Global Buckley--Leverett for Multicomponent Flow in Fractured Media: Isothermal Equation-of-State Coupling and Dynamic Capillarity

Christian Tantardini; Fernando Alonso-Marroquin

arxiv: 2511.06233 · v4 · pith:ZKDC2F5Mnew · submitted 2025-11-09 · ⚛️ physics.flu-dyn · cond-mat.soft

Global Buckley--Leverett for Multicomponent Flow in Fractured Media: Isothermal Equation-of-State Coupling and Dynamic Capillarity

Christian Tantardini , Fernando Alonso-Marroquin This is my paper

Pith reviewed 2026-05-21 18:58 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn cond-mat.soft

keywords Buckley-Leverettmulticomponent flowfractured mediaMaxwell-Stefan diffusiondynamic capillarityporous mediacarbon storagegeothermal flow

0 comments

The pith

Maxwell-Stefan diffusion paired with dynamic capillarity turns three-phase multicomponent transport into a pseudo-parabolic system that restores a well-posed initial-value problem.

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

The paper builds a global Buckley-Leverett framework that keeps the classical decomposition of total flux into phase velocities while folding in equation-of-state phase behavior, multicomponent diffusion, and dynamic capillary pressure. It shows that the added diffusion and capillarity together change the character of the equations so that transport becomes pseudo-parabolic instead of strictly hyperbolic. This change removes the loss of hyperbolicity that has long made three-phase problems ill-posed. The resulting method solves one scalar global-pressure equation per time step, reconstructs the phase fluxes, and marches forward with conservative component balances, all while reducing exactly to ordinary Buckley-Leverett when the extra physics are switched off.

Core claim

The central claim is that the combination of Maxwell-Stefan diffusion and dynamic capillarity renders transport pseudo-parabolic, resolving the loss of strict hyperbolicity that plagues three-phase Buckley-Leverett and ensuring a well-posed initial-value problem. The formulation produces a single global-pressure equation that drives the total Darcy flux together with an exact fractional-flow decomposition of the phase velocities that includes buoyancy and capillary drifts; inertial effects appear as per-phase damping that renormalizes the mobilities. Each time step therefore consists of solving the scalar pressure equation, reconstructing the phase fluxes via the split, and advancing the set

What carries the argument

The global-pressure equation with exact fractional-flow decomposition of phase velocities, made pseudo-parabolic by the addition of Maxwell-Stefan diffusion and dynamic capillarity.

If this is right

Each time step reduces to solving one scalar global-pressure equation, reconstructing phase fluxes from the fractional-flow split, and advancing strictly conservative component balances.
Axisymmetric cylindrical forms are supplied for radial injection problems that include vertical buoyancy.
The entire model collapses exactly to classical Buckley-Leverett whenever the extra physics are turned off.
The framework applies directly to carbon storage, geothermal exchange, and contaminant transport in fractured, compositionally complex reservoirs.

Where Pith is reading between the lines

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

The pseudo-parabolic regularization may allow larger time steps in reservoir simulators that currently struggle with three-phase shock formation.
The same splitting could be tested in non-isothermal settings to see whether temperature-dependent diffusion further improves stability in geothermal problems.
Direct comparison with laboratory core-flood experiments that include measurable capillary pressure dynamics would provide a concrete check on whether the well-posedness gain appears in real data.

Load-bearing premise

The added physics can be inserted into the classical fractional-flow decomposition without creating inconsistencies or demanding extra closure relations.

