Eindhoven University of Technology

MASTER

Upscaling of Richards' equation in fractured porous media

List, F.

Award date:2017

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Master Thesis

Upscaling of Richards’ Equation in FracturedPorous Media

Florian List

Supervisors:

Prof. Dr. Sorin PopFaculty of SciencesDiscipline group Mathematics and statisticsHasselt University

Prof. Dr. Florin RaduDepartment of Mathematics

University of Bergen

Prof. Dr. Christian RohdeInstitute of Applied Analysis andNumerical SimulationUniversity of Stuttgart

Prof. Dr. ir. Barry KorenDepartment of Mathematics and Computer Science

Eindhoven University of Technology

August 2017

Abstract

In this thesis, we consider fluid flow through a two-dimensional porous medium containing a fracture. In boththe solid matrix blocks and the fracture, the flow is described by Richards’ equation, equipped with differingparametrisations of the hydraulic conductivity and the water content in the solid matrix and the fracture,respectively. For the coupling between the matrix blocks and the fracture, we impose the continuity of thepressure and of the flux at the interface. The geometry is chosen rectangular for the sake of simplicity.At first, we fix the fracture width and apply Rothe’s method, that is an implicit Euler discretisation in time, inorder to show the convergence of the time-discrete solutions towards the solution to the time-continuous prob-lem as the time step vanishes. To do so, we make use of compactness arguments based on a priori estimatesfor the solution. It will prove crucial to overcome the difficulties posed by the nonlinearity of Richards’ equa-tion. Rothe’s method yields the existence of a solution to the coupled problem provided that the time-discreteproblem admits a solution. For the existence of solutions of the time-discrete problem, we refer the reader toprevious works.Then, we view the fracture width as a parameter and apply rigorous upscaling to prove the convergence to-wards solutions of different effective models as the fracture width goes to zero. The scaling of permeabilitiesand porosities with respect to the fracture width is decisive for determining the respective effective model.Finally, we present numerical simulations which confirm our theoretical upscaling results.

Zusammenfassung

In dieser Thesis betrachten wir die Stromung durch ein zweidimensionales poroses Medium, das eine Spalteenthalt. Die Stromung wird in der porosen Matrix und in der Spalte durch die Richards-Gleichung beschrieben,wobei sich die Parametrisierungen der hydraulischen Leitfahigkeit und des Wassergehalts im porosen Medi-um und in der Spalte unterscheiden durfen. Die Stromung in der porosen Matrix und in der Spalte werdengekoppelt, indem wir die Stetigkeit des Drucks und des Flusses an der Grenzflache fordern. Der Einfachkeithalber wahlen wir eine rechteckige Geometrie.Zunachst halten wir die Spaltbreite fest und wenden Rothes Methode an, sprich die Zeitdiskretisierung mitHilfe des impliziten Euler-Verfahrens, um die Konvergenz der zeitdiskreten Losungen gegen die Losung deszeitkontinuierlichen Problems zu zeigen, wenn der Zeitschritt gegen null geht. Dazu benutzen wir Kompakt-heitsargumente, die auf A-priori-Abschatzungen fur die Losung beruhen. Es wird sich als entscheidend her-ausstellen, die aus der Nichtlinearitat der Richards-Gleichung resultierenden Schwierigkeiten zu bewaltigen.Rothes Methode liefert die Existenz einer Losung des gekoppelten Problems unter der Annahme, dass eineLosung des zeitdiskreten Problems existiert. Fur die Existenz von Losungen des zeitdiskreten Problems ver-weisen wir den Leser auf bereits bestehende Arbeiten.Danach fassen wir die Spaltbreite als einen Parameter auf und wenden rigoroses Upscaling an, um fur ver-schwindende Spaltbreite die Konvergenz gegen Losungen von unterschiedlichen effektiven Modellen zu be-weisen. Die Skalierung der Permeabilitaten und Porositaten mit der Spaltbreite legt das jeweilige effektiveModell fest. Zum Abschluss prasentieren wir numerische Simulationen, die unsere theoretischen Upscaling-Resultate bestatigen.

Contents

Introduction 3

1 Modelling of subsurface flow 51.1 Basics of subsurface flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Richards’ equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3 Van Genuchten–Mualem parametrisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4 Fractures in porous media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.5 Dimensional model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.6 Dimensionless model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Mathematical foundations 152.1 Function spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1.1 Lebesgue spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.1.2 Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.1.3 Bochner spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Theorems from functional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.3 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.4 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Existence 213.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Weak solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.3 Time discretisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.4 A priori estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.5 Interpolation in time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.6 Essential bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4 Upscaling 354.1 Upscaling formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.2 Effective models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2.1 Strong formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.2.2 Weak formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.3 Uniform estimates with respect to ε . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.4 Upscaling theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.5 The case λ < −1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5 Numerical simulations 555.1 Models with continuous pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.2 Models with discontinuous pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.3 Additional models for no flow conditions in the fracture . . . . . . . . . . . . . . . . . . . . . . . 685.4 Discussion of the numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

Discussion and outlook 71

Introduction

Water is a fundamental requirement of life on earth. Still, two-third of the world population face severe waterscarcity at least one month of the year [38] and predictions foresee a rapid decline in the availability of drinkingwater in the coming decades caused by the increase of the world population, a rising water consumption percapita, and the climate change amongst other reasons. This scenario not only incites politics to act, but also en-courages engineers and scientists to develop water-saving technologies and to gain a deeper understanding offlowing processes in order to optimise the positioning of wells, prevent groundwater from being polluted, andcarry out water management on larger scales, to name a few challenges. During the recent decades, numericalsimulations based on mathematical modelling have become a powerful tool to gain insight in numerous ap-plications of fluid mechanics at high accuracy.In this thesis, we consider a mathematical model for unsaturated subsurface flow in fractured porous media,in which the flow is governed by Richards’ equation. Fractures play a crucial role in subsurface flow since thepermeability in fractures can significantly exceed the one in the surrounding soil and hence lead to massivelyaltered flow patterns. Besides natural processes, fractured porous media occur in various technical applica-tions such as carbon capture and storage (CCS), where CO2 is permanently stored in geological formations toprevent it from being released into the atmosphere, or fracking, that is pumping a pressurised liquid into rockto create fractures that facilitate the flow of gas or petroleum.However, since our mathematical model uses Richards’ equation for the description of the fluid flow, which issuitable for the modelling of liquid flow close to the earth’s surface, we focus on the application of water flowthrough fractured soil. We consider a single fracture embedded in two surrounding matrix blocks. The use ofRichards’ equation in the fracture requires the fracture to be filled with a porous medium yielding sufficientlysmall flow velocities, which we assume to hold true herein.When it comes to modelling of thin fractures in a d-dimensional domain, the fracture width is often neglectedand the fracture is viewed as a (d− 1)-dimensional quantity. It is interesting to justify these models mathemat-ically as limit models of models containing a d-dimensional fracture in the limit when the fracture becomesinfinitely thin. This is commonly done by so-called upscaling techniques, consisting in rescaling the fracturedomain and the corresponding variables and then either employing a series expansion with respect to thefracture width as an ansatz for the solution (formal upscaling) or showing the convergence towards the limitmodel by strict means, harnessing results from functional analysis (rigorous upscaling). Here, we opt for rig-orous upscaling. The central results of this work are the analysis of the model for fixed fracture width and therigorous derivation of limit models for vanishing fracture width.The structure will be as follows: Chapter 1 is dedicated to the mathematical modelling of unsaturated flowthrough porous media. We introduce the basic physical quantities related to porous media and we give thederivation of Richards’ equation. Then, we define our mathematical model for a two-dimensional porousmedium crossed by a fracture.In Chapter 2, we present the mathematical foundations, which will be required in the subsequent chapters.Since these results are well-known, we merely state the results and omit the proofs, which can be found instandard literature.Chapter 3 is concerned with the existence of solutions. Rothe’s method is applied to the model equations lead-ing to time-discrete equations. We proceed by proving essential bounds for the solution in L∞ and a prioriestimates for the solution and its derivatives in L2. Making use of compactness arguments, specifically theAubin–Lions–Simon theorem, we obtain a strongly convergent subsequence as the time step vanishes and weshow that the limit function is a solution to our model. The existence of solutions to the time-discrete model isdiscussed e.g. in [27] and will be assumed in this work.In Chapter 4, we investigate the behaviour of solutions as the fracture width goes to zero. Similar estimatesas in the previous chapter allow us to prove the convergence towards effective models in the limit by usingrigorous upscaling techniques. Depending on the scaling of the permeability and the porosity with respect to

3

the fracture width, various effective models can arise, which strongly differ from each other: the fracture canbecome impermeable or remain permeable, and the subdomains on both sides of the fracture may completelydecouple or stay connected. The remaining equation for the fracture in the effective model can be an interfacecondition or a differential equation.In Chapter 5, we examine the upscaling results from the previous chapter by means of numerical simulations.We end this work with a discussion of the results and an outlook to further research and possible extensions.

4

Chapter 1

Modelling of subsurface flow

In this chapter, we introduce the fundamentals of mathematical modelling of subsurface flow through porousmedia. We define the basic physical quantities and derive Richards’ equation, which is widely employed inthe modelling of unsaturated water flow near the surface, by the principle of mass conservation together withDarcy’s law for multiphase flow.When it comes to subsurface flow, several entities play a role typically, such as a liquid phase, a gas phase anda rigid or deformable solid matrix interacting with the flowing fluids. The respective phases can be composedof several components, e.g. the gas phase could be made up of air and water steam. The interaction among theconstituents on a microscopic level is highly complex and since the pore structure of the solid matrix is mostlyunknown anyway, one commonly resorts to the description of the flow processes by averaged equations, notleast because numerical simulation on the microscale is often not feasible due to computational limitations.We start with defining macroscopic quantities characterising subsurface flow.

1.1 Basics of subsurface flow

Subsurface flow in general describes fluid flow beneath the earth’s surface. Here, the considered fluid is waterflowing in the vadose zone, that is the unsaturated zone between the earth’s surface and the groundwater level,or the groundwater filtering through water-conducting aquifers. The vadose zone and the aquifers consist ofpermeable rock or unconsolidated materials such as silt, loam, sand or gravel, which possess pores allowingthe flow of water. Such media are called porous media.In order to view a porous medium as a single entity without distinguishing the pore space and the solid ma-trix, a representative elementary volume (REV) is introduced. The REV should be chosen large enough torepresent a typical sample of the heterogeneous medium and small enough to capture spatial inhomogeneities(see Figure 1.1). For comprehensive introductions to REVs, we refer the reader to [5, chap. 1] and [25, chap. 2].

sample volume

Prop

erty

aver

age

inhomogeneous media

REV

Figure 1.1: Definition of the REV [5, 25]

On the REV scale, physical quantities that vary rapidly within the REV due to the heterogeneity of the mediumcan be meaningfully defined and the discontinuities at the transition between the pore space and the solid ma-trix are no longer observable. We specify the most important quantities in the context of subsurface flow inwhat follows. Note that we omit any temperature related quantities as we assume isothermal conditions.

5

AQ

solid matrix

fluid phase possible REV

not representative

Figure 1.2: Schematic representation of flow through a porous medium and the REV scale

Porosity

The ratio between the pore volume Vpore [m3], which can be occupied by a fluid, and the total volume Vtotal [m3]of a material sample is called the porosity φ [−] of a porous medium, in formulae

φ :=Vpore

Vtotal, (1.1)

and thus lays between 0 and 1. Whereas the porosity oscillates rapidly if the sample under consideration ischosen too small, it varies slowly when a sample at the REV scale is translated in space (see Figure 1.2). As re-gards fluid flow, one usually considers the effective porosity, which disregards closed pores for the calculationof the pore volume.

Saturation and water content

When several immiscible phases are present in the pore space, the saturation S [−] of a phase α measures thevolume fraction of the pore space which is taken by the respective phase,

Sα :=Vα

Vpore, (1.2)

where Vα [m3] denotes the volume occupied by phase α. The system is then called a multiphase system.Recalling the definition of porosity the volume content θα [−] of phase α can be defined as

θα :=Vα

Vtotal= φSα. (1.3)

In the case of the water phase, the volume content is commonly referred to as the water content.Let the number of phases being present in the pore space be given as nphase. Since the pore space is entirelyfilled, the following equivalent restrictions hold:

nphase

∑α=1

Vα = Vpore,

nphase

∑α=1

θα = φ,

nphase

∑α=1

Sα = 1.

(1.4)

In general, the phases can be made up of several components. Moreover, in several applications phase transi-tions have to be taken into account, e.g. when a liquid evaporates.

6

Darcy velocity

Let the total volume discharge of phase α be given by Qα [m3 s−1], and let A [m2] be the total flow cross-section(comprising solid matrix and pore space). The Darcy velocity (also discharge velocity, specific discharge)vα [m s−1] of the phase is defined as

vα :=Qα

A. (1.5)

In order to obtain the average flow velocity vav,α [m s−1] of individual particles from this, one must relate thetotal discharge to the area accessible to fluid flow, that is

vav,α :=vα

φ. (1.6)

All velocities and fluxes are vectorial if a multidimensional problem is considered.

Permeability and hydraulic conductivity

The intrinsic permeability k [m2] is a macroscopic measure that characterises how easily fluids can pass througha porous medium. It depends only on the porous medium and is independent of the fluid. In the case ofanisotropic media, the intrinsic permeability is a tensorial quantity. From the intrinsic permeability, the fluiddensity ρα [kg m−3] and the dynamic viscosity µα [kg m−1 s−1] of a phase α, another property can be derivedwhich is called the hydraulic conductivity K [m s−1], defined as

Kα := kkr,α(θα)ραgµα

. (1.7)

Here, g [m s−2] is the gravitational constant and kr,α = kr,α(θα) [−] is the relative permeability. The ideabehind introducing the relative permeability is the following: when several phases occupy the pore space, thepresence of one phase will impede the flow of the other phases and vice versa. This is due to a smaller cross-sectional area available for each phase and due to the increased tortuosity of the flow paths, since particles of aphase have to flow around accumulations of particles of the other phases. In order to express this microscopicobservation on the macroscale, it is required to link the relative permeability to another macroscopic quantity.It turns out that the volume content (or the saturation) suits this purpose: the larger the volume content of aphase, the larger is the cross-sectional area through which this phase flows and the less must the particles ofthis phase give way to clusters of other phases. Thus, one can experimentally determine a parametrisation ofthe relative permeability as a function of the water content. Clearly, its values lay between 0 and 1, where 0represents the case when the phase is not present in the pore space and 1 describes the case when the entirepore space is filled with the respective space, i.e. kr,α(θα = 0) = 0 and kr,α(θα = φ) = 1. Typically, kr,α(θα) isa monotonically increasing, convex function. A common parametrisation due to Mualem and van Genuchtenwill be presented in Section 1.3.

Capillary pressure

We introduce the capillary pressure for the case of two fluids present in the system, the extension to three ormore fluids is plain and can be found in [25, sec. 2.4]. When considering fluid flow through porous mediaon the microscale, the interfacial tensions between phases play a crucial role. They are caused by cohesiveforces between molecules of the same phase and adhesive forces between molecules of different phases. Theequilibrium of these forces results in a curved interface, which minimises the total potential energy of thesystem. Thus, the boundary angle in the equilibrium, which is referred to as the wetting angle γ [], is ingeneral not right-angled (see Figure 1.3). One calls the fluid with the acute wetting angle the wetting fluid(denoted with the subscript w here) and the fluid with the obtuse wetting angle the non-wetting fluid (denotedwith the subscript n here). Notice that the notions of wetting and non-wetting depend on all the phases presentin the system: oil is typically the wetting phase in gas-oil systems whereas it is typically the non-wetting phasein water-oil systems, to give an example. The jump in pressure p [kg m−1 s−2] between the fluid phases whichoriginates from the equilibrium of forces is named the capillary pressure pc [kg m−1 s−2]:

pc := pn − pw. (1.8)

7

γ

non-wetting phase

solid phase

wetting phase

Figure 1.3: Wetting angle γ at the interface of two fluid phases and one solid phase

It can be computed from the capillary tube diameter d [m], the wetting angle and the interfacial tensionσ [kg s−2] by Laplace’s equation

pc =4σ cos γ

d. (1.9)

A survey on the interfacial tensions for different fluid combinations can be found e.g. in [48]. Laplace’s equa-tion (1.9) shows that the smaller the pores in the system are, the higher is the capillary pressure. This gives riseto the following observations: when the wetting phase is drained, it leaves the larger pores first and remainsin the smaller pores where capillary action is the highest. On the other hand, when imbibition takes place andwetting phase fluid is injected, the wetting phase fills larger pores where the capillary pressure is lower.This behaviour can be modelled on the macroscale by introducing a relation between the capillary pressureand the volume content (or the saturation respectively), which captures the findings on the microscale in thesense that a higher volume content of the wetting phase leads to a lower capillary pressure and vice versa:

pc = pc(θw). (1.10)

This relation cannot be obtained analytically due to the complexity of the pore space geometry, but severalempirical relations have been proposed based on a multitude of experiments, the most famous among thesebeing due to Leverett [34], Brooks and Corey [10], and Mualem and van Genuchten [54]. We will describe thevan Genuchten–Mualem model in Section 1.3, which we will use in the numerical simulations at the end ofthis thesis.The parametrisation of the capillary pressure only as a function of the water content as in equation (1.10) islimited to creeping flow, however, and deviations from experiments are observed when it comes to transientflow. To overcome this, the capillary pressure parametrisation may be supplemented with dynamic capillarityeffects in order to obtain a model of the form

pc = pc(θw, ∂tθw) = pn − pw + τ(θw)∂tθw, (1.11)

where τ = τ(θw) [kg m−1 s−1] is a non-equilibrium coefficient. Such models are used in e.g. [23, 28]. Besides,hysteresis effects expressing different soil behaviours for imbibition and drainage can be taken into account,see e.g. [13].

1.2 Richards’ equation

Richards’ equation was first presented in 1931 by Lorenzo A. Richards [45]. It is used for the macroscalemodelling of unsaturated water flow close to the earth’s surface and is based upon the assumptions that thenon-wetting gas phase in the porous medium, which is commonly air, is interconnected and connected to theatmosphere. Then, pressure fluctuations of the gas phase are small and can be neglected. In what follows, wewill present the derivation of Richards’ equation from the balance of mass in conjunction with the extendedDarcy law, which provides a suitable form of the momentum balance in porous media.We start by taking an arbitrary control volume Ω ⊂ Rd for d ∈ 1, 2, 3 with smooth boundary ∂Ω. Note thatin this derivation, we assume all properties to be sufficiently smooth for our purposes. The conservation of thewater mass mw [kg] contained in Ω reads as

∂tmw = ∂t

∫Ω

ρwφSw dV +∫

∂Ω(ρwφ vav,w) ·~n dA =

∫Ω

qw dV, (1.12)

where ~n is the outward pointing normal vector on ∂Ω and qw [kg m−3 s−1] is a mass source or sink term, forexample due to wells or pumps in the domain. Interchanging the time derivative and the integral and applying

8

the divergence theorem gives∫Ω

∂t (ρwφSw) dV +∫

Ω∇ · (ρwφ vav,w) dV =

∫Ω

qw dV. (1.13)

As the control volume Ω has been chosen arbitrarily, we demand the equality of the integrands to obtain

∂t (ρwφSw) +∇ · (ρwφ vav,w) = qw. (1.14)

Making use of the relations θw = φSw and vw = φvav,w, we can rewrite this as

∂t (ρwθw) +∇ · (ρwvw) = qw. (1.15)

Now, we neglect the low compressibility of water and assume that the water density is temporally and spatiallyconstant, namely ρw(t,~x) ≡ ρw > 0. Consequently, we get

∂tθw +∇ · vw =qw

ρw. (1.16)

At this point, we are required to relate both the Darcy velocity vw and the water content θw to the pressure.This is where Darcy’s law for multiphase flow comes into play. In 1856, Henry Darcy was concerned withthe drafts for a water system in the French city of Dijon [14]. In this context, he discovered experimentallya proportionality between the Darcy velocity and the negative pressure gradient. Originally, Darcy statedthe law only for saturated single-phase flow – Edgar Buckingham extended it to unsaturated flow, where thecrucial supplement is the concept of relative permeability. By now, several mathematical derivations of Darcy’slaw from the Navier–Stokes equation exist, based on volume averaging as in [56, chaps. 4–5], homogenisationas in [26, chap. 3], or thermodynamically constrained averaging theory as in [22, chap. 9].Darcy’s law for multiphase flow through an isotropic medium writes as

vw = −Kw(θw)∇ψw, (1.17)

where ψw [m] is the pressure head, related to the pressure pw [kg m−1 s−2] by the equation

ψw =pw

ρwg. (1.18)

We neglect gravitational effects herein.In order to obtain a relation between the water content and the pressure, recall the definition of the capillarypressure pc(θw) = pn − pw. Taking now a constant air pressure pn ≡ 0, justified by the assumption that theair is infinitely mobile and can equilibrate pressure fluctuations instantaneously because the flow domain isinterconnected and connected to the surface [40, chap. 3], we get

pc(θw) = −pw. (1.19)

Since the macroscopic relation between the capillary pressure and the water content embodies the microscalebehaviour, it is commonly strictly monotonically increasing, which allows us to invert equation (1.19). Thisprovides us with an explicit parametrisation

θw = θw(ψw). (1.20)

Equations (1.16), (1.17), and (1.20) leave us with Richards’ equation in the mixed pressure-saturation form

∂tθw(ψw)−∇ · [Kw(θw(ψw)) ∇ψw] = fw, (1.21)

where we set fw = qwρw

.

1.3 Van Genuchten–Mualem parametrisation

The van Genuchten–Mualem model is one of the most widely employed empirical models for the relationsbetween the capillary pressure and the volume content, and between the hydraulic conductivity and the water

9

content. The main difference in comparison with the Brooks–Corey model consists in the capillary pressurewhen the wetting phase saturation approaches 1: whereas the capillary pressure in the Brooks–Corey modeltakes a positive value (the so-called entry pressure) in the case of the porous medium being fully saturatedwith the wetting phase, the capillary pressure in the van Genuchten–Mualem model converges smoothly to 0as the wetting phase saturation goes to 1.In the setting of Richards’ equation where pc(θw) = −pw and with pc strictly monotonically increasing, we canexpress the water content as a function of the pressure head and hence the hydraulic conductivity as a functionof the pressure head as well. The van Genuchten–Mualem parametrisation writes as

θw(ψw) =

θR + (θS − θR)[

11+(−αψw)n

] n−1n , ψw ≤ 0,

θS, ψw > 0,

Kw(ψw) = Kw(θw(ψw)) =

KSθw(ψw)12

[1−

(1− θw(ψw)

nn−1

) n−1n]2

, ψw ≤ 0,

KS, ψw > 0,

(1.22)

where θS and KS stand for the water content and the hydraulic conductivity of the fully saturated porousmedium, respectively, θR denotes the residual water content, and α and n are curve fitting parameters express-ing the soil properties. Figure 1.4 shows typical curves. When the pressure head approaches negative infinity,the water content converges to the residual water content θR that remains in the pores and the hydraulic con-ductivity converges to Kw(θR).Note that in the fully saturated regime we retrieve the single-phase Darcy law from Richards’ equation, sincethe water content is then constant, and the hydraulic conductivity is independent of the pressure. This makesRichards’ equation a nonlinear degenerate elliptic-parabolic partial differential equation, which becomes ellip-tic wherever the flow is saturated.

