Analytical and numerical modelling of complex multiphase flow processes in geological environments

Abstract

Most geological flow and transport models describe the flow and convective-dispersive spreading of one or more components entirely dissolved in the most common geological fluid, water. However, many other processes—such as extraction of petroleum products, groundwater contamination by toxic oils, secondary migration of hydrocarbons, migration of CO? and other seasonal gases in storage reservoirs, steam migration in geothermal reservoirs, and contamination of coastal regions by seawater and toxic oils—are examples that need to be treated as multiphase flow processes. Multiphase flow and transport, unlike miscible transport, require additional considerations of interface dynamics. Basic mathematical difficulties in both numerical and analytical solutions of multiphase flow arise from the strong non-linear coupling of flow models. Such systems do not admit exact solutions, which necessitates analysis of flow equations using self-similar or travelling wave approaches. Solutions of non-linear saturation transport problems lead to the formation of shock fronts and steep gradients in the saturation profile. The present study is primarily concerned with the analytical and numerical modeling of multiphase processes in geological environments. The fractional flow approach, which leads to a fully non-linear saturation transport equation, is adopted to develop self-similar and travelling wave solutions of immiscible displacement. The transport equation for viscous flow in the absence of capillary and gravity forces represents the classical Buckley-Leverett equation. In general, the presence of gravity and capillarity may develop counter-current flow systems. The classical approach treats capillarity as a diffusive process with infinite velocity of propagation, whereas the non-classical approach considers capillarity as responsible for advection of the interface, with a finite capillary velocity. In this context, the present study assumes a constant capillary pressure gradient within the transition zone, and the meniscus concentration is defined by the viscosity ratio and relative permeability function. This assumption leads to a first-order hyperbolic equation even in the presence of capillarity. Since the classical Buckley-Leverett approach does not provide a solution for counter-current systems, a modified Welge-Tangent method, based on hyperbolic theory, produces a discontinuous solution. These solutions are in the form of a sequence of rarefaction waves in the transition zone and shock waves at the tails of the transition zone. Physically, shock waves represent immobile zones, whereas rarefaction waves represent mobile zones. In the classical approach, where capillarity appears as a parabolic term, the flow equation can still be analyzed using a generalized travelling wave approach. Travelling wave solutions of this transient convective-diffusive equation provide an exact expression for the saturation gradient in the flow field. However, the explicit expression for saturation either requires numerical treatment or analysis in sequences of saturation zones. Analytical modeling provides detailed characteristics of immiscible displacement. Numerical modeling is essential to understand immiscible displacement in two or higher dimensions under generalized conditions. To study multiphase processes under generalized conditions, two-dimensional numerical models of two- and three-phase flow, miscible-immiscible flow, and non-isothermal flow have been developed. Numerical modeling of multiphase flow is computationally intensive and highly sensitive to numerical methodology. In this study, flow models are solved using finite difference schemes. The numerical model is sensitive to coefficient approximation; in this study, the harmonic mean of intrinsic permeability with upstream weighting of phase mobility is used to provide stable results for multiphase flow models. Miscible and non-isothermal models are solved using finite volume approximation, which ensures better conservative estimates due to its integral approach. A hybrid scheme is used as the coefficient approximation for transport models. The hybrid scheme is marginally more accurate than the upwind scheme, as a second-order central difference scheme is applied in regions of low Peclet number. A second-order central difference scheme is used in regions of low Peclet number (Pe). In transport models, apart from their respective primary variables (mass fractions and temperature), multiphase velocities have been provided from flow models by coupling the models using a sequential approach. The basic theme of the thesis is to understand and analyze multiphase processes in geological systems and then study these processes in some specific systems, such as seawater–freshwater, seawater–freshwater–oil, CO?