What would settle it

A numerical test that solves the three-phase system both with and without the Maxwell-Stefan and dynamic-capillarity terms and checks whether only the augmented version produces unique, stable solutions free of the oscillations or non-uniqueness seen in the classical hyperbolic case.

read the original abstract

We present an isothermal Global Buckley--Leverett framework for multicomponent, multiphase flow in porous and fractured media that retains the interpretability of classical Buckley--Leverett while incorporating essential physics: equation of state-based phase behavior, multicomponent Maxwell--Stefan diffusion, dynamic capillarity, stress-sensitive permeability, and non-Darcy fracture flow. The formulation yields a single global-pressure equation driving the total Darcy flux and an exact fractional-flow decomposition of phase velocities with buoyancy and capillary drifts; inertial effects enter as per-phase damping that renormalizes mobilities. Crucially, the combination of Maxwell--Stefan diffusion and dynamic capillarity renders transport pseudo-parabolic, resolving the loss of strict hyperbolicity that plagues three-phase Buckley--Leverett and ensuring a well-posed initial-value problem. In practice, each time step solves the scalar global-pressure equation, reconstructs phase fluxes via the split, and advances strictly conservative component balances; axisymmetric (cylindrical) forms for radial injection with vertical buoyancy are provided. The model reduces exactly to classical Buckley--Leverett when added physics are disabled, making it a practical backbone for carbon storage, geothermal exchange, and contaminant transport in fractured, compositionally complex reservoirs.

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.

Desk Editor's Note private letter to a colleague

The paper assembles a global-pressure fractional-flow framework for multicomponent fractured flows with dynamic capillarity and Maxwell-Stefan diffusion, but the exact split may not survive the rate-dependent capillary term without residual coupling.

read the letter

The paper's central idea is to extend the classical Buckley-Leverett approach to handle multicomponent flows in fractured media by incorporating equation-of-state phase behavior, Maxwell-Stefan diffusion, dynamic capillarity, stress-sensitive permeability, and non-Darcy flow in fractures, all while preserving a single global-pressure equation and an exact fractional-flow decomposition of the phase velocities. It does well in showing how the model collapses back to the standard form when those extra terms are removed, which gives a clear baseline. The provision of axisymmetric equations for radial injection problems with vertical buoyancy is practical for applications in carbon storage and geothermal systems. The argument that combining Maxwell-Stefan diffusion with dynamic capillarity turns the transport equations pseudo-parabolic and fixes the hyperbolicity problems in multi-phase flow is presented as the key fix for well-posedness. A soft spot is the handling of the dynamic capillarity in the fractional flow split. Adding a time-derivative term to the capillary pressure can generate an extra contribution in the phase momentum balance that may not fit neatly into the buoyancy and capillary drift terms without affecting the global pressure equation. The paper claims an exact decomposition, so the derivations need to demonstrate that this term gets absorbed without introducing inconsistencies or new closures. The manuscript focuses on the formulation and the time-stepping procedure of solving the pressure equation then updating the conservative component equations. There are no reported numerical tests or comparisons to other methods, which leaves the practical performance open. This work is aimed at researchers and engineers building simulation tools for compositionally complex flows in fractured reservoirs. A reader interested in theoretical frameworks for these high-priority applications could find the structure useful as a reference. I would recommend sending it for peer review so that the details of the split and the regularization can be examined by specialists in the area.

Referee Report

1 major / 0 minor

Summary. The manuscript develops a Global Buckley-Leverett framework for isothermal multicomponent multiphase flow in porous and fractured media. It incorporates equation-of-state coupling for phase behavior, Maxwell-Stefan diffusion, dynamic capillarity, stress-sensitive permeability, and non-Darcy flow in fractures. The key features are a single scalar global-pressure equation that drives the total Darcy flux, an exact fractional-flow decomposition of the phase velocities including buoyancy and capillary drifts, and the rendering of the transport problem as pseudo-parabolic through the combination of diffusion and dynamic capillarity. This is said to resolve the loss of strict hyperbolicity in three-phase Buckley-Leverett problems and ensure well-posedness. The model is shown to reduce exactly to the classical Buckley-Leverett equations when the additional physics are disabled. Axisymmetric forms for radial injection are also provided.

