Deriving the maximum-mobility principle for immiscible viscous fingering in porous media

When water displaces a much more viscous oil in a porous medium, the displacement is unstable and breaks into viscous fingers. Sorbie, Al Ghafri, Skauge and Mackay (2020) proposed a now widely used simulation methodology in which an effective fractional flow $f_w^*$ is chosen to reproduce the high water saturations observed inside fingers, and the relative permeability pair is then selected from the infinite family consistent with $f_w^*$ as the one that maximizes the total mobility $\lambda_T$ (equivalently, minimizes pressure drop). This selection rule was stated only as a conjecture, with a fuller derivation and its connection to linear stability analysis promised for later work. Follow-up publications have validated the method against X-ray visualized slab-flood experiments, delivered the promised treatment of wettability and capillarity, and extended the approach to polymer flooding, CO2 storage, and field scale — but the maximum-mobility principle itself remains underived, its precise relation to stability theory is only partially established, and the effective fractional flow still has to be history-matched to each experiment rather than predicted. The open problem, explicitly left by Sorbie et al. (2020), is to complete this theory: derive the principle (or refute it and identify the correct selection rule) from the underlying two-phase flow equations and make precise its relation to linear stability analysis. The effective fractional flow $f_w^*$ is chosen by history matching in the source methodology itself, which recommends matching $f_w^*$ to observed finger saturations; whether an ab initio predictive determination (e.g. of the shock-front saturation $S_{wf}$) is possible is a related open direction raised by the follow-up literature, not part of the source's stated open question.

Background

Viscous fingering is the interfacial instability that develops when a low-viscosity fluid pushes a much more viscous one through a porous medium; it degrades sweep efficiency in oil recovery, contaminant remediation, and CO2 storage. At the Darcy scale — the continuum description in which flow obeys two-phase Darcy law — the unknowns are the phase pressures and saturations, coupled through relative permeabilities (RP), i.e. dimensionless functions $k_{rw}(S_w)$, $k_{ro}(S_w)$ of water saturation $S_w$ that reduce each phase's mobility, and a capillary pressure $P_c(S_w)$ relating the phase pressures. The phase mobilities are $\lambda_w = k_{rw}/\mu_w$ and $\lambda_o = k_{ro}/\mu_o$ (viscosities $\mu_w$, $\mu_o$), the total mobility is $\lambda_T = \lambda_w + \lambda_o$, and the fractional flow of water is $f_w = \lambda_w/\lambda_T$. In one dimension with $P_c = 0$, the classical Buckley–Leverett theory (Buckley and Leverett 1942) gives a shock-front saturation $S_{wf}$ by the tangent construction on $f_w(S_w)$. Instability is conventionally gauged by the mobility ratio $M = \lambda_w/\lambda_o$ across the front. A long-standing puzzle is what Sorbie et al. (2020) call the M-paradox: with conventional RP functions, e.g. of the Brooks–Corey power-law family, 1D Buckley–Leverett theory assigns the more viscous-oil case a lower shock-front mobility ratio $M(S_{wf})$ than a mild-viscosity case, contradicting the fact that only the former fingers violently — so conventional RP functions cannot describe unstable displacement. X-ray imaging of 2D sandstone-slab floods by Skauge and co-workers (e.g. Skauge et al. 2012) shows finger water saturations that are high (roughly $S_w \approx 0.3$–0.8), a fact 1D theory with conventional RPs cannot reproduce. Sorbie et al. (2020) therefore inverted the logic: choose the fractional flow $f_w^*$ so that its shock saturation matches observed finger saturations; among the infinitely many RP pairs $(k_{rw}^*, k_{ro}^*)$ consistent with that $f_w^*$ (they fix only the ratio $k_{ro}^*/k_{rw}^*$), select the pair that maximizes $\lambda_T(S_w)$, which minimizes the pressure drop across the fingering system; impose a random correlated permeability field with a chosen correlation length; and simulate on a sufficiently fine grid with simple upstream weighting. This produced realistic finger patterns, recoveries, watercuts and pressure drops, and — with no further assumptions — reproduced the large oil-recovery gains observed when the injected water is viscosified with polymer. Linear stability analysis (LSA), the study of the growth of infinitesimal perturbations of a base state, has long been the main theoretical tool for fingering onset; Chikhliwala and Yortsos (1988) derived LSA conditions on the RP curves for the onset of immiscible instability, and Sorbie et al. noted (but did not prove) that these conditions point to the same maximum-mobility choice. Subsequent work by the same group validated the method quantitatively against four slab floods with viscosity ratios $\mu_o/\mu_w \approx 400$–7000 (Salmo et al. 2022), delivered the promised study of wettability and capillarity — showing that the nonlinear capillary diffusion term $D(S_w) \sim dP_c/dS_w$ acting near the shock front suppresses fingers in water-wet but not oil-wet systems, with fingers re-emerging at field scale by Rapoport-type scaling — the classical capillary-to-viscous rescaling of laboratory displacements to field conditions (Beteta et al. 2024) — and demonstrated experimentally and numerically that it is the whole total-mobility profile, not just the endpoint viscosity ratio, that controls the instability in both miscible and immiscible systems (Beteta and Sorbie 2026). Recent work has also used LSA operationally to guide RP and $P_c$ selection when matching a specific experiment (Chang et al. 2026), while stating that a unique predictive model remains out of reach.

