arxiv: 2604.05609 · v1 · submitted 2026-04-07 · ⚛️ physics.geo-ph

Recognition: 2 theorem links

· Lean Theorem

Simulating Subterranean Fluid Injection through Iteration on the VirtualQuake Model

John Rundle, Spence Norwood

Pith reviewed 2026-05-10 19:09 UTC · model grok-4.3

classification ⚛️ physics.geo-ph

keywords fluid injectioninduced seismicityearthquake simulationfault stabilitystress modelinginvasion percolationVirtualQuakesubsurface pressure

0 comments

The pith

Repeated fluid injections build persistent high-pressure zones that progressively destabilize faults more than single injections do.

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

The paper modifies the VirtualQuake earthquake simulator so it can handle fluid being pumped into the ground. Instead of modeling faults as fixed rectangular blocks, it switches to point sources of stress that can be placed more freely. These sources are spread using invasion percolation to mimic how injected fluid moves through rock pores. The central result is that one round of injection creates only small, short-lived stress shifts on nearby faults, but repeated rounds leave behind lasting high-pressure pockets. Those pockets keep pushing on faults over time, raising the odds that they will slip and trigger earthquakes. The work aims to give a practical way to judge the long-term seismic risks that come with ongoing commercial injection projects.

Core claim

By converting VirtualQuake to a stress-point-source formulation and distributing inflationary sources according to invasion percolation, the simulation reproduces both the immediate stress and deformation caused by fluid injection and the slower post-injection pressure decay. Single injection episodes produce only limited changes in fault stability. In contrast, successive injection cycles leave behind persistent high-pressure regions that continue to load nearby faults, steadily increasing the probability of induced seismic events. The same framework also captures hydraulic-fracturing effects during active injection.

What carries the argument

Invasion-percolation distribution of inflationary stress point sources inside the reworked VirtualQuake model, which replaces rectangular fault segments with freely placed point sources to allow flexible geometry and cross-fault interactions while simulating fluid-driven stress changes.

If this is right

Operators can use the model to assess cumulative rather than single-event risks from long-running injection programs.
The framework can track both the stress loading during active pumping and the lingering effects after injection ceases.
Hydraulic fracturing and pressure dissipation are now included in the same simulation loop as fault-slip calculations.
The point-source approach restores cross-fault stress interactions that were previously difficult to represent.

Where Pith is reading between the lines

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

Risk assessments for injection sites should weight the entire history of pumping rather than the most recent operation alone.
The same percolation-based placement method could be tested against wastewater-disposal or geothermal-injection records to check consistency.
If the pressure-buildup pattern holds, seismic monitoring networks might prioritize cumulative volume thresholds over instantaneous rate limits.

Load-bearing premise

That placing stress sources by invasion percolation correctly reproduces how real injected fluids spread, how they alter stress on faults, and how pressure fades after pumping stops.

What would settle it

Field measurements at an active injection site showing that cumulative injected volume does not correlate with rising seismicity rates or that pore-pressure anomalies decay much faster than the simulation predicts would disprove the central claim.

Figures

Figures reproduced from arXiv: 2604.05609 by John Rundle, Spence Norwood.

**Figure 2.** Figure 2: Percent difference in magnitude between a tiled fault with 4 and 16 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: The behavioral differences of Fractional Brownian Motion with varying [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** Figure 4: The result of a two-dimensional Brownian transformation on a sample [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

**Figure 5.** Figure 5: Pressure dynamics at the beginning of a series of fracking cycles[1] [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗

**Figure 6.** Figure 6: An example injection simulation, with associated spatial pressure drop [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗

**Figure 7.** Figure 7: The beginning of pressure equalization over the course of a year, post [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗

**Figure 8.** Figure 8: The cumulative effects on CF F from a sample fluid injection. Each voxel square is mapped to an inflationary s In this figure, a sample of a typical fracking sequence is displayed, an iteration of 10 fracking cycles. The sequence leads to repeated bursts of injected fluid, the center moving laterally at the given depth. A nearby fault section is rendered, with the net change in the Coulomb Failure Function… view at source ↗

read the original abstract

This work extends the VirtualQuake earthquake simulation framework to incorporate the effects of fluid injection on fault stability and induced seismicity. Reworking VirtualQuake into a system using stress point sources, instead of rectangular segments, the new model offers increased geometric flexibility, greater stability, and the re-addition of cross-fault interactions. This approach is paired with the addition of fluid injection modeling, through the distribution of inflationary stress sources, according to invasion percolation, simulating both the stress effect of the injection on nearby faults, and deformation from the injection itself. The model captures both immediate and long-term impacts of injection cycles, including hydraulic fracturing processes and post-injection pressure dissipation. Results show that while single injections produce limited stress changes, repeated injections generate persistent high-pressure regions that progressively destabilize nearby faults, increasing the likelihood of seismic events. This model evolution offers a tool for the evaluation and characterization of long term risks from commercial injection.

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 / 1 minor

Summary. The manuscript extends the VirtualQuake earthquake simulation framework by replacing rectangular fault segments with stress point sources, enabling greater geometric flexibility and cross-fault interactions. Fluid injection is modeled by distributing inflationary stress sources according to invasion percolation, which is claimed to capture both immediate stress effects on faults and deformation. The central result is that single injections produce only limited stress changes, while repeated injections generate persistent high-pressure regions that progressively destabilize nearby faults and increase seismic event likelihood; the model also incorporates post-injection pressure dissipation and hydraulic fracturing processes.

Significance. If the modeling assumptions hold and are validated, this extension could supply a flexible simulation tool for assessing long-term induced-seismicity risks from commercial fluid injection. The point-source reformulation restores cross-fault interactions and improves stability. However, the current presentation supplies no quantitative metrics, error analysis, or comparisons to continuum poroelastic models, so the practical significance remains provisional.

major comments (2)

[Abstract] Abstract: the headline claim that repeated injections create persistent high-pressure regions progressively destabilizing faults is generated solely by placing inflationary point sources via invasion percolation; no comparison is shown to the continuum diffusion equation or to poroelastic coupling, so it is unclear whether the reported difference versus single injections is physical or an artifact of the discrete distribution rule.
[Modeling Approach] Modeling section (implied by abstract description): the switch from rectangular segments with explicit stress kernels to point sources plus an unspecified post-injection dissipation rule introduces two untested approximations whose failure would directly undermine the single-versus-repeated injection contrast; quantitative validation against observed injection-induced seismicity or against established codes is absent.

minor comments (1)

[Abstract] Abstract: the phrase 're-addition of cross-fault interactions' is stated without indicating how the point-source formulation restores them or quantifying their contribution to the reported stress changes.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the constructive and detailed comments on our manuscript. We address each major comment below and outline the revisions we will make to improve clarity and rigor.

read point-by-point responses

Referee: [Abstract] Abstract: the headline claim that repeated injections create persistent high-pressure regions progressively destabilizing faults is generated solely by placing inflationary point sources via invasion percolation; no comparison is shown to the continuum diffusion equation or to poroelastic coupling, so it is unclear whether the reported difference versus single injections is physical or an artifact of the discrete distribution rule.

Authors: The abstract reports the central simulation outcome obtained with the invasion-percolation distribution of inflationary sources. This discrete rule is adopted precisely because it generates heterogeneous, path-dependent pressure accumulation that persists across injection cycles when dissipation is incomplete; a single injection permits more complete relaxation. We acknowledge that the manuscript does not include a direct numerical comparison against the continuum diffusion equation or full poroelastic coupling. In the revised version we will add a dedicated discussion paragraph that contrasts the percolation-based localization with continuum expectations and explicitly states the modeling choice and its limitations. revision: partial
Referee: [Modeling Approach] Modeling section (implied by abstract description): the switch from rectangular segments with explicit stress kernels to point sources plus an unspecified post-injection dissipation rule introduces two untested approximations whose failure would directly undermine the single-versus-repeated injection contrast; quantitative validation against observed injection-induced seismicity or against established codes is absent.

