Thermohydraulic Flow Problem in Unsaturated Porous Media: FDM Computational Model
Publication: International Journal of Geomechanics
Volume 20, Issue 7
Abstract
The moisture and heat fluxes in undeformable unsaturated porous media involve the movement of water, air, and heat that are induced by thermal and pressure gradients to which the porous medium is subjected under environmental conditions. Herein, the flow of the liquid phase is governed by the advective flow due to the hydraulic gradient and by the convective heat transfer due to the thermal gradient. The flow of the gas phase is governed by the advective flow due to the pressure gradient and the nonadvective flow of dry air and water vapor diffusion. The heat transport can be carried out by conduction, convection, and advection due to the pressure gradient. The mathematical model includes the air mass conservation, water mass conservation, and thermal energy conservation equations. This paper presents a detailed computational model based on the finite difference method (FDM) for one-dimensional analysis of the flow problem of heat and moisture in undeformable unsaturated porous media. Verification examples involving unsaturated flow analysis in isothermal and nonisothermal conditions are presented, highlighting the importance of having a relatively simple computational model to analyze a very complex physical problem.
Get full access to this article
View all available purchase options and get full access to this article.
Data Availability Statement
All data and the Fortran code used in this study are available from the corresponding author by request.
Acknowledgments
The authors are grateful for the financial support received from Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), CNPq, FAPEMIG, Fundação Gorceix, PPGEM/UFOP, PROPEC/UFOP, and PROPP/UFOP. The first author is grateful for the grant received by CAPES. They also acknowledge Harriet Konkel Reis and Paper True for the English review of this text. Special thanks to Professor Bertrand François from Université libre de Bruxelles (ULB) for his hospitality during the manuscript's review at BATir/ULB.
References
Bittelli, M., F. Ventura, G. S. Campbell, R. L. Snyder, F. Gallegati, and P. R. Pisa. 2008. “Coupling of heat, water vapor, and liquid water fluxes to compute evaporation in bare soils.” J. Hydrol. 362 (3–4): 191–205. https://doi.org/10.1016/j.jhydrol.2008.08.014.
Collin, F., X. L. Li, J. P. Radu, and R. Charlier. 2002. “Thermo-hydro-mechanical coupling in clay barriers.” Eng. Geol. 64 (2–3): 179–193. https://doi.org/10.1016/S0013-7952(01)00124-7.
De Vries, D. A. 1958. “Simultaneous transfer of heat and moisture in porous media.” Trans. Am. Geophys. Union 39 (5): 909–916. https://doi.org/10.1029/TR039i005p00909.
Dupray, F., B. François, and L. Laloui. 2013. “Analysis of the FEBEX multi-barrier system including thermoplasticity of unsaturated bentonite.” Int. J. Numer. Anal. Methods Geomech. 37 (4): 399–422. https://doi.org/10.1002/nag.1103.
François, B., and L. Laloui. 2011. “Non-isothermal infiltration in plastic bentonite.” In Unsaturated soils, edited by E. E. Alonso, and A. Gens, 1375–1382. London: Taylor & Francis Group.
Fredlund, D. G., and H. Rahardjo. 1993. Soil mechanics for unsaturated soils. New York: John Wiley & Sons.
Gardner, W. R. 1958. “Some steady-state solutions of the unsaturated moisture flow equation with application to evaporation from a water table.” Soil Sci. 85 (4): 228–232. https://doi.org/10.1097/00010694-195804000-00006.
Groenevelt, P. H., and B. D. Kay. 1974. “On the interaction of water and heat transport in frozen and unfrozen soils: II. The liquid phase.” Soil Sci. Soc. Am. J. 38 (3): 400–404. https://doi.org/10.2136/sssaj1974.03615995003800030012x.
Hansson, K., J. Šimůnek, M. Mizoguchi, L. C. Lundin, and M. T. van Genuchten. 2004. “Water flow and heat transport in frozen soil.” Vadose Zone J. 3 (2): 693–704. https://doi.org/10.2136/vzj2004.0693.
Lewis, R. W., and B. A. Schrefler. 1998. The finite element method in the static and dynamic deformation and consolidation of porous media. New York: John Wiley & Sons.
Lu, N., and W. J. Likos. 2004. Unsaturated soil mechanics. New York: John Wiley & Sons.
McCartney, J. S., M. Sánchez, and I. Tomac. 2016. “Energy geotechnics: Advances in subsurface energy recovery, storage, exchange, and waste management.” Comput. Geotech. 75: 244–256. https://doi.org/10.1016/j.compgeo.2016.01.002.
