Non-linearities and upscaling in porous media

Transcription

1 Non-linearities and upscaling in porous media Master s Thesis Two phase flow in homogeneous porous media - The role of dynamic capillary pressure in modeling gravity driven fingering David Kraus Supervisors: Rainer Helmig Philipp Nuske Yufei Cao Preprint 2011/3

2 GRK 1398/1 (DFG) DN (NWO) - Geschäftsstelle - Pfaffenwaldring Stuttgart Telefon: 0711/ Telefax: 0711/ maria.costa@iws.uni-stuttgart.de

3 I hereby certify that I have prepared this thesis independently, and that only those sources, aids and advisors that are duly noted herein have been used and / or consulted. Signature Date

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5 Acknowledgement We thank the German Research Foundation (DFG) for the funding within the International Research Training Group Non-Linearities and Upscaling in Porous Media (NUPUS).

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7 Thanks for nice discussions and help to Philipp, Rainer, Yufei, Andreas, Anozie, Bernd, Alexander, Frederike,... Thanks for proofread to Henny, Philipp, Stefan, Jotta. Thanks for everything else to my family

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9 Contents Contents Nomenclatures ii vii 1 Introduction Motivation Structure of the thesis Physical and mathematical model for classical two-phase flow Porous media system Thermodynamic equilibrium Scales Porosity Phases Material properties Density Viscosity Capillary pressure Permeability Classical multiphase flow models Conservation of mass Darcy s law for single and multiphase flow Multiphase flow equations Physical and mathematical model for gravity driven fingering Instable displacement phenomena Fluid movement on the microscale Hysteretic effects Microscale considerations Macroscale models for hysteresis Dynamic effects Microscale considerations Macroscale model for p c S relationship after Stauffer Macroscale model for p c S relationship after Kalaydjian Macroscale model for p c S relationship after Hassanizadeh and Gray Macroscale model for k r /p c S relationship after Barenblatt Gravity driven fingering Initialization of gravity driven fingering Saturation overshoot Stabilization of gravity driven fingering Further preferential flow mechanisms Numerical models Finite differences approximation ii

10 4.2 Fully-implicite euler discretization for coupled two phase flow formulation Semi-implicite euler discretization for 1-D saturation formulation Solution Algorithm Numerical simulations with extended two-phase model D Simulations Comparison simulation with Manthey Grid Convergence verification Neumann inflow boundaries Neumann and Dirichlet inflow boundaries Influence of initial water saturation DiCarlo experiment Second order time discretization Comparison of numerical models D Simulations Summary of simulation results Conclusion 98 Literaturverzeichnis 100 iii

11 List of Figures 2.1 Illustration of different length scales including representative elementary volume scale (REV) Different types of packing for porous media Illustration of the interface between two liquid phases Laminar flow field of newtonian fluids Illustration of surface tension for fluid bubbles Illustration of capillary pressure in narrow tubes Bundle of tubes model for macroscale p c S models Van Genuchten and Brooks-Corey p c - S parameterization curves Illustration of tortuosity leading to enlarged flow paths Relative permeability parameterization Illustration of material volume Stable displacement front compared to an unstable displacement front Illustration of vertical displacement process Illustration of displacement process with different wave numbers Illustration of infiltration fronts for classical infiltration and gravity driven fingering Illustration of the jumping behaviour of infiltration processes Illustration of film flow in porous media Contact angle for imbibition and drainage processes Illustration of anisotropic porous medium Illustration of inkbottle effect Illustration of dead-end pore causing S w,r Phase trapping due to snap-off and bypassing effects Macroscale hysteresis models: Parker and Lenhard and play type approach Pressure drop and inversion of interface curvature for dynamic infiltration process (taken from Weitz et al., 1987 [70]) Interfacial behaviour for different capillary numbers Pore redistribution process for fast infiltration process Reduced relative permeability for transient drainage processes Linear relationship for retardation coefficient after Hassanizadeh and Gray, 1993 [31] Illustration of saturation overshoot profile Longitudinal profile of a finger with different saturation regions Illustration of BOX method discretization Discretized domain for finite differences scheme Illustration of Newton method Implementation of play-type hysteresis model General boundary conditions for 1-D simulations Boundary and initial conditions for numerical simulation example after Manthey, 2006 [42] iv

12 5.3 Simulation results from Manthey, 2006 [42]. Saturation profile plotted over time Comparison simulation for numerical simulation example after Manthey, 2006 [42]. Saturation profile plotted over time Linear approximating initial conditions for Neumann/Dirichlet boundary conditions Look at particular grid points for grid convergence test simulations Boundary and initial conditions for first grid convergence test simulation with two Neumann boundary conditions Simulation results for grid convergence test with two Neumann boundary conditions. Saturation profile plotted over length Boundary and initial conditions for second grid convergence test simulation with one Neumann boundary condition and one Dirichlet boundary condition Simulation results for grid convergence test with Neumann/Dirichlet boundary conditions. Saturation profile plotted over length (1) Simulation results for grid convergence test with one Dirichlet and one Neumann boundary condition. Saturation profile plotted over length (2) Simulation results for two Neumann boundary conditions at the inflow boundary (setup 1) Simulation results for two Neumann boundary conditions at the inflow boundary (setup 2) Simulation results for two Neumann boundary conditions at the inflow boundary (setup 3) Pressure profile for two Neumann boundary conditions without accounting for gravity Simulation results for two Neumann boundary conditions at the inflow boundary (setup 4) Simulation results for Neumann/Dirichlet boundary conditions at the inflow boundary (1) Simulation results for Neumann/Dirichlet boundary conditions at the inflow boundary (2) Simulation results for Neumann/Dirichlet boundary conditions at the inflow boundary (3) Simulation results for Neumann/Dirichlet boundary conditions at the inflow boundary (4) Simulation results for Neumann/Dirichlet boundary conditions with classical p c S model to improve comparison Simulation results for Neumann/Dirichlet boundary conditions at the inflow boundary neglecting gravity (1) Simulation results for Neumann/Dirichlet boundary conditions at the inflow boundary neglecting gravity (2) Simulation results for different initial water saturations Comparison between water content overshoot size and initial water content for experimental as well as simulated data Quantification of the saturation overshoot by experimental results. Experiment conducted by DiCarlo, Results for the numerical simulation of the experimental quantification of the saturation overshoot from DiCarlo, Simulation results for the experiment after DiCarlo, 2004 [11] (1) Simulation results for the experiment after DiCarlo, 2004 [11] (2) v

13 5.30 Simulation results for the comparison of first and second order time discretization Boundary and initial conditions for the comparison simulations of different numerical models Simulation results for saturation equation compared to the fully coupled pressuresaturation formulation Van Genuchten parameterization curves for play type hysteresis model Boundary and initial conditions for numerical simulation in 2-D domain Simulation results for the 2-D simulation without implemented play type hysteresis model Simulation results for the 2-D simulation including the play type hysteresis model. 95 vi

14 List of Tables 2.1 Types of thermodynamic equilibrium states Material and fluid properties for numerical simulation example after Manthey, 2006 [42] Material and fluid properties after DiCarlo, Setup 1 for numerical simulation with two Neumann boundary conditions Setup 2 for numerical simulation with two Neumann boundary conditions Setup 3 for numerical simulation with two Neumann boundary conditions Setup 4 for numerical simulation with two Neumann boundary conditions Setup 1 for numerical simulation with Neumann/Dirichlet boundary conditions Setup for numerical simulation with Neumann/Dirichlet boundary conditions neglecting gravity Setup for numerical simulation with Neumann/Dirichlet boundary conditions for numerical simulations concernig different initial water saturations Corresponding Sw,b co in numerical simulation for infiltration rates after DiCarlo, Setup for numerical simulation with Neumann/Dirichlet boundary conditions for numerical simulations concernig second order time discretization Parameters for play type hysteresis model vii

15 Nomenclatures Symbol Definition Dimension Greek letters: Increment [ ] Γ General boundary [ ] Λ Entropy [ J /K] Φ General unknown [ ] Ω General area [ ] Θ c Contact angle [grad, ] Θ Approximation error [ ] α Van Genuchten parameter [ 1 /Pa] α S Stauffer scaling parameter [m s] β Compressibility/expansion coefficient [ ] β S Stauffer scaling parameter [ ] γ Van Genuchten parameter [ ] δ Kronecker-delta [ ] η Specific entropy J [ kg K ] ε Van Genuchten parameter [ ] ε General residuum [ ] ɛ Residual variance [ ] λ Brooks-Corey parameter [ ] λ Mobility [ m s /kg] λ Fractional flow coefficient [ ] µ Dynamic fluid viscosity [Pa s] ν Kinematic fluid viscosity [ m2 /s] π Mathematical constant (3,14159) [ ] ψ Pressure head [m] ψ Total potential [P a] ϱ Density [ kg /m 3 ] φ Porosity [ ] σ Surface tension [ N /m] θ Water content [ ] τ Retardation coefficient after Hassanizadeh and Gray [Pa s] τ K Retardation coefficient after Kalaydjian [Pa s] τ S Retardation coefficient after Stauffer [Pa s] τ Shear stress [N m 2 ] viii

