In water-based EOR methods, active chemical or biological substances are added to modify the physical properties of the fluids or/and the porous media at the interface between oil and water. The resulting displacement processes are governed by complex interplays between the transport of chemical substances, which is largely linear and highly affected by numerical diffusion, and how these substances affect the flow by changing the properties of the fluids and the surrounding rock. These effects are highly nonlinear and highly sensitive to threshold parameters that determine sharp transitions between regions of very different behavior. Unresolved simulation can therefore lead to misleading predictions of injectivity and recovery profiles.Use of higher-order spatial discretization schemes have been proposed by many authors as a means to reduce numerical diffusion and grid-orientation effects. Most higher-order simulators reported in the literature are based on explicit time stepping, and only a few are implicit. One reason that fully implicit formulations are not widely used might be that it becomes quite involved to compute the necessary linearizations for modern high-resolution discretizations of TVD and WENO type. Herein, we solve this problem by using automatic differentiation. We also demonstrate that using

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Many authors have used higher-order spatial discretizations to reduce numerical diffusion, which can be particularly pronounced when simulating EOR processes involving active chemical substances that are transported by linear or weakly nonlinear waves. Most high-resolution methods reported in the literature are based on explicit temporal discretizations. This imposes severe time-step restrictions when applied to the type of grids seen in industry-standard simulation models of real assets, which usually have ordersof-magnitude variations in porosities and Darcy velocities that necessitate the use of implicit discretization. Herein, we propose a second-order WENO discretization suitable for complex grids with polyhedral cell geometries, unstructured topologies, large aspect ratios, and large variations in interface areas. The WENO scheme is developed as part of a standard, fully implicit formulation that solves for pressure and transported quantities simultaneously. We investigate the accuracy and utility of the WENO scheme for a series of test cases that involve corner-point and 2D/3D Voronoi grids and black-oil and compositional flow models

This dissertation consists of two parts. The first part provides the background theory for the papers included in the second part. The background theory in Part I is structured as follows. First, the motivation and scope of this thesis are given in Chapter 1. Next, different properties and concepts of flow in a reservoir and the applications polymer flooding and CO 2 sequestration are introduced. The governing equations for two-phase flow and the mentioned applications are derived in Chapter 2. These equations are on the general form of hyperbolic conservation laws. General theory and background on this class of equations are given in Chapter 3. In Chapter 4 a general framework for solving hyperbolic conservation laws numerically is presented, with focus on the numerical schemes used in the included papers. Finally, in Chapter 5 the included papers found in Part II are summarized and discussed.

Carbon-neutral oil production is one way to improve the sustainability of petroleum resources. The emissions from produced hydrocarbons can be offset by injecting capture CO$$_{2}$$ 2 from a nearby point source into a saline aquifer for storage or a producing oil reservoir. The latter is referred to as enhanced oil recovery (EOR) and would enhance the economic viability of CO$$_{2}$$ 2 sequestration. The injected CO$$_{2}$$ 2 will interact with the oil and cause it to flow more freely within the reservoir. Consequently, the overall recovery of oil from the reservoir will increase. This enhanced oil recovery (EOR) technique is perceived as the most cost-effective method for disposing captured CO$$_{2}$$ 2 emissions and has been performed for many decades with the focus on oil recovery. The interaction between existing oil and injected CO$$_{2}$$ 2 needs to be fully understood to effectively manage CO$$_{2}$$ 2 migration and storage efficiency. When CO$$_{2}$$ 2 and oil mix in a fully miscible setting, the density can change non-linearly and cause density instabilities. These instabilities involve complex convective-diffusive processes, which are hard to model and simulate. The interactions occur at the sub-centimeter scale, and it is important to understand its implications for the field scale migration of CO$$_{2}$$ 2 and oil. In this work, we simulate gravity effects, namely gravity override and convective mixing, during miscible displacement of CO$$_{2}$$ 2 and oil. The flow behavior due to the competition between viscous and gravity effects is complex, and can only be accurately simulated with a very fine grid. We demonstrate that convection occurs rapidly, and has a strong effect on breakthrough of CO$$_{2}$$ 2 at the outlet. This work for the first time quantifies these effects for a simple system under realistic conditions.