Report
Teaser, summary, work performed and final results
Periodic Reporting for period 2 - PoroFrac (A high-fidelity isogeometric simulation methodology for fracture in porous media)
Teaser
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...
Summary
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.
Work performed
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 has been done on each of the four work packages, and progress is as follows.
Work Package I: A framework has been developed for upscaling the pressure difference at the sub-grid scale to the macroscopic scale and the physical implications of different possibilities for the macroscopic interpolation of the pressure at the crack have been highlighted. This is considered to be an important achievement. The derivations and implementation of a power-law (non-Newtonian) fluid in a fractured porous medium has been completed, including verification and a significant improvement of the numerical formulation, and resulting in a much faster convergence.
Work Package II: An isogeometric analysis model using NURBS has been developed to describe a cohesive crack along a predefined path. This formulation includes adaptive hierarchical refinement, which is a novel element. Building on this advance, excellent progress has been made on hierarchical refinement of Locally Refined (LR)-splines and T-splines. The successful completion of this new and highly efficient methodology has enabled the proper and accurate simulation of 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, a major issue noted in the DoA. The use of B-splines that are based on triangles, so-called Powell-Sabin B-splines, removes this restriction. The technology has been developed and implemented successfully. The proper modelling of fluid-saturated porous media normally requires the separate interpolation of the displacements of the solid and the pressure in the fluid. Moreover, numerical stability imposes restrictions on the interpolation order of both types of variables. This issue of such a so-called consistent u/p interpolation has now also been solved for isogeometric analysis.
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 in the DoA, a continuum description of the fracture model has been chosen for the deterministic description. Reasons for this are two-fold: (i) the difficulties that have been encountered in describing a discrete crack using isogeometric finite element analysis (Work Package II), and (ii) the recent emergence of a powerful continuum description, namely the phase-field approach, which may prove a viable alternative to discrete crack modelling in a number of scenarios.
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. So far, investigations have been done for a solid without interstitial fluid and using a standard finite element framework. Quasi-static and dynamic (earthquake) loading conditions are being considered. Calculations show that, depending on the precise values of the material properties, (severe) convergence difficulties can be encountered.
Final results
Work Package I: Currently, work is under way to implement these different possibilities and, subsequently, to quantify the implications. Also, first steps have been made to multi-phase (e.g., liquid and gas) flows in cracked porous media.
Work Package II: Work is now under way to solve the simulation of crack branching, and preliminary results using Powell-Sabin B-splines look promising. So far, the efforts to describe crack propagation using isogeometric analysis are confined to non-porous materials without interstitial fluid. A first development has been done towards the extension to fluid-transporting cracks in a fluid-saturated porous medium, although it is confined to existing, stationary cracks. An extended isogeometric analysis technology, which was anticipated in the DoA as a fall-back scenario, has also been developed, with a novel implementation based on Bézier extraction.
Work Package III: The next step in this work package is to combine the deterministic fracture model with a reliability method to assess the probability that fracture occurs and in which direction fracture propagates in a medium where the material properties exhibit a scatter.
Work Package IV: There are indications that a major source of the convergence problems resides in the plasticity part, where a non-associated flow rule, which is physically correct for geological materials, may be the main source of convergence problems.