Soil Science Society of America Journal 64:561-564 (2000)
© 2000 Soil Science Society of America
DIVISION S-1-SOIL PHYSICS
Exact solution for horizontal water redistribution by general similarity
Mingan Shaoa and
Robert Hortonb
a National Lab. of Soil Erosion and Dryland Agriculture, Inst. of Soil and Water Conservation, Chinese Academy of Sciences and Ministry of Water Resources, Yangling, Shaanxi, People's Republic of China 712100
b Dep. of Agronomy, Iowa State Univ., Ames, IA 50011 USA
rhorton{at}iastate.edu
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ABSTRACT
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This paper presents an exact solution to horizontal water redistribution by using general similarity theory. A power function of soil water diffusivity is used to derive the exact solution. The similarity solution contains initial wetted length, amount of water present, and coefficients of water diffusivity. A similarity solution for a step function initial condition is compared with a corresponding numerical solution. Error analysis indicates that the maximum global error in water content is within 2%. The general similarity theory provides an approach that exactly solves horizontal water redistribution with a variable first-type boundary of a specific form of time dependence and initial conditions; the Boltzmann transformation case is restricted to a horizontal infiltration problem with constant first-type boundary and initial conditions.
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INTRODUCTION
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THE UNDERSTANDING AND PREDICTION OF THE REDISTRIBUTION OF WATER that has infiltrated soil is just as important as the infiltration process itself (Philip, 1991). Redistribution determines the quantity of water stored in the root zone of crops or natural vegetation and how long this water remains available for uptake by plant roots (Sander et al., 1991). Knowledge of water redistribution is also needed to determine whether or not water or solutes penetrate the root zone, which is useful for agricultural chemical management.
This paper includes the derivation of an exact solution for nonlinear, nonhysteretic redistribution of water in a horizontal soil column by using general similarity theory. The nonlinear water diffusivity is a power function that has been used for more than two decades by a number of soil physicists (Parlange et al., 1980; Parlange and Fleming, 1984; Ross and Parlange, 1994a, 1994b). Philip (1991) gave an analytical solution to the redistribution of water in a horizontal column of infinite dimension. A key for his solution was the similarity character of a horizontal column with two parts, x < 0 and x > 0, at uniform large and small moisture contents. He used the Boltzmann transformation and assumed power law flux-concentration relations to solve the problem. Philip's solution is an implicit integral that requires iterative numerical integrations to have sorptivity be equal to desorptivity. In our analysis, the column does not necessarily need to have equal distance between the wet and dry parts (i.e., the length of the initially wet soil can be arbitrary).
Shao and Horton (1996) showed that the Boltzmann transformation method is a specific form of the general similarity theory. Parlange and Hogarth (1997) extended the work of Shao and Horton (1996) to provide approximate solutions, including other forms of the diffusivity function. Shao and Horton (1996, 1997) used general similarity to show that measurement of the advance of the wetting front with time for redistributing water can be used to determine soil water diffusivity.
The purpose of this paper is to improve the existing Boltzmann transformation method by presenting an exact solution to horizontal water redistribution using general similarity theory. It will be shown that general similarity theory can be used to describe soil water content distributions during redistribution of water. The theory is quite flexible for much more general initial conditions as compared to the Boltzmann transformation case. An example is presented that compares the exact solution by general similarity with a numerical solution for the special case of a step function initial condition. The step function initial condition is used to represent the physical experiment of joining together a wet soil sample and a dry soil sample.
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Theory
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The equation for one-dimensional horizontal flow is given by Bruce and Klute (1956)
 | (1) |
where 
is volumetric water content (m3 m-3), t is time (s), x is distance (m), and D is soil water diffusivity (m2 s-1).
The initial and boundary conditions for the horizontal redistribution are
 | (2a) |
 | (2b) |
 | (3) |
 | (4) |
where x0 is the length of the wet part of the horizontal flow system, f(x) is the water content distribution of the wet part (if the water content is uniform then f(x) is a constant), i is the initial water content in the dry part, q(0,t) is the flux density at the zero-position boundary, and xf is the position for the leading edge of the wetting front. Water is redistributed from the wet part to the dry part and no water flows in or out of the system.
