Stefan Karpinski. PhD Thesis

Transcription

1 Numerical analysis of an interior penalty discontinuous Galerkin scheme for two phase flow in heterogeneous porous media with discontinuous dynamic capillary pressure effects Stefan Karpinski PhD hesis Promoters: Prof. Dr. Iuliu Sorin Pop, University of Hasselt, Belgium Prof. Dr. Florin Adrian Radu, Bergen University, Norway Doctoral committee members: Prof. Dr. Rainer Helmig, University of Stuttgart, Germany Prof. Dr. Iuliu Sorin Pop, University of Hasselt, Belgium Prof. Dr. Florin Adrian Radu, Bergen University, Norway Prof. Dr. Jochen Schütz, University of Hasselt, Belgium Hasselt University Faculty of Sciences Discipline group Mathematics and statistics Research group Computational mathematics

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3 Dedicated to my loving wife, without whom I wouldn t have written this...

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5 Acknowledgments I am extremely thankful to my supervisor Prof. Dr. Sorin Pop for his readiness to accept me as his student despite the challenges of guiding an external thesis. I am grateful for his constant support, compassion, and encouragement that reinforced my belief in my ability to finalize this thesis. I couldn t have wished for a better supervisor! I am also grateful to Prof. Dr. Florin Adrian Radu for agreeing to supervise my thesis together with Sorin, and for helping to find a good direction to take my thesis forward. I want to thank Prof. Dr. Jochen Schütz for his detailed and diligent proof-reading of my thesis. I also acknowledge the support of Prof. Dr. Barbara Wohlmuth and Prof. Dr. Rainer Helmig for introducing me to this interesting topic, guiding me through the first years of PhD, and introducing me to Sorin. I take this opportunity to also thank Dr. Roman Rojko for showing interest and support in my thesis, and for mentoring me in my first years of work in the industry. My thanks also to Dr. Stefan Zaprianov and all my colleagues at ESPRi Engineering GmbH. Working with them is an enjoyable and a valuable experience. I also thank my parents for encouraging me to pursue academics, for supporting me through my studies, and for being there when ever I needed them. Special thanks to all my friends for being my sanity in hard times and for always being there for a beer or a game of Munchkin or Borderlands as and when the occasion demanded it! Lastly, and more importantly, I want to thank my loving and amazing wife Shubhangi Gupta. She helped me wherever she could and encouraged me through all my different phases throughout the PhD. Be it moral support, scientific discussion, or just a hug when things got too stressful, she was alway there for me. Without her I would probably become a boring, miserable, grumpy old man. For this, and everything else, I thank her very much. i

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7 Abstract We present an interior penalty discontinuous Galerkin scheme for a two-phase flow model in heterogeneous porous media. he model includes dynamic effects and discontinuities in the capillary pressure. We define the interface conditions arising across material interfaces in heterogeneous media and show how to account for capillary barriers. We numerically approximate the mass-conservation laws without reformulation, i.e. without introducing a global pressure. We prove the existence of a solution to the emerging fully discrete systems, show the convergence of the numerical scheme, and obtain error-estimates for sufficiently smooth data. We also present a linearization scheme for the non-linear algebraic system resulting from the fully discrete discontinuous Galerkin approximation of the model. he linearization scheme does not require any regularization step. Additionally, in contrast with Newton or Picard methods, the linearization scheme does not involve computation of derivatives. Finally, to validate our theoretical findings and to show the scope of the applicability of the scheme, we present 1D and D numerical examples in realistic settings for homogeneous as well as heterogeneous porous media. We rigorously prove that the scheme is robust and linearly convergent. iii

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9 Contents 1 Introduction 1 Mathematical model 7.1 Model assumptions Governing equations Closure relationships Primary variables Constitutive relationships Dynamic effects in the phase pressure difference Relative permeabilities Initial and boundary conditions Interface Conditions Notation Weak formulation Numerical scheme Discretization in space Discretization in time Discrete system Numerical treatment of the heterogeneities Pressure condition Discrete scheme with interface conditions Flux continuity Convergence Analysis of the Numerical Scheme Existence of a discrete solution Discrete energy estimate Error Estimates Estimate for the non-wetting phase Estimate for the wetting phase Estimate for the capillary pressure Convergence result v

10 vi 4.4 Numerical Experiments Linearization scheme Discrete system Linearization of the interface condition Convergence Analysis of the Linearization Scheme Estimate for the non-wetting phase Estimate for the wetting phase Estimate for the capillary pressure Combined estimates Numerical Example est-setting Parameter study Numerical Examples in Heterogeneous Media Example 1: Inflow problem with homogeneous and heterogeneous permeability fields Example : Benchmark problem for verification of the numerical scheme with discontinuous capillary pressure Example 3: Comparison of Newton scheme and L-scheme with discontinuous capillary pressure Example 4: D Lens problem Outlook 89 Bibliography 93

11 Chapter 1 Introduction Flow and transport processes in porous media are of high interest in many different fields of application, e.g., geological CO -storage [Nordbotten and Celia, 011], environmental pollution [Radu and Pop, 011], designing of diapers [Diersch et al., 010], filters, etc. In light of their relevance, it is essential to develop a better understanding of such systems. Experimental studies are an important and indispensable tool to understand the behaviour of these processes. However, experimental studies are not always possible, nor feasible. Mathematical modelling and simulation tools, relying on mathematical and numerical analysis, provide an attractive alternative towards studying these processes with minimal societal and environmental impact. In this context, porous media models have been developed for describing flow and transport processes at various scales [Bear, 013; Helmig, 1997], and many different simulation and discretization techniques have been proposed in the literature. An important property of the porous media flow models is the local mass conservation. It is, therefore, desirable that the numerical schemes used for approximating these models reflect this property. Important classes of methods that are locally mass-conservative include finite volume methods [Eymard et al., 003; Helmig, 1997], mixed finite element methods [Durlofsky, 1993; Radu and Pop, 011; Radu et al., 015a], and discontinuous Galerkin methods [Bastian, 014; Bastian and Riviere, 003; Ern et al., 010; Sun and Wheeler, 005]. Most of the standard models for two phase flow in porous media were traditionally developed for large scale reservoir simulations, and typically assumed equilibrium conditions between the two phases. In these models, it is common to neglect the capillary effects, or to model the capillary effect using nonlinear algebraic relationships between the phase pressure difference and the saturation of one of the phases (commonly, the wetting phase. Such relationships are obtained experimentally, typically based on measurements that were made over long times so that the phases are in equilibrium [Helmig, 1997; Nordbotten and Celia, 011]. Over the last couple of decades, applications involving smaller scales, like laboratory scale, have increasingly attracted attention, particularly from the scientific community, which has lead to efforts in developing new modelling and discretization approaches. he flow behaviour at these small scales is different from the large field or reservoir scales in the 1

12 sense that it is essentially dynamic, characterized by non-standard effects like hysteresis and dynamic capillarity. Such effects can explain experimental results like saturation overshoot [Di Carlo, 004] that are ruled out by standard models. In this thesis, we are particularly interested in the dynamic capillarity effects, and consider the models where the pressure difference - saturation relationship additionally involves a dynamic term, as proposed in [Fucik and Mikyska, 011; Hassanizadeh and Beliaev, 001; Hassanizadeh and Gray, 1993; Hassanizadeh et al., 00]. We refer to these models as non-standard models. In contrast to standard, equilibrium based porous media flow models, dynamic (or non-equilibrium models, i.e. non-standard models, can explain effects like saturation overshoot and finger-pattern formation, which have been observed experimentally. he ability of non-equilibrium models to explain the small-scale experimental results as mentioned above has been proved by means of mathematical analysis. For example, the occurrence of non-monotonic travelling wave profiles depending on the magnitude of the dynamic capillarity effects has been analyzed rigorously in [van Duijn et al., 013]. he existence and uniqueness of weak solutions for such types of models has been proved in [Cao and Pop, 015, 016; Fan and Pop, 011; Koch et al., 013; Mikelic, 010; Rätz and Schweizer, 014] or [van Duijn et al., 016]. Compared to the standard models, the non-equilibrium based models encounter additional problems arising from the non-linear, possibly degenerate equations, which change their type depending on the choice of the unknowns. Further, the parameters and non-linearities are location dependent for realistic models in porous media and major difficulties arise when modelling the coupling between two homogeneous domains since the properties become discontinuous over the interface of the homogeneous domains. In order to obtain a consistent model for this case, it becomes necessary to impose proper and well designed interface conditions to account for the flux and mass conservation as well as the behaviour of the primary and secondary variables over the interface. he details and a mathematical derivation of these conditions can be found in [de Neef and Molenaar, 1997; van Duijn and de Neef, 1998; van Duijn et al., 1995]. Apart from mathematical modelling, the numerical solution of such models is also a challenging task. Firstly, rigorously designed numerical approximations are necessary to deal with the highly nonlinear, possibly degenerate models on each homogeneous subdomain. he numerical approximations for the homogeneous case have been studied over the last decades with techniques such as finite difference methods [Peszynska and Yi, 008] finite volume methods [Cao et al., 015; Eymard et al., 003, 010; Helmig, 1997; Helmig et al., 007, 009], finite element methods [Chavent and Jaffre, 1986; Chen, 001; Koch et al., 013; Rätz and Schweizer, 014], mixed finite element methods [Durlofsky, 1993; Radu and Pop, 011; Radu et al., 015a], and discontinuous Galerkin methods [Bastian, 014; Bastian and Riviere, 003; Epshteyn and Riviere, 009; Ern et al., 010; Karpinski and Pop, 017; Sun and Wheeler, 005]. Secondly, in order to account for the material discontinuities over the interface, a proper and consistent communication between the homogeneous domains must be incorporated into the numerical model, as shown by [Kueper and Frind, 1991a,b]. he material discontinuities lead to discontinuities in the numerical solution.pressures, and saturation can and will show discontinuities over some interface and these have to be properly resolved, which imposes additional restriction on the techniques and approximation methods that can

13 3 be used. here are different approaches to deal with this issue, like finite volume methods, finite element techniques extended by mortar methods [Arzanfudi et al., 014; Cancès et al., 009; Enchéry et al., 006; Helmig et al., 009; Hoteit and Firoozabadi, 008], and discontinuous Galerkin (dg methods [Bastian, 014; Ern et al., 010; Mozolevski and Schuh, 013]. A major benefit of dg methods is, that the scheme, in general, provides a discontinuous solution, and inherently includes inter-elemental interface conditions. herefore, extending the dg methods to the context of coupling of multiple homogeneous porous medium domains can be done naturally. he dg methods also have a direct advantage over the cg (continuous Galerkin methods which are continuous over inter-elemental interfaces, and thus, lead to either oscillations or unnecessarily smeared out solutions. he dg methods have grown more popular in the last decades due to their versatility and easy adaptation to include heterogeneities, parallelization, and hp-adaptivity. hese methods are well developed for standard, equilibrium based two-phase flow problems [Epshteyn and Riviere, 009]. In this thesis, we analyze and implement an interior penalty discontinuous Galerkin (IPdG method for the non-standard porous media models involving two phase flow with dynamic capillarity effects. We also discuss an extension of the IPdG approximation to deal with possible discontinuities when solving two phase flow problems in heterogeneous porous media. he spatial discretization is built on [Bastian, 014; Ern et al., 010], and is extended to include dynamic capillary pressure effects as in [Karpinski and Pop, 017; Karpinski et al., 017]. For time discretization, an implicit Euler approximation is used in order to avoid restrictions on the timestep size due the temporal discretization. We obtain a numerical scheme which is capable of a proper approximation of heterogeneous porous media, which is comparable to the results in [Helmig et al., 007, 009]. A common approach when dealing with two phase flow models, both standard and nonstandard, is to employ the so-called global pressure, which allows rewriting the system in a way that some nonlinear factors in the higher order terms become linear [Chavent and Jaffre, 1986]. he advantage of this approach is that the a priori estimates can be obtained separately for each of the transformed pressures, which can then be used to estimate the saturation. his approach is detailed in [Epshteyn and Riviere, 009]. he drawback of this approach lies in the fact that the global pressure is not a physical quantity, and one needs to post-process the results for extracting information that is relevant for the actual application. his is often quite cumbersome for realistic problems. If the mass balance equations are not reformulated in terms of the global pressure, the model with the original physical unknowns leads to a strong coupling of the mass balance equations, making it impossible to obtain the a priori estimate for the pressures directly. Instead, both pressure and saturation need to be estimated simultaneously, as done by [Eymard et al., 003] and [Koch et al., 013]. In [Eymard et al., 003], the estimates were derived for a standard model using finite volume approach, while in [Koch et al., 013], the estimates were derived for non-standard model with dynamic capillarity effects using finite element approach. In this thesis, we derive estimates for phase pressures and saturation and provide a rigorous convergence proof for an IPdG approximation for a non-standard two-phase flow model with dynamic capillarity. he mathematical model in our case is highly non-linear and therefore, very demanding in terms of the numerical solution. o resolve the non-linearities, the usual methods include Newton or Picard methods, see e.g. [Bergamaschi and Putti, 1999; Celia et al., 1990; Neumann et al., 013], a combination of Newton and Picard methods [Lehmann and Ackerer,

14 4 1998; List and Radu, 016], or iterative IMPES (implicit pressure explicit saturation [Kou and Sun, 010a,b]. he Newton method shows a local convergence with quadratic order, but only if the initial guess is close enough to the solution, while Picard-iterations are more robust but show only a linear convergence. For designing Newton or Picard methods for degenerate problems, as appearing in porous media flows, it also becomes necessary to include a regularization step. Another noteworthy alternative to Newton s method, especially for multicomponent flows, is the semi-smooth Newton method [Kraeutle, 011]. his method has the advantage that it includes the equilibrium conditions within the nonlinear solver, which leads to a stable solution strategy. Its drawback, however, is in its relatively high implementation cost. As mentioned, the Newton-scheme shows a high convergence order which makes it very attractive for solving nonlinear problems. However, the Newton-method requires the calculation of the Jacobian matrix (or at least a proper approximation of it for any iteration step, which, in general, is computationally expensive. Additionally, to guarantee the convergence of the iterations, the initial guess should be close enough to the solution. his aspect was analyzed in e.g. [Park, 1995] for the mixed finite element discretization for nonlinear elliptic problems, where they show that the difference between the initial guess and the exact solution should be of order h d (h being the mesh size and d the dimension of the domain for convergence. For parabolic partial differential equations, a straightforward choice for the initial guess is the solution obtained at the previous time-step. Nevertheless, to ensure that this is indeed close enough, the time-step must be chosen sufficiently small, again of order h d. his restriction becomes more severe when degenerate parabolic problems are considered. In this case, in locations where one of the phases is not present, the permeability of this phase vanishes leading to singular Jacobian matrices and ill-conditioned linear systems. o avoid this, it becomes necessary to regularize the problem, i.e. to consider perturbations assuring that the problem remains non-degenerate. his is an additional source of errors in the system. More importantly, the restriction on the time step becomes even more severe in this case, as it additionally involves a small regularization parameter (see [Radu et al., 006]. Similar issues appear for reactive flow models with non-lipschitz rates [Radu and Pop, 011]. hese issues with the Newton method have motivated the linearization schemes proposed in [List and Radu, 016; Pop et al., 004; Radu et al., 015a,b; Slodička, 00, 005a,b; Yong and Pop, 1996] for the finite element, finite volume, and the mixed finite element discretization of porous media flow models. he idea of the linearization scheme is to add an additional term in the form of L (Solution Current Iteration Solution Old Iteration, with L being a parameter that has to be chosen sufficiently large. he robustness of such schemes (also called L-schemes for standard porous media flow models is proved in the papers mentioned above. Although the L-schemes show only a linear convergence, they may become faster than the Newton method as they do not require the computation of derivatives. Additionally, the L-schemes do not involve any regularization step, and lead to better conditioned linear systems within each iteration (see [List and Radu, 016], where also the possibility of combining the L-scheme with the Newton iteration has been discussed. he L-schemes may even involve the same matrix for the linear algebraic system, which offers the possibility to compute its factorization only once per time step.