Authors: The point-source reformulation is fully specified in the Modeling section, where we demonstrate that it recovers cross-fault stress interactions while increasing numerical stability and geometric flexibility. The post-injection dissipation is implemented as an exponential decay whose time constant is chosen to reproduce typical field pressure-decline rates; the functional form and parameter values appear immediately after Equation (2). We agree that quantitative checks are needed. The revised manuscript will include error metrics for the point-source stress approximation and a side-by-side comparison against a standard poroelastic finite-element benchmark. Full validation against specific observed induced-seismicity catalogs lies beyond the scope of the present methodological study. revision: yes

standing simulated objections not resolved

Quantitative validation against observed injection-induced seismicity catalogs cannot be completed in the current revision.

Circularity Check

0 steps flagged

No significant circularity in the derivation or results

full rationale

The paper extends the existing VirtualQuake framework by switching to point stress sources and adding fluid injection via invasion percolation. The headline results (limited stress changes from single injections versus persistent destabilizing high-pressure zones from repeated injections) are produced by running the updated simulation under different injection schedules; they are not equivalent to the model inputs by definition, nor are they obtained by fitting parameters to data and relabeling the fit as a prediction. No load-bearing step reduces to a self-citation chain, an ansatz smuggled via prior work, or a uniqueness theorem imported from the same authors. The derivation chain is therefore self-contained as a forward simulation exercise whose outputs can be compared against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review is based on abstract only, so ledger entries are inferred from the described approach. No explicit free parameters or new entities are named. The model rests on standard geophysical assumptions about stress representation and fluid flow.

axioms (2)

domain assumption Stress point sources can represent fault mechanics, stability, and cross-fault interactions with greater flexibility than rectangular segments.
Invoked when reworking VirtualQuake to achieve increased geometric flexibility and stability.
domain assumption Invasion percolation provides a suitable distribution for inflationary stress sources that simulates fluid injection, hydraulic fracturing, and pressure dissipation.
Used to model both immediate stress effects on faults and long-term deformation from injection cycles.

pith-pipeline@v0.9.0 · 5454 in / 1495 out tokens · 80108 ms · 2026-05-10T19:09:57.533799+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
distribution of inflationary stress sources, according to invasion percolation... W_ijk = P_ijk d³ ϕ ... ΔP(δ₀,δ) = C H_r E √t₀ / (H β_s) G(δ,δ₀)
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear
point source solutions for an elastic half space... tiled fault space by assigning rectangular zones to point sources

Reference graph

Works this paper leans on

24 extracted references · 15 canonical work pages

[1]

Springer Science & Business Media, 1997

Bernard Amadei and Ove Stephansson.Rock stress and its measurement. Springer Science & Business Media, 1997

1997
[2]

Probability of Induced Seismicity Associated with Large-Scale Underground Hydrogen Storage in Salt Formations of Northwestern Europe

N. Van Den Ameele et al. “Probability of Induced Seismicity Associated with Large-Scale Underground Hydrogen Storage in Salt Formations of Northwestern Europe”. In: 2025.1 (2025), pp. 1–5.issn: 2214-4609.doi: https://doi.org/10.3997/2214-4609.202521116.url:https://www. earthdoc.org/content/papers/10.3997/2214-4609.202521116

work page doi:10.3997/2214-4609.202521116.url:https://www 2025
[3]

Economides and Kenneth G

Michael J. Economides and Kenneth G. Nolte.Reservoir stimulation. Wi- ley, 2000

2000
[4]

Evidence of fractional-Brownian-motion-type asperity model for earthquake generation in candidate pre-seismic electromagnetic emissions

K. Eftaxias et al. “Evidence of fractional-Brownian-motion-type asperity model for earthquake generation in candidate pre-seismic electromagnetic emissions”. In:Natural Hazards and Earth System Sciences8.4 (2008), pp. 657–669.doi:10.5194/nhess-8- 657-2008.url:https://nhess. copernicus.org/articles/8/657/2008/

work page doi:10.5194/nhess-8- 2008
[5]

Object-oriented programming for engineering soft- ware development

Gregory L. Fenves. “Object-oriented programming for engineering soft- ware development”. In:Engineering with Computers6.1 (Dec. 1990), pp. 1– 15.doi:10 . 1007 / BF01200200.url:https : / / doi . org / 10 . 1007 / BF01200200

1990
[6]

Dynamics of Invasion Percolation

Liv Furuberg et al. “Dynamics of Invasion Percolation”. In:Phys. Rev. Lett.61 (18 Oct. 1988), pp. 2117–2120.doi:10.1103/PhysRevLett.61. 2117.url:https://link.aps.org/doi/10.1103/PhysRevLett.61. 2117

work page doi:10.1103/physrevlett.61 1988
[7]

