Vanishing-capillarity limit of gravity-driven two-phase flow with capillary hysteresis: non-play-type closures, relative-permeability hysteresis, and rigorous convergence

For gravity-driven counter-current flow of two immiscible phases in a one-dimensional porous column, the saturation obeys a Buckley–Leverett equation with a non-monotone fractional-flow function. Mitra and van Duijn (Comput. Geosci. 26:101–114, 2022) showed that when the capillary-pressure relation is hysteretic (play-type model), the vanishing-capillarity (hyperbolic) limit of the parabolic regularization need not be the classical entropy solution: stationary shocks occur, and in one regime the solution is frozen in a semi-infinite half of the domain. Their construction is restricted to play-type closures (or closures satisfying a directional-selection property), omits hysteresis in the relative permeabilities, is supported by travelling-wave admissibility arguments and numerics rather than a convergence theorem, and covers only homogeneous one-dimensional media. The open problem is to complete this selection theory: determine the admissible vanishing-capillarity solutions for hysteresis closures outside the directional-selection class (4.11), incorporate relative-permeability hysteresis with nonlinear pressure–saturation relations, prove convergence of the capillarity-regularized solutions for non-monotone fluxes in the structural hypothesis class of the source, and (optionally) extend the theory beyond one-dimensional homogeneous columns. No work resolving any of these cores is known in the literature to date.

Background

When two immiscible fluid phases (a wetting and a non-wetting phase, e.g. water and methane) flow through a porous medium with no injection at the boundaries, the flow is purely counter-current and driven by gravity: the heavier phase migrates downward, the lighter upward. Writing $S(x,t)\in[0,1]$ for the saturation of the wetting phase (the fraction of pore space it occupies), the hyperbolic limit of this process is the Buckley–Leverett equation

$$\partial_t S+\partial_x h(S)=0,$$

where the fractional-flow function $h$ encodes the relative permeabilities (the phase-dependent effective conductivities of the medium, one for each phase) and the viscosities of the two phases. In the zero-total-flux, gravity-driven case $h$ is non-monotone: it increases, reaches a maximum, and decreases again. With Riemann initial data — constant saturations $S_T$ for $x<0$ and $S_B$ for $x>0$ — the physically relevant solution is selected by the vanishing capillarity method: one solves the capillary Buckley–Leverett equation

$$\partial_t S+\partial_x\bigl[h(S)(1+\delta\,\partial_x p)\bigr]=0,$$

where $p$ is the pressure difference between the phases and $\delta>0$ is a dimensionless capillary number (inversely proportional to the column length), and studies the limit $\delta\to 0$. In the classical equilibrium approach $p=p_c(S)$ with a single-valued capillary pressure function, this reproduces the standard entropy (Oleinik-admissible) solution, and the method parallels the classical vanishing viscosity method for conservation laws; the limit solution consists of shocks, rarefaction waves and constant states. Capillary pressure, however, is hysteretic: the pressure–saturation relation differs between imbibition (wetting, $\partial_tS>0$) and drainage ($\partial_tS<0$), with $p_c^{(i)}(S)<p_c^{(d)}(S)$ bounding a hysteresis region $\mathcal{H}$ in the $(S,p)$-plane. The play-type model (Beliaev–Hassanizadeh) prescribes $p=p_c^{(i)}(S)$ during imbibition, $p=p_c^{(d)}(S)$ during drainage, and $p\in[p_c^{(i)}(S),p_c^{(d)}(S)]$ when $\partial_tS=0$ (the 'undetermined' state); switching between branches occurs along vertical scanning curves. Extended models such as the Lenhard–Parker and extended play-type closures describe the interior of the hysteresis loop by non-vertical scanning curves with their own internal dynamics; nonetheless, both satisfy the directional-selection property (4.11) quoted below and are therefore already covered by the source's theory (Remark 4.2 of the source). Mitra and van Duijn (2022) constructed the $\delta\to0$ limit of the Riemann problem with play-type hysteresis for non-monotone $h$. The hysteretic limit differs sharply from the classical one: stationary shocks (discontinuities fixed at the initial discontinuity $x=0$) become possible, and in one identified regime the solution is frozen in time in one semi-infinite half of the domain. The authors derived an admissibility condition for such shocks from travelling-wave analysis (the flux $h$ and the pressure corresponding to the hysteretic state must remain continuous across the shock) and verified the constructions numerically. They proved no convergence theorem, and their travelling-wave argument relies on the play-type structure through a directional-selection property (their Eq. (4.11): $p=p_c^{(i)}(S)$ implies $\partial_tS\ge0$, $p=p_c^{(d)}(S)$ implies $\partial_tS\le0$), satisfied by play-type, extended play-type and Lenhard–Parker closures. Related prior results: van Duijn and Mitra (2018) analysed hysteretic horizontal redistribution (no gravity); Mitra et al. (2020) treated travelling-wave fronts (propagating fronts of fixed profile, analysed in a frame moving with the front) for monotone fluxes with hysteresis; Andreianov and Cancès (2013) obtained vanishing-capillarity solutions with stationary discontinuities for heterogeneous media composed of two homogeneous rock types ('two-rocks' media) without hysteresis; Schaerer et al. (2006) analysed gravity counterflow segregation with permeability hysteresis but only for a linear pressure–saturation relation; Mitra (2021) proved existence for the extended play-type regularization of the parabolic problem.

