VALIDATION OF THE KOOL AND PARKER CONSTITUTIVE MODEL IN HYDRUS-2D: DEPTH-DEPENDENT HYDRAULIC HYSTERESIS AND DRAINAGE DYNAMICS IN UNSATURATED SOILS
DOI:
https://doi.org/10.11113/aej.v16.24826Keywords:
Hydraulic hysteresis, Constitutive model, Volumetric water content, Suction dynamics, HYDRUS-2DAbstract
Hydraulic hysteresis in unsaturated soils significantly influences subsurface flow dynamics, yet its integration into hydrological models remains limited. This study advances the characterization of hysteresis effects by validating the Kool and Parker (1987) constitutive model within the HYDRUS-2D framework, while systematically comparing hysteresis-inclusive and hysteresis-ignoring simulations in a clay loam soil column. Through two-dimensional unsaturated flow simulations, we quantify depth-dependent suction and volumetric water content (VWC) dynamics under controlled boundary conditions, revealing critical insights into drainage behavior. Results demonstrate that exponential decay and power-law models robustly describe suction evolution (R² > 0.95), with deeper layers exhibiting faster drainage rates (e.g., −0.095 cm/min at 51.17 cm depth) due to proximity to the seepage face and steeper hydraulic gradients. Neglecting hysteresis overestimates suction magnitudes by 10–20% and underestimates equilibrium VWC, particularly in shallow zones where capillary retention dominates. Statistically significant discrepancies (RMSE up to 0.48 cm; p < 0.05) highlight hysteresis-induced delays in drainage, with shallower layers retaining 4–9% higher moisture under cyclic conditions. Methodologically, the integration of depth-specific metrics and hysteresis parameters enhances model fidelity, capturing non-unique soil-water retention relationships critical for real-world applications. Practical implications for slope stability and irrigation planning are emphasized: overestimating suction decline in hysteresis-ignoring models may compromise infrastructure safety assessments, while delayed drainage predictions improve water management strategies in fine-textured soils. This work bridges theoretical and computational hydrology, advocating for hysteresis-aware models to refine predictions in heterogeneous, unsaturated environments. The findings underscore the necessity of vertical stratification and hysteresis algorithms in hydrological simulations, offering a validated framework for geotechnical engineers and earth scientists addressing dynamic soil-water interactions.
References
Al-Soudany, K. Y. H., Fattah, M. Y., & Rahil, F. H. 2024. Enhancement of Expansive Soil Properties by Water Treatment Sludge Ash in Landfill Liners. Civil Engineering Journal, 10(11): 3508–3530. DOI: https://doi.org/10.28991/cej-2024-010-11-04.
Baum, R. L., Godt, J. W., & Savage, W. Z. 2010. Estimating the timing and location of shallow rainfall-induced landslides using a model for transient, unsaturated infiltration. Journal of Geophysical Research: Earth Surface, 115(F3): 1-26. DOI: https://doi.org/10.1029/2009JF001321.
Beriozkin, A., & Mualem, Y. 2018. Comparative analysis of the apparent saturation hysteresis approach and the domain theory of hysteresis in respect of prediction of scanning curves and air entrapment. Advances in Water Resources, 115: 253-263. DOI: https://doi.org/10.1016/j.advwatres.2018.01.016.
Beville, S. H., Mirus, B. B., Ebel, B. A., Mader, G. G., & Loague, K. 2010. Using simulated hydrologic response to revisit the 1973 Lerida Court landslide. Environmental Earth Sciences, 61(6): 1249-1257. DOI: https://doi.org/10.1007/s12665-010-0448-z.
Bikerman, J. J. 1974. Capillary Attraction and Hysteresis of Wetting. The Journal of Adhesion, 6(4): 331-336. DOI: https://doi.org/10.1080/00218467408075036.
Bond, W. J., & Collis-George, N. 1981. Ponded infiltration into simple soil systems: 1. The saturation and transition zones in the moisture content profiles. Soil Science, 131(4): 202-209. DOI: https://doi.org/10.1097/00010694-198104000-00002.
