The Role of Wettability Alteration in Subsurface CO2 Storage: Modeling and Numerical Analysis

This dissertation is aimed to provide mathematical frameworks to assess the large-scale deployment of CO2 in a subsurface formation, where the formation wettability is assumed to be altering through exposure time to the wettability-altering agent. Particularly, this thesis addresses: upscaling the pore-scale process to the macroscale laws, developing an alternative time-stepping method, and quantifying the upscaled models for the subsurface CO2 storage technology. These three components are organized to investigate and understand the effect of the wettability change on the interaction of CO2-water in a porous medium. Wettability refers to the tendency of a fluid to be in contact with the solid surface over the other fluid. This property changes due to many factors (e.g., reservoir temperature, pressure, pH, fluid compositions, and exposure time to the reactive fluid), and the change in wettability is known as wettability alteration. Wettability alteration (WA) takes place at the pore scale, but strongly controls the fluid-fluid interaction that is observed at the macroscale. One of the goals of this thesis is to develop a mathematical framework that upscales the effect of exposure time-dependent WA process to Darcy-scale models such as capillary pressure and relative permeability functions also known as saturation or \textit{flow functions. The upscaling processes introduce a pore-scale WA mechanism that follows a sorption-based model as a function of WA agent and exposure time to a WA agent. This model is then coupled with a bundle-of-tubes (BoT) model to simulate time-dependent WA-induced capillary pressure and relative permeability data. The resulting saturation functions are then used to quantify the WA-induced dynamic components of the saturation functions. More importantly, this part of the study also draws a clear relationship between the pore-scale and upscaled model's behaviors. The developed saturation functions are non-local in time. Coupling these functions adds extra complexity and non-linearity to the solution process of the multi-phase flow model. This thesis develops a monotone fixed-point iterative linearization scheme to approximate the solution for the resulting non-standard model. The scheme treats the capillary pressure function semi-implicitly in time and introduces an L-scheme type stabilization term in both the saturation and pressure equations. The convergence of the scheme is proved theoretically. The theoretical convergence analysis and numerical results show that the scheme is linearly convergent. However, the proposed linearization scheme shows flexibility for the choice of time-step size for reasonably large alteration (possibly jumps) in the capillary pressure function (i.e., saturation discontinuity). Furthermore, the scheme is designed so that it can be combined with Newton's method in a straightforward manner. This may improve the convergence rate of the scheme. The third part of this study concerns the full compositional flow model, where the saturation functions are dependent on solvents (e.g., dissolved CO2 in water), phase saturation, and exposure time to the solvent. Here, we quantify the role of the exposure time-dependent WA processes on the applicability of CO2 storage in saline aquifers. To do so, we design horizontal and vertical CO2-water flow scenarios. For the horizontal flow scenario, we compare the CO2-water front locations for static (i.e., initial-wet condition) and WA induced dynamic saturation functions based on the capillary number. The analysis shows that the CO2 front scales well with the capillary number. More precisely, the effect of WA on the CO2 front movement decreases while the capillary number increases. On the other hand, the integrity of the caprock is evaluated with and without WA effects in the saturation functions for the vertical flow scenario. We design a correlation model that can be used to forecast the total CO2, caused by WA, in the caprock for a given rate of WA dynamics, caprock permeability, entry pressure, and of course time.Doktorgradsavhandlin

A locally adaptive time-stepping algorithm for\ud petroleum reservoir simulations

An algorithm for locally adapting the step-size for large scale finite volume simulations of multi-phase flow in petroleum reservoirs is suggested which allows for an “all-in-one” implicit calculation of behaviour over a very large time scale. Some numerical results for simple two-phase flow in one space dimension illustrate the promise of the algorithm, which has also been applied to very simple 3D cases. A description of the algorithm is presented here along with early results. Further development of the technique is hoped to facilitate useful scaling properties

Development of a Mathematical Model for Electrically Assisted Oil Transport in Porous Media

