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arxiv: 2605.14397 · v1 · submitted 2026-05-14 · ⚛️ physics.geo-ph · cs.NA· math.NA· physics.comp-ph· physics.flu-dyn

Recognition: no theorem link

Three dimensional simulation of fluid-driven frictional and tensile ruptures on existing discontinuities

Brice Lecampion , Sylvain Brisson , Antareep Sarma , Ankit Gupta , Alexis S\'aez , Regina Fakhretdinova
Authors on Pith no claims yet

Pith reviewed 2026-05-15 01:48 UTC · model grok-4.3

classification ⚛️ physics.geo-ph cs.NAmath.NAphysics.comp-phphysics.flu-dyn
keywords 3D hydro-mechanical simulationfluid-driven rupturefrictional sliptensile openingcohesive zone modelboundary element methodhydraulic fractureintersecting discontinuities
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The pith

An implicit coupled solver simulates three-dimensional fluid-driven ruptures combining frictional slip and tensile opening on intersecting discontinuities.

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

The paper introduces a numerical framework to model how fluid pressure drives failures along networks of pre-existing fractures and faults in three dimensions. This capability matters for applications such as hydraulic fracturing, geothermal stimulation, and induced seismicity, where both shear sliding and tensile opening can occur together on the same discontinuities. The method pairs a boundary element discretization of the elastic rock response with a finite element treatment of fluid diffusion inside the fractures. Failure is controlled by a cohesive interface law that incorporates slip-weakening friction, dilatancy, permeability changes, and tensile strength loss, all solved implicitly with adaptive time stepping. The approach is verified on analytical benchmarks ranging from simple frictional ruptures to penny-shaped hydraulic fractures and is demonstrated on multi-fracture geometries.

Core claim

We present an implicit, fully-coupled hydro-mechanical solver for the three dimensional simulation of fluid-driven rupture propagation along existing discontinuities. The solver handles simultaneously frictional slip (shear failure) and tensile opening (hydraulic fracture) along arbitrary intersecting fractures and faults in a linearly elastic and impermeable rock matrix. Spatial discretization combines a collocation displacement discontinuity boundary element method for quasi-static elasticity with a Galerkin finite element method for nonlinear pore-fluid diffusion along the discontinuities. Frictional and tensile failure are governed by a poro-elastoplastic cohesive zone like interface law

What carries the argument

Poro-elastoplastic cohesive zone interface law with slip-weakening friction, dilatancy and tensile strength degradation, coupled to a collocation displacement discontinuity boundary element method for elasticity and a Galerkin finite element method for fluid diffusion, solved with adaptive implicit time-stepping and block preconditioning using hierarchical matrices.

Load-bearing premise

The rock matrix is linearly elastic and impermeable, with all failure localized along pre-existing discontinuities according to the poro-elastoplastic cohesive zone law.

What would settle it

A laboratory or field fluid-injection experiment on a rock mass with known intersecting discontinuities in which the observed rupture timing, extent or mode deviates from the simulated pattern when matrix permeability is non-negligible.

Figures

Figures reproduced from arXiv: 2605.14397 by Alexis S\'aez, Ankit Gupta, Antareep Sarma, Brice Lecampion, Regina Fakhretdinova, Sylvain Brisson.

