pith. sign in

arxiv: 2510.19502 · v2 · submitted 2025-10-22 · 🧮 math.AP · math-ph· math.MP

Correct mathematical models of joint filtration of two immiscible viscous liquids

Anvarbek Meirmanov This is my paper

Pith reviewed 2026-05-18 04:43 UTC · model grok-4.3

classification 🧮 math.AP math-phmath.MP
keywords joint filtrationimmiscible liquidshomogenizationporous mediaoil displacementfree boundaryNewtonian mechanicsBuckley-Leverett model
0
0 comments X

The pith

Joint filtration of two immiscible viscous liquids requires modeling from microscopic Newtonian mechanics equations followed by homogenization.

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

The paper claims that existing macroscopic models such as Buckley-Leverett treat joint filtration as a set of axioms and fail to capture free boundaries or fluid interactions that actually occur at the scale of individual pores. It argues instead for starting with the classical equations of Newtonian mechanics for continuous media at the microscopic level of tens of micrometers and then applying homogenization to obtain effective macroscopic descriptions. This matters because oil displacement by suspension is an economically significant process whose accurate simulation currently relies on models that ignore the underlying physics at the relevant scale. A reader would see the proposal as a way to derive rather than postulate the governing equations while still allowing computations over large domains.

Core claim

Exact mathematical models of joint filtration of two immiscible viscous liquids are obtained by writing the full system of equations from classical Newtonian mechanics of continuous media at the microscopic pore scale of tens of micrometers and then applying homogenization to produce the corresponding macroscopic model.

What carries the argument

Homogenization method of Keller and Sánchez-Palencia, which averages the microscopic equations over the pore geometry to derive effective macroscopic equations that retain free boundaries and interaction details.

If this is right

  • Macroscopic equations will distinguish the free boundary between the two liquids and the detailed interaction between them.
  • Hydrodynamic simulators for oil displacement by suspension will rest on derived equations rather than axiomatic assumptions.
  • Computations over domains of hundreds of meters will become feasible without requiring years of runtime.
  • The resulting models will preserve the essential physics that occurs at the average pore size while operating at engineering scales.

Where Pith is reading between the lines

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

  • The same microscopic-to-macroscopic route could be tested on other multiphase flows where free boundaries matter.
  • Small-scale direct simulations could serve as benchmarks to quantify the accuracy of the homogenized model before large-scale use.
  • The approach might clarify which microscopic details survive averaging and which can be safely discarded in similar filtration problems.
  • Reservoir engineering codes could incorporate the derived models once the homogenization step is verified for representative pore geometries.

Load-bearing premise

The homogenization techniques developed for other problems can be applied directly to the joint filtration of two immiscible liquids without new mathematical difficulties or loss of essential microscopic features.

What would settle it

Perform a side-by-side numerical comparison of the homogenized macroscopic equations against a direct microscopic simulation on a small domain containing a visible free boundary and check whether the interface motion and velocity fields agree within the expected error.

read the original abstract

Mathematical models of joint filtration of liquids are the main part of mathema-tical models of oil displacement by suspension. Since mining is a very important and urgent economic task, exact modeling of joint filtration of two different fluids is also an urgent economic task. For example, mathematical models of oil displacement by suspension are needed to create a hydrodynamic simulator of oil by suspension. All the existing simulators are based on the macroscopic Buckley-Leverett model, which does not distinguish between the free boundary separating liquids, and the details of liquid interaction. All these fundamental processes occur at a microscopic level corresponding to the average size of pores, while all proposed macroscopic models operate on completely different orders of magnitude and do not distinguish between free boundaries or the characteristics of fluid interactions and are simply a set of axioms. Exact modelling involves describing the process using equations from classical Newtonian mechanics of continuous media at the microscopic level (average size of tens of micrometers) followed by homogenization. The only obstacle to using such models is that, in areas measuring hundreds of metres, any numerical implementation would take years. A solution to this problem (the homogenisation method) was proposed in the papers of J. Keller and E. S\'anchez-Palencia.

