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arxiv: 2604.18462 · v1 · submitted 2026-04-20 · ❄️ cond-mat.soft · physics.flu-dyn

Diffusion compaction coupling controls pore pressure dynamics in granular fluid flows

Eric C.P. Breard , Claudia Elijas Parra , Mattia de' Michieli Vitturi This is my paper

Pith reviewed 2026-05-10 03:37 UTC · model grok-4.3

classification ❄️ cond-mat.soft physics.flu-dyn
keywords pore pressuregranular flowsdiffusioncompactiontwo-phase flowexcess pressuremobilityporosity
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0 comments X

The pith

Pore-pressure evolution in granular-fluid flows is controlled by the coupling of diffusion and granular compaction, so that apparent diffusivity depends on flow thickness.

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

The authors start from two-phase mass conservation in a deformable gas-saturated granular assembly and derive an evolution equation for excess pore pressure that includes both diffusive drainage and a forcing term arising from changes in the granular skeleton. In the thin-flow, small-excess-pressure limit this reduces to a one-dimensional diffusion-compaction equation whose solution is governed by a single dimensionless ratio that measures the relative strength of the compaction source and diffusive relaxation. Modal analysis of that equation produces a reduced basal model whose effective diffusivity collapses data from high-resolution two-fluid simulations across nearly two orders of magnitude in bed height. When the same closure is inserted into a depth-averaged flow model it reproduces the observed dependence of pore-pressure decay and runout distance on flow thickness.

Core claim

Starting from two-phase mass conservation for a deformable, gas-saturated granular assembly, the authors obtain an evolution equation for excess pore pressure that retains the deformation of the granular skeleton. In the thin-flow, small-excess-pressure limit this equation reduces to a one-dimensional diffusion-compaction equation containing a time-dependent source term controlled by porosity changes. A modal analysis then yields a reduced basal equation that cleanly separates diffusive drainage from compaction-driven forcing and identifies the corresponding timescales; the resulting dimensionless source-to-diffusion ratio collapses effective diffusivities measured in simulations over a wide

What carries the argument

The one-dimensional diffusion-compaction equation with a time-dependent source term from porosity changes, together with the dimensionless source-to-diffusion ratio that governs the competition between the two processes.

If this is right

  • Effective diffusivity is no longer an intrinsic material property but scales with flow thickness through the competition between diffusion and compaction.
  • Pore-pressure decay times and therefore flow mobility become predictable functions of bed height once the source-to-diffusion ratio is known.
  • Depth-averaged models that incorporate the derived closure reproduce the thickness dependence of runout distance seen in experiments without additional fitting parameters.
  • The framework supplies a physically based replacement for the constant-diffusivity assumption used in many existing granular-flow models.

Where Pith is reading between the lines

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

  • The same coupling could be used to estimate how mobility changes when the same granular mixture is released on slopes of different lengths, providing a scale-aware prediction for natural debris flows.
  • Direct measurement of local porosity evolution during flow would furnish an independent test of the source term strength assumed in the derivation.
  • Extension of the modal analysis to two-dimensional or three-dimensional geometries would show whether lateral drainage pathways alter the thickness scaling in wider channels.

Load-bearing premise

The thin-flow, small-excess-pressure limit in which two-phase mass conservation reduces to the one-dimensional diffusion-compaction equation.

What would settle it

Perform laboratory or numerical experiments that systematically vary bed height while holding all other parameters fixed and measure whether the observed pore-pressure decay rates collapse onto a single curve when plotted against the predicted source-to-diffusion ratio.

Figures

Figures reproduced from arXiv: 2604.18462 by Claudia Elijas Parra, Eric C.P. Breard, Mattia de' Michieli Vitturi.

