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arxiv: 2506.08152 · v1 · pith:BQRBIJ3Unew · submitted 2025-06-09 · 🧮 math.AP · math-ph· math.MP· physics.flu-dyn

The Riemann problem for three-phase foam flow in porous media

Luis Fernando Lozano , Grigori Chapiro , Dan Marchesin This is my paper

Pith reviewed 2026-05-22 00:09 UTC · model grok-4.3

classification 🧮 math.AP math-phmath.MPphysics.flu-dyn
keywords foam flowporous mediaRiemann problemthree-phase flowconservation lawsenhanced oil recoveryoil bank formationnon-classical shocks
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The pith

A methodology classifies all solutions to the Riemann problem for three-phase foam flow when gas viscosity exceeds that of oil and water.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops a method to solve the Riemann problem for three-phase foam displacement in porous media under the condition that gas is more viscous than oil and water. The approach uses the assumption of local equilibrium to treat foam strength as fixed, which simplifies the model and permits a full classification of wave structures for injecting foamed gas and water mixtures across many initial conditions. The classification is carried out inside non-classical conservation law theory and includes identification of cases that produce an oil bank. Such results matter because they supply explicit predictions for foam-assisted recovery processes and supply exact benchmarks that numerical simulators can be checked against.

Core claim

The authors present a methodology to solve the Riemann problem for three-phase foam displacement in porous media in the case when gas viscosity exceeds that of oil and water. Assuming foam in local equilibrium with a constant mobility reduction factor, they classify possible solutions for the injection of foamed gas and water mixtures under a wide range of initial conditions within the framework of non-classical conservation law theory. The work also identifies the conditions that result in oil bank formation and validates the analytical estimates by numerical simulation.

What carries the argument

Classification of admissible wave structures for the three-phase foam system with constant mobility reduction factor inside non-classical conservation law theory.

If this is right

  • The classified solutions directly predict when an oil bank forms during foam injection.
  • Analytical wave structures supply reference data for calibrating numerical simulators of foam flow.
  • The results enable uncertainty quantification for recovery forecasts in applications that use foam.
  • The classification advances the physical description of how foam controls gas mobility in porous rock.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same classification approach could be tested on models that allow foam texture to evolve with time rather than stay fixed.
  • Wave-structure results may guide the choice of injection rates that maximize oil-bank size in field-scale planning.
  • Analogous Riemann-problem techniques could be applied to other multiphase systems that rely on mobility-control additives.

Load-bearing premise

The analysis rests on the assumption that foam remains in local equilibrium, which keeps the mobility reduction factor constant everywhere.

What would settle it

Numerical simulation of a foam injection problem with chosen initial saturations that produces wave speeds or structures different from those listed in the classification would show the method misses solutions.

Figures

Figures reproduced from arXiv: 2506.08152 by Dan Marchesin, Grigori Chapiro, Luis Fernando Lozano.

