Uma solução para a equação de Richards em perfis de solos em camadas com a abordagem de domínio único
Resumo
O presente artigo tem como objetivo desenvolver uma solução para equação de Richards para o escoamento unidimensional de água através de perfis de solo em camadas, usando uma abordagem em domínio único. Foi proposto uma formulação matemática em que uma função de transição é inserida na modelagem para suavizar as descontinuidades presentes nas interfaces. O Método das Linhas (MOL), juntamente com um esquema de volumes finitos, é usado para resolver o problema. Uma comparação é realizada entre os resultados obtidos e aqueles da literatura para verificação do modelo. Para todos os casos estudados, efetuou-se uma análise de convergência e experimentos numéricos foram realizados para analisar as influências dos parâmetros físicos Ks e α. Resultados foram obtidos em termos de teor de umidade (θ) e convertidos em termos de pressão hidráulica (ψ), apresentando boa concordância com os valores da literatura, mostrando que a abordagem do problema em domínio único pode lidar com a descontinuidade presente na interface, e que a função de transição sugerida é um caminho viável para a solução da equação de Richards.
Referências
ALMEIDA, A.P.; NAVEIRA-COTTA, C.P.; COTTA, R.M. Transient Multidimensional Heat Conduction in Heterogeneous Me-dia: Integral Transform with Single Domain Formulation. Int. Comm. Heat & Mass Transfer, v.117, p. 104792 , 2020. https://doi.org/10.1016/j.icheatmasstransfer.2020.104792
BERARDI, M.; DIFONZO, F.; NOTARNICOLA, F.;, VURRO, M. A transversal method of lines for the numerical modeling of vertical infiltration into the vadose zone. Applied Numerical Mathematics, v. 135, p. 264-275, 2019. https://doi.org/10.1016/j.apnum.2018.08.013
BERARDI, M.; DIFONZO, F.;VURRO, M.; LOPEZ L. The 1D Ri-chards’ equation in two layered soils: a Filippov approach to treat discontinuities. Advances in water resources, v. 115, p. 264-272, 2018. https://doi.org/10.1016/j.advwatres.2017.09.027
BROADBRIDGE, P.; DALY, E.; GOARD, J. Exact Solutions of the Richards Equation with Nonlinear Plant-Root Extraction. Water Resources Research, v. 53, n. 11, p. 9679-9691, 2017. https://doi.org/10.1002/2017WR021097
BRUNONE, B.; FERRANTE, M.; ROMANO, N.; SANTINI, A. Nu-merical simulations of one-dimensional infiltration into layered soils with the Richards equation using different estimates of the interlayer conductivity. Vadose Zone Jour-nal, v. 2, n 2, p. 193-200, 2003. https://doi.org/10.2113/2.2.193
CELIA, A,; BOULOUTAS, T.; ZARBA, L. A general massa-conservative numerical solution for the unsaturated flow equation. Water Resources Research. v.26, n 7, p.1483-1496, 1990. https://doi.org/10.1029/WR026i007p01483
COTTA, R.M.; LISBOA. K.M.; ZOTIN, J.L.Z.; Integral transforms for flow and transport in continuum and discrete models of fractured heterogeneous porous media, Water Resources, v. 142, n. 2, p.1-17, 2020a. https://doi.org/10.1016/j.advwatres.2020.103621
COTTA, R.M.; NAVEIRA-COTTA,; C.P.; VAN GENUCHTEN, M.T.; SU, J.; QUARESMA, J.N.N.; Integral Transform Analysis of Radionuclides Migration in Variably Saturated Media with Physical Non-equilibrium Model: Application to Solid Waste Leaching in Uranium Mining. Annals of the Brazilian Aca-demy of Sciences, v.92, n.1, p. 1-28, 2020b. https://doi.org/10.1590/0001-3765202020190427
FARTHING, M.W.; OGDEN, F.L.; Numerical Solution of Ri-chards’ Equation: A Review of Advances and Challenges. Soil Science Society of America Journal, v. 81, n. 6, p. 1257-1269, 2017. https://doi.org/10.2136/sssaj2017.02.0058
GARDNER, W. Some steady-state solutions of the unsatura-ted moisture flow equation with application to evaporation from a water table. Soil science, v. 85, n. 4, p. 228-232, 1958. https://doi.org/10.1097/00010694-195804000-00006
HILSS R; PORRP I; HUDSON B; WIERENGA J. Modeling one-dimensional infiltration into very dry soils 1. Model deve-lopment and evaluation. Water Resources Research, v.25, n 6, p.1259-1269, 1989. https://doi.org/10.1029/WR025i006p01259
HIRATA, S.; GOYEAU, B.; GOBIN, D.; NAVEIRA-COTTA, C.; COT-TA, R. M. Linear stability of natural convection in superposed fluid and porous layers: Influence of the interfacial model-ling. International Journal of Heat and Mass Transfer. v.50, n 7-8, p.1356–1367, 2007. https://doi.org/10.1016/j.ijheatmasstransfer.2006.09.038
IMSL Library, MATH/LIB, Houston, TX, 1994.
