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Iterative coupling of flow, geomechanics and adaptive phase-field fracture including level-set crack width approaches
Sanghyun Lee, Thomas Wick, Mary F. Wheeler
TL;DR
Reliable and efficient coupling of flow and geomechanics with propagating fractures remains computationally challenging. This paper studies fixed-stress splitting for phase-field fractures and proposes a level-set-based width computation used in fracture permeability. Numerical examples report convergence of selected quantities and examine solver iterations, while identifying harder phase-field convergence in heterogeneous, multiple-fracture cases.
Problem
Reliable and efficient numerical methods for coupled poromechanics with fractures remain computationally challenging, and fixed-stress performance has lacked systematic investigation.
Source: §1 · p. 1Method
The paper applies fixed-stress splitting to flow and phase-field geomechanics and computes fracture width through a level-set approach with finite-element interpolation.
Source: §1 · p. 2Results
The numerical examples show spatial and temporal convergence of selected quantities and efficient solver iteration numbers for the tested settings.
Source: §7 · pp. 23–24Takeaways & Limitations
The level-set width computation provides accurate width values inside fractures, particularly in the homogeneous test cases.
Source: §3 · p. 8, §7 · p. 24For multiple fractures and heterogeneous media, phase-field convergence is harder and can dominate convergence while pressure and displacement converge well.
Source: §6.4 · p. 20, §7 · p. 24
Abstract
from arXiv · showhide
In this work, we present numerical studies of fixed-stress iterative coupling for solving flow and geomechanics with propagating fractures in a porous medium. Specifically, fracture propagations are described by employing a phase-field approach. The extension to fixed-stress splitting to propagating phase-field fractures and systematic investigation of its properties are important enhancements to existing studies. Moreover, we provide an accurate computation of the fracture opening using level-set approaches and a subsequent finite element interpolation of the width. The latter enters as fracture permeability into the pressure diffraction problem which is crucial for fluid filled fractures. Our developments are substantiated with several numerical tests that include comparisons of computational cost for iterative coupling and nonlinear and linear iterations as well as convergence studies in space and time.
1. Introduction
The paper studies iterative flow–geomechanics coupling with propagating phase-field fractures, addressing the computational challenges of reliable and efficient methods for fractured porous media. It investigates fixed-stress splitting and proposes a level-set-based fracture-width computation relevant to fluid-filled fracture permeability.
1. Introduction
- Phase-field fracture approximates displacement discontinuities with an auxiliary indicator function and a diffusive transition zone of half-bandwidth ε.The paper identifies changes in fracture width and fracture length as distinct situations of interest.Source: §1 · p. 1
- Fracture evolution imposes an irreversibility constraint, while the variational approach includes nucleation, propagation, kinking, and crack morphology without explicit remeshing.The phase-field formulation is treated quasi-monolithically for mechanics and coupled to flow by splitting.Source: §1 · pp. 1–2
- The paper systematically investigates fixed-stress iterative coupling for phase-field fracture and flow using benchmarks and mesh-refinement studies.The authors identify this performance investigation as a step toward future extensions to practical three-dimensional problems.Source: §1 · p. 2
- The proposed width method uses the phase field to obtain fracture normals and interpolates width values inside the diffusive fracture zone.The width is important for computing permeability in fluid-filled fracture propagation.Source: §1 · p. 2
2. Mathematical models for flow and mechanics of porous media and fractures
The model treats the fracture as a time-dependent region within a porous domain and couples reservoir and fracture flow with poromechanics. Fractures are represented by a phase field subject to irreversibility, with degradation acting on the tensile part of the elastic energy.
2.1. Preliminaries
- The computational domain contains a thin, time-dependent fracture region whose boundary separates fracture and reservoir domains.The preliminaries illustrate a penny-shaped fracture and distinguish propagating from fixed-fracture settings.Source: §2.1 · p. 3, Figure 1 · p. 3, Figure 2 · p. 3
2.2. A pressure diffraction flow system
- The pressure-diffraction formulation combines reservoir and fracture Darcy flow, with fracture permeability determined by the aperture w(u)=[u·n].The model derives a weak pressure formulation after combining the subdomain equations; the leak-off term is assumed zero.Source: §2.2 · pp. 3–4
2.3. Geomechanics and phase-field fracture equations
- The mechanics model couples displacement and a phase field, where ϕ=0 denotes the crack and ϕ=1 the unbroken material, with crack irreversibility imposed over time.Energy degradation acts only on tensile stress, and κ represents residual material stiffness.Source: §2.3 · pp. 5–6
3. The fracture width computation using a level-set approach
The section develops a level-set-based procedure to estimate fracture width, which supplies permeability values for fluid-filled fracture pressure calculations. It computes width at the fracture boundary and interpolates values inside the fracture, with an alternative level-set construction that avoids an additional problem.
