Thesis (MEng)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: Current numerical methods in the eld of hydraulic fracturing are based mainly on continuum methods, such as the Finite Element Method (FEM) and the Boundary Element Method (BEM). These methods are governed by Linear Elastic Fracture Mechanics (LEFM) criteria, which su er from the inherent aw of a non-physical stress representation at the fracture tip. In response to this, a non-local method is proposed, namely the peridynamic theory, to model sleeved hydraulic fracture. A 2D implicit quasi-static ordinary state based peridynamic formulation is implemented on various benchmark problems, to verify the ability to capture constitutive behaviour in a linear elastic solid, as well as, the quanti cation of adverse e ects on the accuracy of the displacement solution, due to the nature of the non-local theory. Benchmark tests consist of a plate in tension, where convergence to the classical displacement solution, non-uniform re nement and varying cell sizes are tested, as well as, a thick walled cylinder with internal pressure, where three di erent loading techniques are tested. The most accurate loading technique is applied to the sleeved fracture model, in order to simulate fracture initiation and propagation. This model is then veri ed and validated by using the Rummel & Winter hydraulic fracturing model and experimental results, respectively. Displacement error minimisation methods are implemented and as a result, the displacement solutions for a plate in tension converges to the analytical solution, while the thick walled cylinder solutions su er from inaccuracies due to an applied load on an irregularly discretized region. The fracture initiation test captures the fracture tip behaviour of the Rummel & Winter model and the fracture propagation test show good correlation with experimental results. This research shows that the peridynamic approach to sleeved hydraulic fracture can yield a realistic representation of fracture initiation and propagation, however, further research is needed in the area of a pressure load application on a solid using the peridynamic approach. / AFRIKAANSE OPSOMMING: Huidige numeriese metodes in die veld van hidrouliese breking is hoofsaaklik gebaseer op kontinuum metodes, soos die Eindige Element Metode (EEM) en die Rand Element Metode (REM). Hierdie metodes word beheer deur Linie^ere Elastiese Breukmeganika (LEB) kriteria, wat ly aan die inherente gebrek van 'n nie- siese voorstelling van die spanning by die fraktuur punt. Om hierdie probleme aan te spreek, word 'n nie-lokale metode voorgestel, naamlik die peridinamiese teorie, om gehulsde hidrouliese breking te modelleer. 'n 2D implisiete kwasi-statiese ordin^ere toestand gebaseerde peridinamika formulering word ge mplimenteer op verskeie norm probleme, om te veri eer of dit oor die vermo e beskik om die konstitutiewe gedrag van 'n linie^ere elastiese soliede materiaal te modeleer, asook die kwanti sering van nadelige e ekte op die verplasings oplossing as gevolg van die natuur van die nie-lokale teorie. Normtoetse bestaan uit 'n plaat in trek spanning, waar konvergensie na die klassieke verplasings oplossing, nie-uniforme verfyning en vari^eerende sel groottes getoets word, asook 'n dikwandige silinder onder interne druk, waar drie verskillende belasting aanwendingstegnieke getoets word. Die mees akkurate belasting aanwendingstegniek word dan gebruik in die gehulsde hidrouliese breking model, om fraktuur aanvangs en uitbreiding na te boots. Die model word dan geveri- eer deur die Rummel & Winter hidrouliese breking model en eksperimentele resultate, onderskeidelik. Fout minimering metodes word toegepas en as 'n resultaat, konvergeer die verplasing oplossing vir die plaat na die analitiese oplossing, terwyl die oplossing van die dikwandige silinder onakuraathede toon as gevolg van 'n toegepaste belasting op 'n onre elmatig gediskretiseerde gebied. Die modellering van die fraktuur inisi ering by die fraktuur punt, stem goed ooreen met die Rummel en Winter voorspelling en die fraktuur uitbreiding stem goed ooreen met eksperimentele resultate. Hierdie navorsing toon dat die peridinamiese benadering tot gehulsde hidrouliese breking wel die fraktuur inisi ering en uitbreiding realisties kan modelleer, maar nog navorsing word wel benodig in die area waar 'n druk belasting op 'n peridinamiese soliede model toegepas word.

The thesis is devoted to study of continuum mechanics and thermodynamics and the related mathematical analysis. It consists of four self-contained chapters dealing with different aspects. The first chapter focuses on peridynamics, a non-local theory of continuum mechanics, and its relation to conventional local theory of Cauchy-Green elasticity. Similar compar- isons has been used for proving consistency and for determining some of the material coefficients in peridynamics, provided the material parameters in the local theory are known. In this chapter the formula for the non-local force-flux is computed in terms of the peridynamic interaction, relating the fundamental concepts of these two theories and establishing hence a new connection, not present in the previous works. The second and third chapters are both devoted to Rate-Independent Systems (RIS) and their applications to continuum mechanics. RIS represents a suitable approximation when the internal, viscous, and thermal effects can be neglected. RIS has been proven to be useful in modeling hysteresis, phase transitions in solids, elastoplasticity, damage, or fracture in both small and large strain regimes. In the second chapter the existence of solutions to an evolutionary rate-independ- ent model of Shape Memory Alloys (SMAs) is proven. The model...