Figure 1.4: Typical profiles of θ(ψ) and K(θ(ψ)) as given by the van Genuchten–Mualem model

1.4 Fractures in porous media

Fractures and fracture networks in porous media influence the flow properties significantly. They are formedwhen mechanical stresses in the porous medium become too high, causing the porous medium to crack orexisting fractures to expand. Fractures divide the medium into several essentially disjoint matrix blocks. Inthe fracture, the permeability is typically higher than in the surrounding matrix blocks, for which reason theprerequisites for using Darcy’s law or Richards’ equation for the description of the flow in the fracture maynot always be given: if the fracture is hollow and the water can flow freely through the fracture, Richards’equation can certainly not be employed. In this case, the flow in the fracture is more accurately described bythe (Navier–)Stokes equation or by the Hagen–Poiseuille equation (see e.g. [39] for the coupling of two-phaseflow and free flow, and [47] for the coupling of Richards’ equation and free flow).

10

Herein, we assume the fracture to consist of a porous medium with possibly different properties than theporous medium in the matrix blocks, which still allows for the use of Richards’ equation for the fracture flow.Examples for this scenario are fractures in rock filled with silt or loam. Several approaches for the relationbetween relative permeability and volume content in fractures have been proposed (see e.g. [25, sec. 2.7],[46]),which have applications in simulations of fractured oil reservoirs [20]. In order to model the capillarity in frac-tures, a model for the relation between capillary pressure and volume content was presented by Pruess andTsang [44], which is also introduced in [25, sec. 2.4]. For a single idealised fracture it is possible to explicitlycalculate the equilibrium of forces [55] and to give a formula for the dependence of the volume content onthe capillary pressure and the pore geometry. As a matter of fact, the interaction between the fracture and thematrix blocks is highly complex – for some aspects of it we refer the reader to [25, sec. 2.7].For the sake of simplicity, we use the van Genuchten–Mualem model in the fracture for our simulations,equipped with van Genuchten parameters differing from the ones used for the matrix blocks.

1.5 Dimensional model

We consider a mathematical model of fluid flow through a fractured porous medium, in which the flow isgoverned by Richards’ equation in the fracture (denoted with the subscript f ) and in the matrix blocks (denotedwith the subscript m). For the ease of presentation, we resort to a simple geometry consisting of two squaresolid matrix blocks with edge length L [m] separated by a fracture of width l [m]. The geometry of the fractureis depicted in Figure 1.5. We use hats on top to indicate quantities associated with the dimensional model todistinguish them from the dimensionless quantities which will be introduced subsequently.

(−L− l2, 0) (− l

2, 0) ( l2, 0) (L + l

2, 0)

(−L− l2, L) (− l

2, L) ( l2, L) (L + l

2, L)

Ωm1 Ωm2Ω f

Γ1 Γ2

y

x

~n ~n

Figure 1.5: Geometry of the fracture and the surrounding matrix blocks

The model defined on this geometry is given by

Problem PD :

∂t θm(ψmj) + ∇ · vmj = fmj in ΩTmj

,

vmj = −Km(θm(ψmj))∇ψmj in ΩTmj

,

∂t θ f (ψ f ) + ∇ · v f = f f in ΩTf ,

v f = −K f (θ f (ψ f ))∇ψ f in ΩTf ,

ψi = 0 on ∂Ωi \ Γ× (0, T],

vmj ·~n = v f ·~n on Γj × (0, T],

ψmj = ψ f on Γj × (0, T],

ψi(0) = ψi,I in Ωi,

for i ∈ m1, m2, f , j ∈ 1, 2, where Ωm := Ωm1 ∪ Ωm2 , Γ := Γ1 ∪ Γ2, ΩTi := Ωi × (0, T] for a given time T > 0,

~n is a normal vector (e.g. with unit length) pointing from Ωmj into Ω f , θ is the water content, v is the flux asgiven by Darcy’s law for unsaturated flow, ψ is the pressure height, K is the hydraulic conductivity, ψi,I is agiven initial condition, and f is a source/sink term.

11

In words, the model consists of Richards’ equation in the fracture and the matrix blocks, supplemented withthe continuity of flux and pressure as interface conditions, and initial conditions. The continuity of the fluxacross the interfaces accounts for the conservation of mass, and the continuity of the pressures can be derivedfrom the conservation of momentum (see e.g. [24]).Note that we neglect gravitational forces in this thesis.

1.6 Dimensionless model

We carry out a non-dimensionalisation procedure in order to render problem PD dimensionless. We define theratio of the fracture width to its length as

ε :=lL

. (1.23)

Moreover, we define reference length scales x = L, y = L, and introduce dimensionless spatial coordinatesx = x/L, y = y/L. The partial derivatives transform according to ∂x = L ∂x, ∂y = L ∂y by the chain rule.Figure 1.6 shows the resulting geometry.

(−1− ε2, 0) (− ε

2, 0) ( ε2, 0) (1 + ε

2, 0)

(−1− ε2, 1) (− ε

2, 1) ( ε2, 1) (1 + ε

2, 1)

Ωm1 Ωm2Ω f

Γ1 Γ2

y

x

~n ~n

Figure 1.6: Dimensionless geometry of the fracture and the surrounding matrix blocks

Since the pressure is continuous at the interfaces, we define a reference pressure head for the entire domainψ = L. For the matrix blocks and the fracture we use different reference hydraulic conductivities Km and K f ,respectively. As the reference time scale, we set

T :=L2

Kmψ=

LKm

. (1.24)

The dimensionless pressure heads are then given as ψmj = ψmj /L and ψ f = ψ f /L, the dimensionless timeas t = t/T, and the final time as T = T/T. The derivative with respect to the dimensionless time becomes∂t = T ∂t. As regards the source terms, let fmj = fmj T and f f = f f T. We define dimensionless hydraulicconductivities Km = Km/Km and K f = K f /K f .Substituting Darcy’s law for multiphase into the mass balance in Problem PD, we obtain for the matrix blocks

1T

∂t(φmSm(ψmj))−1L∇ ·

(KmKm(θm(ψmj))

1L∇(Lψmj)

)=

1T

fmj . (1.25)

Making use of the relation in equation (1.24), this yields

∂t(φmSm(ψmj))−∇ ·(

Km(θm(ψmj))∇ψmj

)= fmj . (1.26)

12

We relate the reference values for the porosity and the hydraulic conductivity in the fracture and the matrixblocks to each other by setting

φ f

φm= εκ ,

K f

Km= ελ.

(1.27)

This is where the scaling parameters κ, λ ∈ R come into play, which will prove crucial in determining effectivemodels in the limit ε→ 0, as we will see later. The value of κ determines the storage capacity of the fracture: forκ < 0, the reference porosity of the fracture increases for decreasing ε as compared to the reference porosity ofthe matrix blocks, and for κ being sufficiently small, the fracture can maintain its ability to store water as ε goesto zero. For κ = 0, no scaling occurs, and for κ > 0, the storage ability of the fracture decreases for ε → 0 dueto the decline of both the fracture volume and the fracture porosity. The parameter λ regulates the scaling ofthe flux term: for λ < 0, the reference hydraulic conductivity of the fracture grows for decreasing ε, assuminga fixed reference conductivity of the matrix blocks. Thus, the flow along the fracture can be sustained whenε goes to zero. For even smaller λ, pressure differences in the fracture are instantaneously equilibrated in theresulting effective models. The choice λ = 0 implies no scaling of the flux term in the fracture, and for λ > 0,the fracture can become so impervious that a discontinuity in pressure arises at the fracture in the effectivemodels, or the matrix blocks can fully decouple from each other.From the fracture equations, we get

1T

∂t(φ f S f (ψ f ))−1L∇ ·

(K f K f (θ f (ψ f ))

1L∇(Lψ f )

)=

1T

f f , (1.28)

and inserting equations (1.24) and (1.27) yields

∂t(φmεκS f (ψ f ))−∇ ·(

ελK f (θ f (ψ f ))∇ψ f

)= f f . (1.29)

The flux interface condition transforms into

Km(θm(ψmj))∇ψmj ·~n = ελ K f (θ f (ψ f ))∇ψ f ·~n on Γj. (1.30)

For the sake of simplicity, we will slightly abuse notation and write θm = φmSm and θ f = φmS f in the following.We treat the dimensionless fracture width ε > 0 as a model parameter. For each ε, the model reads

Problem P :

∂tθm(ψεmj) +∇ · vε

mj= fmj in ΩT

mj,

vεmj

= −Km(θm(ψεmj))∇ψε

mjin ΩT

mj,

∂t(εκθ f (ψ

εf )) +∇ · v

εf = f ε

f in ΩTf ,

vεf = −ελK f (θ f (ψ

εf ))∇ψε

f in ΩTf ,

ψεi = 0 on ∂Ωi \ Γ× (0, T],

vεmj·~n = vε

f ·~n on Γj × (0, T],

ψεmj

= ψεf on Γj × (0, T],

ψεi (0) = ψi,I in Ωi.

13

Chapter 2

Mathematical foundations

In this chapter, we gather mathematical results, in particular from the field of functional analysis, which wewill employ in the subsequent chapters when it comes to the existence of a solution for fixed fracture widthand to the upscaling results. Since these results are standard and can be found in a multitude of introductorybooks, we omit the proofs here and refer the interested reader to [9, 16, 17].

2.1 Function spaces

First of all, we introduce Lebesgue and Sobolev spaces, which are fundamental in the theory of partial differ-ential equation. For the sake of simplicity, we consider real vector spaces here, the extension to complex vectorspaces is plain.

2.1.1 Lebesgue spaces

Let 1 ≤ p ≤ ∞ and let (Ω,A, µ) be a measure space. Then, the set

Lp(Ω,A, µ) :=

f : Ω→ R : f is measurable, ‖ f ‖Lp(Ω) < ∞

(2.1)

is a vector space with seminorm ‖ · ‖Lp(Ω), defined as

‖ · ‖Lp(Ω) : f 7→(∫

Ω | f |p dµ

) 1p , 1 ≤ p < ∞,

ess supx∈Ω| f (x)|, p = ∞.(2.2)

The map ‖ · ‖Lp(Ω) is only a seminorm because every function which is zero everywhere except on a set ofmeasure zero has norm zero. In order to obtain a normed space, one identifies functions that are equal almosteverywhere with each other. To be more specific, we define the subspace

N := f ∈ Lp(Ω,A, µ) : f = 0 µ-a.e. (almost everywhere) = f ∈ Lp(Ω,A, µ) : ‖ f ‖Lp(Ω) = 0, (2.3)

and introduce the quotient space Lp(Ω,A, µ) = Lp(Ω,A, µ)/N , called Lp space or Lebesgue space. All func-tions that differ only on a set of measure zero from each other are represented by one element in Lp(Ω,A, µ).One can show that the space Lp(Ω,A, µ) with the norm ‖ · ‖Lp(Ω) = ‖ · ‖Lp(Ω) is complete and therefore aBanach space.The space L2(Ω,A, µ) plays a special role since it is the only Hilbert space among the Lp spaces, equipped withthe inner product

( f , g)L2(Ω) :=∫

Ωf g dµ. (2.4)

If (Ω,A) is a separable measurable space, the space Lp(Ω,A, µ) is separable as well for 1 ≤ p < ∞, whereasL∞(Ω,A, µ) is not separable in general.For 1 < p < ∞ the dual space of an Lp space is again an Lp space, namely

Lp(Ω,A, µ)∗ = Lq(Ω,A, µ), (2.5)

15

where 1p + 1

q = 1. Therefore, the spaces Lp(Ω,A, µ) are reflexive for 1 < p < ∞, but one can show that thisdoes not hold true for p ∈ 1, ∞.In this thesis, the measure space (Ω,A, µ) is composed of a bounded open domain Ω ⊂ Rd for d ∈ 1, 2, 3with Lipschitz boundary ∂Ω, the Lebesgue σ-algebra Σ on Ω and the Lebesgue measure µ = λ. Then, we justwrite Lp(Ω) = Lp(Ω, Σ, λ).

2.1.2 Sobolev spaces

The Lp spaces can be viewed as a special case of a larger class of spaces, the so-called Sobolev spaces. Sobolevspaces are the natural spaces for the solutions of partial differential equations. In order to introduce them, wefirst need to define the concept of weak derivatives. Let Ω ⊂ Rd be an open set, let L1

loc(Ω) be the set of locallyintegrable functions, that is

L1loc(Ω) := f : Ω→ R : f is measurable, f |K ∈ L1(Ω) for all compact K ⊂ Ω, (2.6)

and let α ∈Nd0 be a multi-index. Then we call g ∈ L1

loc(Ω) the weak α-derivative of f ∈ L1loc(Ω) if∫

Ωf Dαφ dλ = (−1)|α|

∫Ω

gφ dλ, (2.7)

for all φ ∈ C∞c (Ω), which is the space of all infinitely differentiable functions with compact support in Ω. Here,

we set

Dα =∂|α|

∂α1 x1 . . . ∂αd xdand |α| =

d

∑i=1

αi. (2.8)

Then, we write g = Dα f , which is justified as it is easy to show that the weak derivative is unique in the senseof an Lp function. The weak derivative is an extension of the classical derivative in view of the fact that forevery classically differentiable function the weak derivative exists and equals the classical derivative almosteverywhere. Exploiting the concept of weak derivatives, we define for 1 ≤ p ≤ ∞ the Sobolev spaces

Wk,p(Ω) := f ∈ Lp(Ω) : Dα f ∈ Lp(Ω) for all |α| ≤ k, (2.9)

and we endow these spaces with the norm

‖ · ‖Wk,p(Ω) : f 7→

(

∑|α|≤k ‖∂α f ‖pLp(Ω)

) 1p , 1 ≤ p < ∞,

max|α|≤k ‖∂α f ‖L∞(Ω), p = ∞.(2.10)

The Sobolev spaces are Banach spaces, and moreover separable for 1 ≤ p < ∞ and reflexive for 1 < p < ∞.The space Wk,2(Ω) is a Hilbert space, where the inner product is given by

( f , g)Wk,2(Ω) := ∑|α|≤k

(∂α f , ∂αg)L2(Ω) . (2.11)

When it comes to boundary value problems, we are required to evaluate functions on the boundary. Sincethe boundary ∂Ω is of lower dimension than the set Ω itself, ∂Ω is a null set with respect to the Lebesguemeasure λ, and evaluating Lp functions on ∂Ω makes no sense initially. On Sobolev spaces, the trace operatorremedies this issue. We present the trace theorem in a form that we will use later for the case of the Hilbertspace W1,2(Ω), but generalisations to other Sobolev spaces are straightforward and can be found in literature:

THEOREM 2.1.1 (Trace theorem) Let Ω ⊂ Rd be a bounded domain with Lipschitz boundary ∂Ω. Then, there exists abounded linear operator

T : W1,2(Ω)→ L2(∂Ω), (2.12)

such thatT f = f |∂Ω, for all f ∈ C∞(Ω). (2.13)

Furthermore, there exists a C = C(Ω) such that for all f ∈W1,2(Ω)

‖T f ‖2L2(∂Ω) ≤ C‖ f ‖L2(Ω)‖ f ‖W1,2(Ω). (2.14)

16

The above inequality can be proven using the interpolation inequality (see e.g. [8, 21]). We will henceforthomit the trace operator T in the notation when it is clear that a Sobolev function is evaluated in the sense oftraces.Moreover, let Wk,p

0 (Ω) be the closure of C∞c (Ω) in Wk,p(Ω). One can show that if Ω has a Lipschitz boundary

∂Ω the statement f |∂Ω = 0 is equivalent to f ∈ W1,p0 (Ω), that is, the functions in W1,p

0 (Ω) are exactly thefunctions in W1,p(Ω) with zero boundary conditions in the sense of traces.The dual space of W1,p

0 (Ω) is known as the space W−1,2(Ω). It is equipped with the usual dual space norm

‖ f ‖W−1,2(Ω) := supg∈W1,2

0 (Ω)‖g‖W1,2(Ω)

=1

∣∣∣ 〈 f , g〉W−1,2(Ω),W1,20 (Ω)

∣∣∣, (2.15)

where 〈·, ·〉W−1,2(Ω),W1,20 (Ω)

denotes the dual pairing.

For functions that vanish on parts of the boundary in the sense of traces, the following useful variant ofPoincare’s inequality holds:

Lemma 2.1.1 (Poincare’s inequality) Let Ω ⊂ Rd be a bounded and connected set with Lipschitz boundary ∂Ω suchthat ∂Ω can be written as the disjoint union of two sets ∂Ω = Γ1 ∪ Γ2, where Γ1 has positive Lebesgue measure. Then,there exists a constant Cp = Cp(Ω, Γ1) such that

‖ f ‖L2(Ω) ≤ Cp‖∇ f ‖L2(Ω),∥∥∥g− 1λ(Ω)

∫Ω

g dΩ∥∥∥

L2(Ω)≤ Cp‖∇g‖L2(Ω),

(2.16)

for all f ∈W1,2(Ω) such that f = 0 on Γ1 in the sense of traces, and for all g ∈W1,2(Ω).

There exists a variety of results concerning the continuous or compact embedding of Sobolev spaces into otherSobolev spaces or into Holder spaces, some of which can be found e.g. in [33, chap. 11].

2.1.3 Bochner spaces

In the context of partial differential equations, we can view a function f (t, x) as a map from the time t toa function ( f (t))(x) in an appropriate Sobolev space. In order to make this idea rigorous, the definition ofBochner spaces turns out beneficial. Let (X, ‖ · ‖X) be a Banach space, let (I,B, ν) be a measure space, and1 ≤ p ≤ ∞. Then, we can define the vector space

Lp(I,B, ν; X) :=

f : I → X : f is measurable , ‖ f ‖Lp(I;X) < ∞

, (2.17)

where we defined

‖ · ‖Lp(I;X) : f 7→(∫

I‖ f ‖p

X dν

) 1p

. (2.18)

Note that the measurability in equation (2.17) is with respect to the Borel σ-algebra of the norm topologygenerated by ‖ · ‖X on X. Again, the map ‖ · ‖Lp(I;X) defines a only a seminorm on Lp(I,B, ν; X), for whichreason one introduces the Bochner space Lp(I,B, ν; X) as the quotient space which results when functions thatare equal on null sets are identified with each other, just as for the Lebesgue spaces.The dual space of Lp(I,B, ν; X) is given by

Lp(I,B, ν; X)∗ = Lq(I,B, ν; X∗), (2.19)

where 1p +

1q = 1. Since X is a Banach space it follows that Lp(I,B, ν; X) is a Banach space, too. Properties such

as separability and reflexivity hold under the same conditions as for Lebesgue spaces, provided that the spaceX has the respective property as well.Let ν be the Lebesgue measure λ. Then, one defines weak derivatives of Bochner functions similarly as forLebesgue functions by replacing Ω by I, and the space L1

loc(Ω) by L1loc(I; X) for the functions in equation (2.7).

We will use the Bochner spaces for the following setting: the domain I is a time interval [0, T] and the Ba-nach space X is a Lebesgue or Sobolev space. An important space that we will use in particular is the space

17

L2(0, T; X), where X = L2(Ω) or X = W1,2(Ω). This space is a Hilbert space with the inner product

( f , g)L2(0,T;X) :=∫ T

0( f (t), g(t))X dt, (2.20)

since X is a Hilbert space.In the course of this work, we will sometimes consider closed subspaces of Hilbert spaces in order to accountfor the boundary conditions – these subspaces are then Hilbert spaces, too, with the same inner product re-stricted to the subspace.Moreover, let us mention that since in our case the domain I = [0, T] is one-dimensional the space W1,p(0, T; X)consists of time-continuous functions. To make this more precise, we define the space C(0, T; X) as the set ofall measurable functions f : [0, T] → X such that for any t ∈ [0, T] and any sequence tnn∈N ⊂ [0, T] withtn → t it holds

limn→∞

‖ f (t)− f (tn)‖X = 0. (2.21)

This space becomes a normed space with the norm

‖ · ‖C(0,T;X) : f 7→ maxt∈[0,T]

‖ f (t)‖X . (2.22)

We state the following lemma:

Lemma 2.1.2 Let 1 ≤ p ≤ ∞ and f ∈W1,p(0, T; X). Then, there exists an f ∈ C(0, T; X) such that

f = f a.e. (2.23)

In this case, we will tacitly redefine the function f on a null set and write f ∈ C(0, T; X).

2.2 Theorems from functional analysis

We will state some useful theorems from functional analysis in what follows. For a finite-dimensional Eu-clidean space, the Bolzano–Weierstrass theorem ensures that every bounded sequence possesses a convergentsubsequence. This theorem does not generalise to infinite-dimensional spaces, however, in reflexive Banachspaces, one obtains at least the existence of a weakly convergent subsequence by the following theorem:

THEOREM 2.2.1 (Eberlein–Smulian) Let X be a reflexive Banach space and let xn be a bounded sequence in X. Then,the following statements hold:

1. There exists a subsequence xnk of xn and an x ∈ X such that

xnk x weakly in X as k→ ∞. (2.24)

2. If every weakly convergent subsequence of xn has the same limit x, then the entire sequence xn convergesweakly to x.

In some cases, it is possible to extract a strongly convergent subsequence in an Lp space by establishing uniformbounds for the sequence in a Sobolev norm. A classical compact embedding theorem providing this is:

THEOREM 2.2.2 (Rellich–Kondrachov) Let Ω ⊂ Rd be an open, bounded Lipschitz domain. Then, the embedding

W1,p(Ω) → Lq(Ω), 1 ≤ q < p∗, (2.25)

is compact for

p∗ =

dpd−p , for p < d,

∞, for p ≥ d.(2.26)

The following compactness result for Bochner spaces from [4, 49] will prove useful:

THEOREM 2.2.3 (Aubin–Lions–Simon) Let X, B, Y be Banach spaces such that X is compactly embedded in B and B iscontinuously embedded in Y. Suppose fn is a sequence of functions in L2(0, T; B), and assume that fn is bounded

18

in L2(0, T; X) and that the (weak) derivatives ∂t fn are uniformly bounded in L2(0, T; Y).Then, there exists a subsequence fnk of fn such that

fnk → f , strongly in L2(0, T; B). (2.27)

In order to handle products of (weakly) convergent sequences, we state the following result:

THEOREM 2.2.4 (Strong-weak convergence) Let 1 < p < ∞, let fn be a sequence in Lp(Ω) and gn a sequencein Lp∗(Ω) for 1

p + 1p∗ = 1. Suppose that

fn f , weakly in Lp(Ω),

gn → g, strongly in Lp∗(Ω).(2.28)

Then, we havefngn f g, weakly in L1(Ω). (2.29)

2.3 Inequalities

We continue with some fundamental inequalities which will be harnessed to obtain estimates in the subsequentchapters:

Lemma 2.3.1 (Triangle inequality) Let (X, ‖ · ‖) be a normed vector space. Then, it holds for all x, y ∈ X

‖x + y‖ ≤ ‖x‖+ ‖y‖. (2.30)

Lemma 2.3.2 (Cauchy–Schwarz inequality) Let (X, (·, ·)) be an inner product space. Then, it holds for all x, y ∈ X

(x, y) ≤ ‖x‖‖y‖, (2.31)

where ‖x‖ :=√(x, x) is the norm induced by the inner product.