Figure 1
Figure 1. Figure 1: Saturation triangle Ω. The point U represents the umbilic point for the parameter values in [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Rarefaction curves for the parameter values in [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Graphical conventions for depicting wave curves in figures. Dashed curves indicate shocks. The dash-dotted curve corresponds to a composite curve. Continu￾ous curves with arrows represent rarefaction curves. Arrows indicate the direction of increasing characteristic velocity. The colors blue, red, and green represent slow, fast, and undercompressive families. 5. Riemann solutions We aim to solve the Rieman… view at source ↗
Figure 4
Figure 4. Figure 4: Bifurcation loci in the Ω for the viscosity parameters presented in [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Subdivision of the region Γ into eight R-regions Γi , where i ∈ {1, 2, . . . , 8}. (a) Zoom of the region Γ. (b) Zoom of the region close to the cor￾ner O. (c) Zoom of the region close to the corner W [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The blue dashed curves (respectively, red) represent s-shock curves (re￾spectively, f-shock curves). The blue continuous curve (respectively, red) represents the s-rarefaction curves (respectively, the f-rarefaction curves). The arrows indicate increasing characteristic velocity. (a) The black curves [X, T ∗ ] and [G, T ∗ ] repre￾sent the s-right extensions of Wf (R). (b) The black curves [X, T1]ext ∪ [T1,… view at source ↗
Figure 7
Figure 7. Figure 7: Structure of the Riemann solution for R ∈ Γ3. The blue dashed curve (respectively red) represents s-shock curves (respectively f-shock curves). The blue continuous curve (respectively red) represents s-rarefaction curves (respectively the f-rarefaction curve). The arrows indicate the increasing characteristic velocity. The black curve [X, Mc]ext represents the s-right extension of Wf (R). The black curve [… view at source ↗
Figure 8
Figure 8. Figure 8: The blue dashed curves (respectively, red) represent s-shock curves (re￾spectively, f-shock curves). The blue continuous curve (respectively, red) represents the s-rarefaction curves (respectively, the f-rarefaction curves). The arrows indicate increasing characteristic velocity. The black curve [X, TS]ext represents the s-right extension of Wf (R). The green dotted segment [F, Z∗ ] ⊂ [G, D] identifies adm… view at source ↗
Figure 9
Figure 9. Figure 9: The blue dashed curves (respectively, red) represent s-shock curves (re￾spectively, f-shock curves). The blue continuous curve (respectively, red) represents the s-rarefaction curves (respectively, the f-rarefaction curves). The arrows indicate increasing characteristic velocity. The black curve [X, Mc]ext represents the s-right extension of Wf (R). The black curve [G, U]ext represents the s-right extensio… view at source ↗
Figure 10
Figure 10. Figure 10: The blue dashed curves (respectively, red) represent s-shock curves (re￾spectively, f-shock curves). The blue continuous curve (respectively, red) represents the s-rarefaction curves (respectively, the f-rarefaction curves). The arrows indicate increasing characteristic velocity. The black curve [X, TS]ext represents the s-right extension of Wf (R). The green dotted segment [F, Z∗ ] ⊂ [G, D] identifies ad… view at source ↗
Figure 11
Figure 11. Figure 11: The blue dashed curve (respectively red) represents s-shock curves (re￾spectively f-shock curves). The blue continuous curve (respectively red) represents s-rarefaction curves (respectively the f-rarefaction curve). The arrows indicate the increasing characteristic velocity. (a) The black curve [X, Mc]ext represents the s-right extension of Wf (R). The black curve [G, U]ext represents the s-right extensio… view at source ↗
Figure 13
Figure 13. Figure 13: The remaining boundaries of Γ7 are: • The blue curve [I 5 s , I 6 s ], which is part of the s-inflection locus; • The purple curve [I 6 s , VD], which marks the boundary where the admissibility of u-shocks changes; and [PITH_FULL_IMAGE:figures/full_fig_p022_13.png] view at source ↗
Figure 12
Figure 12. Figure 12: The blue dashed curve (respectively red) represents s-shock curves (re￾spectively f-shock curves). The blue continuous curve (respectively red) represents s-rarefaction curves (respectively the f-rarefaction curve). The arrows indicate the increasing characteristic velocity. (a) The black curve [X, Mc]ext represents the s-right extension of Wf (R). The black curve [G, U]ext represents the s-right extensio… view at source ↗
Figure 13
Figure 13. Figure 13: The blue dashed curve (respectively red) represents s-shock curves (re￾spectively f-shock curves). The blue continuous curve (respectively red) represents s-rarefaction curves (respectively the f-rarefaction curve). The arrows indicate the increasing characteristic velocity. The green dotted segment (Z, Z∗ ] ⊂ [G, D] iden￾tifies admissible u-shocks with M. The black curve (TZ, TS]ext is the s-extension of… view at source ↗
Figure 14
Figure 14. Figure 14: Structure of the Riemann solution for R ∈ Γ8. The blue dashed curve (respectively red) represents s-shock curves (respectively f-shock curves). The blue continuous curve (respectively red) represents s-rarefaction curves (respectively the f-rarefaction curve). The arrows indicate the increasing characteristic velocity. The black curve [X, TR]ext ∪ [TR, A3]ext and [G, X2]ext represents the s-right extensio… view at source ↗
Figure 15
Figure 15. Figure 15: Oil bank formation. (a) A solution of the Riemann problem with two shocks enclosing a constant state of elevated oil saturation. (b) Validity region of Theorem 1, highlighting the right states R and its interval IR for which oil bank formation occurs. The black curves represent the composition path for the Riemann problem solution for the left state L [PITH_FULL_IMAGE:figures/full_fig_p028_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Analytical solution for case 1 with R ∈ Γ6 and L ∈ [LS, LR]. (a) The composition path in the saturation triangle that consists of an s-rarefaction from L to T follows for an s-shock from T to M and an f-shock connecting M to R. (b) Analytical profiles (solid curves) compared to numerical simulations (dashed curves). R ∈ Γ3 and L ∈ [L1, LX]. The solution features an s-composite wave from L to M (L p Rs −−→… view at source ↗
Figure 17
Figure 17. Figure 17: Analytical solution for case 2 with R ∈ Γ3 and L ∈ [L1, LX]. (a) The composition path in the saturation triangle that consists of an s-rarefaction from L to T follows for an s-shock from T to M follows for an f-rarefaction from M to A1 and an f-shock connecting A1 to R. (b) Analytical profiles (solid curves) compared to numerical simulations (dashed curves). The solution features a u-composite wave from L… view at source ↗
Figure 18
Figure 18. Figure 18: Analytical solution for Case 3 with R ∈ Γ6 and L = G. (a) The compo￾sition path in the saturation triangle that consists of an f-rarefaction from G to F follows for a u-shock from F to M and an f-shock connecting M to R. (b) Analytical profiles (solid curves) compared to numerical simulations (dashed curves). The solution features an s-composite wave from L to N (L p Rs −−→ T Ss −→ N), followed by an u-co… view at source ↗
Figure 19
Figure 19. Figure 19: Analytical solution for case 4 with R ∈ Γ3 and L ∈ [G, LF ]. (a) The composition path in the saturation triangle that consists of an s-rarefaction from L to T follows for an s-shock from T to N follows for an f-rarefaction from N to F follows for a u-shock from F to M and an f-shock connecting M to R. (b) Analytical profiles (solid curves) compared to numerical simulations (dashed curves). Our solution is… view at source ↗
Figure 20
Figure 20. Figure 20: The colored region indicates the area within the saturation triangle where the umbilic point U is located to satisfy the inequalities in (15). Points corre￾spond to: A(blue) - [PITH_FULL_IMAGE:figures/full_fig_p033_20.png] view at source ↗
read the original abstract