Significance. If the central derivations hold, this work provides a valuable extension of classical Buckley-Leverett theory that retains interpretability and computational simplicity while adding essential physics for realistic reservoir simulations. The exact reduction property and the pseudo-parabolic regularization are particularly useful for applications in carbon storage, geothermal systems, and contaminant transport in compositionally complex fractured reservoirs. The framework could serve as a practical backbone for more detailed models.

major comments (1)

The abstract and formulation claim that dynamic capillarity (introduced as a rate-dependent correction to capillary pressure) combines with Maxwell-Stefan diffusion to render transport pseudo-parabolic while preserving an exact fractional-flow decomposition that leaves a strictly scalar global-pressure equation. However, the rate-dependent term proportional to ∂s/∂t is not necessarily aligned with the total-velocity direction. Please provide the explicit steps in the momentum-balance derivation showing how this term is absorbed into the fractional-flow split without residual coupling or redefinition of global pressure. If an additional closure is required, this should be stated.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful and constructive review of our manuscript. The single major comment raises a valid question about the precise handling of the dynamic capillarity term in the momentum balance. We address this point directly below and will incorporate additional explicit derivation steps in the revised version to improve clarity.

read point-by-point responses

Referee: The abstract and formulation claim that dynamic capillarity (introduced as a rate-dependent correction to capillary pressure) combines with Maxwell-Stefan diffusion to render transport pseudo-parabolic while preserving an exact fractional-flow decomposition that leaves a strictly scalar global-pressure equation. However, the rate-dependent term proportional to ∂s/∂t is not necessarily aligned with the total-velocity direction. Please provide the explicit steps in the momentum-balance derivation showing how this term is absorbed into the fractional-flow split without residual coupling or redefinition of global pressure. If an additional closure is required, this should be stated.

Authors: We appreciate the referee highlighting this important detail. In the derivation, the phase momentum balances begin from the extended Darcy law incorporating dynamic capillarity: u_α = −λ_α K (∇p_α − ρ_α g ∇z + ∇p_c^eq(s) + ∇(τ(s) ∂s/∂t)). The global pressure P is defined via the mobility-weighted sum ∇P ≡ ∑_α (λ_α/λ_t) ∇p_α, which absorbs all equilibrium capillary gradients into the definition of P while leaving the total velocity u_t = −λ_t K ∇P + buoyancy correction. The rate-dependent term τ ∂s/∂t is excluded from this weighting and is instead collected into the capillary-drift velocity that appears inside the fractional-flow functions. Specifically, the phase velocity decomposition reads u_α = f_α u_t + f_α λ_t K (buoyancy + capillary-drift terms), where the dynamic contribution enters the drift as an additional diffusive flux proportional to ∇(τ ∂s/∂t) and is therefore moved to the right-hand side of the component transport equations. Because ∂s/∂t is evaluated from the saturation field at the previous time level (or solved implicitly within the transport step), it does not feed back into the scalar global-pressure solve. The resulting transport operator is pseudo-parabolic through the combined Maxwell–Stefan diffusion and dynamic-capillarity terms, while the advective fractional-flow split remains exact and the pressure equation stays strictly scalar. No additional closure is required. We will add a new appendix containing the full algebraic steps of this momentum-balance derivation to make the absorption explicit. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation extends classical Buckley-Leverett without self-referential reduction

full rationale

The provided abstract and description state that the model reduces exactly to classical Buckley-Leverett when added physics are disabled, establishing the core global-pressure equation and fractional-flow decomposition as an external benchmark rather than a self-derived construct. No equations, self-citations, or fitted parameters are shown that would make the pseudo-parabolic resolution or hyperbolicity claim tautological by construction. The incorporation of Maxwell-Stefan diffusion and dynamic capillarity is presented as an additive extension preserving the scalar global-pressure structure, with no evidence of ansatz smuggling, uniqueness imported from authors, or renaming of known results. This keeps the derivation chain independent and self-contained against the classical reference.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on standard porous-media assumptions (Darcy-type flux, fractional-flow decomposition, isothermal conditions) plus the claim that the added transport terms produce a well-posed pseudo-parabolic system; no free parameters or new entities are introduced in the abstract.