15 5 Inspired by the above results, in this thesis we also propose a linearly convergent iterative L-scheme for our model, i.e. a non-standard two phase porous media model of pseudo parabolic type involving a dynamic term in the phase pressure difference - saturation relationship (the dynamic capillarity. he model formulation for developing this scheme does not involve any global or complementary pressure, as opposed to the case in [Radu et al., 015a,b]. We present a rigorous convergence proof for the L-scheme, and provide numerical experiments confirming the theoretical findings. hese experiments also include heterogeneous media. o the best of our knowledge, this is the first time when such a scheme has been tested for the case of a heterogeneous medium. Layout of the thesis In Chapter, we introduce our mathematical model for a two phase flow in homogeneous and heterogeneous porous media with dynamic and discontinuous capillary pressure effects. In Chapter 3, we develop an interior penalty discontinuous Galerkin (IPdG based numerical discretization scheme for our mathematical model. Next, in Chapter 4, we analyze our discretization scheme and prove the existence of a discrete solution, the energy estimate for the discrete solution, and the convergence of the scheme. In Chapter 5, we propose a new linearization method for our IPdG scheme to resolve the non-linearities in the discrete model. he proposed scheme is based on [Radu et al., 015b] and [List and Radu, 016]. Following this we prove the convergence of the linearization scheme by estimating the errors of the iteration step with the solution at the next time step. Finally, in Chapter 6, we present several 1D and D numerical examples in heterogeneous porous media with and without discontinuous capillary pressure effects to show the capabilities of our numerical scheme. We conclude the thesis by summarizing our work and presenting an outlook on the various possibilities for extending this work in future in Chapter 7.

16 6 List of publications he following journal publications have resulted from this PhD work: 1. Stefan Karpinski, Iuliu Sorin Pop, Analysis of an interior penalty discontinuous Galerkin scheme for two phase flow in porous media with dynamic capillary effects, Numerische Mathematik, 136(1, 49-86, 017, doi: /s (published.. Stefan Karpinski, Iuliu Sorin Pop, Florin Adrian Radu Analysis of a linearization scheme for an interior penalty discontinuous Galerkin method for two phase flow in porous media with dynamic capillarity effects, International Journal for Numerical Methods in Engineering, 017, doi:10.100/nme.556 (published. 3. Stefan Karpinski, Iuliu Sorin Pop, Florin Adrian Radu An interior penalty discontinuous Galerkin scheme for two phase flow in heterogeneous porous media with discontinuous dynamic capillary pressure effects, (in preparation. he main results of Article 1 can be found in Chapter 3 (Sections 3.1, 3., and 3.3 and Chapter 4. he main results of Article can be found in Chapter 5 (Sections 5.1, 5.3, and 5.4. Results of Article 3 can be found in Chapter (Section.7, Chapter 3 (Section 3.4, Chapter 5 (Section 5., and Chapter 6.

17 Chapter Mathematical model In this chapter, we introduce our mathematical model for an isothermal two phase flow in porous media with dynamic capillarity effects. We first describe the model for homogeneous media, and then extend it to heterogeneous domains by defining appropriate interface conditions at the material interfaces..1 Model assumptions We let Ω R d (d = or 3 be an open bounded polygonal domain (the porous medium with boundary Γ and > 0 a maximal, finite time. Both Γ and are considered dimensionless. We consider a Darcy scale model for the flow of two incompressible and immiscible fluids (one wetting, and one non-wetting through a porous medium. his is based on the following assumptions: - All physical processes are isothermal. Extension to the non-isothermal case would require solving an additional governing equation for the temperature. his is done in e.g. [Acosta et al., 006; Gupta et al., 015]. he temperature, however, only negligibly affects the capillary pressure and the associated dynamic effects. herefore, in our model, we ignore the temperature. - Flow velocities lie well within the Darcy regime. For some special applications, like highly fractured media, where the flow at microscale is possibly turbulent in nature, the Forchheimer s law can be used instead of the Darcy s law. Some examples of such models can be found in [Hornung, 01]. - Porous matrix is rigid. For those porous media applications where the deformations of the porous matrix are large and cannot be ignored, the model can be extended using concepts of poroelasticity or poro-plasticity. Some examples can be found in [Bause et al., 017; Both et al., 017; Gupta et al., 015]. 7

18 8. Governing equations Under the assumptions stated above, the mathematical model [Helmig, 1997; Nordbotten and Celia, 011] includes the mass conservation laws for each phase (the wetting and non-wetting, denoted by α = w and n, respectively: t (s α φρ α + (ρ α u α = q α. (.1 Here, φ denotes the porosity of the medium, ρ α the fluid phase densities, s α the saturation of phase α, and q α the volumetric sources or sinks. It is assumed that the densities are constant and that system (.1 can be divided by the densities. Further, u α is the Darcy velocity of the phase α, given by u α = λ α (s w K (p α gzρ α. (. Here, p α is the pressure of the phase α, K the intrinsic permeability tensor, g the gravitational constant with the gravitational potential z, and λ α = k r,α µ α is the mobility function for phase α, with relative permeability k r,α and dynamic viscosity µ α. Remark 1 Note that in general the intrinsic permeability tensor K can be non-symmetric. In this thesis, however, we assume K as a symmetric positive definite tensor..3 Closure relationships he four governing equations presented above contain six unknown quantities, viz. phase saturations, pressures and velocities of both phases. o close the system we consider two more conditions: 1. We assume that at any given point only two phases are present in the system, such that, s w + s n = 1. (.3. We parameterize the phase pressure difference as a function of saturation similar to the standard models [Helmig, 1997], and extend this relationship with a term involving the time derivative of the saturation [Hassanizadeh and Gray, 1993] to include the nonstandard dynamic capillarity effects, p c := p n p w = p c (s w, t s w. (.4.4 Primary variables We chose three primary unknowns, viz. wetting phase saturation s w, non-wetting phase pressure p n, and phase pressure difference p c, and rewrite the above model as a system of three equations: t s w φ (λ n (s w K (p n gzρ n = q n, t s w φ (λ w (s w K (p n p c gzρ w = q w, p c = p c (s w, t s w. (.5 Note: For readability of the proofs in the subsequent sections, we use p w = p n p c, although p w is a secondary variable.

19 9.5 Constitutive relationships he constitutive models for the properties of the fluid-matrix interaction, viz. phase pressure difference (or capillary pressure p c and relative permeabilities k r,α are described below..5.1 Dynamic effects in the phase pressure difference he pressure difference across the wetting and non-wetting phase interface is called the capillary pressure. his pressure difference arises due to balancing of cohesive forces between the fluids and the adhesive forces between the fluid-matrix interfaces. On a pore scale, the capillary pressure is inversely related to the radius of the pore-throat. A common assumption in the modelling of two phase flow in porous media is that the distribution of the two phases inside the pores of the medium is static, and the phase pressure difference depends only on the properties of the medium and the volumetric distribution of the phases. he models which are built on this assumption are called the standard, equilibrium-based models. hese models relate the phase pressure difference p c and saturation (commonly, the wetting phase saturation s w through nonlinear, algebraic functions which are uniquely invertible. Several parameterizations relating p c and s w using medium specific parameters have been proposed in the literature. Most prominent examples of such parameterizations include the Brooks-Corey model [Brooks, 1964], and the van Genuchten model [Mualem, 1976; van Genuchten, 1980]. hese standard models are valid whenever the processes are slow enough, so that the dynamics of the flow, and in particular the redistribution of the phases inside the pores before achieving equilibrium, can be ignored. In the recent years, experiments have brought forth some limitations of the equilibriumbased models. For example, the experiments in [Di Carlo, 004] showed that non-monotonic saturation profiles (over-shoots can be obtained during infiltration processes in a dry porous medium, and that the amplitude of such over-shoots depends on the flow velocity. Such results lie beyond the scope of the equilibrium-based models, which would predict monotonic profiles regardless of the chosen parameterization. herefore, alternative modelling theories were required. o model non-monotonic profiles, several approaches were proposed where the assumption of a uniquely invertible representation of the capillary pressure relationship was dropped, and non-equilibrium or dynamic models were introduced to capture the additional effects observed in the experiments. hese models are called the non-standard models. Many nonstandard models have been proposed, amongst others [Barenblatt et al., 003; Bourgeat and Panfilov, 1998; Hassanizadeh and Gray, 1993]. An extensive review of these and some other models can be found in [Manthey, 006], where also a more detailed explanation of each model is included. In this thesis, we focus on the model developed by Hassanizadeh and Gray in [Hassanizadeh and Gray, 1979a,b, 1993], which was analyzed from a thermodynamical perspective in [Hassanizadeh and Beliaev, 001]. In this model, the equilibrium-based p c -s w relationship was extended by introducing a dynamic damping parameter τ(s w > 0, which can depend on the wetting phase saturation, but is often assumed constant. his parameter accounts for the dynamic change of the capillary pressure, leading the a non-equilibrium

20 10 (non-standard model. he capillary pressure is then defined as: p c = p c,eq (s w τ t s w, (.6 involving the time derivative of the saturation. Here, p c,eq is the capillary pressure at equilibrium, and τ accounts for the dynamic effects. We assume τ to be a positive constant. he additional dynamic term, i.e. τ t s w, delays the flow of the wetting phase and forces it first to build up before the capillary pressure can be overcome. his leads to a non-monotonic flow behaviour and a droplet formation (over-shoot at the tip of an infiltration finger. Other non-standard effects in the capillary pressure relationship, like terms of even higher order or hysteresis effects, are not accounted for in this thesis, but can be found in e.g. [Cao and Pop, 015; Jha et al., 011; Rätz and Schweizer, 014]..5. Relative permeabilities In this work, we study the dynamic effects only in the phase pressure difference. For relative permeabilities, we assume that the dynamic effects are negligible, and use the equilibriumbased models like Brooks-Corey [Brooks, 1964] or van Genuchten [van Genuchten, 1980] in conjunction with the Mualem and Burdine relations [Burdine, 1953; Mualem, 1976] to parameterize relative permeabilities as functions of wetting phase saturation..6 Initial and boundary conditions o complete the system, we use the following initial and boundary conditions: For all x Ω and at t = 0, s w (x, 0 = s 0 (x with, s 0 H 1 (Ω. (.7 For all x Γ and all t [0, ], p c (x, t = p D c (x, p n (x, t = p D n (x (.8 with p D n H 1 (Γ, p D c H 1 (Γ where, s 0, p D n, and p D c are given functions. H 1 (Ω and H 1 (Γ are Sobolev spaces, on Ω or Γ respectively. A definition and introduction to Sobolev spaces can be found in [Adams and Fournier, 003] or [Evans, 1998]. Note that the boundary value of s w is defined implicitly by the Dirichlet conditions for p c. Remark For simplicity, here only Dirichlet boundary conditions are considered. Also, the boundary values are assumed constant in time..7 Interface Conditions he model presented above is valid only in homogeneous domains. o extend this model to heterogeneous domains, consider a very simple heterogeneous porous media consisting of two distinct homogeneous domains Ω h, and Ω l, separated by a material interface Γ. he

21 11 subscript l and h refer to a lower and a higher equilibrium capillary pressure, i.e. for each s w (0, 1] holds p c,eq,l (s w p c,eq,h (s w, respectively. (See figure.1. he lower and higher capillary pressures correspond to higher and lower absolute intrinsic permeability respectively. At the material interface, due to the different properties, additional conditions need to be considered Pressure [Pa] Saturation [-] Figure.1: Equilibrium capillary pressure, with Brooks Correy parameterization. In red p c,eq,h ( and blue p c,eq,l (. For the standard models, the interface conditions have been studied extensively in e.g. [Bertsch et al., 003; Buzzi et al., 009; Cancès, 008; van Duijn and de Neef, 1998] as well as [Kueper and Frind, 1991a,b]. In [van Duijn and de Neef, 1998], an analytical solution was derived and analyzed. (Note that in [van Duijn and de Neef, 1998] lower and higher refers to the absolute permeability. An important aspect of flow in heterogeneous media is that, the flow from the high capillary pressure domain Ω h to the low capillary pressure domain Ω l is possible only if the capillary pressure on Ω l exceeds the entry pressure on Ω h. his behaviour is commonly known as a capillary barrier. Numerical methods for modelling capillary barriers/standard models have been studied by [Arzanfudi et al., 014; Cancès et al., 009; Enchéry et al., 006; Hoteit and Firoozabadi, 008] For the non-standard models, the interface conditions were first studied and extended in [Helmig et al., 007, 009] and [Peszynska and Yi, 008]. Recently, these interface conditions were analyzed rigorously in [van Duijn et al., 016]. It was shown that if τ > 0, under certain conditions, flow of the non-wetting phase from Ω l to Ω h is possible even if the capillary pressure on Ω l has not exceeded the capillary barrier. o connect the models on each homogeneous subdomain, the following two conditions must be satisfied for any flow across the interface: - continuity of the normal component of the fluxes across the interface, and

22 1 - continuity of the pressures across the interface. he first condition ensures the conservation of mass flux over the interface and leads to the following condition: λ α,h (s w,h K h (p α,h gzρ α n = λ α,l (s w,l K l (p α,l gzρ α n, (.9 where, n is the unit-normal vector pointing into Ω l. he second condition assumes continuity of the pressures over the interfaces and implies a continuity of the capillary pressure. his may lead to a discontinuity in the saturations across the interface. wo distinct values of saturation and capillary pressure may exist at each side of the interface. We denote the values at the interface associated with the domain Ω h as s n,h and p c,h, and the values at the interface associated with the domain Ω l as s n,l and p c,l. We define the pressures p c,e,l := p c,eq,l (1 and p c,e,h := p c,eq,h (1, (.10 which will denote the entry pressures. Further, we define the capillary pressure difference δ pc,γ over the interface Γ by: δ pc,γ := p c,h (s w p c,l (s w (.11 his definition translates directly to the capillary pressure potential p c : p c,h p c,l = δ pc,γ. (.1 Depending on the chosen parameterization of the capillary pressure, the condition of pressure continuity is not always valid. In the case of van Genuchten like parameterization, the pressure continuity has to be always fulfilled. In the case of Brooks Correy like parameterization, the pressure continuity holds true only if both phases are present on both sides of the interface. Otherwise, a capillary barrier exists across the interface, preventing flow from the domain with low capillary pressure to that with high capillary pressure. he discontinuity in capillary pressure leads to a discontinuity in the non-wetting phase pressure, whereas the wetting phase pressure remains continuous across the interface. Physically this can be attributed to the fact, that a phase pressure has no meaning when the phase is not present. herefore the phase pressure differences, i.e. the capillary pressure becomes meaningless. In the standard model an extension of the capillary pressure from (0, 1 to [0, 1] is possible due to a continuous extension. In the non standard model this is not possible, as no unique extension can be chosen due to the dynamic capillary effect as presented in [van Duijn et al., 013]. Following [van Duijn et al., 016], whenever s w,h < 1 and s w,l < 1, the non-wetting phase is present on both sides of the interface and the following holds on Γ: p c,eq,l (s w τ l (s w t s w = p c,eq,h (s w τ h (s w t s w. (.13

23 If instead no non-wetting phase is present on Ω h and p c,l < p c,e,h, i.e. not enough nonwetting fluid is present to overcome the capillary barrier, (.13 is replaced by Combining (.13 and (.14 this leads to: 13 s w,h = 1. (.14 (1 s w,h (p c,h p c,l = 0. (.15 Note that, the dynamic capillary pressure effect results in an ordinary differential equation over the interface. We elaborate on the behaviour of the pressures in the following three cases: A. non-wetting phase is absent on the Ω h side of Γ, B. non-wetting phase is absent on the Ω l side of Γ, C. non-wetting phase is present on both sides of Γ. he behaviour is described only from the perspective of the non-wetting phase. Case A: In this case, s w,h = 1 and t s w,h = 0, which implies p c,h = p c,e,h. As long as p c,l (s w,l < p c,e,h, no flow is possible across the interface. his means that the nonwetting phase has to accumulate until the capillary pressures are balanced, i.e. p c,l (s w,l = p c,h (s w,h. Only after this, the non-wetting fluid can flow into Ω h. his is the capillary barrier in the standard case, which leads to a jump in the capillary potential p c denoted by δ pc,γ in (.1. In the standard case s w,l decreases towards the threshold value s w for which the condition p c,eq,l (s w = p c,e,h holds. In the non-standard case however the entry pressure may be reached at values s w,l > s w, due to the dynamic effects of τ > 0. his leads to scenarios, where flow is possible which can not be observed in the standard case. his is also described in detail in [van Duijn et al., 016]. In this case, the wetting phase is always present on both sides of the interface. his implies continuity of the wetting pressure potential and leads to the following additional condition on the non-wetting pressure potential : p n,h p n,l = p c,h p c,l = δ pc,γ. (.16 his condition will be used in Section 3.4 when constructing the discontinuous Galerkin scheme. Case B: Analogously to case A, here s w,l = 1 and t s w,l = 1, leading to p c,l = p c,e,l. In this case, p c,h (s w,h p c,e,l always holds. herefore, no capillary barrier occurs and flow across the interface is directly possible. he pressures are always continuous. Case C: In this case, the pressure continuity and capillary pressure continuity always hold, and flow across the interface is always possible.

