Helge K. Dahle

Dynamic Effect in the Capillary Pressure–Saturation Relationship and its Impacts on Unsaturated Flow

Capillary pressure plays a central role in the description of water flow in unsaturated soils. While capillarity is ubiquitous in unsaturated analyses, the theoretical basis and practical implications of capillarity in soils remain poorly understood. In most traditional treatments of capillary pressure, it is defined as the difference between pressures of phases, in this case air and water, and is assumed to be a function of saturation. Recent theories have indicated that capillary pressure should be given a more general thermodynamic definition, and its functional dependence should be generalized to include dynamic effects. Experimental evidence has slowly accumulated in the past decades to support a more general description of capillary pressure that includes dynamic effects. A review of these experiments shows that the coefficient arising in the theoretical analysis can be estimated from the reported data. The calculated values range from 10 4 to 10 7 kg (m s) −1 . In addition, recently developed pore-scale models that simulate interface dynamics within a network of pores can also be used to estimate the appropriate dynamic coefficients. Analyses of experiments reported in the literature, and of simulations based on pore-scale models, indicate a range of dynamic coefficients that spans about three orders of magnitude. To examine whether these coefficients have any practical effects on larger-scale problems, continuum-scale simulators may be constructed in which the dynamic effects are included. These simulators may then be run to determine the range of coefficients for which discernable effects occur. Results from such simulations indicate that measured values of dynamic coefficients are within one order of magnitude of those values that produce significant effects in field simulations. This indicates that dynamic effects may be important for some field situations, and numerical simulators for unsaturated flow should generally include the additional term(s) associated with dynamic capillary pressure.

Non-equilibrium effects in capillarity and interfacial area in two-phase flow: dynamic pore-network modelling

Current macroscopic theories of two-phase flow in porous media are based on the extended Darcys law and an algebraic relationship between capillary pressure and saturation. Both of these equations have been challenged in recent years, primarily based on theoretical works using a thermodynamic approach, which have led to new governing equations for two-phase flow in porous media. In these equations, new terms appear related to the fluid–fluid interfacial area and non-equilibrium capillarity effects. Although there has been a growing number of experimental works aimed at investigating the new equations, a full study of their significance has been difficult as some quantities are hard to measure and experiments are costly and time-consuming. In this regard, pore-scale computational tools can play a valuable role. In this paper, we develop a new dynamic pore-network simulator for two-phase flow in porous media, called DYPOSIT. Using this tool, we investigate macroscopic relationships among average capillary pressure, average phase pressures, saturation and specific interfacial area. We provide evidence that at macroscale, average capillary pressure–saturation–interfacial area points fall on a single surface regardless of flow conditions and fluid properties. We demonstrate that the traditional capillary pressure–saturation relationship is not valid under dynamic conditions, as predicted by the theory. Instead, one has to employ the non-equilibrium capillary theory, according to which the fluids pressure difference is a function of the time rate of saturation change. We study the behaviour of non-equilibrium capillarity coefficient, specific interfacial area, and its production rate versus saturation and viscosity ratio. A major feature of our pore-network model is a new computational algorithm, which considers capillary diffusion. Pressure field is calculated for each fluid separately, and saturation is computed in a semi-implicit way. This provides more numerical stability, compared with previous models, especially for unfavourable viscosity ratios and small capillary number values.

Quantitative estimation of CO2 leakage from geological storage: Analytical models, numerical models, and data needs

Publisher Summary This chapter focuses on development of large-scale modeling tools to quantify potential CO2 leakage along existing wells. Geological storage of CO2 is emerging as one of the most promising options for carbon mitigation. While this approach appears to be technically feasible, a comprehensive risk assessment is required to determine the overall effectiveness and possible environmental consequences of this approach. One important part of such a risk assessment is an analysis of potential leakage of injected CO2 from the formation into which is injected, to other permeable formations or to the atmosphere. Such leakage is a concern because it may contaminate existing energy, mineral, and/or groundwater resources, it may pose a hazard at the ground surface, and it will contribute to increased concentrations of CO2 in the atmosphere.

Eulerian-Lagrangian Localized Adjoint Methods for a Nonlinear Advection-Diffusion Equation

Eulerian-Lagrangian localized adjoint methods (ELLAM) are developed for the non-linear Buckley-Leverett equation, which is characterized by degenerate diffusion and sharpening near-shock solutions. The ELLAM methodology employs space-time finite elements with edges oriented along flow paths, and space-time test functions that satisfy a local adjoint condition. This combination extends Eulerian-Lagrangian concepts in a systematic mass-conservative fashion to problems with general boundary conditions. Various kinds of boundary conditions are considered, and a local time-stepping procedure is developed for a no-flow outlet condition that leads to a boundary layer. Numerical experiments illustrate the potential of these methods.

A dynamic network model for two‐phase immiscible flow

A dynamic pore‐scale network model is formulated for two‐phase immiscible flow. Interfaces are tracked through the pore throats using a modified Poiseuille equation, whereas special displacement rules are used at the pore bodies. The model allows interfaces to move over several pore‐lengths within a time step. Initial computational results are presented for a drainage experiment to demonstrate some of the features of the model.