Milly, P. C. D. 1984. “A simulation analysis of thermal effects on evaporation from soil.” Water Resour. Res. 20 (8): 1087–1098. https://doi.org/10.1029/WR020i008p01087.
Milly, P. C. D., and P. S. Eagleson. 1980. The coupled transport of water and heat in a vertical soil column under atmospheric excitation. Technical Rep. No. 258. Cambridge: MIT.
Nobre, R. C. M., and N. R. Thomson. 1993. “The effects of transient temperature gradients on soil moisture dynamics.” J. Hydrol. 152 (1–4): 57–101. https://doi.org/10.1016/0022-1694(93)90141-U.
Philip, J. R., and D. A. de Vries. 1957. “Moisture movement in porous materials under temperature gradients.” Transa. Am. Geophys. Union 38 (2): 222–232. https://doi.org/10.1029/TR038i002p00222.
Qi, S., and S. K. Vanapalli. 2018. “Simulating hydraulic and mechanical responses of unsaturated expansive soil slope to rainfall: Case study.” Int. J. Geomech. 18 (6): 05018002. https://doi.org/10.1061/(ASCE)GM.1943-5622.0001106.
Sánchez, M., A. Gens, M. V. Villar, and S. Olivella. 2016. “Fully coupled thermo-hydro-mechanical double-porosity formulation for unsaturated soils.” Int. J. Geomech. 16 (6): D4016015. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000728.
Smith, G. D. 1985. Numerical solution of partial differential equations: Finite difference methods. Oxford, UK: Oxford Univ. Press.
Sophocleous, M. 1979. “Analysis of water and heat flow in unsaturated-saturated porous media.” Water Resour. Res. 15 (5): 1195–1206. https://doi.org/10.1029/WR015i005p01195.
Souza, K. B. 2018. “Análise Acoplada Termo-Hidráulica de Problemas de Fluxo em Meio Poroso Não Saturado.” M.Sc. thesis, Departamento de Engenharia de Minas, Universidade Federal de Ouro Preto.
Souza, K. B., C. L. Nogueira, and D. M. Armond. 2016. “Análise Numérica do Processo de Infiltração de Água num Sistema de Cobertura com Barreira Capilar.” In Proc., 18th Congresso Brasileiro de Mecânica dos Solos e Engenharia Geotécnica, 1–6. Belo Horizonte, MG: ABMS. https://doi.org/10.20906/CPS/CB-08-0049.
Srivastava, R., and T.-C. J. Yeh. 1991. “Analytical solutions for one-dimensional, transient infiltration toward the water table in homogeneous and layered soils.” Water Resour. Res. 27 (5): 753–762. https://doi.org/10.1029/90WR02772.
Thomas, H. R., and S. D. King. 1992. “Coupled heat and mass transfer in unsaturated soil—a potential-based solution.” Int. J. Numer. Anal. Methods Geomech. 16 (10): 757–773. https://doi.org/10.1002/nag.1610161005.
van Genuchten, M. T. 1980. “A closed form equation for predicting the hydraulic conductivity of unsaturated soils.” Soil Sci. Soc. Am. J. 44 (5): 892–898. https://doi.org/10.2136/sssaj1980.03615995004400050002x.
Vilarrasa, V., J. Rutqvist, L. B. Martin, and J. Birkholzer. 2016. “Use of a dual-structure constitutive model for predicting the long-term behavior of an expansive clay buffer in a nuclear waste repository.” Int. J. Geomech. 16 (6): D4015005. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000603.
Wilson, G. W. 1991. “Soil evaporative fluxes for geotechnical engineering problems.” Ph.D. thesis, Dept. of Civil Engineering, Univ. of Saskatchewan.
Zagorščak, R., M. Sedighi, and H. R. Thomas. 2017. “Effects of thermo-osmosis on hydraulic behavior of saturated clays.” Int. J. Geomech. 17 (3): 04016068. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000742.
Zhou, Y., R. K. N. D. Rajapakse, and J. Graham. 1998. “Coupled heat-moisture-air transfer in deformable unsaturated media.” J. Eng. Mech. 124 (10): 1090–1099. https://doi.org/10.1061/(ASCE)0733-9399(1998)124:10(1090).
Zhou, Y., and R. K. Rowe. 2005. “Modeling of clay liner desiccation.” Int. J. Geomech. 5 (1): 1–9. https://doi.org/10.1061/(ASCE)1532-3641(2005)5:1(1).
Information & Authors
Information
Published In
Copyright
© 2020 American Society of Civil Engineers.
History
Received: Jul 2, 2019
Accepted: Jan 16, 2020
Published online: May 5, 2020
Published in print: Jul 1, 2020
Discussion open until: Oct 5, 2020
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.