In this study, it was attempted to develop a mathematical model for electrically assisted oil transport in porous media. The main implementation of the model would be in predicting the oil recovery in electrically enhanced oil recovery method. First, the contributing factors to the electrically assisted oil transport in porous media were investigated through laboratory experiments. Some of these contributing factors were found to be (i) viscous drag of oil with electro-osmosis of the water phase, primarily controlled by the oil/water ratio and the hydraulic and electro-osmotic permeability of the formation; (ii) reduction of oil-water interfacial tension due to electrochemical transformation of oil that affect its viscosity, hence increases its mobility, and (iii) increase in the permeability of the formation rock under applied electric field. A mathematical model that couples the pressure and electric gradients applied to the porous medium and incorporates the viscous drag of water on the oil phase under applied electric gradient was developed. Implicit Pressure Explicit Saturation (IMPES) solution strategy was used to solve the set of governing equations and Finite Volume Method (FVM) was used to solve the model numerically and to run several simulations. One of the most important constitutive relationships necessary for solution of the model, relative permeability coefficients as a function of water saturation, were introduced and evaluated under applied electric field in this study. Although it was attempted to capture the most important parameters in the development of the mathematical model, to have a model that more realistically represent the electrically assisted oil recovery phenomena in reservoir scale, the transient change in the viscosity of formation oil and the non-isothermal effects should also be considered in the model for future researches

Two-phase flow properties upscaling in heterogeneous porous media

The groundwater specialists and the reservoir engineers share the same interest in simulating multiphase flow in soil with heterogeneous intrinsic properties. They also both face the challenge of going from a well-modeled micrometer scale to the reservoir scale with a controlled loss of information. This upscaling process is indeed worthy to make simulation over an entire reservoir manageable and stochastically repeatable. Two upscaling steps can be defined: one from the micrometer scale to the Darcy scale, and another from the Darcy scale to the reservoir scale. In this thesis, a new second upscaling multiscale algorithm Finite Volume Mixed Hybrid Multiscale Methods (Fv-MHMM) is investigated. Extension to a two-phase flow system is done by weakly and sequentially coupling saturation and pressure via IMPES-like method

A finite volume approach for the numerical analysis and solution of the Buckley-Leverett equation including capillary pressure

The study of petroleum recovery is significant for reservoir engineers. Mathematical models of the immiscible displacement process contain various assumptions and parameters, resulting in nonlinear governing equations which are tough to solve. The Buckley-Leverett equation is one such model, where controlling forces like gravity and capillary forces directly act on saturation profiles. These saturation profiles have important features during oil recovery. In this thesis, the Buckley-Leverett equation is solved through a finite volume scheme, and capillary forces are considered during this calculation. The detailed derivation and calculation are also illustrated here. First, the method of characteristics is used to calculate the shock speed and characteristics curve behaviour of the Buckley-Leverett equation without capillary forces. After that, the local Lax-Friedrichs finite-volume scheme is applied to the governing equation (assuming there are no capillary and gravity forces). This mathematical formulation is used for the next calculation, where the cell-centred finite volume scheme is applied to the Buckley- Leverett equation including capillary forces. All calculations are performed in MATLAB. The fidelity is also checked when the finite-volume scheme is computed in the case where an analytical solution is known. Without capillary pressure, all numerical solutions are calculated using explicit methods and smaller time steps are used for stability. Later, the fixed-point iteration method is followed to enable the stability of the local Lax-Friedrichs and Cell-centred finite volume schemes using an implicit formulation. Here, we capture the number of iterations per time-steps (including maximum and average iterations per time-step) to get the solution of water saturation for a new time-step and obtain the saturation profile. The cumulative oil production is calculated for this study and illustrates capillary effects. The influence of viscosity ratio and permeability in capillary effects is also tested in this study. Finally, we run a case study with valid field data and check every calculation to highlight that our proposed numerical schemes can capture capillary pressure effects by generating shock waves and providing single-valued saturation at each position. These saturation profiles help find the amount of water needed in an injection well to displace oil through a production well and obtains good recovery using the water flooding technique

Adaptive mesh optimization for simulation of immiscible viscous fingering

Viscous fingering can be a major concern when waterflooding heavy-oil reservoirs. Most commercial reservoir simulators use low-order finite-volume/-difference methods on structured grids to resolve this phenomenon. However, this approach suffers from a significant numerical-dispersion error because of insufficient mesh resolution, which smears out some important features of the flow. We simulate immiscible incompressible two-phase displacements and propose the use of unstructured control-volume finite-element (CVFE) methods for capturing viscous fingering in porous media. Our approach uses anisotropic mesh adaptation where the mesh resolution is optimized on the basis of the evolving features of flow. The adaptive algorithm uses a metric tensor field dependent on solution-interpolation-error estimates to locally control the size and shape of elements in the metric. The mesh optimization generates an unstructured finer mesh in areas of the domain where flow properties change more quickly and a coarser mesh in other regions where properties do not vary so rapidly. We analyze the computational cost of mesh adaptivity on unstructured mesh and compare its results with those obtained by a commercial reservoir simulator on the basis of the finite-volume methods