16 Symbol Definition Dimension Latin letters: A Area [m 2 ] A Free Helmholtz energy [J] B General boundary of box [ ] Bo Bond number [ ] Ca Capillary number [ ] E Energy [J] F General forces dependent on the context [N] F c Capillary forces [N] F v Viscous forces [N] K Intrinsic permeability [m 2 ] K f Hydraulic conductivity [ m /s] K f,u Unsaturated hydraulic conductivity [ m /s] N Ansatz function [ ] R i Individual gas constant [ J /kg K] R Universal gas constant [ J /mol K] Re Reynolds number [ ] S Saturation [ ] T Temperature [K] U Surface [m 2 ] V Volume [m 3 ] W Weight function [ ] a Acceleration [ m /s 2 ] b General area of box [ ] e Intensive fluid property [ ] e Extensive fluid property [ ] f Fractional flow coefficient [ ] f Body force [ ] f f Specific frictional force [ kg /m 2 s 2 ] f g Specific gravitational force [ kg /m 2 s 2 ] f p Specific pressure force [ kg /m 2 s 2 ] g Gravity acceleration [ m /s 2 ] h Piezometric or hydraulic head [m] i Diffusion term [ ] k r Relative permeability [ ] l Length scale [m] m Mass [kg] m Van Genuchten parameter [ ] n Van Genuchten parameter [ ] p Pressure [Pa] p atm Atmospheric pressure [Pa] p en Entry pressure [Pa] r Radius [m] t Time [s] t r Characteristic time after Barenblatt [s] u Specific internal energy [ J /kg] v Velocity [ m /s] x, y, z Coordinates [m ] ix

17 Subscripts: α αβ β b c e h i j k n l l lg p p s s sg sl t u w Phase, either wetting (w) or non-wetting (n) Interface between phase α and β Phase, either wetting (w) or non-wetting (n) Boundary Capillary Effective Hydrostatic Initial Index Index Non-wetting phase Liquid Part of subcontrol volume boundary Liquid/gas Peak Pore Seepage Solid Solid/gas Solid/liquid Total Unsaturated Wetting phase Superscripts: T co cr d e n li Transpose Constant Critical Dynamic Equilibrium Time step Limit Abbreviations: AV BC MDC MIC IC RE REV Average volume Boundary condition Main drainage curve for Van Genuchte parameterization Main imbibition curve for Van Genuchte parameterization Initial condition Richards equation Representative elementary volume Tensorial as well as vectorial quantities are written in bold symbols. x

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19 1 Introduction 1.1 Motivation The study of infiltration processes in porous media multiphase flow systems involves divers applications in environmental and technical applications such as the recharge of natural groundwater aquifers, CO 2 injection into geological formations or enhanced oil recovery. The peculiarities and problems that emerge by describing such matters are manifold. In recent decades, many scholars have focused their attention on the so-called fingering phenomenon. This phenomenon owes his name to the fact that under certain circumstances, the infiltration front of liquids is not plane but exhibits preferential flow paths, which propagate much faster into the domain than homogeneous infiltration fronts would predicted it. Fingering phenomena can be subdivided into viscous fingering and gravity driven fingering. In accordance with the names, the difference between these two types is mainly given by the participating forces. It has been shown that especially in gravity driven fingering, many highly complex flow processes take place, which could up to now not fully be explained. Therefore, the investigations of this work mainly are focused on a better understanding of this case. Gravity driven fingering is a subcategory of general preferential flow and refers primarily to homogeneous porous media. Further preferential flow mechanisms are given by flow through macropores and redirections caused by heterogeneity effects. Hydrologists are highly interested in these flow phenomena, since they play an important role in explaining everyday occurences. For example, infiltrating surface water will reach the groundwater table faster and also with contaminants in a higher concentration, because they are less adsorbed on the soil matrix. This is the reason, why this work will mainly consider water/air systems with a special focus on the infiltration of water into air filled porous media. For gaining a better understanding of such infiltration processes and improving their predictability, we will discuss the development and improvement of so-called porous media multiphase flow models. Such models are commonly based on conservation of mass equations for the participating fluids of the flow scenario combined with the description of the respective flow velocities. An expression for a representative capillary pressure of the considered system serves us as essential constitutive relation. It has been already shown by several scholars that these models in their classical form are not sufficiently accurate to describe the above mentioned fingering phenomena for all relevant cases. In particular for gravity driven fingering these model are incapable of representing the respective flow behaviour. Hence, endeavours have been made to obtain new models or model extensions, which in turn focus primarily on the improvement of the representative capillary pressure model concepts. Most of them see the major weak point of the model in the assumption of so-called local equilibria. This means that inside a certain control volume, there is a state of thermodynamic equilibrium adjusted instantaneously. As a consequence, the model extension applied in this work will address precisely this point, in order to account for the dynamics of the system by the introduction of the so-called dynamic capillary pressuresaturation relationship. This relationship manifests itself in a new term that is attached into 1

20 the otherwise unchanged classical multiphase flow model. This term further will consist of a third order mixed derivative with respect to space and time and has the property of changing the behaviour of the classical multiphase flow models particularly at steep fronts between the moving fluids in the respective porous media system. The extended model is based on fundamental thermodynamics and has been derived from first principles. The most apparent difference to the classical models is the consideration of the interfaces between the fluids among themselves and among the solid matrix of the porous medium. Scientists from a variety of disciplines including experimental work, physics, mathematics and numerics are still dealing with this extended dynamic model and seek for further improvements. When testing the model extension, this work will primarily focus on 1-D infiltration processes of water into a porous medium that is almost completely filled with air. The whole system will be further assumed to be isothermal and both fluids are considered to be incompressible. Beyond that, also the viscosities and the porosity will be applied as constants. By this means, we can exclude a variety of interacting influences and analyze the dependencies of the flow behaviour with regard to the newly implemented term only. In terms of subsequent practical applications of dynamic multiphase flow models, it is expected to extend the present knowledge of the mathematical and numerical behaviour of the here applied model and thus, contribute to a better understanding and assessing of its physical relevance and correctness. 1.2 Structure of the thesis This work consists basically of 6 main Sections. Subsequent to these introductory words, Sec. 2 gives an overview on the most commonly used physical and mathematical expressions and model concepts relating to multiphase flow in porous media in order to give the reader a basic understanding of the phenomenon. Special emphasize will be given to the two model concepts for the so-called relative permeability and the aforementioned representative capillary pressure. By introducing such physical quantities, we will try to give a detailed derivation and illustrative explanation of all relevant formulas. In doing so, we will approach step by step to one of the most general classical porous media multiphase flow equations. Based thereon, we will additionally derive some adequately simplified formulations, which then will be further used either for the main numerical study in Sec Sec and Sec. 5.2 or for some explicative and comparative examples in Sec and Sec Apart from the very general multiphase flow equation, we thereby will restrict ourselves to two phase formulations. In the following section, we will outline the shortcomings of the classical multiphase flow models and show, how the model can be extended to include in Sec. 2 neglected physical aspects. The starting point will be an overview on stability analysis concernig multiphase flow in general and based thereon multiphase flow in porous media in particular. By this means, we will have to take another closer look at very fundamental physical flow processes. Thus motivated, we will analyze especially two phase flow systems on a much smaller length scale than it is commonly applied within the actual flow problem consideration. In doing so, we will clearly examine that there are effects, which have been neglected so far in the classical models presented in Sec. 2. These effects will be given by on the one hand more or less well-known capillary pressure hysteresis and by on the other hand so-called dynamic effects. For both categories, we will introduce several different model approaches that can be implemented into 2

21 the classical multiphase equations from Sec. 2. However, according to the motivation of this work at the beginning, the main focus of attention will lie with the dynamic description of the capillary pressure. Finally, the introduced dynamic and hysteretic model concepts will be used qualitatively to explain the aforementioned gravity driven fingering phenomenon for infiltration processes in homogeneous porous media. Thereafter Sec. 4 will show, how the extended multiphase flow equations can be discretized in order to attain numerically resolvable systems of equations. This procedure will be based on a previously introduced very general discretization technique given by the so-called finite differences method. The discretization of the multiphase flow formulation mainly used in this work then will be conducted very thoroughly. Thereby, we will apply a finite elements scheme that is further characterized through the chosen ansatz function and, which will lead up to the so-called BOX-method. In addition, having in mind the comparative simulations in Sec , we further will give a finite differences discretization scheme for the so-called saturation equation, which has been derived in Sec. 2. Finally, we will present the solution algorithm that is used for the multiphase flow formulation, which has been discretized by the finite elements method. In the following, Sec. 5 then will apply the newly implemented extended model to a variety of different flow scenarios. In order to be able to do this properly, we will primarily perform some test simulations to verify the correct implementation and further check the model heuristically for grid convergence. Subsequently, we will mainly perform 1-D simulations directing the focus on the influences on the flow behaviour by different setups of boundary conditions. However, we constantly will try to address the simulation results to the physical topics dealed with in Sec. 3. Finally, we will also conduct some 2-D simulations to see whether the extended model is capable of reproducing the main characteristics of gravity driven fingering. Sec. 5.3 then will briefly summarize the main results of the complete numerical study conducted in Sec. 5. As a conclusion to this work, Sec. 6 will again summarize the obtained results of Sec. 5 in the context of the whole work and in addition give an outlook on potential further scholars considering dynamic multiphase flow models. 3

