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Hybrid FEM and peridynamic simulation of hydraulic fracture propagation in saturated porous media
Tao Ni, Francesco Pesavento, Mirco Zaccariotto, Ugo Galvanetto, Qi-Zhi Zhua, Bernhard A. Schrefler
TL;DR
The paper addresses limitations in numerical hydraulic-fracture approaches, including fixed Poisson’s ratio constraints. It proposes a hybrid FEM and peridynamic approach, whose results agree with analytical solutions and reproduce observed fluid-pressure oscillations.
Problem
Existing numerical hydraulic-fracture approaches have limitations, including restricting Poisson’s ratio to a fixed value.
Method
The paper presents a hybrid FEM and peridynamic modelling approach and uses Ordinary State-Based peridynamics.
Results
The numerical results agree well with analytical solutions, and fluid-pressure oscillations are observed in fluid-driven hydraulic-fracture examples.
Takeaways & Limitations
The presented method demonstrates capability for solving multiphysics problems involving discontinuities in hydraulic-fracture propagation.
Takeaways & Limitations
The approach has difficulty representing the response when different constitutive models are used, and peridynamic approaches are generally more computationally expensive.
Abstract
from arXiv · showhide
This paper presents a hybrid modeling approach for simulating hydraulic fracture propagation in saturated porous media: ordinary state-based peridynamics is used to describe the behavior of the solid phase, including the deformation and crack propagation, while FEM is used to describe the fluid flow and to evaluate the pore pressure. Classical Biot poroelasticity theory is adopted. The proposed approach is first verified by comparing its results with the exact solutions of two examples. Subsequently, a series of pressure- and fluid-driven crack propagation examples are solved and presented. The phenomenon of fluid pressure oscillation is observed in the fluid-driven crack propagation examples, which is consistent with previous experimental and numerical evidences. All the presented examples demonstrate the capability of the proposed approach in solving problems of hydraulic fracture propagation in saturated porous media.
1. Introduction
Hydraulic fracture simulation requires handling coupled deformation, fracture evolution, and fluid flow, while existing methods face challenges with complex crack configurations and three-dimensional conditions. The paper proposes combining FEM and ordinary state-based peridynamics, with demonstrated applications to hydraulic-fracture propagation and bifurcation.
- Hydraulic fracture is a hydro-mechanical problem involving solid deformation and fracture evolution together with fluid flow in fractured regions.
- Finite-element approaches can require expensive remeshing, fine meshes, or constitutive choices that affect hydraulic-fracture simulations.
- XFEM and phase-field approaches provide crack-simulation capabilities but face difficulties with multi-cracks, three-dimensional non-planar cracks, or accurate displacement discontinuities.
- Existing hydraulic-fracture methods still face challenges with multi-cracks, crack bifurcation, and leak-off, particularly in three-dimensional conditions.
- The proposed approach combines FEM and peridynamics, details their hydro-mechanical coupling, and introduces a simpler crack-aperture algorithm than the normal displacement-jump approach.
- The method successfully simulates hydraulic-fracture propagation and bifurcation, while fluid-driven cases reproduce pressure oscillations discussed in prior work.
2. Theoretical basis
The theoretical basis combines ordinary state-based peridynamics for solid deformation and fracture with Biot-based pore-pressure coupling and finite-element flow treatment. It defines bond kinematics, constitutive response, damage, failure, effective stress, and reservoir–fracture flow properties.
- 2.1.1. Basic concepts: Ordinary state-based peridynamics models material points interacting within a prescribed horizon, with bond kinematics defined from reference positions, displacements, and deformed configurations.The formulation introduces vector and scalar states, force-density states, and an equation of motion over each point’s neighborhood.
- 2.1.2. Linear isotropic solid material: The linear isotropic constitutive model separates force density into dilatational and deviatoric contributions governed by bulk and shear moduli.The dilatational term uses volume dilatation, while the deviatoric term uses deviatoric extension and weighted-volume quantities.
- 2.1.3. Failure criterion: The model represents opening fracture with a critical bond-stretch criterion and tracks bond connection status through a scalar damage variable.The critical stretch is related to the mode-I critical energy release rate, although its use in state-based peridynamics is described as not fully justified but usually accepted.
- 2.1.3. Failure criterion: Damage at a material point is represented by the ratio of the crack-side neighborhood area to the total neighborhood area, with φx theoretically ranging from 0 to 0.5.The damage evolution is illustrated as the point position changes relative to a crack crossing its neighborhood.
- 2.1.4. Effective stress principle in ordinary state-based peridynamics: Biot effective stress incorporates pore pressure into the peridynamic dilatational response, while flow properties are interpolated between reservoir, transition, and fracture domains using indicator functions.The fracture domain is assumed fully fluid-filled with porosity nf = 1; incompressible fracture fluid removes the volumetric strain term from the flow equation.
3. Discretization and numerical implementation
The method couples peridynamic solid mechanics with finite-element fluid flow through bidirectional coupling matrices and crack-aperture-dependent permeability updates. Relative bond displacement is decomposed to distinguish opening from dislocation, improving permeability updates near cracks.