Shao and Horton (1996) used simplifying assumptions to solve the nonlinear flow equation analytically for this particular flow problem. First they assumed a power function of the water diffusivity:
 | (5) |
where D0 and 
are constants.
Shao and Horton (1996) gave the similarity solutions for the case i = 0 as
 | (6) |
 | (7) |
where 
= D0 t.
Inserting Eq. [5], [6], and [7] into Eq. [1] shows that the power of matches only if 
and ß are related by
 | (8) |
and the resulting equation for 
(
) is then
 | (9) |
According to the mass balance:
 | (10) |
where H0 (a constant) is the amount of water in the wet part (m).
Equation [10] obviously requires = -ß, which, together with Eq. [8], determines and ß to be
 | (11) |
With and ß given by Eq. [11], Eq. [9] can be integrated explicitly to yield
 | (12) |
in which the characteristic wetting depth, f, is related to the integration constant, 0, by
 | (13) |
Furthermore, the integration constant, 0, is determined by
 | (14) |
where I is a constant [i.e.,
is a beta function whose value is found in Abramowitz and Stegun (1972)].
By combining Eq. [13] and [14] f is obtained as
 | (15) |
and the wetting front, xf, is written as
 | (16) |
The solution to the original wetting front, xf(t), is then
 | (17) |
From Eq. [17] it is obvious that D0 and can be obtained by fitting Eq. [17] to observed wetting front data. With D0 and soil water diffusivity can be estimated using Eq. [5].
Collecting our results, we find that (x,t) can be written as
 | (18) |
 | (19) |
where 0(t) is a decaying maximum water content at x = 0. Equation [16], [18], and [19] complete the analytical solution to this problem. The only step remaining is to incorporate the finite length of the wet part of the soil column. This can be done by relating the initial wetted length of the column with an arbitrarily constant time (0). Then a more general similarity solution that incorporates the initial wetted length is obtained through arbitrary time translations of the previous solution because Eq. [1], with Eq. [5], [6], and [7], is invariant under such time translations:
 | (20) |
From Eq. [7], the arbitrary time constant, 0, can be related to the length of the wet part, x0, initial water, H0, and water diffusivity coefficient, . When
 | (21) |
With Eq. [16], [20], and [21] the general similarity solution to the redistribution problem of soil water is complete. In the following part of this paper, the general similarity solution is compared with a numerical solution for a step function initial condition of soil water distribution. The numerical solution is obtained by using CSMP (continuous system modeling program), a specially designed language that allows users to simulate all types of physical systems with a minimum of programming effort (Speckhart and Green, 1976). In this numerical modeling method the governing partial differential equation (Eq. [1]) is approximated by a set of ordinary differential equations, which describe spatial flow distribution at a given time. The flow rates are integrated numerically over a time step to calculate a new water content distribution. The time step is 0.1 min and the grid spacing is 0.4 cm. The integration method follows the trapezoidal rule.
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Results and Discussion
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A step function initial condition is selected to demonstrate the use of general similarity theory for describing redistribution of soil water. This example can represent the physical experiment for the redistribution of soil water in a two-part soil column. One part of the soil column is uniformly wet and the other part is uniformly dry. Physically, one column is wetted and then connected with a dry soil column. This allows us to consider the case of water redistribution in the combined column. This initial condition is shown in Fig. 1
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For an example calculation the coefficients of water diffusivity were taken to be D0 = 0.12 cm2 min-1 and = 0.71. The comparison of analytical and numerical decaying maximum water contents (0[t]) at x = 0 cm is shown in Fig. 2
. The general similarity (referred to as analytical) solution 0(t) is in very close agreement with the numerical values. This indicates the left boundary water content (wetted boundary at x = 0) can be well-described by the analytical solution. The comparison of soil water profiles obtained from general similarity theory and the numerical solution is shown in Fig. 3
. It can be seen that at the times indicated the water content profiles obtained by general similarity theory and by the numerical solution are almost the same. The area under each curve is also very similar (i.e., both analytical and numerical solutions behave well in accordance with mass conservation by maintaining the initial amount of water [0.63 cm] during the redistribution process).
