Explicit Approximate Analytical Solution of the Horizontal Diffusion Equation

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 finite element methods, finite control volume method, and flux concentration method. This article produces an improved approximate analytical solution for nonlinear diffusion, in terms of the Boltzmann similarity variable, that has the advantages of being explicit, accurate, and relatively simple 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 proposed explicit approximate analytical solution has the Boltzmann transformation 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.