3. The fracture width computation using a level-set approach
- The fracture width w(u) enters the pressure diffraction problem as fracture permeability, motivating a robust computation both at the boundary and inside the fracture.The level-set construction provides fracture-boundary normals for the local width calculation.Source: §3 · pp. 6–8
- The fracture boundary is defined by the phase-field threshold C_LS = 0.1, and the level-set field can be computed by solving an additional problem or by shifting the phase field.The threshold defines the boundary within the phase-field diffusion zone; the authors state that its choice has minimal dependency because ε is very small.Source: §3 · p. 7
- The local boundary-width calculation assumes symmetric displacements, u+ · n = −u− · n, justified for tensile stresses and homogeneous isotropic media.The width values on ΓF are then used as the basis for interpolation inside the fracture.Source: §3 · pp. 7–8
- The interpolation procedure produces fracture-width values inside the crack, where the direct evaluation gives nearly zero values, and the reported result is an accurate width approximation.The comparison evaluates width along the fracture’s middle line in the y-direction.Source: Figure 4 · p. 8, §3 · p. 8
- Formulation 3 adds two scalar, linear, elliptic problems but provides accurate width and a global finite-element function; Formulation 4 can reduce computational cost.The procedure can also be used for multiple non-planar fractures.Source: §3 · p. 8
4. Discretization of all sub-systems
The section discretizes the pressure and displacement–phase-field systems, then formulates finite-element problems for domain indicators, level-set, and fracture width. The pressure model uses phase-field-based interpolation of permeability near the fracture tip.
4. Discretization of all sub-systems
- A shape-regular mesh family partitions the computational domain into squares in 2D or cubes in 3D, with a time grid defined by step size δt.The mesh elements are assumed to exactly approximate the domain.Source: §4 · pp. 8–9
4.1. Decomposing the domain Λ into ΩR and ΩF
- Linear indicators χF and χR distinguish fracture from reservoir, with thresholds c1 = 0.4 and c2 = 0.6 defining the respective zones.The thresholds follow from cx = 0.1; the indicator functions vary linearly in the diffusive zone.Source: §4.1 · p. 9, Figure 5 · p. 9
4.2. Temporal and spatial discretization of the pressure diffraction equation
- The pressure field is approximated in a continuous piecewise-polynomial finite-element space and advanced in time by solving the pressure diffraction formulation.The phase-field-defined cake region interpolates the permeability-viscosity ratio; the resulting effective ratio is Lipschitz-continuous in space and time. For transport coupling, enriched Galerkin approximation is used to ensure local and global flux conservation.Source: §4.2 · pp. 9–10
4.3. Spatial discretization of the incremental displacement-phase-field system
- The displacement–phase-field system is discretized as a quasi-monolithic nonlinear variational inequality, with the phase field constrained by crack irreversibility.The phase field is extrapolated in the displacement equation to avoid an indefinite Hessian matrix.Source: §4.3 · p. 10
4.4. Variational formulations of the level-set and the width problems
- The level-set and width problems use continuous piecewise-polynomial finite-element approximations, with the width system formed from the given displacement and phase field.The level-set interface is identified from the phase field, and the formulation uses a large penalty parameter θ ≈ 10^3.Source: §4.4 · p. 11
5. Solution algorithms for fluid-filled phase-field fractures
The algorithm computes level-set and width before each pressure solve, then couples pressure and displacement–phase-field through fixed-stress iterations. Its solver scheme combines nonlinear and constrained solves with mesh adaptivity.