Figure 1
Figure 1. Figure 1: FIG. 1. One-dimensional MFIX-TFM simulations of a 0 [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Comparison between the 1-D diffusion-only solver (no dilatancy, [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Comparison between MFIX–TFM basal excess pore pressure and reduced 1-D models [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Convergence of the Fourier-series reconstruction of the basal excess pressure for a 1 [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Porosity change ∆ [PITH_FULL_IMAGE:figures/full_fig_p027_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Ψ [PITH_FULL_IMAGE:figures/full_fig_p030_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Representation of the inhibit factor, [PITH_FULL_IMAGE:figures/full_fig_p034_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The classical [PITH_FULL_IMAGE:figures/full_fig_p039_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. (a-f) Diffusion of basal excess pore pressure ( [PITH_FULL_IMAGE:figures/full_fig_p041_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. IMEX simulations reproducing the dam break experiment presented in [10, 13, 14] for [PITH_FULL_IMAGE:figures/full_fig_p042_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Simulations to showcase the effect on flow dynamics of a range of features available in [PITH_FULL_IMAGE:figures/full_fig_p044_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Contour map of the normalized diffusivity ratio [PITH_FULL_IMAGE:figures/full_fig_p047_12.png] view at source ↗
read the original abstract

Excess pore pressure in granular--fluid mixtures can transiently suppress frictional contacts and dramatically enhance flow mobility, yet its evolution is commonly modeled using constant effective diffusivities. Here we show that the apparent diffusivity is not intrinsic but emerges from the coupling between pore-pressure diffusion and granular compaction. Starting from two-phase mass conservation for a deformable, gas-saturated granular assembly, we derive an evolution equation for excess pore pressure that captures deformation of the granular skeleton. In the thin-flow, small-excess-pressure limit, this reduces to a one-dimensional diffusion--compaction equation with a time-dependent source term controlled by porosity changes. A modal analysis yields a reduced basal equation that separates diffusive drainage from compaction-driven forcing and identifies the corresponding timescales. This framework introduces a dimensionless source-to-diffusion ratio, $\Psi_0$, which governs the competition between these processes and collapses effective diffusivities obtained from high-resolution two-fluid simulations over nearly two orders of magnitude in bed height. This scaling implies that the apparent diffusivity, and thus flow mobility, is not intrinsic but depends on flow thickness through the competition between diffusion and compaction. Incorporating this physics into a depth-averaged model demonstrates that the resulting closure reproduces the thickness dependence of pore-pressure decay and runout observed in experiments. These results provide a physically grounded description of pore-pressure evolution in granular--fluid flows and clarify how diffusion--compaction coupling controls their mobility.

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 / 4 minor

Summary. The manuscript derives an evolution equation for excess pore pressure in gas-saturated granular assemblies from two-phase mass conservation. In the thin-flow, small-excess-pressure limit this reduces to a one-dimensional diffusion-compaction equation with a time-dependent source term arising from porosity changes. Modal analysis yields a reduced basal equation that isolates diffusive drainage from compaction forcing and defines the dimensionless source-to-diffusion ratio Ψ₀. This parameter collapses effective diffusivities extracted from high-resolution two-fluid simulations across nearly two orders of magnitude in bed height. The resulting closure is inserted into a depth-averaged model that reproduces the observed thickness dependence of pore-pressure decay and runout in experiments.

Significance. If the thin-flow reduction and the resulting Ψ₀ scaling hold, the work establishes that apparent pore-pressure diffusivity is not an intrinsic material constant but emerges from the competition between diffusion and granular compaction, thereby explaining thickness-dependent flow mobility. This supplies a physically derived, falsifiable closure for depth-averaged models of granular-fluid flows. The manuscript is credited for starting from standard conservation laws, demonstrating a clean data collapse over a wide range of bed heights, and closing the loop with experimental validation.

