Estimation of tortuosity coefficient under unsaturated conditions based on fractal concepts

Subject Areas : Article frome a thesis

Maysam Majidi Khalilabad ¹ , shiva gholizadeh sarabi ² , bijan ghahraman ³ , Hadi Memarian Khalilabad ⁴

1 - Assistant Professor, Water Engineering Department, Kashmar Higher Education Institutue, Kashmar, Iran
2 - Researcher, East Water & Environmental Research Institute, Mashhad, Iran
3 - Professor, Water Engineering Department, Ferdowsi university of Mashhad, Mashhad, Iran
4 - Associated Professor, Water Engineering Department, Kashmar Higher Education Institutue, Kashmar, Iran

Received: 2019-08-22 Accepted : 2021-08-02 Published : 2021-07-23

Keywords: Hydraulic tortuosity, Shepard’s model, Van Gneuchten’s model, Unsaturated hydraulic conductivity,

Abstract :

Complex nature of porous media complicates any prediction of their hydraulic properties. To demonstrate shortcoming of hydraulic conductivity models predictions, the concept of tortuosity was introduced. Since there is no measured data of tortuosity, and tortuosity has a direct relationship to hydraulic conductivity, so in this study we aimed to develop a general mathematical relationship to determine tortuosity. An optimization code were run in MATLAB R2014a software, using Monte Carlo algorithm, aimed to minimize Root Mean Square of Logarithmic Deviation (RMSLD) between calculated hydraulic conductivity values based on Shepard (1993) and van Genuchten (1980) models, to determine tortuosity on different water contents for 69 soil samples of UNSODA database with a wide range of soil textures. Considering fractal concepts, we developed a linear equation empirically to determine hydraulic tortuosity as a function of effective saturation, pore fractal dimension, porosity, inverse of air entry pressure and soil water content, covering whole ranges of degree of saturation. Based on results, calculated values of tortuosity were greater than proposed values by Shepard about 30%. To evaluate developed equation, statistical parameters of Root Mean Square of Logarithmic Deviation (RMSLD) and Akaike’s Information Criterion (AICc) was adopted for 17 different soil samples. According to the calculated statistical parameters, using developed equation to estimate tortuosity has improved the results of Shepard’s method significantly. Totally, the results show that, despite the developed equation has a relatively complicated structure, in terms of the compromise between accuracy and complexity has an acceptable performance.

References:

1. Adler, P.M. 1992. Porous media: Geometry and transports. Butterworth–Heinemann, Stoneham, MA. https://doi.org/10.1002/aic.690400220


2. Bear, J. 1972. Dynamics of fluids in porous media. Elsevier, New York. https://books.google.com/books?hl=en&lr=&id=fBMeVSZ_3u8C&oi=fnd&pg=PP1&ots=mhcyzf_JDF&sig=OhDzQnNKGDb7NQLw-aeDdx9Tm8I


3. Brooks, R. H., and A. T. Corey.  Hydraulic  properties  of  porous media. Hydrology  Paper No. 3, Colorado State Univ., Fort Collins, Colorado. https://mountainscholar.org/bitstream/handle/10217/61288/HydrologyPapers_n3.pdf?se


4. Burdine, N.T. 1953. Relative permeability calculations from pore-size distribution data. Pet. Trans. Am. Inst. Min. Metall. Eng. 198:71–77. http://dx.doi.org/10.2118/225-g


5. Carman, P.C. 1937. Fluid flow through granular beds. Trans. Inst. Chem. Eng. 15:150–166. https://www.scirp.org/(S(i43dyn45teexjx455qlt3d2q))/reference/ReferencesPapers.aspx?ReferenceID=1975478


6. Clennell, M.B. 1997. Tortuosity: A guide through the maze. In: M.A. Lovell and P.K. Harvey, editors, Developments in petrophysics. Geol. Soc., London. p. 299–344. https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.907.1114&rep=rep1&type=pdf


7. Du Plessis, J.P., and J.H. Masliyah. 1991. Flow through isotropic granular porous media. Transp. Porous Media 6:207–221. https://doi.org/10.1007/BF00208950


