<br />tributed,
<br />
<br />(6) Raindrops failing on the soil surface will be
<br />treated as a continuous medium of water.
<br />
<br />(7) Pore air pressure is assumed to remain
<br />essentially atmospheric.
<br />
<br />Based on the preceding assumptions, the mathe-
<br />matical description of rain infiltratiOn is:
<br />
<br />~Il. . .3__ (K(a) et1"J,) + 'K(a)
<br />at az 3z az
<br />
<br />(Richards Equation)
<br />
<br />6(z.O) = eo
<br />
<br />[K(a) 3~~') + K(a}]!
<br />
<br />- r(t}
<br />
<br />zoO
<br />
<br />(O~ t S t )
<br />p
<br />
<br />. , , , , , . (3)
<br />
<br />~(O.t) - hIt)
<br />
<br />(4)
<br />
<br />(t ~ tp)
<br />
<br />[K(a) 3t~a} + K(O}] \ z~'- - Ko
<br />
<br />(5)
<br />
<br />where {} = soil moisture content; t = time; z ~ vertical
<br />coordinate positive upward; K( 8) = hydraulic conduc-
<br />tivity; 1/.(8) = soil capillary potential; 80 = initial
<br />moisture content; r(t) = rainfall intensity; t = time
<br />of ponding; h(t) = depth of water ponding 011' the soil
<br />surface (z = 0); and Ko = initial hydraulic
<br />conductivity corresponding to the initial moisture
<br />content, 80, Equation 5 is the "hypothetical" lower
<br />boundary condition that is needed in order to solve
<br />the problem,
<br />
<br />At the time of ponding (t = tp.!, the soil
<br />moisture content, 8 (z, t), just becomes saturated
<br />(8 s)' In the mathematical expression, it is
<br />
<br />a(O,r)-O
<br />p s
<br />
<br />, , , (6)
<br />
<br />Equation 6 is a criterion used for the computation
<br />of t p and is valid only if the air entry value of the soil
<br />is zero.
<br />
<br />The mathematical model consists of Eqs, 1
<br />through 6 and the 1/1<0)- and the K(8)-relationships of
<br />the glven soil. Although the preceding set of equa-
<br />tions, Eqs. I through 6, is formulated in one-space
<br />dimension, z, only the rainfall intensity, r, and the
<br />ponding depth, h, may vary independently, in addi-
<br />tion to time, t, with another space-dimension, x, in
<br />the direction of surface water flow if Eqs. 1 through
<br />6 are coupled with a surface runoff model for the
<br />surface runoff computation.
<br />
<br />Before ponding, it is apparent from Eq, 3 that
<br />the infiltration rate, f(t), is equal to the rainfall
<br />intensity, r(t). However, rain infiltration can continue
<br />indefinitely without ponding if r "Ks, where K is
<br />the saturated hydraulic conductivity corresponding to
<br />the moisture content at saturation, (J . Furthermore,
<br />if r<I(", the total soil moisture ctntent may de-
<br />crease, TIris does not sound logical, but can readily be
<br />seen from
<br />
<br />(I)
<br />
<br />SO 30
<br />r(t.),. _CD at dz + Ko
<br />
<br />.. . .. .. (7)
<br />
<br />(2)
<br />
<br />which was derived by integrating Eq, I with respect
<br />to z from -~ to 0 and then having its results
<br />substituted by Eq, 3 and the "hypothetical" lower
<br />boundary condition, Eq. 5, Consequently, if r<1<o'
<br />Eq, 7 yields
<br />
<br />S.03O
<br />-dz =
<br />_,.,at
<br />
<br />r -
<br />
<br />K '0
<br />o
<br />
<br />. ' , , , , , (8)
<br />
<br />IS
<br />
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