Soil Science Society of America Journal 65:957-959 (2001)
© 2001 Soil Science Society of America
COMMENTS & LETTERS TO THE EDITOR
Comments on "Exact Solution for Horizontal Redistribution by General Similarity"
J.-Y. Parlangea,b,
D. Lockington,
G. Sander,
W.L. Hogarth,
D.A. Barry,
L. Li,
M.B. Parlangea,b and
R.S. Govindaraju
a Department of Agricultural and Biological Engineering Riley-Robb Hall Cornell University Ithaca, NY 14853
b Department of Civil Engineering University of Queensland Brisbane 4072 Australia
c Department of Civil and Building Engineering Loughborough University Loughborough, Leics LE11 3TU United Kingdom
d Faculty of Environmental Science Griffith University Nathan, Brisbane, Queensland 4111 Australia
e School of Civil and Environmental Engineering The University of Edinburgh Edinburgh, Scotland EH9 3JN United Kingdom
f Department of Geography and Environmental Engineering Johns Hopkins University Baltimore, MD 21218
g School of Civil Engineering Purdue University West Lafayette, IN 47907
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INTRODUCTION
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Obtaining exact solutions of Richards' equation, with or without gravity, is of great use to validate numerical procedures, as well as to gain more physical insight in the structure of solutions. In their study, Shao and Horton (2000) apply such a valuable solution for those purposes very effectively. The following examines the relation of their solution to that of other problems, and in the process extends the application of their result.
Parlange et al. (1997) proposed a solution which in general provides an accurate approximation to solutions of Richards' equation, with or without gravity. It is interesting that all the exact solutions discussed here for a power law diffusivity and without gravity follow the general form proposed by Parlange et al. (1997).
In the case when the diffusivity is proportional to the water content, the resulting Richards' equation without gravity reduces to the Boussinesq equation. Barenblatt et al. (1990) obtained two particular exact solutions in that case. One is quadratic in the distance; Shao and Horton's solution is a generalization of that case for any power law diffusivity. The other is linear in the distance, and that particular solution was also generalized for any power law diffusivity (see for instance Parlange and Fleming, 1984 and Gilding, 1982). More generally, the linear solution can be obtained from the travelling wave solution for any diffusivity, either without gravity (Gilding, 1982) or with gravity (Fleming et al., 1984). Parlange et al. (2000) showed that for the Boussinesq equation the two Barenblatt's solutions can be combined in such a way that his two exact cases are limiting cases of the combination. Similarly, Shao and Horton's solution and Parlange and Fleming's solution can be combined so that those two solutions are limiting cases of the combination.
Following Shao and Horton's notations we solve
 | (1) |
where 
and D0 are constants, 
is the water content, x the position, and t the time. The initial condition and boundary conditions will be imposed by the solution itself.
Following Parlange et al. (1997)(2000), we look for an exact solution of the form
 | (2) |
where A, B, and t0 are constants; t0 
0 is arbitrary and in particular could be zero, but clearly the solution could become singular at t = 0 (Parlange et al., 2000). This is theoretically acceptable for the Boussinesq equation, as the height of the aquifer could approach a delta function at t = 0, but it is not acceptable here as the water content cannot exceed its finite saturated value. 0(t) is the value of at the surface x = 0. Introducing from Eq. [2] in Eq. [1] we find at once
 | (3) |
and
 | (4) |
where C is an arbitrary constant. If A = 0 the formula obviously reduces to that of Shao and Horton (2000).
If A/t0 is a negative constant and A 
- 
and with C = A2D0
t-2/
0/2, the solution reduces to that of Parlange and Fleming (1984) (which is exactly what happened in Barenblatt's case, i.e., for = 1, Parlange et al., 2000) and in that limit, Eq. [2] becomes
 | (5) |
In the case of A = 0, Shao and Horton (2000) looked at water redistribution in a semi-infinite medium impervious at x = 0. If we take A = 2BL, the solution of Eq. [2] for x L is basically that of Shao and Horton (2000) for x 0. It is clear that the results in Fig. 3 of Shao and Horton (2000) and those of Parlange et al. (2000) for x > L = 3 are qualitatively the same. However the solution of Eq. [2] is not limited to water redistribution and can be used to model absorption in or out of finite or semi-finite columns. For instance, Fig. 3 of Parlange et al. (2000) shows absorption and then desorption of water in a finite column for x < L (0 
x 3).
The general case, Eq. [2], could be formally obtained by a translation in space of the particular solution for A = 0, then the other limiting solution can be obtained when the translation is infinite as explained above. However, both in the case = 1 and 
1, the two limiting solutions were derived originally independently by different similarities, that is, without recognizing their fundamental link, see for instance Gilding (1982).
Received for publication July 5, 2000.
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REFERENCES
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- Barenblatt, G.L., V.M. Entov, and V.M. Ryzhik. 1990. Theory of fluid flows through natural rocks. Kluwer Academic Publishers, Dordrecht, the Netherlands.