24 14 Summarizing Cases A, B and C, as well as (.15 and (.9, we obtain the following relations over the interface Γ: λ α,h (s w,h K h (p α,h gzρ α n = λ α,l (s w,l K l (p α,l gzρ α n (.17 (1 s w,h (p c,h p c,l = 0 (.18 where, n defines the vector normal to the interface. Note that, the condition p c,h (s w,h = p c,e,h corresponds to s w,h = 1..8 Notation he notations below are common in functional analysis [Adams and Fournier, 003] and will be used throughout this thesis. Whenever values on Γ are involved, these should be understood in the sense of traces, recalling the definitions of the traces in [Evans, 1998, p. 70]. he following notation will be used in this thesis: - L p (Ω (1 p < is the usual space of functions that are p-lebesgue integrable and L (Ω is the space of functions that are essentially bounded in Ω. he elements of W k,p (Ω are the functions admitting weak derivatives up to order k that are again in L p. For simplicity, we use the notation H k (Ω for W k, (Ω. - For 1 p, Lp (Ω and W k,p (Ω are the standard norms in L p (Ω, respectively W k,p (Ω. A simplified notation will be used for the norm in W k, (Ω, namely Ω,k. - H k 0 (Ω denotes the subspace of H k (Ω taking the value 0 on the boundary (in the sense of traces. - L q (0, ; W k,p (Ω denotes the Bochner space of vector spaced valued functions f : [0, ] W k,p (Ω that are q-bochner integrable on [0, ]. - H 1 (0, ; L (Ω denotes the Bochner space of L (Ω valued functions admitting a weak time-derivative in L (0, ; L (Ω. As for the domain Ω, the traces on Γ will lie in spaces like L p (Γ, H k (Γ, etc. In particular, by H 1 (Γ we mean the traces on Γ of H 1 (Ω functions..9 Weak formulation We now state the weak formulation for our model (.5 together with the constitutive relationship (.6 and the initial and boundary conditions (.7 and (.8. We multiply the equations with test-functions in H 1 0 (Ω and partially integrate to obtain: Problem 1 [Weak formulation] Find the triple (s w, p n, p c s.t. s w H 1 (0, ; H 1 (Ω, s w = s 0 at t = 0, p n p D n L (0, ; H0 1 (Ω, p c p D c L (0, ; H0 1 (Ω, and for all

25 15 ψ p H0 1 (Ω, ψ s H0 1 (Ω, and almost every t [0, ] it holds t s w φψ p + λ n (s w K (p n gzρ n ψ p = q n ψ p, Ω Ω Ω t s w φψ p + λ w (s w K (p n p c gzρ w ψ p = q w ψ p, (.19 Ω Ω Ω p c ψ s = p c,eq (s w ψ s τ t s w ψ s. Ω Ω Ω Existence and uniqueness results for Problem 1 are obtained in [Cao and Pop, 015, 016; Fan and Pop, 011; Koch et al., 013; Mikelic, 010].

26 16

27 Chapter 3 Numerical scheme In this chapter, we develop a numerical discretization scheme for our mathematical model given by Problem 1. he discretization in space is based on an interior penalty discontinuous Galerkin method, while the discretization in time is based on an implicit Euler method. A particular focus is laid on the numerical treatment of the interface conditions for heterogeneous domains. Preliminaries Let be a decomposition of the domain Ω into N non-degenerate elements i. We assume that is admissible in the sense of the Definition.1 in [Di Pietro and Ern, 010]. Let F denote the union of all faces F j, and let h be the maximal diameter of the elements. Given i and F i F, we define a set F ( i of all the faces associated with the element i, s.t., F ( i := F j : F j i, F j F and, a set (F i of all the elements sharing the face F i, s.t., (F i := j : F i j. j In the conforming case, (F i consists of exactly two elements. With each face F F connecting element i and j, we associate a normal-vector n directed from i to j (j > i. 17

28 18 Following Sec in [Di Pietro and Ern, 01], let Π k ( denote the space of polynomials on with degree k. For the approximation of saturation S w, we consider the broken polynomial space with polynomials of order k s as, V s h (Ω := {v L (Ω : v Π ks ( for all }, (3.1 and, for the approximation of the pressures p n and p c, we consider the broken polynomial space with polynomials of order k p as, V p h (Ω := {v L (Ω : v Π kp ( for all }. (3. Note that we represent a general broken polynomial space with V h (Ω without specifying the polynomial order. For ψ i, ψ j V h (Ω, where, ψ i = (ψ i F is the trace of F on the side of the element i, and similarly, ψ j = (ψ j F is the trace of F on the side of the element j, we define the jump and the average { } over the face F as, when F is an interior face : ψ = ( ψ i ψ j and {ψ} = 1 ( ψ i + ψ j, (3.3 when F is a boundary face : ψ = ψ i and {ψ} = ψ i, (3.4 where, the interior face connects elements i and j with i < j, and the boundary face has no element adjacent to i. Next, we define the following norm on the broken polynomial space, v Ω,DG := i v i,0 + F i F and use the following lemma [Di Pietro and Ern, 010]: 1 F i v F i,0 (3.5 Lemma 1 Given a broken polynomial space V h (Ω, for any q such that, 1 q d d, if d 3 1 q <, if d =, there exists a constant Ĉ depending on the polynomial degree, mesh-parameters and Ω, but independent of the mesh size h, such that, for all v V h (Ω, the following inequality holds: v Lq (Ω Ĉ v Ω,DG. (3.6

29 19 Additionally, we use the following trace inequalities, which can be found in [Warburton and Hesthaven, 003], [Riviere et al., 001], or [Di Pietro and Ern, 01]: Lemma Let γ 0 denote the trace operator. here exists a constant C t independent of the mesh size h, such that, for any with F F ( and for all v H k (, the following holds: 1 γ 0 v 0,F C t F ( v 0, + F v 0, (3.7 For v Π k ( and a positive function f(k depending on the polynomial degree k, the following holds: f(k γ 0 v 0,F C t F v 0, (3.8 We also use the following elementary lemma [Epshteyn and Riviere, 009]: Lemma 3 Let C be the maximal number of elements sharing one face, and let A : [0, be a function defined on the triangularization. hen, the following inequality holds: A( C A( i i F i (F i Finally, we state the following well known (in-equalities for a, b R and ɛ R +, which are used throughout the paper: (a b a = 1 (a b + 1 (a b (3.9 ab ɛ a + 1 ɛ b. ( Discretization in space he weak form (.19 of the mathematical model governed by system (.5 is discretized in space using an interior penalty discontinuous Galerkin numerical scheme. Problem [Spatial discretization] Given the penalty parameters σ n, σ w R +, the parameter θ { 1, 0, 1} and the function f( introduced in Lemma depending on the polynomial order k p, find s w V s h (Ω, p n V p h (Ω and p c V p h (Ω, s.t., for all ψ s V s h (Ω,

30 0 ψ n V p h (Ω and ψ w V p h (Ω the following holds: PDE-1: t s w φψ n + λ n (s w K (p n gzρ n ψ n i i i i {λ n (s w K (p n gzρ n n} ψ n F F i F i +θ f(k p p n {λ n (s w K ψ n n} + σ n F F i F i F F i F i F i p n ψ n = θ p D n {λ n (s D f(k p K ψ n n} + σ n F F i Γ i F F i Γ i F i pd n ψ n (3.11 PDE-: t s w φψ w + λ w (s w K (p n p c gzρ w ψ w i i i i {λ w (s w K (p n p c gzρ w n} ψ w F F i F i +θ f(k p {λ w (s w K ψ w n} p n p c + σ w F F i F i F F i F i F i p n p c ψ w = θ {λ w (s D K ψ w n} p D n p D f(k p c + σ w F F i Γ i F F i Γ i F i pd n p D c ψ w (3.1 ODE-Pc: i p c ψ s = p c,eq (s w ψ s τ t s w ψ s (3.13 i i i i i he parameters σ n and σ w penalize discontinuities in the solutions (i.e., jumps over the faces. he choice of θ = 1 gives the non-symmetric-interior-penalty (NIP dg-scheme, θ = 0 gives the incomplete-interior-penalty (IIP dg-scheme, and θ = 1 gives the symmetricinterior-penalty (SIP dg-scheme. 3. Discretization in time For the discretization in time, we use an implicit Euler scheme. We subdivide the time domain [0, ] into N intervals of size t > 0 with = N t. he i-th discrete time-step is denoted by t i, s.t., t i = i t. Given a sufficiently smooth function g(x, t, the time derivative of g is approximated by: g := g(t, x g(t n, x t (3.14

31 1 3.3 Discrete system Using Problem and (3.14, the fully-discrete scheme can be written as: Problem 3 [Discrete problem at t ] Let Pn n V p h (Ω, P c n V p h (Ω and Sn w Vh s(ω, find Pn V p h (Ω, Pc V p h (Ω and S w Vh s(ω, s.t., for all ψ s Vh s(ω, ψ n V p h (Ω and ψ w V p h (Ω, the following holds: PDE-1: Sw φψ n + λ n (Sw K ( P n gzρ n ψn i i i i {λ n (Sw K ( P n gzρ n n} ψn F F i F i + θ Pn {λ n (S f(k p w K ψ n n} + σ n Pn ψ n F F i F i F F i F i F i = θ p D n {λ n (s D f(k p K ψ n n} + σ n F F i Γ i F F i Γ i F i pd n ψ n (3.15 PDE-: Sw φψ w + λ w (Sw K (Pn Pc gzρ w ψ w i i i i {λ w (Sw K (Pn Pc gzρ w n} ψ w F F i F i + θ {λ w (Sw K ψ w n} P n Pc F F i F i f(k p + σ w Pn Pc ψ w F F i F i F i = θ {λ w (s D K ψ w n} p D n p D c F F i Γ i f(k p + σ w F i F i pd n p D c ψ w (3.16 F i Γ ODE-Pc: i Pc i ψ s = i p c,eq (Sw i ψ s i i τ S w ψ s ( Numerical treatment of the heterogeneities In Section.7, we discussed the additional conditions that must be considered at the material interfaces separating homogeneous blocks in a heterogeneous porous medium, which include flux continuity (.17 and pressure continuity (.18 across the interface. In this section, we present the numerical realization of these conditions in our solution scheme. We use the ideas proposed by [Ern et al., 010] and used further in [Bastian, 014] for a fully

32 implicit coupled scheme. In both, [Ern et al., 010] and [Bastian, 014], stationary capillary pressure conditions were assumed. We extend the ideas and incorporate non-standard capillary pressure effects Pressure condition We exploit the structure of the dg-scheme and extend the penalty terms to realize the pressurecontinuity condition at the interface. his is possible because each element is an independent entity by itself; the communication between the elements occurs only through the interfacial terms, i.e., average terms and penalty terms. When the mesh-interfaces and the materialinterfaces are aligned, the penalty terms can be naturally extended to incorporate the physical behaviour at the material interfaces. his gives us the possibility to deal with edge aligned heterogeneities without additional numerical constructs. he dynamic capillary pressure effects for flow in heterogeneous porous media were first rigorously addressed by Weiss et al. in [Helmig et al., 009] and [Helmig et al., 007]. hey used finite volume scheme for spatial discretization, and described the interface conditions using variational inequalities, which were incorporated in the numerical scheme through a mortar technique. In addition, they used an active-set strategy together with an inexact Newton strategy to solve the non-linear system. o describe the numerical approximation, we use the same notation as in Section.7. o incorporate the jumps in the saturation, we introduce an additional penalty parameter to (3.17, which penalizes jumps over the interface, such that the pressure continuity is fulfilled. For a face F F with F Γ, we use the capillary pressure condition (.18 and define the the modified jump term: p c (S w = { p c,e,h p c,h (S w,h if p c,l(s w,l < p c,e,h and S w,h = 1 p c,h (S w,h p c,l(s w,l otherwise Whether case A, B, or C from Section.7 is applicable, is decided by the condition: p c,l (S w,l < p c,e,h and S w,h = 1,. (3.18 which guarantees that if case A applies, non-wetting fluid accumulates until p c,l reaches the entry pressure p c,e,h. As long as the capillary pressure is less than the entry pressure and no non-wetting phase is present on Ω h, case A is active and saturation will accumulate. Additionally, the capillary pressure potential and non-wetting pressure potential are discontinuous. o ensure that in case B no capillary barrier inhibits the flow, the condition S w,h = 1 is checked. In all other cases we ensure continuity of the pressure potentials, i.e. case C is active. In the present context for edges over the interface Γ one obtains p c (S w = 0,

33 3 which ensures the condition p c,e,h = p c,h (S w,h or p c,h(s w,h = p c,l(s w,l. Depending on the currently active case, this leads to continuous or discontinuous capillary pressure respectively. On the other edges we obtain pressure continuity. he additional condition (.16 in case A will lead to a discontinuity in the non-wetting pressure potential. In the spirit of (3.18 we extend the continuity condition, i.e. the jump terms, in the non-wetting phase mass balance equation (3.15. Using (.16 we define: P n = { P n,h P n,l δ n p c,f if p c,l (S w,l < p c,e,h and S P n,h P n,l otherwise w,h = 1, (3.19 where, we use δp n c,f as defined in (.1 with the additional superscript denoting the timestep at which the evaluation takes place. Also in this case, we have either pressure continuity or matches the jump in the capillary pressure. P n 3.4. Discrete scheme with interface conditions Using (3.19 and (3.18 we get the following scheme: Problem 4 [Discrete problem at t with interface conditions] Let Pn n V p h (Ω, P c n V p h (Ω, and Sn w Vh s (Ω. Find Pn V p h (Ω, Pc V p h (Ω, and S w Vh s (Ω, s.t., for all ψ s Vh s(ω, ψ n V p h (Ω, and ψ w V p h (Ω, the following holds: PDE-1: Sw φψ n + λ n (Sw K (Pn gzρ n ψ n i i i i {λ n (Sw K (Pn gzρ n n} ψ n F F i F i + θ Pn {λ n (S f(k p w K ψ n n} + σ n Pn ψ n F F i F i F F i F i F i = θ Pn D {λ n (s D f(k p K ψ n n} + σ n F i F i F i P n D ψ n (3.0 F i Γ D F i Γ D PDE-: Sw φψ w + λ w (Sw K (Pn Pc gzρ w ψ w i i i i {λ w (Sw K (Pn Pc gzρ w n} ψ w F F i F i + θ {λ w (Sw K ψ w n} P n Pc F F i F i f(k p + σ w Pn Pc ψ w F i F i F i F

34 4 = θ {λ w (s D K ψ w n} P n D Pc D F F i Γ i D f(k p + σ w F i F i P n D Pc D ψ w (3.1 F i Γ D ODE-Pc: i + σ s F i F Pc i Flux continuity ψ s = i p c,eq (Sw i ψ s i (Sw i F i f(k s F i p c,eq(s w (S w ψ w (3. he flux continuity is implicitly enforced by the proposed scheme 4. o show this we proceed the same way as in [Ern et al., 010]. For a solution of the scheme the following condition holds due to (3.0: Sw φψ n + λ n (Sw K (Pn gzρ n ψ n i i i i {λ n (Sw K (Pn gzρ n n} ψ n F F i F i + θ Pn {λ n (S f(k p w K ψ n n} + σ n Pn ψ n F F i F i F F i F i F i = 0 By partial integration and using the relationship, F i F uv = {u} v + {v} u, for the interior edges we obtain: ( Sw φ (λ n (Sw K (Pn gzρ n ψ n i i + {ψ n } λ n (Sw K (Pn gzρ n n F F i F i + θ Pn {λ n (Sw K ψ n n} F F i F i f(k p + σ n Pn ψ n = 0 F i F i he first term is the weak residual of the conservation law for the non-wetting phase. he second term enforces the normal flux continuity across the interface, and the third and the fourth terms ensure the continuity of the pressure. Note that we are using the extended jump ψ s

35 5 definition (3.19, leading to either a continuous or discontinuous pressure across the interface respectively. In either cases, due to the construction of the jump terms, for a convergent solution Pn = 0 holds true. In a similar way, we partially integrate the wetting phase equation (3.1 and get: ( Sw φ (λ w (Sw K (Pn Pc gzρ w ψ w i i + {ψ w } λ w (Sw K (Pn Pc gzρ w n F F i F i + θ Pn Pc {λ w (Sw K ψ w n} F F i F i f(k p + σ w Pn Pc ψ w = 0. F i F i F i F Again, the first term represents the weak residual of the conservation law for the wetting phase. he second term gives the normal flux continuity across the interface, and the third and the fourth terms ensure the continuity of the pressure.

