A-posteriori error estimation in axisymmetric geotechnical analyses

In this paper, an a-posteriori error estimator suitable for use in axisymmetric geotechnical analyses has been developed. The consolidation superconvergent patch recovery with equilibrium and boundaries (CSPREB) method, developed for plane-strain coupledconsolidation problems, is extended to axisymmetric analyses. The use of pore pressures in the error estimator was found to improve results when predicting consolidation. Collapse loads under undrained soil conditions are known to be over-predicted due to “locking”, especially in axial symmetry where there are further displacement constraints. The proposed solution technique reduced locking slightly, but could not eliminate it, as it is inherent in the displacement formulation for lower order elements

Development of robust and efficient solution strategies for coupled problems

Det er mange modeller i moderne vitenskap hvor sammenkoblingen mellom forskjellige fysiske prosesser er svært viktig. Disse finner man for eksempel i forbindelse med lagring av karbondioksid i undervannsreservoarer, flyt i kroppsvev, kreftsvulstvekst og geotermisk energiutvinning. Denne avhandlingen har to fokusområder som er knyttet til sammenkoblede modeller. Det første er å utvikle pålitelige og effektive tilnærmingsmetoder, og det andre er utviklingen av en ny modell som tar for seg flyt i et porøst medium som består av to forskjellige materialer. For tilnærmingsmetodene har det vært et spesielt fokus på splittemetoder. Dette er metoder hvor hver av de sammenkoblede modellene håndteres separat, og så itererer man mellom dem. Dette gjøres i hovedsak fordi man kan utnytte tilgjengelig teori og programvare for å løse hver undermodell svært effektivt. Ulempen er at man kan ende opp med løsningsalgoritmer for den sammenkoblede modellen som er trege, eller ikke kommer frem til noen løsning i det hele tatt. I denne avhandlingen har tre forskjellige metoder for å forbedre splittemetoder blitt utviklet for tre forskjellige sammenkoblede modeller. Den første modellen beskriver flyt gjennom deformerbart porøst medium og er kjent som Biot ligningene. For å anvende en splittemetode på denne modellen har et stabiliseringsledd blitt tilført. Dette sikrer at metoden konvergerer (kommer frem til en løsning), men dersom man ikke skalerer stabiliseringsleddet riktig kan det ta veldig lang tid. Derfor har et intervall hvor den optimale skaleringen av stabiliseringsleddet befinner seg blitt identifisert, og utfra dette presenteres det en måte å praktisk velge den riktige skaleringen på. Den andre modellen er en fasefeltmodell for sprekkpropagering. Denne modellen løses vanligvis med en splittemetode som er veldig treg, men konvergent. For å forbedre dette har en ny akselerasjonsmetode har blitt utviklet. Denne anvendes som et postprosesseringssteg til den klassiske splittemetoden, og utnytter både overrelaksering og Anderson akselerasjon. Disse to forskjellige akselerasjonsmetodene har kompatible styrker i at overrelaksering akselererer når man er langt fra løsningen (som er tilfellet når sprekken propagerer), og Anderson akselerasjon fungerer bra når man er nærme løsningen. For å veksle mellom de to metodene har et kriterium basert på residualfeilen blitt brukt. Resultatet er en pålitelig akselerasjonsmetode som alltid akselererer og ofte er svært effektiv. Det siste modellen kalles Cahn-Larché ligningene og er også en fasefeltmodell, men denne beskriver elastisitet i et medium bestående av to elastiske materialer som kan bevege seg basert på overflatespenningen mellom dem. Dette problemet er spesielt utfordrende å løse da det verken er lineært eller konvekst. For å håndtere dette har en ny måte å behandle tidsavhengigheten til det underliggende koblede problemet på blitt utviklet. Dette leder til et diskret system som er ekvivalent med et konvekst minimeringsproblem, som derfor er velegnet til å løses med de fleste numeriske optimeringsmetoder, også splittemetoder. Den nye modellen som har blitt utviklet er en utvidelse av Cahn-Larché ligningene og har fått navnet Cahn-Hilliard-Biot. Dette er fordi ligningene utgjør en fasefelt modell som beskriver flyt i et deformerbart porøst medium med to poroelastiske materialer. Disse kan forflytte seg basert på overflatespenning, elastisk spenning, og poretrykk, og det er tenkt at modellen kan anvendes i forbindelse med kreftsvulstmodellering.There are many applications where the study of coupled physical processes is of great importance. These range from the life sciences with flow in deformable human tissue to structural engineering with fracture propagation in elastic solids. In this doctoral dissertation, there is a twofold focus on coupled problems. Firstly, robust and efficient solution strategies, with a focus on iterative decoupling methods, have been applied to several coupled systems of equations. Secondly, a new thermodynamically consistent coupled system of equations is proposed. Solution strategies are developed for three different coupled problems; the