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

1 major / 2 minor

Summary. The paper claims that correct mathematical models of joint filtration of two immiscible viscous liquids (e.g., oil displacement by suspension) must begin with the microscopic Newtonian continuum mechanics of continuous media at the pore scale (tens of micrometers), followed by homogenization via the methods of Keller and Sánchez-Palencia to obtain usable macroscopic equations; existing Buckley-Leverett-type simulators are criticized as axiomatic and blind to free boundaries and microscale fluid interactions.

Significance. If the microscopic-to-homogenized route can be carried through rigorously, the work would supply a first-principles alternative to purely phenomenological macroscopic models, potentially improving the physical fidelity of hydrodynamic simulators for economically important applications in oil recovery and mining. The explicit identification of the scale gap and the computational obstacle addressed by homogenization is a constructive framing.

major comments (1)
  1. [Abstract / main text] The central assertion that Keller–Sánchez-Palencia homogenization can be applied directly to the two-phase immiscible case to produce correct macroscopic models is unsupported by any derivation, asymptotic expansion, or two-scale convergence argument in the manuscript. The standard single-phase fixed-geometry Stokes homogenization does not automatically accommodate a time-evolving free boundary, dynamic contact lines, or surface tension; the required additional steps (phase-indicator transport, effective capillary pressure, or upscaled contact-line conditions) are not supplied, which is load-bearing for the claim that microscopic Newtonian mechanics plus homogenization yields usable macroscopic models.
minor comments (2)
  1. [Abstract] Hyphenation artifact: 'mathema-tical' should read 'mathematical'.
  2. [Abstract / main text] No equations, no explicit microscopic two-fluid system, and no references to specific theorems or results from Keller or Sánchez-Palencia are provided, making the technical route impossible to verify from the text alone.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful and constructive review. Our manuscript is a conceptual call to derive macroscopic filtration models from microscopic Newtonian mechanics via homogenization rather than relying on axiomatic macroscopic closures. We address the single major comment below.

read point-by-point responses

  1. Referee: The central assertion that Keller–Sánchez-Palencia homogenization can be applied directly to the two-phase immiscible case to produce correct macroscopic models is unsupported by any derivation, asymptotic expansion, or two-scale convergence argument in the manuscript. The standard single-phase fixed-geometry Stokes homogenization does not automatically accommodate a time-evolving free boundary, dynamic contact lines, or surface tension; the required additional steps (phase-indicator transport, effective capillary pressure, or upscaled contact-line conditions) are not supplied, which is load-bearing for the claim that microscopic Newtonian mechanics plus homogenization yields usable macroscopic models.

    Authors: We agree that the manuscript contains no explicit asymptotic expansion, two-scale convergence argument, or full derivation for the immiscible two-phase problem. The paper is framed as an argument that correct models must originate at the pore-scale Newtonian description and that homogenization (as developed by Keller and Sánchez-Palencia for Stokes-type problems) supplies the systematic upscaling route; it does not claim to have already executed that upscaling for evolving free boundaries. The additional mathematical ingredients required—transport of a phase indicator, derivation of an effective capillary pressure, and upscaled contact-line conditions—are indeed necessary and are not supplied. In the revised version we will (i) state explicitly that the present work identifies the modeling gap and the appropriate mathematical framework rather than carrying out the complete homogenization, and (ii) outline the principal additional steps that must be developed to treat dynamic interfaces and surface tension within the homogenization procedure. This clarification will remove any impression that the two-phase case follows immediately from the classical single-phase theory. revision: yes