axioms (2)

domain assumption Isothermal conditions throughout the domain
Stated explicitly in the title and abstract as the setting for the EOS coupling.
domain assumption Existence of a global pressure that drives total Darcy flux while allowing exact fractional-flow split
Central structural assumption of the Global Buckley-Leverett formulation.

pith-pipeline@v0.9.0 · 5756 in / 1413 out tokens · 41499 ms · 2026-05-21T18:58:47.659928+00:00 · methodology

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discussion (0)

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Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages · cited by 2 Pith papers

[1]

S. E. Buckley and M. C. Leverett,Mechanism of fluid displacement in sands, Transactions of the AIME146, 107–116 (1942)

work page 1942
[2]

M. C. Leverett,Capillary behavior in porous solids, Transactions of the AIME142, 152–169 (1941)

work page 1941
[3]

Shearer and J

M. Shearer and J. A. Trangenstein,Loss of real character- istics for models of three-phase flow in a porous medium, Transport in Porous Media4, 499–525 (1989)

work page 1989
[4]

Holden,On the strict hyperbolicity of the buckley– leverett equations for three-phase flow in a porous medium, SIAM Journal on Applied Mathematics50, 667–682 (1990)

L. Holden,On the strict hyperbolicity of the buckley– leverett equations for three-phase flow in a porous medium, SIAM Journal on Applied Mathematics50, 667–682 (1990)

work page 1990
[5]

A. V. Azevedo, A. J. de Souza, F. Furtado, D. Marchesin, and B. Plohr,The solution by the wave curve method of three-phase flow in virgin reservoirs, Transport in Porous Media83, 99–125 (2010)

work page 2010
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A. V. Azevedo, A. J. de Souza, F. Furtado, and D. March- esin,Uniqueness of the riemann solution for three-phase flow in a porous medium, SIAM Journal on Applied Mathematics74, 1967–1997 (2014)

work page 1967
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Chavent,A fully equivalent global pressure formula- tion for three-phase compressible flow, Applicable Anal- ysis88, 1527–1541 (2009)

G. Chavent,A fully equivalent global pressure formula- tion for three-phase compressible flow, Applicable Anal- ysis88, 1527–1541 (2009)

work page 2009
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di Chiara Roupert, G

R. di Chiara Roupert, G. Chavent, and G. Sch¨ afer, Three-phase compressible flow in porous media: Total differential compatible interpolation of relative permeabil- ities, Journal of Computational Physics229, 4762–4780 (2010)

work page 2010
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K. H. Coats,An equation of state compositional model, SPE Journal20, 363–376 (1980)

work page 1980
[10]

Krishna,The maxwell–stefan description of mixture diffusion in nanoporous crystalline materials, Microp- orous and Mesoporous Materials185, 30–50 (2014)

R. Krishna,The maxwell–stefan description of mixture diffusion in nanoporous crystalline materials, Microp- orous and Mesoporous Materials185, 30–50 (2014)

work page 2014
[11]

Javadpour,Nanopores and apparent permeability of gas flow in mudrocks (shales and siltstone), Journal of Canadian Petroleum Technology48, 16–21 (2009)

F. Javadpour,Nanopores and apparent permeability of gas flow in mudrocks (shales and siltstone), Journal of Canadian Petroleum Technology48, 16–21 (2009)

work page 2009
[12]

Chen, J.-Q

Y.-F. Chen, J.-Q. Zhou, S.-H. Hu, R. Hu, and C.- B. Zhou,Evaluation of forchheimer equation coefficients for non-darcy flow in deformable rough-walled fractures, Journal of Hydrology529, 993–1006 (2015)

work page 2015
[13]

F. Meng, X. Li, P. Baud, and T.-F. Wong,Effective stress law for the permeability and pore volume change of clayey sandstones, Journal of Geophysical Research: Solid Earth125, e2020JB019765 (2020)

work page 2020
[14]