Gas injection in the context of the three-phase flow in porous media appears in applications such as Enhanced Oil Recovery, aquifer remediation, and carbon capture, utilization, and storage (CCUS). In general, this technique suffers from a difficulty related to excessive gas mobility, which can be circumvented by using foam. This study addresses the non-linear system of differential equations describing the three-phase foam flow based on Corey relative permeability functions. A major obstacle is an umbilic point, where the characteristic wave velocities for different families coincide, complicating the identification of stable wave structures. We developed a methodology to solve the Riemann problem describing the three-phase foam displacement in the case when the gas viscosity exceeds that of oil and water. To allow the analysis, we assume foam in local equilibrium (or maximum foam texture), resulting in a constant mobility reduction factor (MRF). These simplifications allowed the classification of possible solutions for the injection of foamed gas and water mixtures under a wide range of initial conditions within the framework of non-classical Conservation Law Theory. As a relevant industrial application of the proposed solution, we investigate the conditions resulting in oil bank formation. Besides improving the general physical understanding of foam flow in a porous medium, this analysis can be applied to calibrate numerical simulators and perform uncertainty quantification. Our analytical estimates were validated through numerical simulations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a methodology for solving the Riemann problem in three-phase foam flow through porous media modeled by a 2x2 system of conservation laws with Corey-type relative permeabilities. Under the local-equilibrium assumption that fixes the mobility reduction factor (MRF) as a constant, the authors classify admissible wave structures for foamed-gas/water injection into a range of initial states, with emphasis on conditions that produce an oil bank; the classification is framed in non-classical conservation-law theory and is supported by numerical simulations.