Lemma 2.3.3 (Young’s inequality) Let a and b be non-negative real numbers. Then, we have for all ε > 0

ab ≤ a2

2ε+

εb2

2. (2.32)

Lemma 2.3.4 (Jensen’s inequality) Let f be an integrable function on a probability space (X,A, µ), and let g be areal-valued function which is convex in the range of f . Then, it holds

g(∫

Ωf dµ

)≤∫

Ωg f dµ. (2.33)

2.4 Notations

We will briefly introduce notation which will be used in the subsequent chapters. Since the space L2 is ofoutstanding importance, we set for a domain Ω ⊂ Rd and a time interval [0, T]

‖ · ‖Ω := ‖ · ‖L2(Ω), and ‖ · ‖ΩT := ‖ · ‖L2(0,T;L2(Ω)),

(·, ·)Ω := (·, ·)L2(Ω) , and (·, ·)ΩT := (·, ·)L2(0,T;L2(Ω)) .(2.34)

In view of the particular Problem P , we use the following conventions:

• i is the index for the subdomain and takes values in m1, m2, f ,

• j is the index to specify the matrix block subdomain and takes values in 1, 2,

• for functions f which are the same in both matrix block subdomains (such as θ, K, . . .), we use thenotation fm1 = fm2 =: fm, thus allowing to write e.g. θi(ψi).

Moreover, C ≥ 0 is a generic constant.

19

Chapter 3

Existence

This chapter is concerned with the existence of a (weak) solution to Problem P for a fixed fracture widthε > 0. For the Richards equation considered on a single domain, several results concerning the existence anduniqueness of solutions exist, amongst others in the much-quoted article by Alt and Luckhaus [2]. For worksdiscussing the existence of solutions, we mention [3, 52], and uniqueness results can be found in e.g. [41, 42].Analysis for the full two-phase framework is carried out in e.g. [11, 12, 18, 30].In view of the analysis, difficulties arise from the non-linearity and from the degeneracy of Richards’ equation.Degeneracy of Richards’ equation occurs in the slow diffusion case, that is when K(θ(ψ)) → 0, and in the fastdiffusion case, when θ′(ψ) = 0. Since we focus on the upscaling, we exclude degeneracy in this work, whichwill be expressed by the assumptions below.In order to prove the existence of solutions of Problem P , we apply the Rothe method (for an introduction onthe application of the Rothe method to time-dependent differential equations see e.g. [29]), and discretise thetime derivative using an implicit Euler scheme. Under appropriate smoothness assumptions, the existence anduniqueness of a solution to the discretised problem follow from general works on coupled elliptic differentialequations, see e.g. [27]. With the discrete solution at hand, time-interpolated functions can be defined, whichare shown to converge towards a solution to Problem P as the time step ∆t approaches zero. To do so, wemake use of compactness arguments. Moreover, an L∞ estimate is proven for the time-discrete solution, whichthen translates to an L∞ estimate for the time-continuous solution to Problem P . We proceed in the spirit of[43], but observe that a linear partial differential equation is considered in the latter work.For the sake of readability, we drop the superscript ε since it is fixed throughout this chapter.

3.1 Assumptions

For the analysis, we assume in this work that the following conditions are satisfied:

(AD f ) fi ∈ C(0, T; L2(Ωi)).

(A f ) There exists M f > 0 such that | fi| ≤ M f a.e. in ΩTi .

(ADθ) θi ∈ C1(R).

(Aθ) There exist mθ , Mθ > 0 such that 0 < mθ ≤ θ′i(ψi) ≤ Mθ for all ψi ∈ R.

(ADK ) Ki ∈ C2(R) and K′i(θi) > 0 for all θi ∈ R.

(AK) There exist mK, MK > 0 such that 0 < mK ≤ Ki(θi) < MK for all θi ∈ R.

(AI) There exists MI > 0 such that |ψi,I | ≤ MI a.e. in Ωi.

Remark 3.1.1 (Assumptions)

• The time continuity of the source term in Assumption (AD f ) allows us to evaluate it at all times t ∈ (0, T].

• Note that due to Assumption (Aθ), we consider the strict parabolic case only here as θ′i(ψi) is boundedaway from zero, which corresponds to a situation where the porous medium is unsaturated everywhereat all times because θ′i(ψi) = 0 in the saturated regime.

21

• When applying the Kirchhoff transformation to Problem P , the flux term becomes linear at the cost of anon-linear Dirichlet condition at the interface. The latter non-linearity is required to be a C3-function forthe existence result in [27]. This regularity is provided by Assumption (ADK ). However, the smoothnessassumptions on the boundary of the domain in [27] are not satisfied in our setting, for which reason wecan only show the existence of a weak solution to Problem P if the existence of a weak solution to thetime-discrete problem is assumed.

• Assumption (AK) excludes the slow diffusion case and guarantees the existence of a minimum positivepermeability everywhere. J

Moreover, for the sake of presentation, we make the assumption θi(0) = 0, which can easily be achieved byredefining θi(u) = θi(u)− θi(0).

3.2 Weak solution

First, we establish a suitable notion of a solution to Problem P . For this purpose we define the function spaces

Vmj := u ∈W1,2(Ωmj) : u = 0 on ∂Ωmj \ Γj,

V f := u ∈W1,2(Ω f ) : u = 0 on ∂Ω f \ Γ,(3.1)

for j ∈ 1, 2. Note that the function spaces depend on the fixed fracture width ε.We obtain the definition of a weak solution by substituting Darcy’s law into the mass balance equation inProblem P for every subdomain, multiplying with appropriate test functions and applying integration byparts. It is straightforward to show that every strong solution to Problem P is also a weak solution, andconversely every weak solution which is sufficiently regular is a strong solution.

Definition 3.2.1 (Weak solution) A triple (ψm1 , ψm2 , ψ f ) ∈ L2(0, T;Vm1)× L2(0, T;Vm2)× L2(0, T;V f ) is called aweak solution to Problem P if

ψm1 = ψ f on Γ1 and ψm2 = ψ f on Γ2 for a.e. t ∈ [0, T], (3.2)

in the sense of traces, and

− (θm(ψm1), ∂tφm1)ΩTm1− (θm(ψm2), ∂tφm2)ΩT

m2− εκ

(θ f (ψ f ), ∂tφ f

)ΩT

f

+ (Km(θm(ψm1))∇ψm1 ,∇φm1)ΩTm1

+ (Km(θm(ψm2))∇ψm2 ,∇φm2)ΩTm2

+ ελ(

K f (θ f (ψ f ))∇ψ f ,∇φ f

)ΩT

f

= ( fm1 , φm1)ΩTm1

+ ( fm2 , φm2)ΩTm2

+(

f f , φ f

)ΩT

f

+(θm(ψm1,I), φm1(0)

)Ωm1

+(θm(ψm2,I), φm2(0)

)Ωm1

+ εκ(

θ f (ψ f ,I), φ f (0))

Ω f,

(3.3)

for all (φm1 , φm2 , φ f ) ∈W1,2(0, T;Vm1)×W1,2(0, T;Vm2)×W1,2(0, T;V f ) satisfying

φm1 = φ f on Γ1 and φm2 = φ f on Γ2 for a.e. t ∈ [0, T], (3.4)

andφi(T) = 0, for i ∈ m1, m2, f . (3.5)

Note that in the above definition, it makes sense to evaluate the test functions φi at the time t = T since thespace W1,p(0, T;Vi) is embedded in C(0, T;Vi) due to Lemma 2.1.2.Define in each subdomain the Kirchhoff transform by

Ki : R→ R, ui := Ki(ψi) =∫ ψi

0Ki(θi(ϕ)) dϕ. (3.6)

Due to the assumptions on Ki and θi, the Kirchhoff transformation is invertible and we can define the property

bi(ui) := θi(K−1i (ui)). (3.7)

22

By the chain rule we have∇ui = Ki(θi(ψi))∇ψi, (3.8)

which transforms Problem P into a semi-linear problem. For an introduction on the Kirchhoff transform forRichards’ equation see e.g. [7]. Since Ki is Lipschitz continuous, the transformed problem is equivalent to theoriginal problem [36]. Note in particular that ψi = 0 if and only if ui = 0.By definition of the Kirchhoff transform and Assumption (AK), we obtain bounds from both sides for theKirchhoff transformed function and its derivative, provided that the function is sufficiently regular. The nec-essary regularity assumptions will be proven later.

Lemma 3.2.1 For almost all t ∈ (0, T], the following estimates hold in each subdomain Ωi:

mK‖ψi(t)‖Ωi ≤ ‖ui(t)‖Ωi ≤ MK‖ψi(t)‖Ωi ,

mK‖∇ψi‖Ωti≤ ‖∇ui‖Ωt

i≤ MK‖∇ψi‖Ωt

i,

mK‖∂tψi‖Ωti≤ ‖∂tui‖Ωt

i≤ MK‖∂tψi‖Ωt

i.

(3.9)

Let ui,I := K(ψi,I). The weak formulation of the problem in terms of the Kirchhoff transformed variableswrites as

Definition 3.2.2 (Weak solution Kirchhoff transformed) A triple (um1 , um2 , u f ) ∈ L2(0, T;Vm1)× L2(0, T;Vm2)×L2(0, T;V f ) is called a weak solution to the Kirchhoff transformed problem if

K−1m (um1) = K

−1f (u f ) on Γ1 and K−1

m (um2) = K−1f (u f ) on Γ2 for a.e. t ∈ [0, T], (3.10)

in the sense of traces, and

− (bm(um1), ∂tφm1)ΩTm1− (bm(um2), ∂tφm2)ΩT

m2− εκ

(b f (u f ), ∂tφ f

)ΩT

f

+ (∇um1 ,∇φm1)ΩTm1

+ (∇um2 ,∇φm2)ΩTm2

+ ελ(∇u f ,∇φ f

)ΩT

f

= ( fm1 , φm1)ΩTm1

+ ( fm2 , φm2)ΩTm2

+(

f f , φ f

)ΩT

f

+(bm(um1,I), φm1(0)

)Ωm1

+(bm(um2,I), φm2(0)

)Ωm2

+ εκ(

b f (u f ,I), φ f (0))

Ω f

(3.11)

for all (φm1 , φm2 , φ f ) ∈W1,2(0, T;Vm1)×W1,2(0, T;Vm2)×W1,2(0, T;V f ) satisfying

φm1 = φ f on Γ1 and φm2 = φ f on Γ2 for a.e. t ∈ [0, T], (3.12)

andφi(T) = 0 for i ∈ m1, m2, f . (3.13)

The advantage of the Kirchhoff transformed formulation is the linear flux term, which comes at the cost of anon-linear interface condition. We will use either formulation depending on what seems more suitable for aspecific purpose.

3.3 Time discretisation

In what follows, we discretise the problem in time using an implicit Euler approach, which gives ellipticequations at every discrete time tk = k∆t, for k ∈ 0, . . . , N, where N ∈ N. We assume without loss ofgenerality that N∆t = T. Here, ∆t > 0 denotes the fixed time step. Choose ψ0

i = ψi,I and let the sequence ofsolutions to the time-discrete problem be given as ψk

i . Let moreover f ki := fi(tk). We define the notion of

a weak solution to the time-discrete problem in the original formulation – the definition of a weak solution tothe Kirchhoff transformed problem is analogous and one takes u0

i = K(ψi,I) initially.

Definition 3.3.1 Let k > 0 and let (ψk−1m1

, ψk−1m2

, ψk−1f ) ∈ Vm1 × Vm1 × V f be given. We call (ψk

m1, ψk

m2, ψk

f ) ∈Vm1 × Vm1 × V f a weak solution to the time-discrete problem at time tk if it satisfies

ψkm1

= ψkf on Γ1 and ψk

m2= ψk

f on Γ2 (3.14)

23

in the sense of traces, and(θm(ψ

km1), φm1

)Ωm1

+(

θm(ψkm2), φm2

)Ωm2

+ εκ(

θ f (ψkf ), φm1

)Ω f

+ ∆t(

Km(θm(ψkm1))∇ψk

m1,∇φm1

)Ωm1

+ ∆t(

Km(θm(ψkm2))∇ψk

m2,∇φm2

)Ωm2

+ελ∆t(

K f (θ f (ψkf1))∇ψk

f1,∇φ f1

)Ω f

= ∆t(

f km1

, φm1

)Ωm1

+ ∆t(

f km2

, φm2

)Ωm2

+ ∆t(

f kf , φ f

)Ω f

+(

θm(ψk−1m1

), φm1

)Ωm1

+(

θm(ψk−1m2

), φm2

)Ωm2

+ εκ(

θ f (ψk−1f ), φm1

)Ω f

,

(3.15)for all (φm1 , φm2 , φ f ) ∈ Vm1 × Vm2 × V f satisfying φmj = φ f on Γj for j ∈ 1, 2.

3.4 A priori estimates

Define in each domain the energy functional

Wi(ψi) =∫ ψi

0θ′i(ϕ)ϕ dϕ, (3.16)

which we will require in the proof of the following a priori estimate. First, we gather some properties ofWi ina simple lemma, which is based on Assumption (Aθ):

Lemma 3.4.1 The functionWi satisfies the following inequalities:

Wi(ψi) ≥ 0,

Wi(ψi)−Wi(ξi) ≤ ψi(θi(ψi)− θi(ξi)),

mθψ2

i2≤ Wi(ψi) ≤ Mθ

ψ2i

2,

(3.17)

for all ψi, ξi ∈ R.

Proof. The non-negativity follows immediately from θ′i(ψi) > 0 due to Assumption (Aθ): for ψi ≥ 0, we have

Wi(ψi) =∫ ψi

0θ′i(ϕ)ϕ︸ ︷︷ ︸≥0

dϕ ≥ 0,

and in case ψi < 0

Wi(ψi) = −∫ 0

ψi

θ′i(ϕ)ϕ︸ ︷︷ ︸≤0

dϕ ≥ 0.

The second inequality holds because of

Wi(ψi)−Wi(ξi) =∫ ψi

ξi

θ′i(ϕ)ϕ dϕ ≤ ψi

∫ ψi

ξi

θ′i(ϕ) dϕ = ψi(θi(ψi)− θi(ξi)),

for ψi ≥ ξi and

Wi(ψi)−Wi(ξi) = −∫ ξi

ψi

θ′i(ϕ)ϕ dϕ ≤ −ψi

∫ ξi

ψi

θ′i(ϕ) dϕ = ψi(θi(ψi)− θi(ξi)),

for ψi < ξi. Finally, we have

mθψ2

i2

= mθ

∫ ψi

0ϕ dϕ ≤

∫ ψi

0θ′i(ϕ)ϕ dϕ =Wi(ψi) ≤ Mθ

∫ ψi

0ϕ dϕ = Mθ

ψ2i

2.

We obtain the following estimate for the time-discrete solution:

24

Lemma 3.4.2 (A priori estimate) The solution (ψkm1

, ψkm2

, ψkf ) to the time-discrete problem in Definition 3.3.1 satisfies

maxl∈1,...,N

∫Ωm1

Wm(ψlm1) d~x + max

l∈1,...,N

∫Ωm2

Wm(ψlm2) d~x + εκ max

l∈1,...,N

∫Ω f

W f (ψlf ) d~x

+∆t mK

2

N

∑k=1‖∇ψk

m1‖2

Ωm1+

∆t mK2

N

∑k=1‖∇ψk

m2‖2

Ωm2+ ελ ∆t mK

2

N

∑k=1‖∇ψk

f ‖2Ω f

≤∫

Ωm1

Wm(ψm1,I) d~x +∫

Ωm2

Wm(ψm2,I) d~x + εκ∫

Ω f

W f (ψ f ,I) d~x

+∆t Cpm

2mK

N

∑k=1‖ f k

m1‖2

Ωm1+

∆t Cpm

2mK

N

∑k=1‖ f k

m2‖2

Ωm2+ ε−λ

∆t Cp f

2mK

N

∑k=1‖ f k

f ‖2Ω f

.

(3.18)

Proof. We test equation (3.15) with the triple (φm1 , φm2 , φ f ) = (ψkm1

, ψkm2

, ψkf ), which yields

(θm(ψkm1)− θm(ψ

k−1m1

), ψkm1)Ωm1

+ (θm(ψkm2)− θm(ψ

k−1m2

), ψkm2)Ωm2

+ εκ(θ f (ψkf )− θ f (ψ

k−1f ), ψk

f )Ω f

+ ∆t(Km(θm(ψkm1))∇ψk

m1,∇ψk

m1)Ωm1

+ ∆t(Km(θm(ψkm2))∇ψk

m2,∇ψk

m2)Ωm2

+ ελ∆t(K f (θ f (ψkf ))∇ψk

f ,∇ψkf )Ω f

= ∆t( f km1

, ψkm1)Ωm1

+ ∆t( f km2

, ψkm2)Ωm2

+ ∆t( f kf , ψk

f )Ω f .

Poincare’s inequality gives

‖∇ψki ‖2

Ωi≥‖∇ψk

i ‖2Ωi

2+‖ψk

i ‖2Ωi

2Cpi

,

for i ∈ m1, m2, f , where Cpi > 0 denotes the Poincare constant of the respective subdomains. Note that thegeometries of Ωm1 and Ωm2 are the same, and so are the Poincare constants hence, for which reason we setCpm := Cpm1

= Cpm2.

Making use of this together with Assumption (AK) and equation (3.17)2 in Lemma 3.4.1, we estimate

2

∑j=1

∫Ωmj

Wm(ψkmj) d~x + εκ

∫Ω f

W f (ψkf ) d~x +

∆t mK2

(2

∑j=1‖∇ψk

mj‖2

Ωmj+ ελ‖∇ψk

f ‖2Ω f

)

+∆t mK2Cpm

2

∑j=1‖ψk

mj‖2

Ωmj+ ελ

∆t mK‖ψkf ‖

2Ω f

2Cp f

≤2

∑j=1

∫Ωmj

Wm(ψk−1mj

) d~x + εκ∫

Ω f

W f (ψk−1f ) d~x + ∆t

2

∑j=1

( f km1

, ψkm1)Ωm1

+ ∆t( f kf , ψk

f )Ω f

≤2

∑j=1

∫Ωmj

Wm(ψk−1mj

) d~x + εκ∫

Ω f

W f (ψk−1f ) d~x +

∆t Cpm

2mK

2

∑j=1‖ f k

mj‖2

Ωmj+ ε−λ

∆t Cp f

2mK‖ f k

f ‖2Ω f

+∆t mK2Cpm

2

∑j=1‖ψk

mj‖2

Ωmj+ ελ

∆t mK‖ψkf ‖

2Ω f

2Cp f

,

where we applied the Cauchy–Schwarz inequality and Young’s inequality. Summing over k from 1 to l for1 ≤ l ≤ N leaves us with

2

∑j=1

∫Ωmj

Wm(ψlm1) d~x + εκ

∫Ω f

W f (ψlf ) d~x +

∆t mK2

2

∑j=1

l

∑k=1‖∇ψk

mj‖2

Ωmj+ ελ ∆t mK

2

l

∑k=1‖∇ψk

f ‖2Ω f

≤2

∑j=1

∫Ωmj

Wm(ψmj ,I) d~x + εκ∫

Ω f

W f (ψ f ,I) d~x +∆t Cpm

2mK

2

∑j=1

l

∑k=1‖ f k

mj‖2

Ωmj+ ε−λ

∆t Cp f

2mK

l

∑k=1‖ f k

f ‖2Ω f

,

which finishes the proof.

Remark 3.4.1 (Non-degenerate case) Note that we did not use Assumption (Aθ) in the proof above. Here inthe strictly parabolic case, where we have an mθ > 0 such that 0 < mθ ≤ θ′i(ψi) and all ψi ∈ R, we immediately

25

obtain an L2 bound for ψki from the first two terms in Lemma 3.4.2 by Lemma 3.4.1:

∫Ωmk

Wm(ψkmj) d~x ≥ mθ

‖ψkmj‖2

Ωmj

2, and

∫Ω f

W f (ψkf ) d~x ≥ mθ

‖ψkf ‖

2Ω f

2. J

3.5 Interpolation in time

Now, we define functions on a continuous time domain by interpolating the solutions of the time-discreteproblem in time. Since Richards’ equation contains the time derivative of the water content, we bound thetime derivative of the water content and transfer the results later to the pressure head. In order to obtaintime-continuous functions, we introduce piecewise linearly interpolated functions in addition to the piecewiseconstant functions: for almost every t ∈ (tk−1, tk] set

Ψi∆t(t) := ψk

i ,

Θi∆t(t) := θ(ψk

i ),

Θi∆t(t) := θ(ψk−1

i ) +t− tk−1

∆t(θ(ψk

i )− θ(ψk−1i )).

(3.19)

Moreover, we need the piecewise constant interpolation of the source term

f i∆t(t) = f k

i . (3.20)

In view of the a priori estimate in Lemma 3.4.2, we obtain the following result for the interpolated functions:

Lemma 3.5.1 For each ε > 0 fixed, the functions Ψi∆t, Θi

∆t, and Θi∆t are bounded uniformly with respect to ∆t in

L∞(0, T; L2(Ωi)) ∩ L2(0, T;Vi).

Proof. We present the proof for the piecewise linear functions Θi∆t, the proof for Θi

∆t and Ψi∆t is similar and

easier. The first statement follows from the a priori estimate in Lemma 3.4.2 from

ess supt∈(0,T]

‖Θi∆t(t)‖Ωi = max

k∈0,...,N‖θ(ψk

i )‖Ωi ≤ Mθ maxk∈0,...,N

‖ψki ‖Ωi ≤ C(ε, T),

where we used the Lipschitz continuity of θi and the technical assumption θi(0) = 0. Notice that C(ε, T) doesin particular not depend on ∆t due to the time-continuity of the source term fi.Moreover, we estimate

‖Θi∆t‖2

ΩTi=∫ T

0‖Θi

∆t(t)‖2Ωi

dt ≤ maxk∈0,...,N

‖θ(ψki )‖2

ΩiT ≤ C(ε, T),

and

‖∇Θi∆t‖2

ΩTi≤ M2

θ

N

∑k=1

∫ tk

tk−1

(‖∇ψk−1

i ‖Ωi + ‖∇ψki ‖Ωi

)2dt

≤ 2M2θ ∆t

N

∑k=1

(‖∇ψk−1

i ‖2Ωi

+ ‖∇ψki ‖2

Ωi

)≤ C(ε, T),

where we used Assumption (Aθ), the triangle inequality, (a + b)2 ≤ 2(a2 + b2), and Lemma 3.4.2.

By the Eberlein–Smulian theorem 2.2.1, Lemma 3.5.1 gives rise to the existence of a Ψi ∈ L2(0, T;Vi) and asubsequence of ∆t→ 0 along which we obtain the weak convergence

Ψi∆t∆t Ψi in L2(0, T;Vi). (3.21)

In order to get strong convergence in L2(0, T; L2(Ωi)), which we require to show the convergence of the non-linear function θi(Ψi

∆t) to the desired limit function as ∆t → 0, we need the following estimate for the timederivative of the water content:

26

Lemma 3.5.2 For each ε > 0, there exists a constant C = C(ε, T) such that

‖∂tΘi∆t‖L2(0,T;W−1,2(Ωi))

≤ C. (3.22)

Proof. Since the function Θi∆t(t) is piecewise linear, its weak time derivative exists, is piecewise constant, and

for almost every t ∈ (tk−1, tk] given by

∂tΘi∆t(t) =

θi(ψki )− θi(ψ

k−1i )

∆t.