major comments (1)
  1. [§2 (Derivation)] §2 (Derivation): The central claim that the two-phase mass-conservation equations reduce to the one-dimensional diffusion-compaction equation with time-dependent source rests on the thin-flow and small-excess-pressure approximations. The manuscript provides no quantitative estimate of the size of the neglected vertical-velocity or finite-porosity-change terms across the simulated bed-height range (nearly two orders of magnitude). Because any O(1) violation would alter the source term and therefore the modal decomposition that produces the Ψ₀ scaling, this omission directly affects the load-bearing step of the argument.
minor comments (4)
  1. [Simulation methods] The two-fluid simulation protocol (numerical scheme, grid resolution, boundary conditions, and precise procedure for extracting effective diffusivities) is described only at a high level; additional detail is required for reproducibility.
  2. [Modal analysis] Ψ₀ is introduced in the modal-analysis section; its explicit algebraic definition in terms of the source and diffusion coefficients should appear at first use, together with a short statement of how bed height enters through the competition of the two timescales.
  3. [Results (simulation collapse)] The figure showing the collapse of effective diffusivities versus Ψ₀ would benefit from error bars or uncertainty bands on the data points so that the quality of the collapse can be assessed quantitatively.
  4. [Experimental validation] In the depth-averaged model comparison, state whether the parameters were fixed from independent measurements or adjusted to the experimental data; if the latter, quantify the sensitivity of the runout predictions to those choices.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for identifying a point that will strengthen the presentation of our derivation. We address the major comment below.

read point-by-point responses

  1. Referee: §2 (Derivation): The central claim that the two-phase mass-conservation equations reduce to the one-dimensional diffusion-compaction equation with time-dependent source rests on the thin-flow and small-excess-pressure approximations. The manuscript provides no quantitative estimate of the size of the neglected vertical-velocity or finite-porosity-change terms across the simulated bed-height range (nearly two orders of magnitude). Because any O(1) violation would alter the source term and therefore the modal decomposition that produces the Ψ₀ scaling, this omission directly affects the load-bearing step of the argument.

    Authors: We agree that quantitative estimates of the neglected terms would provide stronger support for the approximations. In the revised manuscript we will add an appendix that computes the relative magnitudes of the vertical-velocity contributions and the finite-porosity-change terms directly from the two-fluid simulation data. These estimates will be reported across the full range of bed heights (nearly two orders of magnitude), confirming that both classes of terms remain small compared with the retained terms and do not alter the source term or the resulting Ψ₀ scaling. revision: yes

Circularity Check

0 steps flagged

No significant circularity: derivation from standard conservation laws yields emergent scaling

full rationale

The paper starts from two-phase mass conservation equations for a deformable gas-saturated granular assembly (standard and independent of the target result), derives the excess pore-pressure evolution equation, reduces it under the explicitly stated thin-flow/small-excess-pressure limit to a 1D diffusion-compaction equation with time-dependent source, performs modal analysis to obtain a reduced basal equation, and defines the dimensionless source-to-diffusion ratio Ψ₀ directly from the coefficients of that derived equation. The subsequent collapse of effective diffusivities across bed heights in two-fluid simulations is a consistency check on the derived model rather than a fit; Ψ₀ is not adjusted to the data and the central scaling follows from the modal decomposition of the independently derived PDE. No step reduces by construction to a fitted parameter, self-citation, or renamed input.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Central claim rests on two-phase mass conservation for deformable granular assembly and the thin-flow small-pressure approximation; no free parameters or invented entities are introduced in the abstract description.

axioms (2)
  • domain assumption Two-phase mass conservation for a deformable, gas-saturated granular assembly
    Starting point stated in abstract for deriving the pore-pressure evolution equation.
  • domain assumption Thin-flow, small-excess-pressure limit
    Required for reduction to the one-dimensional diffusion-compaction equation.

pith-pipeline@v0.9.0 · 5559 in / 1374 out tokens · 34699 ms · 2026-05-10T03:37:00.821974+00:00 · methodology

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Reference graph

Works this paper leans on

49 extracted references · 49 canonical work pages

  1. [1]