Figure 1
Figure 1. Figure 1: Schematic of a discontinuity Γf filled with fluid at pressure p, with the definition of the local cartesian frame (s1, s2, n), traction and displacement discontinuity vectors. Partial triangulated mesh as example. We solely focus our discussion on the limiting case where the rock matrix can be considered impermeable at the scale of the duration of the fluid injection. This limit is commonly encountered in … view at source ↗
Figure 2
Figure 2. Figure 2: Differences between a fracture (left) and a fault/shear zone (right) with a principal slip zone (PSZ). [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Combined yield functions for Coulomb frictional and tensile failure with softening for a pre [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Schematic of the 3D problem setup. A polygonal fault is embedded in a three-dimensional domain [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: (a) Evolution of the rupture radius with time obtained from the 3D numerical results and analytical [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: (a) Comparison between analytical and numerical slip profiles (along a radial direction) at three [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Non-circular rupture in a critically stressed fault with [PITH_FULL_IMAGE:figures/full_fig_p023_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: 3D circular fluid-driven rupture with linear slip-weakening friction. Evolution of the normalized [PITH_FULL_IMAGE:figures/full_fig_p024_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Evolution of the rupture radius with time over 6 decades of time for a constant friction dilatant [PITH_FULL_IMAGE:figures/full_fig_p026_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Scaled profiles as function of dimensionless radial distance at different dimensionless times. Top [PITH_FULL_IMAGE:figures/full_fig_p027_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Evolution of the fracture radius of a penny-shaped hydraulic fracture, spanning part of the [PITH_FULL_IMAGE:figures/full_fig_p028_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Profiles along the dimensionless radius ρ = r/R of dimensionless pressure Π and opening Ω for three values of dimensionless toughness K = (t/tmk) 1/9 . We observe the transition from the viscosity solution, denoted K = 0, to the toughness solution with a first order correction for viscosity, denoted K ≳ 0.7. We also show the numerical results obtained by (Madyarova 2003) for reference. The position of the… view at source ↗
Figure 13
Figure 13. Figure 13: Relative difference in pressure and opening between our results and the high fidelity numerical [PITH_FULL_IMAGE:figures/full_fig_p030_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Evolution of the yield function and fluid pressure in the three-fracture configuration at [PITH_FULL_IMAGE:figures/full_fig_p033_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Schematic view of the simulation setup for modeling the intersection of a vertical height contained [PITH_FULL_IMAGE:figures/full_fig_p034_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Numerical results of the height contained HF interacting with a strike slip fault. Opening on [PITH_FULL_IMAGE:figures/full_fig_p035_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Profiles of slip, opening, and pressure taken along the intersections of the fault and HF planes [PITH_FULL_IMAGE:figures/full_fig_p035_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Numerical solver statistics for the three fracture simulation: i) time-step size evolutions, and the [PITH_FULL_IMAGE:figures/full_fig_p036_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Numerical solver statistics for the HF / fault interaction simulation: i) time-step size evolutions, [PITH_FULL_IMAGE:figures/full_fig_p036_19.png] view at source ↗
read the original abstract

We present an implicit, fully-coupled hydro-mechanical solver for the three dimensional simulation of fluid-driven rupture propagation along existing discontinuities. The solver handles simultaneously frictional slip (shear failure) and tensile opening (hydraulic fracture) along arbitrary intersecting fractures and faults in a linearly elastic and impermeable rock matrix. The spatial discretization combines a collocation displacement discontinuity boundary element method for quasi-static elasticity with a Galerkin finite element method for nonlinear pore-fluid diffusion along the discontinuities. Frictional and tensile failure are governed by a poro-elastoplastic cohesive zone like interface law with slip-weakening friction, dilatancy, and tensile strength degradation, integrated via an elastic predictor-plastic corrector scheme. The strong nonlinear coupling between mechanical deformation and fracture permeability is handled via adaptive implicit time-stepping. Efficient block preconditioning of the coupled tangent system, leveraging hierarchical matrix representations of the boundary element operator, is essential to achieve robustness across the full range of fracture behaviors. Accuracy and convergence are demonstrated against a comprehensive suite of analytical and semi-analytical solutions of increasing complexity: fluid-driven frictional ruptures under constant and slip-weakening friction, dilatant ruptures with permeability changes, and penny shaped hydraulic fractures spanning the viscosity-to-toughness transition. The solver is further assessed on two multi-fracture configurations: injection into three intersecting fractures, and a height-confined hydraulic fracture intersecting a strike-slip fault. The proposed framework simultaneously captures frictional slip, dilatancy, permeability evolution, and tensile opening.

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 / 2 minor

Summary. The manuscript presents an implicit, fully-coupled hydro-mechanical solver for three-dimensional simulation of fluid-driven ruptures along existing discontinuities. It combines a collocation displacement discontinuity BEM for quasi-static elasticity with a Galerkin FEM for nonlinear pore-fluid diffusion, governed by a poro-elastoplastic cohesive-zone interface law incorporating slip-weakening friction, dilatancy, and tensile strength degradation. The solver is validated against analytical and semi-analytical solutions for frictional ruptures, dilatant cases, and penny-shaped hydraulic fractures, then demonstrated on multi-fracture configurations such as intersecting fractures and a hydraulic fracture intersecting a strike-slip fault.

Significance. If the central claim holds, the work offers a robust framework for simultaneously modeling frictional slip, dilatancy, permeability evolution, and tensile opening in 3D, which is significant for applications in hydraulic fracturing, fault reactivation, and induced seismicity. The comprehensive validation suite against analytical solutions across viscosity-toughness regimes and the use of hierarchical matrix preconditioning for the coupled system are notable strengths supporting numerical reliability.

