arxiv: 2603.26169 · v2 · submitted 2026-03-27 · ⚛️ physics.flu-dyn

Recognition: 2 theorem links

· Lean Theorem

Porous-Medium Scaling of CO₂ Plume Footprint Growth

Fernando Alonso-Marroquin , Christian Tantardini

Authors on Pith no claims yet

Pith reviewed 2026-05-14 23:23 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn

keywords CO2 plumeporous mediumnonlinear diffusionBarenblatt solutionseismic monitoringaquifer storageplume footprint

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The pith

CO2 plumes in aquifers grow with radii matching Barenblatt solutions of porous-medium nonlinear diffusion.

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

The paper checks whether real CO2 plumes spreading underground follow the scaling rules of nonlinear diffusion through porous rock. It pulls equivalent radii from published seismic maps at the Sleipner, Aquistore, and Weyburn-Midale sites and finds that the measured growth rates line up with the slow axisymmetric porous-medium regime. The authors then treat the plume as a thin, vertically segregated CO2 layer whose thickness varies with radius and time inside a fixed aquifer height. They solve the governing equation exactly and obtain simple formulas for the layer thickness profile, the outer edge of the plume, and the inner zone where the layer fills the whole aquifer. This supplies a clear baseline for predicting how far plumes spread and how their internal structure changes under constant injection or after shut-in.

Core claim

Field measurements of CO2 plume footprints are shown to be consistent with Barenblatt similarity solutions of the porous-medium equation in radial geometry. The plume is represented as a CO2 layer of thickness b(r,t) inside an aquifer of thickness H, leading to explicit expressions for the normalized thickness profile b(r,t)/H, the compactly supported outer radius R(t), and the time-dependent inner radius a(t) that bounds the region of full aquifer occupation. Under steady injection the inner core radius stays constant while the outer edge advances proportionally to the square root of time; once injection stops the core contracts until it disappears, after which the entire plume follows the

What carries the argument

Barenblatt-type similarity solution for the vertically segregated CO2 layer thickness b(r,t) inside aquifer height H, which directly supplies the compact-support outer edge R(t) and the transient inner core radius a(t).

If this is right

Under constant injection the outer plume edge grows proportionally to the square root of time while an inner core of full aquifer thickness persists.
After injection ceases the inner core shrinks and eventually vanishes, after which the plume follows the pure Barenblatt solution.
The same closed-form expressions let observers compare plume evolution across sites using only seismic-derived equivalent radii.
The model framework can be extended to non-local effects by replacing the diffusion term with a fractional derivative.

Where Pith is reading between the lines

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

The scaling could give a fast estimate of when a plume might reach a monitoring boundary without running full numerical simulations.
The predicted post-shut-in shrinkage of the inner core radius offers a new field-checkable quantity for validating simulators.
The segregated-layer approach may transfer to other buoyancy-driven flows such as oil migration or contaminant spreading in aquifers.

Load-bearing premise

The CO2 plume behaves as a vertically segregated layer that obeys porous-medium nonlinear diffusion, and the area-based radius taken from seismic images accurately reflects the true footprint.

What would settle it

Measured plume radii at another CO2 storage site that grow faster or slower than the square-root-of-time law under steady injection, or an inner core that fails to shrink after injection stops.

Figures

Figures reproduced from arXiv: 2603.26169 by Christian Tantardini, Fernando Alonso-Marroquin.

**Figure 2.** Figure 2: FIG. 2. Normalized plume-thickness profiles [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Equivalent plume radius [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