Analysis of fluid injection-induced fault reactivation and seismic slip in geothermal reservoirs

Quan Gan and Derek Elsworth. “Analysis of fluid injection-induced fault reactivation and seismic slip in geothermal reservoirs”. In:Journal of Geo- physical Research: Solid Earth119.4 (2014), pp. 3340–3353.doi:https:// doi.org/10.1002/2013JB010679. eprint:https://agupubs.onlinelibrary. wiley.com/doi/pdf/10.1002/2013JB010679.url:https://agupubs. onlinelibr...

work page doi:10.1002/2013jb010679 2014
[8]

The object oriented model and its advan- tages

Jos´ e de Oliveira Guimar˜ aes. “The object oriented model and its advan- tages”. In:SIGPLAN OOPS Mess.6.1 (Jan. 1995), pp. 40–49.issn: 1055- 6400.doi:10.1145/209866.209872.url:https://doi.org/10.1145/ 209866.209872

work page doi:10.1145/209866.209872.url:https://doi.org/10.1145/ 1995
[9]

On the Coulomb-Mohr failure criterion

John Handin. “On the Coulomb-Mohr failure criterion”. In:Journal of Geophysical Research (1896-1977)74.22 (1969), pp. 5343–5348.doi:https: / / doi . org / 10 . 1029 / JB074i022p05343. eprint:https : / / agupubs . onlinelibrary.wiley.com/doi/pdf/10.1029/JB074i022p05343.url: https : / / agupubs . onlinelibrary . wiley . com / doi / abs / 10 . 1029 / JB074i0...

work page doi:10.1029/jb074i022p05343.url: 1977
[10]

Electrical properties of lunar soil dependence on frequency, temperature and moisture,

J. Huang and D.L. Turcotte. “Fractal distributions of stress and strength and variations of b-value”. In:Earth and Planetary Science Letters91.1 (1988), pp. 223–230.issn: 0012-821X.doi:https://doi.org/10.1016/ 0012 - 821X(88 ) 90164 - 1.url:https : / / www . sciencedirect . com / science/article/pii/0012821X88901641

work page arXiv 1988
[11]

Ang Li et al. “Investigation of pore structure and fractal characteristics of organic-rich shale reservoirs: A case study of Lower Cambrian Qiongzhusi formation in Malong block of eastern Yunnan Province, South China”. In:Marine and Petroleum Geology70 (Nov. 2015).doi:10 . 1016 / j . marpetgeo.2015.11.004

2015
[12]

Effect of longitudinal proppant placement and engineering parameters on fracture conductivity

Zhi Lin et al. “Effect of longitudinal proppant placement and engineering parameters on fracture conductivity”. In:Petroleum Science and Tech- nology0.0 (2025), pp. 1–24.doi:10 . 1080 / 10916466 . 2024 . 2448538. eprint:https : / / doi . org / 10 . 1080 / 10916466 . 2024 . 2448538.url: https://doi.org/10.1080/10916466.2024.2448538

work page doi:10.1080/10916466.2024.2448538 2025
[13]

Geologic characteristics, exploration and produc- tion progress of shale oil and gas in the United States: An overview

T P MCMAHON et al. “Geologic characteristics, exploration and produc- tion progress of shale oil and gas in the United States: An overview”. In: Petroleum Exploration and Development51.4 (2024), pp. 925–948.issn: 1876-3804.doi:https://doi.org/10.1016/S1876- 3804(24)60516- 1.url:https : / / www . sciencedirect . com / science / article / pii / S18763804246...

work page doi:10.1016/s1876- 2024
[14]

Loopless non- trapping invasion-percolation model for fracking

J. Quinn Norris, Donald L. Turcotte, and John B. Rundle. “Loopless non- trapping invasion-percolation model for fracking”. In:Phys. Rev. E89 (2 Feb. 2014), p. 022119.doi:10.1103/PhysRevE.89.022119.url:https: //link.aps.org/doi/10.1103/PhysRevE.89.022119

work page doi:10.1103/physreve.89.022119.url:https: 2014
[15]

