RICHARDS–KLUTE EQUATION: THE STATE OF THE ART
The article is dedicated to the Richards–Klute equation. A derivation of this equation and several forms of its notation are given. Analytical methods for solving the equation are analyzed. The current state and directions of theoretical research are described. The main numerical methods for solving the equation are presented and the methods of time and space discretization used in them are analyzed. The list of programs for numerical modeling of the Richards–
Klute equation is given. Their comparative analysis was carried out. Possible areas of further research are mentioned.
Richardson L. F. Weather prediction by numerical process. University Press, Cambridge. 1922. p. 262. https://doi.org/10.1002/qj.49704820311
Richards L. Capillary conduction of liquids through porous mediums. Physics. 1931. Vol. 1(5). P. 318–333. https://doi.org/10.1063/1.1745010
Darcy H. The public fountains of the city of Dijon: Exposition and application of the principles to be followed and the formulas to be employed in questions of water distribution; work ended with an appendix relating to the water supplies of several cities to the filtering of water and the manufacture of cast iron, lead, sheet metal and bitumen pipes. Paris: V. Dalmont, 1856. 647 p. (in French)
Bahvalov N. S., Panasenko G. P. Averaging processes in periodic media. Moscow: Nauka, 1984. P. 164–169. (in Russian)
Sanchez-Palencia E. Non homogeneous media and vibration theory. Lecture notes in Physics. 1980. Vol. 127. 398 p.
Belyaev A. Y. Averaging in filtration theory problems. Moscow: Nauka, 2004. P. 76–127. (in Russian)
Leontyev N. E. Filtration theory basics. Moscow: Izd-vo CPI pri mechanikomatematicheskom facultete MGU, 2009. P. 24–29. (in Russian)
List F., Radu F. A. A study on iterative methods for solving Richards’ equation. Comput. Geosci. 2016. Vol. 20. P. 341–353. https://doi.org/10.1007/s10596-016-9566-3
Rubinstein L. I. Stefan problem. Riga: Zwaigzne, 1967. 458 p. (in Russian)
Alt H. W., Luckhaus S. Quasilinear elliptic-parabolic differential equations. Math. Z. 1983. Vol. 183. No. 1. P. 311–341.
Van Duyn C. J., Peletier L. A. Nonstationary filtration in partially saturated porous media. Arch. Rational Mech. Anal. 1982. Vol. 78. No. 2. P. 173–198.
Van Duyn C. J. Nonstationary filtration in partially saturated porous media: contunuity of the free boundary. Arch. Rational Mech. Anal. 1982. Vol. 79. No. 3. P. 261–265.
Van Duyn C. J., Hulshof J. An elliptic-parabolic with a nonlocal boundary condition. Arch. Rational Mech. Anal. 1987. Vol. 99. No. 1. P. 61–73.
Bertsch M., Hulshof J. Regularity results for an elliptic-parabolic free boundary problem. Trans. Amer. Math. Soc. 1986. Vol. 297. No. 1. P. 337–350.
Di Benedetto E., Gariepy R. Local behavior of solutions of an elliptic-parabolic equation. Arch. Rational. Mech. Anal. 1987. Vol. 97. No. 1. P. 1–17.
Hulshof J., Peletier L. A. An elliptic-parabolic free boundary problem. Nonlinear Anal: Theory, Method Appl. 1986. Vol. 10. No. 12. P. 1327–1346.
Hulshof J. An elliptic-parabolic free boundary problem: continuity of the interface. Proc. Royal Soc. Edinburg. 1987. Vol. 106A. No. 3. P. 327–339.
Chen X., Friedman A., Kimura T. Nonstationary filtration in partially saturated porous media. Eur. J. Appl. Math. 1994. Vol. 5. No. 3. P. 405–429.
Mannucci P., Vazquez J. L. Viscosity solutions for elliptic-parabolic problems. Nonlinear Differ. Equ. Appl. 2007. Vol. 14. No. 1–2. P. 75–90.