quasi-static linearized Biot equations that couples flow through porous materials and elastic deformation of the solid medium, variational phase-field models for brittle fracture that couple a phase-field equation for fracture evolution with linearized elasticity, and the Cahn-Larché equations that model elastic effects in a two-phase elastic material and couples an extended Cahn-Hilliard phase-field equation and linearized elasticity. Finally, the new system of equations that is proposed models flow through a two-phase deformable porous material where the solid phase evolution is governed by interfacial forces as well as effects from both the fluid and elastic properties of the material. In the work that concerns the quasi-static linearized Biot equations, the focus is on the fixed-stress splitting scheme, which is a popular method for sequentially solving the flow and elasticity subsystems of the full model. Using such a method is beneficial as it allows for the use of readily available solvers for the subproblems; however, a stabilizing term is required for the scheme to converge. It is well known that the convergence properties of the method strongly depend on how this term is chosen, and here, the optimal choice of it is addressed both theoretically and practically. An interval where the optimal stabilization parameter lies is provided, depending on the material parameters. In addition, two different ways of optimizing the parameter are proposed. The first is a brute-force method that relies on the mesh independence of the scheme's optimal stabilization parameter, and the second is valid for low-permeable media and utilizes an equivalence between the fixed-stress splitting scheme and the modified Richardson iteration. Regarding the variational phase-field model for brittle fracture propagation, the focus is on improving the convergence properties of the most commonly used solution strategy with an acceleration method. This solution strategy relies on a staggered scheme that alternates between solving the elasticity and phase-field subproblems in an iterative way. This is known to be a robust method compared to the monolithic Newton method. However, the staggered scheme often requires many iterations to converge to satisfactory precision. The contribution of this work is to accelerate the solver through a new acceleration method that combines Anderson acceleration and over-relaxation, dynamically switching back and forth between them depending on a criterion that takes the residual evolution into account. The acceleration scheme takes advantage of the strengths of both Anderson acceleration and over-relaxation, and the fact that they are complementary when applied to this problem, resulting in a significant speed-up of the convergence. Moreover, the method is applied as a post-processing technique to the increments of the solver, and can thus be implemented with minor modifications to readily available software. The final contribution toward solution strategies for coupled problems focuses on the Cahn-Larché equations. This is a model for linearized elasticity in a medium with two elastic phases that evolve with respect to interfacial forces and elastic effects. The system couples linearized elasticity and an extended Cahn-Hilliard phase-field equation. There are several challenging features with regards to solution strategies for this system including nonlinear coupling terms, and the fourth-order term that comes from the Cahn-Hilliard subsystem. Moreover, the system is nonlinear and non-convex with respect to both the phase-field and the displacement. In this work, a new semi-implicit time discretization that extends the standard convex-concave splitting method applied to the double-well potential from the Cahn-Hilliard subsystem is proposed. The extension includes special treatment for the elastic energy, and it is shown that the resulting discrete system is equivalent to a convex minimization problem. Furthermore, an alternating minimization solver is proposed for the fully discrete system, together with a convergence proof that includes convergence rates. Through numerical experiments, it becomes evident that the newly proposed discretization method leads to a system that is far better conditioned for linearization methods than standard time discretizations. Finally, a new model for flow through a two-phase deformable porous material is proposed. The two poroelastic phases have distinct material properties, and their interface evolves according to a generalized Ginzburg–Landau energy functional. As a result, a model that extends the Cahn-Larché equations to poroelasticity is proposed, and essential coupling terms for several applications are highlighted. These include solid tumor growth, biogrout, and wood growth. Moreover, the coupled set of equations is shown to be a generalized gradient flow. This implies that the system is thermodynamically consistent and makes a toolbox of analysis and solvers available for further study of the model.Doktorgradsavhandlin