read the original abstract

Building on porous-medium-type nonlinear diffusion, we compare analytical Barenblatt-type similarity solutions with plume's radii from digital analysis of published seismic monitoring images, to quantify field-scale CO$_2$ plume-footprint growth. Using an area-based equivalent radius extracted from time-lapse plume maps at Sleipner, Aquistore, and Weyburn--Midale, we obtain effective plume-growth exponents that are broadly compatible with slow porous-medium scaling in axisymmetric geometry. We then interpret the plume as a vertically segregated CO$_2$ layer of thickness $b(r,t)$ within an aquifer of thickness $H$, and derive closed-form expressions for the normalized thickness $b(r,t)/H$, the compact-support plume edge $R(t)$, and a transient inner core radius $a(t)$ that marks the region where the plume occupies the full aquifer thickness. In the shut-in case, the core radius decreases with time and eventually vanishes, after which the plume recovers the pure Barenblatt regime; under constant injection, the model predicts an injection-controlled core and a plume edge that grows with the square-root law. This framework provides a physically transparent baseline for comparing plume-radius evolution, internal plume structure, and core development across sites, and establishes a consistent route for incorporating non-local effects by fractional derivatives in future extensions.

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 derives usable closed-form expressions for CO2 plume thickness and radii under porous-medium scaling and checks them against seismic data from three sites, but the radius extraction lacks error analysis.

read the letter

The core contribution is the set of explicit formulas for normalized plume thickness b(r,t)/H, outer radius R(t), and inner core radius a(t) that follow from treating the CO2 as a vertically segregated layer obeying nonlinear diffusion. For the shut-in case the core shrinks until it vanishes and the plume reverts to the standard Barenblatt profile; under constant injection the core remains fixed by the injection rate while the edge still grows as the square root of time. These expressions are straightforward once the segregation assumption is accepted and give a transparent way to estimate both footprint and internal structure without full numerical runs. The comparison to area-derived radii from Sleipner, Aquistore, and Weyburn-Midale shows exponents that sit near the expected slow porous-medium values, which is a reasonable first check against field data. The derivations themselves look standard and reproducible. The weak point is the empirical step. The radii are taken from published seismic maps with no reported uncertainty from threshold choice, resolution limits, or interpreter variability. Because the exponents are fitted to those radii, even modest systematic shifts in area can change whether a site looks compatible or not, and the paper does not show sensitivity tests. That leaves the claimed compatibility thinner than it first appears. The work is aimed at researchers who model or monitor geological CO2 storage and want an analytical baseline rather than purely computational results. A reader needing quick estimates of plume extent and thickness will find the formulas practical. It is worth sending to peer review because the analytical part is clear and the field comparison is a useful starting point, provided the uncertainty in the radius data is addressed in revision.

Referee Report

2 major / 2 minor

Summary. The paper applies porous-medium nonlinear diffusion to model CO2 plume footprint growth, extracting area-based equivalent radii from time-lapse seismic maps at Sleipner, Aquistore, and Weyburn-Midale to obtain effective growth exponents broadly compatible with axisymmetric Barenblatt scaling. It then derives closed-form expressions for the normalized plume thickness b(r,t)/H, the compact-support edge R(t), and a transient inner core radius a(t) that marks full aquifer occupation, with distinct behaviors under constant injection (injection-controlled core, sqrt(t) edge) and shut-in (core shrinkage to pure Barenblatt regime).

Significance. If the field comparison is robust, the closed-form expressions for b(r,t)/H, R(t), and a(t) supply a transparent analytical baseline for interpreting plume structure and evolution across sites, facilitating quick comparisons with monitoring data and extensions to fractional-derivative non-local effects; this is a strength relative to purely numerical approaches for CO2 storage assessment.

major comments (2)

[Field data analysis section] Field data analysis section: The effective growth exponents are obtained by fitting area-based equivalent radii extracted from published seismic maps, but no uncertainty quantification, sensitivity to saturation thresholding, or resolution limits is reported; modest systematic biases in measured areas (tens of percent) can shift a site from compatible to incompatible with the theoretical 1/4 or 1/2 power, directly undermining the central compatibility claim.
[Derivation of closed-form expressions] Derivation of closed-form expressions (following the vertically segregated assumption): While the expressions for b(r,t)/H, R(t), and a(t) follow standard similarity solutions of the nonlinear diffusion equation, the paper does not clarify whether the exponents are purely predictive or post-hoc fitted to the same radius data used for validation, creating potential circularity that weakens the claim that the model provides an independent baseline.

minor comments (2)