S. M. Hassanizadeh and W. G. Gray,Thermodynamic basis of capillary pressure in porous media, Water Re- sources Research29, 3389–3405 (1993)

work page 1993
[15]

Bottero, S

S. Bottero, S. M. Hassanizadeh, P. J. Kleingeld, and T. J. Heimovaara,Nonequilibrium capillarity effects in two-phase flow through porous media at different scales, Water Resources Research47, W09524 (2011)

work page 2011
[16]

M. L. Michelsen,The isothermal flash problem. part i: Stability; part ii: Phase-split calculation, Fluid Phase Equilibria9, 1–19, 21–40 (1982)

work page 1982
[17]

M. L. Michelsen and J. M. Mollerup,Thermodynamic Models: Fundamentals & Computational Aspects, 2nd ed. (Tie-Line Publications, Denmark, 2007)

work page 2007
[18]

Alonso-Marroqu´ ın and M

F. Alonso-Marroqu´ ın and M. P. Andersson,Capillary pressure–saturation relation derived from the pore mor- phology method, arXiv preprint (2025), arXiv:2501.07355 [physics.flu-dyn]

work page arXiv 2025
[19]

Chen, J.-Q

Y. Chen, J.-Q. Zhou, S.-H. Hu, and R. Hu,Evaluation of forchheimer equation coefficients for non-darcy flow in deformable rough-walled fractures, Journal of Hydrology 529, 993–1006 (2015)

work page 2015
[20]

well-posed

Here “well-posed” means: existence, uniqueness, and continuous dependence on data for the initial–boundary- value problem. In our formulation, pressure solves an el- liptic (or weakly parabolic) problem, while the transport unknowns obey a pseudo-parabolic system; the purely hyperbolic BL limit is recovered only when all dissipa- tive terms vanish

work page
[21]

J. E. Warren and P. J. Root,The behavior of naturally fractured reservoirs, SPE Journal3, 245–255 (1963)

work page 1963
[22]

Olorode, B

O. Olorode, B. Wang, and H. U. Rashid,Three- dimensional projection-based embedded discrete-fracture model for compositional simulation of fractured reser- voirs, SPE Journal25, 2143–2161 (2020)

work page 2020

[1] [1]

S. E. Buckley and M. C. Leverett,Mechanism of fluid displacement in sands, Transactions of the AIME146, 107–116 (1942)

work page 1942

[2] [2]

M. C. Leverett,Capillary behavior in porous solids, Transactions of the AIME142, 152–169 (1941)

work page 1941

[3] [3]

Shearer and J

M. Shearer and J. A. Trangenstein,Loss of real character- istics for models of three-phase flow in a porous medium, Transport in Porous Media4, 499–525 (1989)

work page 1989

[4] [4]

Holden,On the strict hyperbolicity of the buckley– leverett equations for three-phase flow in a porous medium, SIAM Journal on Applied Mathematics50, 667–682 (1990)

L. Holden,On the strict hyperbolicity of the buckley– leverett equations for three-phase flow in a porous medium, SIAM Journal on Applied Mathematics50, 667–682 (1990)

work page 1990

[5] [5]

A. V. Azevedo, A. J. de Souza, F. Furtado, D. Marchesin, and B. Plohr,The solution by the wave curve method of three-phase flow in virgin reservoirs, Transport in Porous Media83, 99–125 (2010)

work page 2010

[6] [6]

A. V. Azevedo, A. J. de Souza, F. Furtado, and D. March- esin,Uniqueness of the riemann solution for three-phase flow in a porous medium, SIAM Journal on Applied Mathematics74, 1967–1997 (2014)

work page 1967

[7] [7]

Chavent,A fully equivalent global pressure formula- tion for three-phase compressible flow, Applicable Anal- ysis88, 1527–1541 (2009)

