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Stable Propagation of Saturation Overshoots for Two-Phase Flow in Porous Media

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Abstract

Propagation of saturation overshoots for two-phase flow of immiscible and incompressible fluids in porous media is analyzed using different computational methods. In particular, it is investigated under which conditions a given saturation overshoot remains stable while moving through a porous medium. Two standard formulations are employed in this investigation, a fractional flow formulation and a pressure–saturation formulation. Neumann boundary conditions for pressure are shown to emulate flux boundary conditions in homogeneous media. Gravity driven flows with Dirichlet boundary conditions for pressure that model infiltration into heterogeneous media with position-dependent permeability are found to exhibit pronounced saturation overshoots very similar to those seen in experiment.

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References

  • Alt, H., Luckhaus, S.: Quasilinear elliptic–parabolic differential equations. Math. Z. 183, 311 (1983)

    Article  Google Scholar 

  • Alt, H., Luckhaus, S., Visintin, A.: On nonstationary flow through porous media. Ann. Mat. Pura Appl. 136, 303 (1984)

    Article  Google Scholar 

  • Bastian, P., Heimann, F., Marnach, S.: Generic implementation of finite element methods in the distributed and unified numerics environment (Dune). Kybernetika 46(2), 294–315 (2010)

    Google Scholar 

  • Beliaev, A.Y., Hassanizadeh, S.M.: A theoretical model of hysteresis and dynamic effects in the capillary relation for two-phase flow in porous media. Transp. Porous Media 43(3), 487–510 (2001)

    Article  Google Scholar 

  • Boudet, H., Clarke, C., Bugden, D., Maibach, E., Roser-Renouf, C., Leiserowitz, A.: Fracking controversy and communication: using national survey data to understand public perceptions of hydraulic fracturing. Energy Policy 65, 57–67 (2014)

    Article  Google Scholar 

  • Briggs, J., Katz, D.: Drainage of water from sand in developing aquifer storage. Paper SPE1501 presented 1966 at the 41st Annual Fall Meeting of the SPE, Dallas, USA (1966)

  • Charbeneau, R.: Groundwater Hydraulics and Pollution Transport. Prentice Hall, Upper Saddle River (2000)

    Google Scholar 

  • Cueto-Felgueroso, L., Juanes, R.: Stability analysis of a phase-field model of gravity-driven unsaturated flow through porous media. Phys. Rev. E 79, 036301 (2009)

    Article  Google Scholar 

  • DiCarlo, D.: Experimental measurements of saturation overshoot on infiltration. Water Resour. Res. 40, W04215 (2004)

    Article  Google Scholar 

  • DiCarlo, D.: Stability of gravity-driven multiphase flow in porous media: 40 years of advancements. Water Resour. Res. 49, 4531 (2013)

    Article  Google Scholar 

  • DiCarlo, D., Mirzaei, M., Aminzadeh, B., Dehghanpur, H.: Fractional flow approach to saturation overshoot. Transp. Porous Media 91, 955 (2012)

    Article  Google Scholar 

  • Doster, F., Zegeling, P., Hilfer, R.: Numerical solutions of a generalized theory for macroscopic capillarity. Phys. Rev. E 81, 036307 (2010)

    Article  Google Scholar 

  • Dullien, F.A.L.: Porous Media—Fluid Transport and Pore Structure. Academic Press, San Diego (1992)

    Google Scholar 

  • Egorov, A., Dautov, R., Nieber, J., Sheshukov, A.: Stability analysis of gravity-driven infiltrating flow. Water Resour. Res. 39, 1266 (2003)

    Article  Google Scholar 

  • Eliassi, M., Glass, R.: On the continuum-scale modeling of gravity-driven fingers in unsaturated porous media: The inadequacy of the Richards equation with standard monotonic constitutive relations and hysteretic equations of state. Water Resour. Res. 37, 2019 (2001)

    Article  Google Scholar 

  • Eliassi, M., Glass, R.J.: On the porous-continuum modeling of gravity-driven fingers in unsaturated materials: Extension of standard theory with a hold-back-pile-up effect. Water Resour. Res. 38(11), 16-1–16-11 (2002)