Domain Decomposition And Time-Splitting Methods For The Biot System Of Poroelasticity

In this thesis, we develop efficient mixed finite element methods to solve the Biot system of poroelasticity, which models the flow of a viscous fluid through a porous medium along with the deformation of the medium. We study non-overlapping domain decomposition techniques and sequential splitting methods to reduce the computational complexity of the problem. The solid deformation is modeled with a mixed three-field formulation with weak stress symmetry. The fluid flow is modeled with a mixed Darcy formulation. We introduce displacement and pressure Lagrange multipliers on the subdomain interfaces to impose weakly the continuity of normal stress and normal velocity, respectively. The global problem is reduced to an interface problem for the Lagrange multipliers, which is solved by a Krylov space iterative method. We study both monolithic and split methods. For the monolithic method, the cases of matching and non-matching subdomain grid interfaces are analyzed separately. For both cases, a coupled displacement-pressure interface problem is solved, with each iteration requiring the solution of local Biot problems. For the case of matching subdomain grids, we show that the resulting interface operator is positive definite and analyze the convergence of the iteration. For the non-matching subdomain grid case, we use a multiscale mortar mixed finite element (MMMFE) approach. We further study drained split and fixed stress Biot splittings, in which case we solve separate interface problems requiring elasticity and Darcy solves. We analyze the stability of the split formulations. We also use numerical experiments to illustrate the convergence of the domain decomposition methods and compare their accuracy and efficiency in the monolithic and time-splitting settings. Finally, we present a novel space-time domain decomposition technique for the mixed finite element formulation of a parabolic equation. This method is motivated by the MMMFE method, where we split the space-time domain into multiple subdomains with space-time grids of different sizes. Scalar Lagrange multiplier (mortar) functions are introduced to enforce weakly the continuity of the normal component of the mixed finite element flux variable over the space-time interfaces. We analyze the new method and numerical experiments are developed to illustrate and confirm the theoretical results

Hybrid-Trefftz finite elements for elastostatic and elastodynamic problems in porous media

The displacement and stress models of the hybrid-Trefftz finite element formulation are applied to the elastostatic and elastodynamic analysis of two-dimensional saturated and unsaturated porous media problems. The formulation develops from the classical separation of variables in time and space, but it leads to two time integration strategies. The first is applied to periodic problems, which are discretized in time using Fourier analysis. A mixed finite element approach is used in the second strategy for discretization in time of non-periodic/transient problems. These strategies lead to a series of uncoupled problems in the space dimension, which is subsequently discretized using either the displacement or the stress model of the hybrid-Trefftz finite element formulation. The main distinction between the two models is in the way that the interelement continuity is enforced. The displacement model enforces the interelement compatibility, while the stress model enforces the interelement equilibrium. As is typical of Trefftz methods, for both models, the approximation bases are constrained to satisfy locally the homogeneous form of the domain (Navier) equations. The free-field solutions of these equations are derived in cylindrical coordinates and used to construct the domain approximations of the hybrid-Trefftz displacement and stress elements. If the original equations are non-homogeneous, the influence of the source terms is modelled using Trefftz-compliant solutions of the corresponding static problem. For saturated porous media, the finite element models are based on the Biot's theory. It assumes an elastic solid phase fully permeated by a compressible liquid phase obeying the Darcy's law. For the modelling of unsaturated porous media, the finite elements are formulated using the theory of mixtures with interfaces. The model is thermodynamically consistent and considers the full coupling between the solid, fluid and gas phases, including the effects of relative (seepage) accelerations. Small displacements and linear-elastic material behaviour are assumed for all models