Degtyarev S. P. Elliptic-parabolic equation and the corresponding free boundary problem I: Elliptic problem with a parameter. Ukr. Math. Vystnyk. 2014. Vol. 11. No. 1. P. 15–48. (in Russian)
Degtyarev S. P. Elliptic-parabolic equation and the corresponding free boundary problem II: smooth solution. Ukr. Math. Vystnyk. 2014. Vol. 11. No. 4. P. 447–479. (in Russian)
Bazaliy B. V., Degtyarev S. P. On the classical solvability of the multidimensional Stefan problem in the case of convective motion of a viscous incompressible fluid. Mat. Sbornyk. 1987. Vol. 132(174). No. 1. P. 3–19. (in Russian)
Bazaliy B. V., Degtyarev S. P. Solvability of a problem with an unknown boundary between the domains of parabolic and elliptic equations. Ukr. Mat. Zhurnal. 1989. Vol. 41. No. 10. P. 1343–1349. (in Russian)
Bazaliy B. V., Degtyarev S. P. On the Stefan problem with kinematic and classical conditions on the free boundary. Ukr. Mat. Zhurnal. 1992. Vol. 44. No. 2. P. 155–166. (in Russian)
Tymoshenko A. A. Optimal point control of mass transfer in porous media. PhD Dissertation, Taras Shevchenko National University of Kyiv. 2021. 149 p. (in Ukrainian)
Berninger H., Loisel S., Sander O. The 2-Lagrange multiplier method applied to nonlinear transmission problems for the Richards equation in heterogeneous soil with cross points. SIAM Journal of. Scientific Computing. 2014. Vol. 36. № 5. P. 2166–2198.
Pop I. S., Schweizer B. Regularization schemes for degenerate Richards equations and outflow conditions. Mathematical Models and Methods in the Applied Sciences. 2011. Vol. 21. № 8. P. 1685–1712.
Zha Y., Shi L., Ye M., Yang J. A generalized Ross method for two- and threedimensional variably saturated flow. Advances in Water Resources. 2013. Vol. 54(4). P. 67–77. https://doi.org/10.1016/j.advwatres.2013.01.002
Williams G. A., Miller C. T., Kelley C. T. Transformation approaches for simulating flow in variably saturated porous media. Water Resources Research. 2000. Vol. 36(4). P. 923–934. https://doi.org/10.1029/1999WR900349
Kostyerina E. A., Lapin A. V. Solving the problem of saturated-unsaturated fluid filtration in soil with monitoring of the saturation front. Isv. Vuzov. Matem. 1995. Vol. 6. P. 42–50. (in Russian)
Rogers C., Stallybrass M. P., Clements D. L. On two phase filtration under gravity and with boundary infiltration: Application of a Backlund transformation. Nonlin. Anal. Theory Meth. Appl. 1983. Vol. 7(7). P. 785–799. https://doi.org/10.1016/0362-546X(83)90034-2
Broadbridge P., White I. Modelling solute transport, chemical adsorption and cation exchange. Int. Hydrology and Water Resources Symp. Nat. Conf. Publ. No. 92/19 (Preprints of Papers. 1988. Vol. 3. P. 924–929).
