# DISCONTINUOUS GALERKIN METHODS FOR COMPRESSIBLE MISCIBLE DISPLACEMENTS AND APPLICATIONS IN RESERVOIR SIMULATION

2024

## Document Type

Open Access Dissertation

## Degree Name

Doctor of Philosophy in Mathematical Sciences (PhD)

## Administrative Home Department

Department of Mathematical Sciences

Yang Yang

Zhengfu Xu

## Committee Member 2

Alexander E. Labovsky

Qingli Dai

## Abstract

This dissertation contains research on discontinuous Galerkin (DG) methods applied to the system of compressible miscible displacements, which is widely adopted to model surfactant flooding in enhanced oil recovery (EOR) techniques. In most scenarios, DG methods can effectively simulate problems in miscible displacements.
However, if the problem setting is complex, the oscillations in the numerical results can be detrimental, with severe overshoots leading to nonphysical numerical approximations. The first way to address this issue is to apply the bound-preserving
technique. Therefore, we adopt a bound-preserving Discontinuous Galerkin method
with a Second-order Implicit Pressure Explicit Concentration (SIPEC) time marching
method to compute the system of two-component compressible miscible displacement in our first work. The Implicit Pressure Explicit Concentration (IMPEC) method is one of the most prevalent time marching approaches used in reservoir simulation for solving coupled flow systems in porous media. The main idea of IMPEC is to treat the pressure equation implicitly and the concentration equations explicitly. However, this treatment results in a first-order accurate scheme. To improve the order of accuracy of the scheme, we propose a correction stage to compensate for the second-order accuracy in each time step, thus naming it the SIPEC method. The SIPEC method is a crucial innovation based on the traditional second-order strong-stability-preserving Runge-Kutta (SSP-RK2) method. However, the SIPEC method is limited to second-order accuracy and cannot efficiently simulate viscous fingering phenomena. High-order numerical methods are preferred to reduce numerical artifacts and mesh dependence. In our second work, we adopt the IMPEC method based on the implicit-explicit Runge-Kutta (IMEX-RK) Butcher tableau to achieve higher order temporal accuracy while also ensuring stability. The high-order discontinuous Galerkin method is employed to simulate the viscous fingering fluid instabilities in a coupled nonlinear system of compressible miscible displacements. Although the bound-preserving techniques can effectively yield physically relevant numerical approximations, their success depends heavily on theoretical analysis, which is not straightforward for high-order methods. Therefore, we introduce an oscillation-free damping term to effectively suppress the spurious oscillations near discontinuities in high-order DG methods. As indicated by the numerical experiments, the incorporation of the bound-preserving DG method with SIPEC time marching and high-order OFDG with IMPEC time marching provides satisfactory results for simulating fluid flow in reservoirs.

Yue_Kang_MTU_PhD .pdf (3004 kB)
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#### ORCID

0000-0001-5540-4705