Discontinuous Galerkin methods for convection-diffusion equations and applications in petroleum engineering

This dissertation contains research in discontinuous Galerkin (DG) methods applying to convection-diffusion equations. It contains both theoretical analysis and applications. Initially, we develop a conservative local discontinuous Galerkin (LDG) method for the coupled system of compressible miscible displacement problem in two space dimensions. The main difficulty is how to deal with the discontinuity of approximations of velocity, u, in the convection term across the cell interfaces. To overcome the problems, we apply the idea of LDG with IMEX time marching using the diffusion term to control the convection term. Optimal error estimates in Linfinity(0, T; L2) norm for the solution and the auxiliary variables will be derived. Then, high-order bound-preserving (BP) discontinuous Galerkin (DG) methods for the coupled system of compressible miscible displacements on triangular meshes will be developed. There are three main difficulties to make the concentration of each component between 0 and 1. Firstly, the concentration of each component did not satisfy a maximum-principle. Secondly, the first-order numerical flux was difficult to construct. Thirdly, the classical slope limiter could not be applied to the concentration of each component. To conquer these three obstacles, we first construct special techniques to preserve two bounds without using the maximum-principle-preserving technique. The time derivative of the pressure was treated as a source of the concentration equation. Next, we apply the flux limiter to obtain high-order accuracy using the second-order flux as the lower order one instead of using the first-order flux. Finally, L2-projection of the porosity and constructed special limiters that are suitable for multi-component fluid mixtures were used. Lastly, a new LDG method for convection-diffusion equations on overlapping mesh introduced in [J. Du, Y. Yang and E. Chung, Stability analysis and error estimates of local discontinuous Galerkin method for convection-diffusion equations on overlapping meshes, BIT Numerical Mathematics (2019)] showed that the convergence rates cannot be improved if the dual mesh is constructed by using the midpoint of the primitive mesh. They provided several ways to gain optimal convergence rates but the reason for accuracy degeneration is still unclear. We will use Fourier analysis to analyze the scheme for linear parabolic equations with periodic boundary conditions in one space dimension. To investigate the reason for the accuracy degeneration, we explicitly write out the error between the numerical and exact solutions. Moreover, some superconvergence points that may depend on the perturbation constant in the construction of the dual mesh were also found out

HIGH ORDER BOUND-PRESERVING DISCONTINUOUS GALERKIN METHODS AND THEIR APPLICATIONS IN PETROLEUM ENGINEERING

This report contains researches in the theory of high-order bound-preserving (BP) discontinuous Galerkin (DG) method and their applications in petroleum engineering. It contains both theoretical analysis and numerical experiments. The compressible miscible displacements and wormhole propagation problem, arising in petroleum engineering, is used to describe the evolution of the pressure and concentrations of different components of fluid in porous media. The important physical features of concentration and porosity include their boundedness between 0 and 1, as well as the monotone increasing for porosity in wormhole propagation model. How to keep these properties in the simulation is crucial to the robustness of the numerical algorithm. In the first project, we develop high-order bound-preserving discontinuous Galerkin methods for the coupled system of compressible miscible displacements on triangular meshes. We consider the problem with multi-component fluid mixture and the (volumetric) concentration of the jth component,cj, should be between 0 and 1. The main idea is stated as follows. First, we apply the second-order positivity-preserving techniques to all concentrations c′ js and enforce P jcj= 1 simultaneously to obtain physically relevant boundedness for every components. Then, based on the second-order BP schemes, we use the second-order numerical fluxes as the lower order one to combine with high-order numerical fluxes to achieve the high-order accuracy. Finally, since the classical slope limiter cannot be applied to polynomial upper bounds, we introduce a new limiter to our algorithm. Numerical experiments are given to demonstrate the high-order accuracy and good performance of the numerical technique. In our second project, we propose high-order bound-preserving discontinuous Galerkin methods to keep the boundedness for the porosity and concentration of acid, as well as the monotone increasing for porosity. The main technique is to introduce a new variable r to replace the original acid concentration and use a consistent flux pair to deduce a ghost equation such that the positive-preserving technique can be applied on both original and deduced equations. A high-order slope limiter is used to keep a polynomial upper bound which changes over time for r. Moreover, the high-order accuracy is attained by the flux limiter. Numerical examples are given to demonstrate the high-order accuracy and bound-preserving property of the numerical technique

On Discontinuous Galerkin Methods for Singularly Perturbed and Incompressible Miscible Displacement Problems

This thesis is concerned with the numerical approximation of problems of fluid flow, in particular the stationary advection diffusion reaction equations and the time dependent, coupled equations of incompressible miscible displacement in a porous medium. We begin by introducing the continuous discontinuous Galerkin method for the singularly perturbed advection diffusion reaction problem. This is a method which coincides with the continuous Galerkin method away from internal and boundary layers and with a discontinuous Galerkin method in the vicinity of layers. We prove that this consistent method is stable in the streamline diffusion norm if the convection field flows non-characteristically from the region of the continuous Galerkin to the region of the discontinuous Galerkin method. We then turn our attention to the equations of incompressible miscible displacement for the concentration, pressure and velocity of one fluid in a porous medium being displaced by another. We show a reliable a posteriori error estimator for the time dependent, coupled equations in the case where the solution has sufficient regularity and the velocity is bounded. We remark that these conditions may not be attained in physically realistic geometries. We therefore present an abstract approach to the stationary problem of miscible displacement and investigate an a posteriori error estimator using weighted spaces that relies on lower regularity requirements for the true solution. We then return to the continuous discontinuous Galerkin method. We prove in an abstract setting that standard (continuous) Galerkin finite element approximations are the limit of interior penalty discontinuous Galerkin approximations as the penalty parameter tends to infinity. We then show that by varying the penalization parameter on only a subset of the domain we reach the continuous discontinuous method in the limit. We present numerical experiments illustrating this approach both for equations of non-negative characteristic form (closely related to advection diffusion reaction equations) and to the problem of incompressible miscible displacement. We show that we may practically determine appropriate discontinuous and continuous regions, resulting in a significant reduction of the number of degrees of freedom required to approximate a solution, by using the properties of the discontinuous Galerkin approximation to the advection diffusion reaction equation. We finally present novel code for implementing the continuous discontinuous Galerkin method in C++