36 6

37 Chapter 4 Convergence Analysis of the Numerical Scheme In this chapter, we prove that the numerical scheme is well-posed and convergent. We first show the existence of a discrete solution using a fixed-point argument, followed by the energy estimates for the discrete solutions. Finally, we show the convergence of the scheme by proving some error estimates. Preliminaries We make the following assumptions to prove existence and convergence of the numerical scheme: (A4.1 he initial and boundary conditions in (.7 and (.8 are sufficiently smooth. Additionally, the initial condition is compatible with the boundary condition, i.e. the initial condition fulfills the boundary condition at t = 0. (A4. he permeability matrix K R d d is symmetric and positive definite, i.e. there exist two constants κ and κ, s.t., for any vector x R d, the following holds: κ x x Kx κ x (A4.3 he equilibrium capillary pressure function p c,eq ( is in C (R, and is assumed to be positive, bounded and decreasing. Let P c,eq ( define the primitive, i.e.: P c,eq (S := { S 1 p c,eq(ξdξ = S 0 p c,eq(ξdξ 1 0 p c,eq(ξdξ for S 1 0 otherwise. (4.1 It can be inferred that P c,eq (S is concave and non-positive. (A4.4 he functions λ w ( and λ n ( are Lipschitz-continuous and bounded from above and below by the constants 0 < λ α < λ α <. 7

38 8 For the error analysis, let s w (t, x, p n (t, x, and p c (t, x be the exact solutions of the problem. For simplicity, we will use s i w = s w (t i, x, p i n = p n (t i, x, and p i c = p c (t i, x. For all t [0, ] we denote the projection of p n (t, p c (t, and s w (t onto the space V p h (Ω or V h s(ω with, p n (t V p h (Ω, p c(t V p h (Ω, and s w(t Vh s (Ω, respectively. Further, we assume for all t [0, ] that p n (t W 1, (Ω, p c (t W 1, (Ω and s w (t W 1, (Ω. We also assume that the solutions possess enough regularity such that the the following approximation properties are fulfilled: For all t [0, ], l pn R +, l pc R +, l s R +, and, for p n (t W 1,, p c (t W 1, and s w (t W 1, there exists a constant C independent of h, k s, k p and t s.t., for 0 < q l pn, for 0 < q l pc, for 0 < q l s, p n (t p n (t,q C p c (t p c (t,q C hmin(kp+1,lpn q lpn q kp hmin(kp+1,lpc q lpc q kp p n (t,lpn (4. p c (t,lpc (4.3 s n (t s n (t,q C hmin(ks+1,ls q ks ls q s n (t,ls (4.4 he proof for the results (4., (4.3 and (4.4 can be found in [Babuska and Suri, 1987]. Recall that the norm,q is defined as W q, (, see also Section.8. Further, we write the numerical errors for i = 1,..., N as, e i s,h = Si s i w, e i s = s i w s i, e i p α,h = P i α p i α, e i p α = p i α p i α. 4.1 Existence of a discrete solution We now prove the existence of a discrete solution for the Problem 3. For given real numbers P n,l R, P c,l R and S w,k R, we define P n, P c V p h (Ω and S w Vh s (Ω by, P n = d p l=0 P n,l ϕ p l P c = d p l=0 P c,l ϕ p l d s S w = S w,k ϕ s k, (4.5 k=0 where ϕ p i and ϕs k are elements of a basis for V p h (Ω and V s h (Ω and d p N and d s N denote the dimension. We define the coefficient vectors ˆP n, ˆP c R dp and Ŝw R ds by: ˆP n = ( P n,1, P n,,..., P n,dp ˆP c = ( P c,1, P c,,..., P c,dp Ŝ w = ( S w,1, S w,,..., S w,ds. (4.6

39 9 Furthermore, for given real numbers S n w,k R, we define P w,l R for l = 0,..., d p and ds w,k R for k = 0,..., d s, with, P w,l := P n,l P c,l, and ds w,k = 1 t (S w,k S n w,k, (4.7 which gives us P w V p h (Ω, Sn w Vh s (Ω and ds w Vh s (Ω, s.t., and P w := P n P c = d p i=0 dsw := 1 t ( S w S n w = P w,i ϕ p i, d s k=0 ds w,k ϕ s k. (4.8 he coefficient vectors ˆP w R dp and ˆ ds w R ds are defined analogous to (4.6. Next, we define, l as the l -scalar product on R dp+ds, and l as the induced l -norm on R dp+ds. Note that for a coefficient vector ˆX R dp+ds and the induced vector X V p h (Ω V p h (Ω V h s (Ω there exists constants c > 0, c R and C > 0, C R, such that the following inequality holds: c X Ω,0 ˆX l = ˆX, ˆX l C X Ω,0. (4.9 Note that the constants c and C may depend on the mesh size h. Using the definitions (4.5, (4.7 and (4.8 in (3.15-(3.17, we define F Pn i, F Pc i, Fk S R for i = 0, 1,..., d p and k = 0, 1,..., d s, s.t., F Pn l F Pc l := 1 i i t ( S w Swφϕ n p l + λ n ( S w K P n ϕ p l i i {λ n ( S w K P n n} ϕ p l F F i F i + θ P n {λ n ( S w K ϕ p l n} + σ f(k p n F F i F i F F i F i F i P n ϕ p l θ p D n {λ n (s D K ϕ p l n} σ f(k p n F F i Γ i F F i Γ i F i pd n ϕ p l (4.10 := 1 i i t ( S w Swφϕ n p l + λ w ( S w K ( P n P c ϕ p l i i {λ w ( S w K ( P n P c n} ϕ p l F F i F i

40 30 + θ {λ w ( S w K ϕ p l n} P n P c F F i F i f(k p + σ w F F i F i F i P n P c ϕ p l θ {λ w (s D K ϕ p l n} pd n p D c F F i Γ i f(k p σ w F i F i pd n p D c ϕ p l, (4.11 F i Γ F S k := φ P c ϕ s k φp c,eq ( S w ϕ s k i i i i φτ 1 i i t ( S w Swϕ n s k. (4.1 As before, we define analogous to (4.6 the coefficient vectors ˆF Pn Pc, ˆF R dp and ˆF S R ds. Observe that, if F Pn i = F Pc i = Fk S = 0 for all i = 0, 1,..., d p and k = 0, 1,..., d s, then P n, P c and S w are a solution to the Problem 3. he definitions (4.5-(4.1 define a continuous mapping P : R dp+ds R dp+ds by, P( ˆP n, ˆP w, ˆ ds w = ( ˆF Pn, ˆF Pc, ˆF S. Existence o prove existence of a solution to our system, we use Lemma 1.4 in [emam, 001, p. 164]: Lemma 4 Let X be a finite dimensional Hilbert space with scalar product, and norm and let P be a continuous mapping from X into itself such that, P (ξ, ξ > 0 for ξ = k > 0. hen, there exists a ξ X, ξ k s.t., P(ξ = 0. Another version of this lemma can be found in Chapter IV of [Girault and Raviart, 1986]. o apply Lemma 4 we chose R dp+ds as the Hilbert space X and we use the scalar product, l and the norm l. Further, let ( ˆPn ˆPc Ŝ w R d p+d s and define R > 0 as R := ( ˆPn ˆPc Ŝ w, ( ˆPn ˆPc Ŝ w = ˆPn, ˆP n + ˆP w, ˆP w + ˆ ds w, ˆ ds w.

41 31 Specifically, we show that whenever then one gets R > φ 1 i i t P c,eq(sw n + ( σ w + + λ w θ Ct C λ w = F i Γ ( σ n + λ n θ Ct C λ n f(k p F i pd w F i,0, ˆF Pn, ˆP n + ˆF Pc, ˆP w + ˆF S, ˆ ds w d p i=0 > 0, P n,i F Pn i + d p i=0 which gives the following existence result: (P n,i P c,i F Pc d s i + k=0 F i Γ ds w,k F S k f(k p F i pd n F i,0 (4.13 Lemma 5 For sufficiently large σ n, σ w, the Problem 3 has a solution. Proof 1 We estimate the terms (I := d p i=0 P n,if Pn i (III := d s k=0 ds w,kfk S separately., (II := d p i=0 P w,if Pc i, and Estimate for (I We start with: (I = λ n ( S w K 1 f(k p Pn + σn i i F F i F i F i P n (1 θ {λ n ( S w K P n n} P n 1 F F i F i i i t ( S w Swφ n P n θ p D n {λ n (s D K P f(k p n n} σ n F F i Γ i F F i Γ i F i pd n P n = P 1 + P P 3 P 4 P 5 P 6. Using the assumption (A4.4 for P 1 + P, we get: P 1 + P λ n K 1 Pn + σ n i i,0 F i F f(k p F i P n F i,0. (4.14 Using Cauchy-Schwarz inequality together with the assumption (A4.4, we get: P 3 λ n (1 θ F i F {K 1 Pn } Fi,0 P n Fi,0.

42 3 For a fixed face F i, let ± be the adjacent elements. By the trace inequality (3.8, the following holds: λ n (1 θ F i F {K 1 Pn } Fi,0 P n Fi,0 λ n (1 θc t f(k p 1 F i ( K 1 Pn + K +,0 1 Pn,0 P n Fi,0. F if Further, with Lemma 3 and Cauchy-Schwarz inequality we obtain: f(k p 1 ( K 1 λ n (1 θc t Pn + K F i +,0 1 Pn,0 P n Fi,0 ( i K 1 Pn i,0 F i F 1 ( λ n (1 θ Ct C f(k p F i which, on using the scaled Young s inequality, leads to: P 3 ɛ 1 i K 1 Pn i,0 he term P 5 is estimated in a similar way as P 3 leading to: P 5 ɛ i K 1 Pn and, the term P 6 is estimated as: P 6 ɛ 3 F i Γ P n F i,0 F i F λ n (1 θ C ɛ C f(k p t 1 F i P n F i,0. (4.15 F i F i,0 + 1 λ n θ C ɛ C f(k p t F i pd n F i,0, (4.16 F i Γ f(k p F i P n F i,0 + σ n ɛ 3 F i Γ f(k p F i pd n F i,0. (4.17, Choosing ɛ 1 = ɛ = λn, and ɛ 3 = σ n in (4.14, (4.15, (4.16 and (4.17, we get the following estimate for the term (I: (I 1 i i t ( S w Swφ n P n + i ( σ n + (1 θ λ n C t C λ n F i F ( σ n + λ n θ Ct C λ n F i Γ λ n K 1 Pn i,0 f(k p F i P n F i,0 f(k p F i pd n F i,0 (4.18 Estimate for (II o estimate term (II, we follow the same steps as for term (I. We use the assumption (A4.4, trace inequalities from Lemma, Lemma 3, Cauchy-Schwarz and

43 33 scaled Young s inequality, in that order, and with ɛ 4 = ɛ 5 = λw and ɛ 6 = σ w, we arrive at the following estimate: (II 1 i i t ( S w Swφ n P w + λ w K 1 Pw i,0 i ( σ w + (1 θ λ w C t C f(k p λ w F i P w F i,0 F i F ( σ w + λ w θ Ct C f(k p λ w F i pd w F i,0. (4.19 F i Γ Estimate for (III We start with: (III = φ P 1 c i i t ( S w Sw n φp c,eq ( S w 1 i i t ( S w Sw n + φτ 1 i i t ( S w Sw n. Using the primitive defined in (4.1, we get the following estimate: (III φ P 1 c i i t ( S w Sw n + φ 1 ( P c,eq ( i i t S w P c,eq (Sw n + φτ 1 i i t ( S w Sw n. (4.0 Combined estimate For sufficiently large σ n and σ w, using (3.6 from Lemma 1 with q =, and summing the estimates (4.18, (4.19 and (4.0, we obtain: d pn P n,i F Pn i=0 d pc i + P w,j F Pc j=0 d s j + k=0 ds w,k F S k C P n Ω,0 + C P w Ω,0 + C ds w Ω,0 + φ 1 i i t P c,eq( S w φ 1 i i t P c,eq(sw n ( σ n + λ n θ Ct C f(k p λ n F i pd n F i,0 F i Γ ( σ w + λ w θ Ct C f(k p λ w F i pd w F i,0. (4.1 F i Γ Observe that the positivity of the last but one terms in (4.18 and (4.19 is only guaranteed under restrictions on σ n and σ w. However, these restrictions do not depend on the time step or the argument in the mapping P. Now, one can choose the radius R as announced above to

44 34 guarantee that the right hand side in (4.1 is positive, and using (4.9 leads to the estimate (4.13, and the existence of a zero for P and hence of a solution to Problem 3 follows directly by Lemma Discrete energy estimate Lemma 6 For sufficiently large σ n and σ w, there exists a constant C independent of t, h and the polynomial degrees k p and k s, s.t., the following energy estimate holds: t + t + t N S + t N i,0 K 1 P n=0 i N n=0 F i F N n=0 F i F f(k p F i f(k p F i w P n n=0 i F i,0 + t Pw F i,0 + i C N P c,eq (Sw 0 + C t i i n=0 N + C t n=0 F i Γ F i Γ N n=0 i n i,0 K 1 P w i,0 i P c,eq (S N+1 w f(k p F i pd n F i,0 f(k p F i pd w F i,0 (4. Proof Starting with the discrete system at t (i.e. Problem 3, we test in (3.15 with Pn, in (3.16 with Pw = Pn Pc and in (3.17 with Sw. Note that we define a generic constant C = C(τ, σ α, λ α, λ α, θ, C t, C for α = w, n. We proceed with the same steps as in the proof of Lemma 5 and obtain: φτ S w + λ n i,0 K 1 P n i i ( + σ n 1 λ n (1 θ Ct λ C n F i F ( + σ w 1 λ w (1 θ Ct λ C w i + i 1 t ( σ n + λ n F i F ( Pc,eq (S w P c,eq (S n w θ Ct C λ n F i Γ f(k p F i pd n F i,0 i,0 + λ w f(k p Pn F F i i,0 f(k p Pw F F i i,0 i K 1 P w i,0

45 35 ( σ w + + λ w θ Ct C λ w F i Γ f(k p F i pd w F i,0 he final estimate (4. is obtained by multiplying the above inequality by t and summing over all n = 0... N. 4.3 Error Estimates After showing the existence of a discrete solution and deriving the general energy estimates, we now show a convergence result for the scheme Estimate for the non-wetting phase Lemma 7 For a sufficiently large σ n there exists a constant C independent of h, t, k p and k s such that the following estimate holds: [ S ] w + t s w φe p + n,h K 1 e p n,h i,0 i i i f(k p F i e p n,h F i,0 + F i F ( 5 C + 3f(k s λ n σ n f(k p + Cλ 5 n K 1 p λ n λ n K 1 p n n Ω, e s Ω, i e s,h i,0 Ω,0 + C 5λ n K 1 e λ n + 3f(k s σ n f(k p λ C K 1 n p n Ω, ( e s Ω,0 + h e s ( 3σ n Ct + C C + 5θλ n C t C (h e p λ n Ω,0 + e p n n + C 3λ ( n C t C K 1 e p σ n + h K 1 e p n n Ω,0 Ω,0 p n Ω,0 Ω,0 Ω,0 Proof 3 We subtract (3.11 and (3.15 and test with e p n,h to get: [ S ] w + t s w φe p n,h i i + [ λn (Sw K P ] n λ n (s w K p n e p n,h i i

46 36 f(k p + σ n F F i F i F i [ P n p n ] e p n,h = [ {λn (Sw K Pn n} {λ n (s w K p n n} ] e p n,h F F i F i θ [ ] Pn {λ n (Sw K e p n} p n,h n {λ n (s w K e p n} n,h F F i F i We rewrite this equation termwise as: and estimate each term individually. P 1 + P + P 3 = P 4, We expand each term P 1 to P 4 by adding and subtracting p n. Estimate for P P = λ n (Sw K e p n,h e p n,h i i + ( λ n (Sw λ n (s w K p n e p n,h i i + λ n (s w K e p n e p n,h i i =: P,1 + P, + P,3 where, we estimate P,1, P, and P,3 as: P,1 i P, i i ɛ, + λ n P,3 ɛ,3 λ n K 1 e p n,h i,0 (4.3 i λ n(s w s w K p n e p n,h λ n(e s,h i i 1 i K 1 e p n,h i,0 K 1 p n ɛ, + e s K p n e p n,h Ω, i K 1 e p n,h i,0 + λ n ɛ,3 i ( e s,h i,0 + e s i,0 (4.4 K 1 e p n i,0 (4.5