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23 2 Physical and mathematical model for classical two-phase flow In order to realize numerical simulations dealing with multiphase flow in porous media, the strategy is first to identify then to understand and finally to abstract the relevant physical processes, which determine the respective flow scenarios. Since these processes mostly find their very origin on the molecular scale, this level also poses the natural starting point of general multiphase flow analysis. In consecutive steps the abstraction is then carried out on larger scales by the help of different so-called upscaling methods. According to these introductory sentences, the second Section of this work consists of detailed descriptions concernig classical and well-established physical quantities with the aim of setting up a closed mathematical model representing porous media multiphase flow. 2.1 Porous media system Subject to this work, a porous media system is defined as a more encompassing thermodynamic system, which in turn is given by a distinct material object further characterized by its state variables and boundary conditions. Such state variables are well-defined system properties, which usually go along with a distinct length scale. Since this section considers the state variables prior in states of thermodynamic equilibrium, we will start with defining this term properly. Subsequently we then introduce several useful length scales and give fundamental definitions for porous media systems and their particular constituents. The behaviour of these systems in terms of multiphase flow applications will then be analyzed in the ensuing sections Thermodynamic equilibrium The state of thermodynamic equilibrium can be divided into thermal, mechanical, chemical and electrical equilibrium and is always related to a distinct thermodynamic system. In the state of thermodynamic equilibrium, there occurs no more temporal change of state variables and thus, there exist no more potential gradients. The particular potentials are given in Tab Equilibrium type Thermal equilibrium Mechanical equilibrium Chemical equilibrium Electrical equilibrium Potential Temperature Pressure Chemical potential Electric potential Table 2.1: Types of thermodynamic equilibrium states. Electrical, chemical and thermal potentials will be completely neglected in the course of this work, since we will only consider isothermal porous media multiphase flow with a constant composition of each phase. Usually, mechanical equilibrium between two phases prevails in the case of isobar conditions across the interface. However, in porous media we have to account for surface tension, which will lead to a pressure discontinuity across the interface. This fact will be enlightened in detail in Sec

24 2.1.2 Scales In general, fluid dynamics are considered on manifold length scales that reach from molecular observations up to field observations with an extent of meters or even kilometers. Neglecting quantum mechanic effects, it is classical mechanics based on Newton s three main axioms [50], which is the basic physical concept of multiphase flow in porous media: 1. Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon. 2. The alteration of motion is ever proportional to the motive force impressed and is made in the direction of the right line in which that force is impressed. 3. To every action, there is always opposed an equal reaction or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts. One of the most famous equation, which was first formulated by Euler, 1750 [21], is based on the second axiom above: F = m a [N]. (2.1) Therein, F is a vector and represents the force acting on a body with mass m [kg] and acceleration a [ m /s 2 ], which also represents a vector. With the following supplementing fundamental equations of motion and keeping Newton s axioms in mind, it is possible to solve particle movements on the molecular scale: s = 1 2 a t2 [m], (2.2) v = s t [ m /s]. (2.3) However, there are mainly two aspects that make it quasi impossible to calculate fluid flows on larger scales by the simply use of these equations. The first point is that usually the initial configuration of each particle is not known. This means that neither the molecular structure of the solid matrix nor the position of individual fluid particles can be determined exactly. The second point concerns the huge amount of particles that emerge as soon as larger systems are considered. Although the first point could in theory be solved by statistical allocations, it is primarily the limited capacity of computer processing speed and memory requirements that forces us to look for alternative solutions. Since in most situations macroscopic dynamics are not influenced by individual particle behaviour at the molecular scale, it is well-established to average physical quantities up to larger scales. Therefore, material properties like density, viscosity, capillary pressure and (relative) permeability have to be defined. However, before this will be done in the following sections, we will first introduce several useful length scales, which will be used in the further course of this work. The length scales will be identified by a virtual increasing average volume (AV) and resulting characteristics of fluid properties like density and viscosity. 6

25 Molecular scale: The AV comprises individual identifiable molecules of one phase. In molecular dynamic simulations the classical equations of motion can be solved for each molecule. Averaged quantities like density and viscosity cannot be defined reasonable due to strongly varying values by increasing the AV. Continuum scale: The AV still comprises onyl one single phase. However, from this scale on a further increase of the AV inside a phase does no longer cause varying values of density and viscosity. Fluids are now considered as continua. Microscale: The AV now comprises different phases. Individual phases and interfaces are still identifiable. An further increasing AV once more causes varying values of density and viscosity due to the phase overlapping consideration. The flow field can be described by the Navier-Stokes equations. REV scale: Although the AV comprises different phases, there occurs no more varying fluid properties like density and viscosity by its increase (see Fig. 2.1). The flow field description is based on the representative elementary volume (REV) concept after Bear, 1972 [4]. In this work we will use the term macroscale as equivalent. Figure 2.1: Illustration of different length scales including the representative elementary volume scale (REV) based on the concept after Bear, 1972 [4]. Field scale: Expresses the dimension of the particular considered problem that is intended to be solved. Fluid properties like density and viscosity might change gradually. Their determination can be part of the desired solution Porosity A porous medium consists of a solid matrix with a certain amount of interspaces. These interspaces are called pores as far as they are connected with each other and thus form possible flow paths for fluids. Interspaces which are not connected with each other are considered to be part of the solid matrix. The amount of pores is accounted for by the porosity, which is a dimensionless quantity defined by φ = V p V p + V s [ ]. (2.4) Thereby, V p [m 3 ] is the volume of the pores and V s [m 3 ] the respective volume of the solid matrix. Depending on the packing the porosity has a possible theoretical range between zero and one. For instance, the porosity of a cubic packing of equal balls is A more denser packing would be the rombohedral one with a porosity of 0.26 [58]. 7

26 Figure 2.2: Cubic (left) and rombohedral (right) packing of balls of same size Phases In the scope of this work we do not consider compositional effects of phases. We define a phase as homogeneous material with constant chemical and physical composition. There are solid, liquid, gaseous and supercritical phases, which are conventionally considered in multiphase flow in porous media. However, supercritical phases will not be regarded in the following. The solid phase is given by the before mentioned solid matrix and therefore per definition part of every porous media multiphase system. The pores of the solid matrix can be filled by various fluid phases. The following specifications refer to the molecular scale in order to give the reader a better understanding of phases and phase states. Every material consists of myriads of molecules. A single molecule in turn is a distinct combination and formation of atoms. An atom exists of a positively charged nucleus which is surrounded by negatively charged electrons. The different charges are the reason for attractive as well as repulsive forces of molecules on each other. For polar molecules like water the attractive forces are a result of permanent dipoles. The distance dependency of the potential energy between permanent dipoles is r 3 (r distance) [1]. However, also unpolar molecules exhibit dipoles yet, in contrast to polar molecules these dipoles are only temporarily induced. For the resulting Van-der-Waals-interactions the distance dependency is r 6. As can be seen from these potential dependencies, attractive forces loose importance very fast with increasing distance. Contrary to attractive forces, repulsive forces are mainly given by electrostatical interactions of overlapping electron clouds. These forces become dominating for the molecules coming very close together. In gases the molecular distances are large and thus intermolecular forces are weak. As a consequence, different gas phases always mix up perfectly and hence there is maximal one gas phase in the porous media system. In a solid phase attractive and repulsive forces are in equilibrium and force a molecule to remain in a fixed arrangement with its neighbours. Hence there is quasi no mixing between a solid phase and any other phase. In liquid phases molecules are able to change their arrangement inside a phase. Whether a mixing process between different liquids takes place strongly depends on the particular attractive and repulsive forces of the molecules on each other. On the microscale, there exists a clear identifiable interface between different phases. However, on the molecular scale it can be justified to consider the interface as its own phase [58]. In figure 2.3 we take a look at two phases and the interface among them at the molecular scale. 8

27 Figure 2.3: Interface between two fluids. Although the phases are not miscible with each other, there is always a certain length r where the composition of the phases gradually does change. On the left and the right side, there exists a particular homogeneous phase. Between these two phases, there is a zone with a gradual change from one fluid to the next. This is the so-called interfacial zone. As soon as we want to define the interface as its own phase, the first difficulty that arises is the determination of distinct values for the fluid properties. We will have another look at interfaces in Sec For the classical multiphase model that will be derived in the following sections, interphases are not considered as additional independent phases. The amount of phase α then either is expressed by its volumetric content θ α = V α V p + V s [ ], (2.5) wherein V α [m 3 ] is the volume of the phase α, or its saturation S α = V α V p [ ]. (2.6) The sum over all saturations in multiphase systems at all times is given by S α = 1. (2.7) 2.2 Material properties α As already mentioned, porous media systems are certain types of more encompassing thermodynamic systems. Following the definitions of different length scales in Sec , the fluids and the soil matrix will be exclusively considered as continua. Then, the here considered systems can be fully described by their inherent state variables and distinct boundary conditions. The common form of an equation of state for a state variable Y is given by Y = f(x 1, X 2,...X n ). (2.8) Therein, f depicts the dependencies of Y from n further independent state variables X n. State variables can be classified into intensive (i.e., independent of the extent of the system), as for example the temperature or extensive (i.e., proportionally dependent on the extent of the system), as in the case of the internal energy. Since in the scope of this work only isothermal and incompressible multiphase flow will be considered, we will only give a rough description of the 9