Nonstandard Finite Element Methods

[no abstract available

Locally Mass-Conservative Method With Discontinuous Galerkin In Time For Solving Miscible Displacement Equations Under Low Regularity

The miscible displacement equations provide the mathematical model for simulating the displacement of a mixture of oil and miscible fluid in underground reservoirs during the Enhance Oil Recovery(EOR) process. In this thesis, I propose a stable numerical scheme combining a mixed finite element method and space-time discontinuous Galerkin method for solving miscible displacement equations under low regularity assumption. Convergence of the discrete solution is investigated using a compactness theorem for functions that are discontinuous in space and time. Numerical experiments illustrate that the rate of convergence is improved by using a high order time stepping method. For petroleum engineers, it is essential to compute finely detailed fluid profiles in order to design efficient recovery procedure thereby increase production in the EOR process. The method I propose takes advantage of both high order time approximation and discontinuous Galerkin method in space and is capable of providing accurate numerical solutions to assist in increasing the production rate of the miscible displacement oil recovery process

Coupled space-time discontinuous Galerkin method for dynamical modeling in porous media

This thesis deals with coupled space-time discontinuous Galerkin methods for the modeling of dynamical phenomena in fluid saturated porous media. The numerical scheme consists of finite element discretizations in the spatial and in the temporal domain simultaneously. In particular, two major classes of approaches have been investigated. The first one is the so-called time-discontinuous Galerkin (DGT) method, consisting of discontinuous polynomials in the temporal domain but continuous ones in space. A natural upwind flux treatment is introduced to enforce the continuity condition at discrete time levels. The proposed numerical approach is suitable for solving first-order time-dependend equations. For the second-order equations, an Embedded Velocity Integration (EVI) technique is developed to degenerate a second-order equation into a first-order one. The resulting first-order differential equation with the primary variable in rate term (velocity) can in turn be solved by the time-discontinuous Galerkin method efficiently. Applications concerning both the first- and second-order differential equations as well as wave propagation problems in porous materials are investigated. The other one is the coupled space-time discontinuous Galerkin (DGST) method, in which neither the spatial nor the temporal approximations pocesses strong continuity. Spatial fluxes combined with flux-weighted constraints are employed to enforce the interelement consistency in space, while the consistency in the time domain is enforced by the temporal upwind flux investigated in the DGT method. As there exists no coupling between the spatial and temporal fluxes, various flux treatments in space and in time are employed independently. The resulting numerical scheme is able to capture the steep gradients or even discontinuities. Applications concerning the single-phase flow within the porous media are presented.Im Rahmen dieser Arbeit werden gekoppelte Raum-Zeit Finite-Element-Methoden für die Simulation dynamischer Effekte in fluid-gesättigten porösen Materialien entwickelt und numerisch umgesetzt. Dazu wird eine gekoppelte Diskretisierung des räumlichen und zeitlichen Gebietes vorgenommen. Insbesondere werden zwei Klassen von Verfahren untersucht. Die erste Methode ist ein sogenanntes zeitlich-diskontinuierliches Galerkin Verfahren (DGT-Methode). Hierbei werden diskontinuierliche Ansätze in der Zeit und kontinuierliche Ansätze im Raum verwendet. Die Kontinuitätsnebenbedingung in der Zeit wird durch einen upwind-Flussterm erzwungen. Der Flussterm unterliegt mathematischen Restriktionen und daher ist das resultierende Finite Element Verfahren nur für Gleichungen mit zeitlichen Ableitungen erster Ordnung geeignet. Um auch Gleichungen zweiter Ordnungen mit dem entwickelten DGT-Verfahren behandeln zu können, ist die EVI-Methode (Embedded Velocity Integration method) entwickelt worden. Im Rahmen der EVI-Technik wird die Geschwindigkeit als Primärvariable gewählt und im Bezug auf die gewählten zeitlichen Ansätze integriert. Die auf der Geschwindigkeit basierenden schwachen Formen können wiederum mit der DGT-Methode gelöst werden. Die entwickelten numerischen Raum-Zeit Finite-Elemente-Methoden werden sowohl für elastische Wellenausbreitungsprobleme als auch für gekoppelte Fragestellungen in porösen Medien angewendet. Abschließend wird ein räumlich diskontinuierliches Finite-Element-Verfahren entwickelt und mit den bereits entwickelten zeitlich-diskontinuierlichen Methoden gekoppelt. Die räumliche Kontinuitätsbedingung wird durch die Entwicklung eines speziellen Flusstermes erzwungen. Es wird gezeigt, dass sich das Verfahren mit den bereits entwickelten Flusstermen für die zeitliche Kontinuität koppeln lässt. Dies wird durch die Entkopplung der räumlichen und zeitlichen Flussterme möglich. Das resultierende Raum-Zeit diskontinuierliche Finite-Element-Verfahren wird wiederum auf Strömungsprobleme mischbarer Fluide in porösen Medien angewendet und mit klassischen Methoden verglichen