Problem Statement

Establish the missing theoretical foundation of the fractional-flow/maximum-mobility description of immiscible viscous fingering in porous media, as sketched by Sorbie et al. (2020), for water displacing oil at adverse viscosity ratio ($\mu_o/\mu_w \gg 1$) under the two-phase Darcy-scale equations:

Derive or refute the maximum-mobility principle. Among the infinite family of relative permeability pairs $(k_{rw}^*, k_{ro}^*)$ consistent with a prescribed effective fractional flow $f_w^*(S_w)$ — $f_w^*$ fixes only the ratio $k_{ro}^*/k_{rw}^*$ — the methodology selects the pair maximizing the total mobility $\lambda_T(S_w)$, i.e. minimizing the pressure drop across the fingered system. In the construction of Sorbie et al. (2020), the admissible pairs are parametrized by generalized Brooks–Corey-type forms whose RP end points are fixed at the initial water saturation $S_{wi}$ and the residual oil saturation $S_{or}$, with all other RP parameters varied until a maximum $\lambda_T$ consistent with the given $f_w^*$ is found; strictly, the quantity to be minimized is the pressure-drop functional $\int dx/\lambda_T$ (in two dimensions, the corresponding pressure equation), which the source equates with maximizing $\lambda_T$. Provide a first-principles derivation of this selection rule — for example, from a variational or stability argument applied to the coupled pressure–saturation equations; the derivation must cover at least the homogeneous case with no capillary pressure ($P_c = 0$), while coverage of capillary-pressure and heterogeneous (random correlated) permeability fields is encouraged but not required — or refute it by exhibiting a documented displacement that some other, named selection rule describes better, and identify that rule. Whether the principle is sustained or refuted, make precise the relation between the resulting selection rule (the maximum-mobility principle, or the identified replacement if the principle is refuted) and classical linear stability results, e.g. the Chikhliwala–Yortsos conditions on relative permeabilities at the onset of immiscible instability, clarifying whether that rule is implied by, equivalent to, or independent of them.

This derivation and its connection to stability theory constitute the detailed theory whose absence Sorbie et al. (2020) explicitly noted: they presented the maximum-mobility rule as a conjecture and promised a forthcoming paper with supporting calculations and its relation to linear stability analysis. A complete answer replaces the conjectural selection rule with a theoretically grounded one.

References

1. K. S. Sorbie, A. Y. Al Ghafri, A. Skauge, E. J. Mackay, "On the Modelling of Immiscible Viscous Fingering in Two-Phase Flow in Porous Media", Transport in Porous Media 135(2):331–359 (2020), DOI 10.1007/s11242-020-01479-w, https://doi.org/10.1007/s11242-020-01479-w
2. I. C. Salmo, K. S. Sorbie, A. Skauge, M. A. Alzaabi, "Immiscible Viscous Fingering: Modelling Unstable Water–Oil Displacement Experiments in Porous Media", Transport in Porous Media 145(2):291–322 (2022), DOI 10.1007/s11242-022-01847-8, https://doi.org/10.1007/s11242-022-01847-8
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