Dynamic Capillary Pressure Mechanism for Instability in Gravity-Driven Flows; Review and Extension to Very Dry Conditions

Nieber, John L.; Егоров, А. Г.; Даутов, Р. З.; Sheshukov, Aleksey Y.

doi:10.1007/1-4020-3604-3_8

Cited by 36 publications

(73 citation statements)

References 45 publications

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“…Such non-equilibrium Richards equations (NERE) are studied e.g. in [10,17]; a low-frequency instability criterion is introduced and used to predict an instability in the NERE model. Once more, a low initial saturation is important for a spatial instability.…”

Section: Gravity Wetting Frontsmentioning

confidence: 99%

Instability of gravity wetting fronts for Richards equations with hysteresis

Schweizer¹

2012

Interfaces Free Bound.

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Abstract:We study the evolution of saturation profiles in a porous medium. When there is a more saturated medium on top of a less saturated medium, the effect of gravity is a downward motion of the liquid. While in experiments the effect of fingering can be observed, i.e. an instability of the planar front solution, it has been verified in different settings that the Richards equation with gravity has stable planar fronts.In the present work we analyze the Richards equation coupled to a play-type hysteresis model in the capillary pressure relation. Our result is that, in an appropriate geometry and with adequate initial and boundary conditions, the planar front solution is unstable. In particular, we find that the Richards equation with gravity and hysteresis does not define an L 1 -contraction.

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Section: Gravity Wetting Frontsmentioning

confidence: 99%

Instability of gravity wetting fronts for Richards equations with hysteresis

Schweizer¹

2012

Interfaces Free Bound.

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“…A strong motivation for these studies were mathematical results excluding the existence of overshoot profiles for the traditional Richards equation from hydrology (Nieber 1996;Otto 1996Otto , 1997Geiger and Durnford 2000;Glass 2001, 2002;Egorov et al 2003;DiCarlo 2005DiCarlo , 2013Duijn et al 2007;Juanes 2008, 2009). It has been conjectured that new theoretical approaches might be unavoidable (Eliassi and Glass 2002;Duijn et al 2007Duijn et al , 2013Juanes 2008, 2009;Hilfer et al 2012;Nieber et al 2005) to reconcile theory and experiment. On the other hand, a recent investigation (Hilfer and Steinle 2014) has uncovered the existence of saturation overshoot profiles within the traditional theory in the hyperbolic limit, while the Richards equation represents a special parabolic limit.…”

Section: Introductionmentioning

confidence: 99%

Non-monotonic Travelling Wave Fronts in a System of Fractional Flow Equations from Porous Media

et al. 2016

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Motivated by observations of saturation overshoot, this article investigates generic classes of smooth travelling wave solutions of a system of two coupled nonlinear parabolic partial differential equations resulting from a flux function of high symmetry. All boundary resp. limit value problems of the travelling wave ansatz, which lead to smooth travelling wave solutions, are systematically explored. A complete, visually and computationally useful representation of the five-dimensional manifold connecting wave velocities and boundary resp. limit data is found by using methods from dynamical systems theory. The travelling waves exhibit monotonic, non-monotonic or plateau-shaped behaviour. Special attention is given to the non-monotonic profiles. The stability of the travelling waves is studied by numerically solving the full system of the partial differential equations with an efficient and accurate adaptive moving grid solver.

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“…A first class of model assumes a Darcy-scale description of the multiphase flow and is based on various extensions of the Richards equation (RE). The stability of the standard RE to fingering has been shown and demonstrated rigorously by Eliassi & Glass (2001), Egorov et al (2003), van Duijn, Pieters & Raats (2004), Nieber et al (2005) and Fürst et al (2009). Several authors have modified RE successfully to reproduce fingering instabilities observed in the context of laboratory experiments (Nieber et al , 2005.…”

Section: Numerical Modelmentioning

confidence: 94%

“…The stability of the standard RE to fingering has been shown and demonstrated rigorously by Eliassi & Glass (2001), Egorov et al (2003), van Duijn, Pieters & Raats (2004), Nieber et al (2005) and Fürst et al (2009). Several authors have modified RE successfully to reproduce fingering instabilities observed in the context of laboratory experiments (Nieber et al , 2005. Chapwanya & Stockie (2010) showed that, in the context of RE, the growth of fingering instabilities requires the inclusion of non-equilibrium effects to correct for the evolution of the capillary pressure in response to saturation changes.…”

Section: Numerical Modelmentioning

confidence: 97%

Pore-scale mass and reactant transport in multiphase porous media flows

et al. 2011

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Reactive processes associated with multiphase flows play a significant role in mass transport in unsaturated porous media. For example, the effect of reactions on the solid matrix can affect the formation and stability of fingering instabilities associated with the invasion of a buoyant non-wetting fluid. In this study, we focus on the formation and stability of capillary channels of a buoyant non-wetting fluid (developed because of capillary instabilities) and their impact on the transport and distribution of a reactant in the porous medium. We use a combination of pore-scale numerical calculations based on a multiphase reactive lattice Boltzmann model (LBM) and scaling laws to quantify (i) the effect of dissolution on the preservation of capillary instabilities, (ii) the penetration depth of reaction beyond the dissolution/melting front, and (iii) the temporal and spatial distribution of dissolution/melting under different conditions (concentration of reactant in the non-wetting fluid, injection rate). Our results show that, even for tortuous non-wetting fluid channels, simple scaling laws assuming an axisymmetrical annular flow can explain (i) the exponential decay of reactant along capillary channels, (ii) the dependence of the penetration depth of reactant on a local Péclet number (using the non-wetting fluid velocity in the channel) and more qualitatively (iii) the importance of the melting/reaction efficiency on the stability of non-wetting fluid channels. Our numerical method allows us to study the feedbacks between the immiscible multiphase fluid flow and a dynamically evolving porous matrix (dissolution or melting) which is an essential component of reactive transport in porous media.

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Dynamic Capillary Pressure Mechanism for Instability in Gravity-Driven Flows; Review and Extension to Very Dry Conditions

Cited by 36 publications

References 45 publications

Instability of gravity wetting fronts for Richards equations with hysteresis

Instability of gravity wetting fronts for Richards equations with hysteresis

Non-monotonic Travelling Wave Fronts in a System of Fractional Flow Equations from Porous Media

Pore-scale mass and reactant transport in multiphase porous media flows

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