The main objective of this dissertation is to quantify petrophysical properties of conventional and unconventional reservoirs using a mechanistic approach. Unconventional transport mechanisms are described from the pore to the reservoir scale to examine their effects on macroscopic petrophysical properties in hydrocarbon-bearing organic shale. Petrophysical properties at the pore level are quantified with a new finite-difference method. A geometrical approximation is invoked to describe the interstitial space of grid-based images of porous media. Subsequently, a generalized Laplace equation is derived and solved numerically to calculate fluid pressure and velocity distributions in the interstitial space. The resulting macroscopic permeability values are within 6% of results obtained with the Lattice-Boltzmann method after performing grid refinements. The finite-difference method is on average six times faster than the Lattice-Boltzmann method. In the next step, slip flow and Knudsen diffusion are added to the pore-scale method to take into account unconventional flow mechanisms in hydrocarbon-bearing shale. The effect of these mechanisms is appraised with a pore-scale image of Eagle Ford shale as well as with several grain packs. It is shown that neglecting slip flow in samples with pore-throat sizes in the nanometer range could result in errors as high as 2000% when estimating permeability in unconventional reservoirs. A new fluid percolation model is proposed for hydrocarbon-bearing shale. Electrical conductivity is quantified in the presence of kerogen, clay, hydrocarbon, water, and the Stern-diffuse layer in grain packs as well as in the Eagle Ford shale pore-scale image. The pore-scale model enables a critical study of the [delta]LogR evaluation method commonly used with gas-bearing shale to assess kerogen concentration. A parallel conductor model is introduced based on Archie's equation for water conductivity in pores and a parallel conductive path for the Stern-diffuse layer. Additionally, a non-destructive core analysis method is proposed for estimating input parameters of the parallel conductor model in shale formations. A modified reservoir model of single-phase, compressible fluid is also developed to take into account the following unconventional transport mechanisms: (a) slip flow and Knudsen diffusion enhancement in apparent permeability, (b) Langmuir desorption as a source of gas generation at kerogen surfaces, and (c) the diffusion mechanism in kerogen as a gas supply to adsorbed layers. The model includes an iterative verification method of surface mass balance to ensure real-time desorption-adsorption equilibrium with gas production. Gas desorption from kerogen surfaces and gas diffusion in kerogen are the main mechanisms responsible for higher-than-expected production velocities commonly observed in shale-gas reservoirs. Slip flow and Knudsen diffusion marginally enhance production rates by increasing permeability during production. / text

We connect the Mimetic Finite Difference method (MFD) with the finite-volume two-point flux scheme (TPFA) for Voronoi meshes. The main effect is reducing the saddle-point system to a much smaller symmetric-positive definite matrix. In addition, the generalization allows MFD to seamlessly integrate with existing porous media modeling technology. The generalization also imparts the monotonicity property of the TPFA method on MFD. The connection is achieved by altering the consistency condition of the velocity bilinear operator. First-order convergence theory is presented as well as numerical results that support the claims. We demonstrate a methodology for using MFD in modeling fluid flow in fractures coupled with a reservoir. The method can be used for nonplanar fractures. We use the method to demonstrate the effects of fracture curvature on single-phase and multi-phase flows. Standard benchmarks are used to demonstrate the accuracy of the method. The approach is coupled with existing reservoir simulation technology. / text