–water, and DNAPL–water. In a coastal aquifer with a large freshwater lens compared to the transition zone, seawater intrusion is modeled as two-phase flow. If a freshwater–seawater system encounters an oil (DNAPL) spillage, which is a frequent occurrence in coastal zones, the process is modeled as immiscible three-phase flow. In a CO? storage reservoir, immiscible displacement of formation water by CO? is the dominating process for short-term analysis. Contaminants like DNAPL flow in a fracture with deformation, which is again a multiphase process. The fracture deformation, in turn, affects multiphase flow and transport processes. Some of these issues are addressed and discussed through analytical and numerical models developed in this study. Thus, to achieve this, the thesis is organized in the following chapters, summarized below: Chapter 1 presents a general overview of multiphase flow in geological environments. A comprehensive literature review on modeling of immiscible flow is discussed. A discussion on partially miscible flow and non-isothermal multiphase flow is presented. Then a brief background of multiphase flow through a fracture under deformation is discussed. Complexities of analytical and numerical approaches for solving multiphase flow equations are addressed. Finally, the organization of the thesis is given in brief at the end of this chapter. Chapter 2 presents the fundamentals of multiphase flow in geological environments. The pore-scale and macro-scale parameters defining capillary pressure and relative permeability functions are discussed. Then the governing multiphase processes under basic driving forces are elaborated Forces are discussed. The conceptual models of multiphase systems are also presented. Chapter 3 presents the pressure–saturation approach for immiscible flows through fractures and porous media. Numerical models are developed for immiscible flow through fractures and porous media. Computational algorithms are sensitive to the way equations are formulated and the choice of primary variables adopted. Many approaches for choosing the formulation and primary variables have been reported and studied in the past. In this study, we are not interested in such comparative studies; rather, we adopt the best choice of wetting phase saturation and wetting phase pressure as primary variables, as reported in the literature. The harmonic mean of intrinsic permeability with upstream weighting of phase mobility, which provides stable results for multiphase flow systems, is used in the flow models as a coefficient approximation. Picard’s method is used to linearize the discretized flow equations. To study interface miscibility, miscible and immiscible fractions are considered as components of a phase. Mathematically, two balance equations for two phases, each having two components, represent the miscible model, finally resulting in four unknowns, which are supplemented by two complementary conditions representing mass fraction constraints. To study non-isothermal displacement in the case of unequal thermal states of displacing and displaced phases, local thermal equilibrium is established by energy exchange across the interface. Mathematically, a single equation representing energy balance over the phases is considered under the assumption of local equilibrium, which is represented by a single temperature over the entire domain. Phase velocities are provided by flow models. Mass and energy transport equations are solved using the integral finite volume approach. Coupling of transport models with flow models is achieved using a sequential approach with an “n”-step fully implicit iterative method. Unknown matrices for flow and transport models are solved using the Incomplete Cholesky Conjugate Gradient method. Selection of spatial and temporal steps in the flow models satisfies the Courant number and tolerance criteria set in the model to ensure numerical stability, and the Peclet number condition is satisfied for transport models. discusses the fractional flow approach to analyze immiscible displacement of two phases. The characteristics and significance of specific flow functions are discussed. The non-dimensional form of the saturation transport equation is presented, and three non-dimensional numbers, namely the Capillary number, Gravity number, and Bond number, are discussed in this context. Two-phase viscous flow in the absence of capillarity and gravity is solved using the classical Buckley–Leverett approach. A simple analysis using a self-similar approach to study drainage and imbibition processes during immiscible displacement is proposed. We suggest an analogy between classical and non-classical capillarity. This work is novel in the sense that this analogy is able to define the non-equilibrium meniscus concentration function in terms of relative permeability and viscosity ratio. For this analogy, we have considered a constant capillary pressure gradient within the transition zone. Basically, capillarity assumes a direction, in contrast to the scalar capillarity in the classical approach. Another important aspect of this approach is the finite capillary flow velocity, in contrast to the infinite capillary flow in the classical equilibrium approach. Viscous or buoyancy-driven flow with capillarity may develop counter-current spontaneous imbibition and advective transport. The displacement fronts show stretching of waves with multiplicity in the formal solution. The multiplicity is resolved by a modified Welge-Tangent method, which leads to drainage and imbibition fronts. We also discuss traveling wave solutions for classical diffusive capillarity, but