Introduce the inner core radius a(t) with a clear definition and diagram in the early methods section to improve readability of the subsequent shut-in and injection cases.
Add a brief note on how the aquifer thickness H is chosen or estimated for each site when normalizing b(r,t)/H.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments, which help clarify the presentation of our field analysis and analytical derivations. We respond to each major comment below and have revised the manuscript to address the concerns.

read point-by-point responses

Referee: [Field data analysis section] Field data analysis section: The effective growth exponents are obtained by fitting area-based equivalent radii extracted from published seismic maps, but no uncertainty quantification, sensitivity to saturation thresholding, or resolution limits is reported; modest systematic biases in measured areas (tens of percent) can shift a site from compatible to incompatible with the theoretical 1/4 or 1/2 power, directly undermining the central compatibility claim.

Authors: We acknowledge the absence of formal uncertainty quantification in the field data section. The equivalent radii are extracted from published seismic maps, which inherently limits our ability to perform full error propagation from raw data. In the revised manuscript we will add a dedicated paragraph discussing resolution limits and potential systematic biases, together with a sensitivity analysis in which the saturation threshold for area extraction is varied over a plausible range (e.g., 10–30 %). This will quantify the resulting spread in effective growth exponents and demonstrate that the broad compatibility with porous-medium scaling remains robust within those bounds. revision: partial
Referee: [Derivation of closed-form expressions] Derivation of closed-form expressions (following the vertically segregated assumption): While the expressions for b(r,t)/H, R(t), and a(t) follow standard similarity solutions of the nonlinear diffusion equation, the paper does not clarify whether the exponents are purely predictive or post-hoc fitted to the same radius data used for validation, creating potential circularity that weakens the claim that the model provides an independent baseline.

Authors: The growth exponents reported in the field-analysis section are obtained solely by fitting the seismic-derived radii in order to test compatibility with the theoretical porous-medium scaling (R ∼ t^{1/4} for shut-in, R ∼ t^{1/2} for constant injection). The closed-form expressions for the normalized thickness b(r,t)/H, the outer edge R(t), and the inner core radius a(t) are derived independently from the self-similar solutions of the nonlinear diffusion equation under the vertically segregated assumption; they employ the appropriate scaling exponents dictated by the injection regime and do not depend on the fitted field values. We will revise the text to state this separation explicitly, thereby removing any ambiguity about circularity and reinforcing that the analytical baseline is predictive from porous-medium theory. revision: partial

Circularity Check

0 steps flagged

No significant circularity; empirical exponents are independent of the closed-form derivations.

full rationale

The paper extracts area-based radii directly from published seismic maps, computes effective growth exponents by power-law fitting to those radii, and reports numerical compatibility with the known Barenblatt exponents of axisymmetric porous-medium flow. This comparison is a post-hoc empirical check, not a fitted parameter renamed as a prediction. The subsequent closed-form expressions for normalized thickness b(r,t)/H, compact-support edge R(t), and inner-core radius a(t) are obtained by solving the nonlinear diffusion equation under the vertically segregated layer assumption; these algebraic steps follow from the governing PDE and do not reference or depend on the fitted exponents. No self-citation chain, uniqueness theorem, or ansatz smuggling is invoked to close the derivation. The central mathematical results therefore remain independent of the data-reduction step.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The model rests on standard assumptions of porous media flow and similarity solutions, with growth exponents fitted to seismic data. No new entities are postulated.

free parameters (1)

effective plume-growth exponents
Obtained from digital analysis of seismic images to match Barenblatt solutions.

axioms (2)

domain assumption Porous-medium-type nonlinear diffusion governs the plume spread
Building on nonlinear diffusion for porous media flow as stated in the abstract.
domain assumption Vertically segregated CO2 layer of thickness b(r,t) within aquifer of thickness H
Assumed in the derivation of thickness and radii expressions.

pith-pipeline@v0.9.0 · 5534 in / 1587 out tokens · 44688 ms · 2026-05-14T23:23:05.103370+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the scalar q–PME ... ∂ρ/∂t = D0 r^{1-d} ∂/∂r (r^{d-1} ρ^{1-q} ∂ρ/∂r) ... Barenblatt solution ρ(r,t)=t^{-α}[C-K r² t^{-2β}]^{1/(1-q)} with α=d/[d(1-q)+2], β=1/[d(1-q)+2]
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

vertically segregated CO2 layer of thickness b(r,t) ... composite profile with transient full-thickness core a(t)