8. Duda, A., Z. Koza, and M. Matyka. 2011. Hydraulic tortuosity in arbitrary porous media flow. Phys. Rev. E 84:036319. https://doi.org/10.1103/PhysRevE.84.036319


9. Dullien, F.A.L. 1979. Porous media: Fluid transport and pore structure. Academic Press, San Diego. https://www.elsevier.com/books/porous-media-fluid-transport-and-pore-structure/dullien/978-0-12-223650-1


10. Epstein, N. 1989. On tortuosity and the tortuosity factor in flow and diffusion through porous media. Chem. Eng. Sci. 44:777–779. https://doi.org/10.1016/0009-2509(89)85053-5


11. Fatt, I., and H. Dykstra. 1951. Relative permeability studies. Trans. Am. Inst. Min. Metall. Pet. Eng. 192:249–255. https://doi.org/10.2118/951249-G.


12. Feder, J. 1988. Fractals. Plenum Press, New York. https://www.springer.com/gp/book/9780306428517


13. Ghanbarian, B. Hunt, A.G. Ewing, R.P. and Sahimi, M. 2013. Tortuosity in Porous Media: A Critical Review. Soil Sci. Soc. Am. J. 77: 1461-1477. https://doi.org/10.2136/sssaj2012.0435


14. Hager, J., M. Hermansson, and R. Wimmerstedt. 1997. Modeling steam drying of a single porous ceramic sphere: Experiments and simulations. Chem. Eng. Sci. 52:1253–1264. https://doi.org/10.1016/S0009-2509(96)00493-9


15. Hillel, D. 2004. Introduction to environmental soil physics. Academic Press, San Diego. https://www.elsevier.com/books/introduction-to-environmental-soil-physics/hillel/978-0-12-348655-4


16. Koponen, A., M. Kataja, and J. Timonen. 1996. Tortuous flow in porous media. Phys. Rev. E 54:406–410. https://doi.org/10.1103/PhysRevE.54.406


17. Kravchenko, A., Zhang, R., 1997. Estimating soil hydraulic conductivity from soil particle-size distribution. Proceedings of the International Workshop on Characterization and Measurement of the Hydraulic Properties of Unsaturated Porous Media.


18. Leij, F.J., Alves, W.J., Van Genuchten, M.Th., and Williams, J.R. 1999. “The UNSODA unsaturated soil hydraulic database”. p. 1269-1281. In M.Th. van Genuchten et al. (ed.) Characterization and measurement of the hydraulic properties of unsaturated porous media. Univ. of California, Riverside, CA. https://data.nal.usda.gov/dataset/unsoda-20-unsaturated-soil-hydraulic-database-database-and-program-indirect-methods-estimating-unsaturated-hydraulic-properties


19. McQuarrie, A. D. R. and Tsai, C.-L. 1998. Regression and time series model selection. World Scientific, London, UK.
    455 pp. https://doi.org/10.1142/3573


20. Millington, R.J., and J.P. Quirk. 1961. Permeability of porous solids. Trans. Faraday Soc. 57:1200–1206. https://doi.org/10.1039/TF9615701200


21. Moldrup, P., T. Olesen, J. Gamst, P. Schjønning, T. Yamaguchi, and D.E. Rolston. 2000a. Predicting the gas diffusion coefficient in repacked soil: Water-induced linear reduction model. Soil Sci. Soc. Am. J. 64:1588–1594. https://doi.org/10.2136/sssaj2000.6451588x


22. Mualem, Y. 1976. A new model for predicting the hydraulic conductivity of unsatuwww.soils.org/publications/sssaj 1477 rated porous media. Water Resour. Res 12:2187–2193


23. Rawls, W.J., D.L. Brakensiek, and S.D. Logsdon. 1993. Predicting saturated hydraulic conductivity utilizing fractal principles. Soil Sci. Soc. Am. J. 57: 1193-1197. https://doi.org/10.2136/sssaj1993.03615995005700050005x


24. Rieu, M. and G. Sposito. 1991. Fractal fragmentation soil porosity and soil water properties: 1. Theory. Soil Sci. Soc. Am. J. 55: 1231- 1238. https://doi.org/10.2136/sssaj1991.03615995005500050007x