    The hydraulic permeability may be dynamically updated depending on the porosity and the equivalent particle diameter, according to the Carman-Kozeny Eq

    Physical Formulation and Depth-Averaging The evolution of excess pore pressureP exc(t, z) at any depth within the granular mixture is governed by a 1D vertical diffusion process: ∂Pexc ∂t =D ∂2Pexc ∂z 2 (63) wheretis the temporal coordinate,zis the vertical coordinate parallel to gravity, andDis the hydraulic diffusivity coefficient, defined as Iverson [2...

    work page

  2. [2]

    We quantify this mass loss using Darcy’s Law, evaluated at the free surface (z=h)

    Mass Loss due to Pore Pressure Dissipation The dissipation of excess pore pressure implies a physical loss of gas from the top of the flow. We quantify this mass loss using Darcy’s Law, evaluated at the free surface (z=h). The volumetric flux of gas leaving the flow per unit area,v loss,gas, depends on the gradient of theabsolutepressureP=P exc +P atm: vl...

    work page

  3. [3]

    Effect of Maximum Particle Packing The dissipation of pore pressure and the associated gas loss are physically limited by the porosity of the granular mixture. As the solid volume fraction,α s = P αs,i, approaches the maximum random close packing fraction,α max, the gas phase loss must decrease and eventually go to zero to guarantee the maximum limit on t...

    work page

  4. [4]

    (66) possesses the important mathematical property ofboundedness

    Conservative Formulation for Numerical Implementation As explained by Greenshields and Weller [22], the advective form of Eq. (66) possesses the important mathematical property ofboundedness. This is a physically desirable char- acteristic for an intensive quantity like pressure, which represents a state of the material rather than a conserved quantity li...

    work page

  5. [5]

    Our model accounts for this by incorporating a specific formulation for the pore pressure within the mass source terms

    Pore Pressure in Source Terms In many scenarios the granular material can be initially fluidized, entering the simulation domain with a high pore pressure. Our model accounts for this by incorporating a specific formulation for the pore pressure within the mass source terms. Let ˙hbe the volumetric flux per unit area (with units of velocity, m s −1) repre...

    work page

  6. [6]

    This is incorporated into the model via the principle of effective stress

    Coupling of Pore Pressure with Basal Friction The primary physical effect of pore pressure is the reduction of inter-particle friction at the base of the flow. This is incorporated into the model via the principle of effective stress. To derive this relationship clearly, we first consider the case of a flow on a horizontal surface before generalizing to c...

    work page

  7. [7]

    Partially regularisedµ(I)rheology Theµ(I) or inertial rheology is a phenomenological model that describes the steady flow of a quasi-monodisperse granular mixture made of small, hard, spherical particles under a finite pressure and shear rate [23]. It predicts a one-to-one relationship between coefficient of friction (µ, the ratio between shear to normal ...

    work page

  8. [8]

    overpressured

    showed that fluid flow equations that incorporateµ(I) are ill posed when the inertial number is too high or too low, due to this lack of internal length scale. The ill-posedness results in unbounded growth rates for high-wavenumber perturbations. Consequently, the numerical solution depends inherently on the grid resolution. To address this, [26] proposes...

    work page 2021

  9. [9]

    in air–particle gravity currents. In those experiments, thin, initially fluidised columns produce highly mobile currents whose interiors remain weakly supported by gas overpres- sures, whereas thick columns rapidly lose mobility as pore pressure diffuses away. Our decomposition of the basal pressure into a sum of diffusion and compaction contributions (Fi...

    work page 2025

  10. [10]

    Initial fundamental decay rate s −1 Grain Deborah number (scale analysis) Ded =d 2/(D t0) Grain Deborah number (ratio of grain diffusion time to deformation time) — dGrain diameter (in De d context) m t0 Characteristic deformation timescale s 1D numerical solver ∆zUniform spatial grid step m Nactive Number of active (in-bed) grid nodes at timet— βloc = 1/...