Cihan, A., Birkholzer, J., Illangasekare, T. H., & Zhou, Q. 2014. A modeling approach to represent hysteresis in capillary pressure-saturation relationship based on fluid connectivity in void space. Water Resources Research, 50(1): 119-131. DOI: https://doi.org/10.1002/2013WR014280.
Chen, P., Mirus, B., Lu, N., & Godt, J. W. 2017. Effect of hydraulic hysteresis on stability of infinite slopes under steady infiltration. Journal of Geotechnical and Geoenvironmental Engineering, 143(9): 04017041. DOI: https://doi.org/10.1061/(ASCE)GT.1943-5606.0001724.
Chen, P., & Wei, C. 2016. Numerical procedure for simulating the two phase flow in unsaturated soils with hydraulic hysteresis. International Journal of Geomechanics, 16(1): 04015030. DOI: https://doi.org/10.1061/(ASCE)GM.1943-5622.0000505.
Chen, P., Wei, C., Liu, J., & Ma, T. 2013. Strength theory model of unsaturated soils with suction stress concept. Journal of Applied Mathematics, 2013: 1-11. DOI: https://doi.org/10.1155/2013/756854.
Chen, P., Wei, C., & Ma, T. 2015. Analytical model of soil-water characteristics considering the effect of air entrapment. International Journal of Geomechanics, 15(6): 04014102. DOI: https://doi.org/10.1061/(ASCE)GM.1943-5622.0000462.
Chen, P., & Wei, C. 2016. Numerical Procedure for Simulating the Two-Phase Flow in Unsaturated Soils with Hydraulic Hysteresis. International Journal of Geomechanics, 16(1): 04015030. DOI: https://doi.org/10.1061/(ASCE)GM.1943-5622.0000505.
Ebel, B. A., Loague, K., & Borja, R. I. 2010. The impacts of hysteresis on variably saturated hydrologic response and slope failure. Environmental Earth Sciences, 61(6): 1215-1225. DOI: https://doi.org/10.1007/s12665-009-0445-2.
Gillham, R. W., Klute, A., & Heermann, D. F. 1976. Hydraulic properties of a porous medium measurement and empirical representation. Soil Science Society of America Journal, 40(2): 203-207. DOI: https://doi.org/10.2136/sssaj1976.03615995004000020008x.
Gillham, R. W., Klute, A., & Heermann, D. F. 1979. Measurement and numerical simulation of hysteretic flow in a heterogeneous porous medium. Soil Science Society of America Journal, 43(6): 1061-1067. DOI: https://doi.org/10.2136/sssaj1979.03615995004300060001x.
Hoa, N. T., Gaudu, R., & Thirriot, C. 1977. Influence of hysteresis effect on transient flows in saturated-unsaturated porous media. Water Resources Research, 13(6): 992-996. DOI: https://doi.org/10.1029/WR013i006p00992.
Huang, H.-C., Tan, Y.-C., Liu, C.-W., & Chen, C.-H. 2005. A novel hysteresis model in unsaturated soil. Hydrological Processes, 19(8): 1653-1665. DOI: https://doi.org/10.1002/hyp.5594.
Jeong, S., Kim, Y., & Park, H. 2019. Influences of rainfall infiltration and hysteresis SWCC of unsaturated soil on settlement of shallow foundations. Japanese Geotechnical Society Special Publication, 7(2): 569-575. DOI: https://doi.org/10.3208/jgssp.v07.088.
Kechavari, C., Soga, K., & Illangasekare, T. H. 2005. Two-dimensional laboratory simulation of LNAPL infiltration and redistribution in the vadose zone. Journal of Contaminant Hydrology, 76(Feb): 211-233. DOI: https://doi.org/10.1016/j.jconhyd.2004.09.001.
Khalili, N., Habte, M. A., & Zargarbashi, S. 2008. A fully coupled flow deformation model for cyclic analysis of unsaturated soils including hydraulic and mechanical hysteresis. Computers and Geotechnics, 35(6): 872-889. DOI: https://doi.org/10.1016/j.compgeo.2008.08.003.