Disertacijoje išnagrinėta trečiosios eilės vienmatė pseudoparabolinė lygtis su dviejų tipų nelokaliosiomis sąlygomis. Šiems uždaviniams spręsti sudarytos skirtuminės schemos, kurių stabilumas tiriamas, taikant skirtuminių operatorių su nelokaliosiomis sąlygomis spektro struktūrą. Trečiosios eilės vienmatėms ir dvimatėms pseudoparabolinėms lygtims su integralinėmis sąlygomis sudarytos ir išnagrinėtos padidinto tikslumo skirtuminės schemos. Išnagrinėta dvimatė pseudoparabolinė lygtis su nelokaliosiomis integralinėmis sąlygomis viena koordinačių kryptimi. Tokiam uždaviniui spręsti pritaikytas ir išnagrinėtas lokaliai vienmatis metodas, ištirtos šio metodo stabilumo sąlygos. Taip pat išnagrinėtos: trisluoksnės skirtuminės schemos vienmatei pseudoparabolinei lygčiai su įvairiomis, taip pat ir nelokaliosiomis, sąlygomis; trisluoksnių išreikštinių skirtuminių schemų stabilumo sąlygos. / The thesis analyzes the third-order one-dimensional pseudoparabolic equations with two types of nonlocal conditions. The stability of difference schemes for this problem was studied using the analysis of the spectrum structure of a difference operator with nonlocal conditions. The analysis of the increased accuracy difference schemes for third-order one-dimensional and two-dimensional pseudoparabolic equations with integral conditions has been made. The thesis considers a two-dimensional pseudoparabolic equation with nonlocal integral conditions in one coordinate direction. This problem was solved by a locally one-dimensional method. The stability of a difference scheme has been investigated based on the spectrum structure. The doctoral disertation investigates three-layer difference schemes for one-dimensional pseudoparabolic equations with various, including nonlocal, conditions. Also, the conditions for the stability of three-layer explicit difference schemes have been explored.

Disertacijoje sprendžiama lėtinimo sistemų kraštinių ir galinių laidininkų įtakos modelių tikslumui įvertinimo problema. Pagrindiniai tyrimo objektai – daugialaidžių ir meandrinių mikrojuostelinių linijų matematiniai modeliai, skaitiniai analizės metodai. Darbo tikslas – sukūrus modelius, grįstus baigtinių skirtumų laiko srities metodu, ištirti galinių ir kraštinių laidininkų netolygumų įtaką meandrinių mikrojuostelinių vėlinimo linijų laiko ir dažninėms charakteristikoms, pasiūlyti meandrinių vėlinimo linijų konstrukcijų tobulinimo priemones. Darbe sprendžiami uždaviniai: matematinių pavienės, susietųjų ir daugialaidžių mikrojuostelinių linijų modelių sudarymas ir savybių tyrimas, taikant baigtinių skirtumų bei baigtinių skirtumų laiko srities analizės metodus; daugialaidžių mikrojuostelinių linijų sintezės ir meandrinių mikrojuostelinių vėlinimo linijų analizės algoritmų bei jų elektrinių charakteristikų skaičiavimo metodikų sudarymas. Disertaciją sudaro įvadas, keturi skyriai, bendrosios išvados, naudotos literatūros ir autoriaus publikacijų disertacijos tema sąrašai. Įvadiniame skyriuje formuluojama tiriamoji problema, aptariamas darbo aktualumas, aprašomas tyrimų objektas, formuluojamas darbo tikslas bei uždaviniai, aprašoma tyrimų metodika, darbo mokslinis naujumas, rezultatų praktinė reikšmė, ginamieji teiginiai. Įvado pabaigoje pristatomi pranešimai konferencijose disertacijos tema bei pateikiama disertacijos struktūra. Pirmajame skyriuje pateikiama literatūros... [toliau žr. visą tekstą] / There are investigated accuracy issues of edges and ends evaluation problems of meander slow-wave systems in the dissertation. Objects of research – mathematical models of multiconductor and meander microstrip delay lines, numerical analysis methods. Aim of the work – after creating mathematical models, based on finite difference time domain method, explore ends and edges discontinuity effects on meander microstrip delay lines time and frequency characteristics, propose structural improvement measures of meander delay line. The dissertation approaches major tasks such as: mathematical individual, coupled, multiconductor microstrip lines models composition for performance and their properties investigation using finite difference and finite difference time domain methods of analysis; synthesis algorithm of multiconductor microstrip lines and analysis algorithm of meander microstrip delay line and methodology of their electrical characteristics calculation creation. The thesis consists of four parts including introduction, 4 chapters, conclusions, references. The introduction reveals investigated problem, importance of the thesis and object of research. It also describes the purpose and tasks of the dissertation, research methodology, scientific novelty and the practical significance of results examined in the thesis and defended statements. The introduction ends in presenting the author’s publications on the subject of the defended dissertation, offering the material of made... [to full text]