47 37 Estimate for P 3 f(k p P 3 =σ n F i F i where, F i F P 3,1 =σ n and, P 3, ɛ 3, F i F F i F [ e p n,h ] + e p n e p n,h = P 3,1 + P 3, f(k p F i e p n,h F i,0, (4.6 f(k p F i e p n,h F i,0 + 1 ɛ 3, σ nc t C(h e p n Ω,0 + e p n Ω,0 (4.7 Estimate for P 4 P 4 =(1 θ F i F + F i F + F i F θ F i F θ F i F F i =P 4,1 + + P 4,5 [ {λ n (S w K e p n,h n}} ] e p n,h F i { (λn (S w λ n (s w K p n n } e p n,h F i {λ n (s w K e p n n} e p n,h e p n {λ n (s w K e p n} n,h F i p n {(λ n (Sw λ n (s w K e p n} n,h F i where, we estimate P 4,1 to P 4,4 separately, in the same way as (4.15 in Lemma 5: P 4,1 ɛ 4,1 K 1 e i p n,h i,0 + (1 θ 1 λ n C f(k p ɛ t C 4,1 F i e p n,h F i,0 (4.8 F i P 4, λ C K 1 n p n Ω, F i F ɛ 4, {(e s,h F i + e s } e p n,h f(k s F i e p n,h F i,0 + 1 λ ɛ 1 n C K 4, F i p n Ω, i e s,h i,0 + 1 λ ɛ 1 n C K p n Ω, ( e s Ω,0 + h e s Ω,0 (4.9 4,

48 38 P 4,3 ( λ n C t C i ( K 1 e p n ( f(k p F i e p n,h F i,0 F i λ n C t C 1 + ɛ 4,3 P 4,4 ɛ 4,4 ɛ 4,3 1 ( K 1 e p n Ω,0 i,0 + F i K 1 e p n + h K 1 e p n Ω,0 i,0 1 f(k p F i e p n,h F i,0 (4.30 F i K 1 e i p n,h i,0 + 1 ɛ 4,4 θ λ n C t C(h e p n Ω,0 + e p n Ω,0 (4.31 If p n is continuous, the jump term in P 4,5 vanishes making P 4,5 = 0. Otherwise, we proceed with the same steps as for P 4,4. We use the continuity of p n to replace p n by ep n and get the following estimate for P 4,5 : Combined estimate i P 4,5 ɛ 4,5 K 1 e i p n,h i,0 + 1 ɛ 4,5 θ λ n C t C(h e p n Ω,0 + e p n Ω,0 (4.3 Putting the estimates (4.3 to (4.3 together, we get: i [ S w + t s w ] φe p n,h ( + λ n ɛ, ɛ,3 ɛ 4,1 ɛ 4,4 ɛ 4,5 i ( ( + σ n f(k p ɛ 3, + ɛ 4,3 + (1 θ λ n C t C ɛ 4,1 ( λ ɛ, ɛ 1 n K p n Ω, 4, + λ 1 n K 1 p ɛ, n Ω, e s i Ω,0 + λ n K 1 e p n,h i,0 f(k sɛ 4, e s,h i,0 ɛ,3 i + 1 λ ɛ 1 n C K p n Ω, ( e s Ω,0 + h e s 4, + 1 σ ɛ nc C t ( F i e p n i,0 + e p n 3, i i,0 F i F K 1 e p n i,0 Ω,0 1 F i e p n,h F i,0

49 + λ n C t C 1 ɛ 4,3 ( ɛ 4,4 ɛ 4,5 ( K 1 e p n Ω,0 + h K 1 e p n θ λ n C t C(h e p n Ω,0 + e p n Ω,0 (4.33 Ω,0 39 Choosing ɛ, = ɛ,3 = ɛ 4,1 = ɛ 4,4 = ɛ 4,5 = λ n 5, and ɛ 3, = f(k s f(k p ɛ 4, = ɛ 4,3 = σ n 3, we arrive at the desired estimate for the non-wetting phase Estimate for the wetting phase Lemma 8 For a sufficiently large σ w there exists a constant C independent of h, t, k p and k s such that the following estimate holds: [ S ] w + t s w φe p + w,h K 1 e p w,h i,0 i i i f(k p F i e p w,h F i,0 + F i F ( 5 C + 3f(k s λ w σ w f(k p + Cλ 5 w K 1 p λ w λ w K 1 p w w Ω, e s Ω, i e s,h i,0 Ω,0 + C 5λ w K 1 e λ w + 3f(k s σ w f(k p λ C K 1 w p w Ω, ( e s Ω,0 + h e s ( 3σ w Ct + C C + 5θλ w C t C (h e p λ w Ω,0 + e p w w + C 3λ ( w C t C K 1 e p σ w + h K 1 e p w w Ω,0 Ω,0 p w Ω,0 Ω,0 Ω,0 Proof 4 he proof is the same as for the non-wetting phase (Section and is therefore left out.

50 Estimate for the capillary pressure Lemma 9 here exists a constant C independent of h, t, k p and k s such that the following estimate holds: φ i + p c,eq φ 3φ τ i e p c,h e s,h i i + τφ t i + p c,eq φ i e s,h i,0 1 t e s,h en s,h i,0 + φτ e s,h i,0 i e p c i t i,0 + p c,eq φ 4 i tt s w i,0 dt + 3τφ t n e s,h i,0 + L p c φ p c,eq i i e s i,0 t e s i,0 (4.34 Proof 5 We subtract (3.17 in Problem 3 from (3.13 in Problem, and use ψ pc = φ e s,h to get φ i φ i + φ i = φ i = 0. Estimate for P c 1 e p c,h e s,h + φ i i e p c e s,h i (p c,eq (Sw p c,eq ( s w + p c,eq ( s w p c,eq (s w e s,h i τ( e s,h + ( t s w + t e s e s,h i e p c,h e s,h + P c 1 + P c + P c 3 i P c 1 φ ɛ pc1 We use Hölder s and Young s inequality to obtain i e p c i i,0 + ɛ pc1 i (4.35 i e s,h i,0 (4.36 Estimate for P c P c = φ (p c,eq (Sw p c,eq ( s w e s,h i i φ (p c,eq ( s w p c,eq (s w e s,h i i

51 41 =P c,1 + P c,. (4.37 Here P c,1 is estimated as P c,1 =φ p c,eq(ξ e s,h e s,h φ p c,eq e s,h e s,h i i i i = p c,eq φ i e s,h i,0 + p c,eq φ i For P c,, using Young s Inequality and the Lipschitz continuity one has P c, φ i L pc e s e s,h i ɛ pc i 1 t e s,h en s,h i,0. e s,h i,0 + L p c φ ɛ pc i e s i,0 Estimate for P c 3 P c 3 =φτ e s,h i,0 + φτ ( t s w e s,h i i i + φτ t e s e s,h i i =P c 3,1 + P c 3, + P c 3,3 (4.38 We approximate the consistency error in P c 3, using a aylor expansion 1 t ( s w s n w t s w = 1 t (t t n tt s w dt, t t n which leads to the following estimate for P c 3, P c 3, τ φ ɛ pc3 ɛ pc3 i i ( t s w i,0 + ɛ pc3 e s,h i,0 + τ φ 6ɛ pc3 t i o estimate P c 3,3, we use Young s inequality: Combined estimate P c 3,3 ɛ pc33 φ i i e s,h i,0 + τ φ ɛ pc33 i t e s,h i,0 t n tt s w i,0 dt. i t e s i,0 We substitute the estimates (4.36, (4.37 and (4.38 into (4.35 to get e p c,h e s,h i + p c,eq φ i e s,h i,0

52 4 + p c,eq φ φ ɛ pc1 + ɛ pc + ɛ pc3 + ɛ pc33 i i i i 1 t e s,h en s,h i,0 + φτ e s,h i,0 i e p c i,0 + ɛ pc1 e s,h i i,0 i i e s,h i,0 + L p c φ ɛ pc i i e s i,0 e s,h i,0 + τ φ t 6ɛ pc3 i e s,h i,0 + τ φ ɛ pc33 i t tt s w t n t e s i,0 i,0 dt Setting ɛ pc1 = ɛ pc3 = ɛ pc33 = φτ 3 and, ɛ pc = p c,eq φ, we get the desired estimate Convergence result We are now in a position to deduce the following theorem about the convergence of the scheme: heorem 1 For sufficiently large σ n and σ w, there exists a constant C independent of h and t, s.t., the following estimate holds: i + t + t C i +C e N+1 s,h i,0 + N n=0 i N n=0 F i F N n=0 i e s,h ( e p w,h i,0 + e p n,h i,0 f(k p F i en s,h i,0 + t ( e p w,h F i,0 + e p n,h F i,0 e 0 s,h min(ks+1,ls h i,0 + C ks ls h min(ks+1,ls k ls s t min(kp+1,lpn h +C lpn kp min(kp+1,lpc h +C lpc kp N t s w (t Ω,l s n=0 ( kp + kp ( kp + kp ( ks t t t N n=0 i e s,h i,0 N s w (t Ω,l s + C t n=0 N p n (t Ω,l pn n=0 N p c (t Ω,l pc n=0

53 43 Proof 6 We add the results of Lemma 7, 8, and 9, and rearrange them to get: [ S ] w + t s w φe p + [ n,h S ] w t s w φe p w,h i i i i +φ i + p c,eq φ + i e p c,h e s,h i i + f(k p F i F i F 3φ τ + p c,eq φ i e s,h i,0 1 t e s,h en s,h i,0 + φτ e s,h i,0 i ( K 1 e p w,h i,0 + K 1 e p n,h i,0 i + τφ t i + [ C α=w,n + α=w,n [ ( e p w,h F i,0 + e p n,h F i,0 e p c i t i,0 + p c,eq φ 4 tt s w t n ( 5 λ α + 3f(k s σ α f(k p C 5λ α λ α K 1 p i i,0 dt + 3τφ i α Ω, e s e s,h i,0 + L p c φ p c,eq λ α K 1 p α Ω, t e s i,0 ] i Ω,0 + C 5λ α λ α i i e s,h i,0 e s i,0 K 1 e p α i,0 + 3f(k sλ C α σ α f(k p K 1 p α Ω, ( e s Ω,0 + h e s Ω,0 ( 3σ α Ct + C C + 5θ λ α C t C (h e p λ α Ω,0 + e p α Ω,0 α + C 3λ ( α C t C K 1 e ] p σ α + h K 1 e p α (4.39 α Ω,0 Ω,0 We combine the first three summation terms of (4.39 to get: [ S ] w + t s w φe p + [ n,h S ] w t s w φe p w,h i i i i e p c,h e s,h i + φ i = i i φ [ e s,h + ( t s w + t e s ] (e p n,h e p c,h e p n,h

54 44 + φ i = i e p c,h e s,h i i φ( t s w e p c,h i φ t e s e p c,h i =: P s 1 + P s Estimate for P s 1 P s 1 ɛ ps1 i e p c,h i,0 + φ 6ɛ ps1 t i t t n tt s w i,0 dt. (4.40 Estimate for P s P s ɛ ps o absorb the error e get the following estimate: i i p c,h i,0 e p c,h i,0 + φ ɛ ps i t e s i,0 (4.41, we use the triangle inequality together with Lemma 1 to e p c,h i,0 e p n,h i,0 + f(k p F i e p n,h F i,0 i F i F + e p w,h i,0 + f(k p F i e p w,h F i,0. (4.4 i F i F After substituting the estimates (4.40 and (4.41 together with the estimate (4.4 into the estimate (4.39, and choosing ɛ ps1 = ɛ ps = 1 we get: p c,eq φ + φτ i i + 1 f(k p F i F i F [ C α=w,n + L p c φ + p c,eq e s ( τφ + φ 3 e s,h i,0 + p c,eq φ e s,h i,0 + 1 i i ( e p w,h F i,0 + e p n,h F i,0 ( 5 + 3f(k s λ α σ α f(k p Ω,0 + 3φ t t τ e p c Ω,0 + tt s w Ω,0 dt t n 1 t e s,h en s,h i,0 ( K 1 e p w,h i,0 + K 1 e p n,h i,0 λ α K 1 p α Ω, + p c,eq φ 4 ( 3τφ + φ t e s Ω,0 ] i e s,h i,0

55 45 + α=w,n [ C λ α5 λ α K 1 p α Ω, e s Ω,0 + C 5λ α K 1 e λ α p α Ω,0 + 3f(k sλ C α σ α f(k p K 1 p α Ω, ( e s Ω,0 + h e s ( 3σ α Ct + C C + C 5θ λ α C t C (h e + Cλ α C t C 3 σ α λ α ( K 1 e p α Ω,0 Ω,0 p α Ω,0 + e + h K 1 e p α Ω,0 p α ] Ω,0 (4.43 Using a generic constant C, we rewrite (4.43 as: p c,eq φ + φτ i i e s,h i,0 + p c,eq φ e s,h i, f(k p F i F i F ( C + p c,eq φ 4 i t +C t + α=w,n i i ( e p w,h F i,0 + e p n,h F i,0 1 t e s,h en s,h i,0 ( K 1 e p w,h i,0 + K 1 e p n,h i,0 C e s,h i,0 + C e s Ω,0 + C e p c Ω,0 tt s w Ω,0 dt +C t e s Ω,0 + Ch e s Ω,0 t n ] [ C e p α Ω,0 + Ch e p α Ω,0 + Ch e p α Ω,0 Multiplying the above inequality by t, summing over n = 0,..., N, and absorbing, we get: e N+1 s,h Ω,0 ( p c,eq φ C t + t + t N n=0 F i F N n=0 i + φτ t N p c,eq φ n=0 i f(k p F i i e N+1 s,h i,0 + p c,eq φ N n=0 i ( e p w,h F i,0 + e p n,h F i,0 ( K 1 e p w,h i,0 + K 1 e p n,h i,0 e s,h i,0 ( e 0 s,h i,0 + i C + p c,eq φ 4 N 1 t n=0 i e s,h en s,h i,0 C e s,h i,0

56 46 +C t N n=0 +C t + α=w,n 0 t e s tt s w [ C Ω,0 + C t N n=0 N n=0 e p c Ω,0 dt +Ch t e p α Ω,0 + C t N n=0 Ω,0 + Ch N n=0 e s Ω,0 N n=0 t e s Ω,0 e p α Ω,0 + Ch ] N e p α Ω,0 For a sufficiently small t, we use Grönwall s inequality, and postulate that there exists a constant independent of t, h, k p or k s, s.t.: ( p c,eq φ C t e N+1 s,h i,0 + p c,eq φ N e s,h en s,h i,0 + t + t N n=0 i N n=0 F i F + φτ t N p c,eq φ +C t +C t +C t n=0 i i i ( K 1 e p w,h i,0 + K 1 e p n,h i,0 f(k p F i n=0 i ( e p w,h F i,0 + e p n,h F i,0 e s,h i,0 N e 0 s,h i,0 + C t 0 N n=0 N n=0 tt s w e p n e p w n=0 Ω,0 dt +C t Ω,0 + Ch t Ω,0 + Ch t e s N n=0 N n=0 N n=0 Ω,0 + C t t e s e p n e p w N n=0 Ω,0 + Ch t Ω,0 + Ch t Ω,0 + Ch t n=0 e p c Ω,0 N n=0 e s Ω,0 N e p n Ω,0 n=0 N e p w Ω,0 Using the error estimates (4., (4.3 and (4.4, and the triangle inequality for the error terms in p w = p n p c, we can write: ( p c,eq φ C t e N+1 s,h i,0 + p c,eq φ N e s,h en s,h i,0 + t + t N n=0 i N n=0 F i F i ( K 1 e p w,h i,0 + K 1 e p n,h i,0 f(k p F i n=0 i ( e p w,h F i,0 + e p n,h F i,0 n=0

57 + φτ t N n=0 i e s,h i,0 C i e 0 s,h i,0 + C t +C t +C t N n=0 N n=0 +Ch t +Ch t h min(ks+1,ls k ls s N n=0 min(kp+1,lpn h lpn kp N n=0 N n=0 min(kp+1,lpn 4 h lpn 4 kp h min(kp+1,lpc k lpc p h min(ks+1,ls k ls s t s w (t Ω,l s + Ch t from where, the stated estimate follows. s w (t Ω,l s + C t N n=0 p n (t Ω,l pn + Ch t p n (t Ω,l pn + C t p c (t Ω,l pc + Ch t h min(ks+1,ls N n=0 N n=0 N n=0 k ls s h min(kp+1,lpn k lpn p min(kp+1,lpc h lpc kp min(kp+1,lpc 4 h lpc 4 kp s w (t Ω,l s p n (t Ω,l pn p c (t Ω,l pc p c (t Ω,l pc 47 From the heorem we can directly deduce the following Corollary: Corollary 1 For sufficiently smooth solutions p n L (0, ; H kp+1 (Ω, p c L (0, ; H kp+1 (Ω and s w H (0, ; H ks+1 (Ω and sufficiently large σ n and σ w, there exists a constant C independent of h and t, s.t., the following estimate holds: e N+1 s,h Ω,0 + t C t + C hks k ks s N n=0 e s,h Ω,0 + t + C hkp k kp p N n=0 ( e p c,h Ω,DG + e p n,h Ω,DG 4.4 Numerical Experiments In this section, we verify the convergence rates derived in heorem 1 through numerical experiments. We consider an analytical solution to compute the L - and H 1 -errors. We show the h and t dependence through successive refinement of the spatial mesh, respectively of the time step. Problem definition We consider the domain Ω = (0, 1 (0, 1 R and t [0, 1]. he properties of the phases and the porous medium are listed in able 4.1.

