Evaporation-Induced Capillary Siphoning Through Hydraulically Connected Porous Domains: The Vedernikov–Bouwer Model Revisited

  • A. R. KacimovEmail author
  • Yu. V. Obnosov
  • D. Or


Evaporation-driven wicking of soil water through porous domains with contrasting hydraulic properties is studied analytically by conformal mappings and compared to numerical solutions. Initially, the connected rectangular domains are fully saturated. The first rectangle, Gp, is comprised of a coarse-textured porous medium with large permeability and low capillary forces. Evaporation-induced capillary flow pulls water horizontally across the domains to the surface of fine-textured rectangular domain, Gz, through an interfacial hydraulic exchange region that shrinks with time. The flow field in Gz is 2-D and is analytically expressed by the Vedernikov–Bouwer model that assumes a constant hydraulic conductivity for pressure heads higher than the air-entry value. The rate of 1-D drop of the phreatic surface in Gp is proportional to the evaporation rate (decreasing with time) from the Gz surface. The complex potential domain Gw “shrinks” with time, and at any time instance, it is conformally mapped onto Gz via two auxiliary planes using the Schwarz–Christoffel and Mobius transformations. The resulting Cauchy problem for an integro-differential equation with respect to an affix of the conformal mapping is solved using numerical algebra routines. A similar capillary coupled flow problem was numerically simulated using HYDRUS2D considering 2-D flow in both Gp and Gz. New insights into process dynamics are gained from a solution of an auxiliary optimization for a vertical imbibition in a column brought in contact with a water table where particle size (linking capillarity and permeability) is used as a control variable and counter-gravity front propagation dynamics as criteria.


Counter-gravity imbibition Stages of evaporation 2-D potential flow Complex potential HYDRUS simulations 





Reference to the last (Russian) edition of the Polubarinova-Kochina (1977) book


Supplementary Electronic File


Vedernikov and Bouwer



This work was funded by a grant from the Sultan Qaboos Higher Center for Culture and Science – Diwan of Royal Court and the Research Council of Oman (TRC) [RC/AGR/SWAE/17/01], SQU Grant IG/CAMS/SWAE/18/01 and by the subsidy allocated to Kazan Federal University for the state assignment in the sphere of scientific activities, Project No. 1.12878.2018/12.1. Helpful comments by three anonymous referees are greatly appreciated.

Data Sharing

The Mathematica codes (Sects. 2, 4) and HYDRUS project (Sect. 3) are open files and available to the referees and readers upon request.

Supplementary material

11242_2019_1285_MOESM1_ESM.pptx (4.1 mb)
Supplementary material 1 (PPTX 4188 kb)


