Entropy solutions and traveling-wave stability for two-phase porous-media fronts with hysteresis and dynamic capillarity: the non-monotone flux regime and the large-dynamic-capillarity instability

The 2020 analysis of Mitra, Köppl, Pop, van Duijn and Helmig established traveling-wave (TW) solutions and the entropy solutions they induce for one-dimensional two-phase flow through a porous column when capillary hysteresis, dynamic capillarity, and (in one scenario) hysteresis in the relative permeabilities are included, but only in the regime where the fractional-flow function is monotone (small gravity number) and, when permeabilities are hysteretic, only for sufficiently small dynamic-capillary number. Two directions were explicitly left open: (i) in Scenario A (no hysteresis in the relative permeabilities), when the gravity number is large so that the flux is non-monotone, a systematic classification of the admissible entropy solutions — patterns with up to three shocks moving both forward and backward, rarefaction waves propagating in opposite directions, and stationary shocks, together with their wave speeds and selection rules — and (ii) the onset and mechanism of the numerically observed instability of the traveling waves at large values of the dynamic-capillary number when hysteresis is present in the permeabilities. Both directions remain open: the closest follow-up (Mitra and van Duijn 2021) resolves the stationary-shock admissibility question only for the simpler gravity-segregation hyperbolic limit without dynamic capillarity or injection, and no later work settles either the full classification or the instability onset and mechanism.

Background

In two-phase flow through a porous medium, the state of the system is described by the saturation $S\in[0,1]$ of the wetting phase (the fraction of pore volume it occupies). Extending Darcy's law to two immiscible phases and eliminating the phase velocities in favor of a prescribed total velocity yields, in the limit of vanishing capillary diffusion, the Buckley–Leverett equation: a scalar conservation law $\partial_t S+\partial_z F(S)=0$ whose flux $F(S)=f(S)+N_g h(S)$ combines a viscous fractional-flow contribution $f(S)$ with a gravity contribution $h(S)$ ; the dimensionless coefficient $N_g$ is the gravity number. Solving this equation with piecewise-constant initial data (a Riemann problem) produces shock waves (jump discontinuities whose speed is fixed by the Rankine–Hugoniot condition, i.e. conservation of mass across the jump) and rarefaction waves (continuous self-similar fans); the physically admissible solution is singled out by the classical Oleinik entropy condition, obtained as the vanishing-capillarity limit of the diffusive model.

The standard equilibrium model, however, ignores two well-documented non-equilibrium effects. First, the capillary pressure (the pressure difference between the phases) is hysteretic: the pressure–saturation curve measured during infiltration (wetting) differs from that measured during drainage (drying). The play-type hysteresis model idealizes this by two boundary curves $p_c^{(i)}(S)$ and $p_c^{(d)}(S)$ with vertical scanning curves: during infiltration the pressure follows $p_c^{(i)}(S)$ , during drainage $p_c^{(d)}(S)$ , and when the process switches the pressure may take any value in the hysteresis interval between the two curves. Second, dynamic capillarity: the Hassanizadeh–Gray nonequilibrium closure replaces the algebraic capillary-pressure relation by $p_c(S)-\tau\,\partial_t S$ , where $\tau>0$ is the dynamic capillary number quantifying the relaxation of the capillary pressure toward equilibrium. Hysteresis also arises, more weakly, in the relative permeabilities (the phase mobilities appearing in $f$ ).

The mathematical engine for these effects is the traveling-wave (TW) approximation: a front connecting a left state $S_B$ and a right state $S_T$ is sought as a profile depending on $z-ct$ , which turns the PDE system into a dynamical system in the $(S,p)$ plane. As capillary diffusion tends to zero, the existence and shape of such profiles dictate which shocks of the hyperbolic limit are admissible, in the same way that viscous profiles select Oleinik entropy solutions. Without hysteresis, van Duijn, Peletier and Pop (2007) showed that a non-monotone flux (large gravity) yields a new class of entropy solutions with nonclassical shocks, and Spayd and Shearer (2011) and Shearer, Spayd and Swanson (2015) constructed and linearly stabilized TW profiles for dynamic capillary pressure. With play-type hysteresis combined with dynamic capillarity, Behi-Gornostaeva, Mitra and Schweizer (2019) and Mitra and van Duijn (2019) analyzed wetting fronts for the unsaturated (Richards) case, and Mitra (2021) proved well-posedness of an extended play-type model.