28 respective state variables density and viscosity and their particular thermodynamic dependencies on pressure and temperature. We again emphasize that in the following sections especially for the description of capillary pressure and relative permeability a state of thermodynamic equilibrium will be presumed Density The density of a substance is defined as the relation between its mass m and its occupying volume V : ϱ = m V [ kg /m 3 ]. (2.9) In single component systems, density usually is characterized by its dependency on temperature T [K] and pressure p [Pa]. The total derivative is then given by dϱ = β T dp + β p dt, (2.10) with β T = ϱ p [ ] (2.11) T as the isothermal compressibility and β p = ϱ T [ ] (2.12) p as the coefficient of thermal expansion. The density of liquids in isothermal multiphase flow applications often is assumed to be constant due to mostly very low isothermal compressibilities. For instance the isothermal compressibility of water at 20 C is in the order of magnitude of about 10 6 [8]. For ideal gases the dependency of density on temperature and pressure is given by the so-called ideal gas law: with as the specific gas constant and ϱ = p R i T, (2.13) R i = R M i [ J kg /K], (2.14) R = [ J mol /K], (2.15) as the universal gas constant. M i [ kg /mol] depicts the molar mass of substance i. For isothermal multiphase flow in porous media the density of the gaseous phase is often considered to be constant due to very low viscosity and therefore low pressure gradients inside the phase. 10

29 2.2.2 Viscosity On the molecular scale particles in a flow field transport momentum orthogonal to their streamline because of random thermal movement. Thereby, the braking effect of slow particles hitting fast particles is interpreted as viscosity [1]. For the quantification of this resistance on the macroscale let us consider two plates separated by a liquid. The upper plate is moving with a constant velocity v while the lower plate remains on its position. The flow field is assumed to be laminar with a constant velocity gradient dv dy see Fig Figure 2.4: Laminar flow field of newtonian fluids. It has been prooved by experiments that the force F, which is needed for the maintenance of the constant velocity v, is proportional to the surface A and velocity gradient dv dy : F v Av y. (2.16) The proportinality coefficient of (2.16) is defined as dynamic viscosity µ [Pa s] and the fraction of the term Fv A is expressed as shear stress τ [N /m 2 ]. Then (2.16) can be written as τ = µ dv dy. (2.17) Usually in the literature a simple τ for shear stress is used. However, for a clear notation we introduce an underline because of a later on definition of τ as retardation coefficient. For newtonian fluids µ is independent of v also for non-laminar flow fields. The viscosity of gases increases with temperature because of higher thermal movement of the molecules and therefore a higher transfer of momentum. In liquids this effect is superimposed by a higher probability of molecules to change their molecular arrangement. Therefor the viscosity of liquids decreases with higher temperature (see Sec for a molecular description of gas and liquid phases). The pressure dependency of the viscosity in general is neglected for both phase states. Since in the scope of this work the viscosity is held constant, we refer to the literature for any thermodynamic equations of state. Instead of the dynamic viscosity µ, the literature sometimes makes use of the kinematic viscosity ν = µ ϱ [ m2 /s]. (2.18) 11

30 2.2.3 Capillary pressure Since the improvement of a representative capillary pressure model is the main topic of this work, we restrain this section solely to the most commonly applied models. These models will be based on surface tension considerations in combination with narrow glas tubes representing idealized pores. In the following, we will mostly use the term macroscale pressure instead of representative capillary pressure. Surface tension Liquids surrounded by gases tend to minimze their surface U [m 2 ] due to their stronger intermolecular attractive forces related to the interactions with the respective gas phase. For isochore and isothermal conditions, a spherical shape provides the formation of minimal free Helmholtz energy A [J] and accordingly is formed by liquids as soon as no other or only minor forces are present [1] (see Fig. 2.5). The free Helmholtz energy, which is then required to change the surface U is proportional to the amount of surface expansion du. The proportionality factor is called surface tension σ [ N /m]. da = σ du. (2.19) Figure 2.5: At the liquid/gaseous interface the attractive forces on a liquid molecule inside the liquid phase are not balanced by the forces in the gas phase. The result is a spherical form of the liquid phase. With (2.19) and neglecting dr 2 terms (r [m] is the radius of the spherical surface), the free Helmholtz energy da for an infinitesimal alteration dr of a spherical surface is given by da = σ4π(r + dr) 2 4πr 2 = σ8πrdr. (2.20) Employing further a balance of forces between the pressure forces and surface forces gives 4πr 2 p in = 4πr 2 p out + σ8πr. (2.21) Therein, p in denotes the pressure on the convex side of the interface and p out the pressure on the concave side accordingly. As a consequence of (2.21), p in > p out. Rearrangemnet of (2.21) leads to Laplace equation: p in = p out + 2σ r. (2.22) 12

31 Capillary pressure on the microscale Having the molecular forces that act inside and between particular fluid phases in mind, we will now expand these considerations to the interaction of fluids and solids. In narrow glas tubes (capillary tubes) which are dipped into water an ascent of the water level above the surrounding water table is observable. This phenomenon is called capillarity and is caused by higher adhesion forces between the water phase and the solid phase than cohesion forces inside the water phase (see Fig. 2.6). Alike in the previous case, the interfacial forces can be treated by the concept of respective surface tensions. Figure 2.6: The water level inside a narrow tube increases due to the surface tension. The liquid/gas interface is curved with p out < p in. The respective subscripts stand for: g := gas phase, l := liquid phase, s := solid phase. Θ c is the so-called contact angle. Below the meniscus the water pressure can be estimated by equation (2.22) (note: here the pressure in the water phase is lower than in the gas phase unlike in the previous example of surface tension between only fluids). The pressure of the air phase is given by the atmospheric pressure p atm and for the liquid phase by the hydrostatic pressure p h = ϱgh. (2.23) Due to the mechanical equilibrium, the atmospheric pressure p atm has to equal p h at the water table. The height h [m] of the water level inside the glas tube (piezometric head) is then determined by the equilibrium between hydrostatic pressure and surface tension: h 2σ ϱ w gr. (2.24) This height or the respective pressure discontinuity represents the capillary pressure in narrow glas tubes. Furthermore, the contact angle between the glas tube and the water/gas surface can be calculated by Young equation: cos(θ c ) = σ sg σ sl σ lg. (2.25) Liquids whose contact angle is 0 < Θ c < 90 are called wetting fluids, otherwise they are called non-wetting fluids. Whether a fluid is wetting or non-wetting depends on the combination of the participating phases. For instance, a system with mercury and air in the same glas tube would cause the mercury level to decrease below the surrounding mercury level. Then the contact angle for air is smaller than 90 and thus air is now the wetting phase. 13

32 Capillary pressure on the macroscale So far we have detected that a liquid in contact with a gas phase or a gas phase in contact with a solid phase gives rise to a curved surface. This in turn leads to a discontinuity in pressure across the interface, which in general is called capillary pressure. The dimension of capillary pressure in glas tubes can be obtained by the Laplace equation and is reciprocally proportional to its diameter. For multiphase flow in porous media the flow inside each pore is basically determined by the dimension of the particular capillary pressure. However, since a natural porous medium consists of many different pores with varying diameters (and therefore varying p c ), we have to determine a representing capillary pressure on the macroscale. Let us assume a porous medium that consists of a bundle of tubes with different diameters. These diameters are supposed to represent the pore size distribution of the porous medium. Surrounded by air (non-wetting phase) and dipped into water (wetting phase), the water level in each tube will increase due to capillary rise. At equilibrium, we observe a higher water level in narrow tubes than in broad tubes and therefore also a higher capillary pressure p c. The summed up water content in all tubes (related to a considered height) then corresponds to a distinct capillary pressure (see Fig. 2.7). Figure 2.7: Bundle of tubes model for p c S parameterization. For a sufficient number of considered glas tubes, we obtain a unique function between the water saturation (content) and the capillary pressure. By this means, we are able to define an averaged equilibrium capillary pressure for a REV that depends on saturation. In real porous media the pore geometry usually is more complex and p c S curves are usually obtained by empirical parameterization models, fitted with experimental data. Similarly to the bundle of tube model, these models commonly assume a saturation dependency. In this work we will use the so-called Van Genuchten parameterization: Van Genuchten p c - S parameterization: p c (S e ) = 1 α [ ] S 1 1 n m e 1 for p c > 0, (2.26) with the effective saturation S e given by S e = S w S w,r 1 S w,r. (2.27) 14