Simulation of fracture slip and propagation in hydraulic stimulation of geothermal reservoirs

Rollen til hydraulisk stimulering i å øke produksjonen fra geotermiske reservoarer, og muliggjøre kommersiell utnyttelse av et større spekter av geotermiske ressurser, har fått økt oppmerksomhet de siste tiårene. Under stimulering kan eksisterende sprekker sideforskyves, forplante seg og koble seg til andre sprekker og der igjennom øke permeabiliteten i reservoaret. Prosessene er preget av sterke hydromekaniske interaksjoner, som vi har begrensede muligheter til å overvåke. Numeriske simuleringer er derfor et viktig verktøy for å hjelpe oss til å bedre forstå mekanismene som er i spill. Avhandlingen tar sikte på å utvikle en omfattende matematisk modell og en numerisk tilnærming for å analysere bruddmekanismer og undersøke koblede hydromekaniske prosesser som forekommer i oppsprukne porøse medier. Den foreslåtte modellen benytter en blandet-dimensjonal konseptuell modell, som inkluderer porelastisitet i det porøse mediet og kontaktmekanikk for sprekkene. Modellen tillater også forplantning og koalescens av eksisterende sprekker. Et nytt diskretiseringsskjema for å løse den foreslåtte matematiske modellen presenteres. Den foreslåtte metoden bruker en to-nivå simuleringstilnærming, kategorisert i grove og fine nivåer, for å redusere beregningskostnader og sikre nøyaktighet. En endelig volummetode kombineres med en aktiv-sett løsningsstrategi for å diskretisere porelastisitet og bruddkontaktmekanikk på det grove nivået. Sprekkeforplantning betraktes på et fint nivå, der en endelig elementmetode kombineres med kollapsede kvartpunktselementer for å approksimere singulariteten i spenningen ved enden av sprekkene. Adaptiv gitring basert på en feilestimator og Laplace-glatting av gitteret introduseres på begge nivåer for effektivt å håndtere sprekkepropagering og koalescens. Simuleringene utført i denne avhandlingen forbedrer vår forståelse av hydraulisk stimulering og dens effekt på forbedring av sprekkepermeabilitet og konnektivitet i geotermiske reservoarer.The role of hydraulic stimulation in enhancing geothermal reservoir production and allowing for commercial exploitation of a larger range of geothermal resources has attracted attention from researchers in recent decades. During stimulation, preexisting fractures may slip, propagate, and connect to other fractures to enhance permeability. The processes are characterized by strong hydromechanical interactions, which have limited monitoring opportunities. Therefore, numerical simulations provide a powerful tool to help us better understand the mechanisms. This thesis aims to develop a comprehensive mathematical model and a numerical approach to analyze fracture mechanisms, and to investigate the coupled hydromechanical processes occurring in fractured porous media. The proposed model will employ a mixed-dimensional conceptual model, incorporating the concepts of poroelasticity and fracture contact mechanics. The model will also allow for the growth and coalescence of preexisting fractures. A novel discretization scheme for solving the proposed mathematical model is presented. The proposed scheme employs a two-level simulation approach, categorized into coarse and fine levels, to reduce the computational costs and ensure accuracy. A finite volume method is combined with an active set strategy to discretize poroelasticity and fracture contact mechanics on the coarse level. Fracture propagation is considered on a fine level, in which a finite element method is combined with collapsed quarter-point elements to capture the stress singularity at the fracture tips. Adaptive remeshing based on an error estimator and Laplacian smoothing is introduced on both levels to effectively capture fracture propagation and coalescence in the computational grid. The simulations conducted in this thesis improve our understanding of hydraulic stimulation and its effect on enhancing fracture permeability and connectivity in geothermal reservoirs.Doktorgradsavhandlin