Problem Statement

The overarching objective is to complete the selection theory for the vanishing-capillarity ($\delta\to0$) limit of the Riemann problem for gravity-driven two-phase flow in a porous column when the pressure–saturation relation is hysteretic. Mitra and van Duijn solved the one-dimensional homogeneous, play-type case by travelling-wave admissibility arguments and numerical verification; the problem as left by that work has the following linked components. Components (1)–(3) all concern the same model system (the Buckley–Leverett equation $\partial_tS+\partial_xh(S)=0$ with non-monotone $h$, its capillary regularization $\partial_tS+\partial_x[h(S)(1+\delta\partial_xp)]=0$ coupled with a hysteretic pressure–saturation relation, and Riemann data); component (4) goes beyond this one-dimensional homogeneous setting:

(1) Hysteresis closures beyond the directional-selection class. For closures that do not satisfy the directional-selection property (4.11) — note that the play-type, extended play-type and Lenhard–Parker closures all satisfy (4.11) (Remark 4.2 of the source) and are therefore already covered by the existing theory, even though the extended models have non-vertical scanning curves — determine the admissible shocks and the vanishing-capillarity solutions of the Riemann problem. Do the stationary shocks and the frozen-half solutions persist, and does the continuity-of-flux-and-hysteretic-pressure criterion derived for the play-type case still characterize admissible shocks; if not, what replaces it?

(2) Relative-permeability hysteresis. Incorporate hysteresis in the relative-permeability functions $k_\alpha(S)$, together with nonlinear capillary-pressure–saturation relations, into the hyperbolic theory, and determine how such hysteresis participates in selecting the vanishing-capillarity solutions and the admissible stationary shocks. Prior treatments of permeability hysteresis in gravity counterflow cover only linear pressure–saturation relations; the nonlinear case is open.

(3) Rigorous convergence. Prove (or disprove) existence, uniqueness, and convergence, as $\delta\to0$, of solutions of the capillarity-regularized hysteretic problem to the constructed vanishing-capillarity solutions, replacing the current travelling-wave heuristics and numerical evidence with a theorem. The object is the rate-independent regularized problem $\partial_tS+\partial_x[h(S)(1+\delta\partial_xp)]=0$ coupled with the hysteretic closure — not the $\tau$-relaxed closure $p=p_c^{(j)}(S)-\tau\,\partial_tS$ used only in the source's numerical computations — with $h$ and $p_c^{(i)},p_c^{(d)}$ ranging over the structural hypothesis class of the source, for which the shock catalogue and the constructed limit solutions are defined: (P1) $p_c^{(i)},p_c^{(d)}\in C^1$, strictly decreasing, with $p_c^{(i)}(S)<p_c^{(d)}(S)$ on $(0,1)$ and $p_c^{(i)}(1)=p_c^{(d)}(1)=0$; (P2) $h$ smooth with $h(0)=h(1)=0$, exactly one interior maximum $S_M$ ($h'>0$ on $(0,S_M)$, $h'<0$ on $(S_M,1)$) and exactly two inflection points $0<S_1<S_M<S_2<1$ ($h''>0$ on $(0,S_1)\cup(S_2,1)$, $h''<0$ on $(S_1,S_2)$). Uniqueness is requested for closures for which the regularized problem is well posed (currently known for play-type and extended play-type closures).

(4) Beyond one-dimensional homogeneous media (an optional extension, not part of the mandatory core). Extend the admissibility theory and the vanishing-capillarity constructions to heterogeneous media and to higher spatial dimensions, where geometric heterogeneity and multidimensional flow patterns may alter which shocks are admissible and may produce additional non-classical phenomena. In this component the shared model system of (1)–(3) no longer applies: heterogeneity and higher dimensions require their own model, data, and notion of admissible shock.

Components (1)–(3) constitute the mandatory core of the one-dimensional hyperbolic-limit theory; a solution to the task is complete when it resolves all three, while resolving a proper subset of them constitutes a partial solution. Component (4) is an optional extension beyond this core and is not required for completion.

References

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2. B. Andreianov and C. Cancès, 'Vanishing capillarity solutions of Buckley–Leverett equation with gravity in two-rocks' medium', Computational Geosciences 17(3):551–572 (2013). DOI: 10.1007/s10596-012-9329-8. https://doi.org/10.1007/s10596-012-9329-8
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