33 The coefficient α is a scaling parameter and related to the inverse of the entry pressure p c,en, which in turn is the pressure needed for the non-wetting phase to penetrate into the pore with the largest diameter. The coefficient n accounts for the pore size distribution. A large n stands for little distributed porous media. According to Mualem, 1976 [49], m is defined as m = 1 1 n. (2.28) Fig. 2.8 (left) illustrates a typical p c S curve according to the Van Genuchten parameterization. Figure 2.8: Van Genuchten (left) and Brooks-Corey (right) p c S parameterization curves with residual saturation S w,r and capillary entry pressure p c,en As we can see, the capillary pressure is zero for a fully saturated porous medium (S w = 1) and tends to infinity for S w S w,r. The residual saturation S w,r stands for the fraction of water that is no more reducable by hydraulic processes. This effect will be further enlightened in Sec Since the Van Genuchten parameterization is strongly related to the Brooks-Corey model, the latter is shortly outlined in the following. The Brooks-Corey parameterization incorporates the entry pressure p c,en directly into the mathematical description. In comparison to the Van Genuchten function, the main difference is that p c equals p c,en, for a fully saturated porous medium (S w = 1) (see Fig. 2.8 (left)). Brooks-Corey p c - S parameterization: p c (S e ) = p c,en S 1 λ e for p c > p c,en. (2.29) Although the Van Genuchten parameterization and similar models are commonly applied in porous media multiphase flow applications, it has turned out that there are some facts, which have been ignored by these models. As Sec. 3.3 will demonstrate, there is an decisive influence of the pore geometry that makes the p s S relationship a hysteretic function of saturation. Furthermore, later on in Sec. 3.4, we will discuss the main topic of this work that concerns potential dynamic effects in the p c S relationship (note: until now we have always assumed local equilibrium states). 15

34 2.2.4 Permeability Due to the second law of thermodynamics, there is a transformation of potential energy into energy of molecular thermal movement on each flow process. Such dissipation processes inside and between phases determine the resistance, that is exerted within the flow field. The saturated hydraulic conductivity K f [ m /s] displays the reciprocal value of this resistance for single phase flow. The application of this resistance coefficient in terms of the complete flow field description will be done in Sec It has been shown by experimental work, that K f is proportional to the specific weight of the respective fluid and reciprocally proportional to its viscosity [58]: K f = K ϱg µ [ m /s]. (2.30) In (2.30), K [ ] is the intrinsic permeability, which is only a characteristic of the porous medium. The higher the amount of fluid in contact with the solid matrix, the higher are dissipation processes due to frictional forces. For porous media with low porosity, high tortuosity and a grain size distribution that provides only a small number of large pores, K becomes very small [58]. Thereby, the tortuosity is defined as length of a curve (flow path) related to the distance of its end points. As soon as the hydraulic conductivity takes different values for different directions, this has to be taken into account by a tensorial notation, K f,xx K f,xy K f,xz K f = K f,yx K f,yy K f,yz. (2.31) K f,zx K f,zy K f,zz In the case of an isotropic porous medium with different hydraulic conductivities orientated along the coordinate system, (4.40) turns into a diagonal matrix. For multiphase systems, there are different hydraulic conductivities for each fluid phase α. In addition to different viscosities and densities, there is a disturbance of the phases on each other. To account for this, a dimensionless coefficient - relative permeability k rα [ ] - is introduced, which is usually seen to be a nonlinear function on the saturations of the phases. For a better understanding of this disturbance depending on saturation, let us consider a water/air two-phase flow scenario on the microscale. Due to the wetting character of the water phase related to the air phase, water occupies preferentially smaller pores. Thus, assuming a low water saturation, only the small pores provide possible flow paths for the water phase, whereas bigger pores provide the flow path for the gas phase (see Fig. 2.9). By this means, the tortuosity is enhanced and both fluids have to get over a longer distance in flow direction as it would be in the case of fully saturated conditions. Accordingly this leads to higher dissipation and therefore to a reduced permeability of the system for both phases. Figure 2.9: Enhanced tortuosity leading to enlarged flow path for the wetting as well as for the non-wetting phase. 16

35 Fig illustrates typical k r S parameterization curves for water/air systems. As can be seen, the slope for k r,w (S w ) at high water saturations is steeper than the slope for k r,n (S w ) at high air saturations. If we think of pores as tubes, their flow field can be described by the Hagen-Poiseuille concept for laminar flow. Therein the flow velocity in x direction is given by v x (r) = 1 4µ (R2 r 2 ) p x. (2.32) In (2.32) the inner radius of the considered tube R determines the flow velocity quadratically. Thereby, the faster decrease of k r,w (S w ) for drainage can be explained by the fact that water retracts from bigger to smaller pores which exhibit higher flow resistance. Figure 2.10: Typical relative permeability saturation relationship for wetting and non-wetting phase, parameterized by the Van Genuchten approach according to Mualem, 1976 [49]. Furthermore, the disturbance of the fluid phases on each other in Fig is reflected by the fact that the sum of k r,w and k r,n is less than one in the range where both fluids are seen to be mobile. However, fluids can also become completely inmobile, which is the case by loosing their hydraulic connection. Then, the relative permeability of the respective phase equals zero. Recalling Sec , this refers to the state of residual saturation. Taking relative permeabilities into account, the Van Genuchten parameterization according to Mualem, 1976 [49], is given by Van Genuchten k r parameterization: and k r,w = S ε e [ ( ) m ] S 1 m e (2.33) k r,n = (1 S e ) γ [1 S 1 m e ] 2m. (2.34) The parameters ε and γ are usually set to ε = 1 2 and γ = 1 3 (see Helmig, 1997 [33]). 17

36 2.3 Classical multiphase flow models In the following, we will derive differential equations which describe multiphase flow in porous media. The various equations are quasilinear of first, second and third order. Furthermore, they can be classified by a parabolic, hyperbolic and elliptic character. In order to demonstrate what this means, let us consider the general form of a 2-D partial differential equation of second order with Φ(x, y) as unknown depending on x and y: a 2 Φ(x, y) x 2 + 2b 2 Φ(x, y) + c 2 Φ(x, y) Φ(x, y) Φ(x, y) x y y 2 + d + e + fφ(x, y) + g = 0 (2.35) x y Equation (2.35) is linear with constant coefficients for linear for and quasilinear for a R, b R, c R, (2.36) a = a(x, y), b = b(x, y), c = c(x, y), (2.37) a = a(x, y, Φ), b = b(x, y, Φ), c = c(x, y, Φ). (2.38) For the linear case with constant coefficients, equation (2.35) is named Calssification of second order partial differential equations: parabolic for b 2 ac = 0, (2.39) hyperbolic for b 2 ac > 0, (2.40) elliptic for b 2 ac < 0. (2.41) As in the quasilinear case, the values of the coefficients depend on x, y and Φ, the character of those partial differential equations is subject to change and thus, can only be considered locally. For each type of equation, there prevail certain characteristics which are outlined in the following: Parabolic In parabolic flows, there is a dominant flow direction with perturbations only propagating downstream. A prominent representative is the advection-diffusion equation. In this work the general used multiphase flow formulation will be parabolic. Hyperbolic For hyperbolic flows we assume a distinct point A inside the flow domain. There are restricted areas upstream and downstream that can only be affected by perturbations at point A. This area is enveloped by the two characteristics of the flow equation. In particular differential equations with wave character are hyperbolic. Elliptic Elliptic equations primarily describe stationary states. In porous media multiphase flow fields the description of the pressure term exhibits an elliptic character. 18

37 2.3.1 Conservation of mass There are two common ways of observing fluid flow in classical mechanics, either by tracking a fluid volume on its way through the domain (Lagrangian point of view) or by observing a fixed fluid volume in space (Eulerian point of view). Let t R + 0 be the temporal and x = (x 1,..., x d ) R d, d {1, 2, 3} be the spatial coordinates. Then (x(t), t) in R d R + 0 denotes the Langrangian coordinate system that moves along with a considered fluid volume. Contrary to this (x, t) in R d R + 0 denotes the Eulerian coordinate system, which is fixed in space. Thermodynamic and mechanic fluid properties are always related to a distinct material object. Therefore, the physical axiom of mass conservation is naturally described by the Langrangian point of view, considering a discrete fluid volume that constantly contains exact the same amount of particles. Let us assume a continuum, where the whole space is filled by particles, which all possess a defined place at time t 0. The mass in a discrete control volume Ω(t 0 ) R d is then given by m Ω (t 0 ) = Ω(t 0 ) ϱ(x, t 0 ) dω. (2.42) At time t 1 > t 0 each particle may have moved with a different velocity along its trajectory. The result is a deformed control volume Ω(t 1 ), which, however, still contains the same amount of particles (see Fig. 2.11). Figure 2.11: Material volume deformed over time Such control volumes are called material volumes and represent the Lagrangian point of view. Mathematically the conservation of mass can then be written as d dt Ω(t) ϱ(x, t)dω = 0. (2.43) 19

