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On the hydraulic fracturing in naturally-layered porous media using the phase field method
Xiaoying Zhuang, Shuwei Zhou, Mao Sheng, Gensheng Li
TL;DR
Predicting hydraulic fractures across naturally layered rock formations remains an open problem. This study applies a phase field framework whose energy-minimization formulation produces fracture patterns and predicts multiple fracture scenarios, while further validation is planned.
Problem
Predicting hydraulic fractures across different rock formations remains an open problem, with hydraulic fracturing also carrying potential controversy.
Method
The study applies a phase field framework in which fracture patterns arise from minimization of an energy functional and the fracture pattern follows fluid-pressure evolution.
Results
Penetration, singly-deflected, and doubly-deflected fracture scenarios can be predicted, including penetration with widening fractures in the stiff-to-soft configuration.
Takeaways & Limitations
The framework predicts varied fracture scenarios in naturally layered media, including widening penetration fractures when fractures enter the soft layer.
Takeaways & Limitations
Further validation of the phase field method is planned.
Abstract
from arXiv · showhide
In the hydraulic fracturing of natural rocks, understanding and predicting crack penetrations into the neighboring layers is crucial and relevant in terms of cost-efficiency in engineering and environmental protection. This study constitutes a phase field framework to examine hydraulic fracture propagation in naturally-layered porous media. Biot's poroelasticity theory is used to couple the displacement and flow field, while a phase field method helps characterize fracture growth behavior. Additional fracture criteria are not required and fracture propagation is governed by the equation of phase field evolution. Thus, penetration criteria are not required when hydraulic fractures reach the material interfaces. The phase field method is implemented within a staggered scheme that sequentially solves the displacement, phase field, and fluid pressure. We consider the soft-to-stiff and the stiff-to-soft configurations, where the layer interface exhibits different inclination angles $θ$. Penetration, singly-deflected, and doubly-deflected fracture scenarios can be predicted by our simulations. In the soft-to-stiff configuration, $θ=0^\circ$ exhibits penetration or symmetrical doubly-deflected scenarios, and $θ=15^\circ$ exhibits singly-deflected or asymmetric doubly-deflected scenarios. Only the singly-deflected scenario is obtained for $θ=30^\circ$. In the stiff-to-soft configuration, only the penetration scenario is obtained with widening fractures when hydraulic fractures penetrate into the soft layer.
1 Introduction
Hydraulic fracture propagation across naturally layered rocks remains difficult to predict, especially at material interfaces where penetration mechanisms lack consensus. The study applies a phase field framework to investigate these patterns without imposing penetration criteria.
- Motivation: Naturally layered reservoirs contain discontinuities and interfaces that hydraulic fractures may interact with, producing penetration, singly-deflected, or doubly-deflected patterns.These three scenarios are identified when hydraulic fractures reach layer interfaces.
- Research gap: Predicting fracture networks across reservoir and cap layers remains challenging, and prior studies have not reached consensus on the mechanisms governing different penetration patterns.This uncertainty keeps hydraulic fracture prediction in naturally layered media an open research topic.
- Approach: The study applies a phase field model to hydraulic fracture propagation in naturally layered geological formations, considering soft-to-stiff and stiff-to-soft configurations with varying interface inclinations.The framework is based on a quasi-static formulation for fluid-driven fracture.
- Scope: The framework is used to examine how fracture patterns relate to layer stiffness contrast, interface inclination, displacement, and stress characteristics.The study also investigates interface cracks and crack appearances in neighboring layers.
- Contribution: Phase field simulations automatically reveal penetration, singly-deflected, and doubly-deflected fracture scenarios through energy minimization rather than imposed penetration criteria.This addresses the prior use of mandatory or “man-made” penetration criteria at layer interfaces.
2 Mathematical models for fracture propagation
The model couples poroelastic deformation and fluid flow with a phase-field representation of fracture growth in two bonded porous layers. Fractures evolve through energy minimization and phase-field equations, while phase-field thresholds distinguish intact, transition, and fractured domains.
- Domain and boundary conditions: The calculation domain contains two bonded layers, ΩA and ΩB, with prescribed displacement, traction, body-force, and internal-fracture conditions.Continuity conditions are naturally fulfilled across the fully bonded interface.
- Energy functional: The variational formulation combines elastic energy, fracture dissipation, external work, and a fluid-pressure contribution for porous media.The Biot coefficient links fluid pressure changes to perturbations in total stress.
- Phase field description: The phase field varies from 0 for unbroken material to 1 for a fully broken region and regularizes the fracture surface over a continuous domain.Its gradient represents crack-surface density, with l0 as the length-scale parameter.
- Phase field description: Tensile and compressive strain components are decomposed separately so the elastic energy can drive fracture evolution without producing unrealistic fracture patterns.A positive stability parameter avoids singularity in the stiffness matrix and improves simulation convergence.
- Fracture evolution: A history field enforces fracture irreversibility by storing the maximum tensile elastic energy density and ensuring monotonic phase-field growth during compression or unloading.The initial phase field is zero, and a pre-existing crack is artificially introduced.
- Flow field: Darcy flow and mass conservation are coupled to phase-field-defined reservoir, transition, and fracture domains, with linearly interpolated properties in the transition region.The study presents one feasible phase-field–flow coupling and notes that more general flow descriptions remain possible.
3 Numerical implementation
The governing multi-field fracture equations are solved with finite elements using stabilized time discretization and a staggered solution strategy. Displacement and fluid pressure are coupled within one staggered step, while the fields are solved sequentially.
- Spatial and temporal discretization: Finite elements solve the weak forms of the coupled displacement, phase-field, and flow problem.The displacement field uses a modified stiffness matrix.