5. Solution algorithms for fluid-filled phase-field fractures
- Before each pressure solve, the algorithm computes the level-set and fracture width to supply fracture permeability.The algorithm repeats these solves within the fixed-stress iteration.Source: §5 · p. 12, §5.1 · p. 12
5.1. Solution algorithms and solver details
- The overall scheme solves level-set, width, pressure, and nonlinear displacement–phase-field problems in sequence, with Newton, line search, and a primal-dual active-set method for mechanics.Local predictor-corrector mesh refinement restores ε > h while keeping ε fixed; pressure uses a direct solver, while level-set and width use parallel CG with SSOR preconditioning.Source: §5.1 · p. 12
5.2. A fixed-stress algorithm for fluid-filled phase-field fractures
- Each fixed-stress iteration solves pressure using the current displacement and phase field, then solves displacement and phase field using the updated pressure until the stopping criterion is met.The level-set and width are computed before pressure to provide fracture permeability.Source: §5 · p. 12, §5.2 · p. 13
6. Numerical tests
The five numerical examples examine fixed-stress coupling and fracture behavior from a fluid-filled Sneddon test through propagating and interacting fractures, including a 3D case. They report convergence, iteration behavior, pressure patterns, and comparisons of level-set formulations.
6. Numerical tests
- Across five examples of increasing complexity, tests focus on fixed-stress iteration numbers and refinement studies, while examining α = 0 versus α = 1, negative fracture-tip pressure, propagating-fracture pressure drop, and multiple-fracture interaction in heterogeneous media.Source: §6 · p. 13
6.1. Example 1: Extension of Sneddon’s test to a fluid-filled fracture in a porous medium
- The fluid-filled Sneddon test recovers pressure near 1 × 10^-3 Pa and a similar crack opening for α = 0; α = 1 yields four times higher pressure and slightly higher opening.The finite-element width agrees excellently with the symmetric-test displacement corresponding to crack opening.Source: §6.1 · pp. 15–16, Figure 6 · p. 15
6.2. Example 2: A fluid-filled fracture with emphasis on the pressure at the fracture tips.
- Negative pressures appear around fracture tips during early injection, consistent with fluid lag when the advancing crack outpaces fluid flow into the new space.At the beginning of injection, fluid enters the fracture from the porous medium at its tips.Source: §6.2 · p. 16, Figure 9 · p. 17
6.3. Example 3: A propagating fluid-filled fracture in a porous medium
- Temporal convergence of pressure and fracture length is observed but is slow; Formulations 3 and 4 give almost identical results, while Formulation 3 costs more because it solves an additional scalar problem.The example also reports averages of 4–8 Newton iterations per mesh per time step and 10–50 GMRES iterations, with occasional increases to 100–150.Source: §6.3 · p. 17, Figure 11 · p. 18
6.4. Example 4: Fracture networks in homogeneous and heterogeneous porous media
- Across the three fracture-network cases, nonlinear iterations average 8–10 and GMRES iterations 10–20, with no significant difference between homogeneous and heterogeneous media.Both level-set formulations produce the same crack pattern, but Formulation 4 reaches the final shape earlier; the authors call for further investigation given the test complexity.Source: §6.4 · p. 20, Figure 15 · p. 21, Figure 16 · p. 22
6.5. Example 5: Propagating penny-shaped fracture in 3D
- Formulations 3 and 4 produced almost identical computational results in the previous example; for propagating fracture, Formulation 4 is more efficient, especially in three-dimensional cases.The passages show fracture propagation and crack-opening displacement and fixed-stress iteration counts over time.Source: §6.5 · p. 23
7. Conclusions
The proposed fixed-stress splitting for phase-field fractures showed spatial and temporal convergence of selected quantities and typical fluid-filled-fracture pressure behavior. Level-set width computation produced accurate fracture widths in homogeneous tests, while its applicability to arbitrary heterogeneous materials remains open.
7. Conclusions
- Selected quantities showed spatial and temporal convergence, while negative pressure at fracture tips appeared when Biot’s coefficient was nonzero.The negative tip pressure was not present when Biot’s coefficient was zero.Source: §7 · p. 23
- For heterogeneous Example 4, convergence was dominated by the phase-field variable, while pressure and displacement converged well.The study also reports efficient GMRES, Newton, and fixed-stress iteration counts for its settings.Source: §7 · p. 24
- Level-set width computation gave accurate fracture widths, particularly in homogeneous Examples 1–3 and 5.Results were also good for heterogeneous materials and multiple fractures in Example 4.Source: §7 · p. 24
- Whether the level-set width methodology works for arbitrary heterogeneous materials remains an open question for future research.Source: §7 · p. 24