38 Equation (2.43) reflects the plausible physical statement that mass does not change when the same amount of particles is considered for different times and spatial arrangements. Note that this assumption only holds when neglecting source/sink terms due to chemical reactions or the like. In order to obtain a more useful description of mass conservation based on the Eulerian point of view, we refer to the Reynolds Transport Theorem that describes the temporal change of an integral over a material volume [7]. Reynolds Transport Theorem: Let e : R d R + 0 R be a differentiable scalar function that represents an arbitrarily intensive fluid property and Ω(t) a material volume whose deformation can be described by a bijective continuous function. With v s as the seepage velocity of the particles the following equation holds: d dt Ω(t) e(x, t) dx = Ω ( ) e t + (v se) (x, t) dx. (2.44) In (2.44), Ω denotes that the control volume is fixed in space and time. Note that for the time when the transformation (2.44) is made, Ω(t) equals Ω. For the derivation of the conservation of mass equation of phase α, let e be substituted by (S α φϱ α ). Therein, ϱ α is the mass density of phase α, φ is the porosity and S α is the volume fraction of pores filled with phase α. In doing so, we only consider the volume fraction of the control volume that is filled by the phase α. Substituting e in (2.44) by (S α φϱ α ) and taking account for (2.43) finally gives Ω ( (Sα φϱ α ) t ) + (v s,α (S α φϱ α )) (x, t) dx = 0. (2.45) For the consideration of any source/sink terms we can easily add an additional term (ϱq). Consequently (2.45) becomes Ω ( (Sα φϱ α ) t ) + (v s,α (S α φϱ α )) (ϱq) (x, t) dx = 0. (2.46) Postulating that (2.46) holds for every control volume inside the domain, it implicates that its integrand becomes zero in any point. This way we obtain a partial differential equation describing the conservation of mass for multiphase flow: Conservation of mass: ( ) (Sα φϱ α ) + (v s,α (S α φϱ α )) (ϱq) (x, t) = 0. (2.47) t 20

39 2.3.2 Darcy s law for single and multiphase flow Single phase flow In this section we discuss Darcy s law for isothermal single phase flow. Thereby, single phase flow implies that the porous medium is fully saturated by the considered fluid. In the following, Darcy s law will be derived from the Navier-Stokes equations for incompressible fluids neglecting inertial forces. However, it should be mentioned that Darcy s law primarily is an empirically found relationship based on experimental measurements by Henry Darcy, 1856 [10]. Darcy found that the flux of water related to the cross-section of the porous media is proportional to the applied potential gradient. The definition of this potential will be given in the course of this section. After Bear, 1972 [4], Darcy s law holds for Reynolds numbers Re smaller than one. The Reynolds number is a dimensionless number defined by the ratio of inertial and viscous forces: Re = vlϱ µ [ ]. (2.48) In (2.48), l is a characteristic length of the system that is here represented by the average grain size of the porous medium. The Navier-Stokes equations for incompressible fluids are given by Navier-Stokes equations for incompressible fluids: ϱ v t + ϱ(v )v µ v = p ϱg. (2.49) In (2.49) we can identify the following terms for gravitational forces for an imposed pressure gradient and for frictional forces f g = ϱg, (2.50) f p = p, (2.51) f f = µ v. (2.52) Expression (2.50) - (2.52) all represent specific forces. The vector g is given by (0, 0, g) T, wherein g is further defined as gravity acceleration (g 10 m /s 2 ). Neglecting inertial forces, (2.49) simplifies to µ v = p ϱg. (2.53) Assuming further a potential for gravitational forces, (2.53) in vertical z-direction can be written as ( ) 1 ν v z + ϱ p + gz = 0. (2.54) 21

40 Subsequently, applying the conservation of mass equation gives ( ) 1 ϱg p + z = 0. (2.55) The term inside the brackets in equation (2.55) is the hydraulic head for which (2.55) satisfies the Laplace equation h = 1 p + z [m], (2.56) ϱg h = 0. (2.57) One solution of (2.57) is the primarily empirically found Darcy s equation for isotropic homogeneous porous media: Darcy s law for single phase flow: v f = K f h. (2.58) In (2.58), K f represents the hydraulic conductivity (see Sec ). The Darcy velocity v f is a fictive velocity that is related to the cross-section of the porous media. To obtain the seepage velocity v s one has to divide v f by the porosity: Multi phase flow v s = v f 1 φ. (2.59) In the unsaturated zone we have to deal with the behaviour of different fluid phases on each other and on the solid matrix. Therefore, in Sec. 2.2 we introduced the effective parameters capillary pressure and relative permeability. Taking these two parameters into account, Buckingham (1907) [6], extended Darcy s law for single phase flow to multiphase flow: Extended Darcy s Law: v f,α = k r,α µ α K( p α ϱ α g). (2.60) The quotient of relative permeability k r,α and dynamic viscosity µ α for a phase α is defined as its mobility λ α [ m s /kg]. The seepage velocity for multiphase flow is given by v s = v f,α 1 φs α. (2.61) 22

41 phy Summarizing the results in Sec. 2 up to now, we have developed several concepts concernig relevant physical characteristics of porous media multiphase flow systems. This has included the introduction of appropriate material properties like density, viscosity, capillary pressure and relative permeability. Based thereon, we continued with the derivation of the well-founded physical principle of conservation of mass. Then, in the previous section, we finally gave a theoretically derivable and experimentally observable description of single and multiphase flow fields. As mentioned in Sec. 1, we will now put these achievements together, which then will come up with the attempted classical porous media multiphase flow models Multiphase flow equations In the scope of this work only isothermal, immiscible two phase flow in non-deformable porous media is considered. Nevertheless, we will start this section with the derivation of the general porous media multiphase flow equation, based on the conservation of mass equation and the extended form of Darcy s law. Thereby, we will obtain a fully coupled system of nonlinear partial differential equations that will be closed by two constitutive relationships. For this system of equations we will give two different formulations. Furthermore, we will derive a degenerated partial differential equation describing porous media multiphase flow sufficiently good for certain flow scenarios, which we outline in the following: Considering a natural aquifer in the field, various zones can be identified. In the fully saturated zone we do not have to account for any multiphase flow effects. There, Darcy s law for single phase flow is sufficient to describe the flow field. Close to this fully saturated zone the air saturation increases gradually (vadose zone). Beneath a critical value for the air saturation that is on the order of magnitude of about 0.1 [61], the water phase and the air phase are strongly coupled to each other. There, it is reasonable to use the fully coupled multiphase system of equations to describe the flow field. However, far enough from the groundwater table the water saturation can be very low. Air S cr a as the non-wetting phase then is continuously distributed within the pore matrix. Due to its high mobility and low density, the pressure gradient of the air phase p a then is very small in horizontal as well as in vertical direction. This can justify the assumption of a degenerated flow regime, wherein p a is seen to be constant. For modeling this region one commonly uses the so-called Richards equation, a degenerated form of the general multiphase flow equations. As mentioned, we will also make reference to this equation because it is used in many scientific works including analysis of dynamic capillary pressure. In particular, it will be used in Sec to demonstrate and simultaneously compare alternative ways of handling dynamic multiphase flow phenomena. Coupled multiphase regime Combining the extended Darcy s law (2.60) with the conservation of mass equation (2.47) yields General form of the multiphase flow equation: φϱ α S α t φ + ϱ α S α t + φs ϱ α α t { k r,α ϱ α µ α K( p α ϱ α g) } ϱ α q α = 0. (2.62) 23

42 Considering an isothermal two phase system we obtain two equations with the saturations and pressures of the phases as unknowns. The system can be closed by the constitutive relations and S w + S n = 1, (2.63) p c = p n p w. (2.64) Recalling Sec , we are able to express the capillary pressure by terms dependent on the saturation. Substituting (2.63) and (2.64) in (2.62) and assuming further a non-deformable porous medium we obtain the pressure-saturation formulation of two-phase flow in porous media: Pressure-saturation formulation with p w and S n as unknowns: φ (ϱ { } ws n ) ϱ w k r,w K( p w ϱ w g) ϱ w q w = 0, t µ w φ (ϱ { } ns n ) ϱ w k r,n K( p w + p c ϱ n g) ϱ n q n = 0. t µ n (2.65) In (2.65), all coefficients can be calculated by the primary variables p w and S n. The pressuresaturation formulation will be mainly used for the numerical study in this work. However, for later on comparison simulations we also derive the so-called saturation equation. Therefore, we define the total velocity as v t = v w + v n. (2.66) Furthermore, we introduce the fractional flow coefficient f w for the wetting phase and f n for the non-wetting phase, and f w = λ w λ w + λ n, f n = λ = λ w f n = λ n f w = The velocities for both phases is given by Darcy s law: λ n λ n + λ w, (2.67) λ wλ n λ w + λ n. (2.68) v w = k r,w µ w K( p w + ϱ w g), v n = k r,n µ n K( p w + p c + ϱ w g). (2.69) Combining (2.67) - (2.69) the velocities can be expressed by v w = f w v t + λk( p c + ϱ w g ϱ n g), v n = f n v t λk( p c + ϱ w g ϱ n g). (2.70) 24

