An upscaling approach to simulate unsaturated flow in horizontally heterogeneous soils at field scale under flood irrigation

An upscaling approach was developed for simulating soil water flow in horizontal heterogeneous unsaturated zone at field scale under flood irrigation. Based on the assumption of stream tube model and the van Genuchten-Mualem soil hydraulic function with five parameters Ks, [alpha], n, [theta]r and [theta]s, the Richards equation was transformed into a dimensionless form by using the dimensionless forms of temporal and spatial variables, and pressure head. Although a strong dependency of parameters Ks and [alpha] on scale does not exist in the transformed Richards equation, the parameter n, which is slightly dependent on scale, still exists in the transformed equation, and the parameter [alpha] is introduced in the transformed initial and boundary conditions. Therefore, a power law averaging technique was also included in our upscaling approach. Compared with traditional numerical methods, the distribution of pressure head of each soil column, and the mean and variance of soil water dynamics in all the soil columns can be obtained by numerically solving the transformed Richards equation only one time and by returning to the dimensionless variable expressions, while the traditional Richards equation must be solved once for every soil column. The new approach was calibrated by two numerical experiments, and the soil water content as a function of time and depth was reasonably well simulated for the two experiments which involved different soil textures. Different combinations of [alpha] and n were applied for comparing the accuracy of numerical simulations: n had little effect on the simulated results whereas [alpha] had some significant effect on the simulated results. To further verify the numerical efficiency of the new approach, we adopted the effective [alpha] values corresponding to the exponent p of the power law averaging technique [Eq. (16)] when p approaches 0, p = ± 1 and ± 0.5 to conduct more numerical experiments. It showed that with an initial pressure head profile of equilibrium for Example 1, and with an initial pressure head profile of constant for Example 2, the simulated results of the wetting front at the specific depths were in agreement with that of traditional numerical method, and the errors in simulated pressure heads were small. In summary, the proposed approach can be useful for upscaling unsaturated water flow characteristics in horizontally heterogeneous soils.