25. Sahimi, M. 1993. Flow phenomena in rocks: From continuum models to fractals, percolation, cellular automata, and simulated annealing. Rev. Mod. Phys. 65:1393–1534. https://doi.org/10.1103/RevModPhys.65.1393


26. Schaap, M.G., and F.J. Leij. 2000. Improved prediction of unsaturated hydraulic conductivity with the Mualem–van Genuchten model. Soil Sci. Soc. Am. J. 64:843–851. https://doi.org/10.2136/sssaj2000.643843x


27. Scheidegger, A.E. 1974. Te physics of flow through porous media. 3rd ed. Univ. of Toronto Press, Toronto. https://utorontopress.com/us/the-physics-of-flow-through-porous-media-3rd-edition-1


28. Shepard, J.S. 1993. Using a fractal model to calculate the hydraulic conductivity function. Soil Sci.Soc.Am.J. 57: 300-307. https://doi.org/10.2136/sssaj1993.03615995005700020002x


29. Shinomiya, Y., K. Takahashi, M. Kobiyama, and J. Kubota. 2001. Evaluation of the tortuosity parameter for forest soils to predict unsaturated hydraulic conductivity. J. For. Res. 6:221–225. https://doi.org/10.1007/BF02767097


30. Tye, F.L. 1983. Tortuosity. J. Power Sources 9:89–100.


31. Tyler, S.W., and S.W. Wheatcraft. 1990. Fractal processes in soil water retention. Water Resour. Res. https://doi.org/10.1029/WR026i005p01047


32. Van Gneuchten,  Th.  1980.  A closed-form  equation  for  predicting  the  hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J. 44: 892-898. https://doi.org/10.2136/sssaj1980.03615995004400050002x


33. van Damme H. Scale invariance and hydric behavior of soils and clays. CR Acad Sci Paris 1995, 320: 665–81. https://www.semanticscholar.org/paper/Scale-invariance-and-hydric-behaviour-of-soils-and-Damme/5d9b585189bd74ddd9649ae47dca8dff3e68aadd


34. Van Genuchten, M.Th., F.J. Leij, , and S.R. Yates.  The RETC code for quantifying the hydraulic functions of unsaturated soils. Project summary, EPA’S Robert S. Kerr Environmental Research Lab., Ada ,OK, USA. https://nepis.epa.gov/Exe/ZyNET.exe/30003U6Q.TXT?ZyActionD=ZyDocument&Client=EPA&Index=1991+Thru+1994&Docs=&Query=&Time=&EndTime=&SearchMethod=1&TocRestrict=n&Toc=&TocEntry=&QField=&QFieldYear=&QFieldMonth=&QFieldDay=&IntQFieldOp=0&ExtQFieldOp=0&XmlQuery=&File=D%3A%5Czyfiles%5CIndex%20Data%5C91thru94%5CTxt%5C00000002%5C30003U6Q.txt&User=ANONYMOUS&Password=anonymous&SortMethod=h%7C-&MaximumDocuments=1&FuzzyDegree=0&ImageQuality=r75g8/r75g8/x150y150g16/i425&Display=hpfr&DefSeekPage=x&SearchBack=ZyActionL&Back=ZyActionS&BackDesc=Results%20page&MaximumPages=1&ZyEntry=1&SeekPage=x&ZyPURL


35. Wheatcraft, S.W., and S.W. Tyler. 1988. An explanation of scale-dependent dispersivity in heterogeneous aquifers using concepts of fractal geometry. Water Resour. Res. 24:566–578. https://doi.org/10.1029/WR024i004p00566


36. Xu, Y. 2004. Calculation of unsaturated hydraulic conductivity using a fractal model for the pore-size distribution. Computers and Geotechnics. 31:549-557. http://dx.doi.org/10.1016%2Fj.compgeo.2004.07.003


37. Xu, Y.F., and D.A. Sun. 2002. A fractal model for soil pores and its application to determination of water permeability. Physica. 316(1-4): 56–64. http://dx.doi.org/10.1016/S0378-4371(02)01331-6


38. Zhang, X., and M.A. Knackstedt. 1995. Direct simulation of electrical and hydraulic tortuosity in porous solids. Geophys. Res. Lett. 22:2333–2336. https://doi.org/10.1029/95GL02230