  1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1969)Google Scholar
  2. Alyafei, N., Blunt, M.J.: Estimation of relative permeability and capillary pressure from mass imbibition experiments. Adv. Water Resour. 115, 88–94 (2018)CrossRefGoogle Scholar
  3. Al-Ismaily, S.S., Al-Maktoumi, A.K., Kacimov, A.R., Al-Saqri, S.M., Al-Busaidi, H.A., Al-Haddabi, M.H.: A morphed block-crack preferential sedimentation: a smart design and evolution in nature. Hydrol. Sci. J. 58(8), 1779–1788 (2013)CrossRefGoogle Scholar
  4. Al-Maktoumi, A., Al-Ismaily, S., Kacimov, A., Al-Busaidi, H., Al-Saqri, S., Al-Haddabi, M.: Soil substrate as a cascade of capillary barriers for conserving water in a desert environment: lessons learned from arid nature. J. Arid Land 6(6), 690–703 (2014)CrossRefGoogle Scholar
  5. Angulo-Jaramillo, R., Bagarello, V., Iovino, M., Lassabatere, L.: Infiltration Measurements for Soil Hydraulic Characterization. Springer, Berlin (2016)CrossRefGoogle Scholar
  6. Assouline, S., Narkis, K., Gherabli, R., Lefort, P., Prat, M.: Analysis of the impact of surface layer properties on evaporation from porous systems using column experiments and modified definition of characteristic length. Water Resour. Res. 50(5), 3933–3955 (2014)CrossRefGoogle Scholar
  7. Babaev, A.G.: Landscapes of Turkmenistan. In: Fet, V., Atamuradov, K.I. (eds.) Biogeography and Ecology of Turkmenistan. Monographiae Biologicae, vol. 72, pp. 5–22. Springer, Dordrecht (1994)CrossRefGoogle Scholar
  8. Babaev, A.G. (ed.): Desert Problems and Desertification in Central Asia: The Researchers of the Desert Institute. Springer, Berlin (2012)Google Scholar
  9. Bagnold, R.A.: The Physics of Blown Sand and Desert Dunes. Methuen, London (1941)Google Scholar
  10. Benner, E.M., Petsev, D.N.: Evaporation effect on two-dimensional wicking in porous media. J. Colloid Interface Sci. 514, 21–29 (2018)CrossRefGoogle Scholar
  11. Bechtold, M., Haber-Pohlmeier, S., Vanderborght, J., Pohlmeier, A., Ferré, T.P.A., Vereecken, H.: Near-surface solute redistribution during evaporation. Geophys. Res. Lett. 38(17), L17404 (2011)CrossRefGoogle Scholar
  12. Bergstad, M., Or, D., Withers, P.J., Shokri, N.: Evaporation dynamics and NaCl precipitation on capillarity-coupled heterogeneous porous surfaces. Water Resour. Res. 54(6), 3876–3885 (2018)CrossRefGoogle Scholar
  13. Beyhaghi, S., Geoffroy, S., Prat, M., Pillai, K.M.: Wicking and evaporation of liquids in porous wicks: a simple analytical approach to optimization of wick design. AIChE J. 60(5), 1930–1940 (2014)CrossRefGoogle Scholar
  14. Blunt, M.J.: Multiphase Flow in Permeable Media: A Porescale Perspective. Cambridge University Press, Cambridge (2017)CrossRefGoogle Scholar
  15. Breckle, S.W., Yair, A., Veste, M. (eds.): Arid Dune Ecosystems: The Nizzana Sands in the Negev Desert, vol. 200. Springer, Berlin (2008)Google Scholar
  16. Bouwer, H.: Unsaturated flow in ground-water hydraulics. J. Hydraul. Div. ASCE 90(5), 121–144 (1964)Google Scholar
  17. Chen, L., Wang, W., Zhang, Z., Wang, Z., Wang, Q., Zhao, M., Gong, C.: Estimation of bare soil evaporation for different depths of water table in the wind-blown sand area of the Ordos Basin, China. Hydrogeol. J. 26, 1693–1704 (2018)CrossRefGoogle Scholar
  18. El-Sheikh, M.A., Abbadi, G.A., Bianco, P.M.: Vegetation ecology of phytogenic hillocks (nabkhas) in coastal habitats of Jal Az-Zor National Park, Kuwait: role of patches and edaphic factors. Flora 205(12), 832–840 (2010)CrossRefGoogle Scholar
  19. Elizalde, E., Urteaga, R., Berli, C.L.: Rational design of capillary-driven flows for paper-based microfluidics. Lab Chip 15(10), 2173–2180 (2015)CrossRefGoogle Scholar
  20. Erly, L.M.: Reclaiming native soil: cultural mythologies of soil in Russia and Its Eastern Borderlands from the 1840s to the 1930s. Ph.D. dissertation, UC Berkeley (2012). Accessed 13 April 2019
  21. Gael, A.G., Smirnova, L.F.: Sands and Sandy Soils. GEOS, Moscow (1999). (in Russian) Google Scholar