major comments (2)
  1. [Section 4] Validation suite (Section 4 and multi-fracture examples): the reported benchmarks test frictional slip, dilatancy/permeability changes, and tensile opening in separate analytical cases of increasing complexity before moving to multi-fracture configurations; no single test activates all four mechanisms concurrently under the full poro-elastoplastic coupling. This leaves the robustness of the elastic predictor-plastic corrector and block-preconditioned tangent system under simultaneous activation less directly evidenced, which is load-bearing for the abstract claim that the framework 'simultaneously captures' all effects.
  2. [§3] §3 (interface law and time-stepping): the adaptive implicit scheme is stated to handle the strong nonlinear coupling between deformation and permeability, yet the convergence behavior and stability of the block-preconditioned system are not quantified for cases where slip-weakening, dilatancy, and tensile degradation are all active at once; this requires explicit demonstration to support the central claim.
minor comments (2)
  1. [Figures 8-10] Figure captions for the multi-fracture configurations could more explicitly label the regions of frictional versus tensile failure to illustrate the simultaneous mechanisms.
  2. [Notation and tables] Notation for dilatancy and permeability evolution parameters should be checked for consistency between the interface law definition and the numerical results tables.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive review of our manuscript. The comments highlight important aspects of validation and numerical robustness that we address below. We propose targeted revisions to strengthen the evidence for the coupled solver's performance.

read point-by-point responses

  1. Referee: [Section 4] Validation suite (Section 4 and multi-fracture examples): the reported benchmarks test frictional slip, dilatancy/permeability changes, and tensile opening in separate analytical cases of increasing complexity before moving to multi-fracture configurations; no single test activates all four mechanisms concurrently under the full poro-elastoplastic coupling. This leaves the robustness of the elastic predictor-plastic corrector and block-preconditioned tangent system under simultaneous activation less directly evidenced, which is load-bearing for the abstract claim that the framework 'simultaneously captures' all effects.

    Authors: We agree that presenting a single benchmark activating all mechanisms (frictional slip, dilatancy, permeability evolution, and tensile opening) concurrently would provide stronger direct evidence for the coupled scheme. Although the multi-fracture examples (injection into intersecting fractures and hydraulic fracture intersecting a strike-slip fault) already involve concurrent activation of tensile opening, frictional slip, and dilatancy-driven permeability changes under the full poro-elastoplastic interface law, we will add a new dedicated test case in the revised Section 4. This case will simulate a fluid-driven rupture where slip-weakening, dilatancy, tensile degradation, and permeability evolution are all active simultaneously, including explicit reporting of nonlinear iteration counts and residual convergence for the elastic predictor-plastic corrector and block-preconditioned system. revision: yes

  2. Referee: [§3] §3 (interface law and time-stepping): the adaptive implicit scheme is stated to handle the strong nonlinear coupling between deformation and permeability, yet the convergence behavior and stability of the block-preconditioned system are not quantified for cases where slip-weakening, dilatancy, and tensile degradation are all active at once; this requires explicit demonstration to support the central claim.

    Authors: We acknowledge that explicit quantification of convergence metrics for the fully coupled nonlinear system under simultaneous activation of slip-weakening, dilatancy, and tensile degradation would better support the claims in the abstract and Section 3. In the revised manuscript, we will expand Section 3 to include a new figure or table reporting the number of nonlinear iterations per time step, adaptive time-step sizes, and residual norms for a representative simulation where all three nonlinear mechanisms are active concurrently. This will demonstrate the stability and robustness of the block-preconditioned tangent system and adaptive implicit time-stepping scheme. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation combines established BEM/FEM with independent interface law

full rationale

The paper describes a numerical framework that discretizes elasticity via collocation displacement discontinuity BEM and fluid diffusion via Galerkin FEM, then couples them through an elastic predictor-plastic corrector scheme applied to a poro-elastoplastic cohesive-zone law. All load-bearing steps (spatial discretization, time integration, block preconditioning) are standard techniques whose correctness is verified against external analytical solutions rather than being presupposed by the target result. No equation reduces to a fitted parameter renamed as prediction, no uniqueness theorem is imported from self-citation, and the simultaneous-capture claim follows directly from the coupled residual formulation rather than from any self-definitional loop.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The model rests on standard domain assumptions in linear elasticity and poroelasticity plus a cohesive zone interface law; no new entities are postulated.

free parameters (1)
  • slip-weakening and dilatancy parameters
    Parameters in the poro-elastoplastic cohesive zone law are introduced to govern failure behavior.
axioms (2)
  • domain assumption Rock matrix is linearly elastic and impermeable.
    Stated explicitly as the setting for the boundary element discretization.
  • domain assumption Quasi-static elasticity applies.
    Used for the displacement discontinuity boundary element method.

pith-pipeline@v0.9.0 · 5603 in / 1222 out tokens · 41139 ms · 2026-05-15T01:48:38.659472+00:00 · methodology

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Reference graph

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