MATHEMATICAL MODELLING OF WATER REGULATION PROCESSES ON DUAL-ACTION DRAINAGE SYSTEMS
Abstract
The solution of the problem of increasing water regulation areas in the Polissia zone of Ukraine requires investigation and development of new, more effective methods for determining structural parameters of drainage systems when developing projects for their reconstruction in accordance with the requirements aimed at ensuring water regulation during systems’ operation. The paper considers the problem of improving the efficiency of water regulation on dual-action drainage systems by using mathematical modelling tools to determine the structural parameters of the systems and the parameters of their operational management. The proposed means are based on the use of Richards equation stated in terms of water head. As a tool for scenario modelling, an initial-boundary value problem of modelling moisture transfer on dual-action systems is formulated and a finite-difference scheme for obtaining its numerical solution is given. We consider the problem of determining the depth of drains installation and the distance between them at which the system provides not only the drainage of soil’s surface layer, but also the maintenance of its moisture supply level in a given range with a minimum need for irrigation during the growing season. The algorithm for solving such a problem is presented. It is based on the construction of a set of admissible values of system’s parameters using, in particular, the bisection method, followed by the minimization of an objective function on this set. Under the conditions when the implementation of underground water supply technology is economically impractical, the possibility of supplementing the drainage system with an irrigation system is considered. In this case, the cost of building a drainage system and an additional irrigation system is a criterion for the optimality of system’s parameters. Additionally, we consider the problem of operational management of water regulation, i.e., the determination, given the initial distribution of moisture, of the optimal control influences necessary to ensure an acceptable level of moisture availability during a given period of time. This minimization problem is proposed to be solved by a genetic algorithm. The results of modelling the operation of a dual-action system and the optimization of its parameters under the conditions of drained peat soils of the Panfyly Research Station (Ukraine, Kyiv region) are presented.
References
2. Parlange, J.-Y., Hogarth, W.L., Barry, D.A., Parlange, M.B., Haverkamp, R., Ross, P.J., Steenhuis, T.S., DiCarlo, D.A., & Katul, G. (1999). Analytical approximation to the solutions of Richards' equation with applications to infiltration, ponding, and time compression approximation. Advances in Water Resources, 23(2), 189-194, https://doi.org/10.1016/S0309-1708(99)00022-6
3. Zlotnik, V.A., Wang, T., Nieber, J.L., & Šimunek, J. (2007). Verification of numerical solutions of the Richards equation using a traveling wave solution. Advances in Water Resources, 30(9), 1973-1980. https://doi.org/10.1016/j.advwatres.2007.03.008
4. Bomba, A., Tkachuk, M., Havryliuk, V., Kyrysha, R., Gerasimov, I., & Pinchuk, O. (2018). Mathematical modelling of filtration processes in drainage systems using conformal mapping. Journal of Water and Land Development, 39, 11–15. https://doi.org/10.2478/jwld-2018-0054
5. Garcia, L. A., Gillham, D., Patterson, D., & Harma, B. (2002). A decision support system for field drainage management. Energy. Climate, Environment and Water-Issues and Opportunities for Irrigation and Drainage, 49.
6. Romashchenko M. I., Bohaienko V. O., Matiash T. V., Kovalchuk V. P., Krucheniuk A. V. (2021). Numerical simulation of irrigation scheduling using fractional Richards equation. Irrigation Science, 39(3), 385-396.
7. Romashchenko, M., Bohaienko, V., Matiash, T., Sardak, A., & Nykytiuk, O. (2022). Usage of Mathematical Modelling for Minimizing Ecological Risks in Irrigation. In 16th International Conference Monitoring of Geological Processes and Ecological Condition of the Environment, 1, 1-5. EAGE Publications BV.
8. Bonyadi, M. R., & Michalewicz, Z. (2017). Particle swarm optimization for single objective continuous space problems: a review. Evolutionary Computation, 25 (1), 1–54. https://doi.org/10.1162/EVCO_r_00180
9. Maslov, B.S., Stankevitch, V.S., & Chernenok, V.Ia. (1981). Osushitel'no-uvlazhnitel'nyye sistemy. [Drainage-irrigation systems]. Moscow, Kolos [in Russian]
10. Dogan, A., & Motz, L.H. (2005). Saturated-Unsaturated 3D Groundwater Model. I: Development. Journal of Hydrologic Engineering, 10(6), 492-504. https://doi.org/10.1061/(ASCE)1084-0699(2005)10:6(492)
11. Van Genuchten, M.T. (1980). A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J., 44, 892±8.
12. Mualem, Y. (1976). A new model for predicting the hydraulic conductivity of unsaturated porous media. Water Resour. Res., 12 (3), 513–522. https://doi.org/10.1029/ WR012i003p00513
13. Shokri, A., & Bardsley, W. E. (2015). Enhancement of the Hooghoudt Drain-Spacing Equation. Journal of Irrigation and Drainage Engineering, 141(6), 04014070. https://doi.org/10.1061/(ASCE)IR.1943-4774.0000835
14. Romashchenko, M., Bohaienko, V., & Bilobrova, A. (2021). Dvovymirne matematychne modelyuvannya vodnoho rezhymu gruntu za kraplynnoho zroshennya [Two-dimensional mathematical modelling of soil water regime under drip irrigation]. Bulletin of Agricultural Science, 99(4), 59-66. https://doi.org/10.31073/agrovisnyk202104-08 [in Ukrainian]
15. Bohaienko, V., & Gladky, A. (2022). Multithreading performance simulating fractional-order moisture transport on AMD EPYC. Journal of Numerical and Applied Mathematics, 2, 174-182. https://doi.org/10.17721/2706-9699.2022.2.20
16. Goldberg, D.E. (1989). Genetic algorithms in search, optimization and machine learning. Boston, USA. Addison-Wesley Professional. ISBN:978-0-201-15767-3, p.372.
17. Van Dam, J.C., & Feddes, R.A. (2000). Numerical simulation of infiltration, evaporation and shallow groundwater levels with the Richards equation. Journal of Hydrology, 233(1), 72-85. https://doi.org/10.1016/S0022-1694(00)00227-4.
18. Molz, F.J., & Remson, I. (1970). Extraction term models of soil moisture use by transpiring plants. Water Resour. Res., 6, 1346-1356. https://doi.org/10.1029/WR006i005p01346.
19. Samarskii, A.A. (2001). The theory of difference schemes. New York (NY): CRC Press.
20. Freund, R.W. (1993). A transpose-free quasi-minimal residual algorithm for non-hermitian linear systems. SIAM J. Sci. Comput., 14(2), 470–482.
