Difference approximation of soil heat and water flow equations
To calculate the flow between two adjacent compartments, a finite difference approximation is made. The governing gradients of temperature and total water potential are calculated linearly between the mid points of consecutive compartments. The flow, q, is given by:
(8.14)
where i designates the layer number, Φ the appropriate potential, ∆z the layer thickness and k is conductivity as a function of soil moisture, θ.
For water flow the total potential is the sum of both matric potential and the gravity potential. The gravity potential is directed from the soil surface downwards which justify the use of a single ended approximation of the inter-block conductivity between compartments. Thus the water flow may be given as:
The numerical solution is sensitive to the choice of inter-block conductivity (Haverkamp & Vauclin, 1979). The solution used by the CoupModel is obtained by defining conductivity at the boundary between two bordering compartments. States, and parameters defining conductivities, are assumed to vary linearly between mid points of compartments. Water content at the boundary between two compartments is, thus, given by:
The only exception to this procedure is the gravity generated flow of water which is using the water content of the upper compartment instead of the boundary water content.