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<br />BOUNDARY-VALUE PROBLEM OF RAIN INFILTRATION
<br />AND ITS NUMERICAL SOLUTIONS
<br />
<br />Because the solution to the boundary.value
<br />problem of rain infiltration can produce the infiltra-
<br />tion-capacity decay curve, though not in closed form,
<br />as mentioned previously, a mathematical infiltration
<br />model consisting of the Richards equation and
<br />appropriately prescribed inItial and boundary condi-
<br />tions was formulated in this study and solved
<br />numerically on a digital computer. The infiltration-
<br />capacity decay curve so determined did help evaluate
<br />the effect of each significant factor on infiltration
<br />capacity without resorting to very extensive labora-
<br />tory experIments, Thus, it was hoped that the
<br />mathematical evaluation of the significant factors
<br />might help lead to the establishment of relationships,
<br />if any, between the significant factors and the model
<br />parameters used in each of the algebraic infiltration
<br />equations.
<br />
<br />No analytical solution to the rain infiltration
<br />problem has been devised for actual solis with the
<br />time-varying soil-surface condition, Le., changing
<br />from a flux condition (before ponding) to a con.
<br />centtation condition (after ponding). (The tlme of
<br />ponding is defined herein as an instant at which the
<br />soil surface becomes saturated.) For a concentration
<br />condition on the soli surface, Philip (1957a) ex-
<br />pressed the infiltration flux in a power series expan-
<br />sion of time. For the same concentration condition as
<br />specified by Philip (1957a), Parlange (1971) has
<br />recendy developed an altemate method to obtain an
<br />approximate (or "quasi.analytieal") solution,
<br />Patlange (1972) applied his method to the problem
<br />with a flux condition, Knight and Philip (1973)
<br />critically studied the appllcabllity and limitatlons of
<br />the Parlange method and offered a new quasi-
<br />analytical technique which, however, has affinities
<br />with Patlange's method (Philip and Knight, 1974),
<br />
<br />In view of the difficulty in obtaining the
<br />analytical solutlon to the infiltration problem, almost
<br />every investigator resorted to a numerical method for
<br />various boundary conditions and solved it on a
<br />computer. For example, Hanks and Bowers (1962)
<br />obtained numerical solutions for a zero ponding-
<br />depth condition, Wang and Lakshminarayana (1968)
<br />for a saturated moisture-content condition, Rubin
<br />and Steinhardt (1963) and Rubin (1966a) for a
<br />constant water-flux condition, and Bruce and Whisler
<br />
<br />(1973) for nonconstant water-flux (raInfall) condi.
<br />tion.
<br />
<br />Numerical solutions to various types of the
<br />boundary-value problems of infiltration have been
<br />obtained by many investigators using the implicit
<br />numerical scheme (e,g" Hanks and Bowers, 1962;
<br />Rubin and Steinhardt. 1963; Rubin, 1966b; Freeze,
<br />1969) whlle other investigators have used the explicit
<br />numerical scheme fur solution (e.g" Staple and
<br />Lehane, 1954; Gupta and Staple, 1964; Staple, 1966
<br />and 1969; Wang and Lakshminarayana, 1968), Both
<br />numerical schemes have been used with success in
<br />solving the boundary.value problem of rain infiltra-
<br />tion although most of those solved are subject to a
<br />less restrictive boundary condition than those pre-
<br />scribed in the present study, Generally speaking,
<br />expllcit-difference methods are less efficient but
<br />easier to program than implicit methods. In this
<br />study, partly due to a complex and variable upper-
<br />boundary condition (1.e" a change from a constant
<br />flux to constant head condition) to be imposed on
<br />the soil surface, the explicit difference method is
<br />adopted herein,
<br />
<br />No effort was spent on development of an
<br />unsaturated flow equation of a general type for a soil
<br />containing swelling clay as well as for infIltration with
<br />counter flow of air. As mentioned previously, several
<br />investigators have already started formulating such a
<br />general flow equation (or a set of flow equatlons in
<br />the case with air counterflow), but no solution to
<br />such a complex ptoblem is avallable as yet. Making
<br />the mathematical modeling of the natural rain in-
<br />ftItration ptoblem more eompllcated is the fact that
<br />during heavy rainstorms, a lens of air may be trapped
<br />between the advancing wetting front and a lower
<br />layer with high resistance to flow of ait due to a high
<br />water content and/or a small porosity. When air
<br />ahead of the wetting front is compressed because of
<br />no access to the atmosphere, the wetting front
<br />becomes unstable, Raats (1973) derived criteria for
<br />instabllity of the wetting front on the basis of a
<br />simple hydraulic model due to Green and Ampt
<br />(1911). Several investigators (e,g" WlIson and Luthin,
<br />1963; Peck 1965; Dixon and Linden, 1972; Vachaud,
<br />Gaudet, and Kuraz, 1974) have measured large
<br />increases in air pressure during ponded infiltration in
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