    work page

  11. [11]

    R. M. Iverson, Reviews of Geophysics35, 245 (1997)

    work page 1997

  12. [12]

    G. Lube, E. C. P. Breard, T. Esposti Ongaro, J. Dufek, and B. D. Brand, Nature Reviews Earth & Environment1, 348 (2020)

    work page 2020

  13. [13]

    R. M. Iverson, Journal of Geophysical Research: Earth Surface110(2005), https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2004JF000268

    work page doi:10.1029/2004jf000268 2005

  14. [14]

    E. C. P. Breard, G. Lube, J. R. Jones, J. Dufek, S. J. Cronin, G. A. Valentine, and A. Moebis, Nature Geoscience9, 767 (2016)

    work page 2016

  15. [15]

    G. Lube, E. C. Breard, J. Jones, L. Fullard, J. Dufek, S. J. Cronin, and T. Wang, Nature Geoscience12, 381 (2019)

    work page 2019

  16. [16]

    R. M. Iverson, M. E. Reid, M. Logan, R. G. LaHusen, J. W. Godt, and J. P. Griswold, Nature Geoscience4, 116 (2011)

    work page 2011

  17. [17]

    R. M. Iverson and D. L. George, Proceedings of the Royal Society A470, 20130819 (2014)

    work page 2014

  18. [18]

    R. M. Iverson and R. P. Denlinger, Journal of Geophysical Research: Solid Earth106, 537 (2001)

    work page 2001

  19. [19]

    Montserrat, A

    S. Montserrat, A. Tamburrino, O. Roche, and Y. Ni˜ no, Journal of Geophysical Research: 60 Earth Surface117, F02034 (2012)

    work page 2012

  20. [20]

    Roche, S

    O. Roche, S. Montserrat, Y. Ni˜ no, and A. Tamburrino, Journal of Geophysical Research: Solid Earth115, 10.1029/2009JB007133 (2010)

    work page doi:10.1029/2009jb007133 2010

  21. [21]

    Roche, Bulletin of Volcanology74, 1807 (2012)

    O. Roche, Bulletin of Volcanology74, 1807 (2012)

    work page 2012

  22. [22]

    E. C. P. Breard, J. R. Jones, L. Fullard, G. Lube, C. Davies, and J. Dufek, Journal of Geophysical Research: Solid Earth124, 1343 (2019)

    work page 2019

  23. [23]

    Roche, Bulletin of Volcanology74(2012)

    O. Roche, Bulletin of Volcanology74(2012)

    work page 2012

  24. [24]

    Aravena, L

    ´A. Aravena, L. Chupin, T. Dubois, and O. Roche, Bulletin of Volcanology83, 10.1007/s00445- 021-01507-7 (2021)

    work page doi:10.1007/s00445- 2021

  25. [25]

    E. C. P. Breard, J. Dufek, S. Charbonnier, V. Gueugneau, T. Giachetti, and B. Walsh, Nature Communications14, 2079 (2023)

    work page 2079

  26. [26]

    E. C. P. Breard, J. Dufek, and O. Roche, Journal of Geophysical Research: Solid Earth124, 5557 (2019)

    work page 2019

  27. [27]

    Bouchut, E

    F. Bouchut, E. D. Fern´ andez-Nieto, A. Mangeney, and G. Narbona-Reina, Journal of Fluid Mechanics801, 166 (2016)

    work page 2016

  28. [28]

    Garres-D´ ıaz, F

    J. Garres-D´ ıaz, F. Bouchut, E. D. Fern´ andez-Nieto, A. Mangeney, and G. Narbona-Reina, Journal of Computational Physics419, 109699 (2020)

    work page 2020

  29. [29]

    Goren, E

    L. Goren, E. Aharonov, D. Sparks, and R. Toussaint, Journal of Geophysical Research: Solid Earth115(2010), https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2009JB007191

    work page doi:10.1029/2009jb007191 2010

  30. [30]

