Explicit Approximate Analytical Solution of the Horizontal Diffusion Equation Christos Tzimopoulos, 1 Chris Evangelides, 1 and George Arampatzis 2 Abstract: The solution of unsaturated flow is a never-ending quest for many scientists. Many methods exist with their corresponding advantages and disadvantages, such as semianalytic methods, finite difference and fi- nite element methods, finite control volume method, and flux concentra- tion method. This article produces an improved approximate analytical solution for nonlinear diffusion, in terms of the Boltzmann similarity vari- able, that has the advantages of being explicit, accurate, and relatively sim- ple to evaluate. It is assumed that the diffusivity can be described with an exponential function, the profiles of soil water content are of finite extent, the concentration at the boundaries is constant, and the reduced flux of Philip (1973) is of the form of Vauclin and Haverkamp (1985). The pro- posed explicit approximate analytical solution has the Boltzmann trans- formation as the dependent variable and the soil water moisture as the independent variable. The solution is presented in normalized form as a function of normalized diffusivity and normalized soil moisture. It is tested with 12 soils and shows an excellent agreement with Philip’s method. Key Words: Moisture profile, diffusivity, sorptivity, vadose zone (Soil Sci 2015;180: 47–53) I nfiltration of soil water is an important process in the field water cycle. Knowledge of sorptivity is also required to determine whether water can propagate into the soil and reach the root zone. The nonlinear diffusion equation is important for the formulation of many problems concerning unsaturated horizontal flow in po- rous media and the horizontal transfer of heat: ∂θ ∂t ¼ ∂ ∂x D ∂θ ∂x   ; ð1Þ subject to the conditions: θ¼θ r ; x>0; t¼0; θ¼θ s ; x¼0; t >0; ð1aÞ where D = D(θ), θ r = the residual water content, and θ s = the water content at saturation. Many attempts have been made to solve this equation through analytical methods. The pioneer was Storm (1951), who solved a more complicated type of diffusion equation, with differ- ent forms of the coefficients: S θ ðÞ ∂ θ ∂t ¼ ∂ ∂x K θ ðÞ ∂ θ ∂x   ; ð2Þ which after certain transformations became a linear equation. Later, Knight and Philip (1974) arrived at the same linear equa- tion and showed that the solution of Storm also solved the usual nonlinear diffusion equation with the Fujita diffusivity. Fujita (1952a), arrived to an analytical solution of Equation (1) for the following D(θ) function: D¼ D0 ðÞ 1-λθ ; ð3Þ where λ is a parameter and D(0) is the diffusivity for θ = 0 at x > 0 and t = 0. This form of the diffusion coefficient, when λ is prop- erly chosen, gives a graph showing a very rapid increase of D in a considerably narrow range of θ. Later, Fujita (1952b, 1954) pre- sented two other forms of diffusivity: D¼ D0 ðÞ 1-λθ ð Þ 2 ; D¼ D0 ðÞ 1þ2aθþbθ 2   ð4Þ which lead to an exact, rather than approximate, solution that is still easy to evaluate, but it is a parametric solution, not an explicit one. In 1960, Philip presented a general method of obtaining exact solutions, making the substitution φ = xt - 1/2 (Boltzmann transfor- mation) in Equation (1) obtaining the transformed equation: d dφ D dθ dφ   þ φ 2 dθ dφ ¼0; θ r <θ<θ s : ð5Þ Philip (1960) presented a group of exact solutions for the transformed profile φ for 14 different forms of diffusivity D(θ). Babu (1976a) presented a perturbation method in the special case of an exponential function of the diffusivity under the following assumptions: a) the concentration at the boundaries is constant, and b) the profiles of soil water content have finite extent; in other words, he accepted no moisture flow beyond a distance μ, desig- nated as the wetting front. Babu (1976b) applied the perturba- tion method under the general diffusivity function, subject again to constant boundary concentration. His solution presented the Bolztmann transformation as the dependent variable and the water content as the independent variable. He presented the solution in terms of integrals involving the diffusivity function. Parlange and Babu (1976) proved that Babu’s perturbation solution becomes identical to an earlier iterative solution, using Gisler’s correction (Gisler, 1974). Liu (1976) developed a perturbation solution using the concept of diffusion front and presented results for two specific diffusivities: a) diffusivity as a power function of the concentration, and b) diffusivity as one of the above-mentioned exact solutions. 1 Department of Hydraulics and Transportation Engineering, Aristotle Univer- sity of Thessaloniki, Thessaloniki, Greece. 2 Hellenic Agricultural Organization, Land Reclamation Institute, GR-57400, Sindos, Greece. Address for correspondence: Chris Evangelides, PhD, Aristotle University of Thessaloniki, Department of Hydraulics and Transportation Engineering, GR-54124 Thessaloniki, Greece. E-mail: evan@vergina.eng.auth.gr Financial Disclosures/Conflicts of Interest: None reported. Received August 21, 2014. Accepted for publication March 23, 2015. Copyright © 2015 Wolters Kluwer Health, Inc. All rights reserved. ISSN: 0038-075X DOI: 10.1097/SS.0000000000000113 TECHNICAL ARTICLE Soil Science • Volume 180, Number 2, February 2015 www.soilsci.com 47 Copyright © 2015 Wolters Kluwer Health, Inc. All rights reserved.