Simulation of multiphase flow in natural subsurface formations include selection of time-step size, i.e., the discrete snapshots (steps) over which the time-dependent nonlinear process is investigated. The simulation results would naturally depend on the size of the discrete time step, according to the order of accuracy of the implemented numerical scheme. For accurate analyses the required time steps might be needed to be very small, which then leads to very costly simulations, which are often times impractical in real-world simulation applications. Taking coarse-scale time-step sizes can reduce the computational costs, however, it results in lower simulation accuracy. Note that a common simulation practice is to take constant time-step everywhere in the computational domain. To resolve this challenge, this MSc thesis report presents a novel local time-stepping (LTS) method for multiphase flow in porous media. In this method, different time-steps can be applied in one computational domain; allowing for employing small time steps in sub-domains with higher sensitivity to the time step size and bigger time steps elsewhere. The LTS method is developed for all types of sequentially-coupled and fully-coupled simulation strategies. For all of these developments, the interfacial flux continuity is the key requirement to connect the sub-domains of different time zones. The fluxes chosen for the inner stages of the coarse time-step are consistent with the chosen simulation strategy. More precisely, in the case of the implicit-pressure-explicit-saturation (IMPES) method these fluxes are lagged in time (explicit), while for the sequential implicit and fully implicit (FIM) methods they are calculated at the current (implicit) time-step. For implicit transport method, the refined zone partition matrices are developed to algebraically form the refined zone Jacobian matrix and refined zone residual vector. The numerical results show that only a fraction of the domain can be simulated with small time steps; while the rest can be simulated at a bigger one. Local time-stepping method provides accurate simulation results compared with the fine-scale time-step simulation results for all three simulation strategies. Specially, the local time-stepping for FIM simulations, which is the mostly common commercial-grade simulation approach, can preserve the accuracy of the results and also the computational efficiency. For the studied cases, the computational efficiency gain of LTS for sequential implicit method is not as significant as that of the FIM. To exploit full adaptivity in simulation, as presented in Appendix B, one can directly apply the developed method of this thesis to adaptive implicit method (AIM); where sequential and fully implicit coupling approaches are applied in different sub-domains of the reservoir. Overall, LTS casts a promising approach to optimize accuracy-efficiency tradeoff when it comes to time step selection.

One of the most important challenges facing academic, industrial and policy-making sectors is meeting with the increasing energy demand while preserving the affordability of the energy and maintaining the quality of the planet earth (including the environment, specially by reducing its greenhouse gas footprint). This global challenge demands for exploiting the subsurface formations as giant storage space for industrial by prod- ucts, e.g., CO2, while a future utilisation is found for them. Exploitation of subsurface formations for CO2 storage depends on our capacity to “accurately” and “efficiently” simulate the multiphase nonlinear flow in heterogeneous large-scale natural formations. An accurate and efficient simulation would allow for proper predictive understanding for several operational aspects including: (1) capacity of the storage; (2) life-time long behaviour of the injected CO2 in the subsurface formation, and (3) safety and integrity of the capturing procedure. The challenge of extreme scale dissimilarity between the heterogeneous coefficients (rock con- ductivity and fluid physics, e.g., mixing) and the reservoir has been known in the scientific literature as one of the main simulation challenges. Classically, to resolve this mismatch, reservoir models have been exces- sively upscaled to a resolution which can be solved affordably with the state-of-the-art commercial-grade simulators. However, as the rapid extension of the computational capacity, as well as the newly developed multiscale scalable nonlinear solvers, one needs to revisit our simulation frameworks to provide a scalable (to field scale) and accurate CO2 simulation framework. This thesis work focuses on extension of the recently developed Algebraic Dynamic Multilevel Method (ADM) to highly-nonlinear simulations of CO2-Brine multi- phase flow problems. ADM maintains simulation scalability by imposing fine-scale grids only where needed; and imposing coarser grids paired with accurate prolongation operators far from the sub-domains with sharp gradients. In this work, to provide a systematic study, the developed ADM framework is formulated such that it allows for both multiscale-based basis functions and also upscaling-based procedure. This is crucially im- portant as it reveals how the "solution-based” interpolations based on fine-scale heterogeneity can impact the simulation results; compared with the “effective-coefficient-based” approach.
In brief, the novelty and contribution of this research is two fold: (1) development of a multiscale-based ADM method for CO2-Brine simulator and (2) systematic study to find the best model order reduction strat- egy when it comes to complex multiphase characteristics and rock heterogeneity. Two different classes of simulations are studied: (1) injection and evolution of the injected CO2 phase into the brine with capillary effects; and (2) miscible mixing process in which resolving front fingers into the residing less-mobile fluid imposes computational challenges. The study, as such, intends to reveal the details in “upscaling-based” vs. “multiscal-based” approach; and tends to provide a generic framework in which the scalability of the simu- lations are preserved by employing fine-scale grids only where and when needed.
Several challenging sensitivity analysis are performed in which fine-scale solutions are compared with the multi-scale ADM and upscaling ADM solutions for immiscible and miscible fluid flow displacement. From these numerical experiments can be concluded that Multiscale-ADM outperforms the Upscaling-based ADM approach. More precisely with similar active grid cells, it provides a more accurate simulation method. As such, Multiscale-ADM casts a promising approach for next-generation simulators for CO2-Brine systems.

This thesis elaborates on a newly developed Fully Implicit simulation model that captures the full coupling of reaction kinetics and fluid dynamics during carbonate matrix acidizing at an extensive (field) scale. The framework describes the physical processes at reservoir scale, by accounting for upscaled pore-scale-based quantities for reaction rates and reservoir flow properties (e.g. permeability and porosity). The coupled Darcy-Stokes system of equation, as represented in a unified Brinkman formulation, is incorporated in the model to account for free flow (wormholes) and flow trough porous media. The model is built to investigate the subsurface behavior of the specifc (Dissolvine Stimwell) acid developed by AkzoNobel Co. Dissolvine Stimwell includes chelating agents which have, under specific conditions, significant advantages over conventional acids. Generally, carbonate acidizing treatment experiments are conducted at the core scale. However, in Salt Lake City Utah a carbonate acidizing experiment was carried out at unprecedented scale. The corresponding results are compared with the developed model to validate the dependency of the wormhole formation with the injection rate. In addition, the reduction in the skin factor is briefly addressed. Apart from the FIM simulation development of this thesis work for coupled reactive transport equation with Brinkman formula for flow, its other valuable contribution lies on the validation with experimental results at the unprecedented scale.