  22. Gardner, W.R., Fireman, M.: Laboratory studies of evaporation from soil columns in the presence of a water table. Soil Sci. 85(5), 244–249 (1958)CrossRefGoogle Scholar
  23. Geology of the USSR: Volume 22. Turkmenskaya SSR. Part 1, Geological Description. Ed. N.P.Louppov, Moscow, GNTILGON (1957). (in Russian) Google Scholar
  24. Hellwig, D.H.R.: Evaporation of water from sand, 1: experimental set-up and climatic influences. J. Hydrol. 18(2), 93–108 (1973)CrossRefGoogle Scholar
  25. Ijjas, I.: Effect of compactness and initial moisture content of the soil on the process of capillary rise. In: Rijtema, P.E., Wassink, H. (eds.) Water in the Unsaturated Zone, Proceedings of the Wageningen Symposium, pp. 547–559. UNSECO-IAHS, Printed by Ceuterick, Louvain (1966)Google Scholar
  26. Kacimov, A.R.: Analytic solution for transient flow into a hemispherical auger hole. J. Hydrol. 2000(228), 1–9 (2000)CrossRefGoogle Scholar
  27. Kacimov, A.R.: Capillarity and evaporation exacerbated seepage losses from unlined channels. J. Irrig. Drain. Eng. ASCE 132(6), 623–626 (2006)CrossRefGoogle Scholar
  28. Kacimov, A., Al-Issai, J., Al-Amri, M., Al-Balushi, M.: Green-roof project in Oman: capillary siphoning as a novel and thrifty irrigation technique. In: Proceedings of the 4rd International Conference on Water Resources and Arid Environments, Riyad, Saudi Arabia, 6–8 Dec, 2010, pp. 479–487 (2010)Google Scholar
  29. Kacimov, A., Al-Maktoumi, A., Al-Ismaily, S., Al-Busaidi, H.: Moisture and temperature in a proppant-enveloped silt block of a recharge dam reservoir. J. Agric. Mar. Sci. 21(1), 8–17 (2017)Google Scholar
  30. Kacimov, A.R., Lapin, A.V.: Determination of seepage parameters from data of a quick water intake test and solution of the Boussinesq equation. Power Technol. Eng. 27(10), 600–607 (1993)Google Scholar
  31. Kacimov, A.R., Obnosov, YuV: Strip-focused phreatic surface flow driven by evaporation: analytical solution by the Riesenkampf function. Adv. Water Resour. 29, 1565–1571 (2006)CrossRefGoogle Scholar
  32. Kacimov, A.R., Obnosov, YuV: Analytical solutions for seepage near material boundaries in dam cores: the Davison-Kalinin problems revisited. Appl. Math. Model. 36, 1286–1301 (2012)CrossRefGoogle Scholar
  33. Kacimov, A.R., Obnosov, YuV: Pseudo-hysteretic double-front hiatus-stage soil water parcels supplying a plant-root continuum: the Green-Ampt–Youngs model revisited. Hydrol. Sci. J. 58(1), 237–248 (2013)CrossRefGoogle Scholar
  34. Kacimov, A., Obnosov, YuV, Simunek, J.: Steady flow from an array of subsurface emitters: Kornev’s irrigation technology and Kidder’s free boundary problems revisited. Transp. Porous Media 121(3), 643–664 (2018)CrossRefGoogle Scholar
  35. Kacimov, A.R., Sherif, M.M., Perret, J.S., Al-Mushikhi, A.: Control of sea-water intrusion by salt-water pumping: coast of Oman. Hydrogeol. J. 17, 541–558 (2009)CrossRefGoogle Scholar
  36. Kacimov, A.R., Yakimov, N.D.: Moving phreatic surface in a porous slab: an analytical solution. J. Eng. Math. 40, 399–411 (2001)CrossRefGoogle Scholar
  37. Kornev, V.G.: Subsurface Irrigation. Selhozgiz, Moscow-Leningrad (in Russian) (1935)Google Scholar
  38. Kunin, V.N.: Local Waters in Deserts and Problems in Their Utilization. Akad. Nauk SSSR, Moscow (1959). (in Russian) Google Scholar
  39. Lehmann, P., Assouline, S., Or, D.: Characteristic lengths affecting evaporative drying of porous media. Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 77(5), 056309 (2008)CrossRefGoogle Scholar
  40. Lehmann, P., Or, D.: Evaporation and capillary coupling across vertical textural contrasts in porous media. Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 80(4), 046318 (2009)CrossRefGoogle Scholar
  41. Maas, K.: Lek langs de stijgbuis van een put via een slecht afgedichte kleilaag. Water Cycle Research Institute, Nieuwegein (2011). (in Dutch) Google Scholar