Khoury, N., Brooks, R., Khoury, C., & Yada, D. 2012. Modeling resilient modulus hysteretic behavior with moisture variation. International Journal of Geomechanics, 12(5): 519-527. DOI: https://doi.org/10.1061/(ASCE)GM.1943-5622.0000140.
Khoury, C. N., & Miller, G. A. 2012. Influence of hydraulic hysteresis on the shear strength of unsaturated soils and interfaces. Geotechnical Testing Journal, 35(1): 135-149. DOI: https://doi.org/10.1520/GTJ103616.
Koestoer, R. H., Ligayanti, T., Kartohardjono, S., & Susanto, H. 2024. Down-streaming Small-Scale Green Ammonia to Nitrogen-Phosphorus Fertilizer Tablets for Rural Communities. Emerging Science Journal, 8(2): 625-643. DOI: https://doi.org/10.28991/esj-2024-08-02-016.
Kool, J. B., & Parker, J. C. 1987. Development and evaluation of closed form expressions for hysteretic soil hydraulic properties. Water Resources Research, 23(1): 105-114. DOI: https://doi.org/10.1029/WR023i001p00105.
Li, Y., Wu, P., Xia, Z., Yang, Q., Flores, G., Jiang, H., Kamon, M., & Yu, B. 2014. Changes in residual air saturation after thorough drainage processes in an air-water fine sandy medium. Journal of Hydrology, 519: 271-283. DOI: https://doi.org/10.1016/j.jhydrol.2014.07.019.
Liu, C., & Muraleetharan, K. 2012. Coupled hydro-mechanical elastoplastic constitutive model for unsaturated sands and silts. I: Formulation. International Journal of Geomechanics, 12(3): 239-247. DOI: https://doi.org/10.1061/(ASCE)GM.1943-5622.0000146.
Liu, J., Chen, P., & Li, W. 2018. Assessing Hydraulic Hysteresis Models to Characterize Unsaturated Flow Behavior under Drying and Wetting Conditions. International Journal of Geomechanics, 18(7): 04018084. DOI: https://doi.org/10.1061/(ASCE)GM.1943-5622.0001205.
Liu, K., Vardon, P., Arnold, P., & Hicks, M. A. 2015. Effect of hysteresis on the stability of an embankment under transient seepage. IOP Conference Series: Earth and Environmental Science, 26: 012013. DOI: https://doi.org/10.1088/1755-1315/26/1/012013.
Lu, N., Kaya, M., Collins, B. D., & Godt, J. W. 2013. Hysteresis of unsaturated hydrochemical properties of a silty soil. Journal of Geotechnical and Geoenvironmental Engineering, 139(3): 507-510. DOI: https://doi.org/10.1061/(ASCE)GT.1943-5606.0000786.
Ma, T., Wei, C., Wei, H., & Li, W. 2016. Hydraulic and mechanical behavior of unsaturated silt: Experimental and theoretical characterization. International Journal of Geomechanics, 16(6): D4015007. DOI: https://doi.org/10.1061/(ASCE)GM.1943-5622.0000576.
Min, T.-K., & Huy, P. T. 2010. A Soil-Water hysteresis model for unsaturated sands based on fuzzy set plasticity theory. KSCE Journal of Civil Engineering, 14(2): 165-172. DOI: https://doi.org/10.1007/s12205-010-0165-x.
McCord, J. T., Stephens, D. B., & Wilson, J. L. 1991. Hysteresis and state-dependent anisotropy in modeling unsaturated hillslope hydrologic processes. Water Resources Research, 27(7): 1501-1518. DOI: https://doi.org/10.1029/91WR00880.
Miller, G. A., Khoury, C. N., Muraleetharan, K. K., Liu, C., & Kibbey, T. C. G. 2008. Effects of soil skeleton deformations on hysteretic soil water characteristic curves: Experiments and simulations. Water Resources Research, 44(5): W00C06. DOI: https://doi.org/10.1029/2007WR006492.