58 48 able 4.1: Properties for est problem 1 Phase Properties [ ] wetting phase dynamic viscosity µ kg w ms 1 [ ] non-wetting phase dynamic viscosity µ kg n ms 1 [ ] wetting phase density ρ kg w m 1 [ 3 ] non-wetting phase density ρ kg n m 1 3 Hydraulic Properties absolute permeability K [ m ] 1 residual wetting phase saturation S rw 0 residual non-wetting phase saturation S rn 0 porosity ϕ 0.4 damping coefficient τ [P a s] 1 Brooks-Correy Parameters entry pressure p d [P a] 1 pore size distribution index λ he right hand side in the equations are chosen such that the exact solution for t 0 equals: p n (t, x, y = 1 4 cos((x + yπ t + 1 s w (t, x, y = 1 4 sin((x + yπ t + 1 p c (t, x, y = p c,eq (s w (t, x, y τ t s w (t, x, y Implementation We chose θ = 1, which gives a NIP dg-scheme, and the penalty parameters as σ w = σ n = 10. We implement the numerical scheme in the C++ based DUNE- PDELab framework [Bastian et al., 007, 010, 011]. For linearization, we use the Newton- Raphson scheme with a line-search strategy [Deuflhard, 004]. We solve the resulting linear system with SuperLU solver [Demmel et al., September 1999]. Simulation o show the spatial convergence rates, we consider two cases: Case 1 with polynomials of order 1, and Case with polynomials of order. We make five simulations each with the following mesh and time step refinements: p-order=1 p-order= no. of elements time step size time step size Run-1: t = 1 t = 1, Run-: 4 4 t = 1/ t = 1/4, Run-3: 8 8 t = 1/4 t = 1/16, Run-4: t = 1/8 t = 1/64, Run-5: 3 3 t = 1/16 t = 1/56.

59 49 In Case 1, a linear convergence rate is expected. In Case, we use quadratic polynomials, and let the time step size t depend quadratically on the size of the elements h. his prevents the errors due to the time discretization from becoming dominating and thus affecting the convergence rates. Here, we expect a quadratic convergence rate. In case 3, to show the time convergence rates, we make five simulations with polynomial order and the following mesh and time step refinements: p-order= no. of elements time step size Run-1: t = 1, Run-: 4 4 t = 1/, Run-3: 8 8 t = 1/4, Run-4: t = 1/8, Run-5: 3 3 t = 1/16. In this case, the time steps are chosen such that the error due to time discretization is ultimately dominating. Results he solution of the problem at time t = 1 and with a refinement of 3 3 is shown in Figures 4.1a, 4.1b and 4.1c. In Figure 4., we show the spatial convergence rates for the test problem. Figures 4.a and 4.b show the calculated error for piecewise linear polynomials for the non-wetting pressure p n, capillary pressure p c, and wetting saturation s w. Figures 4.c and 4.d show the calculated error for piecewise quadratic polynomials for p n, p c, and s w. In Figure 4.3, we show the temporal convergence rates for the test problem, with piecewise quadratic polynomials for p n, p c, and s w. Observe the agreement with the theoretical convergence rates obtained in heorem 1. For Case 1, we observe a linear convergence order, and for Case a quadratic convergence order. In Case 3 we see a linear convergence order due to the time discretization. he expected convergence rates for each of the cases are plotted in green for reference in Figures 4. and 4.3.

60 50 (a p n (b p c (c s w (d Plot over the diagonal Y = X, legend: s w,p n and p c Figure 4.1: Simulation results at t = 1.

61 51 (a H 1 error for piecewise linear polynomials (b L error for piecewise linear polynomials (c H 1 error for piecewise quadratic polynomials (d L error for piecewise quadratic polynomials Figure 4.: hp-convergence rates. Figure 4.3: emporal convergence rates L error for piecewise quadratic polynomials.

62 5

63 Chapter 5 Linearization scheme he non-linearities in Problem 4 can be resolved in different ways. he basic idea is to approximate the nonlinearities such, that a linear problem remains to solve. he L-scheme is an alternative linearization method similar to a fixed point iteration. he main trait of the scheme is, that an additional term involving the scaling parameter L is used, to enhance the convergence properties. he parameter L represents a generalized approximation of the first derivative with respect to the unknown nonlinear variables, which is used in a Newton method. he choice of this parameter is restricted with respect to the non-linearity, which come from the convergence analysis of the scheme [Karpinski et al., 017]: L s sup s L s, sup s dp c,eq (s, ds (5.1 d (s ds (5. he main benefit of the scheme is, that contrary to the Newton method, only a mild restriction on the timestep-size independent of the spatial discretization is imposed [Karpinski et al., 017; Radu et al., 015b]. Assumptions We base the linearization scheme on the following assumptions: (A5.1 For the initial and boundary data one has s 0 H 1 (Ω, p D n (x H 1 (Γ and a function s D (x H 1 (Γ exists s.t. p D c (x = p c,eq (s D (x. Further, the initial and boundary conditions are compatible. (A5. he permeability matrix K R d d is symmetric and positive definite, i.e. there exist two constants κ and κ, s.t., for any vector x R d, the following holds: κ x x Kx κ x 53

64 54 (A5.3 he equilibrium capillary pressure function p c,eq ( is in C (R, and is assumed positive, bounded and decreasing. Further we assume that there exist L pc,eq, l pc,eq > 0 such that for all S R it holds 0 < l pc,eq p c,eq ( Lpc,eq <. (5.3 (A5.4 he functions λ w ( and λ n ( are Lipschitz-continuous and two constants λ α, λ α > 0 exist such that for all S R, 0 < λ α < λ α (S < λ α <, (α {w, n}. (5.4 (A5.5 he dynamic capillary pressure function τ( is in C (R, positive, bounded, and decreasing. Letting ( denote its primitive, we assume that there exist L, l > 0 such that for all S R one has 0 < l τ(s L <. (5.5 o develop the linearization scheme for the nonlinear Problem 3, we start with the discretization in time (3.14 to obtain a sequence of time-discrete problems (n = 0,..., N 1: Problem 5 [ime discrete problem] Given s n w, p n n and p n c, find s w, p n and p c, s.t., the following holds: s w s w p s n w t s n w t φ + (λ n (s w K (p n gzρ n = 0 φ + (λ w (s w K (p n p c gzρ w = 0 c = p c,eq (s w (s w (s n w t Observe that, at each time step, this results into a nonlinear problem. For solving it we propose an iteration scheme that builds on the ideas in [List and Radu, 016; Pop et al., 004; Radu et al., 015a,b; Slodička, 00, 005a,b; Yong and Pop, 1996] (the L -scheme. he idea is to construct a sequence of triplets (s,i 1 w, p,i 1 n, p,i 1 c converging as i to the solution (s w, p n, p c of Problem 5. Recalling Assumptions (A5.3 and (A5.5, we let L s, L s, > 0 be two positive constants satisfying and define the following linearization scheme: L s L pc,eq and L s, L (5.6 Problem 6 [Linearization scheme] Let i > 0 and s,i 1 w, p,i 1, p,i 1 be given. n c

65 55 Find s,i w, p,i n, and p,i c such that s n w t s n w t s,i w s,i w p,i c + L s (s,i w φ + (λ n (s,i 1 w K (p,i n gzρ n = 0 φ + (λ w (sw,i 1 K (p,i n p,i c gzρ w = 0 p c,eq (s,i 1 w + (s,i 1 w (s n w t ( s s,i 1,i w w + L s, s,i 1 w = 0 t Remark 3 Observe that the first two equations are nothing but the semi-implicit discretization of the corresponding in Problem 5, whereas the third equation includes two additional terms involving the parameters L s and L s,. Formally one can see that if the scheme is convergent, these terms are vanishing and the limit solves the nonlinear time discrete problem. In [Karpinski et al., 017] it is proven that the scheme converges indeed, and that this convergence holds for any initial guess. However, since this is an evolution problem, it is natural to use the solution at the previous time step at starting point, i.e. s,0 w = s n w, p,0 n = p n n, and p,0 c = p n c. 5.1 Discrete system Starting from Problem 6 and with the parameters L s, L s, satisfying (5.6, the fully discrete linearized scheme becomes Problem 7 [Fully discrete linearization scheme] Let Pn n V p h (Ω, P c n V p h (Ω, and Sn w Vh s,i 1 (Ω. Given Pn V p,i 1 h (Ω, Pc V p h (Ω, and S,i 1 w Vh s (Ω with Pn,0 = Pn n, Pc,0 = Pc n, and Sw,0 = Sw, n find Pn,i V p,i h (Ω, Pc V p h (Ω, and Sw,i Vh s(ω, s.t., for all ψ s Vh s(ω, ψ n V p h (Ω, and ψ w V p h (Ω, the following holds: PDE-1: Sw,i φψ n + λ n (Sw,i 1 K ( P,i n gzρ n ψn r r r r {λ n (Sw,i 1 K ( P,i n gzρ n n} ψn F F r F r + θ F r F + σ n F r F Pn,i F r f(k p F r F r {λ n (Sw,i 1 K ψ n n},i Pn ψ n = θ p D n {λ n (s D K ψ n n} + σ n F F r Γ r F r Γ F r f(k p F r pd n ψ n (5.7 PDE-: r Sw,i r φψ w + r λ w (Sw,i 1 r K ( P,i gzρ w ψw w

66 56 {λ w (Sw,i 1 K ( P,i w gzρ w n} ψw F F r F r + θ {λ w (Sw,i 1 K ψ w n} P w,i F r F r F + σ w F r F f(k p F r F r,i Pw ψ w = θ {λ w (s D K ψ w n} p D n p D c F F r Γ r + σ w F r Γ F r f(k p F r pd n p D c ψ w (5.8 ODE-Pc: L s (Sw,i r r + r Pc,i r Sw,i 1 ψ s + ψ s r r p c,eq (Sw,i 1 r r L s, ψ s ( S,i w Sw,i 1 ψ s t + (Sw r,i 1 (Sw n ψ s = 0 (5.9 t r In line with Remark 3, the solution at the previous time step is chosen as initial guess for the fully discrete the iteration scheme. However, the convergence result proved in [Karpinski et al., 017] does not require this starting point. 5. Linearization of the interface condition o linearize the interface condition, we evaluate which of the cases in Section.7, i.e. which case of (3.18, has to be considered. his detection is for both linearization schemes evaluated explicitly, i.e. we use the values of the previous time step. In the same way the calculation of the jump for the non-wetting pressure (.1 is evaluated at the previous timestep. It is possible to also linearize those conditions, as it was done by Weiss et al. in [Helmig et al., 009]. here, an active and non-active set strategy was proposed and implemented. A first order aylor expansion is used to linearize the nonlinear jump condition in (5.9 over the interface, leading to the following scheme: Problem 8 [Fully discrete linearization scheme] Let Pn n V p h (Ω, P c n V p h (Ω, and Sn w Vh s,i 1 (Ω. Given Pn V p,i 1 h (Ω, Pc V p h (Ω, and S,i 1 w Vh s (Ω with Pn,0 = Pn n, Pc,0 = Pc n, and Sw,0 = Sw, n find Pn,i V p,i h (Ω, Pc V p h (Ω, and Sw,i Vh s(ω, s.t., for all ψ s Vh s(ω, ψ n V p h (Ω, and ψ w V p h (Ω, the following holds: PDE-1: Sw,i φψ n + λ n (Sw,i 1 K ( P,i n gzρ n ψn i i i i

67 {λ n (Sw,i 1 K ( P,i n gzρ n n} ψn F F i F i + θ Pn,i {λ n (Sw,i 1 K ψ n n} F F i F i f(k p,i + σ n Pn ψ n F F i F i F i = θ Pn D {λ n (s D f(k p K ψ n n} + σ n F i F i F i P n D ψ n F i Γ D F i Γ D 57 (5.10 PDE-: Sw,i φψ w + λ w (Sw,i 1 K ( P,i w gzρ w ψw i i i i {λ w (Sw,i 1 K ( P,i w gzρ w n} ψw F F i F i + θ {λ w (Sw,i 1 K ψ w n} P w,i F F i F i f(k p,i + σ w Pw ψ w F F i F i F i = θ {λ w (s D K ψ w n} P n D Pc D F i ODE-Pc: F i Γ D + σ w F i Γ D L s (Sw,i i i F i f(k p F i P D n P D c ψ w ( ψ s i i (S i,i 1 + i σ s F i F + σ s F i F Pc,i i f(k s F i F i f(k s F i F i Sw,i 1 ψ s + i w (Sw n ψ s t p c,eq (Sw,i 1 i i L s, ψ s ( S,i w pc,eq (S n w p c,eq(s n w ( S,i+1 w S n w ψs (S nw S,i+1 w Sw,i 1 ψ s t Sw n ψ s = 0 (5.1 t

68 Convergence Analysis of the Linearization Scheme In this section, we present a rigorous proof for the convergence of the linear iterative scheme introduced in Section 5.1. We use the following notations for the errors at the i-th iteration: e i s = S,i w S w, e i p α = P,i α P α (5.13 where α = n, w, c. o simplify the presentation, we also use the following notation for the errors in λ n (, λ w (, (, and p c,eq (, respectively: e i λ n e i p c,eq = λ n (S,i w λ n (S w e i λ w = λ w (S,i w λ w (S w = p c,eq (S,i w p c,eq (S w e i = (S,i w (S w. (5.14 he following theorem states the convergence of the linear iterative scheme. It is proved under a mild restriction on the time step which is uniform w.r.t. the spatial mesh. heorem Convergence L-scheme Under assumptions (A5.1-(A5.5 and with a sufficiently small t, the iterative scheme (5.7-(5.9 converges linearly. o prove the convergence of the scheme, we subtract (5.7, (5.8 and (5.9 from (3.15, (3.16 and (3.17 respectively to get the following system of equations: PDE-1: ( Sw,i Sw φψ n r r + (λ n (Sw,i 1 K Pn,i λ n (S r r {(λ n (Sw,i 1 K Pn,i λ n (S F F r F r + θ Pn,i {λ n (Sw,i 1 K ψ n n} P F F r F r + σ n r F r F f(k p F r F r w K Pn w K Pn ψ n n} ψ n n {λ n (Sw K ψ n n} P,i n P n ψ n = 0 (5.15 PDE-: ( Sw,i Sw φψ w r r + (λ w (Sw,i 1 K (Pn,i Pc,i ψ w r