43 Inserting (2.70) into the conservation of mass equation (2.47) and again assuming non-deformable porous media gives the saturation formulation: Saturation formulation: φ (ϱ ws w ) + { ϱ w [f w v t + t λk( p c + ϱ w g ϱ n g)] } = ϱ w q w, φ (ϱ ns n ) + { ϱ n [f n v t t λk( p c + ϱ w g ϱ n g)] } = ϱ n q n. (2.71) The two saturation equations (2.71) are part of the fractional flow formulation of the general porous media multiphase flow equation (2.62). The complete formulation additionally contains the so-called pressure equation. For the whole fractional flow formulation we refer to Helmig, 1997 [33]. With a known total velocity v t, the only unknown in (2.71) is the saturation S w. In the following we will consider only the 1-D case and further neglect gravity as well as any source/sink terms. The summation of the conservation of mass equation (2.47) over all phases in the incompressible case gives: v t = 0. (2.72) Thus, v t is constant in space. However, the determination of v t might still depend on the temporal evolution of the flow system. To enlighten this dependency, let us consider v t : v t = λ w K d dx (p w) λ n K d dx (p w + p c ) = λ t K d dx p w λ n K d dx p c ( ) d = λ t K dx p d w + f n dx p c. (2.73) wherein λ t has been defined by Integrating (2.73) over the domain from x 0 to x 1 yields λ t = λ w + λ n. (2.74) v t = p w (x 1 ) p w (x 0 ) + x 1 x 0 1 λ t x 1 x 0 f n dx d dx p c dx. (2.75) By (2.75), v t does depend on the boundary values of p w and on the current solution of S w over the whole domain. This fact has to be accounted for in later on comparative simulations between the pressure-saturation formulation and the saturation formulation. The pressure-saturation formulation is a fully coupled system of equations, wherein all equations have to be solved simultaneously. The saturation equation includes only one unknown presuming a predetermined total velocity. Both, the pressure-saturation and the saturation equation are partial differential equations of parabolic type. However, the saturation equation exhibits the form of a classical advection-diffusion equation, where one advective term and one diffusive term can be distinguished. This characteristic can be used for later on adapted 25

44 discretization schemes for each term. In doing so, the physical meaning of the particular terms can be considered more thoroughly. Degenerate multiphase regime For the derivation of the degenerated multiphase regime we will start with the aforementioned general multiphase flow equation (2.62). Thereby, we assume from the beginning onwards a non-deformable porous medium and an incompressible wetting phase. For the wetting phase in the 1-D case, (2.62) turns into φϱ w S w t { } ϱ w k r,w K( p w ϱ w g) ϱ w q w = 0. (2.76) µ w Factoring out (ϱ w g) and dividing the whole equation by ϱ w gives φ S ( w {K f,u p )} w t ϱ w g 1 ϱ w q w = 0, (2.77) wherein K f,u is the unsaturated hydraulic conductivity, which is given by K f,u = k r,w ϱ w g µ w K [ m /s]. (2.78) In the literature (2.77) often is written in a simplified way with the volumetric wetting phase content and the pressure head θ w = S w φ, (2.79) ψ = p w ϱ w g [m]. (2.80) 1-D Richards equation θ w t z [ ( )] K f,u (θ w ) z ψ(θ w) 1 = 0 (2.81) The Richards equation assumes a constant pressure for the non-wetting phase due to the consideration of a degenerated flow field. The pressure of the wetting phase p w is given by a reference pressure p w, which is usually determined by certain boundary conditions and the capillary pressure p c. Hence p w only depends on the saturation: p w (S w ) = p w p c (S w ). (2.82) By the Richards equation, we have obtained a simplified formulation of the general classical porous media multiphase flow equation. In the following sections, we will now discuss cases, where it has been found that neither the simplified nor the general formulation does sufficiently describe the real flow behaviour of the system. In particular this will concern to gravity driven fingering. According to the course so far, we will primarily try to identify potential physical processes in a reasonable and illustrative way before we will subsequently introduce adequate model approaches. 26

45 3 Physical and mathematical model for gravity driven fingering Gravity driven fingering is a special form of general instable fluid displacement that commonly goes along with vertical infiltration processes. Thereby, the term fingering expresses the observed flow pattern at the displacement front, which is not uniform, but exhibits a characteristic formation of finger-like structures. Furthermore, infiltration here means the invasion of a wetting phase into a porous medium, which initially is completely or at least to a very large extent filled by the non-wetting phase. In the following, we will use the term imbibition for a wetting fluid displacing a non-wetting fluid, and the term drainage for the inverse process. For imbibition processes the term displacement front sometimes will be replaced by the term wetting front. Since gravity driven fingering deals with instabilities, this section will start with a brief discussion of general stability criteria for multiphase flow in porous media. Stable flow prevails if any perturbation of the flow field is compensated completely. Hence, one obtains an uniform profile of the displacement front. For the opposite case of unstable flow any perturbation evolves more and more into the domain. This leads to non-uniform profiles of the displacement front such like in the case of gravity driven fingering. However, a flow can also be something between these two cases. Then one talks about restricted stable flow. This means that for instance the propagation of a perturbation depends on its frequency or further impacts. We will see that the classical models, which have been developed in Sec. 2 are not sufficient to describe instabilities related to gravity driven fingering. This in turn means that we have lost essential physical information during the applied upscaling procedure. Therefore, the next step will be one step back in order to take another closer look at multiphase flow on the microscale. By this means, we will detect that the up to now developed classical porous media multiphase flow models are additionally affected by physical processes concerning to the direction, history and dynamics of the system. Having identified additional effects on the microscale, we subsequently will have to improve our current macroscale models. Further investigations have found that mainly the p c S relationship is concerned by these effects. Related to this we will first discuss two model approaches concerning the direction and history of these relationships before we will afterwards deal with its dynamic effects. Especially one approach of a so-called dynamic p d c S relationship will be highlightened, which is one of the most prominent in current literature concerning this topic and, which will be also used for the numerical studies in this work. With the help of the new identified microscale processes and the accordingly improved macroscale models we will finally try to explain gravity driven fingering in a physical founded way. However, we will also see that there are still questions open for a complete and conclusive model about this subject. Since gravity driven fingering in common literature is classified as a distinct form of more general preferential flow, we will close this section for the sake of completeness by a brief look at further preferential flow mechanisms. 27

46 3.1 Instable displacement phenomena Fig. 3.1 illustrates the sketch of a typical flow pattern for instable fluid displacement compared to a typical flow pattern for stable fluid displacement. In the instable case we note that there is an irregular displacement front with the formation of so-called fingers. That would be a classical example of gravity driven fingering. Figure 3.1: (left) Typical flow pattern of unstable fluid displacement. (right) Classical infiltration front without instabilities. Fingering phenomena were first observed by Hill, 1952 [34], and primary analyzed without accounting for porous media characteristics. Saffman and Taylor, 1958 [63], examined a linear stability analysis on the vertical 1-D displacement process of two immiscible fluids in homogeneous porous media. However, in their studies Saffman and Taylor neglected capillary pressure and relative permeability at all. Furthermore, their instability criterion is restricted to only infinitesimal perturbations and a sharp fluid interface without transition zone. Nevertheless it might be useful as a point of departure for further investigations. Let us assume a displacement process where fluid 1 is overlying and displacing fluid 2 (see Fig 3.8). Figure 3.2: Vertical displacement process of two fluids separated by a sharp front without transition zone. Having the conditions just outlined in mind, Saffman and Taylor, 1958 [63], suggested multiphase flow to be stable for and unstable for (ϱ 1 ϱ 2 )Kg v f (µ 1 µ 2 ) < 0, (3.1) (ϱ 1 ϱ 2 )Kg v f (µ 1 µ 2 ) > 0. (3.2) By (3.1) and (3.2), there are certain combinations possible, which would lead to an unstable displacement process. For a more detailed examination of such fingering phenomena we refer to Homsy, 1987 [35]. 28

47 Parlange and Hill, 1972 [55], argument that for two phase flow with water and air as participating fluids, one might simplify the instability criterion (3.2) to v f < ϱ 1Kg µ 1 = K f. (3.3) In (3.3), fluid 1 represents the water phase. This simplification can be justified due to a much higher viscosity and density for the water phase than for the air phase. By (3.3), it is predicted that a flow is unstable for an infiltration rate of water ( q w ) smaller than the saturated hydraulic conductivity, otherwise it is stable (note that q w is a specific flux related to the flow cross-section and thus conforms to v f ). Furthermore, we notice that under said circumstances gravity is a force that benefits instability, whereas viscosity benefits stability. For other fluids and spatial arrangement this can also be exactly vice versa with gravity as stabilizing and viscosity as destabilizing factor. The ratio of both forces can be expressed by the dimensionless bond number, Bo = F g = ϱ wgr 2 F c σ [ ]. (3.4) It turned out that instability criterion (3.3) fails for fluid displacement in porous media with a sufficient high initial water saturation S w,i. Lu et al, 1994 [41], observed for an infiltration rate q w < K f a change from unstable to stable flow by exceeding a critical value Sw,i cr for the initial water saturation. Bauters et al., 2000 [3], conducted a series of infiltration experiments in one sand with different initial water content. They observed a gradual change from unstable flow for initially dry porous medium to stable flow for higher initial water saturations. Due to the neglected influence of capillary pressure and relative permeability in the studies of Saffman and Taylor, 1958 [63], their instability criterion might be seen only as a necessary but not as a sufficient condition for unstable flow phenomena in porous media. Besides initial water saturation, also the wave number ω [ 1 /m] of occuring instabilities can determine stability constraints due to capillarity effects. Thereby, the wave number reflects the quantity of instabilities. In Fig. 3.3 we look at two displacement processes where we assume that (3.3) would predict instabilities for both cases. According to that the fingers in the left sketch will further propagate into the domain. However, in the right sketch where the fingers are very close to each other, lateral diffusion caused by capillary pressure might act as a stabilizing factor and lead to a primarily not predicted stable displacement process [45]. Figure 3.3: Displacement process with different wave numbers. In certain cases with ω > ω cr, the displacement process will be stabilized by lateral diffusion. 29