Barry D. A., Sander G. C. Exact solutions for water infiltration with an arbitrary surface flux or nonlinear solute adsorption. Water Resour. Res. 1991. Vol. 27. P. 2667–2680. https://doi.org/10.1029/91WR01445
Ross P. J., Parlange J.-Y. Comparing exact and numerical solutions of Richards’ equation for one-dimensional infiltration and drainage. Soil Sci. 1994. Vol. 157(6). P. 341–344. https://doi.org/10.1097/00010694-199406000-00002
Pullan A. The quasilinear approximation for unsaturated porous media flow. Water Resources Research. 1990. Vol. 26(6). P. 1219–1234. https://doi.org/10.1029/WR026i006p01219
Zlotnik V. A., Wang T., Nieber J. L., Simunek J. Verification of numerical solutions of the Richards equation using a traveling wave solution. Advances in Water Resources. 2007. Vol. 30. P. 1973–1980. https://doi.org/10.1016/j.advwatres.2007.03.008
De Luca D. L., Cepeda J. M. Procedure to obtain analytical solutions of one-dimensional Richards’ equation for infiltration in twolayered soils. Journal of Hydrologic Engineering. 2016. Vol. 21(7). https://doi.org/10.1061/(ASCE)HE.1943-5584.0001356
Barry D., Parlange J., Sander G., Sivaplan M. A class of exact solutions for Richards’ equation. Journal of Hydrology. 1993. Vol. 142(1–4). P. 29–46. https://doi.org/10.1016/0022-1694(93)90003-R
Menziani M., Pugnaghi S., Vincenzi S. Analytical solutions of the linearized Richards equation for discrete arbitrary initial and boundary conditions. Journal of Hydrology. 2007. Vol. 332(1–2). P. 214–225. https://doi.org/10.1016/j.jhydrol.2006.06.030
Yuan F., Lu Z. Analytical solutions for vertical flow in unsaturated, rooted soils with variable surface fluxes. Vadose Zone Journal. 2005. Vol. 4. P. 1210–1218. https://doi.org/10.2136/vzj2005.0043
Tracy F. T. Clean two- and three-dimensional analytical solutions of Richards’ equation for testing numerical solvers. Water Resources Research. 2006. Vol. 42(8). P. 1–11. https://doi.org/10.1029/2005WR004638
Chen J. M., Tan Y. C., Chen C. H. Multidimensional infiltration with arbitrary surface fluxes. Journal of Irrigation and Drainage Engineering. 2001. Vol. 127(6). P. 370–377. https://doi.org/10.1061/(ASCE)0733-9437(2001)127:6(370)
Broadbridge P., Daly E., Goard J. Exact solutions of the Richards equation with nonlinear plant-root extraction. Water Resources Research. 2017. Vol. 53. P. 9679–9691. https://doi.org/10.1002/2017WR021097
Vabisshevich P. N. Numerical solution of the problem of identification of the right side of the parabolic equation. Izvestiya vysshyh uchebnyh zavedeniy. Matematika. 2003. No. 1. P. 29–37. (in Russian)
Lyashko S. I., Klyshin D. A., Semenov V. V., Shevchenko K. V. Lagrange-Euler approach to solving the inverse problem of convective diffusion. Dopovidi NAN Ukrainy. 2007. No. 10. P. 38–43. (in Ukrainian)
Lions J.-L. Optimal control of systems described by partial differential equations. Moscow: MIR, 1972. 416 p.
Lyashko S. I. Generalized control of linear systems. Kyiv: Naukova Dumka, 1998. 470 p.
Paronetto F. Existence results for a class of evolution equations of mixed type. J. Funct. Anal. 2004. Vol. 212(2). P. 324–356.
Paronetto F. G-convergence of mixed type evolution operators. J. Math. Pures Appl. 2010. Vol. 9(93). P. 361–407.
Paronetto F. A Harnack’s inequality for mixed type evolution equations. J. Differ. Equ. 2016. Vol. 260. P. 5259–5355.
Paronetto F. Further existence results for elliptic–parabolic and forward–backward parabolic equations. Calc. Var. 2020. Vol. 59(137). https://doi.org/10.1007/s00526-020-01793-7
Farthing M. W., Ogden F. L. Numerical solution of Richards’ equation: A review of advances and challenges. Soil Science Society of America Journal. 2017. Vol. 81(6). P. 1257–1269. https://doi.org/10.2136/sssaj2017.02.0058
Zha Y., Yang J., Zeng J., Tso C.-H. M., Zeng W., Shi L. Review of numerical solution of Richardson–Richards equation for variably saturated flow in soils. WIREs Water. 2019. Vol. 6. p. e1364. https://doi.org/10.1002/wat2.1364
Shahraiyni H. T., Ataie-Ashtiani B. Mathematical forms and numerical schemes for the solution of unsaturated flow equations. Journal of Irrigation and Drainage Engineering. 2012. Vol. 138(1). P. 63–72. https://doi.org/10.1061/(asce)ir.1943-4774.0000377
Hills R., Porro I., Hudson D. B., Wierenga P. J. Modeling onedimensional infiltration into very dry soils: 1. Model development and evaluation. Water Resources Research. 1989. Vol. 25(6). P. 1259–1269. https://doi.org/10.1029/WR025i006p01259
Celia M., Bouloutas E., Zarba R. A general mass-conservative numerical solution for the unsaturated flow equation. Water Resources Research. 1990. Vol. 26(1). P. 1483–1496. https://doi.org/10.1029/WR026i007p01483
Zha Y., Yang J., Yin L., Zhang Y., Zeng W., Shi L. A modified Picard iteration scheme for overcoming numerical difficulties of simulating infiltration into dry soil. Journal of Hydrology. 2017. Vol. 551. P. 56–69. https://doi.org/10.1016/j.jhydrol.2017.05.053
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. Riverside: University of California Riverside, 2006.