  42. McKay, C.P., Rask, J.C., Detweiler, A.M., Bebout, B.M., Everroad, R.C., Lee, J.Z., Chanton, J.P., Mayer, M.H., Caraballo, A.A., Kapili, B., Al-Awar, M.: An unusual inverted saline microbial mat community in an interdune sabkha in the Rub’al Khali (the Empty Quarter), United Arab Emirates. PLoS ONE 11(3), e0150342 (2016)CrossRefGoogle Scholar
  43. Meng, Q., Cai, Z., Cai, J., Yang, F.: Oil recovery by spontaneous imbibition from partially water-covered matrix blocks with different boundary conditions. J. Pet. Sci. Eng. 172, 454–464 (2019)CrossRefGoogle Scholar
  44. Mendez, S., Fenton, E.M., Gallegos, G.R., Petsev, D.N., Sibbett, S.S., Stone, H.A., Zhang, Y., López, G.P.: Imbibition in porous membranes of complex shape: quasi-stationary flow in thin rectangular segments. Langmuir 26(2), 1380–1385 (2009)CrossRefGoogle Scholar
  45. Navarro, V., Yustres, A., Cea, L., Candel, M., Juncosa, R., Delgado, J.: Characterization of the water flow through concrete based on parameter estimation from infiltration tests. Cem. Concrete Res. 36(9), 1575–1582 (2006)CrossRefGoogle Scholar
  46. Obnosov, YuV, Kacimov, A.R.: Steady Darcian flow in subsurface irrigation of topsoil impeded by substratum: Kornev–Riesenkampf–Philip legacies revisited. Irrig. Drain. 67(3), 374–391 (2018)CrossRefGoogle Scholar
  47. Ogar, N.P.: Vegetation dynamics on the Syrdarya delta and modern land use. In: Breckle, S.-W., Veste, M., Wucherer, W. (eds.) Sustainable Land Use in Deserts, pp. 74–83. Springer, Berlin (2001)CrossRefGoogle Scholar
  48. Orlovsky, L., Matsrafi, O., Orlovsky, N., Kouznetsov, M.: Sarykamysh lake: collector of drainage water—the past, the present, and the future. In: Zonn, I., Kostianoy, A. (eds.) The Turkmen Lake Altyn Asyr and water resources in Turkmenistan. The Handbook of Environmental Chemistry, vol. 28 pp. 107–140. Springer, Berlin (2012)CrossRefGoogle Scholar
  49. Pachepsky, Y., Hill, R.L.: Scale and scaling in soils. Geoderma 287, 4–30 (2017)CrossRefGoogle Scholar
  50. Pan, N., Zhong, W.: Fluid transport phenomena in fibrous materials. Text. Prog. 38(2), 1–93 (2006)CrossRefGoogle Scholar
  51. Patnaik, A., Rengasamy, R.S., Kothari, V.K., Ghosh, A.: Wetting and wicking in fibrous materials. Text. Prog. 38(1), 1–105 (2006)CrossRefGoogle Scholar
  52. Philip, J.R.: Variable-head ponded infiltration under constant or variable rainfall. Water Resour. Res. 29(7), 2155–2165 (1993)CrossRefGoogle Scholar
  53. Platonov, A.A.: Takyr. Krasnaya Nov’, 9, 82–93 (1934) (in Russian). Accessed 13 April 2019
  54. Polubarinova-Kochina, P.Ya.: Theory of Ground Water Movement. Second Edition published in 1977 in Russian, Nauka, Moscow (First Edition published in 1962 in English, Princeton Univ. Press, Princeton) (1977)Google Scholar
  55. Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishechenko, E.F.: The Mathematical Theory of Optimal Processes. Wiley, New York (1962)Google Scholar
  56. Prudnikov, A., Brychkov, I., Marichev, O.: Integrals and Series: Elementary functions. Vol. 1. Fizmtalit, Moscow (2002), Second Edition. (In Russian, Engl. Translation of the First Edition is: New York: Gordon and Breach Science Publishers, 1986)Google Scholar
  57. Reynolds, W.D.: A generalized variable-head borehole permeameter analysis for saturated, unsaturated, rigid or deformable porous media. Eng. Geol. 185, 10–19 (2015)CrossRefGoogle Scholar
  58. Rode, A.A.: The Soil-forming Process and Soil Evolution. Moscow, OGIZ 1947. (in Russian). (Engl transl.: Israel Program Sci. Transl., Jerusalem)Google Scholar
  59. Ruoff, A.L., Prince, D.L., Giddings, J.C., Stewart, G.H.: The diffusion analogy for solvent flow in paper. Kolloid-Zeitschrift 166(2), 144–151 (1959)CrossRefGoogle Scholar
  60. Samal, K.P., Mishra, G.C.: Analysis of seepage from a triangular furrow considering soil capillarity using inverse hodograph and conformal mapping technique. ISH J. Hydraul. Eng. 23(1), 1–12 (2017)CrossRefGoogle Scholar
  61. Schulz, S., Horovitz, M., Rausch, R., Michelsen, N., Mallast, U., Köhne, M., Siebert, C., Schüth, C., Al-Saud, M., Merz, R.: Groundwater evaporation from salt pans: examples from the eastern Arabian Peninsula. J. Hydrol. 531, 792–801 (2015)CrossRefGoogle Scholar