We view ∂tΘi∆t as an element of L2(0, T; W−1,2(Ωi)), where W−1,2(Ωi) is the dual of W1,2

0 (Ωi) (the latter spacecontaining the W1,2-functions on Ωi with vanishing trace on the entire boundary ∂Ωi). This is clearly possibledue to the natural inclusion L2(Ωi) →W−1,2(Ωi) given by u 7→ (u, ·)Ωi .Testing equation (3.15) with arbitrary φi ∈W1,2

0 (Ωi) and φs ≡ 0 for s 6= i gives the estimate

∣∣∣〈∂tΘi∆t(t), φi〉W−1,2(Ωi),W

1,20 (Ωi)

∣∣∣ = ∣∣∣∣∣(

θi(ψki )− θi(ψ

k−1i )

∆t, φi

)Ωi

∣∣∣∣∣≤∣∣∣ (Ki(θi(ψ

ki ))∇ψk

i ,∇φi

)Ωi

∣∣∣+ ∣∣∣ ( f ki , φi

)Ωi

∣∣∣≤ ‖φi‖W1,2(Ωi)

(MK‖∇ψk

i ‖Ωi + ‖ f ki ‖Ωi

).

Using the a priori estimate in Lemma 3.4.2, we obtain

‖∂tΘi∆t‖L2(0,T;W−1,2(Ωi))

≤ C(ε, T),

which finishes the proof.

The strong convergence of the piecewise linear functions Θi∆t in L2 will be provided by the Aubin–Lions-Simon

theorem 2.2.3:

Lemma 3.5.3 There exists a Θi ∈ L2(0, T; L2(Ωi)) and a subsequence ∆t→ 0 along which we get

Θi∆t∆t → Θi in L2(0, T; L2(Ωi)). (3.23)

Proof. In view of the estimate in Lemma 3.5.2 for the time derivative, the boundedness of Θi∆t∆t in L2(0, T;Vi)

and Lemma 3.5.1, the Aubin–Lions–Simon theorem 2.2.3 provides the existence of a subsequence ∆t→ 0 alongwhich Θi

∆t → Θi in L2(0, T; L2(Ωi)) by choosing X = Vi, B = L2(Ωi), Y = W−1,2(Ωi). Note that L2(Ωi) iscontinuously embedded in W−1,2(Ωi), and Vi is compactly embedded in L2(Ωi) by Theorem 2.2.2.

The following lemma is taken from [32] and transfers the strong convergence result to the piecewise constantlyinterpolated function:

Lemma 3.5.4 Let X be a Hilbert space. Then, the strong convergence

Θi∆t∆t → Θi in L2(0, T; X)

implies the strong convergenceΘi

∆t∆t → Θi in L2(0, T; X).

From this, we obtain immediately

Lemma 3.5.5 Along a subsequence ∆t→ 0, we get

Θi∆t∆t → Θi in L2(0, T; L2(Ωi)). (3.24)

It remains to carry over this strong convergence result for the water content to the pressure head. This is donein the following way: due to Assumption (Aθ), the inverse function θ−1

i exists and is continuous. Note that bydefinition of the piecewise constantly interpolated functions, we have

Ψi∆t = θ−1

i (Θi∆t). (3.25)

27

From the strong convergence in equation (3.24) and the Lipschitz continuity of θ−1i , we deduce:

Lemma 3.5.6 Along a subsequence ∆t→ 0, we get

Ψi∆t∆t → Ψi in L2(0, T; L2(Ωi)). (3.26)

Remark 3.5.1 Applying the previous considerations to the weakly convergent subsequence in equation (3.21),one obtains actually a subsubsequence (still denoted as Ψi

∆t∆t with a slight abuse of notation), which con-verges strongly in L2(0, T; L2(Ωi)) and weakly in L2(0, T;Vi). The limit function Ψi in equation (3.26) is indeedthe same as the limit function in equation (3.21) for which the same notation has been used: since Ψi

∆t∆t Ψiin L2(0, T;Vi), this weak convergence holds true in particular in L2(0, T; L2(Ωi)), and as strong convergenceimplies weak convergence to the same limit function, the two limits are identical by the uniqueness of theweak limit. From now on, we omit these standard arguments. J

It remains to show that the limit functions are a weak solution:

THEOREM 3.5.1 The limit (Ψm1 , Ψm2 , Ψ f ) is a weak solution to Problem P in the sense of Definition 3.2.1.

Proof. Let t ∈ (tk−1, tk) and (φm1 , φm2 , φ f ) ∈ Vm1 × Vm2 × V f . Summing equation (3.15) from 1 to k yields

2

∑j=1

(θm(Ψ

mj∆t (t)), φmj

)Ωmj

+ εκ(

θ f (Ψf∆t(t)), φ f

)Ω f−

2

∑j=1

(θm(ψmj ,I), φmj

)Ωmj

− εκ(

θ f (ψ f ,I), φ f

)Ω f

+2

∑j=1

∫ t

0

(Km(θm(Ψ

mj∆t (τ)))∇Ψ

mj∆t ,∇φmj

)Ωmj

dτ + ελ∫ t

0

(K f (θ f (Ψ

f∆t(τ)))∇Ψ f

∆t,∇φ f

)Ω f

dτ

−2

∑j=1

∫ t

0

(fmj(τ), φmj

)Ωmj

dτ −∫ t

0

(f f (τ), φ f

)Ω f

dτ

=2

∑j=1

∫ tk

t

(fmj(τ), φmj

)Ωmj

dτ +∫ t

0

(f f (τ), φ f

)Ω f

dτ

−2

∑j=1

∫ tk

t

(Km(θm(Ψ

mj∆t (τ)))∇Ψ

mj∆t ,∇φmj

)Ωmj

dτ − ελ∫ tk

t

(K f (θ f (Ψ

f∆t(τ)))∇Ψ f

∆t,∇φ f

)Ω f

dτ.

The terms on the right hand side correct the error made on the left hand side by integrating to t instead of tk.Now, we choose test functions (φm1 , φm2 , φ f ) ∈ L2(0, T;Vm1)× L2(0, T;Vm2)× L2(0, T;V f ) fulfilling φmj = φ fon Γj and integrate in time from 0 to T to get

2

∑j=1

∫ T

0

(θm(Ψ

mj∆t (t)), φmj(t)

)Ωmj

dt + εκ∫ T

0

(θ f (Ψ

f∆t(t)), φ f (t)

)Ω f

dt

−2

∑j=1

∫ T

0

(θm(ψmj ,I), φmj(t)

)Ωmj

dt− εκ∫ T

0

(θ f (ψ f ,I), φ f (t)

)Ω f

dt

+2

∑j=1

∫ T

0

∫ t

0

(Km(θm(Ψ

mj∆t (τ)))∇Ψ

mj∆t (τ),∇φmj(t)

)Ωmj

dτ dt

+ ελ∫ T

0

∫ t

0

(K f (θ f (Ψ

f∆t(τ)))∇Ψ f

∆t(τ),∇φ f (t))

Ω fdτ dt

−2

∑j=1

∫ T

0

∫ t

0

(fmj(τ), φmj(t)

)Ωmj

dτ dt−∫ T

0

∫ t

0

(f f (τ), φ f (t)

)Ω f

dτ dt

=2

∑j=1

N

∑k=1

∫ tk

tk−1

∫ tk

t

(fmj(τ), φmj(t)

)Ωmj

dτ dt +N

∑k=1

∫ tk

tk−1

∫ t

0

(f f (τ), φ f (t)

)Ω f

dτ dt

−2

∑j=1

N

∑k=1

∫ tk

tk−1

∫ tk

t

(Km(θm(Ψ

mj∆t (τ)))∇Ψ

mj∆t (τ),∇φmj(t)

)Ωmj

dτ dt

− ελN

∑k=1

∫ tk

tk−1

∫ tk

t

(K f (θ f (Ψ

f∆t(τ)))∇Ψ f

∆t(τ),∇φ f (t))

Ω fdτ dt.

(4)

28

From the strong L2 convergence from equation (3.24) and the Lipschitz continuity of θm, we infer that∫ T

0

(θm(Ψ

mj∆t (t)), φmj(t)

)Ωmj

dt→∫ T

0

(θm(Ψmj(t)), φmj(t)

)Ωmj

dt.

Furthermore, we have to show that∫ T

0

∫ t

0

(Km(θm(Ψ

mj∆t (τ)))∇Ψ

mj∆t (τ),∇φmj(t)

)Ωmj

dτ dt

→∫ T

0

∫ t

0

(Km(θm(Ψmj(τ)))∇Ψmj(τ),∇φmj(t)

)Ωmj

dτ dt.(∗)

This follows from the following considerations: due to the boundedness of Km, we know that there exists a

~ξmj ∈(

L2(Ωmj))d

such that

Km(θm(Ψmj∆t (τ)))∇Ψ

mj∆t (τ)

~ξmj , weakly in(

L2(Ωmj))d

.

Due to the finite measure of Ωmj , this also implies

Km(θm(Ψmj∆t (τ)))∇Ψ

mj∆t (τ)

~ξmj , weakly in(

L1(Ωmj))d

.

From the Strong-weak convergence theorem 2.2.4, we infer that

Km(θm(Ψmj∆t (τ)))∇Ψ

mj∆t (τ) Km(θm(Ψmj(τ)))∇Ψmj(τ), weakly in

(L1(Ωmj)

)d,

in view of the strong convergence from equation (3.24), the Lipschitz continuity of θm and Km, and the weak

convergence from equation (3.21). From the uniqueness of the weak limit in(

L1(Ωmj))d

, we deduce that

~ξmj = Km(θm(Ψmj(τ)))∇Ψmj(τ),

which proves equation (∗). Moreover, the time-continuity of f gives∫ T

0

∫ t

0

(fmj(τ), φmj(t)

)Ωmj

dτ dt→∫ T

0

∫ t

0

(fmj(τ), φmj(t)

)Ωmj

dτ dt.

For the fracture solution, we obtain analogous results with the same arguments.In what follows, we show that the terms on the right hand side in equation (4) vanish as ∆t approaches zero.For the terms involving the source term fmj , we obtain∣∣∣∣∣ N

∑k=1

∫ tk

tk−1

∫ tk

t

(fmj(τ), φmj(t)

)Ωmj

dτ dt

∣∣∣∣∣ ≤ (∆t)2

2

N

∑k=1‖ f k

mj‖2

Ωmj+

∆t2

∫ T

0‖φmj‖

2Ωmj≤ C∆t,

where we used the Cauchy–Schwarz inequality and Young’s inequality.Furthermore, we get∣∣∣∣∣ N

∑k=1

∫ tk

tk−1

∫ tk

t

(Km(θm(Ψ

mj∆t (τ)))∇Ψ

mj∆t (τ),∇φmj(t)

)Ωmj

dτ dt

∣∣∣∣∣ ≤ (∆t)2MK2

N

∑k=1‖∇ψk

mj‖2

Ωmj+

∆t2

∫ T

0‖∇φmj‖

2Ωmj

≤ C∆t,

using the a priori estimate in Lemma 3.4.2. Estimates for the correspondent terms for the fracture solution

29

follow similarly. Therefore, in the limit ∆t→ 0, we are left with

2

∑j=1

∫ T

0

(θm(Ψmj(t)), φmj(t)

)Ωmj

dt + εκ∫ T

0

(θ f (Ψ f (t)), φ f (t)

)Ω f

dt

+2

∑j=1

∫ T

0

∫ t

0

(Km(θm(Ψmj(τ)))∇Ψmj(τ),∇φmj(t)

)Ωmj

dτ dt

+ ελ∫ T

0

∫ t

0

(K f (θ f (Ψ f (τ)))∇Ψmj(τ),∇φ f (t)

)Ω f

dτ dt

=2

∑j=1

∫ T

0

∫ t

0

(fmj(τ), φmj(t)

)Ωmj

dτ dt +∫ T

0

∫ t

0

(f f (τ), φ f (t)

)Ω f

dτ dt

+2

∑j=1

∫ T

0

(θm(ψmj ,I), φmj(t)

)Ωmj

dt + εκ∫ T

0

(θ f (ψ f ,I), φ f (t)

)Ω f

dt,

()

for all (φm1 , φm2 , φ f ) ∈ L2(0, T;Vm1)× L2(0, T;Vm2)× L2(0, T;V f ) such that φmj = φ f on Γj for j ∈ 1, 2.Note that when choosing test functions (φm1 , φm2 , φ f ) ∈ W1,2(0, T;Vm1)×W1,2(0, T;Vm2)×W1,2(0, T;V f ) sat-isfying φm1(T) = φm2(T) = φ f (T) = 0, integration by parts yields

∫ T

0

∫ t

0

(fmj(τ), ∂tφmj(t)

)Ωmj

dτ dt = −∫ T

0

(fmj(t), φmj(t)

)Ωmj

dt,

∫ T

0

∫ t

0

(Km(θm(Ψmj(τ)))∇Ψmj(τ),∇∂tφmj(t)

)Ωmj

dτ dt = −∫ T

0

(Km(θm(Ψmj(t)))∇Ψmj(t),∇φmj(t)

)Ωmj

dt,

and similarly for the terms associated with the fracture.Thus, selecting φi = ∂tφi in equation () gives

2

∑j=1

∫ T

0

(θm(Ψmj(t)), ∂tφmj(t)

)Ωmj

dt + εκ∫ T

0

(θ f (Ψ f (t)), ∂tφ f (t)

)Ω f

dt

−2

∑j=1

∫ T

0

(Km(θm(Ψmj(τ)))∇Ψmj(τ),∇φmj(t)

)Ωmj

dt

− ελ∫ T

0

(K f (θ f (Ψ f (t)))∇Ψmj(t),∇φ f (t)

)Ω f

dt

= −2

∑j=1

∫ T

0

(fmj(τ), φmj(t)

)Ωmj

dt−∫ T

0

(f f (τ), φ f (t)

)Ω f

dt

−2

∑j=1

(θm(ψmj ,I), φmj(0)

)Ωmj

− εκ(

θ f (ψ f ,I), φ f (0))

Ω f.

Therefore, equation (3.3) holds true for all appropriate test functions.In order to show that the interface conditions are satisfied, we estimate

‖Ψmj −Ψ f ‖ΓTj≤ ‖Ψmj − Ψ

mj∆t‖ΓT

j+ ‖Ψmj

∆t − Ψ f∆t‖ΓT

j+ ‖Ψ f

∆t −Ψ f ‖ΓTj

(?)

and consider the terms on the right hand side individually. The second term is zero by definition of the discreteweak solution. For the first term, we obtain by the trace inequality

‖Ψmj − Ψmj∆t‖

2ΓT

j≤ C(Ωmj)‖Ψmj − Ψ

mj∆t‖ΩT

(‖∇Ψmj −∇Ψ

mj∆t‖ΩT + ‖Ψmj − Ψ

mj∆t‖ΩT

).

By the weak convergence in equation (3.21), the term in brackets is bounded, and from the strong convergencein equation (3.24), we get that ‖Ψmj − Ψ

mj∆t‖ΩT → 0 for ∆t → 0. The third term on the right hand side of

equation (?) vanishes with an analogous argument, which finishes the proof.

30

3.6 Essential bounds

In what follows, we prove the L∞-stability of the solution. For this, we employ techniques in the spirit of[15, 43]. We define the non-negative and non-positive cuts of a function u ∈W1,2(Ω) by

[u]+ := maxu, 0,[u]− := minu, 0.

(3.27)

One can show that [u]+, [u]− ∈W1,2(Ω), and

∇ [u]+ =

∇u, u > 0,0, u ≤ 0,

∇ [u]− =

0, u ≥ 0,∇u, u < 0,

(3.28)

for a proof see e.g. [19, Lemma 7.6]. First, we consider the time-discrete solution:

Lemma 3.6.1 For each ∆t > 0 and k ∈ 1, . . . , N, it holds

‖ψkm1‖L∞(Ωm1 )

+ ‖ψkm2‖L∞(Ωm2 )

+ ‖ψkf ‖L∞(Ω f )

≤ 3Mψ (k∆t + 1) , (3.29)

where

Mψ := max

MI ,M f

mθ

. (3.30)

Proof. The proof is done by induction. For k = 0, the statement holds due to Assumption (AI). Assume nowthat

‖ψk−1i ‖L∞(Ωi)

< Mψ ((k− 1)∆t + 1) .

First, we show that ψki ≤ Mψ (k∆t + 1) almost everywhere in Ωi.

We test equation (3.15) with φi =[ψk

i −Mψ(k∆t + 1)]+

. These test functions satisfy the required interface

conditions because ψkmj

= ψkf on Γj. Adding some terms on both sides of the equation, we obtain

2

∑j=1

(θm(ψ

kmj)− θm

(Mψ(k∆t + 1)

),[ψk

mj−Mψ(k∆t + 1)

]+

)Ωmj

+ εκ

(θ f (ψ

kf )− θ f

(Mψ(k∆t + 1)

),[ψk

f −Mψ(k∆t + 1)]+

)Ω f

+ ∆t2

∑j=1

(Km(θm(ψ

kmj))∇

(ψk

mj−Mψ(k∆t + 1)

),∇[ψk

mj−Mψ(k∆t + 1)

]+

)Ωmj

+ ελ∆t(

K f (θ f (ψkf ))∇

(ψk

f −Mψ(k∆t + 1))

,∇[ψk

f −Mψ(k∆t + 1)]+

)Ω f

=2

∑j=1

(θm(ψ

k−1mj

)− θm(

Mψ(k∆t + 1))

,[ψk

mj−Mψ(k∆t + 1)

]+

)Ωmj

+ εκ

(θ f (ψ

k−1f )− θ f

(Mψ(k∆t + 1)

),[ψk

f −Mψ(k∆t + 1)]+

)Ω f

+ ∆t2

∑j=1

(f kmj

,[ψk

mj−Mψ(k∆t + 1)

]+

)Ωmj

+ ∆t(

f kf ,[ψk

f −Mψ(k∆t + 1)]+

)Ω f

.

31

From Assumptions (Aθ) and (AK), and in particular the monotonicity of θi, we deduce

mθ

2

∑j=1

∥∥∥ [ψkmj−Mψ(k∆t + 1)

]+

∥∥∥2

Ωmj

+ εκmθ

∥∥∥ [ψkf −Mψ(k∆t + 1)

]+

∥∥∥2

Ω f

+ ∆t mK

2

∑j=1

∥∥∥∇ [ψkmj−Mψ(k∆t + 1)

]+

∥∥∥2

Ωmj

+ ελ∆t mK

∥∥∥∇ [ψkf −Mψ(k∆t + 1)

]+

∥∥∥2

Ω f

≤2

∑j=1

(θm(ψ

k−1mj

)− θm(

Mψ((k− 1)∆t + 1))

,[ψk

mj−Mψ(k∆t + 1)

]+

)Ωmj

+ εκ

(θ f (ψ

k−1f )− θ f

(Mψ((k− 1)∆t + 1)

),[ψk

f −Mψ(k∆t + 1)]+

)Ω f

+ ∆t2

∑j=1

((f kmj−mθ Mψ

),[ψk

mj−Mψ(k∆t + 1)

]+

)Ωmj

+ ∆t((

f kf −mθ Mψ

),[ψk

f −Mψ(k∆t + 1)]+

)Ω f

,

(∗)

where we used θi(u) ≥ θi(v) + mθ(u− v) on the right hand side in order to get

θi(

Mψ((k∆t + 1))≥ θi

(Mψ((k− 1)∆t + 1)

)+ mθ Mψ∆t.

Note that the first two terms on the right hand side in equation (∗) are non-positive due to the induction

assumption and the monotonicity of θi. Moreover, since Mψ ≥M fmθ

, the last two terms are non-positive as well,from which we infer that ψk

i ≤ Mθ(k∆t + 1) almost everywhere in Ωi.

Similarly, we test equation (3.15) with φi =[ψk

i + Mψ(k∆t + 1)]−

in order to show that ψki ≥ −Mψ (k∆t + 1)

almost everywhere in Ωi.This gives

2

∑j=1

(θm(ψ

kmj)− θm

(−Mψ(k∆t + 1)

),[ψk

mj+ Mψ(k∆t + 1)

]−

)Ωmj

+ εκ

(θ f (ψ

kf )− θ f

(−Mψ(k∆t + 1)

),[ψk

f + Mψ(k∆t + 1)]−

)Ω f

+ ∆t2

∑j=1

(Km(θm(ψ

kmj))∇

(ψk

mj+ Mψ(k∆t + 1)

),∇[ψk

mj+ Mψ(k∆t + 1)

]−

)Ωmj

+ ελ∆t(

K f (θ f (ψkf ))∇

(ψk

f + Mψ(k∆t + 1))

,∇[ψk

f + Mψ(k∆t + 1)]−

)Ω f

=2

∑j=1

(θm(ψ

k−1mj

)− θm(−Mψ(k∆t + 1)

),[ψk

mj+ Mψ(k∆t + 1)

]−

)Ωmj

+ εκ

(θ f (ψ

k−1f )− θ f

(−Mψ(k∆t + 1)

),[ψk

f + Mψ(k∆t + 1)]−

)Ω f

+ ∆t2

∑j=1

(f kmj

,[ψk

mj+ Mψ(k∆t + 1)

]−

)Ωmj

+ ∆t(

f kf ,[ψk

f + Mψ(k∆t + 1)]−

)Ω f

.

32

With the same arguments as above, we obtain

mθ

2

∑j=1

∥∥∥ [ψkmj

+ Mψ(k∆t + 1)]−

∥∥∥2

Ωmj

+ εκmθ

∥∥∥ [ψkf + Mψ(k∆t + 1)

]−

∥∥∥2

Ω f

+ ∆t mK

2

∑j=1

∥∥∥∇ [ψkmj

+ Mψ(k∆t + 1)]−

∥∥∥2

Ωmj

+ ελ∆t mK

∥∥∥∇ [ψkf + Mψ(k∆t + 1)

]−

∥∥∥2

Ω f

≤2

∑j=1

(θm(ψ

k−1mj

)− θm(−Mψ((k− 1)∆t + 1)

),[ψk

mj+ Mψ(k∆t + 1)

]−

)Ωmj

+ εκ

(θ f (ψ

k−1f )− θ f

(−Mψ((k− 1)∆t + 1)

),[ψk

f + Mψ(k∆t + 1)]−

)Ω f

+ ∆t2

∑j=1

((f kmj

+ mθ Mψ

),[ψk

mj+ Mψ(k∆t + 1)

]−

)Ωmj

+ ∆t((

f kf + mθ Mψ

),[ψk

f + Mψ(k∆t + 1)]−

)Ω f

.

Again, the first two terms on the right hand side are non-positive in view of the induction assumption andthe last two terms are non-positive due to the choice of Mψ. This shows that ψk

i ≥ −Mψ (k∆t + 1) almosteverywhere in Ωi. Altogether, we conclude that

‖ψki ‖L∞(Ωi)

≤ Mψ(k∆t + 1),

which finishes the proof.

These estimates can be translated to the time-continuous solution, as the following lemma shows:

Lemma 3.6.2 For almost all t ∈ (0, T], it holds that

‖Ψm1(t)‖L∞(Ωm1 )+ ‖Ψm2(t)‖L∞(Ωm2 )

+ ‖Ψ f (t)‖L∞(Ω f )≤ 3Mψ(t + 1). (3.31)

Proof. Let ∆t > 0 and t ∈ (tk−1, tk]. Due to Lemma 3.6.1, we have for the piecewise constantly interpolatedfunction

‖Ψi∆t(t)‖L∞(Ωi)

= ‖ψki ‖L∞(Ωi)

≤ Mψ(k∆t + 1) ≤ Mψ(t + 1) + Mψ∆t.

In the limit ∆t→ 0, the strong convergence in Lemma 3.5.5 gives the desired estimate.