  • Evans, L.C.: Partial Differential Equations. Graduate studies in mathematics. American Mathematical Society, Providence (1998)

    Google Scholar 

  • Flemisch, B., Darcis, M., Erbertseder, K., Faigle, B., Lauser, A., Mosthaf, K., Müthing, S., Nuske, P., Tatomir, A., Wolff, M., Helmig, R.: DuMux: DUNE for multi-\(\{\)phase, component, scale, physics,.\(\}\) flow and transport in porous media. Adv. Water Resour. 34(9), 1102–1112 (2011)

    Article  Google Scholar 

  • Fritz, S.: Experimental Investigations of Water Infiltration into Unsaturated Soil—Analysis of Dynamic Capillarity Effects. Master’s thesis, Universität Stuttgart, Department of Hydromechanics and Modelling of Hydrosystems (2012)

  • Fürst, T., Vodak, R., Sir, M., Bil, M.: On the incompatibility of Richards’ equation and finger-like infiltration in unsaturated homogeneous porous media. Water Resour. Res. 45, W03408 (2009)

    Article  Google Scholar 

  • Glass, R., Steenhuis, T., Parlange, J.: Mechanism for finger persistence in homogeneous unsaturated, porous media: theory and verification. Soil Sci. 148, 60 (1989)

    Article  Google Scholar 

  • Helmig, R.: Multiphase Flow and Transport Processes in the Subsurface: A Contribution to the Modeling of Hydrosystems. Springer, Berlin (1997)

    Book  Google Scholar 

  • Hilfer, R., Doster, F., Zegeling, P.: Nonmonotone saturation profiles for hydrostatic equilibrium in homogeneous media. Vadose Zone J. (2012). https://doi.org/10.2136/vzj2012.0021

  • Hilfer, R., Steinle, R.: Saturation overshoot and hysteresis for twophase flow in porous media. Eur. Phys. J. Spec. Top. 223(11), 2323–2338 (2014)

    Article  Google Scholar 

  • Jasak, H., Jemcov, A., Tukovic, Z.: OpenFOAM: A C++ library for complex physics simulations. In: International Workshop on Coupled Methods in Numerical Dynamics, vol. 1000, pp. 1–20 (2007)

  • Kalaydjian, F.J-M. et al. Dynamic capillary pressure curve for water/oil displacement in porous media: theory vs. experiment. In: SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers (1992)

  • Koch, J., Rätz, A., Schweizer, B.: Two-phase flow equations with a dynamic capillary pressure. Eur. J. Appl. Math. 24(01), 49–75 (2013)

    Article  Google Scholar 

  • Lamacz, A., Rätz, A., Schweizer, B.: A well-posed hysteresis model for flows in porous media and applications to fingering effects. Adv. Math. Sci. Appl. 21(01), 33–64 (2011)

    Google Scholar 

  • Luckner, L., Van Genuchten, M.T., Nielsen, D.R.: A consistent set of parametric models for the two-phase flow of immiscible fluids in the subsurface. Water Resour. Res. 25(10), 2187–2193 (1989)

    Article  Google Scholar 

  • Manthey, S., Hassanizadeh, S.M., Helmig, R., Hilfer, R.: Dimensional analysis of two-phase flow including a rate-dependent capillary pressure–saturation relationship. Adv. Water Resour. 31(9), 1137–1150 (2008)

    Article  Google Scholar 

  • Middendorf, J.: Zur Beschreibung des kapillaren Flüssigkeitstransports in Papier. Ph.D. thesis, Fakultät für Maschinenbau und Verfahrenstechnik, Technische Universität Chemnitz, Germany (2000)

  • Nieber, J.L., Dautov, R.Z., Egorov, A.G., Sheshukov, A.Y.: Dynamic capillary pressure mechanism for instability in gravity-driven flows; review and extension to very dry conditions. Transp. Porous Media 58(1–2), 147–172 (2005)