  62. Shokri, N., Lehmann, P., Or, D.: Evaporation from layered porous media. J. Geophys. Res. B Solid Earth 115(6), B06204 (2010)Google Scholar
  63. Shokri, N., Or, D.: What determines drying rates at the onset of diffusion controlled stage-2 evaporation from porous media? Water Resour. Res. 47(9), W09513 (2011)CrossRefGoogle Scholar
  64. Shokri, N., Or, D.: Drying patterns of porous media containing wettability contrasts. J. Colloid Interface Sci. 391(1), 135–141 (2013)CrossRefGoogle Scholar
  65. Shou, D., Fan, J.: Structural optimization of porous media for fast and controlled capillary flows. Phys. Rev. E 91(5), 053021 (2015)CrossRefGoogle Scholar
  66. Shou, D., Ye, L., Fan, J., Fu, K.: Optimal design of porous structures for the fastest liquid absorption. Langmuir 30(1), 149–155 (2013)CrossRefGoogle Scholar
  67. Šimůnek, J., Van Genuchten, M.T., Šejna, M.: Recent developments and applications of the HYDRUS computer software packages. Vadose Zone J. 15(7), 25 (2016)Google Scholar
  68. Singer, A., Banin, A., Poberejzsky, L., Gilenko, M.: Soil crusts in the Amudarya River Delta: properties and formation. In: Breckle, S.W., Veste, M., Wucherer, W. (eds.) Sustainable Land Use in Deserts, pp. 103–114. Springer, Berlin (2001)CrossRefGoogle Scholar
  69. Strack, O.D.L.: Distributed sources for unconfined groundwater flow in a half-space. J. Hydrol. 39(3–4), 239–253 (1978)CrossRefGoogle Scholar
  70. Strack, O.D.L.: Analytical Groundwater Mechanics. Cambridge Univ Press, Cambridge (2017)CrossRefGoogle Scholar
  71. Tóth, J.: Gravitational Systems of Groundwater Flow: Theory, Evaluation, Utilization. Cambridge University Press, Cambridge (2009)CrossRefGoogle Scholar
  72. Vedernikov, V.V.: Theory of Seepage and Its Application in Irrigation and Drainage. Gosstrojizdat, Moscow (1939). (in Russian) Google Scholar
  73. Youngs, E.G.: Effect of the capillary fringe on steady-state water tables in drained lands. J. Irrig. Drain. Eng. ASCE 138(9), 809–814 (2012)CrossRefGoogle Scholar
  74. Youngs, E.G., Elrick, D.E., Reynolds, W.D.: Comparison of steady flows from infiltration rings in “Green and Ampt” and “Gardner” soils. Water Resour. Res. 29(6), 1647–1650 (1993)CrossRefGoogle Scholar
  75. Wang, W., Zhang, Z., Yeh, T.C.J., Qiao, G., Wang, W., Duan, L., Huang, S.Y., Wen, J.C.: Flow dynamics in vadose zones with and without vegetation in an arid region. Adv. Water Resour. 106, 68–79 (2017)CrossRefGoogle Scholar
  76. Willis, W.O.: Evaporation from layered soils in the presence of a water table. Soil Sci. Soc. Am. J. 24(4), 239–242 (1960)CrossRefGoogle Scholar
  77. Wolfram, S.: Mathematica. A System for Doing Mathematics by Computer. Addison-Wesley, Redwood City (1991)Google Scholar
  78. Xiao, J., Stone, H.A., Attinger, D.: Source-like solution for radial imbibition into a homogeneous semi-infinite porous medium. Langmuir 28(9), 4208–4212 (2012)CrossRefGoogle Scholar
  79. Yechieli, Y., Wood, W.W.: Hydrogeologic processes in saline systems: playas, sabkhas, and saline lakes. Earth Sci. Rev. 58(3–4), 343–365 (2002)CrossRefGoogle Scholar
  80. Zhang, Z., Wang, W., Wang, Z., Chen, L., Gong, C.: Evaporation from bare ground with different water-table depths based on an in situ experiment in Ordos Plateau, China. Hydrogeol. J. 26, 1683–1691 (2018)CrossRefGoogle Scholar
  81. Zonn, I.S., Esenov, P.E.: The Karakum Desert. In: Zonn, I.S., Kostianoy, A.G. (eds.) The Turkmen Lake Altyn Asyr and water resources in Turkmenistan, pp. 23–37. Springer, Berlin (2012)CrossRefGoogle Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Soils, Water and Agricultural EngineeringSultan Qaboos UniversityMuscatSultanate of Oman
  2. 2.Institute of Mathematics and MechanicsKazan Federal UniversityKazanRussia
  3. 3.Department of Environmental Systems ScienceETH ZurichZurichSwitzerland

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