33

Chapter 4

Upscaling

An appealing approach for the modelling of thin fractures is to embed the fractures as a (d− 1)-dimensionalmanifold into a d-dimensional porous medium domain. In the fracture domain, a differential equation issolved or physical restrictions are imposed, such as no flow across the fracture (see e.g. [1, 37]).In this chapter, we show that such limit models can be rigorously derived from the original model with positivefracture width ε > 0 in the limit of vanishing ε, provided that the ratios of porosity and permeability betweenthe fracture and the matrix blocks scale appropriately with ε. We follow ideas from [53], where the upscalingwas considered in the context of crystal dissolution and precipitation, and [43], which is concerned with areactive transport model. In order to avoid averaging the non-linearity in the flux term, we consider theKirchhoff transformed version in Definition 4.1.1.

4.1 Upscaling formulation

We rescale the fracture in horizontal direction by defining z = x/ε, and introduce the following notation forthis chapter:

u f (t, z, y) = u f (t, zε, y),

u f ,I(z, y) = u f ,I(zε, y) = K f (ψ f ,I)(zε, y),

f f (t, z, y) = f f (t, zε, y).

(4.1)

To unify the notation, we set z = x in the matrix blocks. For the notation of the domains, we use the followingconventions: since the solid matrix subdomains are merely translated when ε varies, we omit the ε in the nota-tion and write Ωmj without a superscript. The fracture domain will be denoted as Ωε

f := (− ε2 , ε

2 )× (0, 1), and

the shorthand notation Ω f := Ω1f will be used. Moreover, we introduce the one-dimensional fracture domain

for the effective models Γ = (0, 1).Figure 4.1 illustrates the geometry of the problem in rescaled variables and the geometry of the effective mod-els, in which the fracture has become one-dimensional.

(−1.5, 0) (−0.5, 0) (0.5, 0) (1.5, 0)

(−1.5, 1) (−0.5, 1) (0.5, 1) (1.5, 1)

Ωm1 Ωm2Ω f

Γ1 Γ2

y

z

ε→ 0

(−1, 0) (0, 0) (1, 0)

(−1, 1) (0, 1) (1, 1)

Ωm1 Ωm2Γ

y

z

z

0.5−0.5

Γ1 Γ2

Figure 4.1: Geometry with two-dimensional fracture in rescaled variables (left) and upscaled geometry withone-dimensional fracture (right)

35

In some effective models, however, terms remain in the fracture equation which include the coordinate z, stilldefined on the domain (− 1

2 , 12 ). This domain has no longer a physical representation in the upscaled geome-

try, but mathematically, a differential equation with respect to the z-coordinate determines the pressure jumpacross the fracture in these models.For the ease of presentation, we restrict ourselves to the case of ε-independent initial conditionsKi(ψi). As willbe seen later when it comes to the a priori estimates, we require an assumption on the decline of the sourceterm for certain choices of the parameter λ, in addition to the assumptions at the beginning of the previouschapter:

(Aλ)

‖ f ε

f ‖2Ωε,T

f≤ C, if λ ≤ 1,

‖ f εf ‖

2Ωε,T

f≤ Cελ, if λ > 1.

The solution in each domain uεi will be endowed with a superscript ε to emphasise the ε-dependence.

We directly state the weak formulation in terms of the rescaled variables, which allow to eliminate the ε-dependence of the geometry. Instead, the x-argument of the functions associated with the fracture becomesε-dependent:

Definition 4.1.1 (Weak solution for upscaling) A triple (uεm1

, uεm2

, uεf ) ∈ L2(0, T;Vm1)× L2(0, T;Vm2)× L2(0, T;V f )

is called a weak solution to the Kirchhoff transformed upscaling problem if

K−1m (uε

m1) = K−1

f (uεf ) on Γ1 and K−1

m (uεm2) = K−1

f (uεf ) on Γ2 for a.e. t ∈ [0, T], (4.2)

in the sense of traces, and

−(bm(uε

m1), ∂tφm1

)ΩT

m1−(bm(uε

m2), ∂tφm2

)ΩT

m2− εκ+1

(b f (uε

f ), ∂tφ f

)ΩT

f

+(∇uε

m1,∇φm1

)ΩT

m1+(∇uε

m2,∇φm2

)ΩT

m2+ ελ−1

(∂zuε

f , ∂zφ f

)ΩT

f

+ ελ+1(

∂yuεf , ∂yφ f

)ΩT

f

= ( fm1 , φm1)ΩTm1

+ ( fm2 , φm2)ΩTm2

+ ε(

f f , φ f

)ΩT

f

+(bm(um1,I), φm1(0)

)Ωm1

+(bm(um2,I), φm2(0)

)Ωm2

+ εκ+1(

b f (u f ,I), φ f (0))

Ω f,

(4.3)

for all (φm1 , φm2 , φ f ) ∈W1,2(0, T;Vm1)×W1,2(0, T;Vm2)×W1,2(0, T;V f ) satisfying

φm1 = φ f on Γ1 and φm2 = φ f on Γ2 for a.e. t ∈ [0, T], (4.4)

andφi(T) = 0 for i ∈ m1, m2, f . (4.5)

Remark 4.1.1 (Transformation of the terms) When rescaling the fracture domain to Ω f = (0, 1) × (0, 1), theterms in equation (3.2.2) transform due to z = x/ε and the chain rule (yielding ∂z = ε∂x) into

(b f (uεf ), ∂tφ f )Ωε,T

f= ε(b f (uε

f ), ∂tφ f )ΩTf,

(∂yuεf , ∂yφ f )Ωε,T

f= ε(∂yuε

f , ∂yφ f )ΩTf,

(∂xuεf , ∂xφ f )Ωε,T

f=

1ε(∂zuε

f , ∂zφ f )ΩTf,

( f f , φ f )Ωε,Tf

= ε( f f , φ f )ΩTf,

(b f (u f ,I), φ f (0))Ωεf= ε(b f (u f ,I), φ f (0))Ω f ,

(4.6)

36

for example

(∂xuε

f (t), ∂xφ(t))

Ωεf

=∫ 1

0

∫ ε2

− ε2

∂xuεf (t, x, y) ∂xφ(t, x, y) dx dy = ε

∫ 1

0

∫ 12

− 12

∂xuεf (t, εz, y) ∂xφ(t, εz, y) dz dy

=1ε

∫ 1

0

∫ 12

− 12

∂zuεf (t, εz, y) ∂zφ(t, εz, y) dz dy =

1ε

∫ 1

0

∫ 12

− 12

∂zuεf (t, z, y) ∂zφ(t, z, y) dz dy (4.7)

=1ε(∂zuε

f (t), ∂zφ f (t))Ω f . J

4.2 Effective models

We will see that, depending on the scaling parameters for the porosity and the permeability κ and λ, respec-tively, several different effective models occur in the limit ε→ 0 (see Figure 4.2). We will consider the parame-ter range κ ∈ [−1, ∞) and λ ∈ [−1, ∞) for Problem P . In addition, we will discuss the case λ ∈ (−∞,−1) forNeumann no flow conditions at the fracture boundary.The resulting effective models are presented in Figure 4.2 and brief descriptions of the models can be foundin Table 4.1, where the abbreviations PDE and ODE stand for partial and ordinary differential equation, re-spectively. For the case κ = −1 and λ > 1, an equation for the fracture remains in the limit ε → 0, but thetwo matrix blocks decouple due to a no flow condition at the interface, for which reason the fracture equationplays no role. The Effective models VIII and IX arise only if no flow conditions are imposed at the fractureboundary. This is because they incorporate a spatially constant pressure in the fracture, which is prevented ifthe pressure head is fixed at the boundary or if an influx is given, entailing a non-zero pressure gradient.Let us mention that if one allows the medium to be anisotropic by defining a permeability tensor and intro-ducing different scaling powers with respect to different coordinate directions, even more models can arise inthe limit ε→ 0.

Model IModel II

Model III Model IVκ

λModel V

Model VI

Model VII Model VII∗

Model VIII Model IX

κ = −1 κ > −1

λ > 1

λ = 1

λ ∈ (−1, 1)

λ = −1

λ < −1

Figure 4.2: Overview of the resulting effective models for different values of the scaling parameters κ and λ

37

Fracture equation Pressure continuity

Model I Richards’ equation√

Model II elliptic equation√

Model III ODE in time√

Model IV flux continuity between matrix blocks√

Model V parabolic PDE determines pressure jump −Model VI elliptic PDE determines pressure jump −Model VII no flow between matrix blocks −Model VII∗a no flow between matrix blocks −Model VIIIb ODE for spatially constant pressure

√

Model IXb spatially constant pressure√

afracture equation remains without relevance for the decoupled model

bno flow boundary conditions required in the fracture

Table 4.1: Summary of the effective models

4.2.1 Strong formulation

To begin with, we introduce the effective models in the strong form in physical variables. We subdivide theeffective models in models with continuous pressure at the interface (Effective models I – IV), discontinuouspressure at the interface (Effective models V – VII∗), and in additional models which are only possible forNeumann no flow conditions at the fracture boundary (Effective models VIII and IX).

Models with continuous pressure

The first effective model consists of the Richards equation in the matrix block subdomains and the one-dimensional Richards equation in the fracture. It occurs for κ = λ = −1.

Effective model I :

∂tθm(ψmj)−∇ ·(

Km(θm(ψmj))∇ψmj

)= fmj , in ΩT

mj,

∂tθ f (ψ f )− ∂y

(K f (θ f (ψ f ))∂yψ f

)= [~qm]Γ , on ΓT ,

ψmj = ψ f , on ΓT ,

ψmj = 0, on ∂Ωmj \ Γ× (0, T],

ψ f = 0, on ∂Γ× (0, T],

ψmj(0) = ψmj ,I , on Ωmj ,

ψ f (0) = ψ f ,I , on Γ,

(4.8)

where[~qm]Γ := (Km(θm(ψm1))∇ψm1 ·~nm1 + Km(θm(ψm2))∇ψm2 ·~nm2)

∣∣Γ (4.9)

is the flux difference between the two solid matrix subdomains acting as a source term for Richards’ equationin the fracture (note that~nm1 = −~nm2).If the porosity ratio does not change with vanishing fracture width and the permeability ratio is taken to bereciprocally proportional to the fracture width, i.e. κ > −1 and λ = −1, one ends up with an effective modelconsisting of the Richards equation in the matrix blocks and a stationary elliptic equation in the fracture:

Effective model II :

∂tθm(ψmj)−∇ ·(

Km(θm(ψmj))∇ψmj

)= fmj , in ΩT

mj,

−∂y

(K f (θ f (ψ f ))∂yψ f

)= [~qm]Γ , on ΓT ,

ψmj = ψ f , on ΓT ,

ψmj = 0, on ∂Ωmj \ Γ× (0, T],

ψ f = 0, on ∂Γ× (0, T],

ψmj(0) = ψmj ,I , on Ωmj .

(4.10)

38

On the other hand, only the flux term in the fracture vanishes in the limit for κ = −1 and λ ∈ (−1, 1), andan ordinary differential equation remains. This limit case corresponds to a fracture capable of storing andreleasing water from the solid matrix, but which does not conduct fluid along the fracture:

Effective model III :

∂tθm(ψmj)−∇ ·(

Km(θm(ψmj))∇ψmj

)= fmj , in ΩT

mj,

∂tθ f (ψ f ) = [~qm]Γ , on ΓT ,

ψmj = ψ f , on ΓT ,

ψmj = 0, on ∂Ωmj \ Γ× (0, T],

ψmj(0) = ψmj ,I , on Ωmj ,

ψ f (0) = ψ f ,I , on Γ,

(4.11)

For κ > −1 and λ ∈ (−1, 1), an effective model results in which the fracture as a physical entity has disap-peared. In this case, both the pressure and the flux are continuous on Γ.

Effective model IV :

∂tθm(ψmj)−∇ ·(

Km(θm(ψmj))∇ψmj

)= fmj , in ΩT

mj,

[~qm]Γ = 0, on ΓT ,

ψmj = ψ f , on ΓT ,

ψmj = 0, on ∂Ωmj \ Γ× (0, T],

ψmj(0) = ψmj ,I , on Ωmj .

(4.12)

Models with discontinuous pressure

If the permeability ratio is proportional to the fracture width, i.e. λ = 1, effective models result in which thepressure is discontinuous across the interface. For κ = −1, the pressure jump is determined by a parabolicdifferential equation on the spatial domain (− 1

2 , 12 ), which does no longer represent a physical domain:

Effective model V :

∂tθm(ψmj)−∇ ·(

Km(θm(ψmj))∇ψmj

)= fmj , in ΩT

mj,

ψm2 − ψm1 =[ψ f

], on ΓT ,

ψmj = 0, on ∂Ωmj \ Γ× (0, T],

ψmj(0) = ψmj ,I , on Ωmj ,

ψ f (0) = ψ f ,I , for z ∈(−1

2,

12

), y ∈ Γ,

(4.13)

where ψ f solves for each y a parabolic differential equation on a one-dimensional domain:

∂tθ f (ψ f )− ∂z

(K f (θ f (ψ f ))∂zψ f

)= 0, for z ∈

(−1

2,

12

), (t, y) ∈ ΓT ,

ψ f (t,−12

, y) = ψm1(t, 0, y), for (t, y) ∈ ΓT ,(K f (θ f (ψ f ))∇ψ f

)(t,−1

2, y) ·~n = (Km(θm(ψm1))∇ψm1) (t, 0, y) ·~n, for (t, y) ∈ ΓT ,

ψ f (t,12

, y) = ψm2(t, 0, y), for (t, y) ∈ ΓT ,(K f (θ f (ψ f ))∇ψ f

)(t,

12

, y) ·~n = (Km(θm(ψm2))∇ψm2) (t, 0, y) ·~n, for (t, y) ∈ ΓT ,

(4.14)

and where we write [ψ f

]:= ψ f |z= 1

2− ψ f |z=− 1

2. (4.15)

39

Similarly, in the effective model for κ > −1, the solution to a one-dimensional elliptic problem determines thepressure jump across the interface:

Effective model VI :

∂tθm(ψmj)−∇ ·(

Km(θm(ψmj))∇ψmj

)= fmj , in ΩT

mj,

ψm2 − ψm1 =[ψ f

], on ΓT ,

ψmj = 0, on ∂Ωmj \ Γ× (0, T],

ψmj(0) = ψmj ,I , on Ωmj ,

(4.16)

where ψ f solves

−∂z

(K f (θ f (ψ f ))∂zψ f

)= 0, for z ∈

(−1

2,

12

), (t, y) ∈ ΓT ,

ψ f (t,−12

, y) = ψm1(t, 0, y), for (t, y) ∈ ΓT ,(K f (θ f (ψ f ))∇ψ f

)(t,−1

2, y) ·~n = (Km(θm(ψm1))∇ψm1) (t, 0, y) ·~n, for (t, y) ∈ ΓT ,

ψ f (t,12

, y) = ψm2(t, 0, y), for (t, y) ∈ ΓT ,(K f (θ f (ψ f ))∇ψ f

)(t,

12

, y) ·~n = (Km(θm(ψm2))∇ψm2) (t, 0, y) ·~n, for (t, y) ∈ ΓT .

(4.17)

If the permeability ratio decreases even faster as ε vanishes, that is λ > 1, we obtain effective models in whichthe pressure is discontinuous across the interface, and the subsolutions in the two solid matrix blocks and thefracture are entirely decoupled from each other, separated by a no flow condition:

Effective model VII :

∂tθm(ψmj)−∇ ·(

Km(θm(ψmj))∇ψmj

)= fmj , in ΩT

mj,

∂tθ f (ψ f )(t, z, y) = 0, for z ∈(−1

2,

12

), (t, y) ∈ Γ,

Km(θm(ψmj))∇ψmj ·~nmj = 0, on ΓT ,

ψmj = 0, on ∂Ωmj \ Γ× (0, T],

ψmj(0) = ψmj ,I , on Ωmj ,

ψ f (0) = ψ f ,I , for z ∈(−1

2,

12

), y ∈ Γ.

(4.18)

In the above model, which corresponds to κ = −1, physical quantities are assigned to the fracture, whereasthe fracture is merely a geometrical entity that blocks the flow in the effective model for κ > −1:

Effective model VII∗ :

∂tθm(ψmj)−∇ ·(

Km(θm(ψmj))∇ψmj

)= fmj , in ΩT

mj,

Km(θm(ψmj))∇ψmj ·~nmj = 0, on ΓT ,

ψmj = 0, on ∂Ωmj \ Γ× (0, T],

ψmj(0) = ψmj ,I , on Ωmj .

(4.19)

40

Additional models for no flow conditions

If one replaces the homogeneous Dirichlet boundary conditions in the fracture by Neumann no flow condi-tions, the following additional models can arise in the limit ε→ 0 for κ = −1 and λ < −1:

Effective model VIII :

∂tθm(ψmj)−∇ ·(

Km(θm(ψmj))∇ψmj

)= fmj , in ΩT

mj,

ψ f (t, y) = ψ f (t), on ΓT ,

∂tθ f (ψ f )(t) =∫ 1

0[~qm]Γ dy, on ΓT ,

ψmj = ψ f , on ΓT ,

ψmj = 0, on ∂Ωmj \ Γ× (0, T],

ψmj(0) = ψmj ,I , on Ωmj ,

ψ f (0) = ψ f ,I , on Γ.

(4.20)

Note that the pressure in the fracture is spatially constant here, and so is the pressure in the solid matrices atthe interface due to the pressure continuity.For κ > −1 and λ < −1, the pressure in the fracture takes a constant value at each time, in such a way that thetotal flux across the fracture is conserved:

Effective model IX :

∂tθm(ψmj)−∇ ·(

Km(θm(ψmj))∇ψmj

)= fmj , in ΩT

mj,

ψ f (t, y) = ψ f (t), on ΓT ,∫ 1

0[~qm]Γ dy = 0, on ΓT ,

ψmj = ψ f , on ΓT ,

ψmj = 0, on ∂Ωmj \ Γ× (0, T],

ψmj(0) = ψmj ,I , on Ωmj .

(4.21)

4.2.2 Weak formulation

We state the effective models in the weak formulation, which naturally arises when ε goes to zero in Definition4.1.1. For this, we introduce the function space for the fracture solution on the one-dimensional fracture:

V f := u ∈ L2(Γ) : ∂yu ∈ L2(Γ), u = 0 on ∂Γ. (4.22)

The pressure head in Effective models I – IV is continuous at the interface. Therefore, whenever the modelconverges towards one of the these effective models, we expect the solution in the fracture to approach a con-stant value in z-direction as the fracture becomes narrower. For this reason, we define z-averaged quantities,which will prove useful in order to prove convergence towards Effective models I – IV:

Definition 4.2.1 (Averaged quantities) For each ε > 0, we define the horizontally averaged quantities

uεf (t, y) :=

∫ 12

− 12

uεf (t, z, y) dz,

b f (uεf )(t, y) :=

∫ 12

− 12

b f (uεf )(t, z, y) dz,

b f (uεf ,I)(y) :=

∫ 12

− 12

b f (uεf ,I)(z, y) dz,

f f (t, y) :=∫ 1

2

− 12

f (t, z, y) dz.

(4.23)

In Effective models VIII and IX, the pressure head is spatially constant along the entire fracture. Therefore, wedefine the following averages, which will be used for Problem PN in Section 4.5:

41

Definition 4.2.2 (Averaged quantities for Problem PN) For each ε > 0, we define

uεf (t) :=

∫ 1

0

∫ 12

− 12

uεf (t, z, y) dz dy =

∫ 1

0uε

f (t, y) dy,

b f (uεf )(t) :=

∫ 1

0

∫ 12

− 12

b f (uεf )(t, z, y) dz dy =

∫ 1

0b f (uε

f )(t, y) dy,

b f (uεf ,I) :=

∫ 1

0

∫ 12

− 12

b f (uεf ,I)(z, y) dz dy =

∫ 1

0b f (u f ,I)(y) dy

f f (t) :=∫ 1

0

∫ 12

− 12

f (t, z, y) dz dy =∫ 1

0f f (t, y) dy.

(4.24)

From the regularity of uεf , which will be shown later, we obtain uε

f ∈ L2(0, T; V f ) and uεf ∈ L2(0, T).

Models with continuous pressure

Definition 4.2.3 (Effective model I) The triple (Um1 , Um2 , U f ) ∈ L2(0, T;Vm1) × L2(0, T;Vm2) × L2(0, T; V f ) iscalled a solution to Effective model I if

K−1m (Um1)(t, z, y) = K−1

f (U f )(t, y) = K−1m (Um2)(t, z, y) on Γ for a.e. t ∈ [0, T], (4.25)

in the sense of traces, and

− (bm(Um1), ∂tφm1)ΩTm1− (bm(Um2), ∂tφm2)ΩT

m2−(

b f (U f ), ∂tφ f

)ΓT

+ (∇Um1 ,∇φm1)ΩTm1

+ (∇Um2 ,∇φm2)ΩTm2

+(

∂yU f , ∂yφ f

)ΓT

= ( fm1 , φm1)ΩTm1

+ ( fm2 , φm2)ΩTm2

+(bm(Km(ψm1,I)), φm1(0)

)Ωm1

+(bm(Km(ψm2,I)), φm2(0)

)Ωm2

+(

b f (K f (ψ f ,I)), φ f (0))

Γ,

(4.26)

for all (φm1 , φm2 , φ f ) ∈W1,2(0, T;Vm1)×W1,2(0, T;Vm2)×W1,2(0, T; V f ) satisfying

φm1(t, z, y) = φm2(t, z, y) = φ f (t, y) on Γ for a.e. t ∈ [0, T], (4.27)

andφi(T) = 0 for i ∈ m1, m2, f . (4.28)

Definition 4.2.4 (Effective model II) The triple (Um1 , Um2 , U f ) ∈ L2(0, T;Vm1) × L2(0, T;Vm2) × L2(0, T; V f ) iscalled a solution to Effective model II if

K−1m (Um1)(t, z, y) = K−1

f (U f )(t, y) = K−1m (Um2)(t, z, y) on Γ for a.e. t ∈ [0, T], (4.29)

in the sense of traces, and

− (bm(Um1), ∂tφm1)ΩTm1− (bm(Um2), ∂tφm2)ΩT

m2

+ (∇Um1 ,∇φm1)ΩTm1

+ (∇Um2 ,∇φm2)ΩTm2

+(

∂yU f , ∂yφ f

)ΓT

= ( fm1 , φm1)ΩTm1

+ ( fm2 , φm2)ΩTm2

+(bm(Km(ψm1,I)), φm1(0)

)Ωm1

+(bm(Km(ψm2,I)), φm2(0)

)Ωm2

,

(4.30)

for all (φm1 , φm2 , φ f ) ∈W1,2(0, T;Vm1)×W1,2(0, T;Vm2)×W1,2(0, T; V f ) satisfying

φm1(t, z, y) = φm2(t, z, y) = φ f (t, y) on Γ for a.e. t ∈ [0, T], (4.31)

andφi(T) = 0 for i ∈ m1, m2, f . (4.32)

42

Definition 4.2.5 (Effective model III) The triple (Um1 , Um2 , U f ) ∈ L2(0, T;Vm1)× L2(0, T;Vm2)× L2(0, T; V f ) iscalled a solution to Effective model III if

K−1m (Um1)(t, z, y) = K−1

f (U f )(t, y) = K−1m (Um2)(t, z, y) on Γ for a.e. t ∈ [0, T], (4.33)

in the sense of traces, and

− (bm(Um1), ∂tφm1)ΩTm1− (bm(Um2), ∂tφm2)ΩT

m2−(

b f (U f ), ∂tφ f

)ΓT

+ (∇Um1 ,∇φm1)ΩTm1

+ (∇Um2 ,∇φm2)ΩTm2

= ( fm1 , φm1)ΩTm1

+ ( fm2 , φm2)ΩTm2

+(bm(Km(ψm1,I)), φm1(0)

)Ωm1

+(bm(Km(ψm2,I)), φm2(0)

)Ωm2

+(

b f (K f (ψ f ,I)), φ f (0))

Γ,

(4.34)

for all (φm1 , φm2 , φ f ) ∈W1,2(0, T;Vm1)×W1,2(0, T;Vm2)×W1,2(0, T; V f ) satisfying

φm1(t, z, y) = φm2(t, z, y) = φ f (t, y) on Γ for a.e. t ∈ [0, T], (4.35)

andφi(T) = 0 for i ∈ m1, m2, f . (4.36)

Definition 4.2.6 (Effective model IV) The triple (Um1 , Um2 , U f ) ∈ L2(0, T;Vm1)× L2(0, T;Vm2)× L2(0, T; V f ) iscalled a solution to Effective model IV if

K−1m (Um1)(t, z, y) = K−1

f (U f )(t, y) = K−1m (Um2)(t, z, y) on Γ for a.e. t ∈ [0, T], (4.37)

in the sense of traces, and

− (bm(Um1), ∂tφm1)ΩTm1− (bm(Um2), ∂tφm2)ΩT

m2+ (∇Um1 ,∇φm1)ΩT

m1+ (∇Um2 ,∇φm2)ΩT

m2

= ( fm1 , φm1)ΩTm1

+ ( fm2 , φm2)ΩTm2

+(bm(Km(ψm1,I)), φm1(0)

)Ωm1

+(bm(Km(ψm2,I)), φm2(0)

)Ωm2

,(4.38)

for all (φm1 , φm2 , φ f ) ∈W1,2(0, T;Vm1)×W1,2(0, T;Vm2)×W1,2(0, T; V f ) satisfying

φm1(t, z, y) = φm2(t, z, y) = φ f (t, y) on Γ for a.e. t ∈ [0, T], (4.39)

andφi(T) = 0 for i ∈ m1, m2, f . (4.40)

Models with discontinuous pressure

The interfaces Γ1 and Γ2 in Effective models V and VI do no longer represent physical entities, but describe theboundary of the virtual domain along which the pressure jump is computed. Models with discontinuities inthe capillary pressure are considered in e.g. [50, 51].

