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

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-preservingtechnique. Therefore, we adopt a bound-preserving Discontinuous Galerkin methodwith a Second-order Implicit Pressure Explicit Concentration (SIPEC) time marchingmethod 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

Effects of viscous dissipation on miscible thermo-viscous fingering instability in porous media

The thermo-viscous fingering instability associated with miscible displacement through a porous medium is studied numerically, motivated by applications in upstream oil industries especially enhanced oil recovery (EOR) via wells using hot water flooding and steam flooding. The main innovative aspect of this study is the inclusion of the effects of viscous dissipation on thermal viscous fingering instability. An Arrhenius equation of state is employed for describing the dependency of viscosity on temperature. The normalized conservation equations are solved with the finite element computational fluid dynamics code, COMSOL (Version 5) in which glycerol is considered as the solute and water as the solvent and the two-phase Darcy model employed (which couples the study Darcy flow equation with the time-dependent convection-diffusion equation for the concentration). The progress of finger patterns is studied using concentration and temperature contours, transversely averaged profiles, mixing length and sweep efficiency. The sweep efficiency is a property widely used in industry to characterize how effective is displacement and it can be defined as the ratio of the volume of displaced fluid to the total volume of available fluid in a porous medium in the displacement process. The effects of Lewis number, Brinkman number and thermal lag coefficient on this instability are examined in detail. The results indicate that increasing viscous dissipation generates significant enhancement in the temperature and a marked reduction in viscosity especially in the displaced fluid (high viscous phase). Therefore, the mobility ratio is reduced, and the flow becomes more stable in the presence of viscous dissipation

Higher-order conservative interpolation between control-volume meshes: Application to advection and multiphase flow problems with dynamic mesh adaptivity

© 2016 .A general, higher-order, conservative and bounded interpolation for the dynamic and adaptive meshing of control-volume fields dual to continuous and discontinuous finite element representations is presented. Existing techniques such as node-wise interpolation are not conservative and do not readily generalise to discontinuous fields, whilst conservative methods such as Grandy interpolation are often too diffusive. The new method uses control-volume Galerkin projection to interpolate between control-volume fields. Bounded solutions are ensured by using a post-interpolation diffusive correction. Example applications of the method to interface capturing during advection and also to the modelling of multiphase porous media flow are presented to demonstrate the generality and robustness of the approach

The dynamics of miscible viscous fingering from onset to shutdown

We examine the full ‘life cycle’ of miscible viscous fingering from onset to shutdown with the aid of high-resolution numerical simulations. We study the injection of one fluid into a planar two-dimensional porous medium containing another, more viscous fluid. We find that the dynamics are distinguished by three regimes: an early-time linearly unstable regime, an intermediate-time nonlinear regime and a late-time single-finger exchange-flow regime. In the first regime, the flow can be linearly unstable to perturbations that grow exponentially. We identify, using linear stability theory and numerical simulations, a critical Péclet number below which the flow remains stable for all times. In the second regime, the flow is dominated by the nonlinear coalescence of fingers which form a mixing zone in which we observe that the convective mixing rate, characterized by a convective Nusselt number, exhibits power-law growth. In this second regime we derive a model for the transversely averaged concentration which shows good agreement with our numerical experiments and extends previous empirical models. Finally, we identify a new final exchange-flow regime in which a pair of counter-propagating diffusive fingers slow exponentially. We derive an analytic solution for this single-finger state which agrees well with numerical simulations. We demonstrate that the flow always evolves to this regime, irrespective of the viscosity ratio and Péclet number, in contrast to previous suggestions

Accounting for viscous fingering in relative permeability estimation of special core analysis measurements

Relative permeability (kr) is a critical input data for any calculation involving multiphase flow in petroleum reservoirs. Normally, kr curves are obtained by performing coreflood experiments as part of SCAL measurements or EOR studies. The results of the experiments are then used to obtain kr values often by either analytical models (e.g. JBN) or history matching techniques. Most of these models are based on the Buckley-Leverett displacement theory and are not applicable to unstable displacements. Therefore, using these models to describe a core flood experiment involving viscous fingering will result in potentially large errors in the estimation of kr curves. This study focused on the estimation of relative permeability curves for unstable experiments, more specifically in unfavourable mobility corefloods with a tendency to develop viscous fingering. Refined 2D coreflood simulations were used to evaluate the effect of viscous fingering in kr estimation methods. The simulations were performed as immiscible corefloods in homogeneous cores using a Black-oil model in a commercial simulator. The first part of this study, describes the methodology used to generate viscous fingering in numerical corefloods. Instability triggering methods were used with high resolution simulation to generate the viscous fingering. This methodology was then used to generate different numerical experiments with viscous fingering formation. In the second part, the currently widely used oil industry approaches for relative permeability estimation (1D history matching and JBN method) were evaluated for cases with unfavourable mobility. The errors were quantified in order to understand the effect of fingering on these methods and the amount of error one can incur when using them for these cases. In the latter part of the thesis, two novel methods are proposed for estimating relative permeabilities for unfavourable mobility coreflood experiments, namely viscous fingering. These methods are based on the proposed model called ‘stable equivalent model’. This model proposes a correction to the velocity of the fluids in a coreflood affected by viscous fingering, allowing to account for viscous fingering in relative permeability estimation. The model is used to modify the JBN method and 1D history matching, allowing these methods to tackle viscous instability. The integrity of these techniques was validated against published experimental data and numerical data