- Spatial and temporal discretization: The implicit Generalized-α method discretizes time and is used to enhance numerical stability unconditionally.
- Staggered scheme: The three fields are solved in a staggered manner, with displacement and fluid pressure coupled and solved together in one staggered step.Each field solve is independent within one time step.
- Nonlinear solution: Newton–Raphson iterations are applied within each segregated step of the solution flowchart.
4 Validation of PFM for hydraulic fracture
The validation compares phase field simulations with analytical hydraulic-fracture results across toughness-, viscous-, and leak-off-dominated regimes. The simulations reproduce consistent evolution trends, while value differences reflect distinct medium, initiation, flow, and fracture representations.
- Validation setup: A 60 m × 30 m domain contains a 1.2 m × 0.2 m initial injection notch, with permeable outer boundaries and p = 0.The simulations use finite elements with maximum size h = 0.2 m and time increment ∆t = 0.025 s.
- Validation setup: The validation examines toughness-dominated, viscous-dominated, and leak-off toughness-dominated hydraulic-fracture regimes.The regimes differ in permeability and the relative importance of fracturing, viscous, and leak-off dissipation.
- Validation results: PFM reproduces consistent trends in fracture half-length and midpoint fluid pressure compared with the analytical approach.Figures 3 and 4 compare these evolutions across the three flow regimes.
- Validation limitations: Differences in actual values arise because PFM models a permeable medium, a finite initial notch, different flow models, and a smeared finite-width fracture.The analytical approach assumes an impermeable medium, zero initial fracture length, point injection, different leak-off treatment, and a discrete fracture.
5 Soft-to-stiff configuration
The soft-to-stiff simulations show that interface inclination and stiffness contrast govern whether fractures penetrate, deflect once, or branch into doubly-deflected paths. Stress concentrations and fracture patterns are consistent with these propagation modes.
- Model setup: Twelve cases vary the two layers’ Young’s moduli and critical energy release rates while keeping ν1 = ν2 = 0.3.The cases are divided into soft-to-stiff and stiff-to-soft configurations, with θ = 0°, 15°, and 30° tested.
- E2 = 2E1: For E2 = 2E1 and θ = 0°, the fracture penetrates perpendicular to the interface and propagates deeply into layer 2.The stiffer layer does not prevent penetration in this case.
- Stress and displacement: Stress distributions coincide with fracture patterns: penetration and doubly-deflected cases concentrate stress near tips, whereas singly-deflected cases also concentrate stress at the deflection zone.Increasing θ slightly decreases the maximum vertical displacement, and displacement grows with fracture propagation.
6 Stiff-to-soft configuration
In the stiff-to-soft configuration, all tested cases produce penetration into the softer layer, where fractures widen and stresses decrease near the advancing tip. The combined stiffness-ratio and angle map distinguishes penetration, singly-deflected, and doubly-deflected regimes.
- Fracture patterns: Only penetration occurs in the stiff-to-soft configuration, unlike the soft-to-stiff configuration where all three fracture scenarios can occur.The penetration pattern is attributed to the softer layer’s lower tensile strength.
- Fracture patterns: Fracture width increases after penetration into the softer layer and becomes larger when that layer’s stiffness is lower.At θ = 0° the fractures propagate horizontally; at θ = 15° and 30° they deflect slightly after crossing the interface.
- Stress response: Stress concentration remains near the fracture tip during penetration and decreases as the fracture propagates into the softer layer.Stress differences between the layers reflect their different tensile strengths.
- Displacement and pressure: After approximately 65 s, maximum vertical displacement increases at a relatively large rate as the fracture reaches the interface and propagates deeply into layer 2.The displacement-versus-time curves are minimally affected by interface inclination.
- Displacement and pressure: Fluid pressure rises rapidly during the first 65 s, then increases more slowly and becomes nearly stable; the second stage is shorter than in the soft-to-stiff configuration.The pressure-time curve is minimally affected by θ.
- Pattern map: The fracture-pattern map identifies penetration at low E2/E1 or low θ with medium E2/E1, singly-deflected paths at high θ, and doubly-deflected paths at high E2/E1 with low θ.The map summarizes how layer stiffness ratio and interface angle jointly determine the three scenarios.
7 Conclusions
The phase field framework predicts hydraulic-fracture interaction with layered interfaces without separate penetration criteria, while linking fracture patterns to stiffness contrast and interface inclination. Simulations reproduce penetration, singly deflected, and doubly deflected scenarios, but the study assumes perfectly bonded interfaces and uses a smeared fracture representation that limits permeability coupling.
- Framework: The phase field framework predicts fracture propagation at material interfaces without requiring separate penetration criteria.Fracture patterns arise from minimization of the total energy functional.
- Predicted scenarios: Penetration, singly-deflected, and doubly-deflected fracture scenarios are all predicted using PFM.The framework examines both soft-to-stiff and stiff-to-soft configurations with varying interface inclinations.
- Soft-to-stiff configuration: At 30°, the soft-to-stiff configuration produces only singly-deflected fractures.This completes the reported angle-dependent sequence for that configuration.
- Stiff-to-soft configuration: In the stiff-to-soft configuration, only penetration occurs, with fractures widening after entering the softer layer.The fracture in the softer layer deflects at a small angle relative to the fracture in the stiffer layer, and that angle increases with interface angle.
- Limitations and future work: The study assumes perfect bonding at layer interfaces and notes that smeared fractures make direct permeability–crack-opening coupling difficult.Future work is directed toward weakly bonded interfaces, experimental validation, and more accurate permeability models.