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

20 extracted references · 20 canonical work pages · 1 internal anchor

[1]

R. Arts, O. Eiken, A. Chadwick, P. Zweigel, L. van der Meer, and B. Zinszner, Energy29, 1383 (2004)

work page 2004
[2]

R. A. Chadwick, D. Noy, R. Arts, and O. Eiken, Energy Procedia1, 2103 (2009)

work page 2009
[3]

Furre and O

A.-K. Furre and O. Eiken, Geophysical Prospecting62, 1075 (2014)

work page 2014
[4]

L. A. N. Roach, D. J. White, B. Roberts, and D. Angus, Geophysics82, B95 (2017)

work page 2017
[5]

D. J. White, K. Harris, L. Roach,et al., Energy Procedia 114, 4056 (2017)

work page 2017
[6]

White, The Leading Edge28, 838 (2009)

D. White, The Leading Edge28, 838 (2009)

work page 2009
[7]

J. M. Nordbotten, M. A. Celia, and S. Bachu, Transport in Porous Media58, 339 (2005)

work page 2005
[8]

J. M. Nordbotten and M. A. Celia,Geologic Storage of CO2: Modeling Approaches for Large-Scale Simulation (Wiley, 2011)

work page 2011
[9]

G. I. Barenblatt, Prikladnaya Matematika i Mekhanika 16, 67 (1952)

work page 1952
[10]

G. I. Barenblatt,Scaling, Self-Similarity, and Intermedi- ate Asymptotics(Cambridge University Press, 1996)

work page 1996
[11]

J. L. V´ azquez,The Porous Medium Equation: Mathe- matical Theory(Oxford University Press, 2007)

work page 2007
[12]

M. L. Szulczewski, C. W. MacMinn, H. J. Herzog, and R. Juanes, Proceedings of the National Academy of Sci- ences109, 5185 (2012)

work page 2012
[13]

Alonso-Marroquin, K

F. Alonso-Marroquin, K. Arias-Calluari, M. Harr´ e, M. N. Najafi, and H. J. Herrmann, Physical Review E99, 062313 (2019)

work page 2019
[14]

Gharari, K

F. Gharari, K. Arias-Calluari, F. Alonso-Marroquin, and M. N. Najafi, Physical Review E104, 054140 (2021)

work page 2021
[15]

Arias-Calluari, M

K. Arias-Calluari, M. N. Najafi, M. S. Harr´ e, Y. Tang, and F. Alonso-Marroquin, Physica A: Statistical Mechan- ics and its Applications587, 126487 (2022). 10

work page 2022
[16]

Y. Tang, F. Gharari, K. Arias-Calluari, F. Alonso- Marroquin, and M. N. Najafi, Physical Review E109, 024310 (2024)

work page 2024
[17]

Y. Tang, K. Arias-Calluari, M. Nattagh Najafi, M. S. Harr´ e, and F. Alonso-Marroquin, Fractal and Fractional 9, 635 (2025)

work page 2025
[18]

Al-Hussainy, H

R. Al-Hussainy, H. J. Ramey, and P. B. Crawford, Jour- nal of Petroleum Technology18, 624 (1966)

work page 1966
[19]

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

C. Tantardini and F. Alonso-Marroquin, Global buckley– leverett for multicomponent flow in fractured media: Isothermal equation-of-state coupling and dynamic cap- illarity (2025), arXiv:2511.06233 [physics.flu-dyn]

work page internal anchor Pith review Pith/arXiv arXiv 2025
[20]

L. A. N. Roach and D. J. White, International Journal of Greenhouse Gas Control74, 79 (2018). 11 Appendix A: Direct reduction of the GBL transport equation to the axisymmetricq–PME We now show explicitly how the scalarq–PME used in this work arises as a reduced limit of the GBL transport equation [19]. The objective is to identify, within the full multico...

work page 2018