KANZARI, S.; MARIEM, S. Kirchhoff Transformation of Ri-chards
Equation for Simulating Water Flow in Porous Media. System
Science and Apllied Mathematics, v.2, n 2, p. 8-12, 2017.
KNUPP, D.; COTTA, R.; NAVEIRA-COTTA, C.; KAKAÇ, S. Transi-ent conjugated heat transfer in microchannels: Integral transforms with single domain formulation. International Journal of Thermal Sciences, v. 88, p. 248-257, 2015. https://doi.org/10.1016/j.ijthermalsci.2014.04.017
LAIO, F.; PORPORATO, A.; RIDOLFI, L.; RODRIGUEZ I. Plants in water-controlled ecosystems: Active role in hydrologic pro-cesses and response to water stress. Adv. Water Resour. v. 24, n 7, p. 707–723, 2001. https://doi.org/10.1016/S0309-1708(01)00005-7
LIBARDI, P. L. Dinâmica da Água no Solo. v. 61. Edusp, 2005.
LISBOA, K.; SU, J.; COTTA, R. M. Single domain integral trans-form analysis of natural convection in cavities partially filled with heat generating porous medium. Numerical Heat Trans-fer, Part A: Applications. v.74, n 3, p. 1068-1086, 2018. https://doi.org/10.1080/10407782.2018.1511141
MALISKA, C. R. Transferência de calor e mecânica dos flui-dos computacional. 2. ed. Rio de Janeiro: Livros Técnicos e Científicos Editora, v. 200, 2004.
MÄNNICH, M. Desenvolvimento de soluções analítica e nu-mérica da equação de Richards. [Dissertação]. Curitiba (PR): Universidade Federal do Paraná; p.129, 2008.