Significance. If the local-equilibrium regime is representative, the explicit classification of Riemann solutions supplies concrete wave-curve constructions and admissible shock loci that can be used to benchmark simulators and to predict oil-bank formation in foam-assisted gas injection. The work therefore contributes a concrete analytical tool within the restricted modeling framework it adopts.

major comments (2)
  1. The central classification rests on the construction of wave curves through the umbilic point; however, the manuscript provides only a high-level description of this construction (see the paragraph following Eq. (3.2) and the discussion in §4). A step-by-step verification that the proposed rarefaction and shock curves satisfy the Liu entropy condition and the correct ordering of characteristic speeds at the umbilic point is required before the claimed completeness of the solution catalogue can be accepted.
  2. §5, numerical validation: the reported simulations use a fixed MRF value; it is not shown how the admissible wave sequences change when a small but non-zero foam-texture transport equation is restored. A single sensitivity test with a variable-MRF model would quantify the robustness of the oil-bank criterion derived under the constant-MRF assumption.
minor comments (2)
  1. Notation: the symbol for the constant MRF is introduced without an explicit equation reference; add a numbered display equation when it first appears.
  2. Figure 4: the phase-plane trajectories near the umbilic point are difficult to read at the printed scale; enlarge the inset or add a separate zoomed panel.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and positive assessment of the manuscript's significance. We address each major comment point by point below, indicating where revisions will be made.

read point-by-point responses

  1. Referee: The central classification rests on the construction of wave curves through the umbilic point; however, the manuscript provides only a high-level description of this construction (see the paragraph following Eq. (3.2) and the discussion in §4). A step-by-step verification that the proposed rarefaction and shock curves satisfy the Liu entropy condition and the correct ordering of characteristic speeds at the umbilic point is required before the claimed completeness of the solution catalogue can be accepted.

    Authors: We agree that a more detailed verification is needed to substantiate the wave-curve constructions. In the revised manuscript we will expand §4 with an explicit step-by-step derivation of the rarefaction and shock curves through the umbilic point. This will include direct computation of the eigenvalues and eigenvectors, verification that the Liu entropy condition holds for the proposed shocks, and confirmation of the correct ordering of characteristic speeds at the umbilic point. revision: yes

  2. Referee: §5, numerical validation: the reported simulations use a fixed MRF value; it is not shown how the admissible wave sequences change when a small but non-zero foam-texture transport equation is restored. A single sensitivity test with a variable-MRF model would quantify the robustness of the oil-bank criterion derived under the constant-MRF assumption.

    Authors: The constant-MRF assumption is central to reducing the model to a 2×2 system of conservation laws and to the analytical classification we present. Restoring the full foam-texture transport equation would yield a larger, non-strictly hyperbolic system whose Riemann solutions lie outside the scope of the present work. We will add a concise discussion in the revised manuscript explaining this modeling choice and the expected qualitative robustness of the oil-bank criterion under small MRF perturbations, but we do not plan to perform a full variable-MRF sensitivity test. revision: no

Circularity Check

0 steps flagged

No circularity: classification follows directly from governing equations under stated local-equilibrium assumption

full rationale

The paper begins with the standard three-phase conservation laws using Corey relative permeabilities, then explicitly imposes the modeling assumption of local equilibrium (maximum foam texture) to fix the mobility reduction factor as a constant. This reduces the system to a 2x2 hyperbolic conservation law whose umbilic point and wave curves are analyzed via non-classical Riemann theory. The resulting classification of solutions for different injection and initial data follows from the simplified PDE system and standard admissibility criteria; no prediction is obtained by fitting to a subset of the target data, no uniqueness theorem is imported from self-citation, and no ansatz is smuggled in. The constant-MRF choice is presented as an enabling simplification rather than a derived result, leaving the derivation self-contained.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim depends on the local equilibrium assumption for foam and the viscosity ordering; these are domain simplifications that enable the classification but are not independently verified in the abstract.

free parameters (1)
  • constant mobility reduction factor (MRF)
    Assumed constant due to local equilibrium (maximum foam texture) to simplify the non-linear system.
axioms (2)
  • domain assumption Foam in local equilibrium resulting in constant MRF
    Explicitly stated to allow analysis of the three-phase flow equations.
  • domain assumption Corey relative permeability functions for three-phase flow
    Basis for the model of foam, gas, oil, and water interactions.

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Reference graph

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