This thesis develops efficient modeling frameworks via a Partial Differential Equation (PDE) approach for multi-factor financial derivatives, with emphasis on three-factor models, and studies highly efficient implementations of the numerical methods on novel high-performance computer architectures, with particular focus on Graphics Processing Units (GPUs) and multi-GPU platforms/clusters of GPUs. Two important classes of multi-factor financial instruments are considered: cross-currency/foreign exchange (FX) interest rate derivatives and multi-asset options. For cross-currency interest rate derivatives, the focus of the thesis is on Power Reverse Dual Currency (PRDC) swaps with three of the most popular exotic features, namely Bermudan cancelability, knockout, and FX Target Redemption. The modeling of PRDC swaps using one-factor Gaussian models for the domestic and foreign interest short rates, and a one-factor skew model for the spot FX rate results in a time-dependent parabolic PDE in three space dimensions. Our proposed PDE pricing framework is based on partitioning the pricing problem into several independent pricing subproblems over each time period of the swap's tenor structure, with possible communication at the end of the time period. Each of these subproblems requires a solution of the model PDE. We then develop a highly efficient GPU-based parallelization of the Alternating Direction Implicit (ADI) timestepping methods for solving the model PDE. To further handle the substantially increased computational requirements due to the exotic features, we extend the pricing procedures to multi-GPU platforms/clusters of GPUs to solve each of these independent subproblems on a separate GPU. Numerical results indicate that the proposed GPU-based parallel numerical methods are highly efficient and provide significant increase in performance over CPU-based methods when pricing PRDC swaps. An analysis of the impact of the FX volatility skew on the price of PRDC swaps is provided. In the second part of the thesis, we develop efficient pricing algorithms for multi-asset options under the Black-Scholes-Merton framework, with strong emphasis on multi-asset American options. Our proposed pricing approach is built upon a combination of (i) a discrete penalty approach for the linear complementarity problem arising due to the free boundary and (ii) a GPU-based parallel ADI Approximate Factorization technique for the solution of the linear algebraic system arising from each penalty iteration. A timestep size selector implemented efficiently on GPUs is used to further increase the efficiency of the methods. We demonstrate the efficiency and accuracy of the proposed GPU-based parallel numerical methods by pricing American options written on three assets.

<p>In recent years, there has been a great interest towards the higher order numerical methods for singularly perturbed problems. As compared to their lower order counterparts, they provide better accuracy with fewer mesh points. Construction and/or implementation of direct higher order methods is usually very complicated. Thus a natural choice is to use some convergence acceleration techniques, e.g., Richardson extrapolation, defect correction, etc. In this thesis, we will consider various classes of problems described by singularly perturbed ordinary and partial differential equations. For these problems, we design some novel numerical methods and attempt to increase their accuracy as well as the order of convergence. We also do the same for existing numerical methods in some instances. We &macr / nd that, even though the Richardson extrapolation technique always improves the accuracy, it does not perform equally well when applied to different methods for certain classes of problems. Moreover, while in some cases it improves the order of convergence, in other cases it does not. These issues are discussed in this thesis for linear and nonlinear singularly perturbed ODEs as well as PDEs. Extrapolation techniques are analyzed thoroughly in all the cases, whereas the limitations of the defect correction approach for certain problems is indicated at the end of the thesis</p>

Cette thèse participe à l'étude générale et à la compréhension des relations structure- propriétés optiques de couches minces d'oxyde de tungstène, nanostructurées lors de leur dépôt par la technique Glancing Angle Déposition. Cette technique repose sur le contrôle de l'orientation relative du substrat vis à vis de la source de vapeur.[...]