Moore, N. W. 2011. Adhesion Hysteresis from Interdependent Capillary and Electrostatic Forces. Langmuir, 27(7): 3678-3684. DOI: https://doi.org/10.1021/la200043a.
Morita, M. 1996. Numerical Methods for Three-dimensional Unsaturated Subsurface Flow. Proceedings of Hydraulic Engineering, 40: 437-442. DOI: https://doi.org/10.2208/prohe.40.437.
Mualem, Y. 1976. A new model for predicting hydraulic conductivity of unsaturated porous media. Water Resources Research, 12(3): 513-522. DOI: https://doi.org/10.1029/WR012i003p00513.
Mualem, Y., & Dagan, G. 1975. A dependent domain model of capillary hysteresis. Water Resources Research, 11(3): 452-460. DOI: https://doi.org/10.1029/WR011i003p00452.
Muraleetharan, K., & Wei, C. 2001. *U-DYSCA2: Dynamic unsaturated soil analysis code for 2-dimensional problems*. Technical Report, School of Civil Engineering and Environmental Science, University of Oklahoma, Norman, OK.
Muraleetharan, K. K., Liu, C., Wei, C., Kibbey, T. C. G., & Chen, L. 2009. An elastoplastic framework for coupling hydraulic and mechanical behavior of unsaturated soils. International Journal of Plasticity, 25(3): 473-490. DOI: https://doi.org/10.1016/j.ijplas.2008.04.001.
Parker, J. C., & Lenhard, R. J. 1987. A model for hysteretic constitutive relations governing multiphase flow: 1. Saturation-pressure relations. Water Resources Research, 23(12): 2187-2196. DOI: https://doi.org/10.1029/WR023i012p02187.
Parlange, J.-Y. 1976. Capillary hysteresis and relationship between drying and wetting curve. Water Resources Research, 12(2): 224-228. DOI: https://doi.org/10.1029/WR012i002p00224.
Pasha, A. Y., Khoshghalb, A., & Khalili, N. 2017. Hysteretic model for the evolution of water retention curve with void ratio. Journal of Engineering Mechanics, 143(7): 04017030. DOI: https://doi.org/10.1061/(ASCE)EM.1943-7889.0001238.
Pham, H. Q., Fredlund, D. G., & Barbour, S. L. 2003. A practical hysteresis model for the soil-water characteristic curve for soils with negligible volume change. Géotechnique, 53(2): 293-298. DOI: https://doi.org/10.1680/geot.2003.53.2.293.
Pickens, J. F., & Gillham, R. W. 1980. Finite element analysis of solute transport under hysteretic unsaturated flow conditions. Water Resources Research, 16(6): 1071-1078. DOI: https://doi.org/10.1029/WR016i006p01071.
Poulovassilis, A. 1970. The effect of the entrapped air on the hysteresis curves of a porous body and on its hydraulic conductivity. Soil Science, 109(3): 154-162. DOI: https://doi.org/10.1097/00010694-197003000-00003.
Poulovassilis, A., & Childs, E. C. 1971. The hysteresis of pore water: the non-independence of domains. Soil Science, 112(5): 301-312. DOI: https://doi.org/10.1097/00010694-197111000-00002.
Rudiyanto, Sakai, M., Van Genuchten, M. Th., Alazba, A. A., Setiawan, B. I., & Minasny, B. 2015. A complete soil hydraulic model accounting for capillary and adsorptive water retention, capillary and film conductivity, and hysteresis. Water Resources Research, 51(11): 8757-8772. DOI: https://doi.org/10.1002/2015WR017703.
Scarfone, R., Wheeler, S. J., & Lloret-Cabot, M. 2022. A hysteretic hydraulic constitutive model for unsaturated soils and application to capillary barrier systems. Geomechanics for Energy and the Environment, 30: 100224. DOI: https://doi.org/10.1016/j.gete.2020.100224.
Scott, P. S., Farquhar, G. J., & Kouwen, N. 1983. Hysteresis effects on net infiltration. In Proceedings of the National Conference on Advances in Infiltration, 163-170. St. Joseph, MI: American Society of Agricultural and Biological Engineers.