The source paper (Mitra, Köppl, Pop, van Duijn, Helmig, Studies in Applied Mathematics 2020) carried this program to the full two-phase setting under two scenarios. In Scenario A only the capillary pressure is hysteretic; the flux $F(S)$ is monotone because the gravity number is small, and TWs are shown to exist for the whole range of $\tau$ , becoming non-monotone (saturation overshoot, plateau structures between an infiltration and a drainage front) beyond an explicit threshold in $\tau$ . The induced entropy solutions of the hyperbolic limit include infiltration shocks followed by rarefactions of non-matching speed and shock pairs enclosing growing overshoot plateaus — patterns impossible in the classical theory. In Scenario B, hysteresis is included in both the capillary pressure and the relative permeabilities, making the flux depend on the pair $(S,p)$ ; here TW existence is proved (Propositions 11–12) only for $\tau$ below the explicit bounds $\tau^*_i(S_T)$ and $\tau^*_d(S_T)$ , while the stability of the profiles in that range is indicated only by the paper's numerical computations. The paper closes by identifying, in its own words, what is left for future research: the case of large gravity number, where $F(S)$ is non-monotone and wave patterns can involve up to three shocks traveling both forward and backward, rarefaction waves with endpoints moving in opposite directions, and stationary shocks — 'a detailed analysis is much more involved and therefore left for future research' — and the observation (Section 5.1.1) that 'instability of the TWs is observed for large $\tau$ when hysteresis is included in the permeabilities. This requires further attention and we have excluded this possibility in our current analysis and computations.' A first step in this direction was taken by Mitra and van Duijn (2021), who, for the pure gravity-segregation configuration (no injection, zero total flux) in the hyperbolic limit with capillary hysteresis only, derived the admissibility criterion for stationary shocks (continuity of the flux and of the hysteretic pressure across the shock) and classified all admissible stationary shocks, obtaining two Riemann-solution families including solutions frozen in time in a half-line. That analysis, however, contains no dynamic capillarity, no permeability hysteresis, and no infiltration (injection) configuration, so it resolves only the stationary-shock piece of the outlook in a strictly simpler setting.

Problem Statement

Complete the traveling-wave and entropy-solution theory of the one-dimensional two-phase porous-media model with play-type capillary hysteresis, dynamic capillarity (coefficient $\tau$ ), and — in Scenario B — play-type hysteresis in the relative permeabilities, in the two regimes excluded from the analysis of Mitra, Köppl, Pop, van Duijn and Helmig (2020). Both of these linked deliverables are required to resolve the task, and jointly they serve one objective: to extend the TW-based construction of admissible entropy solutions from the monotone-flux, small- $\tau$ regime to the full parameter range of the same model — deliverable (1) classifying the admissible entropy solutions in the non-monotone-flux regime and deliverable (2) settling the large- $\tau$ stability of the traveling waves of Scenario B.

(1) Non-monotone flux regime. In Scenario A of the source (no hysteresis in the relative permeabilities; flux $F(S)=f(S)+N_g h(S)$ with capillary-pressure hysteresis and dynamic capillarity), when the gravity number $N_g$ is large so that the fractional-flow function $F$ is non-monotone, classify the admissible entropy solutions of the Riemann problem for the hyperbolic limit of the system. This requires: determining, for the traveling-wave (dynamical-system) formulation with capillary hysteresis and dynamic capillarity, which multi-wave patterns exist and are admissible — including patterns with up to three shocks traveling both forward and backward, rarefaction waves with endpoints moving in opposite directions, and stationary shocks, as well as their combinations with saturation plateaus — together with the wave speeds (Rankine–Hugoniot speeds and their hysteretic modifications), the admissibility (entropy) criteria for each shock, and the selection rules that pick the physically relevant solution among competing patterns; and proving existence or nonexistence of the corresponding traveling-wave profiles so that the classification is complete rather than conjectural. The known gravity-segregation result of Mitra and van Duijn (2021) covers only the zero-total-flux hyperbolic limit with capillary hysteresis alone; the requested classification concerns the infiltration configuration with dynamic capillarity.

(2) Large- $\tau$ instability in Scenario B. For the scenario with hysteresis in both the capillary pressure and the relative permeabilities, traveling-wave orbits are proved to exist only for sufficiently small $\tau$ ( $\tau<\tau^*_i(S_T)$ , Proposition 11, and $\tau<\tau^*_d(S_T)$ , Proposition 12), while the stability of the profiles in that range is supported only by the paper's numerical computations; those computations also indicate that the traveling waves become unstable for large $\tau$ . The published source states, in its Section 5.1.1: 'instability of the TWs is observed for large $\tau$ when hysteresis is included in the permeabilities. This requires further attention and we have excluded this possibility in our current analysis and computations.' Characterize the onset of this instability — the range of $\tau$ in which the traveling waves lose stability, in relation to the boundary states $S_B$ , $S_T$ and the capillary-pressure and permeability data — and identify the mechanism of the instability (for example, through a linearized stability analysis of the traveling-wave profiles of the coupled hysteresis system, extended where needed beyond small perturbations). Both sub-results of this deliverable are required — the characterization of the onset (the range of $\tau$ in which the traveling waves lose stability) and the identification of the mechanism of the instability; neither alone resolves the deliverable.

References

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