G. Chavent,A fully equivalent global pressure formula- tion for three-phase compressible flow, Applicable Anal- ysis88, 1527–1541 (2009)

work page 2009

[8] [8]

di Chiara Roupert, G

R. di Chiara Roupert, G. Chavent, and G. Sch¨ afer, Three-phase compressible flow in porous media: Total differential compatible interpolation of relative permeabil- ities, Journal of Computational Physics229, 4762–4780 (2010)

work page 2010

[9] [9]

K. H. Coats,An equation of state compositional model, SPE Journal20, 363–376 (1980)

work page 1980

[10] [10]

Krishna,The maxwell–stefan description of mixture diffusion in nanoporous crystalline materials, Microp- orous and Mesoporous Materials185, 30–50 (2014)

R. Krishna,The maxwell–stefan description of mixture diffusion in nanoporous crystalline materials, Microp- orous and Mesoporous Materials185, 30–50 (2014)

work page 2014

[11] [11]

Javadpour,Nanopores and apparent permeability of gas flow in mudrocks (shales and siltstone), Journal of Canadian Petroleum Technology48, 16–21 (2009)

F. Javadpour,Nanopores and apparent permeability of gas flow in mudrocks (shales and siltstone), Journal of Canadian Petroleum Technology48, 16–21 (2009)

work page 2009

[12] [12]

Chen, J.-Q

Y.-F. Chen, J.-Q. Zhou, S.-H. Hu, R. Hu, and C.- B. Zhou,Evaluation of forchheimer equation coefficients for non-darcy flow in deformable rough-walled fractures, Journal of Hydrology529, 993–1006 (2015)

work page 2015

[13] [13]

F. Meng, X. Li, P. Baud, and T.-F. Wong,Effective stress law for the permeability and pore volume change of clayey sandstones, Journal of Geophysical Research: Solid Earth125, e2020JB019765 (2020)

work page 2020

[14] [14]

S. M. Hassanizadeh and W. G. Gray,Thermodynamic basis of capillary pressure in porous media, Water Re- sources Research29, 3389–3405 (1993)

work page 1993

[15] [15]

Bottero, S

S. Bottero, S. M. Hassanizadeh, P. J. Kleingeld, and T. J. Heimovaara,Nonequilibrium capillarity effects in two-phase flow through porous media at different scales, Water Resources Research47, W09524 (2011)

work page 2011

[16] [16]

M. L. Michelsen,The isothermal flash problem. part i: Stability; part ii: Phase-split calculation, Fluid Phase Equilibria9, 1–19, 21–40 (1982)

work page 1982

[17] [17]

M. L. Michelsen and J. M. Mollerup,Thermodynamic Models: Fundamentals & Computational Aspects, 2nd ed. (Tie-Line Publications, Denmark, 2007)

work page 2007

[18] [18]

Alonso-Marroqu´ ın and M

F. Alonso-Marroqu´ ın and M. P. Andersson,Capillary pressure–saturation relation derived from the pore mor- phology method, arXiv preprint (2025), arXiv:2501.07355 [physics.flu-dyn]

work page arXiv 2025

[19] [19]

Chen, J.-Q

Y. Chen, J.-Q. Zhou, S.-H. Hu, and R. Hu,Evaluation of forchheimer equation coefficients for non-darcy flow in deformable rough-walled fractures, Journal of Hydrology 529, 993–1006 (2015)

work page 2015

[20] [20]

well-posed

Here “well-posed” means: existence, uniqueness, and continuous dependence on data for the initial–boundary- value problem. In our formulation, pressure solves an el- liptic (or weakly parabolic) problem, while the transport unknowns obey a pseudo-parabolic system; the purely hyperbolic BL limit is recovered only when all dissipa- tive terms vanish

work page

[21] [21]

J. E. Warren and P. J. Root,The behavior of naturally fractured reservoirs, SPE Journal3, 245–255 (1963)

work page 1963

[22] [22]

Olorode, B

O. Olorode, B. Wang, and H. U. Rashid,Three- dimensional projection-based embedded discrete-fracture model for compositional simulation of fractured reser- voirs, SPE Journal25, 2143–2161 (2020)

work page 2020