Seymour, R. M. 2000. Air entrapment and consolidation occurring with saturated hydraulic conductivity changes with intermittent wetting. Irrigation Science, 20(1): 9-14. DOI: https://doi.org/10.1007/PL00006716.
Serrano, S. E. 1990. Modeling infiltration in hysteretic soils. Advances in Water Resources, 13(1): 12-23. DOI: https://doi.org/10.1016/0309-1708(90)90034-2.
Šimůnek, J., van Genuchten, M. Th., & Šejna, M. 2012. The Hydrus software package for simulating two- and three-dimensional movement of water, heat, and multiple solutes in variably-saturated media, version 2.0. Prague, Czech Republic: PC Progress.
Song, X., & Borja, R. I. 2014. Mathematical framework for unsaturated flow in the finite deformation range. International Journal for Numerical Methods in Engineering, 97(9): 658-682. DOI: https://doi.org/10.1002/nme.4603.
Stonestrom, D. A., & Rubin, J. 1989. Water-content dependence of trapped air in two soils. Water Resources Research, 25(9): 1947-1958. DOI: https://doi.org/10.1029/WR025i009p01947.
Sun, D., Sheng, D., & Sloan, S. W. 2007. Elastoplastic modelling of hydraulic and stress-strain behaviour of unsaturated soils. Mechanics of Materials, 39(3): 212-221. DOI: https://doi.org/10.1016/j.mechmat.2006.05.002.
Sun, D. M., Zang, Y. G., Feng, P. F., & Semprich, S. 2016. Quasi-saturated zones induced by rainfall infiltration. Transport in Porous Media, 112(1): 77-104. DOI: https://doi.org/10.1007/s11242-016-0633-y.
Tami, D., Rahardjo, H., & Leong, E.-C. 2004. Effects of hysteresis on steady-state infiltration in unsaturated slopes. Journal of Geotechnical and Geoenvironmental Engineering, 130(9): 956-967. DOI: https://doi.org/10.1061/(ASCE)1090-0241(2004)130:9(956).
Toyota, H., Nakamura, K., Sakai, N., & Sramoon, W. 2003. Mechanical Properties of Unsaturated Cohesive Soil in Consideration of Tensile Stress. Soils and Foundations, 43(2): 115-122. DOI: https://doi.org/10.1016/S0038-0806(20)30806-4.
Vachaud, G., & Thony, J.-L. 1971. Hysteresis during infiltration and redistribution in a soil column at different initial water contents. Water Resources Research, 7(1): 111-127. DOI: https://doi.org/10.1029/WR007i001p00111.
van Genuchten, M. Th. 1980. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Science Society of America Journal, 44(5): 892-898. DOI: https://doi.org/10.2136/sssaj1980.03615995004400050002x.
Wei, C., & Dewoolkar, M. M. 2006. Formulation of capillary hysteresis with internal state variables. Water Resources Research, 42(7): W07405. DOI: https://doi.org/10.1029/2005WR004594.
Werner, A. D., & Lockington, D. A. 2006. Artificial pumping errors in the Kool-Parker scaling model of soil moisture hysteresis. Journal of Hydrology, 325(1-4): 118-133. DOI: https://doi.org/10.1016/j.jhydrol.2005.10.012.
Wheeler, S. J., Sharma, R. S., & Buisson, M. S. R. 2003. Coupling of hydraulic hysteresis and stress-strain behaviour in unsaturated soils. Géotechnique, 53(1): 41-54. DOI: https://doi.org/10.1680/geot.2003.53.1.41.
Yang, C., Sheng, D., & Carter, J. P. 2012. Effect of hydraulic hysteresis on seepage analysis for unsaturated soils. Computers and Geotechnics, 41: 36-56. DOI: https://doi.org/10.1016/j.compgeo.2011.11.006.
Zhou, A. 2017. Modelling hydro-mechanical behavior for unsaturated soils. Japanese Geotechnical Society Special Publication, 5(2): 79-94. DOI: https://doi.org/10.3208/jgssp.v05.019.