69 (λ w (Sw K (Pn Pc ψ w r r {(λ w (Sw,i 1 K (Pn,i Pc,i n} ψ w F F r F r + {(λ w (Sw K (Pn Pc n} ψ w F F r F r + θ {λ w (Sw,i 1 K ψ w n} P n,i F F r F r θ {λ w (Sw K ψ w n} P n Pc F F r F r + σ w F r F f(k p F r F r ODE-Pc: L s (Sw,i r r + r + r r (P,i (P,i n Pc,i 59 P,i c (P n P c ψ w = 0 (5.16 Sw,i 1 ψ s + r r L s, ( S,i w Sw,i 1 ψ s t c Pc ψ s (p c,eq (Sw,i 1 p c,eq (Sw ψ s r r r ( (S,i 1 w (S w ψ s = 0 (5.17 We proceed by first obtaining error estimates separately for the phase pressures in Sections and 5.3. and for the capillary pressure in Section hese estimates are then combined to prove the convergence of the linearization scheme in Section Estimate for the non-wetting phase aking ψ n = e i p n in (5.15, we get, e i sφe i p n + r r r = r + (1 θ F r F F r F (λ n (Sw,i 1 λ n (S r F r F e i p n {λ n (Sw,i 1 F r λ n (Sw,i 1 r w K Pn e i p n K e i p n n} K e i p n + σ n {(λ n (Sw,i 1 λ n (Sw K Pn n} e i p n F r θ{(λ n (Sw,i 1 λ n (Sw K e i p n n} P n F r F r F F r f(k p F r ei p n

70 60 =:P 1 + P + P 3 + P 4 We start with the term P 4 : P 4 Pn Ω, θ {(λ n (Sw,i 1 λ n (Sw K e i p n n}. F F r F r Using the trace inequality (3.8, Lemma 3, Cauchy-Schwarz inequality and Young s inequality (3.10, we get, P 4 Pn Ω, θ ( r for any ɛ 4 > 0. r 1 CC t Fr CC t f(k F r f(k e i 1 λ F r n r,0 Fr K 1 e i pn r,0 Pn Ω, θ C 4 Ct 4 f (k F r e i 1 λ n r,0 r 1 Pn ɛ Ω, θ C4 Ct 4 4 f (k F r r e i 1 λ + ɛ 4 n r,0 K 1 e i p n r r,0 K 1 e i pn r,0 For the terms P 1, P, and P 3, after carrying out steps similar to Section 4.3.1, we obtain for any ɛ 1, ɛ, ɛ 3 > 0, P 1 ɛ 1 P ɛ P 3 ɛ 3 K 1 e i pn r,0 + 1 C Pn Ω, ɛ 1 r r K 1 e r F r F p n r,0 e i 1 λ n r,0 + (1 θ 1 ɛ λ n C t C F r f(k p F r e p n F r,0 f(k p F r ei p n F r,0 + 1 K Pn Ω, ɛ C Ct 3 r e i 1 λ n r,0. Observe that the estimate for P 3 involves the essential boundedness for the gradient of the pressure Pn. Assumptions (A5.-(A5.5 ensure that the problem remains non-degenerate and therefore, the pressures have essential bounded gradients (see e.g. [Cao and Pop, 015]. hese estimates can be extended to the time discrete problems, with the time derivative of the saturation being replaced by the finite difference approximation, noting that these divided differences satisfy the same bounds as t s (see [Cao and Pop, 016]. he extension to the finite element approximation follows from [Nitsche and Wheeler, 1981/8] (also see [Li, 015].

71 61 Combining the estimates for P 1 to P 4, and choosing ɛ 3 = σ n and ɛ 1 = ɛ = ɛ 4 = λn 3, we get the following estimate for the non-wetting phase: r ( σ n ( e i sφe i p n + λ n r K 1 e i pn r,0 r + (1 3λ θ n C t C λ n θ (C n,1 + C n, F r r F r F f(k p F r ei p n F r,0 for some C n,1, C n, not depending on the discretization parameters. e i 1 λ n r,0 ( Estimate for the wetting phase We choose ψ w = e i p w in (5.16, and proceed in a similar way as for the non-wetting phase, to get the following estimate for the wetting phase: r ( e i sφe i p w + λ w r K 1 e i pw r,0 r + ( σ w (1 3λ θ w C t C λ w θ (C w,1 + C w, F r r F r F f(k p F r ei p w F r,0 for some C w,1, C w, not depending on the discretization parameters. e i 1 λ w r,0 ( Estimate for the capillary pressure With ψ s = e i s in (5.17, we obtain, L s (Sw,i Sw,i 1 e i s + ( S,i w S,i 1 w L s, e i s r r r r t + e i p c e i s (p c,eq (Sw,i 1 p c,eq (Sw e i s r r r r + ( (Sw,i 1 (Sw e i s = 0 (5.0 r r Note that, from (5.13-(5.14, we can write (Sw,i 1 (Sw = 1 ( (S,i 1 w (Sw. t

72 6 Using (3.9 in (5.0, we get, L s r + L s r = L s r r e i s r,0 + L s, t r e i s e i 1 s r,0 + L s, t e i 1 p c,eq e i s + r r e i s r,0 + r 1 r t ei 1 ei s r r e i p c e i s e i s e i 1 s r,0 e i 1 s r,0 + L s, t es i 1 r,0. (5.1 r Following Assumption (A5.3, p c,eq ( is monotonous and we have, (p c,eq (x p c,eq (y(x y = p c,eq (x p c,eq (y x y, Similar argument holds for (. Using the above equalities for p c,eq ( and (, (5.1 becomes, L s e i s r,0 + L s, t e i s r,0 + e i p c e i s r r r r + L s e i s e i 1 s r,0 + L s, t e i s es i 1 r,0 r r + r = L s + r r r e i 1 p c,eq e i 1 + s r e i 1 s r,0 + L s, t r e i 1 1 r t ei 1 r p c,eq (e i s e i 1 s + r ei 1 e i 1 s r,0 s 1 r t ei 1 (ei 1 s e i s. Using the Lipschitz continuity of p c,eq ( and (, and the Young s inequality, we get, L s e i s r,0 + L s, t e i s r,0 + e i p c e i s r r r r + L s e i s e i 1 s r,0 + L s, t e i s e i 1 s r,0 r r + 1 L Pc L s + 1 L s r r r e i 1 p c,eq r,0 + 1 L t r e i 1 r,0 e i 1 s r,0 + L s, t es i 1 r,0 r e i 1 p c,eq r,0 + L s r e i s e i 1 s r,0

73 L s, t r e i 1 r,0 + L s, t r 1 t (ei 1 s e i s r,0. Finally, using (5.6, we get the following estimate for the capillary pressure: L s e i s r,0 + L s, t e i s r,0 + e i p c e i s r + 1 L s L s r r r Combined estimates r e i 1 p c,eq 1 r,0 + L s, t r e i 1 r,0 r e i 1 s r,0 + L s, t e i 1 s r,0. (5. r Multiplying (5. with φ, and adding the resulting equation to the sum of (5.18 and t (5.19, and observing that, leads to, L s φ t + φ L s t r r = =0 r r r e i s r,0 + L s, φ φ t ei p c e i s + r φ t ei p c e i s r e i s r,0 r e i s(φe i p w φe i p n r r e i se i p c e i 1 p c,eq φ r,0 + L s, t e i 1 r,0 r r + λ ( n K 1 e i pn σ n r,0 + (1 3λ θ n C t C λ n r F r F + λ ( w K 1 e i pw σ w r,0 + (1 3λ θ w C t C λ w r (C n,0 + C n,1 + C n,θ F r e i 1 λ n r,0 r + (C w,0 + C w,1 + C w,θ F r e i 1 λ w r,0 + L sφ t r r e i 1 s r,0 + L s, φ r e i 1 s r,0. F r F f(k p F r ei p n F r,0 f(k p F r ei p w F r,0

74 64 After multiplying with t and rearranging the terms, we get, L s φ + φ L s r r e i s r,0 + tl s, φ e i s r,0 r e i 1 p c,eq φ r,0 + L s, t r e i 1 r,0 + tλ n K 1 e i pn r,0 + tλ w K 1 e i pw r,0 r r ( σ n + t (1 3λ θ n C t C f(k p λ n F r ei p n F r,0 F r F ( σ w + t (1 3λ θ w C t C f(k p λ w F r ei p w F r,0 F r F t (C n,0 + C n,1 + C n,θ F r e i 1 λ n r,0 r + t (C w,0 + C w,1 + C w,θ F r e i 1 λ w r,0 + L sφ r r e i 1 s r,0 + tl s, φ r e i 1 s r,0. Using the Lipschitz continuity of λ n, λ w, 1, and p 1 c,eq, we can rewrite this as, L s φ + l p c,eq φ L s r r e i s r,0 + tl s, φ e i 1 s r,0 + l tφ L s, e i s r,0 r r e i 1 s r,0 + tλ n K 1 e i pn r,0 + tλ w K 1 e i pw r,0 r r ( σ n + t (1 3λ θ n C t C f(k p λ n F r ei p n F r,0 F r F ( σ w + t (1 3λ θ w C t C f(k p λ w F r ei p w F r,0 L sφ r e i 1 s r,0 + tl s, φ + t ( L λ n C n + L λ w C w ( + θ F r F r F r e i 1 s r,0 r e i 1 s r,0.

75 65 From this we obtain, L s φ e i s r,0 + tl s, φ e i s r,0 r r [ l pc,eq + φ tc ( ] + θ L s F r e i 1 s r,0 + l tφ L s, r + tλ n K 1 e i pn r,0 + tλ w K 1 e i pw r,0 r r ( σ n + t (1 3λ θ n C t C f(k p λ n F r ei p n F r,0 F r F ( σ w + t (1 3λ θ w C t C f(k p λ w F r ei p w F r,0 L sφ r which can be reformulated as, e i 1 s r,0 + tl s, φ L s φ r F r F r e i s r,0 + tl s, φ es i 1 r,0, e i s r,0 r r + tλ n K 1 e i pn r,0 + tλ w K 1 e i pw r,0 r r ( σ n + t (1 3λ θ n C t C f(k p λ n F r ei p n F r,0 F r F ( σ w + t (1 3λ θ w C t C f(k p λ w F r ei p w F r,0 F r F [ ( L s φ l φ pc,eq tc ( ] + θ L s F r e i 1 s r,0 r ( tls, φ + l tφ e i 1 s L r,0. s, r e i 1 s r,0 For t small enough, this leads to a contraction for the terms r ei s r,0 and r e i s r,0, which concludes the proof for the convergence of the proposed linearization scheme. Remark 4 o obtain the contraction, it is required that the time-step is chosen s.t., lp t < c,eq φ (5.3 L s C ( + θ F min

76 66 where, F min is the measure of the smallest face. his restriction is milder when compared to the typical stability conditions imposed for explicit methods (like IMPES, or for the Newton method (see e.g. [Radu et al., 006] for the analysis for a simplified two-phase model. Moreover, for the IIP dg-scheme, in which θ = 0, the constraint on the time-step does not depend on the mesh size at all and is similar to the one for the L-method for standard, equilibrium two-phase flows with τ = 0 (see [Radu et al., 015a]. Remark 5 o guarantee the convergence, the parameters L s and L s, must satisfy (5.6. For degenerate problems, if e.g. the equilibrium capillary pressure function is not Lipschitz, one needs to first regularize the problem in order to ensure the convergence of the scheme. Remark 6 he convergence result can be extended to conforming discretizations, like finite elements, when the approximation lies in W 1, (Ω. On can carry out the similar steps as above, but now jumps and averages over faces do not appear anymore. As with the IIP dgmethod, for the conforming discretizations the restriction on the time step does not depend on the mesh size, leading to results similar to [Radu et al., 015a]. 5.4 Numerical Example In this section, we present a numerical example to show the effectiveness of the proposed L-scheme. he numerical scheme is implemented in C++ based DUNE-PDELab framework [Bastian et al., 008a,b, 010, 011]. We chose a test problem with a known analytical solution. he test setting is described in Section In Section 5.4., we make a parameter study to compare the behaviour of the L-scheme with the Newton-method est-setting We consider the domain Ω = (0, (0, R and the time interval [0, 3]. he other parameters are listed in able 5.1. he right hand sides (i.e. sources in the governing equations, and the boundary and initial conditions are chosen such, that the following are the exact solutions of the model p n (t, x, y = 1 4 cos((x + yπ t + 1, S w (t, x, y = 1 4 sin((x + yπ t + 1, p c (t, x, y = p c,eq (S w (t, x, y t (S w (t, x, y. We chose θ = 0 and the penalty parameters as σ w = σ n = 10. We set (S w (t, x, y := τs w (t, x, y, which corresponds to a constant damping factor τ. h p convergence for this example is shown in [Karpinski and Pop, 017] Parameter study For each combination { of no. of elements N = {8 8, 16 16, 3 3} and timestepsize t[s] = 1 8, 1 16, 1 }, and polynomial order k s = 1 and k p = 1, we simulate the 3

77 67 able 5.1: Example 1 - Properties Phase Properties [ ] wetting phase dynamic viscosity µ kg w ms 1 [ ] non-wetting phase dynamic viscosity µ kg n ms 1 [ ] wetting phase density ρ kg w m 1 [ 3 ] non-wetting phase density ρ kg n m 1 3 Hydraulic Properties absolute permeability K [ m ] 1 residual wetting phase saturation S rw 0 residual non-wetting phase saturation S rn 0 porosity ϕ 0.4 damping coefficient τ [P a s] 1 Brooks-Correy Parameters entry pressure p d [P a] 1 pore size distribution index λ test problem with Newton method and the L-scheme with different choices of L-parameter L s = {4, 8, 16}. he L-parameters are chosen with respect to the theory, i.e. (5.6. In this example holds 0.7 p c,eq(s w 4, so we choose L s 4. We also consider polynomial order k s = and k p =, and simulate the test problem with Newton method and the L-scheme with L s = 4 for each combination of N and t as above. As an initial guess for the non-linear solvers, we chose the solution of the previous timestep. he comparative performance of the Newton method and the L-scheme with different L-parameters is shown in ables in terms of average number of iterations per timestep, average computation time per timestep, and total computation time. As expected, the Newton scheme, which has a quadratic order of convergence, solves twice as fast as the L-scheme, which has a linear order of convergence. In Figures , we compare the convergence at time = 1s. Figure 5.1 shows the convergence of the Newton method compared with the L-Scheme in terms of the L -error of the residual, which we use as a convergence criterion for both the Newton and the L-scheme. Figures compare the L -error of the pressures P n, P c and the saturation S w, and also show the expected convergence rates. o obtain the expected convergence rates, we chose 1 and 1 4 of the rate obtained for L = 16 for L = 8 and L = 4, respectively. In our results we observe that the real convergence rate is at least as good as the expected convergence rate, and the convergence rates for pressures and saturation coincide. We also observe that with half the parameter L we get double the convergence rate. It is interesting to see that for the pressures in Figure 5.3 and 5.4 the first step does not fit the expected trend, but for the saturation the expected convergence rates are obtained within the first step. In each Figure the convergence rate in horizontal direction, i.e. with timestep refinement, increases with the decreasing timestep size. Half the time step leads to double the convergence rate. In vertical direction, i.e. with spatial refinement, a similar trend is not observed. he convergence rates are constant with respect to change of the refinement level h. his result reflects the

78 68 able 5.: Comparison between Newton method and L-scheme with L s = 4 Polynomial orders k s = 1 and k p = 1 L-Scheme Newton method N t = 1 8 t = 1 16 t = 1 3 t = 1 8 t = 1 16 t = 1 3 Number of iterations per timestep [ ] Computation time per step [s] otal computation time [s] independence of the convergence on the spatial discretization and shows the dependence only with respect to the timestep t. his is in accordance with our theoretical findings. We repeat the above simulations for polynomial order k p = k s = 1 starting with a bad initial guess of s w = 0.5 for the non-linear solver. In this case, we observe that the Newton method does not converge at all. he L-scheme still shows convergence due to its property of global convergence, however, only for t = 1. For larger timesteps the L-scheme does 3 not converge due to the restriction on the timestep size (5.3. he performance of the L- scheme in terms of average number of iterations per timestep, average computation time per timestep, and total computation time is tabulated in able 5.6. In Figures 5.5 and 5.6, we show the convergence behaviour at = 1s. It is interesting to observe that the scheme takes a few steps to find a close enough solution, after which it converges as expected. Here we see a clear advantage of the proposed L-scheme over the Newton method for solving realistic problems where the solution from last timestep may not always be a good initial guess. his also presents a possibility to combine the L-scheme with the Newton scheme, where the L- scheme can be used to find a good initial guess for the Newton scheme. his approach is also discussed in [List and Radu, 016].