this approach also needs to be defined in sequences of saturation zones. Thus, multiphase displacement can be analyzed in sequences of saturation zones defined by self-similar or traveling waves. Such direction-based capillarity is a suitable approach in most geological environments, especially when it encounters very low-permeable thin layers where capillarity drives the interface. Chapter 5 presents immiscible flow in coastal reservoirs. Two-phase flow of seawater and freshwater is discussed through the developed numerical model. Accidental spillage of oil (DNAPL) in the seawater–freshwater zone of a coastal aquifer is modeled as three-phase flow. Existing studies of three-phase flow are mainly conducted in the unsaturated zone with one phase as gas; in contrast, this study considers a system where all the phases are liquids The phases are liquids, which are highly sensitive and demonstrate non-equilibrium processes such as fingering and local blob formation. In this context, we develop numerical modeling for oil (DNAPL) spillage in the freshwater–seawater zone as a three-phase immiscible flow. The interface mechanism depends on the wetting preference of the phases and is defined in terms of capillary pressure and relative permeability. However, in the absence of sufficient retention laws for the class of system considered here, we investigate the study using linear capillary pressure and non-linear relative permeability functions. Based on the two-phase theory, the present study assumes oil as the most non-wetting phase and seawater as the intermediate phase. Most interestingly, the existing seawater–freshwater interface acts like a capillary barrier, or in other words, poses a heterogeneity condition to the invading oil, giving rise to fingering of oil in the three-phase region. Viscous fingering occurs in the high-saturation zones of oil, along with the development of local blobs of oil. Fingering depends on the type of capillary function, relative permeability function, density, and viscosity of oil. In the case of relative permeability functions independent of saturations of other interfering phases, no fingering develops. At low saturation, oil is unable to pass through the top of the wedge, where it pools and migrates downward in some regions, whereas stretching of oil along the seawater–freshwater capillary wedge is observed. Sensitivity studies with density and viscosity of oil demonstrate the occurrence of fingering. This study can be applied in several other geological processes, especially in CO? sequestration, where migration may occur in formations with existing interfaces between varying salinity. Chapter 6 presents the study of immiscible displacement of formation water by injected CO? in a deep reservoir. First, the analytical model is used to study viscous migration of CO? during injection. Then, the decelerating effect of capillary flow is demonstrated, showing how it can arrest buoyant CO? migration and slow the advance under forced injection. The positive front, referring to buoyant CO? migrating vertically upward, becomes a negative front at low Bond numbers. Viscous injection demonstrates a unidirectional single front for large Capillary number flows, whereas small Capillary number flows show double fronts. The results show that this approach can explain co-current and counter-current flow, which in other words defines the imbibition and drainage processes during immiscible displacement. For two-dimensional studies, the developed numerical model is used for short-term migration analysis of CO?. The effect of variable permeability fields on flow processes is studied to demonstrate the development of fingering. The effects of viscosity ratio, relative permeability, and capillarity on the development of viscous fingering are studied qualitatively. For non-linear capillary functions, CO? migration is slower compared to linear capillary functions, whereas fingering is more pronounced in the non-linear case. This study is performed to show the sensitivity of different parameters on fingering of CO? qualitatively. Chapter 7 presents the study of multiphase flow in a fracture with deformation. This chapter focuses on demonstrating immiscible displacement in a fracture under confining stresses and multiphase fluid pressures, and its effect on the travel time of the displacing phase. Sensitivity analyses for physical properties such as fracture aperture and fracture inclination are discussed. Temporal evolution of aperture is obtained, which affects the flow pattern of fluids within a fracture. These studies are relevant for non-wetting phase fluids trapped in fractures with very small apertures, which bring changes to the flow pattern when exposed to deformation. Mass and energy transport profiles are also seen to change due to deformation. This case study is performed under mild conditions, meaning that the fracture lies in a shallow reservoir with small normal stresses and fluid pressures. In the case of fractures in deep reservoirs under high injection pressures, this study becomes even more relevant, as fractures may be exposed to higher external stresses and multiphase fluid pressures. Chapter 8 presents the major conclusions drawn from this study and discusses further studies needed.