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Phase-field modeling of fluid-driven dynamic cracking in porous media
Shuwei Zhou, Xiaoying Zhuang, Timon Rabczuk
TL;DR
Hydraulic fracturing in porous media requires models that can represent complex fracture patterns while accounting for hydro-mechanical coupling and dynamic effects. This paper extends phase-field dynamic-crack modeling to poroelastic media using Biot theory, indicator functions, and a staggered COMSOL implementation. Benchmark comparisons agree well with analytical solutions, while further examples demonstrate dynamic propagation, branching, and interaction with pre-existing natural fractures.
Problem
Accurate prediction of hydraulic-fracture branching, merging, and unintended pathways in porous media remains important, while prior phase-field approaches rarely considered inertial effects.
Method
The approach couples classical Biot poroelasticity with phase-field dynamic fracture, indicator functions for fluid-property transitions, and a staggered COMSOL scheme.
Results
Three benchmark examples agree well with existing analytical results, and additional 2D and 3D examples show dynamic crack branching and interaction with pre-existing natural fractures.
Takeaways & Limitations
The proposed approach handles dynamic fluid-driven fracture propagation, branching, and interaction with pre-existing natural fractures within a succinct implementation.
Takeaways & Limitations
The framework does not yet determine coefficients, including the permeability model describing hydro-mechanical responses and fracture behavior.
Abstract
from arXiv · showhide
A phase field model for fluid-driven dynamic crack propagation in poroelastic media is proposed. Therefore, classical Biot poroelasticity theory is applied in the porous medium while arbitrary crack growth is naturally captured by the phase field model. We also account for the transition of the fluid property from the intact medium to the fully broken one by employing indicator functions. We employ a staggered scheme and implement our approach into the software package COMSOL Multiphysics. Our approach is first verified through three classical benchmark problems which are compared to analytical solutions for dynamic consolidation and pressure distribution in a single crack and in a specimen with two sets of joints. Subsequently, we present several 2D and 3D examples of dynamic crack branching and their interaction with pre-existing natural fractures. All presented examples demonstrate the capability of the proposed approach of handling dynamic crack propagation, branching and coalescence of fluid-driven fracture.
Research highlight
The paper proposes a phase-field model for fluid-driven dynamic fracture in porous media, extending dynamic-crack modeling from single-phase solids. It captures branching and interactions with pre-existing natural fractures, with results agreeing well with analytical solutions.
- The study proposes a phase-field model for dynamic fracture propagation in porous media.
- The method extends phase-field modeling of dynamic cracks in single-phasic solids to fluid-driven dynamic cracks.
- Elastic energy drives crack propagation and branching in the model.
- The presented results agree well with existing analytical results.
- Examples demonstrate dynamic crack branching and interaction with pre-existing natural fractures.
- The paper addresses phase-field modeling of dynamic cracks, hydraulic fractures, poroelasticity, and implementation in COMSOL.
1 Introduction
The introduction motivates accurate modeling of complex hydraulic-fracture behavior in porous media and positions phase-field methods as a suitable coupled approach. The paper extends this framework to dynamic poroelastic fracture propagation while addressing prior limited consideration of inertial effects.
- Hydraulic fracturing creates artificial channels for resource extraction but can also form unintended pathways that contaminate groundwater.
- Predicting branching and merging fracture patterns requires accurate mathematical models and numerical tools for porous media.
- Fracture propagation in porous media is a hydro-mechanical coupling process commonly based on Biot theory and laminar Darcy-type flow.
- Existing fracture approaches include discrete, continuous, and hybrid methods, but discrete methods can depend on initial particle or block arrangements and require complex calibration.
- Phase-field models avoid complex crack-tracking algorithms and are attractive for hydraulic fracturing because cracks emerge naturally in a thermodynamic framework suited to coupled problems.
- Earlier porous-media phase-field approaches rarely considered inertial effects, motivating a dynamic poroelastic formulation.
