Some effects of linearizing the unsaturated soil moisture transfer diffusivity model

Five methods are evaluated for estimating the nonlinear diffusivity coefficient in a finite element Crank‐Nicolson analog of the diffusivity model for moisture transfer in an unsaturated horizontal homogeneous soil column. The Galerkin technique is utilized for the quasi‐linear spatial discretization problem, and the commonly used Crank‐Nicolson time advancement method is used with a diffusivity coefficient update method based upon (1) a constant diffusivity coefficient for all space and time and (2) a quasi‐constant diffusivity coefficient within each element during a Crank‐Nicolson time step. In the case of constant diffusivity coefficient problems a modified capacitance matrix scheme significantly improves results over the standard finite element capacitance matrix. For nonlinear problems, five methods of estimating quasi‐constant soil water diffusivity equivalents were evaluated. These are (1) a straight line average using element endpoint water content in a diffusivity function, (2) an integration of the diffusivity constant over each element, (3) an inverse diffusivity function integration over each element, (4) an evaluation of the diffusivity coefficient as the mean parabolic value over a specified subdomain of each element, and (5) an evaluation of the diffusivity coefficient for each element at one half of the Crank‐Nicolson time step, using method 4. Both the modified and unmodified finite element capacitance matrices were used for each method considered. The unmodified finite element capacitance matrices based upon values of diffusivity coefficients evaluated as the mean parabolic value of the state variable gave the superior results.