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A high-fidelity isogeometric simulation methodology for fracture in porous media

Periodic Reporting for period 4 - PoroFrac (A high-fidelity isogeometric simulation methodology for fracture in porous media)

Reporting period: 2020-07-01 to 2020-12-31

Fracture lies at the heart of many failures of man-made and natural materials. The vast majority of the developments in fracture relate to solid materials. Occasionally, porous materials have been considered, but studies of crack initiation and propagation in porous materials where the pores can be filled with liquid and gas, are rare and the field can be considered as being underdeveloped. Nevertheless, fracture initiation and propagation in (partially) fluid-saturated porous materials occur frequently, indicating that there is a huge practical relevance of having models and predictive simulation technologies for this class of materials. Often, fracture in such materials is unwanted, like cracks that form in (fluid-saturated) human tissues, while at other times cracking is a crucial element in an industrial process, for example hydraulic fracturing in the oil and gas industry.

To have a predictive simulation technology for fracture in porous media is therefore of major importance for many aspects of human life. Regarding energy, the ability to reliably simulate the direction of fracture propagation when pressurising an existing crack in order to induce propagation (hydraulic fracturing) can be a crucial element in the societal acceptance of shale gas exploration, and is highly relevant for geothermal energy. The transport of contaminants in rock faults or fractured salt domes, used for storage of (nuclear) waste or CO2, is a major environmental concern. The understanding of pore pressure generation and stress build-up is an important issue in shear faults in the earth crust, and is underpinning to any methodology for predicting earthquakes. Finally, fracture in human tissues is a major cause of personal discomfort, e.g. lower back pain. It is a major health issue, costing billions of euros in care, rehabilitation, and lost productivity (estimated 1 – 2% of the GDP in developed countries).

The objective of the project is therefore to develop a robust, flexible simulation methodology for existing and propagating fractures in porous media, consisting of (i) a mesoscopic, multi-phase model for fluid transport in the fractures, which is coupled to the macroscopic model for the flow in the porous medium between the cracks via a meso-macro relation for the mass and momentum balances, (ii) complemented by a flexible discretisation method tailored to the application, and (iii) embedded in a stochastic approach to create a high-fidelity simulation technology.
To achieve the objective of the project three scientific pillars have been formulated (work packages I-III), complemented by a fourth work packages which exploits and connects the scientific and technological advances in the first three work packages.

Work Package I: A framework has been developed for upscaling the pressure difference at the sub-grid scale to the macroscopic scale. The physical implications of different possibilities for the macroscopic interpolation of the pressure at the crack have been highlighted, which is considered to be a major achievement. The derivations and implementations of a power-law (non-Newtonian) fluid as well as of multi-phase fluid flow in a fractured porous medium have been completed, including verification and a significant improvement of the numerical formulation. This has resulted in a much faster convergence.

Work Package II: An isogeometric analysis method using a novel formulation of Locally Refined (LR)-splines and T-splines has been developed, including adaptive hierarchical refinement. It enables capturing a propagating cohesive crack along a predefined path. This breakthrough has enabled the development of a methodology to accurately simulate the propagation of (cohesive) cracks along non-predefined paths, one of the ultimate goals of this work package. Limitations have been identified and quantified. Among them is the difficulty to simulate branching. The use of B-splines which are based on triangles, so-called Powell-Sabin B-splines, removes this restriction. The technology has been developed and implemented successfully, including casting the formulation in a standard finite element framework via Bézier extraction. As an alternative formulation, eXtended isogeometric analysis has been developed and successfully implemented, again cast in a standard finite element data structure via Bézier extraction. Another recent development is that phase-field methods have been investigated, where the discrete crack is distributed over a finite width. They have been built on top of adaptive isogeometric analysis. The efficacy achieved through adaptivity is important since phase-field methods are computationally highly demanding. At present, efforts are devoted to incorporate cohesive-zone models and fluid flow in the formulation. Two different avenues are being pursued, one in which the crack width is discretised as an independent field, which provides knowledge of the crack width necessary for the cohesive-zone crack model and the description of fluid flow within fractures. The other approach relies on reconstructing the crack based on the strain field and the regularisation.

Work Package III: The first step towards a reliability-based description of crack propagation is the implementation of a deterministic crack model. Different from the original plan, a continuum description of the fracture model has been chosen for the deterministic description. Then, the combination with a stochastic approach must be realised.

Work Package IV: The main thrust has been to investigate the issue that computations can divergence when several dissipative mechanisms co-exist. The latter typically holds for earthquakes, where there is energy dissipation in the fault, but also plasticity, and hence energy dissipation, in the surrounding bulk material. Recently, it has been proven rigorously that the destabilising effect of a non-associated flow rule, which is ubiquitous in the description of the inelastic behaviour of rocks and soils, is at the root of these convergence difficulties. Moreover, proper regularisation mechanisms have been identified, and it has been proven that quadratic convergence of the Newton-Raphson method can be restored for inelastic media without a crack, with a stationary crack, and with a propagating crack. As a last addition, the role of fluids and fluid inertia has been examined, with an emphasis on shear fracture propagation.
Work Package I: Completed according to original plans, and beyond, since also the extension to multi-phase flows in cracked porous media has been developed and implemented.

Work Package II: So far, the efforts to describe crack propagation using isogeometric analysis have been confined to either non-porous media, or to fluid-saturated porous media with predefined cracks. In the last part of the project, this will be extended to freely propagating cracks, either modelled in a discrete manner, or through a smeared approach, in particular the phase-field method.

Work Package III: Unfortunately, the main objective of this work package, namely to combine the deterministic fracture model with a reliability method to assess the probability that fracture occurs and in which direction fracture propagates, has not been achieved.

Work Package IV: The inclusion of proper regularisation strategies removes a major road block in geodynamics computations (including for earthquakes).
Crack propagation in a double-edge notched specimen tested by M.B. Nooru-Mohamed at TU Delft