Circularity Check

0 steps flagged

No circularity: methodological proposal relies on external citations without self-referential reduction

full rationale

The paper states that exact modeling requires microscopic Newtonian mechanics of continuous media at the pore scale (tens of micrometers) followed by homogenization, citing Keller and Sánchez-Palencia for the homogenization method. This is a high-level methodological claim rather than a derivation chain containing equations that reduce by construction to fitted inputs or self-definitions. No self-citations appear in the load-bearing steps; the cited works are external. No parameters are fitted to data and then relabeled as predictions, and no uniqueness theorems or ansatzes are smuggled via author-overlapping references. The argument is self-contained as an advocacy for applying established homogenization to the two-phase microscopic problem.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central proposal rests on the applicability of existing homogenization techniques to this specific two-liquid filtration setting.

axioms (1)
  • domain assumption Homogenization methods developed by Keller and Sánchez-Palencia apply without modification to joint filtration of two immiscible viscous liquids.
    Cited as the solution to the computational obstacle for large domains.

pith-pipeline@v0.9.0 · 5743 in / 1058 out tokens · 32254 ms · 2026-05-18T04:43:25.232219+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

  1. [1]

    Mechanism of fluid displacements in sands, Transactions of the AIME, 1942, Vol

    Buckley S., and Leverett M. Mechanism of fluid displacements in sands, Transactions of the AIME, 1942, Vol. 146, pp. 107-116

    work page 1942

  2. [3]

    Dardagenian

    S. Dardagenian. The application of the Backley-Leverette frontal advance theory of petroleum recovery, Journal of petroleum technology, 1958, 49-52 pp

    work page 1958

  3. [4]

    A. Farina at al, Special phenomena occoring in small blood vessels: models, inter- pretation and importance for life, Rendiconti Lincei, Matematica e Aplicazioni, 2024, 397-445,DOI 10.4171/RLM/1046

    work page doi:10.4171/rlm/1046 2024

  4. [5]

    Guerillot, M

    D. Guerillot, M. Kadiri, S. Trabelsi, Backley-Leverette theory for two-phase immis- ible fluid flownfrontal model with explicit phase-coupling terms, Water, 2020, 3041, https://doi.org/10.3390/w12113041

    work page doi:10.3390/w12113041 2020

  5. [6]

    Burridge, J

    R. Burridge, J. B. Keller, Poroelasticity equations derived from microstructure, J. Acoust. Soc. Am., V. 70, issue 4, 1981, pp. 1140 – 1146

    work page 1981

  6. [7]

    Sanchez-Palencia, Non-homogeneous media and vibration theory, Lecture Notes in Phys., 127, Springer-Verlag, Berlin–New York, 1980

    E. Sanchez-Palencia, Non-homogeneous media and vibration theory, Lecture Notes in Phys., 127, Springer-Verlag, Berlin–New York, 1980

    work page 1980

  7. [8]

    Sobolev’s spaces, Academic Press, 1975 O

    AdamsR. Sobolev’s spaces, Academic Press, 1975 O. A. Ladyzhenskaja, V. A. Solonnikov, N. N. Uraltseva, Linear and quasilinear equa- tions of parabolic type, Transl. Math. Monogr., V. 23, Amer. Math. Soc., Providence, R.I., 1968

    work page 1975

  8. [9]

    J. L. Lions, Quelques m`etodes de resolution des problemes aux limites non lin`eaire, Dunon Gauthier – Villars, Paris, 1969

    work page 1969

  9. [10]

    Meirmanov

    A. Meirmanov. Mathematical models for poroelastic flow, Paris, Atlantis Press, 2014

    work page 2014

  10. [11]

    Poincare

    H. Poincare. Sur les Equations aux D´ eriv´ ees Partielles de la Physique Math´ ematique, American Journal of Mathematics; 1890; 12 (3)

    work page

  11. [12]

    R. Adams. Sobolev’s spaces, Academic Press, 1975

    work page 1975

  12. [13]

    V. V. Zhikov, S. M. Kozlov, O. A. Oleynik. Homogenization of differential operators and integral functionals, Springer-Verlag, Berlin–New York, 1994. 9

    work page 1994