<p>Probing the subsurface for evidence related to the degradation of porous mediums and the evolution of damage mechanisms has been a long-standing challenge in geophysics. As such imaging and predicting fracture network development has remained a difficult area for subsurface science for decades despite the seminal and significant works put forward by many researchers. While this has provide great understanding about the behaviours and properties of natural porous media, there is still much that needs to be explored particularly in regard to the mineralogical composition and chemistry of clay-rich rocks. Despite the fact that argillaceous rocks which consist of different types of clays and varied mineral composition are ubiquitous in nature and are often the target of several technologies (e.g. geotechnical engineering, nuclear waste storage and disposal,hydrocarbon exploration and extraction, carbon capture and sequestration, etc.), many studies focus primarily on the bulk properties or the percentage of components in the matrix. For these reason and due to the problems that can be encountered with natural rocks that contain a swelling clay component whether randomly distirbuted or localized in consolidated globs in zones of the matrix, the influence of clay chemistry in relation to fracture development which is not well characterized, especially during desaturation is investigated with analogue rock samples which were systematically fabricated for this purpose.</p> <p><br></p> <p>The research performed in this dissertation investigated, the applicability of  the fabrication protocol for developing synthetic rocks with desirable rock like features and behavior, the impact and relationship between the rock properties, the microstructural composition, water loss, and the macroscopic behavior of the analogue rocks, focusing on the structure and chemistry of the constituent clay materials. Synthetic rocks were fashioned with the necessary geometries, properties, and material compositions. On the macroscopic scale the fracture and drying behavior of the synthetic rocks were examined with 3D X-ray microscopy and further evaluated through the utility of acoustic emission monitoring, water loss monitoring, and unconfined compressive testing. On the finer scale (nano-microscale), the chemical and mechanical properties, and behavior of select clays was explored by exploiting several methods of material characterization which also included cation exchange experiments coupled with inductively coupled plasma – optical emission spectrometry (ICP-OES). </p> <p><br></p> <p>For the finer scale, experiments verified that calcined kaolinite clay had a different mineral structure and negligible to non-existence shrinkage abilities. In contrast, the montmorillonite clays possessed higher and similar moisture contents but, owing to the different principal cations these clays interacted a bit differently in the highly akaline environment, experienced varying degrees of shrinkage, and had observedly minor structural dissimilarities. For the relatively larger scale, the emergence of damage, extent of the damage network, and the patterns of the crack network mainly depended on the microstructural composition of the analogue rocks, particularly it's clay chemistry and/ or distribution. The location of damage depended on the emplacement and percentage of swelling clay in the matrix, and numerical investigations with peridynamics revealed that the observed damage was a consequence of the action of the swelling and non-swelling components of the matrix. Furthermore, if the microstructure consisted of no clay or calcined kaolinite the AE activity was solely attributed to interfacial processes that occurred during fluid front movement. If the microstructure consisted of a particular montmorillonite, the cracks propagated in the direction of the drying front. Conversely, for montmorillonite clay predominated by a different principal cation, the crack network developed and propagated differently during water loss. Additionally, on the laboratory core scale, properties and behavior similar to natural rocks were confirmed and the rock strength, porosity, AE activity, and velocities were primarily affected by the microstructural composition of the analogue rocks. </p> <p><br></p> <p>An added challenge for investigating and monitoring evolving systems and processes, whether on the laboratory or field scale, is the problem of extracting useful information from the physical data that can be used to identify signatures of developing processes, and changes in the properties or the behavior of a system. Here, data driven machine learning modeling and clustering techniques were undertaken to build a mechanistic understanding of the AE activity generated during drying. The intent is for this work to add to the fundamental research aimed at developing methods that will robustly detect and extract signatures related to evolutionary processes or features in the AE signals, and group them according to some degree of similarity. Such research will support reliable interpretations of the physical data for predictions of the behavior of systems, development of engineering controls, and improvement of the understanding of intrinsic dynamics related to complex processes particularly those that occur in clay-rich systems.</p> <p><br></p> <p>Combined chemical and mechanical investigations have great potential for unraveling practical challenges in subsurface science, especially regarding damage processes in clay-rich rock systems, and identifying and interpreting the presence of discontinuities from geophysical data. The present findings are useful for establishing a link between the constituent clay and observed damage, and improving our understanding of the development of damage in clay-bearing systems. These results provide insight on the influence of swelling clay and the chemistry of such clays on the generation of cracks and crack networks in rock like materials which can be useful for the characterization of damage in both laboratory and the field. The work presented here can also be a basis for further experiments that aim to uncover methods and protocols that will help with the indirect characterization of evolutionary processes, damage mechanisms, and damage in clay rich porous media. Additionally support for the use of analogue rocks in experimental rock physics, architected with specific material compositions, pore structures, crack systems, or clay fractions, is provided here. </p>