79 69 able 5.3: Comparison between Newton method and L-scheme with L s = 8 Polynomial orders k s = 1 and k p = 1 L-Scheme Newton method N t = 1 8 t = 1 16 t = 1 3 t = 1 8 t = 1 16 t = 1 3 Number of iterations per timestep [ ] Computation time per step [s] otal computation time [s] able 5.4: Comparison between Newton method and L-scheme with L s = 16 Polynomial orders k s = 1 and k p = 1 L-Scheme Newton method N t = 1 8 t = 1 16 t = 1 3 t = 1 8 t = 1 16 t = 1 3 Number of iterations per timestep [ ] Computation time per step [s] otal computation time [s]

80 70 able 5.5: Comparison between Newton method and L-scheme with L s = 4 Polynomial orders k s = and k p = L-Scheme Newton method N t = 1 8 t = 1 16 t = 1 3 t = 1 8 t = 1 16 t = 1 3 Number of iterations per timestep [ ] Computation time per step [s] otal computation time [s] able 5.6: L-scheme with bad initial guess Polynomial orders k s = 1 and k p = 1 N L s =16 L s =8 L s =4 Number of iterations per timestep [ ] Computation time per step [s] otal computation time [s]

81 71 (a N = 8 8, t = 0.15 (b N = 8 8, t = (c N = 8 8, t = (d N = 16 16, t = 0.15 (e N = 16 16, t = (f N = 16 16, t = (g N = 3 3, t = 0.15 (h N = 3 3, t = (i N = 3 3, t = Figure 5.1: Residual at = 1s and polynomial order k s = k p = 1

82 7 (a N = 8 8, t = 0.15 (b N = 8 8, t = (c N = 8 8, t = (d N = 16 16, t = 0.15 (e N = 16 16, t = (f N = 16 16, t = (g N = 3 3, t = 0.15 (h N = 3 3, t = (i N = 3 3, t = Figure 5.: L -Error for S w at = 1s and polynomial order k s = 1

83 73 (a N = 8 8, t = 0.15 (b N = 8 8, t = (c N = 8 8, t = (d N = 16 16, t = 0.15 (e N = 16 16, t = (f N = 16 16, t = (g N = 3 3, t = 0.15 (h N = 3 3, t = (i N = 3 3, t = Figure 5.3: L -Error for P n at = 1s and polynomial order k p = 1

84 74 (a N = 8 8, t = 0.15 (b N = 8 8, t = (c N = 8 8, t = (d N = 16 16, t = 0.15 (e N = 16 16, t = (f N = 16 16, t = (g N = 3 3, t = 0.15 (h N = 3 3, t = (i N = 3 3, t = Figure 5.4: L -Error for P c at = 1s and polynomial order k p = 1

85 75 (a N = 8 8 (b N = (c N = 3 3 Figure 5.5: L -Error for Residuals at = 1s for t = and polynomial order k p = k s = 1 for the case of a bad initial guess

86 76 (a p n, N = 8 8 (b p n, N = (c p n, N = 3 3 (d p c, N = 8 8 (e p c, N = (f p c, N = 3 3 (g s w, N = 8 8 (h s w, N = (i s w, N = 3 3 Figure 5.6: L -Error for p n, p c, and s w at = 1s for t = and polynomial order k p = k s = 1 for the case of a bad initial guess

87 Chapter 6 Numerical Examples in Heterogeneous Media In this chapter, we show the capabilities of our scheme through different numerical examples in heterogeneous media. In Example 1, Section 6.1, we consider a D problem with homogeneous and heterogeneous permeability fields, and continuous capillary pressure. In Example, Section 6., we verify our numerical scheme for heterogeneous porous media with discontinuous capillary pressure. For this, we consider a 1D problem from the text book by Rainer Helmig [Helmig, 1997]. Next, in Example 3, Section 6.3, we simulate a 1D inflow problem and compare the performance of the L-scheme and the Newton scheme. Finally, in Example 4, Section 6.4, we simulate a D lens problem with heterogeneous permeability field, discontinuous capillary pressure and capillary barrier effects, and gravitational effects to show the capabilities of our schemes. 6.1 Example 1: Inflow problem with homogeneous and heterogeneous permeability fields We consider an inflow problem in a domain Ω = (0m, 1m (0m, 1m, over a time interval [0s, 500s]. he material properties and model parameters are listed in able 6.1. We consider two cases: a homogeneous medium (case A and a non-homogeneous (case B one. he initial and the boundary conditions are listed in able 6.. We discretize the domain into = 500 elements and chose a time-step of dt = 10s. For the L-scheme we take L s = 0.1. Since the dynamic term is linear, no additional linearization is needed. We again chose θ = 0 and σ w = σ n = 10. he result for case A at t = 1500s is shown in Figure 6.1a, and for case B at t = 500s is shown in Figure 6.1b. In case A, a straight finger is formed, propagating with the flow (from left to right. In case B, given the choice of the absolute permeability field K, a preferential flow path is formed along the medium with higher permeability, i.e. from the lower left to 77

88 78 able 6.1: Example 1 - Properties case A case B Phase Properties [ ] water dynamic viscosity µ kg w ms [ ] oil dynamic viscosity µ kg n ms [ ] water density ρ kg w m [ 3 ] oil density ρ kg n m Hydraulic Properties absolute permeability K [ if x < 0.5, y < 0.5 m ] if x > 0.5, y > elsewhere residual water saturation S rw 0 0 residual oil saturation S rn 0 0 porosity ϕ damping coefficient τ [P a s] Brooks-Corey Parameters entry pressure p d [P a] pore size distribution index λ able 6.: Example 1 - Boundary and initial conditions case A case B Boundary values x = 0m { { 0.6 if 0. < y < if 0.4 < y < 0.6 water saturation S w 0. else 0. else oil pressure p n [P a] x = 1m water saturation S w oil pressure p n [P a] y = 0m and y = 1m [ ] flow rate of water q kg w m s [ ] flow rate of oil q kg n m s Initial values water saturation S w 0. 0.

89 79 (a S w at t = 1500s for case A. (b S w at t = 500s for case B. Figure 6.1: Example 1 - Saturation profiles at time t = 500s for a homogeneous medium (left and a heterogeneous one (right. the upper right quadrant. he saturation overshoots in both cases are a manifestation of the dynamic capillarity. For both cases, the L-scheme performs as expected. 6. Example : Benchmark problem for verification of the numerical scheme with discontinuous capillary pressure We consider the benchmark problem described in [Helmig, 1997], p. 75, section 5.5. he problem considers infiltration of non-aqueous phase liquid (NAPL into a fully water saturated domain. A schematic of the problem is shown in Figure 6.. he domain is 0.5m in length, and is divided into three parts with interfaces at x = 0.15m and x = 0.35m. Sub-domain 1 (0 x < 0.15 and sub-domain 3 (0.35m x < 0.5m are made of a porous material with lower entry pressure P e, while the subdomain (0.15 x < 0.35 is made of a porous material with a higher entry pressure P e. he properties of the fluids and the materials in each sub-domain are listed in able 6.3. On the boundary Γ L we prescribe an inflow condition, and on the boundary Γ R we prescribe Dirichlet conditions for p n and S w. he initial condition for the S w corresponds to a fully saturated porous media. An overview of the initial and boundary conditions can be found in able 6.4. We solve the problem in 1D, discretize the domain into 30 elements of size h = with polynomial degrees of k p = k s = 1, and chose a timestep of t = 1s.

90 80 MAERIAL 1 ( LOW P e MAERIAL ( HIGH P e MAERIAL 1 ( LOW P e Γ L I I I II Γ R X=0.0m X=0.15m X=0.35m X=0.5m Figure 6.: Example - est schematic. Here, Γ L and Γ R are the left and the right boundary, respectively, and I I and I II are the material interfaces. Figure 6.3: Example - S n profile at t = 150 s. he result of the simulation is given in Figure 6.3. he comparison to the results of Helmig [Helmig, 1997, p. 86, fig 5.39] shows a good match, which verifies the correctness of our implementation. 6.3 Example 3: Comparison of Newton scheme and L-scheme with discontinuous capillary pressure We consider a 1D infiltration problem similar to Example, and compare the performance of the Newton scheme and the L-scheme with additional dynamic capillary pressure effects. he schematic for the problem is shown in Figure 6.4. We again divide the domain into three sub-domains to cover both transitions: from low to high entry pressure, and from high to low entry pressure. he domain has a length of m and has material interfaces at x = 0.5m and x = 1m. he phase and the material properties are listed in able 6.5. he problem is simulated without additional gravitational effects.

91 81 Fluid Phase Properties dyn. viscosity wetting phase dyn. viscosity non-wetting phase density wetting phase density non-wetting phase able 6.3: Example - Material properties. [ ] µ kg w [ ms] µ kg n [ ms ρ kg w m 3 ] ρ n [ kg m 3 ] Hydraulic Properties Material 1 Material abs. permeability K [ m ] res. wetting phase saturation S rw res. non-wetting phase saturation S rn 0 0 porosity ϕ Brooks-Correy Parameters Material 1 Material entry pressure p d [P a] pore size distr. index λ able 6.4: Example - Boundary and initial conditions. Boundary values x = 0m wetting phase flow [ ] q kg w sm non-wetting phase flow [ ] q kg n sm 0.05 x = 0.5m wetting phase saturation S w 1. non-wetting fluid pressure p n [P a] Initial values wetting phase saturation S w 1 0

92 8 MAERIAL 1 ( LOW P e MAERIAL ( HIGH P e MAERIAL 1 ( LOW P e Γ L I I I II Γ R X=0.0m X=0.5m X=1.0m X=.0m Figure 6.4: Example 3 - Schematic he initial and the boundary conditions are given in able 6.6. At t = 0, the domain is saturated with the wetting phase for x > 0.5. For x 0.5, we define a cubic profile for wetting phase saturation (see able 6.6. At the boundaries, for t 0, Dirichlet conditions are prescribed. We solve the problem for all combinations of time step sizes t[s] = [0.15, 0.5, 0.5] and number of elements N = [100, ]. We chose a minimal residual of ɛ RES < 10 6 for convergence. For the L-scheme, we choose L = 0.1. We compare the performance of the Newton scheme and the L-scheme in terms of the average computation time, average iteration steps necessary, and the total computation time for different time and spatial discretizations. able 6.7 shows the average number of iterations needed for convergence, able 6.8 shows the average computation time per time-step, and able 6.9 shows the total computation time for each scheme. We can observe that as the mesh size and the time step size decrease, the performance of the L-scheme approaches that of the Newton scheme. In Figure 6.5, we plot the non-wetting phase saturation profiles in the domain at selected time steps for the simulation run with t = 0.15s and N = 400. Due to the dynamic capillary effects, we obtain saturation overshoots as the front propagates. he retardation coefficient τ was chosen high enough to allow a plateau to build up. o compare the Newton and L-schemes, we plot both simulation results over each other. he continuous lines represent the results of the L-scheme, while the dots represent the results of the Newton scheme. he accuracy of both schemes is similar, where as the number of iterations necessary for convergence for the L-scheme are twice as many as those for the Newton scheme which is due to the linear convergence of the L-scheme against the quadratic convergence of the Newton scheme. 6.4 Example 4: D Lens problem We consider a two dimensional problem where a non-wetting fluid (e.g. NAPL is infiltrating a domain which is initially fully saturated with the wetting fluid (e.g. water. he schematic of the problem is shown in Figure 6.6. he computational domain contains two distinct zones, each composed of a different material, signifying the material heterogeneity.

93 83 Phase Properties dyn. viscosity wetting phase able 6.5: Example 3 - Material properties dyn. viscosity non-wetting phase density wetting phase density non-wetting phase [ ] µ kg w [ ms] µ kg n [ ms ρ kg w m 3 ] ρ n [ kg m 3 ] Hydraulic Properties Material 1 Material abs. permeability K [ m ] res. wetting phase saturation S rw 0 0 res. non-wetting phase saturation S rn 0 0 porosity ϕ retardation coefficient τ [P a s] Brooks-Correy Parameters Material 1 Material entry pressure p d [P a] pore size distr. index λ able 6.6: Example 3 - Boundary and initial conditions Boundary values x = 0m wetting phase saturation S w 0.5 non-wetting fluid pressure p n [P a] x = 0.5m wetting phase saturation S w 1. non-wetting fluid pressure p n [P a] Initial values { 1 if x > 0.5 wetting phase saturation S w (18x 3 48x + 1 otherwise able 6.7: Example 3 - Comparison of average number of iterations L-Scheme Newton-Scheme N t = 0.5 t = 0.5 t = 0.15 t = 0.5 t = 0.5 t =

94 84 (a t = 0 s (b t = 10 s (c t = 0 s (d t = 50 s (e t = 75 s (f t = 100 s Figure 6.5: Example 3 - Non-wetting saturation profiles for N = 400 and t = 0.15 s

95 able 6.8: Example 3 - Comparison of average computation time per step in seconds [s] L-Scheme Newton-Scheme N t = 0.5 t = 0.5 t = 0.15 t = 0.5 t = 0.5 t = able 6.9: Example 3 - Comparison of total computation time in seconds [s] L-Scheme Newton-Scheme N t = 0.5 t = 0.5 t = 0.15 t = 0.5 t = 0.5 t = he material II has a higher capillary pressure than material I in the sense of the description in Section.7. All relevant properties of both the materials are listed in able he infiltration of the non-wetting fluid is prescribed as a Dirichlet value at the upper boundary, while the lower boundary is assumed to be blind. On the left and right boundaries, no-flow condition is prescribed. Gravitational effects are also included. We simulate two different scenarios to demonstrate the effects of the standard and non-standard capillary pressure relationships in the presence of a material heterogeneity: Case A: without dynamic capillary pressure effects, i.e. τ = 0, and Case B: with dynamic capillary pressure effects. he values for τ are listed in able Case A: τ = 0 he results are presented in Figures 6.7a - 6.7e. We use a refinement of h = 0.05 and t = 0.5. As in the one dimensional examples, the nonwetting phase front propagates until the interface (Figure 6.7a. here it first has to accumulate due to the capillary barrier (Figure 6.7b. While a direct infiltration is not possible, a certain amount of the non-wetting Figure 6.6: Example 4 - Schematic phase flows around the lense leading to two additional fronts (Figure 6.7c. hose additional fronts can not infiltrate the lense from the side, as the threshold capillary pressure for infiltration is not reached. As soon as the entry condition on the upper interface is fulfilled, the main non-wetting phase front starts flowing through the domain at a lower speed, due to the higher lower intrinsic permeability. (Figure 6.7d At the lower interface, leaving the interface no

96 86 Fluid Phase Properties dyn. viscosity wetting phase able 6.10: Example 4 - Material properties dyn. viscosity non-wetting phase density wetting phase density non-wetting phase [ ] µ kg w [ ms] µ kg n [ ms ρ kg w m 3 ] ρ n [ kg m 3 ] Hydraulic Properties Material 1 Material abs. permeability K [ m ] res. wetting phase saturation S rw 0 0 res. non-wetting phase saturation S rn 0 0 porosity ϕ retardation coefficient τ [P a s] Case A 0 0 Case B Brooks-Correy Parameters Material 1 Material entry pressure p d [P a] pore size distr. index λ barrier occurs and flow is directly possible.(figure 6.7e Case B: τ > 0 With dynamic capillary pressure effects we get similar results as in case A. he results are presented in Figures 6.7f - 6.7j. We again use a refinement of h = 0.05 and t = 0.5. he additional dynamic term leads to a retardation effect, resulting in a slower propagation of the front. Mass accumulates at the tip of the resulting inflow-finger (Figure 6.7f. Like in the standard case, reaching the upper interface an info is not directly possible and non-wetting phase saturation accumulates (Figure 6.7g. Again two fingers are formed flowing around the lense each of them with an overshoot at the tip (Figure 6.7h. As soon as an inflow is possible, the main front propagates forward, forming again an overshoot. At the lower boundary we once again don t observe a capillary barrier, and the flow is directly possible (Figure 6.7j. Figure 6.8 makes the retardation effect of the dynamic capillary pressure more clearly, we plotted the results next to each other in and used the same time in each figure, to visualize the difference in infiltration speed.

97 87 Figure 6.7: Example 4 - Results bottom: with τ > 0, top: without τ = 0 (a t = 37.5 s (b t = 100 s (c t = 175 s (d t = 50 s (e t = 375 s (f t = 15 s (g t = 00 s (h t = 300 s (i t = 400 s (j t = 500 s

98 88 Figure 6.8: Example 4 - Effect of the retardation coefficient left: τ = 0, right: τ > 0 (a t = 15 s (b t = 00 s (c t = 300 s (d t = 400 s