A pore network model for calculation of interfacial velocities

Abstract Two-phase flow in porous media is characterized by fluid–fluid interfaces that separate fluid phases at the pore scale. These interfaces support pressure differences between phases, and their dynamics lead to changes in phase saturation within the porous medium. Dynamic pore-scale network models mathematically track the dynamic position of each fluid–fluid interface through a pore network, based on imposed boundary conditions, fluid and solid properties, and geometric characteristics of the network. Because these models produce a detailed description of both phase and interface dynamics, results from these models can be volume-averaged to provide values for many upscaled variables. These include traditional variables such as saturation and macroscopic capillary pressure, as well as non-traditional variables such as amount of interfacial area in the averaging volume. With appropriate geometric definitions in the dynamic pore-scale model, a new algorithm may be included in the pore-scale network model to calculate a new variable: average interfacial velocity. This algorithm uses local information in any pore that contains a fluid–fluid interface to estimate the velocity of that interface over a time step. Summation over all interfaces in the network provides a measure of average velocity. Computations for dynamic drainage experiments indicate that this average interfacial velocity is well defined and exhibits distinct behavior for stable and unstable displacements. Comparison of calculated interfacial velocities to a theoretical conjecture on the functional dependence of this macroscopic variable demonstrates another important use of pore-scale model, namely testing of new theories involving non-traditional variables.

The corrected operator splitting approach applied to a nonlinear advection-diffusion problem

So-called corrected operator splitting methods are applied to a 1-D scalar advection-diffusion equation of Buckley-Leverett type with general initial data. Front tracking and a 2nd order Godunov method are used to advance the solution in time. Diffusion is modelled by piece wise linear finite elements at each new time level. To obtain correct structure of shock fronts independently of the size of the time step, a dynamically defined residual flux term is grouped with diffusion. Different test problems are considered, and the methods are compared with respect to accuracy and runtime. Finally, we extend the corrected operator splitting to 2-D equations by means of dimensional splitting, and we apply it to a Buckley-Leverett type problem including gravitational effects.

Eulerian-Lagrangian localized adjoint methods for systems of nonlinear advective-diffusive-reactive transport equations

Abstract The contamination of groundwater by various hazardous materials has emerged as a primary environmental issue. The pollution of oil reservoirs is a closely related problem in that microorganisms are involved in the contaminant process. The mathematical models that describe these phenomena involve a set of nonlinear advective-diffusive-reactive transport equations, which may involve reactions with all the species and are themselves coupled to growth equations for the subsurface bacterial population. In this article, we discuss and compare different mathematical models, present Eulerian-Lagrangian localized adjoint methods (ELLAM) and combine them with specific linearization techniques to solve these nonlinear transport systems. The derived numerical schemes systematically adapt to the changing features of governing equations. The relative importance of advection, diffusion and reaction is directly incorporated into the schemes by judicious choice of the test functions in the variational formulations. Numerical experiments are presented to show the potential of these methods.

A pore-scale network approach to investigate dynamic effects in multiphase flow

Current theories of multiphase flow rely on capillary pressure and saturationships that are traditionally measured under static conditions. To make the description of multiphase flow more complete, new multiphase flow theories have been proposed that include an extended capillary pressure-saturation relationship that is valid under dynamic dynamic capillary pressure, and is assumed to be a function of the saturation and its time rate of change. In this work, this relationship is investigated using a pore-scale network model. This model consists of a three-dimensional network of tubes (pore throats) connected to each other by pore bodies. The pore bodies are spheres and pore throats are cylinders. Numerical experiments are performed wherein typical experimental procedures for both static and dynamic measurements of capillary pressure-saturation curves are simulated. From these, dynamic coefficient τ is found to be a function of wetting fluid saturation.

Vertically integrated models with coupled thermal processes

CO2 storage in geological formations involves coupled processes that affect the migration and ultimate fate of injected CO2 over multiple length and time scales. For example, coupling of thermal and mechanical process has implications for storage security, including thermally induced fracturing and loss of caprock integrity in the near wellbore environment. This may occur when CO2 is injected at a different temperature than reservoir conditions, e.g. Snohvit injection, potentially leading to large temperature, density and volume changes within the plume over space and time. In addition, thermally induced density changes also impacts plume buoyancy that may affect large-scale migration patterns in gravity-driven systems such as Utsira storage site. This interaction becomes particularly important at temperatures and pressures near the critical point. Therefore, coupling thermal processes with fluid flow should be considered in order to correctly capture plume migration and trapping within the reservoir. A practical modeling approach for CO2 storage at the field scale is the vertical-equilibrium (VE) model, which solves partially integrated conservation equations for flow in two lateral dimensions. This class of models is well suited for strongly segregated flows, as can be the case for CO2 injection. In this paper, we extend the classical VE model to non-isothermal systems by vertically integrating the coupled heat transport equations, focusing on the thermal processes that most impact the CO2 plume. The model allows for heat exchange between the CO2 plume and the surrounding environment assuming thermal equilibrium across the plume thickness for relatively thin plumes. We investigate the validity of simplifying assumptions required to reconstruct the fine-scale thermal structure from the coarse-scale model solution. The model concept is verified for relatively simple systems. The results of this work demonstrate the potential for reduced models to advance our understanding of the impact of thermal processes in realistic storage systems.