MATTHEWS, C. J.; COOK, F. J.; KNIGHT, J. H.; BRADDOCK, R. D. Handling the water content discontinuity at the interface between layered soils within a numerical scheme. Soil Re-search, v. 43, n. 8, p. 945-955, 2006. https://doi.org/10.1071/SR05069
MATTHEWS, C.; BRADDOCK, D.; SANDER, G. Modeling Flow Through a One-Dimensional Multi-Layered Soil Profile Using the Method of Lines. Environmental Modeling & Asses-sment. v. 9, n. 2, p.103-113, 2004. https://doi.org/10.1023/B:ENMO.0000032092.10546.c6
ORGOGOZO, L.; RENON, N.; SOULAINE, C.; HÉNON, F.; TO-MER, S.K,; QUINTARD, M. An open source massively parallel solver for Richards equation: Mechanistic modelling of water fluxes at the watershed scale. Computer Physics Communi-cations, v. 185, n. 12, p. 3358-3371, 2014. https://doi.org/10.1016/j.cpc.2014.08.004
ÖZISIK, M.N.; ORLANDE H. R. B.; COLAÇO, M. J.; COTTA, R. M. Finite difference methods in heat transfer. 2. ed. New York: CRC Press. 2017. https://doi.org/10.1201/9781315168784
PHILIP J. Theory of Infiltration:1.the Infiltration Equation and Its Solution. Soil Science, v. 83, n. 5, p. 345-358, 1957.10. Hillel D. Environmental soil physics. San Diego: Academic Press; 1998.144 p. https://doi.org/10.1097/00010694-195705000-00002
RICHARDS LA. Capillary conduction of liquids through porous mediums. Physics Rev. v.54, n 6, p. 318-333, 1931. https://doi.org/10.1063/1.1745010
ROMANO, N.; BRUNONE, B.; SANTINI A. Numerical analysis of one-dimensional unsaturated flow in layered soils. Advances in Water Resources. v. 21, n. 4, p. 315-324, 1998. https://doi.org/10.1016/S0309-1708(96)00059-0
ROMANO, N.; BRUNONE, B.; SANTINI, A. Numerical analysis of one-dimensional unsaturated flow in layered soils. Ad-vances in Water Resources. v.21, n 4, p.315-324, 1998. https://doi.org/10.1016/S0309-1708(96)00059-0
SCHIESSER, William E.; GRIFFITHS, Graham W. A compen-dium of partial differential equation models: method of lines analysis with Matlab. Cambridge University Press, 2009. https://doi.org/10.1017/CBO9780511576270
SIMUNEK, J.; VAN GENUCHTEN, M.; SEJNA M. The HYDRUS software package for simulating the two-and three-dimensional movement of water, heat, and multiple solutes in variably saturated media. Technical manual, version, v. 1, p. 241, 2006.
SRIVASTAVA, R.; YEH, T. Analytical solutions for one-dimensional, transient infiltration toward the water table in homogeneous and layered soils. Water Resources Research, v. 27, n 5, p.753-762,1991. https://doi.org/10.1029/90WR02772
VERSTEEG, Henk Kaarle; MALALASEKERA, Weeratunge. An introduction to computational fluid dynamics: the finite volume method. Pearson education, 2007.
WENDLAND, E.; PIZARRO, M. Modelagem computacional do fluxo unidimensional de água em meio não saturado do solo. Engenharia Agrícola, Associação Brasileira de Enge-nharia Agrícola, v.30, n 3, p 424-434, 2010. https://doi.org/10.1590/S0100-69162010000300007
YEH, G.T.; SHIH,D.S.; CHENG, J.R.C. An integrated media, integrated processes watershed model. Computers and Fluids. Computers & Fluids, v. 45, n. 1, p. 2-13, 2011. https://doi.org/10.1016/j.compfluid.2010.11.018
ZHA, Y.; YANG, J.; ZENG, J.; TSO, C. H.; ZENG, W.; SHI L. Review of numerical solution of Richardson-Richards equa-tion for variably saturated flow in soils. Wiley Interdiscipli-nary Reviews: Water. v.6, n 5, p.1-23, 2019. https://doi.org/10.1002/wat2.1364
ZHA, Y.; YANG, J.; SHI, L.; SONG X. Simulating One-Dimensional Unsaturated Flow in Heterogeneous Soils with Water Content-Based Richards Equation. Vodose Zone Jour-nal, v. 12, n 2, p.1-13, 2013. https://doi.org/10.2136/vzj2012.0109
ZHAN, T. L.T.; NG, C.W.W. Analytical analysis of rainfall infil-tration mechanism in unsaturated soils. International Jour-nal of Geomechanics, v. 4, n. 4, p. 273-284, 2004. https://doi.org/10.1061/(ASCE)1532-3641(2004)4:4(273)
ZHU, H.; LIU, T.; XUE, B.; YINGLAN, A.; WANG G. Modified Richrads’Equation to Improve Estimates of Soil Moisture in Two-Layered Soils after Infiltration. Water Rev, v.10, n 9, p.1174-1190, 2018. https://doi.org/10.3390/w10091174