- Peridynamic discretization: The discretized peridynamic equation is expressed with mass, stiffness, coupling, force, and pressure terms for each current node and its family nodes.The family-node formulation uses the number and volumes of neighboring nodes, while the global form identifies M_PD, KPD, and QPD as mass, stiffness, and coupling matrices.
- Solid–fluid coupling: The coupled formulation uses peridynamics for the solid and finite elements for the fluid, with shared node coordinates and bidirectional interaction between discretizations.The coupling links local fluid equations with non-local peridynamic equations through shared shape functions and coupling matrices.
- Numerical implementation: The implementation uses four-node fluid elements, shared shape functions, and four Gauss points for numerical integration in the normalized natural domain.The solid and fluid discretizations use coincident node coordinates in the coupled formulation.
- Aperture and permeability: Crack aperture is computed from relative displacement components parallel and perpendicular to the original bond direction, representing opening displacement and crack dislocation.The resulting aperture is used to update permeability and storage coefficients around cracked nodes.
- Aperture and permeability: The displacement decomposition avoids permeability updates at closed dislocated cracks and reduces errors at open cracks with substantial dislocation.The procedure evaluates broken bonds connected to a node, averages their bond apertures, and updates node-related permeability and storage coefficients.
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The solution strategy uses a modified staggered scheme that sequentially solves hydraulic diffusion and hydro-driven solid deformation. This sacrifices some robustness to capture hydrodynamic phenomena such as pore-pressure oscillations.
- Time integration: The formulation includes implicit time integration for fluid flow, with the integration parameter typically chosen in the stable range 0.5 ≤ ϑ ≤ 1.The pressure update uses the storage matrix, permeability matrix, and fluid source term.
- Coupling strategy: A monolithic formulation can become memory- and time-intensive because the combined system matrices must be updated during solution.The staggered alternative is used to reduce this computational burden, although the conventional scheme may lose some coupled-system phenomena.
- Coupling strategy: The modified staggered approach is adopted to capture pore-pressure oscillations and other hydrodynamic phenomena, at the expense of some robustness.The coupled system is divided into hydraulic diffusion and hydro-driven deformation of the porous solid.
- Coupling strategy: Each solution sequence first solves the pressure field and then solves the solid displacement field using adaptive dynamic relaxation.Previously computed displacement and pressure are used to evaluate fluid volumetric sources or nodal forces for the next sequence.
- Hydraulic-fracture algorithm: Permeability and storage matrices are updated at every time step when initial or propagating cracks are present.The resulting hydraulic-fracture procedure is summarized in the solution flow chart.
4. Numerical examples
Numerical examples verify the hybrid method against analytical consolidation and fluid-flow solutions, then demonstrate crack propagation, stepwise advancement, pressure oscillations, and crack bifurcation. The method generally agrees closely with analytical results while reproducing experimentally observed hydraulic-fracture features.
- Verification examples: The FEM/PD consolidation results are in excellent agreement with analytical pore-pressure and displacement solutions, while requiring 37.69 s versus 62.85 s for PD-only.The computing-cost comparison reports CPU time for FEM/PD and PD-only methods, respectively.
- Verification examples: The single-crack fluid-flow solution is very close to the analytical solution, with denser grids producing greater accuracy.The pressure distribution is compared along the crack, and the denser-mesh displacement results are closer to the analytical solution.
- Fluid-driven fracture propagation: Crack propagation begins when the applied pressure reaches 59.235 MPa at 2154 s, after which crack patterns and pressure distributions evolve with propagation.The numerical results reproduce stepwise, irregular crack-tip advancement rather than the smooth behavior of the analytical KGD solution.
- Fluid-driven fracture propagation: The hybrid FEM/PD model reproduces experimentally observed stepwise crack advancement and describes spontaneous crack propagation with fluid flow in fractured saturated porous media.The examples also show that assumed solid behavior affects hydraulic-fracture modeling.
- Pressure-driven effects: Pressure initially rises at injection, drops as the central crack opens and permeability increases, then changes sharply when propagating cracks interact with natural cracks or pressure boundaries.The pressure evolution reflects transitions among isotropic diffusion, crack-directed flow, propagation, and boundary interaction.
- Pressure-driven effects: Fluid pressure oscillates at the injection point, with rapid pressure drops during fracture events when induced-crack volume grows faster than the injection rate.At sufficiently high injection rates, hydraulic crack bifurcation is also observed, and fracture-initiation pressure increases with injection rate.
5. Conclusions
The paper proposes and validates a hybrid FEM–peridynamics approach for hydraulic fracture propagation in saturated porous media. Benchmark agreement, pressure- and fluid-driven examples, and observed pressure oscillations demonstrate its reported capabilities.
- FE equations govern fluid flow, while peridynamics describes solid deformation and captures crack propagation.
- Benchmark results agree well with analytical solutions, validating the proposed approach on two examples.
- The approach simulates crack propagation and bifurcation in saturated porous media under pressure- and fluid-driven conditions.
- Fluid pressure oscillations are observed in fluid-driven hydraulic fracture examples, consistent with experimental observations.
- The method addresses multiphysics problems involving discontinuities by coupling local models with peridynamics.
- The coupled framework could be extended to crack propagation driven by electrical, thermal, or chemical fields.