Definition 4.2.7 (Effective model V) The triple (Um1 , Um2 , U f ) ∈ L2(0, T;Vm1) × L2(0, T;Vm2) × L2(0, T;V f ) iscalled a solution to Effective model V if

K−1m (Um1)(t, z, y) = K−1

f (U f )(t, z, y) on Γ1,

K−1m (Um2)(t, z, y) = K−1

f (U f )(t, z, y) on Γ2,for a.e. t ∈ [0, T], (4.41)

43

in the sense of traces, and

− (bm(Um1), ∂tφm1)ΩTm1− (bm(Um2), ∂tφm2)ΩT

m2−(

b f (U f ), ∂tφ f

)ΩT

f

+ (∇Um1 ,∇φm1)ΩTm1

+ (∇Um2 ,∇φm2)ΩTm2

+(

∂zU f , ∂zφ f

)ΩT

f

= ( fm1 , φm1)ΩTm1

+ ( fm2 , φm2)ΩTm2

+(bm(Km(ψm1,I)), φm1(0)

)Ωm1

+(bm(Km(ψm2,I)), φm2(0)

)Ωm2

+(

b f (K f (ψ f ,I)), φ f (0))

Ω f,

(4.42)

for all (φm1 , φm2 , φ f ) ∈W1,2(0, T;Vm1)×W1,2(0, T;Vm2)×W1,2(0, T;V f ) satisfying

φm1 = φ f on Γ1 and φm2 = φ f on Γ2 for a.e. t ∈ [0, T], (4.43)

andφi(T) = 0 for i ∈ m1, m2, f . (4.44)

Definition 4.2.8 (Effective model VI) The triple (Um1 , Um2 , U f ) ∈ L2(0, T;Vm1)× L2(0, T;Vm2)× L2(0, T;V f ) iscalled a solution to Effective model VI if

K−1m (Um1)(t, z, y) = K−1

f (U f )(t, z, y) on Γ1,

K−1m (Um2)(t, z, y) = K−1

f (U f )(t, z, y) on Γ2,for a.e. t ∈ [0, T], (4.45)

in the sense of traces, and

− (bm(Um1), ∂tφm1)ΩTm1− (bm(Um2), ∂tφm2)ΩT

m2

+ (∇Um1 ,∇φm1)ΩTm1

+ (∇Um2 ,∇φm2)ΩTm2

+(

∂zU f , ∂zφ f

)ΩT

f

= ( fm1 , φm1)ΩTm1

+ ( fm2 , φm2)ΩTm2

+(bm(Km(ψm1,I)), φm1(0)

)Ωm1

+(bm(Km(ψm2,I)), φm2(0)

)Ωm2

,

(4.46)

for all (φm1 , φm2 , φ f ) ∈W1,2(0, T;Vm1)×W1,2(0, T;Vm2)×W1,2(0, T;V f ) satisfying

φm1 = φ f on Γ1 and φm2 = φ f on Γ2 for a.e. t ∈ [0, T], (4.47)

andφi(T) = 0 for i ∈ m1, m2, f . (4.48)

Definition 4.2.9 (Effective model VII) The triple (Um1 , Um2 , U f ) ∈ L2(0, T;Vm1)× L2(0, T;Vm2)× L2(0, T;V f ) iscalled a solution to Effective model VII if

− (bm(Um1), ∂tφm1)ΩTm1− (bm(Um2), ∂tφm2)ΩT

m2−(

b f (U f ), ∂tφ f

)ΩT

f

+ (∇Um1 ,∇φm1)ΩTm1

+ (∇Um2 ,∇φm2)ΩTm2

= ( fm1 , φm1)ΩTm1

+ ( fm2 , φm2)ΩTm2

+(bm(Km(ψm1,I)), φm1(0)

)Ωm1

+(bm(Km(ψm2,I)), φm2(0)

)Ωm2

+(

b f (K f (ψ f ,I)), φ f (0))

Ω f,

(4.49)

for all (φm1 , φm2 , φ f ) ∈W1,2(0, T;Vm1)×W1,2(0, T;Vm2)×W1,2(0, T;V f ) satisfying

φm1 = φ f on Γ1 and φm2 = φ f on Γ2 for a.e. t ∈ [0, T], (4.50)

andφi(T) = 0 for i ∈ m1, m2, f . (4.51)

44

Definition 4.2.10 (Effective model VII∗) The pair (Um1 , Um2) ∈ L2(0, T;Vm1)× L2(0, T;Vm2) is called a solution toEffective model VII∗ if

− (bm(Um1), ∂tφm1)ΩTm1− (bm(Um2), ∂tφm2)ΩT

m2+ (∇Um1 ,∇φm1)ΩT

m1+ (∇Um2 ,∇φm2)ΩT

m2

= ( fm1 , φm1)ΩTm1

+ ( fm2 , φm2)ΩTm2

+(bm(Km(ψm1,I)), φm1(0)

)Ωm1

+(bm(Km(ψm2,I)), φm2(0)

)Ωm2

,(4.52)

for all (φm1 , φm2) ∈W1,2(0, T;Vm1)×W1,2(0, T;Vm2) satisfying

φm1(T) = φm2(T) = 0. (4.53)

Additional models for no flow conditions

Models for Neumann no flow boundary conditions in the fracture subdomain in which the fracture solution isspatially constant and the pressure is continuous across the interface read as follows:

Definition 4.2.11 (Effective model VIII) The triple (Um1 , Um2 , U f ) ∈ L2(0, T;Vm1) × L2(0, T;Vm2) × L2(0, T) iscalled a solution to Effective model VIII if

K−1m (Um1)(t, z, y) = K−1

f (U f )(t) = K−1m (Um2)(t, z, y) on Γ for a.e. t ∈ [0, T], (4.54)

in the sense of traces, and

− (bm(Um1), ∂tφm1)ΩTm1− (bm(Um2), ∂tφm2)ΩT

m2−∫ T

0b f (U f ) ∂tφ f dt

+ (∇Um1 ,∇φm1)ΩTm1

+ (∇Um2 ,∇φm2)ΩTm2

= ( fm1 , φm1)ΩTm1

+ ( fm2 , φm2)ΩTm2

+(bm(Km(ψm1,I)), φm1(0)

)Ωm1

+(bm(Km(ψm2,I)), φm2(0)

)Ωm2

+ b f (K f (ψ f ,I)) φ f (0),

(4.55)

for all (φm1 , φm2 , φ f ) ∈W1,2(0, T;Vm1)×W1,2(0, T;Vm2)×W1,2(0, T) satisfying

φm1(t, z, y) = φm2(t, z, y) = φ f (t) on Γ for a.e. t ∈ [0, T], (4.56)

andφi(T) = 0 for i ∈ m1, m2, f . (4.57)

Definition 4.2.12 (Effective model IX) The triple (Um1 , Um2 , U f ) ∈ L2(0, T;Vm1) × L2(0, T;Vm2) × L2(0, T) iscalled a solution to Effective model IX if

K−1m (Um1)(t, z, y) = K−1

f (U f )(t) = K−1m (Um2)(t, z, y) on Γ for a.e. t ∈ [0, T], (4.58)

in the sense of traces, and

− (bm(Um1), ∂tφm1)ΩTm1− (bm(Um2), ∂tφm2)ΩT

m2+ (∇Um1 ,∇φm1)ΩT

m1+ (∇Um2 ,∇φm2)ΩT

m2

= ( fm1 , φm1)ΩTm1

+ ( fm2 , φm2)ΩTm2

+(bm(Km(ψm1,I)), φm1(0)

)Ωm1

+(bm(Km(ψm2,I)), φm2(0)

)Ωm2

(4.59)

for all (φm1 , φm2 , φ f ) ∈W1,2(0, T;Vm1)×W1,2(0, T;Vm2)×W1,2(0, T) satisfying

φm1(t, z, y) = φm2(t, z, y) = φ f (t) on Γ for a.e. t ∈ [0, T], (4.60)

andφi(T) = 0 for i ∈ m1, m2, f . (4.61)

45

4.3 Uniform estimates with respect to ε

In order to prove the convergence of Problem P towards one of the effective models introduced in the previoussection, we establish estimates for the solution and its derivatives which do not depend on ε, similar to theuniform estimates with respect to ∆t in Chapter 3.Testing with z-independent functions φ f (t, z, y) = φ f (t, y) in the fracture (and ergo φm1 and φm2 fulfillingφm1(t, z, y)|Γ1 = φm2(t, z, y)|Γ2 = φ f (t, y) for a.e. t ∈ [0, T]) in Definition 4.1.1 gives

−(bm(uε

m1), ∂tφm1

)ΩT

m1−(bm(uε

m2), ∂tφm2

)ΩT

m2− εκ+1(b f (uε

f ), ∂tφ f )ΓT

+(∇uε

m1,∇φm1

)ΩT

m1+(∇uε

m2,∇φm2

)ΩT

m2+ ελ+1(∂yuε

f , ∂yφ f )ΓT

= ( fm1 , φm1)ΩTm1

+ ( fm2 , φm2)ΩTm2

+ ε( f f , φ f )ΓT ,

+(bm(um1,I), φm1(0)

)Ωm1

+(bm(um2,I), φm2(0)

)Ωm2

+ εκ+1(

b f (u f ,I), φ f (0))

Γ,

(4.62)

for all (φm1 , φm2 , φ f ) ∈W1,2(0, T;Vm1)×W1,2(0, T;Vm2)×W1,2(0, T; V f ) satisfying

φm1(t, z, y)|Γ1 = φm2(t, z, y)|Γ2 = φ f (t, y) for a.e. t ∈ [0, T], (4.63)

andφi(T) = 0 for i ∈ m1, m2, f . (4.64)

The proof of the following lemma provides us with a first restriction on the powers of ε, required to establishuniform bounds with respect to ε:

Lemma 4.3.1 Let κ ≥ −1 and Assumption (Aλ) be satisfied. Then, there exists a C > 0 independent of ε such that

‖uεm1‖2

L2(0,T;Vm1 )+ ‖uε

m2‖2

L2(0,T;Vm2 )+ εκ+1‖uε

f ‖2ΩT

f+ ελ+1‖∂yuε

f ‖2ΩT

f+ ελ−1‖∂zuε

f ‖2ΩT

f≤ C. (4.65)

Proof. Recalling the estimate in Lemma 3.4.2, we find that in view of Remark 3.4.1 for any l ∈ 0, . . . , N thesolution to the (not Kirchhoff transformed) time-discrete problem satisfies

mθ

2max

l∈1,...,N

2

∑j=1‖ψl,ε

mj‖2

Ωmj+ εκ mθ

2max

l∈1,...,N‖ψl,ε

f ‖2Ωε

f+

∆t mK2

2

∑j=1

N

∑k=1‖∇ψk,ε

mj‖2

Ωmj+ ελ ∆t mK

2

N

∑k=1‖∇ψk,ε

f ‖2Ωε

f

≤ Mθ

2

2

∑j=1‖ψmj ,I‖

2Ωmj

+ εκ Mθ

2‖ψ f ,I‖2

Ωεf+

∆t Cpm

2mK

2

∑j=1

N

∑k=1‖ f k

mj‖2

Ωm1+ ε−λ

∆t Cp f

2mK

N

∑k=1‖ f k

f ‖2Ωε

f.

Note that ε was kept fixed in Chapter 3, for which reason it was omitted in the notation. Let f kf (z, y) =

f kf (εz, y) and ψ f ,I(z, y) = ψ f ,I(εz, y). The Poincare constants are bounded uniformly in ε due to the rectangular

geometry. The initial data are assumed to be ε-independent. Using κ ≥ −1 and Assumption (Aλ), the righthand side of the above equation can be bounded uniformly in ε for all ∆t > 0 by

Mθ

2

2

∑j=1‖ψmj ,I‖

2Ωmj

+ εκ Mθ

2‖ψ f ,I‖2

Ωεf+

∆t Cpm

2mK

2

∑j=1

N

∑k=1‖ f k

mj‖2

Ωm1+ ε−λ

∆t Cp f

2mK

N

∑k=1‖ f k

f ‖2Ωε

f

=Mθ

2

2

∑j=1‖ψmj ,I‖

2Ωmj

+ εκ+1 Mθ

2‖ψ f ,I‖2

Ω f+

∆t Cpm

2mK

2

∑j=1

N

∑k=1‖ f k

mj‖2

Ωm1+ ε−λ+1

∆t Cp f

2mK

N

∑k=1‖ f k

f ‖2Ω f

≤ C,

where the constant C ≥ 0 is independent of ε and ∆t. Therefore, this estimate translates to the time-interpolatedfunctions Ψi

∆t by Lemma 3.5.1. Consequently, the weak convergence in equation (3.21) and the weak lowersemicontinuity of the norm yield

2

∑j=1‖ψε

mj‖2

L2(0,T;Vmj )+ εκ‖ψε

f ‖2Ωε,T

f+ ελ‖∇ψε

f ‖2Ωε,T

f≤ C.

46

By the properties of the Kirchhoff transformation from Lemma 3.2.1, analogous estimates for uεi follow imme-

diately:2

∑j=1‖uε

mj‖2

L2(0,T;Vmj )+ εκ‖uε

f ‖2Ωε,T

f+ ελ‖∇uε

f ‖2Ωε,T

f≤ C.

Finally, the identities ‖uεf ‖

2Ωε,T

f= ε‖uε

f ‖2ΩT

f, ‖∂yuε

f ‖2Ωε,T

f= ε‖∂yuε

f ‖2ΩT

f, and ‖∂zuε

f ‖2Ωε,T

f= 1

ε ‖∂zuεf ‖

2ΩT

fgive

2

∑j=1‖uε

mj‖2

L2(0,T;Vmj )+ εκ+1‖uε

f ‖2ΩT

f+ ελ+1‖∂yuε

f ‖2ΩT

f+ ελ−1‖∂zuε

f ‖2ΩT

f≤ C,

which concludes the proof.

This leads us to similar estimates for uεf :

Lemma 4.3.2 Let κ ≥ −1 and Assumption (Aλ) be satisfied. Then there exists a C > 0 independent of ε such that

εκ+1‖uεf ‖

2ΓT + ελ+1‖∂yuε

f ‖2ΓT ≤ C. (4.66)

Proof. By Jensen’s inequality (see Lemma 2.3.4), we have

‖uεf ‖

2ΓT =

∫ T

0

∫ 1

0

(∫ 12

− 12

uεf (t, z, y) dz

)2

dy dt ≤∫ T

0

∫ 1

0

∫ 12

− 12

(uε

f (t, z, y))2

dz dy dt = ‖uεf ‖

2ΩT

f.

The estimate for the second term follows with the same argument, and Lemma 4.3.1 gives the result.

Moreover, the L∞ estimate from Lemma 3.6.2 allows us to bound uεf in L2(ΩT

f ) and uεf in L2(ΓT) uniformly

with respect to ε as the following lemma shows:

Lemma 4.3.3 There exists a constant C > 0 such that for all κ, λ ∈ R

‖uεf ‖L∞(ΩT

f )+ ‖uε

f ‖L∞(ΓT) + ‖uεf ‖ΩT

f+ ‖uε

f ‖ΓT ≤ C. (4.67)

Proof. The constant Mψ := maxMI ,M fmθ in the estimate in Lemma 3.6.2 is independent of ε. Therefore,

uεf and uε

f are uniformly bounded in L∞(ΩTf ) and L∞(ΓT), respectively, with respect to ε. Since the time

interval and the domain have finite measure, this implies uniform estimates for uεf in L2(0, T; L2(Ω f )) and uε

f

in L2(0, T; L2(Γ)) as well.

The estimates in Lemmas 4.3.1 – 4.3.3 give rise to a subsequence of ε→ 0 along which we get

uεmj

Umj , weakly in L2(0, T;Vmj),

uεf U f , weakly in L2(0, T; L2(Ω f )),

uεf U f , weakly in L2(0, T; L2(Γ)),

(4.68)

by the Eberlein–Smulian theorem. Moreover, if λ = −1 we obtain a uniform bound for ‖∂yuεf ‖

2ΓT in Lemma

4.3.2, implying that for a subsequence

uεf U f , weakly in L2(0, T; V f ). (4.69)

The next lemma furnishes the strong convergence of uεmj→ Umj in L2(0, T; L2(Ωmj)) along a subsequence

of ε → 0. This will prove essential for the convergence of the non-linear expression bm(uεmj) and for the

convergence on the boundary.

Lemma 4.3.4 Let κ ≥ −1 and Assumption (Aλ) be satisfied. Then, there exists a subsequence of ε→ 0 along which

uεmj→ Umj , strongly in L2(0, T; L2(Ωmj)). (4.70)

47

Proof. We will show this statement for the untransformed variable Ψεmj

= K−1(uεmj); the result for uε

mjfollows

immediately from the Lipschitz continuity of the Kirchhoff transformation. The capital letter Ψ is used here toconform to the notation in the previous chapter.First, note that the right hand side in the a priori estimate in Lemma 3.4.2 is bounded uniformly with respectto ε as ε decreases to zero due to κ ≥ −1 and Assumption (Aλ). Therefore, the estimate for the time derivative∂tΘ

mj ,ε∆t in Lemma 3.5.2 is independent of ε. From this, we infer in the limit ∆t→ 0 that

‖∂tΘεmj‖2

L2(0,T;W−1,2(Ωmj ))≤ C,

where the constant C is independent of ε. From the strong convergence in equation (3.26) and the Lipschitzcontinuity of θ−1

m , we know thatΨε

mj= θ−1

m (Θεmj).

Hence, we obtain using the properties of the Kirchhoff transformation (see Proposition 3.2.1) together withLemma 4.3.1 and the Lipschitz continuity of θm

‖θm(Ψεmj)‖2

L2(0,T;Vmj )+ ‖∂tθm(Ψε

mj)‖2

L2(0,T;W−1,2(Ωmj ))≤ C.

Thus, the existence of a strongly convergent subsequence θm(Ψεmj)ε in L2(0, T; L2(Ωmj)) in the limit ε → 0

follows from the Aubin–Lions–Simon theorem along the lines of the proof of Lemma 3.5.5, in formulae

θm(Ψεmj)→ θm(Ψmj), strongly in L2(0, T; L2(Ωmj)).

The Lipschitz continuity of θ−1m gives

Ψεmj→ Ψmj , strongly in L2(0, T; L2(Ωmj)),

and the statement of the lemma follows from the Lipschitz continuity of Km.

In order to show that the difference between the traces of uεf and the averaged quantity uε

f on the interfaceconverges to zero as ε→ 0, we will make use of Proposition 4.3 in [53]:

Proposition 4.3.1 Let Ω = (− 12 , 1

2 )× (0, L), f ∈W1,2(Ω), and let f : [0, L]→ R be defined as f (y) =∫ 1

2− 1

2f (ξ, y) dξ.

Then, in the sense of traces‖ f (ξ0, ·)− f ‖(0,L) ≤ ‖∂ξ f ‖Ω, (4.71)

for each ξ0 ∈ [− 12 , 1

2 ].

This proposition together with Lemma 4.3.1 give the following estimate:

Lemma 4.3.5 There exists a C > 0 independent of ε such that for any z0 ∈ [− 12 , 1

2 ] it holds

‖uεf (·, z0, ·)− uε

f ‖ΓT ≤ ε1−λ

2 C. (4.72)

Remark 4.3.1 (λ ≥ 1) Lemma 4.3.1 shows that the requirement λ ≤ 1 is necessary in order to keep the left handside in equation 4.72 bounded when ε vanishes (and convergence is only achieved for λ < 1). However, in thecase of λ ≥ 1, all estimates in this section remain valid due to Assumption (Aλ). Therefore, the solution to theeffective model cannot be expected to be continuous across the interface then, but the existence of convergentsubsequences is still guaranteed. J

For λ ≤ −1, the gradient of the fracture solution∇uεf is bounded in L2(0, T; L2(Ω f )) according to Lemma 4.3.1.

From this, we infer just as for the matrix block solutions in Lemma 4.3.4 that there exists a strongly convergentsubsequence of ε→ 0 along which

uεf → U f , strongly in L2(0, T; L2(Ω f )), (4.73)

making use of the Aubin–Lions–Simon theorem 2.2.3. The case λ ≤ −1 is the most interesting one, since itmeans that the fracture permeability grows at least as fast as the fracture width ε decreases. We analyse theother values of λ as well, but under additional assumptions.

48

For λ ∈ (−1, 1], the estimate in Lemma 4.3.1 does not provide a uniform estimate for ∂yuεf any longer. For this

reason, we have to assume that the y-derivative of uεf remains bounded when ε approaches zero in order to

obtain a strongly convergent subsequence.In the case λ > 1, we cannot bound any of the derivatives, and we need to assume the uniform boundednessof the gradient of uε

f additionally. This is summarised in the following assumption:

Assumption 4.3.1 (Assumption for λ > −1) There exists a constant C > 0 such that for all ε > 0 it holds

(A∇)

‖∂yuε

f ‖ΩTf≤ C, −1 < λ ≤ 1,

‖∂yuεf ‖ΩT

f+ ‖∂zuε

f ‖ΩTf≤ C, λ > 1.

