High-accuracy phase-field models for brittle fracture based on a new family of degradation functions

Phase-field approaches to fracture based on energy minimization principles have been rapidly gaining popularity in recent years, and are particularly well-suited for simulating crack initiation and growth in complex fracture networks. In the phase-field framework, the surface energy associated with crack formation is calculated by evaluating a functional defined in terms of a scalar order parameter and its gradients, which in turn describe the fractures in a diffuse sense following a prescribed regularization length scale. Imposing stationarity of the total energy leads to a coupled system of partial differential equations, one enforcing stress equilibrium and another governing phase-field evolution. The two equations are coupled through an energy degradation function that models the loss of stiffness in the bulk material as it undergoes damage. In the present work, we introduce a new parametric family of degradation functions aimed at increasing the accuracy of phase-field models in predicting critical loads associated with crack nucleation as well as the propagation of existing fractures. An additional goal is the preservation of linear elastic response in the bulk material prior to fracture. Through the analysis of several numerical examples, we demonstrate the superiority of the proposed family of functions to the classical quadratic degradation function that is used most often in the literature.Comment: 33 pages, 30 figure

A fully coupled numerical model of thermo-hydro-mechanical processes and fracture contact mechanics in porous media

Various phenomena in the subsurface are characterised by the interplay between deforming structures such as fractures and coupled thermal, hydraulic and mechanical processes. Simulation of subsurface dynamics can provide valuable phenomenological understanding, but requires models which faithfully represent the dynamics involved; these models therefore are themselves highly complex. This paper presents a mixed-dimensional thermo-hydro-mechanical model designed to capture the process–structure interplay using a discrete–fracture–matrix framework. It incorporates tightly coupled thermo-hydro-mechanical processes based on balance laws for momentum, mass and energy in subdomains representing the matrix and the lower-dimensional fractures and fracture intersections. The deformation of explicitly represented fractures is modelled by contact mechanics relations and a Coulomb friction law, with a novel formulation consistently integrating fracture dilation in the governing equations. The model is discretised using multi-point finite volume methods for the balance equations and a semismooth Newton scheme for the contact conditions and is implemented in the open-source fracture simulation toolbox PorePy. Finally, simulation studies demonstrate the model’s convergence, investigate process–structure coupling effects, explore different fracture dilation models and show an application of the model to stimulation and long-term cooling of a three-dimensional geothermal reservoir.publishedVersio

A fully coupled numerical model of thermo-hydro-mechanical processes and fracture contact mechanics in porous media

A range of phenomena in the subsurface is characterised by the interplay between coupled thermal, hydraulic and mechanical processes and deforming structures such as fractures. Modelling subsurface dynamics can provide valuable phenomenological understanding, but requires models which faithfully represent the dynamics involved; these models, therefore are themselves highly complex. This paper presents a mixed-dimensional thermo-hydro-mechanical model designed to capture the process-structure interplay using a discrete-fracture-matrix framework. It incorporates tightly coupled thermo-hydro-mechanical processes based on laws for momentum, mass and entropy in subdomains representing the matrix and the lower-dimensional fractures and fracture intersections. The deformation of explicitly represented fractures is modelled by contact mechanics relations and a Coulomb friction law, with particular attention on coupling of fracture dilation to the governing equations in both fractures and matrix. The model is discretised using multi-point finite volumes for the balance equations and a semismooth Newton scheme for the contact conditions and is implemented in the open source fracture simulation toolbox PorePy. Finally, simulation studies demonstrate the model's convergence, investigate process-structure coupling effects, explore different fracture dilation models and show an application of the model to a 3d geothermal pressure stimulation and long-term cooling scenario

Finite volume discretisation of fracture deformation in thermo-poroelastic media

This paper presents a model where thermo-hydro-mechanical processes are coupled to a deformation model for preexisting fractures. The model is formulated within a discrete-fracture-matrix framework where the rock matrix and the fractures are considered as individual subdomains, and interaction between them takes place on the matrix-fracture interfaces. A finite volume discretisation implemented in the simulation toolbox PorePy is presented and applied in a simulation showcasing the effects of the different mechanisms on fracture deformation governed by contact mechanics, as well as their different timescales.Comment: 8 pages, 4 figure

Modelling of the Shear Dilation Based Hydraulic Stimulation in Enhanced Geothermal Systems Considering Fractures in Different Scales

A numerical approach for modelling of shear dilation of existing fractures in hydraulic stimulation of geothermal reservoirs at low elevated pressures is presented. The fractured rock in the reservoir is modelled as a combination of explicitly represented fractures and the rock matrix surrounding these fractures. The efficient modelling of slip-induced permeability enhancement requires coupling of the fluid flow in fractured rock with the mechanical deformation of the rock matrix and the shear dilation of the fractures. For flow simulations, conductive fractures are represented in the domain as high-permeable discontinuities; therefore they dominate the overall flow behaviour. The rock matrix is represented by a low permeability, capturing the effect of small-scale fractures. For the mechanical deformation problem, the rock matrix is assumed to be a linear elastic material, while the fractures in the rock matrix are introduced as internal boundaries. The shear dilation of the fractures is calculated by a joint deformation model (JDM), which connects the shear slip in the fracture surfaces and additional permeability caused by shear displacement. The flow simulations and the mechanical deformation of the rock matrix are both obtained by finite volume discretizations. Several numerical experiments designed by resembling realistic reservoir parameters are conducted to provide better understanding of the shear dilation mechanism. Moreover, fractures present in different scales in a geothermal reservoir. Ignoring the effect of small-scale fractures to the fluid flow in the matrix may result in an overestimate of the permeability enhancement. Hence, the influence of rock matrix permeability on fracture aperture and the overall flow behaviour of the reservoir are examined.publishedVersio

Multiscale simulation of injection-induced fracture slip and wing-crack propagation in poroelastic media

In fractured poroelastic media under high differential stress, the shearing of fractures and faults and the corresponding propagation of wing cracks can be induced by fluid injection. Focusing on low-pressure stimulation with fluid pressures below the minimum principal stress but above the threshold required to overcome the fracture's frictional resistance to slip, this paper presents a mathematical model and a numerical solution approach for coupling fluid flow with fracture shearing and propagation. Numerical challenges are related to the strong coupling between hydraulic and mechanical processes, the material discontinuity the fractures represent in the medium, the wide range of spatial scales involved, and the strong effect that fracture deformation and propagation have on the physical processes. The solution approach is based on a multiscale strategy. In the macroscale model, flow in and poroelastic deformation of the matrix are coupled with the flow in the fractures and fracture contact mechanics, allowing fractures to frictionally slide. Fracture propagation is handled at the microscale, where the maximum tangential stress criterion triggers the propagation of fractures, and Paris' law governs the fracture growth processes. Simulations show how the shearing of a fracture due to fluid injection is linked to fracture propagation, including cases with hydraulically and mechanically interacting fractures