Tracking Space and Time Changes of Physical Properties in Complex Geological Media

An important issue in seismology concerns the characterization of the propagation medium, aiming to analyze the behavior of rocks in relation to the generation of earthquakes (both natural and human-made). The basic idea is that seismic waves can be used to image the medium’s physical properties. In this context we placed our research project, concerning the reconstruction of the spatial and temporal changes of physical properties (velocity, attenuation, rock parameters) in complex geological media. In the first part of this thesis we present a detailed description of known and new methodologies useful to track the seismicity, the propagation medium’s features and their temporal variation. In particular, a new rock modelling approach is constructed, allowing the conversion of velocity and attenuation values in rock micro-parameters; and a new equalization procedure for the 4D tomography is developed, allowing at once to optimize the choice of time-windows in the case of massive data-sets and to completely handle seismic tomography issues. In the second part, we show the results obtained by applying this methodologies to three complex areas: the Irpinia fault zones, The Geysers geothermal area and the Solfatara volcano. The relevance of these three areas lies not only in their different physical nature, but also in their different dimension. The obtained results show how the described methodologies can be used in seismogenic and volcanic areas to improve the knowledge of the medium’s properties, in order to mitigate the risk associated to destructive events, and in geothermal areas, to monitor the induced seismicity through the tracking of the medium properties’ temporal variation. Therefore, this thesis represents a useful tool for the characterization of the propagation medium, by providing a compendium of different methodologies and by showing the results of their application to three complex areas characterized by different physical nature and dimensional scale

Microporomechanical modeling of shale

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Civil and Environmental Engineering, February 2010.Cataloged from PDF version of thesis.Includes bibliographical references (p. 401-429).Shale, a common type of sedimentary rock of significance to petroleum and reservoir engineering, has recently emerged as a crucial component in the design of sustainable carbon and nuclear waste storage solutions and as a prolific natural gas source. Despite its importance, the highly heterogeneous and anisotropic nature of shale has challenged the theoretical modeling and prediction of its mechanical properties. This thesis presents a comprehensive microporomechanics framework for developing predictive models for shale poroelasticity and strength. Modeling is accomplished through a multi-scale approach, in which the experimental evidence gathered from novel nanoindentation techniques and conventional macroscopic tests informs the development of a suit of micromechanics tools for linking composition and microstructure to material performance. Based on a closed loop approach of calibration and validation of elastic and strength properties at different length scales, it was possible to deconstruct shale to the scale of an elementary material unit with mechanical behaviors governed by invariant properties, and to upscale these behaviors from the nanoscale to the macroscale of engineering applications. The elementary building block for elasticity is an anisotropic solid characterizing the in situ stiffness of highly consolidated clay.(cont.) This intrinsic behavior represents the composite response of clay platelets, interlayer galleries, and interparticle contacts, yielding an invariant stiffness with respect to clay mineralogy. The anisotropic nanogranular nature of the porous clay in shale as inferred from nanoindentation is confirmed through micromechanics modeling. The intrinsic anisotropy of the clay fabric is suggested as the dominant factor driving the multi-scale anisotropic poroelasticity of unfractured shale compared to the contributions of geometrical sources related to shapes and orientations of particles. For strength properties, the micromechanics approach revealed that the frictional behavior of the elementary unit of compacted clay is scale independent, whereas a scale effect modifies its cohesive behavior. Having established a fundamental material unit and the adequate micromechanics representation for the microstructure, the macroscopic diversity of shale predominantly depends on two volumetric properties derived from mineralogy and porosity: the clay packing density and the silt inclusion volume fraction. The proposed two-parameter microporoelastic and strength models represent appealing alternatives for use in geomechanics and geophysics applications.by J. Alberto Ortega.Ph.D

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described