Scudeler C., Putti M., Paniconi C. Mass-conservative reconstruction of Galerkin velocity fields for transport simulations. Advances in Water Resources. 2016. Vol. 94. P. 470–485.
Mostaghimi P. et al. Anisotropic Mesh Adaptivity and Control Volume Finite Element Methods for Numerical Simulation of Multiphase Flow in Porous Media. Mathematical Geosciences. 2015. Vol 47. No. 4. P. 417–440.
Lai W., Ogden F. L. A mass-conservative finite volume predictor – corrector solution of the 1D Richards’ equation. Journal of Hydrology. 2015. Vol. 523. P. 119–127.
Pop I. S., Radu F., Knabner P. Mixed finite elements for the Richards’ equation: linearization procedure. Journal of Computational and Applied Mathematics. 2004. Vol. 168. P. 365–373.
Dogan A., Motz L. H. Saturated-unsaturated 3D groundwater model. I: Development. Journal of Hydrologic Engineering. 2005. Vol. 10(6). P. 492–504. https://doi.org/10.1061/(ASCE)1084-0699(2005)10:6(492)
Lipnikov K., Moulton D., Svyatskiy D. New preconditioning strategy for Jacobianfree solvers for variably saturated flows with Richards’ equation. Advances in Water Resources. 2016. Vol. 94. P. 11–22.
Zha Y. et al. Comparison of noniterative algorithms based on different forms of Richards’ equation. Environmental Modelling Assessment. 2016. Vol. 21. No. 3. P. 357–370.
Zeng J., Zha Y., Yang J. Switching the Richards’ equation for modeling soil water movement under unfavorable conditions. Journal of Hydrology. 2018. Vol. 563. P. 942–949.
Klyushin D. A., Onotskiy V. V. Numerical modeling of three-dimensional moisture transfer under microirrigation. Zhurnal Obchyslyuvalnoyi ta Prykladnoyi Matematyky. 2016. No. 1. P. 54–64. (in Ukrainian)
Caviedes-Voullieme D., Garcia-Navarro P., Murillo J. Verification, conservation, stability and efficiency of a finite volume method for the 1D Richards equation. Journal of Hydrology. 2013. Vol. 480. P. 69–84. https://doi.org/10.1016/j.jhydrol.2012.12.008
Svyatskiy D., Lipnikov K. Second-order accurate finite volume schemes with the discrete maximum principle for solving Richards’ equation on unstructured meshes. Advances in Water Resources. 2017. Vol. 104. P. 114–126. https://doi.org/10.1016/j.advwatres.2017.03.015
Li H., Farthing M. W., Miller C. T. Adaptive local discontinuous Galerkin approximation to Richards’ equation. Advances in Water Resources. 2007. Vol. 30(9). P. 1883–1901. https://doi.org/10.1016/j.advwatres.2007.02.007
Arbogast T. An error analysis for Galerkin approximations to an equation of mixed ellipticparabolic type. Technical Report TR90-33, Department of Computational and Applied Mathematics. Rice University, Houston, TX. 1990. 28 p.
