Date of Award

2019

Document Type

Open Access Master's Report

Degree Name

Master of Science in Mathematical Sciences (MS)

Administrative Home Department

Department of Mathematical Sciences

Advisor 1

Yang Yang

Committee Member 1

Zhengfu Xu

Committee Member 2

Fengjing Liu

Abstract

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.

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