Using the dual Black-Scholes partial differential equation, Dupire derived an explicit formula, involving the ratio of partial derivatives of the evolving fair value of a European call option (ECO), for recovering information about its variable volatility. Because the prices, as a function of maturity and strike, are only available as discrete noisy observations, the evaluation of Dupire’s formula reduces to being an ill-posed numerical differentiation problem, complicated by the need to take the ratio of derivatives. In order to illustrate the nature of ill-posedness, a simple finite difference scheme is first used to approximate the partial derivatives. A new method is then proposed which reformulates the determination of the volatility, from the partial differential equation defining the fair value of the ECO, as a parameter identification activity. By using the weak formulation of this equation, the problem is localized to a subregion on which the volatility surface can be approximated by a constant or a constant multiplied by some known shape function which models the local shape of the volatility function. The essential regularization is achieved through the localization, the choice of the analytic weight function, and the application of integration-by-parts to the weak formulation to transfer the differentiation of the discrete data to the differentiation of the analytic weight function.

A theory is developed for the analysis of composite systems consisting of steel wide flange sections reinforced with GFRP plates connected to one of the flanges through a layer of adhesive. The theory is based on an extension of the Gjelsvik theory and thus incorporates local and global warping effects but omits shear deformation effects. The theory captures the longitudinal transverse response through a system of three coupled differential equations of equilibrium and the lateral-torsional response through another system of three coupled differential equations. Closed form solutions are developed and a super-convergent finite element is formulated based under the new theory. A comparison to 3D FEA results based on established solid elements in Abaqus demonstrates the validity of the theory when predicting the longitudinal-transverse response, but showcases its shortcomings in predicting the torsional response of the composite system. The comparison sheds valuable insight on means of improving the theory. A more advanced theory is subsequently developed based on enriched kinematics which incorporates shear deformation effects. The shear deformable theory captures the longitudinal-transverse response through a system of four coupled differential equations of equilibrium and the lateral-torsional response through another system of six coupled differential equations. A finite difference approximation is developed for the new theory and a new finite element formulation is subsequently to solve the new system of equations. A comparison to 3D FEA illustrates the validity of the shear deformable theory in predicting the longitudinal-transverse response as well as the lateral-torsional response. Both theories are shown to be computationally efficient and reduce the modelling and running time from several hours per run to a few minutes or seconds while capturing the essential features of the response of the composite system.

This thesis develops and explores two methodologies to assess earthquake ground shaking in southwestern British Columbia based on 1D soil layering and 3D basin structure. To assess site response based on soil layering, microtremor array measurements were conducted at two sites of contrasting geology to estimate Rayleigh-wave dispersion curves. A Bayesian inversion algorithm is developed to invert the dispersion data for the shear-wave velocity (VS) profile together with quantitative uncertainty estimates, accounting rigorously for data error covariance and model parameterization selection. The recovered VS profiles are assessed for reliability by comparison with invasive VS measurements at each site with excellent agreement. Probabilistic site response analysis is conducted based on a sample of VS profiles drawn from the posterior probability density of the microtremor inversion. The quantitative uncertainty analysis shows that the rapid and inexpensive microtremor array method provides sufficient resolution of soil layering for practical characterization of earthquake ground motion. To assess the effects of 3D Georgia basin structure on long-period (> 2 s) ground motion for large scenario earthquakes, numerical 3D finite difference modelling of viscoelastic wave propagation is applied. Both deep (> 40 km) subducting Juan de Fuca plate and crustal (5 km) North America plate earthquakes are simulated in locations congruent with known seismicity. Simulations are calibrated by comparing synthetic waveforms with 36 selected strong- and weak-motion seismograms of the 2001 MW 6.8 Nisqually earthquake. The ratio between predicted peak ground motions in models with and without Georgia basin sediments is applied as a quantitative measure of basin amplification. Steep edges in the upper 1 km of the northwest and southeast extents of the basin are coincident with the appearance of surface waves. Focussing of north-to-northeast propagating surface waves by shallow (< 1 km) basin structure increases ground motion in a localized region of southern Greater Vancouver. This effect occurs for both types of earthquakes located south-southwest of Vancouver at distances greater than ~80 km. The predicted shaking level is increased up to 17 times and the duration of moderate shaking (> 3.4 cm/s) is up to 16 times longer due to the 3D Georgia basin structure. / Graduate