Introducing a Method to Reduce the Mass Balance Error in the Numerical Solution of the Richards Equation

Abstract
Background and Objectives
In many fields of water sciences, the moisture movement in the soil profile is very important. Due to the non-linearity of the governing equation in moisture movement in the soil profile, the comprehensive analytical solution of this equation is impossible, and there is a need to solve it numerically. The abundance of data and simulation steps reduces the flexibility in using existing software to solve the Richards equation and using this data as initial conditions. Therefore, having a numerical solution plan for the Richards equation will be very useful. Among the existing methods, the fully implicit finite difference method can continue the simulation unceasingly due to its unconditional stability in providing a non-iteration solution. However, using the finite difference method to solve the Richards equation leads to significant mass balance error. Some usual methods to reduce the mass balance error could not be applied to the non-reverse solution. By creating the Richards equation numerical solution in MATLAB software, this research introduces a method to reduce mass balance error, which is a simple method to apply to all kinds of numerical solutions.
Methodology
In this research, a Non-Iteration implicit solution for the Richards equation was developed. After creating the Richards equation numerical solution, the difference between the entering flux to the soil column and the added water to the same soil column, is added to each node to reduce the mass balance error. This amount is added to the nodes that their matrix pressure head difference (PD) compared to the initial state, is more than a threshold value. This amount applies as a coefficient proportional to the humidity of each node. To investigate the effect of various factors on the performance of the created numerical solution, a series of 11 experiments were designed with different characteristics in terms of infiltration rate, infiltration duration, duration of moisture redistribution in the soil profile, and initial volumetric water content. The optimal value for PD in each experiment was determined in such a way as to minimize the mass balance error. For the integrity of the results, the average PD values equal to 10 cm were used to simulate all cases. These series of experiments were simulated by two-dimensional HYDRUS software, and the results were used to verification of the created numerical solution.
Findings
The results showed that the most mass balance error is forced in the infiltration phase. Using the proposed method to preserve the mass balance caused a significant reduction in the mass balance error. The absolute value of the mass balance error varied in a narrow range and was always less than 3.5 percent. It is expected that time and space steps have a considerable effect on the mass balance error. The absolute mass balance error decreased linearly from 3.36 to 3.33, with the infiltration duration increasing from 10 to 30 minutes at a constant infiltration rate of 1 mm 〖min〗^(-1). The absolute mass balance error increased linearly from 3.36 to 3.37% by the initial volumetric water content increasing from 13 to 17% and decreased from 3.40 to 3.35 percent by the increase in the duration of the moisture redistribution in the soil profile from 300 to 900 minutes. Also, the Mean Absolute Relative error (MAE) increased from 1.17 to 9.53% by the infiltration time increasing from 10 to 30 minutes; the MAE increased from 1.17 to 8.89% by the infiltration rate increasing from 1 to 3 mm 〖min〗^(-1); and the MAE increased from 1.17 to 6.89% by the volumetric water content increasing from 13 to 17%.
Conclusion
The results showed that this method leads off the simulation to a significant reduction in the mass balance error; so that, in all experiment cases, the mass balance error ever fluctuated in a small range, less than 3.5%. The results comparison of the created simulation and the HYDRUS software showed that this method provided acceptable results, in cases such as summer rains where the infiltration duration is much less than the water redistribution time in the soil profile. Also, the highest value of the MAE in comparing the created simulations with the obtained results from the HYDRUS software is less than 10%. It proves a good agreement between the created simulation and the HYDRUS software.
Keywords: Finite difference, HYDRUS software, Implicitly solution, Non-iteration solution, Simulation.