With the additional Assumption (A∇), we get a strongly convergent subsequence as in equation (4.73) for allvalues of λ ∈ R by the Aubin–Lions–Simon theorem. Similarly, Lemma 4.3.3 together with Assumption (A∇)yields a strongly convergent subsequence

uεf → U f , strongly in L2(0, T; L2(Γ)). (4.74)

4.4 Upscaling theorem

With the tools of the previous chapter at hand, we are able to prove the convergence of Problem P towardsone of the effective models in Section 4.2 for different values of the scaling powers κ and λ:

THEOREM 4.4.1 (Upscaling theorem for Problem P) For the following ranges of κ and λ, these tupels are a solutionto the following effective models:

κ = −1, λ = −1 : (Um1 , Um2 , U f ) Effective model I,

κ ∈ (−1, ∞), λ = −1 : (Um1 , Um2 , U f ) Effective model II,

κ = −1, λ ∈ (−1, 1) : (Um1 , Um2 , U f ) Effective model III,

κ ∈ (−1, ∞), λ ∈ (−1, 1) : (Um1 , Um2 , U f ) Effective model IV,

κ = −1, λ = 1 : (Um1 , Um2 , U f ) Effective model V,

κ ∈ (−1, ∞), λ = 1 : (Um1 , Um2 , U f ) Effective model VI,

κ = −1, λ ∈ (1, ∞) : (Um1 , Um2 , U f ) Effective model VII,

κ ∈ (−1, ∞), λ ∈ (1, ∞) : (Um1 , Um2) Effective model VII∗.

(4.75)

Proof. • λ < 1:Choose arbitrary test functions (φm1 , φm2 , φ f ) ∈ W1,2(0, T;Vm1)×W1,2(0, T;Vm2)×W1,2(0, T; W1,2

0 (Γ)) satisfy-ing equations (4.63) and (4.64), and denote the terms in equation (4.62) by I1, . . . , I12. For all values of κ and λ,the term I9 vanishes in the limit ε→ 0 due to

|I9| = ε∣∣∣( f ε

f , φ f

)ΓT

∣∣∣ ≤ ε‖ f εf ‖ΓT‖φ f ‖ΓT → 0.

The terms I7, I8, I10, I11 do not depend on ε and remain unchanged as ε approaches zero. The strong L2 conver-gence from Lemma 4.3.4 and the Lipschitz continuity of bi give

I1 = −(bm(uε

m1), ∂tφm1

)ΩT

m1→ − (bm(Um1), ∂tφm1)ΩT

m1,

and the convergence of I2 follows with the same argument. Making use of the weak convergence from equation(4.68)1, we obtain

I4 =(∇uε

m1,∇φm1

)ΩT

m1→ (∇Um1 ,∇φm1)ΩT

m1,

and similarly for I5. As regards the fracture solution, we distinguish the cases λ = −1 and λ > −1: in case ofλ = −1, the weak convergence from equation (4.69) yields

I6 =(

∂yuεf , ∂yφ f

)ΓT→(

∂yU f , ∂yφ f

)ΓT

,

49

whereas for λ > −1, we get

|I6| = ελ+1∣∣∣(∂yuε

f , ∂yφ f

)ΓT

∣∣∣ ≤ ελ+1‖∂yuεf ‖ΓT‖∂yφ f ‖ΓT ≤ ε

λ+12 C‖∂yφ f ‖ΓT → 0,

where we made use of the estimate for ∂yuεf in Lemma 4.3.2.

It remains to consider the terms I3 and I12. Here, we make a distinction between the cases κ = −1 and κ > −1.First, consider the case κ = −1, where I3 = −

(b f (uε

f ), ∂tφ f

)ΓT

and I12 is independent of ε.For the term I3, we start by estimating∣∣∣∣∣ (b f (uε

f )− b f (U f ), ∂tφ f

)ΓT

∣∣∣∣∣ ≤∣∣∣∣∣ (b f (uε

f )− b f (uεf ), ∂tφ f

)ΓT

∣∣∣∣∣+∣∣∣∣∣ (b f (uε

f )− b f (U f ), ∂tφ f

)ΓT

∣∣∣∣∣.For the first term on the right hand side we obtain using the Lipschitz continuity of b f and Lemma 4.3.5∣∣∣∣∣ (b f (uε

f )− b f (uεf ), ∂tφ f

)ΓT

∣∣∣∣∣ ≤ ‖b f (uεf )− b f (uε

f )‖ΓT‖∂tφ f ‖ΓT

=∥∥∥ ∫ 1

2

− 12

(b f (uε

f )− b f (uεf ))

dz∥∥∥

ΓT‖∂tφ f ‖ΓT

≤ Mθ

∥∥∥ ∫ 12

− 12

∣∣∣uεf − uε

f

∣∣∣ dz∥∥∥

ΓT‖∂tφ f ‖ΓT

≤ MθC ε1−λ

2 ‖∂tφ f ‖ΓT ,

which goes to zero as ε→ 0.The second term vanishes due to∣∣∣∣∣ (b f (uε

f )− b f (U f ), ∂tφ f

)ΓT

∣∣∣∣∣ ≤ Mθ‖uεf − U f ‖ΓT‖∂tφ f ‖ΓT ,

and the strong convergence in equation (4.74). This shows that

I3 = −(

b f (uεf ), ∂tφ f

)ΓT→ −

(b f (U f ), ∂tφ f

)ΓT

.

For κ > −1, we estimate

|I3| = εκ+1∣∣∣(b f (uε

f ), ∂tφ f

)ΓT

∣∣∣ ≤ εκ+1Mθ‖uεf ‖ΩT

f‖∂tφ f ‖ΓT ≤ εκ+1MθC‖∂tφ f ‖ΓT → 0,

in view of Lemma 4.3.3, and similarly, we get

|I12| = εκ+1∣∣∣(b f (u f ,I), φ f (0)

)Γ

∣∣∣ ≤ εκ+1Mθ‖u f ,I‖Ω f ‖φ f (0)‖Γ ≤ εκ+1MθC‖φ f (0)‖Γ → 0.

Finally, the Dirichlet interface condition for the pressure head has to be proven. It turns out that a weaklyconvergent subsequence in L2(0, T; L2(Γ)) in the fracture suffices for this purpose, and Assumption (A∇) isnot necessary at this point. We present the proof for the interface Γ1, the proof for the interface Γ2 follows alongthe same lines.As the weak convergence of uε

f towards U f does not directly imply the weak convergence of K−1f (uε

f ) towards

K−1f (U f ), we define the function R(um1) := (K f K−1

m )(um1) in order to transform the interface condition

K−1m (Um1) = K

−1f (U f ) on Γ1 into a linear expression in U f , namely

R(Um1) = U f on Γ1.

50

Now we take an arbitrary test function φ ∈ L2(0, T; L2(Γ1)) and estimate∣∣∣ (R(Um1)− U f , φ)

Γ1,T

∣∣∣ ≤ ∣∣∣ (R(Um1)−R(uεm1), φ)

Γ1,T

∣∣∣+ ∣∣∣ (R(uεm1)− uε

f , φ)

Γ1,T

∣∣∣+∣∣∣ (uε

f − uεf , φ)

Γ1,T

∣∣∣+ ∣∣∣ (uεf − U f , φ

)Γ1,T

∣∣∣.Let us denote the terms on the right hand side by J1, . . . , J4. As uε

m1and uε

f satisfy the interface condition,we immediately get J2 = 0. Note that Assumption (AK) implies the Lipschitz continuity of R with Lipschitzconstant MK

mK. Making use of this and the Cauchy–Schwarz inequality yields

J1 ≤MKmK‖Um1 − uε

m1‖Γ1,T‖φ‖Γ1,T ,

and one shows as in the proof of Theorem 3.5.1 that J1 → 0 in the limit ε → 0 using the trace inequality, thestrong convergence in Lemma (4.3.4) and the boundedness of the gradient due to the weak convergence inequation (4.69). For the term J3, we obtain

J3 ≤ ‖uεf − uε

f ‖Γ1,T‖φ‖Γ1,T ,

and from Lemma 4.3.5 we infer that J3 → 0. Finally, the weak convergence in equation (4.69) yields J4 → 0.Since φ ∈ L2(0, T; L2(Γ1)) was arbitrary, one has R(Um1) = U f on Γ1 and therefore K−1

m (Um1) = K−1f (U f ) on

Γ1 in the sense of traces, which concludes the proof in the case λ < 1.

• λ ≥ 1:

We choose arbitrary test functions (φm1 , φm2 , φ f ) ∈W1,2(0, T;Vm1)×W1,2(0, T;Vm2)×W1,2(0, T;V f ) satisfyingthe conditions in Definition 4.1.1, and denote the terms in equation (4.3) by I1, . . . , I13. Note that the testfunction φ f may depend on z now. The term I10 vanishes in the limit ε→ 0 as before because

|I10| = ε∣∣∣( f ε

f , φ f

)ΩT

f

∣∣∣ ≤ ε‖ f εf ‖ΩT

f‖φ f ‖ΩT

f→ 0.

The terms I8, I9, I11, I12 are ε-independent, and the convergence of the terms I1, I2, I4, I5 follows as in the caseλ < 1. The term I7 vanishes thanks to the estimate in Lemma 4.3.1:

|I7| = ελ+1∣∣∣ (∂yuε

f , ∂yφ f

)ΩT

f

∣∣∣ ≤ ελ+1‖∂yuεf ‖ΩT

f‖∂yφ f ‖ΩT

f≤ ε

λ+12 C‖∂yφ f ‖ΩT

f→ 0.

For the term I6, we first consider first the case λ = 1: then, ‖∂zuεf ‖ΩT

fis uniformly bounded with respect to ε

due to Lemma 4.3.1. By the Eberlein–Smulian theorem 2.2.1, there exists a subsequence of ε→ 0 along which

∂zuεf ∂zU f weakly in L2(0, T; L2(Ω f )).

This gives immediatelyI6 =

(∂zuε

f , ∂zφ)

ΩTf

→(

∂zU f , ∂zφ)

ΩTf

.

If λ > 1, we estimate in view of Lemma 4.3.1

|I6| = ελ−1∣∣∣ (∂zuε

f , ∂zφ)

ΩTf

∣∣∣ ≤ ελ−1‖∂zuεf ‖ΩT

f‖∂zφ‖ΩT

f≤ ε

λ−12 C‖∂zφ‖ΩT

f→ 0.

For the terms I3 and I13, we distinguish the cases κ = −1 and κ > −1. For κ = −1, we use the strongconvergence in equation (4.73) and the Lipschitz continuity of b f to obtain

I3 =(

b f (uεf ), ∂tφ f

)ΩT

f

→(

b f (U f ), ∂tφ f

)ΩT

f

,

51

and I13 is independent of ε. For κ > −1, both terms vanish due to

|I3| = εκ+1∣∣∣ (b f (uε

f ), ∂tφ f

)ΩT

f

∣∣∣ ≤ εκ+1‖b f (uεf )‖ΩT

f‖∂tφ f ‖ΩT

f≤ εκ+1Mθ‖uε

f ‖ΩTf‖∂tφ f ‖ΩT

f→ 0,

and similarly

|I13| = εκ+1∣∣∣ (b f (u f ,I), φ f (0)

)Ω f

∣∣∣ ≤ εκ+1‖b f (u f ,I)‖Ω f ‖φ f (0)‖Ω f ≤ εκ+1Mθ‖uεf ,I‖Ω f ‖φ f (0)‖Ω f → 0.

It remains to show the continuity of the pressure head at the interfaces Γ1 and Γ2 for λ = 1, which imposes theboundary conditions for the partial differential equation that determines the pressure jump across the fracture.Again, we show this only for Γ1, the proof for Γ2 is analogous. In contrast to the case λ < 1, we need tomake use of the trace inequality in Theorem 2.1.1, for which reason we explicitly exploit Assumption (A∇) inorder to get a strongly convergent subsequence. We choose an arbitrary test function φ ∈ L2(0, T; L2(Γ1)) andestimate∣∣∣ (R(Um1)−U f , φ

)Γ1,T

∣∣∣ ≤ ∣∣∣ (R(Um1)−R(uεm1), φ)

Γ1,T

∣∣∣+ ∣∣∣ (R(uεm1)− uε

f , φ)

Γ1,T

∣∣∣+ ∣∣∣ (uεf −U f , φ

)Γ1,T

∣∣∣.We denote the terms on the right hand side by J1, J2, and J3. The term J2 equals zero as before due to theinterface condition, and the term J1 can be treated as in the case λ < 1. Finally, the term J3 converges to zerousing the trace inequality, the strong convergence in equation (4.73) and the boundedness of the gradients dueto Assumption (A∇).Hence, we are left with the respective weak formulations of the effective models in the limit ε→ 0.

4.5 The case λ < −1

Consider the case when the permeability ratio between the fracture and the matrix blocks rises faster thanthe fracture width decreases, that is λ < −1. Recall that in order to derive Essential models I – IV, we testedequation (4.3) with z-independent functions. Now, we expect the permeability to increase so rapidly that thefracture solution becomes spatially constant in the limit ε → 0. For showing this, we will use test functionswhich are constant in space, i.e. φ f (t, z, y) = φ f (t). Since we have chosen homogeneous Dirichlet boundaryconditions in Problem P , the only constant function in the test function space V f is the zero function.For this reason, we consider the problem now with Neumann no flow boundary conditions in the fracture.The strong formulation writes as

Problem PN :

∂tθm(ψεmj) +∇ · vε

mj= fmj in ΩT

mj,

vεmj

= −Km(θm(ψεmj))∇ψε

mjin ΩT

mj,

∂t(εκθ f (ψ

εf )) +∇ · v

εf = f ε

f in ΩTf ,

vεf = −ελK f (θ f (ψ

εf ))∇ψε

f in ΩTf ,

ψεmj

= 0 on ∂Ωmj \ Γj × (0, T],

vεf ·~n = 0 on ∂Ω f \ Γ× (0, T),

vεmj·~n = vε

f ·~n on Γj × (0, T],

ψεmj

= ψεf on Γj × (0, T],

ψεi (0) = ψi,I in Ωi.

The altered boundary condition is expressed in the weak formulation by modifying the function spaces for thefracture solution and the fracture test function:

Definition 4.5.1 (Weak solution to Problem PN) A triple (um1 , um2 , u f ) ∈ L2(0, T;Vm1) × L2(0, T;Vm2) ×L2(0, T; W1,2(Ω f )) is called a weak solution to the Kirchhoff transformed Problem PN if

K−1m (um1) = K

−1f (u f ) on Γ1 and K−1

m (um2) = K−1f (u f ) on Γ2 for a.e. t ∈ [0, T], (4.76)

52

in the sense of traces, and

− (bm(um1), ∂tφm1)ΩTm1− (bm(um2), ∂tφm2)ΩT

m2− εκ

(b f (u f ), ∂tφ f

)ΩT

f

+ (∇um1 ,∇φm1)ΩTm1

+ (∇um2 ,∇φm2)ΩTm2

+ ελ(∇u f ,∇φ f

)ΩT

f

= ( fm1 , φm1)ΩTm1

+ ( fm2 , φm2)ΩTm2

+(

f f , φ f

)ΩT

f

+(bm(um1,I), φm1(0)

)Ωm1

+(bm(um2,I), φm2(0)

)Ωm2

+ εκ(

b f (u f ,I), φ f (0))

Ω f

(4.77)

for all (φm1 , φm2 , φ f ) ∈W1,2(0, T;Vm1)×W1,2(0, T;Vm2)×W1,2(0, T; W1,2(Ω f )) satisfying

φm1 = φ f on Γ1 and φm2 = φ f on Γ2 for a.e. t ∈ [0, T], (4.78)

andφi(T) = 0 for i ∈ m1, m2, f . (4.79)

Testing with functions (φm1 , φm2 , φ f ) such that φ f (t, z, y) = φ f (t) and φm1(t, z, y)|Γ1 = φm2(t, z, y)|Γ2 = φ f (t)for a.e. t ∈ [0, T] in Definition 4.1.1 yields

−(bm(uε

m1), ∂tφm1

)ΩT

m1−(bm(uε

m2), ∂tφm2

)ΩT

m2− εκ+1

∫ T

0b f (uε

f ) ∂tφ f dt

+(∇uε

m1,∇φm1

)ΩT

m1+(∇uε

m2,∇φm2

)ΩT

m2

= ( fm1 , φm1)ΩTm1

+ ( fm2 , φm2)ΩTm2

+ ε∫ T

0f f φ f dt

+(bm(um1,I), φm1(0)

)Ωm1

+(bm(um2,I), φm2(0)

)Ωm2

+ εκ+1 b f (u f ,I) φ f (0),

(4.80)

for all (φm1 , φm2 , φ f ) ∈W1,2(0, T;Vm1)×W1,2(0, T;Vm2)×W1,2(0, T) satisfying

φm1(t, z, y)|Γ1 = φm2(t, z, y)|Γ2 = φ f (t) for a.e. t ∈ [0, T], (4.81)

andφi(T) = 0 for i ∈ m1, m2, f . (4.82)

Now, one mimics the previous procedure to obtain the same a priori estimate as in Lemma 3.4.2. Note that thespace for the time-discrete fracture solution W1,2(Ω f ) equals again the space for the time-discrete test functionφ f , for which reason we are allowed to employ the time-discrete solutions as test functions in the time-discreteweak formulation. Thus, the estimates in Lemmas 4.3.1 and Lemma 4.3.3 can be carried over to the currentsituation.Jensen’s inequality immediately gives

‖uεf ‖

2(0,T) ≤ ‖u

εf ‖

2ΓT ≤ ‖uε

f ‖2ΩT

f, (4.83)

just as in Lemma 4.3.2.Therefore, the Eberlein–Smulian theorem 2.2.1 provides a subsequence ε→ 0 along which

uεmj

Umj , weakly in L2(0, T;Vmj),

uεf U f , weakly in L2(0, T).

(4.84)

Due to λ < −1, all derivatives of uεmj

and uεf are bounded uniformly in ε by the estimate in Lemma 4.3.1, and

we obtain in addition by the Aubin–Lions–Simon theorem 2.2.3

uεmj→ Umj , strongly in L2(0, T; L2(Ωmj)),

uεf → U f , strongly in L2(0, T).

(4.85)

We state the upscaling theorem for the case λ < −1.

53

THEOREM 4.5.1 (Upscaling theorem for Problem PN) For the following ranges of κ and λ, these tupels are a solutionto the following effective models:

κ = −1, λ ∈ (−∞,−1) : (Um1 , Um2 , U f ) Effective model VIII,

κ ∈ (−1, ∞), λ ∈ (−∞,−1) : (Um1 , Um2 , U f ) Effective model IX.(4.86)

Proof. We choose arbitrary test functions (φm1 , φm2 , φ f ) ∈ W1,2(0, T;Vm1)×W1,2(0, T;Vm2)×W1,2(0, T) satis-fying equations (4.81) and (4.82) and denote the resulting terms in equation (4.80) by I1, . . . , I11.The terms I1, I2, I4, I5, I6, I7, I9, and I10 are treated exactly as in the upscaling theorem for Problem P , Theorem4.4.1. For the term I8, we calculate

|I8| = ε∣∣∣ ∫ T

0f f φ f dt

∣∣∣ = ε∣∣∣ ( f f , φ f

)ΓT

∣∣∣→ 0, (4.87)

as in Theorem 4.4.1. For the remaining terms, consider first the case κ > −1: then we have

|I3| = εκ+1∣∣∣ ∫ T

0b(uε

f ) ∂tφ f dt∣∣∣ ≤ εκ+1Mθ‖uε

f ‖ΩTf‖∂tφ f ‖(0,T) → 0,

and|I11| = εκ+1|b(u f ,I)||φ f (0)| ≤ εκ+1Mθ‖u f ,I‖Ω f |φ f (0)| → 0.

For κ = −1, the term I11 is independent of ε. To tackle the term I3, we write∫ T

0

(b f (uε

f )− b f (U f ))

∂tφ f dt =∫ T

0

(b f (uε

f )− b f (uεf ))

∂tφ f dt +∫ T

0

(b f (uε

f )− b f (U f ))

∂tφ f dt.

The second term on the right hand side converges to zero due to the strong convergence in equation (4.85)2and the Lipschitz continuity of b f . For the first term on the right hand side, we compute

∣∣∣ ∫ T

0

(b f (uε

f )− b f (uεf ))

∂tφ f dt∣∣∣ ≤ ‖b f (uε

f )− b f (uεf )‖(0,T)‖∂tφ f ‖(0,T)

=∥∥∥ ∫ 1

0

∫ 12

− 12

(b(uε

f )− b(uεf ))

dz dy∥∥∥(0,T)‖∂tφ f ‖(0,T)

≤ Mθ

∥∥∥ ∫ 1

0

∫ 12

− 12

∣∣∣uεf − uε

f

∣∣∣ dz dy∥∥∥(0,T)‖∂tφ f ‖(0,T)

≤ Mθ Cp f ‖∇uεf ‖ΩT

f‖∂tφ f ‖(0,T),

where we made use of Poincare’s inequality. Since we have λ < −1, Lemma 4.3.1 gives

‖∇uεf ‖ΩT

f≤ Cε

−1−λ2 → 0

in the limit ε→ 0.It follows that

I3 =∫ T

0b(uε

f ) ∂tφ f dt→∫ T

0b(U f ) ∂tφ f dt.

The pressure continuity at the interface is shown along the lines of the proof of Theorem 4.4.1 for the caseλ < 1, which concludes the proof.

54

Chapter 5

Numerical simulations

This chapter is dedicated to numerical studies aiming at the numerical validation of the theoretical upscalingresults from the previous chapter. The simulations are carried out using a standard finite volume scheme im-plemented in MATLAB. The code solves the model in physical variables as stated in Problem P . We use amatching grid, composed of uniform rectangular cells for partitioning the two-dimensional subdomains, andintervals of equal size for the one-dimensional fracture in the effective models. The flux is computed with atwo-point flux approximation (TPFA) scheme. We use an implicit Euler discretisation in time with fixed timestep, and the modified Picard scheme for the linearisation (see [31, 35] for a comparison of the classical lineari-sation schemes for Richards’ equation). We employ a monolithic approach and solve the system of equationsfor the entire domain at once.Our numerical example deals with the injection of water into a groundwater aquifer, which is crossed by a frac-ture featuring a higher permeability. Boundary and initial conditions are illustrated in Figure 5.1. Althoughour analysis was limited to homogeneous Dirichlet conditions, we expect the theoretical results to hold for themore interesting boundary conditions in our numerical example as well. All geometric dimensions (includingthe pressure heads) are given in meters. In the simulations with a two-dimensional fracture, the fracture widthtakes the values ε ∈ 1, 0.1, 0.01, 0.001, 0.0001. We impose no flow conditions on the boundary except for theinflow region in the lower edge of the left matrix block subdomain and the right upper boundary, where aDirichlet condition allows for outflow. Thus, the water must enter or cross the fracture to leave the domain.The parameters of the van Genuchten parametrisation of the water content and the hydraulic conductivity arelisted in Figure 5.1. These parameters are taken from [54], and correspond to silt loam and Touchet silt loamin the matrix blocks and in the fracture, respectively. We take the end time of the simulation to be T = 4 dand the time step is chosen as ∆t = 3 h. The grid size is chosen as ∆y = ∆z = 0.025 in the solid matrix and∆y = 0.025 and ∆z ∈ 0.025, 0.005, 0.001, 0.0002 corresponding to the different fracture widths ε.