- Fleming, J.R., J.-Y. Parlange, and W.L. Hogarth. 1984. Scaling of flux and water content relations: Comparison of optimal and exact results. Soil Sci. 137:464–468.
- Gilding, B.H. 1982. Similarity solution of the porous media equation. J. Hydrol. 56:251–263.
- Parlange, J.-Y., D.A. Barry, M.B. Parlange, W.L. Hogarth, R. Haverkamp, P.J. Ross, L. Ling, and T.S. Steenhuis. 1997. New approximate analytical technique to solve Richards equation for arbitrary surface boundary conditions. Water Resour. Res. 33:903–906.
- Parlange, J.-Y., and J.F. Fleming. 1984. First integrals of the infiltration equation: 1. Theory. Soil Sci. 137:391–394.
- Parlange, J.-Y., W.L. Hogarth, R.S. Govindaraju, M.B. Parlange, and D. Lockington. 2000. On an exact analytical solution of the Boussinesq solution. Transp. Porous Media 39:339–345.
- Shao, M., and R. Horton. 2000. Exact solution for horizontal redistribution by general similarity. Soil Sci. Soc. Am. J. 64:561–564.[Abstract/Free Full Text]
Response to "Comments on ‘Exact Solution for Horizontal Redistribution by General Similarity’ by Parlange et al."
Mingan Shao and
Robert Horton
h Shaanxi 712100, China
i Department of Agronomy Iowa State University Ames, IA 50011
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INTRODUCTION
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We thank Parlange et al. (2000) who provided a very important comment on Shao and Horton (2000). We do agree with their statement that our result can be extended to other problems. For example, our solution and the Parlange and Fleming (1984) solution can be combined to obtain the two particular exact solutions of Barenblatt et al. (1990). In Shao and Horton (2000), a step function initial condition was selected to show the general similarity solution for describing redistribution of soil water. The example represents the experiment for the redistribution of a two-part column. However, the redistribution problem after infiltration is more common. Here we show how the solution predicts the redistribution after infiltration problem.
Experimentally, water first infiltrates into a soil column, infiltration ceases, and the inlet boundary is closed while water redistributes. The initial condition is presented in Fig. 1 . The zero water content (corresponding to oven-dried soil) of the dry part of previous cases has been altered to a non-zero water content (corresponding to air-dried soil). The length of the soil column is 40 cm. In Fig. 2
, we show a comparison of water content profiles between similarity prediction and numerical solution. In this case the predictability of the general similarity solution for water redistribution is still good. The absolute value of the maximum error is <0.02. This error is from the approximation of the initial condition. In this case only two parameters, the wetted length (x0) and the amount of infiltrated water (H0), are used to describe the initial water content profile in the general similarity solution. This simple approximation introduces an error especially for early time.
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Fig. 2. The comparison of soil water content profiles predicted by the similarity solution (solid curves) and the numerical solution (symbols) for water redistribution after infiltration.
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Once again, the general similarity solutions are closed form and flexible. The general similarity solution for the initial condition after infiltration compares well with the corresponding numerical solution. However the most important part of the application of the general similarity solution is that the similarity solution itself provides a method of estimating soil water diffusivity (Shao and Horton, 1996, 1997).
Finally, readers interested in the approximate solution for the absorption problem in which soil hydraulic properties are described by van Genuchten (1980) and Mualem (1976) may read Shao and Horton (1998).
Received for publication September 18, 2000.
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REFERENCES
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- Barenblatt, G.I., V.M. Entov, and V.M. Ryzhik. 1990. Theory of fluid flows through natural rocks. Kluwer Academic Publishers, Dordrecht, the Netherlands.
- Mualem, Y. 1976. A new model for predicting the hydraulic conductivity of unsaturated porous media. Water Resour. Res. 12:593–622.
- Parlange, J.-Y., and J.F. Fleming. 1984. First integrals of the infiltration equation: I. Theory. Soil Sci. 137:391–394.
- Parlange, J.-Y., D. Lockington, G. Sander, W.L. Hogarth, D.A. Barry, L. Li, M.B. Parlange, and R.S. Govindaraju. 2001. Comment on an "Exact solution for horizontal redistribution by general similarity". Soil Sci. Soc. Am. J. 65:957–958 (this issue).[Free Full Text]
- Shao, M., and R. Horton. 1996. Soil water diffusivity determination by general similarity theory. Soil Sci. 161:727–734.
- Shao, M., and R. Horton. 1997. Reply to comment on soil water diffusivity determination by general similarity theory. Soil Sci. 162:769–770.
- Shao, M., and R. Horton. 1998. Integral method for estimating soil hydraulic properties. Soil Sci. Soc. Am. J. 62:585–592.[Abstract/Free Full Text]
- Shao, M., and R. Horton. 2000. Exact solution for horizontal redistribution by general similarity theory. Soil Sci. Soc. Am. J. 64:561–564.
- van Genuchten, M.Th. 1980. A closed form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J. 44:892–898.[Abstract/Free Full Text]
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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""
Soil Sci. Soc. Am. J.,
May 1, 2001;
65(3):
957 - 959.
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INTRODUCTION
REFERENCES