    Gueugneau, K

    V. Gueugneau, K. Kelfoun, O. Roche, and L. Chupin, Geophysical Research Letters44, 2194 (2017)

    work page 2017

  31. [31]

    R. M. Iverson, Reviews of geophysics35, 245 (1997)

    work page 1997

  32. [32]

    Greenshields and H

    C. Greenshields and H. Weller,Notes on Computational Fluid Dynamics: General Principles (CFD Direct Ltd, Reading, UK, 2022)

    work page 2022

  33. [33]

    P. Jop, Y. Forterre, and O. Pouliquen, Nature441, 727 (2006)

    work page 2006

  34. [34]

    Lagr´ ee, L

    P.-Y. Lagr´ ee, L. Staron, and S. Popinet, Journal of Fluid Mechanics686, 378 (2011)

    work page 2011

  35. [35]

    Barker, D

    T. Barker, D. Schaeffer, P. Bohorquez, and J. Gray, Journal of Fluid Mechanics779, 794 (2015)

    work page 2015

  36. [36]

    Barker and J

    T. Barker and J. M. N. T. Gray, Journal of Fluid Mechanics828, 5 (2017)

    work page 2017

  37. [37]

    Montserrat, A

    S. Montserrat, A. Tamburrino, O. Roche, and Y. Ni˜ no, Journal of Geophysical Research: Earth Surface117, F02034 (2012). 61

    work page 2012

  38. [38]

    Aravena, L

    ´A. Aravena, L. Chupin, T. Dubois, and O. Roche, Bulletin of Volcanology86, 90 (2024)

    work page 2024

  39. [39]

    S. B. Savage and K. Hutter, Journal of fluid mechanics199, 177 (1989)

    work page 1989

  40. [40]

    Y. Zhu, R. Delannay, and A. Valance, Granular Matter22, 82 (2020)

    work page 2020

  41. [41]

    Xia and Q

    X. Xia and Q. Liang, Engineering Geology234, 174 (2018)

    work page 2018

  42. [42]

    R. P. Denlinger and R. M. Iverson, Journal of Geophysical Research: Solid Earth106, 553 (2001)

    work page 2001

  43. [43]

    R. M. Iverson, M. Logan, R. G. LaHusen, and M. Berti, Journal of Geophysical Research: Earth Surface115(2010)

    work page 2010

  44. [44]

    de’ Michieli Vitturi, T

    M. de’ Michieli Vitturi, T. Esposti Ongaro, and S. Engwell, Geoscientific Model Development Discussions2023, 1 (2023)

    work page 2023

  45. [45]

    de’ Michieli Vitturi, T

    M. de’ Michieli Vitturi, T. Esposti Ongaro, G. Lari, and A. Aravena, Geoscientific Model Development12, 581 (2019)

    work page 2019

  46. [46]

    Neglia, R

    F. Neglia, R. Sulpizio, F. Dioguardi, L. Capra, and D. Sarocchi, Journal of Volcanology and Geothermal Research410, 107146 (2021)

    work page 2021

  47. [47]

    Kelfoun, Journal of Geophysical Research: Solid Earth116(2011), https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2010JB007622

    K. Kelfoun, Journal of Geophysical Research: Solid Earth116(2011), https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2010JB007622

    work page doi:10.1029/2010jb007622 2011

  48. [48]

    Zuccarello, D

    F. Zuccarello, D. Andronico, P. Del Carlo, M. de’Michieli Vitturi, A. Di Roberto, G. Ganci, B. Behncke, T. Esposti Ongaro, F. Ciancitto, and A. Cappello, Communications Earth & Environment6, 495 (2025)

    work page 2025

  49. [49]

    S. E. Ogburn and E. S. Calder, Frontiers in Earth ScienceVolume 5 - 2017, 10.3389/feart.2017.00083 (2017). 62

    work page doi:10.3389/feart.2017.00083 2017