Ωm1 Ωm2Ω f

Γ1 Γ2

y

x

Van Genuchten parameters

Geometry Boundary conditionsΩm1 = (−1− ε/2,−ε/2)× (0, 4)Ωm2 = (ε/2, 1 + ε/2)× (0, 4)

Ωεf = (−ε/2, ε/2)× (0, 4)

Initial condition and source termψI ≡ −2

f ≡ 0

no flow

ψ = −2

q = 10−6

Fracture Solid matrixα [m−1] 0.500 0.423θS [−] 0.469 0.396θR [−] 0.190 0.131nVG [−] 7.09 2.06KS [m s−1] 3.507× 10−5 5.740× 10−7

Figure 5.1: Simulation parameters

55

5.1 Models with continuous pressure

Convergence towards Effective model I: κ = λ = −1

For this choice of κ and λ, we expect the solution to converge towards Effective model I featuring the one-dimensional Richards equation in the fracture. Figure 5.2 depicts the pressure head and the saturation ofEffective model I at time t = 4 d. One observes that the pressure head in the lower left matrix block and in thelower fracture has risen, whereas the pressure head in the right matrix block has little increased. The lower leftcorner of the domain is fully saturated.Figure 5.3 shows that the averages of the pressure head across the fracture width do almost not differ fromeach other for different fracture widths with this choice of κ and λ. Still, one notices that the averages convergetowards the solution to the effective model as ε approaches zero.

Convergence towards Effective model II: κ = 0, λ = −1

When κ = 0 and λ = −1, the time derivative term is expected to vanish in the limit ε→ 0, resulting in Effectivemodel II. The solution to Effective model II, plotted in Figure 5.4, is similar to the solution to Effective model I,but the pressure height in the fracture and the surrounding matrix blocks is slightly higher.The difference is more noticeable in Figure 5.5. The averaged pressure heads in the fracture increase monotoni-cally for decreasing ε. This is because the fracture gradually loses its ability to store water as the fracture widthbecomes smaller. For a fracture width of ε = 0.1, the averaged pressure head is already close to the pressurehead of Effective model II.

Convergence towards Effective model III: κ = −1, λ = 0

For κ = −1 and λ = 0, we expect the flux term in the fracture to vanish and hence the solution to converge to-wards Effective model III, which incorporates an ordinary differential equation in time in the fracture. Figures5.6 and 5.7 show that the solution to Effective model III significantly deviates from the solutions to Effectivemodels I and II. Whereas the maximal pressure head in Effective models I and II was less than −1.65 m in thefracture and the entire fracture is thus clearly unsaturated, the pressure head in Effective model III is positivein the vicinity of the lower domain boundary implying that the fracture is saturated in this region. Since theflux term does not scale with the fracture width in this case, the fracture cannot distribute the water flowingin from the lower left corner fast enough within the fracture and the water is stored in the lower part of thefracture, while the upper fracture half is barely affected by the inflow for small fracture widths. Instead, watercrosses the fracture leading to high saturations in the lower right matrix block in contrast to Effective models Iand II.Notice that although our mathematical analysis is limited to the unsaturated case, the averaged pressure headsevidently converge towards the solution to the effective model. In comparison to Effective models I and II, theconvergence is considerably slower and for a fracture width of 0.01, the fracture is still completely unsaturatedafter 4 days.

Convergence towards Effective model IV: κ = λ = 0

In the case where the fracture equation does not scale with respect to the fracture width, i.e. κ = λ = 0,our theoretical results predict convergence towards Effective model IV, in which the pressure and the flux arecontinuous across the fracture. The solution to the effective model is depicted in Figure 5.8. Large parts of thedomain are fully saturated, in all three subdomains.Figure 5.9 shows that the pressure heads in the fracture are much higher than in Effective model III, due tothe fact that the time derivative term (which is sometimes called storage term) disappears in the fracture, andthe fracture can neither store nor conduct water along the fracture in the effective model. For this reason,the pressure head at the lower boundary of the fracture is more than 1 m higher than in Effective model III.Besides, the influence of the inflow at the lower left matrix block reaches out further and a larger part of thefracture is saturated. The convergence speed towards the effective solution is similarly slow as in Effectivemodel III.

56

Figure 5.2: Solution to Effective model I at t = 4 d

Figure 5.3: z-averaged pressure profile in the fracture for different ε at t = 4 d for κ = λ = −1

57

Figure 5.4: Solution to Effective model II at t = 4 d

Figure 5.5: z-averaged pressure profile in the fracture for different ε at t = 4 d for κ = 0, λ = −1

58

Figure 5.6: Solution to Effective model III at t = 4 d

Figure 5.7: z-averaged pressure profile in the fracture for different ε at t = 4 d for κ = −1, λ = 0

59

Figure 5.8: Solution to Effective model IV at t = 4 d

Figure 5.9: z-averaged pressure profile in the fracture for different ε at t = 4 d for κ = λ = 0

60

Figure 5.10: Convergence towards Effective models I – IV: L2 error of z-averaged pressure head in the fractureat t = 4 d in logarithmic scaling

Figure 5.10 shows the error of the z-averaged pressure heads at the final simulation time. One observes thatfor λ = −1 (corresponding to Effective models I and II), the error between the solutions with positive fracturewidths and the solution to the effective models is smaller than in the case λ = 0 (corresponding to Effectivemodels III and IV). Besides, κ = −1 leads to a smaller error as compared to the choice κ = 0, but the parameterλ plays a bigger role in view of the error curves as compared to the parameter κ. Altogether, the differences ofthe convergence speeds for different choices of the scaling parameters are remarkable: for a fracture width ofε = 0.0001, the L2 error for Effective model I is less than 10−7, whereas the error for Effective model IV is stillgreater than 10−2.

5.2 Models with discontinuous pressure

Convergence towards Model V: κ = −1, λ = 1

We consider now the solutions for the scaling parameters κ = −1 and λ = 1. The choice λ = 1 is a criticalone in that it is the smallest parameter for which we can the solution to the effective model no longer expect tobe continuous across the fracture since the permeability of the fracture tends to zero too fast. The jump in thefracture is determined by a partial differential equation.Figure 5.11 shows the solution to the expected effective model in the limit ε → 0, namely Effective model V.It stands out that the solution closely resembles the solution to Effective model III, where λ = 0 was chosen,and the pressure jump across the fracture is very small. This is due to the high permeability in the fracturethat enters the Richards equation in one “virtual” space dimension, which determines the pressure jump (seeequation (4.14)), because the high permeability yields a low pressure gradient along the virtual dimension. Inthe case of a confining fracture with low permeability, one would expect the pressure jump to be considerablyhigher for this choice κ and λ.The jump of the pressure head across the interface is presented in Figure 5.12 in logarithmic scaling. As for thesolutions corresponding to two-dimensional fractures, the pressure is continuous at the interface by definition.Therefore, the value of ψm1 |Γ1 − ψm2 |Γ2 is evaluated at the centres of the cells adjacent to the fracture. Note thatthe pressure jump is not monotonic in y-direction, but exhibits a bump at the saturation front.

Convergence towards Model VI: κ = 0, λ = 1

For the choice κ = 0, λ = 1, the model is expected to converge towards Effective model VI. The only differencecompared to Effective model V consists in the partial differential equation from which the pressure jump iscomputed, which is now elliptic. Figure 5.13 shows that also in this case, the jump in the pressure head is

61

small, and the solution to Effective model VI differs only slightly from the solution to Effective model IV(reminiscent of the similarity between the solutions of Effective model III and V). Large parts of the domainare saturated after 4 d. As in the previous case, a bump in the pressure jump occurs at the saturation front,which has advanced further, as Figure 5.14 exhibits.

Convergence towards Model VII: κ = −1, λ = 2

For the scaling parameter κ = −1 and λ = 2, the permeability in the fracture depends quadratically on thefracture width and hence decreases rapidly as the fracture width goes to zero. This results in the decoupling ofthe matrix blocks with a no flow condition at the interface in the limit, expressed by Effective model VII. Sincethe water can thus not exit the left matrix block, we stop the simulation at an end time of t = 2 d – afterwards,the pressure rockets upwards because the injection of more water in the already saturated domain requireshuge pressures which are no longer physical.Figure 5.15 shows the solution to Effective model VII at t = 2 d. The fracture saturation is averaged over zin the right subplot. As expected, the water remains in the left matrix block and does not enter or cross thefracture due to the no flow condition. The pressure head at the injection boundary is already higher than 3.5 mat this time.For this large value of λ, the convergence occurs remarkably slowly as Figure 5.16 demonstrates: for a fracturewidth of 0.001, the maximum pressure head in the fracture is not even half as high as in the effective model, andeven for a fracture width of 0.0001, the deviation with respect to the effective model is more than 0.5 m. Figure5.17 reveals that the lower pressure heads at the interface Γ1 for larger fracture widths come with (slightly)higher pressure heads at the interface Γ2. Hence, as long as the fracture still possesses a positive width, thehigh pressures in the left matrix block are relieved by pumping water into the barely permeable fracture.This can also be seen in Figure 5.18, which depicts the absolute values of the normal fluxes along the fracture,summed over both interfaces. The narrower the fracture becomes, the smaller become the fluxes near the lowerboundary of the interfaces, converging to the no flow condition. The flux intensity across the interfaces is notmonotonic along the fracture, but has bumps for ε between 0.1 and 0.001, which move in positive y-directionas the fracture width decreases.

Convergence towards Model VII∗: κ = 0, λ = 2

For the choice of parameters κ = 0, λ = 2, convergence towards the same effective model as in the previouscase is expected – the equation for the fracture which remains in the convergence analysis is meaningless forthe effective model as it decouples anyway in the limit owing to the no flow interface condition.Now, the storage term does not scale with respect to the fracture width, leading to higher pressures on bothinterfaces as the capacity of the fracture to store water gradually decreases (see Figures 5.19 and 5.20). On onehand, this means that the pressure ψm1 on Γ1 approaches the solution to the effective model more rapidly frombelow, whereas on the other hand, the rise in pressure ψm2 on Γ2 over time is noticeably higher than in theprevious case considered, yielding slower convergence towards the effective model on Γ2.Figure 5.21 shows that the flux profile along the fracture is smoother than for κ = −1, but the magnitude ofthe normal fluxes across the interfaces near the lower boundary are similar in both cases for ε ≥ 0.001. Forthe smallest positive fracture width considered, that is ε = 0.0001, the fluxes across the interfaces are lower forκ = 0 than for κ = −1.

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Figure 5.11: Solution to Effective model V at t = 4 d

Figure 5.12: Pressure jump across the fracture in logarithmic scaling for different ε at t = 4 d for κ = −1, λ = 1

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Figure 5.13: Solution to Effective model VI at t = 4 d

Figure 5.14: Pressure jump across the fracture in logarithmic scaling for different ε at t = 4 d for κ = 0, λ = 1

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Figure 5.15: Solution to Effective model VII at t = 2 d

Figure 5.16: Pressure head ψm1 at Γ1 for different ε at t = 2 d for κ = −1, λ = 2

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Figure 5.17: Pressure head ψm2 at Γ2 for different ε at t = 2 d for κ = −1, λ = 2

Figure 5.18: Sum of absolute values of normal fluxes at Γ1 and Γ2 for different ε at t = 2 d for κ = −1, λ = 2

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Figure 5.19: Pressure head ψm1 at Γ1 for different ε at t = 2 d for κ = 0, λ = 2

Figure 5.20: Pressure head ψm2 at Γ2 for different ε at t = 2 d for κ = 0, λ = 2

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Figure 5.21: Sum of absolute values of normal fluxes at Γ1 and Γ2 for different ε at t = 2 d for κ = 0, λ = 2

5.3 Additional models for no flow conditions in the fracture

Convergence towards Model VIII: κ = −1, λ = −2

For the choice κ = −1 and λ = −2, theory predicts convergence towards a spatially constant solution inthe fracture as described by Effective model VIII. This can only happen if Neumann no flow boundary con-ditions are imposed because Dirichlet conditions fix the pressure head inadmissibly and non-zero Neumannconditions imply a pressure gradient contradicting the spatially constant pressure. We hence carry out thesimulation for Problem PN . Since the fracture permeability increases quadratically with respect to decreas-ing fracture widths, we execute simulations only for ε ≥ 0.001 – for smaller fracture widths, the linearisationscheme failed to converge.Figure 5.22 shows the solution to the effective model at t = 4 d. Due to the “infinite” permeability in thefracture, the inflowing water at the lower part of the fracture is immediately distributed along the fracture andis discharged into both matrix blocks at the upper part of the fracture, observable by the pressure gradientpointing towards the fracture in the this region.The z-averaged pressure head profiles in the fracture are plotted in Figure 5.23. While the pressure headslightly decreases along the fracture for ε ≥ 0.1, it is virtually constant for ε ≤ 0.01. However, a small deviationof 0.003 [m] between the value towards which the solutions of Problem PN seem to converge and the solutionto Effective model VIII is noticeable. This flaw occurred for other choices of κ and λ in our simulations as wellwhen Neumann conditions were imposed on both sides of the fracture, for which reason we suspect that thisis due to numerical errors.

Convergence towards Model IX: κ = 0, λ = −2

Finally, we consider the parameters κ = 0 and λ = −2 for Problem PN . This means that the storage capacity ofthe fracture vanishes in the limit ε→ 0, but the pressure within the fracture is instantly equalised when waterflows in or out, as modelled by Effective model IX. For this case, the linearisation scheme only converged forε ≥ 0.01 owing to the fast increase of the fracture permeability.Figure 5.24 shows that the pressure head in the fracture has risen much more than for the case κ = −1 andthe matrix blocks are more saturated, too. Figure 5.25 exhibits the increase in pressure in the fracture as thefracture width becomes smaller due to the decreasing storage capacity.

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Figure 5.22: Solution to Effective model VIII at t = 4 d

Figure 5.23: z-averaged pressure profile in the fracture for different ε at t = 4 d for κ = −1, λ = −2

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Figure 5.24: Solution to Effective model IX at t = 4 d

Figure 5.25: z-averaged pressure profile in the fracture for different ε at t = 4 d for κ = 0, λ = −2

5.4 Discussion of the numerical results

In summary, our simulations show the convergence towards the correct effective models as derived by rigorousupscaling for all choices of κ and λ under consideration. This is despite the fact that fully saturated zonesoccurred in the simulations, which was excluded in the analysis, and despite the more complex boundaryconditions as compared to the analysis. It turns out that the parameters κ and λ are crucial as regards theconvergence speed: for λ = −1, the effective models are an accurate approximation of the original model evenfor wide fractures, whereas the effective models are only suitable to approximate very thin fractures for largervalues of λ. This behaviour is in accordance with the estimate in Lemma 4.3.5, which gives sharper bounds forthe error between the fracture solution and its z-average as λ becomes smaller. In addition, the choice κ = −1resulted in faster convergence compared with κ = 0 in our simulations.

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Discussion and outlook

In this thesis, we considered a two-dimensional model for subsurface flow through a porous medium con-taining a fracture, consisting of Richards’ equation for the description of the flow in both the fracture and thesurrounding matrix blocks. The parametrisation of the hydraulic properties in the fracture and the matrixblocks may differ from one another. The model is supplemented with physical interface conditions expressingthe conservation of mass and momentum.First, we fixed the fracture width and applied Rothe’s method, i.e. an implicit Euler time discretisation, inorder to prove that the coupled model possesses a solution, provided that the time-discrete system has a so-lution. Proving the latter lies out of the scope of this thesis, but existence results for coupled elliptic problemscan be found in e.g. [27]. For the existence proof, we established a priori bounds for the time-discrete solution,independent of the choice of the time step. In a next step, time-interpolated functions were introduced. Byvirtue of the a priori bounds, compactness arguments, namely the Eberlein–Smulian theorem and the Aubin–Lions–Simon theorem, could be applied in order to get convergent subsequences as the time step approacheszero. The limits of these subsequences were shown to be a solution to the model.Then, the fracture width was viewed as a model parameter and by means of rigorous upscaling, we provedthat one obtains different effective models in the limit of vanishing fracture width. Depending on the scalingof the permeability and porosity with respect to the fracture width, a variety of models can arise in the limit.In these effective models, the pressure can be continuous or discontinuous at the fracture, and the flux in thefracture can be zero, given by a differential equation, or described as an interface condition between the sur-rounding matrix blocks. The upscaling was carried out by rescaling the fracture and the variables related to it,by proving uniform a priori bounds with respect to the fracture width, and again by compactness arguments.Finally, the convergence towards the effective models was examined numerically, and the simulation resultswere consistent with our theoretical findings.Many extensions of this work are possible: additional effective models could be derived by allowing a perme-ability tensor and introducing scaling parameters with respect to different coordinate directions. Moreover, themodel could be supplemented by (reactive) transport, or gravity could be incorporated. The analysis in thisthesis was carried out only for the case of a fully unsaturated porous medium – with more elaborate analysis,one could extent the results to the degenerate elliptic-parabolic case. For the upscaling, one could to try toavoid the additional boundedness assumption for the derivatives of the fracture solution in the case λ > −1.As regards the numerics, domain decomposition schemes in the spirit of [6] could be applied in order to ac-celerate the simulations. Besides, it would be interesting to consider the upscaling for more complex fracturegeometries or entire fracture networks.

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Acknowledgements

I would like to thank everybody who supported me in the course of this work: first of all, thanks to mysupervisors Prof. Dr. Sorin Pop, Prof. Dr. Florin Radu, Prof. Dr. ir. Barry Koren, and Prof. Dr. Christian Rohdefor their guidance and numerous valuable advice. Moreover, thank you for making my research stay at theUniversity of Bergen possible. I had a great and productive time there and spent many sporty mornings andmerry evenings with the members of the Porous Media Group, to whom I want to express my thanks as well.I am grateful to Dr. Kundan Kumar for extensive and fertile discussions, to Dr.-Ing. Maren Paul for helpingwith all kinds of issues, and to Dr. Laura Iapichino and Dr. Oliver Tse for participating in the Exam Committee.My gratitude also goes to the University of Bergen, to the Industrial Consortium SimTech e.V., and to theBaden-Wurttemberg-Stipendium for their financial support.Finally, I wish to thank my family and my girlfriend Stephany for their enduring support throughout mystudies.

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List of Figures

1.1 Definition of the REV [5, 25] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Schematic representation of flow through a porous medium and the REV scale . . . . . . . . . . 61.3 Wetting angle γ at the interface of two fluid phases and one solid phase . . . . . . . . . . . . . . 81.4 Typical profiles of θ(ψ) and K(θ(ψ)) as given by the van Genuchten–Mualem model . . . . . . . 101.5 Geometry of the fracture and the surrounding matrix blocks . . . . . . . . . . . . . . . . . . . . . 111.6 Dimensionless geometry of the fracture and the surrounding matrix blocks . . . . . . . . . . . . 12

4.1 Geometry in rescaled variables and upscaled geometry . . . . . . . . . . . . . . . . . . . . . . . . 354.2 Overview of the resulting effective models for different values of the scaling parameters κ and λ 37

5.1 Simulation parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.2 Effective model I: Solution at t = 4 d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.3 Effective model I (κ = λ = −1): z-averaged pressure profile in the fracture at t = 4 d . . . . . . . 575.4 Effective model II: Solution at t = 4 d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.5 Effective model II (κ = 0, λ = −1): z-averaged pressure profile in the fracture at t = 4 d . . . . . 585.6 Effective model III: Solution at t = 4 d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.7 Effective model III (κ = −1, λ = 0): z-averaged pressure profile in the fracture at t = 4 d . . . . . 595.8 Effective model IV: Solution at t = 4 d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.9 Effective model IV (κ = λ = 0): z-averaged pressure profile in the fracture at t = 4 d . . . . . . . 605.10 Convergence towards Effective models I – IV: L2 error of z-averaged pressure head at t = 4 d . . 615.11 Effective model V: Solution at t = 4 d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.12 Effective model V (κ = −1, λ = 1): pressure jump across the fracture at t = 4 d . . . . . . . . . . 635.13 Effective model VI: Solution at t = 4 d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.14 Effective model VI (κ = 0, λ = 1): pressure jump across the fracture at t = 4 d . . . . . . . . . . . 645.15 Effective model VII: Solution at t = 2 d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.16 Effective model VII (κ = −1, λ = 2): pressure head ψm1 at Γ1 at t = 2 d . . . . . . . . . . . . . . . 655.17 Effective model VII (κ = −1, λ = 2): pressure head ψm2 at Γ2 at t = 2 d . . . . . . . . . . . . . . . 665.18 Effective model VII (κ = −1, λ = 2): fluxes across the interfaces at t = 2 d . . . . . . . . . . . . . 665.19 Effective model VII∗ (κ = 0, λ = 2): pressure head ψm1 at Γ1 at t = 2 d . . . . . . . . . . . . . . . 675.20 Effective model VII∗ (κ = 0, λ = 2): pressure head ψm2 at Γ2 at t = 2 d . . . . . . . . . . . . . . . 675.21 Effective model VII∗ (κ = 0, λ = 2): fluxes across the interfaces at t = 2 d . . . . . . . . . . . . . . 685.22 Effective model VIII: Solution at t = 4 d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.23 Effective model VIII (κ = −1, λ = −2): z-averaged pressure profile in the fracture at t = 4 d . . . 695.24 Effective model IX: Solution at t = 4 d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.25 Effective model IX (κ = 0, λ = −2): z-averaged pressure profile in the fracture at t = 4 d . . . . . 70

75

List of physical quantities

Greek lettersα Van Genuchten parameter m−1

Γ Interfaceγ Wetting angle

ε Dimensionless fracture width –θ Volume content –θR Residual water content –θS Saturated water content –κ Scaling parameter for porosity –λ Scaling parameter for permeability –µ Dynamic viscosity kg m−1 s−1

ρ Mass density kg m−3

σ Interfacial tension kg s−2

τ Non-equilibrium coefficient kg m−1 s−1

φ Porosity –ψ Pressure head mΩ Domain, bounded subset of Rd

∂Ω Boundary of domain Ω

Roman lettersA Area m2

b Water content as a function of u –d Capillary tube diameter mf Source / sink term s−1

g Gravitational constant m s−2

K Hydraulic conductivity m s−1

KS Saturated hydraulic conductivity m s−1

k Intrinsic permeability m2

kr Relative permeability –K Kirchhoff transformationL Edge length of a solid block ml Fracture width mm Mass kgN Number of discrete times –n Number of phases –n Van Genuchten parameter –~n Normal vectorp Pressure kg m−1 s−2

pc Capillary pressure kg m−1 s−2

Q Volume discharge m3 s−1

q Source / sink density kg m−3 s−1

S Saturation –T Final time st Time s∆t Time step s

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u Kirchhoff variable m2 s−1

V Volume m3

v Darcy velocity m s−1

vav Average velocity m s−1

W Energy functional m

Subscriptsf Subscript for fractureI Subscript for initial conditioni Subscript for subdomain m1, m2, or fj Subscript for matrix block 1 or 2k Subscript for discrete time tkm Subscript for matrix blockm1 Subscript for left matrix blockm2 Subscript for right matrix blockn Subscript for non-wetting phase∆t Subscript for the time stepw Subscript for wetting phaseα Subscript for phase

Superscriptsd Superscript for spatial dimension, here 1, 2, or 3i Superscript for subdomain m1, m2, or fk Superscript for discrete time tk

78

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