    Article  Google Scholar 

  • Otto, F.: L\(^{1}\)-contraction and uniqueness for quasilinear elliptic–parabolic equations. J. Differ. Equ. 131, 20 (1996)

    Article  Google Scholar 

  • Rätz, A., Schweizer, B.: Hysteresis models and gravity fingering in porous media. ZAMM J. Appl. Math. Mech. 94(7–8), 645–654 (2014)

    Article  Google Scholar 

  • Schröder, N., Javaux, M., Vanderborght, J., Steffen, B., Vereecken, H.: Effect of root water and solute uptake on apparent soil dispersivity: A simulation study. Vadose Zone J. 11(3), 1–16 (2012)

    Article  Google Scholar 

  • Shiozawa, S., Fujimaki, H.: Unexpected water content profiles under flux-limited one-dimensional downward infiltration in initially dry granular media. Water Resour. Res. 40(7), W07404 (2004)

    Article  Google Scholar 

  • Steinle, R., Hilfer, R.: Influence of initial conditions on propagation, growth and decay of saturation overshoot. Transp. Porous Media 111(2), 369–380 (2016)

    Article  Google Scholar 

  • Steinle, R., Hilfer, R.: Hysteresis in relative permeabilities suffices for propagation of saturation overshoot: A quantitative comparison with experiment. Phys. Rev. E 95, 043112 (2017)

    Article  Google Scholar 

  • van Duijn, C., Fan, Y., Peletier, L., Pop, I.: Travelling wave solutions for degenerate pseudo-parabolic equations modelling two-phase flow in porous media. Nonlinear Anal. Real World Appl. 14, 1361–1383 (2013)

    Article  Google Scholar 

  • Van Duijn, C.J., Peletier, L.A., Pop, I.S.: A new class of entropy solutions of the Buckley–Leverett equation. SIAM J. Math. Anal. 39(2), 507–536 (2007)

    Article  Google Scholar 

  • Van Genuchten, M.T.: A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J. 44(5), 892–898 (1980)

    Article  Google Scholar 

  • Weishaupt, K., Beck, M., Becker, B., Class, H., Fetzer, T., Flemisch, B., Futter, G., Gläser, D., Grüninger, C., Hommel, J., Kissinger, A., Koch, T., Schneider, M., Schröder, N., Schwenck, N., Seitz, G.: Dumux 2.9.0 (2016)

  • Weller, H.G., Tabor, G., Jasak, H., Fureby, C.: A tensorial approach to computational continuum mechanics using object-oriented techniques. Comput. Phys. 12(6), 620–631 (1998)

    Article  Google Scholar 

  • Xiong, Y.: Flow of water in porous media with saturation overshoot: a review. J. Hydrol. 510, 353–362 (2014)

    Article  Google Scholar 

  • Youngs, E.G.: Redistribution of moisture in porous materials after infiltration: 2. Soil Sci. 86, 202–207 (1958)

    Article  Google Scholar 

Download references

Acknowledgements

We would like to thank Mike Celia for his comments and many fruitful discussions. This work was partially supported by the Cluster of Excellence in Simulation Technology (EXC 310/2), the Internationales Graduiertenkolleg NUPUS and the Deutsche Forschungsgemeinschaft.

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Authors and Affiliations

  1. Lehrstuhl für Hydromechanik und Hydrosystemmodellierung, Pfaffenwaldring 61, 70569, Stuttgart, Germany

    M. Schneider, T. Köppl & R. Helmig

  2. Institut für Computerphysik, Allmandring 3, 70569, Stuttgart, Germany

    R. Steinle & R. Hilfer

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  1. M. Schneider
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  2. T. Köppl
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  3. R. Helmig
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  4. R. Steinle
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  5. R. Hilfer
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Correspondence to M. Schneider.

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Schneider, M., Köppl, T., Helmig, R. et al. Stable Propagation of Saturation Overshoots for Two-Phase Flow in Porous Media. Transp Porous Med 121, 621–641 (2018). https://doi.org/10.1007/s11242-017-0977-y

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  • DOI: https://doi.org/10.1007/s11242-017-0977-y

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