Internal deformation due to shear and tensile fault in a half space

Yoshimitsu Okada. “Internal deformation due to shear and tensile fault in a half space”. In:Bulletin of the Seismological Society of America92 (Apr. 1992), pp. 1018–1040

1992
[16]

Percolation and invasion percolation as fractal growth problems

L. Pietronero, W. R. Schneider, and A. L. Stella. “Percolation and invasion percolation as fractal growth problems”. In:Phys. Rev. A42 (12 Dec. 1990), pp. 7496–7499.doi:10.1103/PhysRevA.42.7496.url:https: //link.aps.org/doi/10.1103/PhysRevA.42.7496

work page doi:10.1103/physreva.42.7496.url:https: 1990
[17]

Euclidean and fractal models for the description of rock surface roughness

William L. Power and Terry E. Tullis. “Euclidean and fractal models for the description of rock surface roughness”. In:Journal of Geophysical Research: Solid Earth96.B1 (1991), pp. 415–424.doi:https : / / doi . org/10.1029/90JB02107. eprint:https://agupubs.onlinelibrary. wiley . com / doi / pdf / 10 . 1029 / 90JB02107.url:https : / / agupubs . onlinelibra...

work page doi:10.1029/90jb02107 1991
[18]

2024.url:https://gist.github

Stephen Sinclair.BrownianSurface. 2024.url:https://gist.github. com/radarsat1/6f8b9b50d1ecd2546d8a765e8a144631

2024
[19]

Hydraulic fracturing operation for oil and gas production and associated earthquake activities across the USA

Ramesh P. Villa Valeriaand Singh. “Hydraulic fracturing operation for oil and gas production and associated earthquake activities across the USA”. In:Environmental Earth Sciences79.11 (May 2020), p. 271.issn: 1866- 6299.doi:10.1007/s12665-020-09008-0.url:https://doi.org/10. 1007/s12665-020-09008-0

work page doi:10.1007/s12665-020-09008-0.url:https://doi.org/10 2020
[20]

Texier and G

D Wilkinson and J F Willemsen. “Invasion percolation: a new form of percolation theory”. In:Journal of Physics A: Mathematical and General 16.14 (Oct. 1983), p. 3365.doi:10.1088/0305- 4470/16/14/028.url: https://doi.org/10.1088/0305-4470/16/14/028

work page doi:10.1088/0305- 1983
[21]

Analytical interpretation of hydraulic fracturing initi- ation pressure and breakdown pressure

Feipeng Wu et al. “Analytical interpretation of hydraulic fracturing initi- ation pressure and breakdown pressure”. In:Journal of Natural Gas Sci- ence and Engineering76 (2020), p. 103185.issn: 1875-5100.doi:https: / / doi . org / 10 . 1016 / j . jngse . 2020 . 103185.url:https : / / www . sciencedirect.com/science/article/pii/S1875510020300391

2020
[22]

A fault and seismicity based composite simulation in northern California

M. B. Yıkılmaz et al. “A fault and seismicity based composite simulation in northern California”. In:Nonlinear Processes in Geophysics18.6 (2011), pp. 955–966.doi:10 . 5194 / npg - 18 - 955 - 2011.url:https : / / npg . copernicus.org/articles/18/955/2011/

2011
[23]

A Criterion for Non-Darcy Flow in Porous Media

Zhengwen Zeng and Reid Grigg. “A Criterion for Non-Darcy Flow in Porous Media”. In:Transport in Porous Media63.1 (Apr. 2006), pp. 57– 69.issn: 1573-1634.doi:10.1007/s11242- 005- 2720- 3.url:https: //doi.org/10.1007/s11242-005-2720-3

work page doi:10.1007/s11242- 2006
[24]

Mechanisms for Geological Carbon Se- questration

Dongxiao Zhang and Juan Song. “Mechanisms for Geological Carbon Se- questration”. In:Procedia IUTAM10 (2014). Mechanics for the World: Proceedings of the 23rd International Congress of Theoretical and Ap- plied Mechanics, ICTAM2012, pp. 319–327.issn: 2210-9838.doi:https: / / doi . org / 10 . 1016 / j . piutam . 2014 . 01 . 027.url:https : / / www . scienc...

2014