- The proposed approach combines Biot poroelasticity, phase-field dynamic fracture, indicator-function fluid-property transitions, and staggered COMSOL implementation.
- The paper verifies the approach with three examples before presenting 2D and 3D branching and natural-fracture interaction examples.
2 Mathematical models of fracture in porous media
The model combines phase-field fracture mechanics with poroelastic flow to represent dynamic cracking in porous media. It diffuses crack geometry, enforces crack irreversibility, and smoothly interpolates fluid and solid properties between reservoir and fracture domains.
- Poroelastic setting: The formulation applies Biot poroelasticity to a bounded 2D or 3D domain containing an internal crack and time-dependent mechanical and pressure boundary conditions.The porous medium is assumed linear elastic, homogeneous, isotropic, and filled with compressible viscous fluid.
- Fracture energetics: Griffith fracture energy is incorporated through a crack-surface density approximation, while tensile and compressive strains are decomposed so cracking is driven only by tensile energy.The stiffness reduction affects the tensile energy, and a small residual parameter prevents singularity as the phase field approaches one.
- Phase-field fracture: The phase field replaces explicit crack tracking by diffusing a sharp crack over a finite transition region while keeping displacement continuous.The phase field is zero in intact material and one in cracked material; its length scale controls the transition width.
- Fracture evolution: A history field enforces irreversibility by ensuring that the phase field does not decrease during unloading or compression.The history field satisfies a Kuhn–Tucker condition and replaces the tensile elastic-energy contribution in the strong phase-field formulation.
- Fluid pressure: The flow formulation divides the medium into reservoir, fracture, and transition domains, using phase-field indicator functions to interpolate properties and governing equations.The reservoir and fracture domains use distinct mass-conservation and Darcy-flow descriptions, while the transition is smoothed through interpolated permeability and viscosity.
- Fluid pressure: Indicator-based interpolation keeps all fields continuous and avoids special treatment at fracture tips, but the examples retain unchanged fracture-domain fluid properties because crack opening is not directly extracted in COMSOL.The paper identifies permeability models dependent on crack opening or volumetric strain as future work.
3 Numerical implementation
The model is implemented as a staggered finite-element/finite-difference framework in COMSOL, with separate modules exchanging displacement, pressure, phase-field, history-strain, and stored intermediate quantities.
- Coupled formulation: The weak-form system solves displacement u, phase field ϕ, and fluid pressure p using their corresponding trial and test spaces.The formulation derives residuals and force contributions for the coupled fields.
- COMSOL implementation: COMSOL implementation uses five modules: Solid Mechanics, Darcy Flow, Phase Field, History-strain, and Storage.The modules are written in strong form and discretized with standard finite elements in space and finite differences in time.
- Module roles: The Solid Mechanics and Darcy Flow Modules solve displacement and pressure, while the Phase Field and History-strain Modules solve ϕ and H.Boundary and initial conditions are implemented in the mechanics and flow modules, with the history-strain module using COMSOL’s distributed ODE/DAE interface.
- Data exchange: The Storage Module transfers principal strains, elastic energy, and constitutive quantities between staggered updates.Updated history strain drives the phase-field solve, after which volumetric strain and phase field enter the Darcy pressure calculation.
- Scheme choice and limitation: The staggered scheme was selected for ease of implementation in COMSOL, although the paper notes that monolithic methods can be more robust, efficient, and accurate.The authors also state that mathematical validity of the numerical solution has been established for monolithic porous-media phase-field fracture, but not for staggered schemes.
- Staggered scheme: The staggered scheme solves displacement and pressure together, while history strain and phase field are solved in two additional staggered steps.Linear extrapolation supplies the initial guess at each time, and the implicit Generalized-α method provides unconditionally stable time integration.
4 Verification of the proposed approach
Three benchmark problems assess the proposed model against analytical solutions for dynamic consolidation and pressure distribution in cracked or jointed specimens. The numerical results show good agreement, including a pressure error below 0.004 along the pre-existing crack.