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Fig. 2 Comparison of analytical and numerical predictions of the decaying maximum water contents at x = 0
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Fig. 3 Comparison of analytical and numerical solutions for soil water content profiles at selected times for a two-part soil column
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Figure 4
illustrates the global error in water content, described as the difference between the numerical and similarity solutions. The error is small and decreases with time. This indicates the general similarity solution has the ability to predict water redistribution not only for short times (not less than 20 min in this example), but especially for long times. Long-time prediction of water redistribution by numerical solution can require substantial computing time. The general similarity solution overcomes this limitation of numerical solutions. The maximum error for the time concerned is within 0.003. Maximum error for a specific time happens at the wetting front. The wetting front zone of redistribution is the most difficult part to predict.
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Conclusions
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A general similarity solution for redistribution of soil water with certain restrictive boundary conditions, but much more general initial-condition and soil-flow properties (D0 and can be chosen arbitrarily), has been presented. The general similarity solution is closed form and explicit. The similarity solution of water redistribution for a step function initial condition compares well with the corresponding numerical solution. Not only can the similarity solution be used to predict soil water distribution, but Shao and Horton (1996) have shown that the similarity solution itself provides a method of estimating soil water diffusivity. This general similarity solution is useful for checking numerical procedures and can be used to analyze the physical experiment of soil water redistribution.Vachaud Thony 1971; Watson Sardana 1987
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NOTES
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Journal Paper No. 17032 of the Iowa Agric. and Home Econ. Exp. Stn., Ames, IA. Project No. 3262 and 3287, and supported in part by Hatch Act and State of Iowa funds, and in part by Project kz 951-B1-211 of resources, ecological, and environmental research at the Chinese Academy of Sciences.
Received for publication December 21, 1998.
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REFERENCES
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- Abramowitz M., Stegun I.A. Handbook of mathematical functions. New York: Dover Publ, 1972.
- Bruce R.R., Klute A. The measurement of soil moisture diffusivity. Soil Sci. Soc. Am. Proc. 1956;20:458-462.
- Parlange J.-Y., Braddock R.D., Chu B.T. First integrals of the diffusion equation: An extension of the Fujita solutions. Soil Sci. Soc. Am. J. 1980;44:908-911.[Abstract/Free Full Text]
- Parlange J.-Y., Fleming J.F. First integrals of the infiltration equation: I. Theory. Soil Sci. 1984;137:391-394.
- Parlange J.-Y., Hogarth W.L. Comments on soil water diffusivity determination by general similarity theory. Soil Sci. 1997;162:767-768.
- Philip J.R. Horizontal redistribution with capillary hysteresis. Water Resour. Res. 1991;27:1459-1469.
- Ross P.J., Parlange J.-Y. Investigation of a method for deriving unsaturated soil hydraulic properties from water content profiles. Soil Sci. 1994;157:335-340 a.
- Ross P.J., Parlange J.-Y. Comparing exact and numerical solutions of Richards's equation for one-dimensional infiltration and drainage. Soil Sci. 1994;157:341-344 b.
- Sander G.C., Cunning I.F., Hogarth W.L., Parlange J.-Y. Exact solution for nonlinear, nonhysteretic redistribution in vertical soil of finite depth. Water Resour. Res. 1991;27:1529-1536.
- Shao M., Horton R. Soil water diffusivity determination by general similarity theory. Soil Sci. 1996;161:727-734.
- Shao M., Horton R. Reply to comments on soil water diffusivity determination by general similarity theory. Soil Sci. 1997;162:769-770.
- Speckhart F.H., Green W.L. A guide to using CSMP—the continuous system modeling program. Englewood Cliffs, NJ: Prentice-Hall, 1976.
- Vachaud G., Thony J.L. Hysteresis during infiltration and redistribution in a soil column at different initial water contents. Water Resour. Res. 1971;7:111-127.
- Watson, K.K., and V. Sardana. 1987. Numerical study of the effect of hysteresis on post-infiltration redistribution. International Conference on Infiltration Development and Application. 6–9 Jan. 1987. University of Hawaii.
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J.-Y. Parlange, D. Lockington, G. Sander, W.L. Hogarth, D.A. Barry, L. Li, M.B. Parlange, R.S. Govindaraju, M. Shao, and R. Horton
Comments on ""Exact Solution for Horizontal Redistribution by General Similarity""
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ABSTRACT
INTRODUCTION