Miller C., Abhishek C., Farthing M. A spatially and temporally adaptive solution of Richards’ equation. Advances in Water Resources. 2006. Vol. 29(4). P. 525–545. https://doi.org/10.1016/j.advwatres.2005.06.008
Vauclin M., Khanji D., Vachaud G. Experimental and numerical study of a transient, two-dimensional unsaturated-saturated water table recharge problem. Water Resources Research. 1979. Vol. 15(5). P. 1089–1101. https://doi.org/10.1029/WR015i005p01089
Shen C., Phanikumar M. S. A process-based, distributed hydrologic model based on a large-scale method for surface–subsurface coupling. Advances in Water Resources. 2010. Vol. 33(12). P. 1524–1541. https://doi.org/10.1016/j.advwatres.2010.09.002
Twarakavi N. K. C., Simunek J., Seo S. Evaluating interactions between groundwater and vadose zone using the HYDRUS-based flow package for MODFLOW. Vadose Zone Journal. 2008. Vol. 7(2). P. 757–768. https://doi.org/10.2136/vzj2007.0082
Xu X., Huang G., Zhan H., Qu Z., Huang Q. Integration of SWAP and MODFLOW-2000 for modeling groundwater dynamics in shallow water table areas. Journal of Hydrology. 2012. Vol. 412-413. P. 170–181. https://doi.org/10.1016/j.jhydrol.2011.07.002
Jansson P.-E., Karlberg L. Coupled Heat and Mass Transfer Model for Soil-Plant Atmosphere Systems. Royal Institute of Technology. Stockholm. 2010. 484 p.
Jansson P.-E. CoupModel: Model Use, Calibration, and Validation. ASABE. 2012. Vol. 55(4). P. 1337-1346. https://www.coupmodel.com
Sejna M., Simunek J., van Genuchten M. Th. The HYDRUS Software Package for Simulating One-, Two- and Three-Dimensional Movement of Water, Heat, and Multiple Solutes in Variably-Saturated Porous Media, User Manual, Version 5.0, PC Progress, Prague, Czech Republic. 2022. 348 p.
Aquanty Inc. HGS user manual. Waterloo, ON: Aquanty Inc. 2015.
Hammond G. E., Lichtner P. C., Mills R. T. Evaluating the performance of parallel subsurface simulators: An illustrative example with PFLOTRAN. Water Resources Research. 2014. Vol. 50(1). P. 208–228. https://doi.org/10.1002/2012WR013483
Hsieh P. A., Wingle W., Healy R. W. VS2DI—A graphical software package for simulating fluid flow and solute or energy transport in variably saturated porous media. U.S. Geological Survey Water-Resources Investigations Report 99-4130. Lakewood, CO. 2000.
Healy R. W. Simulating water, solute, and heat transport in the subsurface with the VS2DI software package. Vadose Zone J. 2008. Vol. 7. P. 632–639. https://doi.org/10.2136/vzj2007.0075
Howington S. E., Berger R. C., Hallberg J. P., Peters J. F., Stagg A. K., Jenkins E. W., Kelley C. T. A model to simulate the interaction between groundwater and surface water. ADA451802. US Army Engineering Research and Development Center, Vicksburg, MS. 1999. P. 1–12.
Camporese M., Paniconi C., Putti M., Orlandini S. Surface-subsurface flow modeling with path-based runoff routing, boundary condition-based coupling, and assimilation of multisource observation data. Water Resour. Res. 2010. Vol. 46. https://doi.org/10.1029/2008WR007536.
Dirersch H.-J. G. FEFLOW 5.1 user’s manual. Berlin, Germany: WASY Institute for Water Resources Planning and Systems Research Ltd. 2009.
Yeh G.-T., Shih D.-S., Cheng J.-R. C. An integrated media, integrated processes watershed model. Comput. Fluids. 2011. Vol. 45(1). P. 2–13. https://doi.org/10.1016/j.compfluid.2010.11.018
Klyushin D. A., Tymoshenko A. A. Computer program "Algorithm for optimizing the power of point sources in a two-dimensional porous medium": pat. 106332, Ukraine. Registr. Date: 16.07.2021. Publ. Date: 30.09.2021. Bulet. No. 66.