Abstract
A sequence of drainage and imbibition shocks within the traditional theory of two-phase immiscible displacement can give rise to shallow non-monotone saturation profiles as shown in Hilfer and Steinle (Eur Phys J Spec Top 223:2323, 2014). This phenomenon depends sensitively on model parameters and initial conditions. The dependence of saturation overshoot on initial conditions is investigated more systematically in this article. The results allow to determine regions in the parameter space for the observation of saturation overshoot and to explore limitations of the underlying idealized hysteresis model. Numerical solutions of the nonlinear partial differential equations of motion reveal a strong dependence of the overshoot phenomenon on the boundary and initial conditions. Overshoot solutions with experimentally detectable height are shown to exist numerically. Extensive parameter studies reveal different classes of initial conditions for which the width of the overshoot region can decrease, increase or remain constant.
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References
Alt, H., Luckhaus, S., Visintin, A.: On nonstationary flow through porous media. Ann. Mat. Pura Appl. 136, 303 (1984)
Alt, H., Luckhaus, S.: Quasilinear elliptic-parabolic differential equations. Math. Z. 183, 311 (1983)
Beliaev, A., Hassanizadeh, S.: A theoretical model of hysteresis and dynamic effects in the capillary relation for two-phase flow in porous media. Transp. Porous Media 43, 487 (2001)
Cueto-Felgueroso, L., Juanes, R.: Nonlocal interface dynamics and pattern formation in gravity-driven unsaturated flow through porous media. Phys. Rev. Lett. 101, 244504 (2008)
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)
di Carlo, D.: Stability of gravity-driven multiphase flow in porous media: 40 years of advancements. Water Resour. Res. 49, 4531 (2013)
Duijn, C., Peletier, L., Pop, I.: A new class of entropy solutions of the Buckley–Leverett equation. SIAM J. Math. Anal. 39, 507 (2007)
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 (2013)
Egorov, A., Dautov, R., Nieber, J., Sheshukov, A.: Stability analysis of gravity-driven infiltrating flow. Water Resour. Res. 39, 1266 (2003)
Eliassi, M., Glass, R.: 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, 1234 (2002)
Hilfer, R.: Macroscopic capillarity and hysteresis for flow in porous media. Phys. Rev. E 73, 016307 (2006)
Hilfer, R.: Macroscopic capillarity without a constitutive capillary pressure function. Phys. A 371, 209 (2006)
Hilfer, R., Steinle, R.: Saturation overshoot and hysteresis for twophase flow in porous media. Eur. Phys. J. Spec. Top. 223, 2323 (2014)
Otto, F.: L 1-contraction and uniqueness for quasilinear elliptic-parabolic equations. J. Differ. Equ. 131, 20 (1996)
Rätz, A., Schweizer, B., Angew, Z.: Hysteresis models and gravity fingering in porous media. Math. Mech. 94, 645 (2014)
**ong, Y.: Flow of water in porous media with saturation overshoot: a review. J. Hydrol. 510, 353 (2014)
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The authors acknowledge financial support from the Deutsche Forschungsgemeinschaft.
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Steinle, R., Hilfer, R. Influence of Initial Conditions on Propagation, Growth and Decay of Saturation Overshoot. Transp Porous Med 111, 369–380 (2016). https://doi.org/10.1007/s11242-015-0598-2
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DOI: https://doi.org/10.1007/s11242-015-0598-2