48 Motivated by the results so far, let us now look upon the stability critera of complete multiphase flow equations. For this purpose we consult the Richards equation (RE). Due to its parabolic character in space and time the RE predicts for any infiltration scenario a monotonic saturation profile with a constant saturation Sw co that is adjusted behind the front, as soon as the unsaturated hydraulic conductivity K f,u (Sw co ) equals the infiltration rate q w [14]. Thus, we obtain fully saturated conditions behind the front in particular, when q w equals the saturated hydraulic conductivity K f. In any case, the infiltration front is not represented by an abrupt drop from Sw co to the initial saturation S w,i, but has a diffusive character due to capillary forces. Therefore, the front will be sharper for coarse-textured and more diffusive for fine-textured porous media (see Fig. 3.4 (left)). Figure 3.4: (left) Infiltration front is less diffusive for coarse textured porous media (curve 2) than for fine textured porous media (curve 3). Sw co equals one for q w K f (curve 1), otherwise Sw co is determined by an equilibrium between q w and K f,u. (right) Infiltration front for gravity driven fingering exhibits non-monotonic profile. It has been observed in several experimental works (see Sec. 3.4 and Sec ) that infiltration processes attributing gravity driven fingering exhibit a non-monotonic saturation profile. For an imbibing wetting phase this profile is illustrated in Fig. 3.4 (right). In addition, similar to the saturation also the pressure profile shows a non-monotonicity [13]. These non-monotonicities are seen to be the underlying reason for the instable behaviour of the flow field. However, it has been shown by stability analysis that the RE is simply not capable to describe such instabilities. For this, linear stability analysis have been carried out by Diment and Watson, 1983 [15], Ursino, 2000 [67], Kapoor, 1996 [36], Du et al., 2001 [16], and Egorov et al., 2003 [20]. By this means, stability could be proven only for infinitesimal perturbations. Otto (1997) [52], was the first, who showed by nonlinear stability analysis that the RE also is unconditionally stable for finite perturbations. His examinations were restricted to homogeneous porous media and have been further extended by Egorov et al. (2003) [20], to heterogeneous porous media. Although to the authors of this work there is no stability analysis known that deals with the fully coupled equation system (2.65), it is also seen to be not capable to describe gravity driven fingering phenomena. This in turn means that the macroscale multiphase models derived in Sec. 2 do not fully represent all essential physical flow mechanisms, which take place on smaller scales. Therefore, we have to go one step back and take another closer look at multiphase flow on the microscale. The following effort is aimed to improve the present models regarding in particular the observed non-monotonicities and instabilities in saturation and pressure. Thereby, the main focus will lie with the p c S relationship. In order to get a closed picture of all relevant physics we will begin with the very basics of water/air flow in porous media. 30

49 3.2 Fluid movement on the microscale The extended Darcy s law describes fluid flow on the macroscale (see Sec ) with the help of effective parameters. These parameters are introduced to account for processes taking place on smaller scales. Thereby, the two state variables density and viscosity are considered to be constant or only linearly dependent on pressure (see Sec and Sec ). Furthermore, their physical and mathematical models are widely applied and accepted also in other branches of science. In contrast to this, the physically correct description of macroscale capillary pressure and relative permeability are a much greater challenge in porous media science. In Sec , we primarily have assumed that a distribution of uniform capillary tubes reflecting the grain size distribution of porous media is capable to fully describe macroscale capillary pressure. We directly thereafter noticed that non-uniformities of natural pore systems require an experimental fitting of such obtained p c S models. However, we did not perform any theoretical investigations on the form of these non-uniformities. Likewise, we did not consider any dynamic influences on the system. Capillary pressure and relative permeability have been presumed to adjust instantaneously for any flow scenario. In order to understand porous media multiphase flow and in particular gravity driven fingering as well as possible let us take a closer look at the very basics of fluid flow. Roth, 2006 [61], describes imbibition in two phase flow scenarios of water and air as quickly succeeding jumps of the water phase from one pore to the next. On the macroscale these jumps are observed as continuous flux, which can be described by the extended Darcy s law. Fig. 3.5 illustrates the infiltration of water into a fully air saturated region. Figure 3.5: (left) Increasing slighly the pressure of water first leads to a readjustment of the interface from position 1 to position 2. (right) Further increase of the pressure then leads to a sudden jump into the next smaller pores. Assuming position 1 in the left picture to be in equilibrium, a further increase of the pressure of the water phase first leads to a readjustment of the interface (position 2). As soon as the pressure is high enough it provokes a sudden jump into the next smaller pores (position 3). Such jumps are so-called Haines jumps [26]. Lu et al, 1994 [40], [41], studied the behaviour of fluid interfaces for capillary rise and infiltration experiments into glass bead porous media. Thereby, they could observe two different flow characteristics depending on the initial water content. In the case of initially dry as well as in the case of initially wet conditions water moves upwards by fast succeeding jumps as described by Roth, 2006 [61]. However, for initially wet conditions these jumps are associated with a certain degree of film flow. Once more referring to Fig. 3.5, we notice that the jump from position 2 to 3 leads to an 31

50 uniform progression of the wetting front since neighbouring smaller pores are filled very fast due to capillarity. In contrast to that, Fig. 3.6 illustrates an imbibition process into a before drained porous medium that accordingly contains to a certain degree residual saturation of the water phase (position 2). Thinking 3-D, the before mentioned jump behaviour of the main forward moving wetting front can be delayed because a decisive amount of infiltrating water is transported via film flow. As observed by Lu et al, 1994 [40], this film flow might lead to a non-uniform wetting front profile into before drained porous media due to the non-uniform disposition of residual water saturation. Figure 3.6: (left) The situation is similar to the situation in Fig. 3.5 (left) apart from the fact that there now exists residual saturation of the imbibing fluid. (right) The residual saturation benefits film flow and thus delays the occuring jump behaviour. This picture of fluid movement on the microscale will serve us for the analysis of dynamic effects. However, before discussing this, we will first investigate the just mentioned residual saturation and potential different flow behaviour for drainage and imbibtion processes. 3.3 Hysteretic effects The expression hysteresis means that a variable (here: p c and k r ) is not only dependent on the value of an input parameter (here: S w ), but also on its history and direction. In order to be able to develop reasonable macroscale models the first steps again are microscale considerations Microscale considerations In multiphase flow in porous media we have two process directions: drainage and imbibition. In the following, we will identify three microscale effects contributing to different flow behaviour for these two cases. Subsequently to this, these effects will be accounted for by macroscale models. contact angle inkbottle effect phase trapping 32

51 Contact angel Fig. 3.7 illustrates a single pore once for an imbibition and once for a drainage process. For any drainage process the driving force of the fluid flow is contrary to the wetting force (adhesion force). This in turn leads to smaller contact angles for drainage processes than for imbibition processes, where both forces are acting in the same direction. Figure 3.7: Contact angle θ c for imbibition (left) and drainage (righ). Driving forces for flow and wetting forces are illustrated by black arrows. A further effect on the contact angle can occur due to potential anisotropic structures of the solid pore surface (see Fig. 3.8 ). Remembering Sec , we placed the contact angle in narrow glas tubes in the context of the respective capillary pressure. As a consequence, the here considered effects also should be considered for proper macroscale p c S models. Figure 3.8: Anisotropic porous medium Inkbottle effect For the so-called inkbottle effect let us assume a pore with a varying diameter (see Fig. 3.9). On this pore we will apply a virtual flow scenario. Thereby, we start at a fully saturated state at point 0 an primarily consider a drainage process. Therefore, the pressure of the wetting phase decreases and the non-wetting phase penetrates gradually into the pore until point 1 is reached, which is the location with the minimal radius on the left side. Further penetration of the non-wetting phase leads to a sudden depletion of the cavity until a pore radius on the right side is reached, where the pressure difference is about the height of the respective capillary pressure (point 2). At this point we let the drainage process stop and switch to an imbibition process, where the wetting phase repenetrates into the cavity. In contrast to the drainage process, there is no sudden depletion of the non-wetting phase from point 2 to 1. Instead of this the wetting phase gradually needs a higher pressure to reach point 3, which is the place with the maximal radius in the cavity. As soon as point 3 is exeeded, the wetting phase immediately penetrates the whole pore until a place is reached where the radius is again about the height of point 3. 33