Numerical and physical simulations of the displacement of synthetic oil mixtures

Numerical simulation and experiments are often used to study the mixing phenomenon of the fluids during an unstable displacement process. One of the deficiencies of the conventional compositional reservoir simulators for predicting the unstable displacement process is that these simulators all involve the use of the assumption that fluids in each grid block are in a state of thermodynamic equilibrium. In reality, the different fluid phases coexisting in each grid block may not be in equilibrium with each other because of insufficient contact time. The main objective of this study is to develop a non-equilibrium phase behaviour model for compositional simulation of the unstable displacement process and is to verify the simulation results with experimental data. Physical simulations of the displacement process were carried out in a slim-tube apparatus. Four synthetic oil mixtures were used as displaced fluids and four gases were used as displacing fluids. A total of fifteen experiments were performed at displacement pressures ranging from 2390 psia to 3430 psia and with injection rates varying from 0.048 to 0.127 PV/hr. The results of the experiments are presented. A model has been developed to calculate the non-equilibrium phase behaviour of the fluids under displacement process conditions. The model is based on the mixing parameter model proposed by Todd and Longstaff (1972) and the concept of Murphree efficiency commonly used in multicomponent, multistage separation calculations. Phase behaviour calculations are performed for the fluids over the entire grid block under non-equilibrium conditions. The deviation from equilibrium in respect of each component is considered a function of the equilibrium K-value and the effective mobility ratio of the in-situ fluids. Efficient algorithms for phase behaviour calculations (e.g., flash calculations and saturation pressure calculations) generally used in the numerical simulations are presented. An acceleration scheme based on the dominant eigenvalue method coupled with Newton's method is developed for two-phase flash calculations. Effective switching criteria are suggested for the switch over of the acceleration scheme to Newton's method. The proposed method is robust and fast for flash calculations when the specifications are near the critical state values of the fluid mixtures. The performance of the proposed method is compared with those of other improved methods for flash calculations. An algorithm is developed to accelerate the convergence of phase-boundary calculations using Newton's method. The algorithm takes advantage of the history of the iterates and uses the derivative of the iterates to further improve the iterates after three Newton steps. The performance of the proposed algorithm shows its superiority over that of Newton's method particularly when the specifications are near the critical state values of the fluid mixtures. Comparisons of the performance of the proposed algorithm with that of Newton's method and other acceleration algorithms for Newton's method are presented. Comparisons of the numerical simulation results based on the proposed non-equilibrium phase behaviour model with the experimental data obtained from the slim-tube displacement tests for well-defined hydrocarbon systems and with simulation results based on conventional equilibrium phase behaviour model are presented

Pore-scale study on porous media flows with chemical reaction using lattice Boltzmann method

Porous media flows with chemical reaction are common in nature and widely exist in many scientific and industrial applications. However, due to the complexity of coupled mechanisms, numerical modelling and comprehensive understanding of such flows face significant challenges. Therefore, this thesis develops novel lattice Boltzmann (LB) models to undertake pore-scale simulations of porous media flows with chemical reaction. These models, with new reaction source terms and boundary schemes, can describe both homogeneous reaction between two fluids and heterogeneous reaction (dissolution or combustion) at the fluid-solid interface. Unlike previous studies, current models recast heat and mass transfer equations to correctly consider the thermal expansion effects and the conjugate heat transfer and species conservation conditions. Separate LB equations are also developed to include different species properties. Density fingering with homogeneous reaction is studied at the pore scale. By changing species contributions to density, diffusion coefficients, initial concentrations, and medium heterogeneities, results obtained demonstrate that reaction can enhance, suppress, or trigger fingering. Then, pore-scale simulations of viscous fingering with dissolution reaction are performed. Effects of fluid diffusion, chemical dissolution, and viscosity contrast are extensively assessed. Results illustrate four fingering regimes as stable, unstable, reactive stable, and reactive unstable. Finally, pore-scale coke combustion in porous media is studied. General combustion dynamics are correctly produced, verifying the superior performance of the present LB model over previous ones. A parametric study demonstrates that the inlet air temperature and the driving force are influential factors and should be constrained within certain ranges for stable combustion fronts. These pore-scale findings provide valuable insights, like temperature fluctuations at the fluid-solid interface, porous structure evolutions, exact reaction and diffusion rates, and medium heterogeneity effects, which are more precise and explicit than macroscopic results. Furthermore, detailed fingering and combustion dynamics under diverse conditions are helpful in scientific and industrial fields