Conservation Laws for Equations Related to Soil Water Equations

where ψ is soil moisture pressure head, C(ψ) is specific water capacity, K(ψ) is unsaturated hydraulic conductivity, S(ψ) is a sink or source term, t is time, x is the horizontal axis, and z is the vertical axis which is considered positive downward. Equation (1.1) is an approximation within the context of the theory of mixtures. It is studied here in the form (1.1) as a first step. An introduction to the general theory of mixtures can be found in the book by Rajagopal and Tao [17] which contains references to the earlier papers of Truesdell that have had a great impact on the development of the theory. This approach is rather general and takes into account a variety of physical mechanisms and chemical reactions. Also of crucial importance within such a theory is the choice of boundary conditions and the problems associated with the choice of these conditions (see [6]). The book also provides numerous references on flow through porous media and the interested reader can consult the relevant references therein. Group classification of (1.1) with respect to admitted point transformation groups was done in [2]. Some exact solutions were also obtained in the paper just cited.