- Three numerical examples compare phase-field simulations with existing analytical solutions to assess correctness and accuracy.
- 4.1 Dynamic consolidation: Dynamic consolidation produces oscillatory top-edge displacement and bottom fluid pressure ranging from 0 to 1.8 Pa under transient wave reflection.The proposed displacement and pressure results agree well with the 1D analytical solution.
- 4.2 Pressure distribution in a single crack: The single-crack benchmark shows pressure extending farther along the crack over time and penetrating more readily along than perpendicular to it.
- 4.2 Pressure distribution in a single crack: The simulated pressure distribution along the pre-existing crack agrees well with the analytical solution and approaches P0 over time.The numerical error is less than 0.004, with the largest error at the crack’s right end and at later times.
- 4.3 Pressure distribution in a specimen with two sets of joints: In the specimen with two perpendicular joint sets, pressure varies linearly with the y coordinate along the joints, consistent with the analytical water-head solution.
5 Examples of dynamic fluid-driven crack branching
Internal fluid injection drives dynamic crack propagation, branching, and three-dimensional penny-shaped crack growth in porous specimens. The simulations also examine sensitivity to time step, length scale, critical energy release rate, mesh size, and fluid source.
- 5.1 2D examples: Internal injection into a pre-existing crack drives progressive hydraulic crack propagation in the 2D porous specimen.The crack begins propagating at 430 µs, branches at 495 µs, and develops four similar cracks by 510 µs.
- 5.1 2D examples: Crack branching is associated with high fracture-zone stress, insufficient energy dissipation by a single crack, crack speed, loading rate, and energy release.
- 5.1 2D examples: Changing the time step has little effect on the final crack patterns, whereas increasing the length scale produces wider cracks and slower arrival at specimen boundaries.
- 5.1 2D examples: Smaller critical energy release rates produce more branching and faster crack arrival at the left and right boundaries.
- 5.1 2D examples: Mesh size has negligible effect on fracture patterns unless it is too large to characterize the fracture path; decreasing mesh size increases the pressure-curve slope.
- 5.1 2D examples: Fluid source magnitude has little effect on final crack patterns or maximum pressure, but larger sources reach maximum pressure sooner.
- 5.2 3D example: In 3D, the penny-shaped crack propagates parallel to the x-y plane and begins branching at 770 µs before advancing toward specimen boundaries.
6 Interaction of hydraulic fracturing with natural cracks
The model captures interaction between an injected hydraulic crack and a pre-existing natural crack across four spacing configurations. Depending on the configuration, hydraulic cracks capture and coalesce with the natural crack, followed by propagation from its tips.
- 6.1 Interaction cases: Four interaction cases vary horizontal spacing sh between 0.1 and 0.2 m and vertical spacing sv between 0 and 0.1 m.
- 6.1 Interaction cases: In Cases 1 and 4, the newly generated hydraulic crack captures the natural crack and subsequently propagates from the natural crack’s tips.
- 6.1 Interaction cases: Hydraulic-natural crack coalescence requires no special treatment because the continuous phase-field formulation automatically completes the calculation while ensuring fluid mass balance.
- 6.1 Interaction cases: Stress concentrates around crack tips, and fluid pressure drops suddenly when the hydraulic crack joins the natural crack; otherwise, pressure curves are similar.
7 Conclusions
The paper extends phase-field modeling of dynamic fracture from single-phasic solids to poroelastic media using Biot coupling and phase-field-controlled crack evolution. COMSOL implementation and benchmark agreement support its use for dynamic branching and interaction with natural fractures.
- The approach extends a dynamic fracture phase-field method for single-phasic solids to poroelastic media using classical Biot poroelasticity.
- Elastic energy drives fracture propagation, while indicator functions transition fluid properties between intact and fully broken regions.
- A staggered COMSOL implementation solves displacement, pressure, and phase fields independently, and three verification examples agree well with analytical results.
- Additional 2D and 3D examples demonstrate dynamic crack branching and interaction with pre-existing natural fractures.