Examination of the assumption of thermal equilibrium on the fluid distribution and front stability

Transcription

1 Universität Stuttgart - Institut für Wasserbau Lehrstuhl für Hydromechanik und Hydrosystemmodellierung Prof. Dr.-Ing. Rainer Helmig Diplomarbeit Examination of the assumption of thermal equilibrium on the fluid distribution and front stability Submitted by Andreas Geiges Matrikelnummer Stuttgart, July 2009 Examiners: Prof. Dr.-Ing. Rainer Helmig, Dr.-Ing. Holger Class Supervisor: Dipl.-Ing Klaus Mosthaf

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3 Author s statement I herewith certify that I prepared this diploma thesis independently. All used sources are duly referenced, if not wished differently by the contributor. Andreas Geiges, Juli 2009

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5 Contents 1 Introduction Climate modelling Purpose of this work Basic concepts Terms and definitions Scales Fundamentals Moist air Local thermodynamic equilibrium Dimensionless numbers Physical model Mass transfer Diffusion Advection Types of energy transfer Energy conduction Convective energy transport Energy exchange between phases Mathematical model 30

6 CONTENTS I 4.1 Conservation of momentum Conservation of mass Conservation of energy General Model 1 - thermal equilibrium Model 2 - thermal non-equilibrium Model 3 - thermal non-equilibrium and separated soil conduction 36 5 Numerical Model Box method Model results Parameter analysis of the effect on the local thermal equilibrium Analytical analysis Numerical analysis Example: Infiltration in hot soil Example: Evaporation in the subsurface Example: Evaporation affected by solar radiation Discussion and outlook 72

7 List of Figures 2.1 Schematic averaging of the microscale Transfer processes in multiphase flow Relative permeability-saturation relationship Interfacial tension and wetting angle p c S w -relation after Brooks-Corey and Van Genuchten Specific internal energy and specific enthalpy of water Evaporation enthalpy of different fluids Saturation vapor pressure for water after equation Correlations for the effective conductivity for a water saturated soil Schematic structure of porous contact areas at different saturations Velocity and thermal boundary layer Packing types with different porosities Specific interfacial area for different sphere diameters Schematic energy transfer processes for model Schematic energy transfer processes for model Schematic energy transfer processes for model Schematic energy transfer area for model Dependency of the correction factor C Example of a box mesh II

8 LIST OF FIGURES III 6.1 Results of the analytical analysis Comparison of the fluid temperature in the standard case Comparison of the water saturation in the standard case Results of the numerical study Conceptional setup of the infiltration examples Fluid temperature distribution for the infiltration example Distribution of the water saturation for the infiltration example Conceptional setup of the evaporation example Overview of the simulation progress Velocity distribution of the water phase for homogenous permeability Fluid temperature distribution in the evaporation example Evaporation rate of the evaporation case Conceptional setup of the time dependent evaporation problem Temperature oscillation of a representative node in the system Overview of the simulation progress Evaporation rate of the three compared models Differences of the non-equilibrium model over time

9 List of Tables 3.1 Thermal energy conductivities of selected soil types Varried parameters in the analytical analysis Characteristic values of the analytical analysis Variation of the parameters Boundary condition of the infiltration example Soil parameter for the infiltration example Boundary condition of evaporation example Soil parameter for the evaporation example Boundary conditions of solar dependent evaporation Soil parameters for the solar dependent evaporation IV

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11 Chapter 1 Introduction 1.1 Climate modelling Climate change and global warming is one important challenge for research and science. In the field of climate modeling, the constant increase of computing power allows more complex models with a better description of the influencing factors. Most important for the modeling of global climate is the global water circuit. The interaction between the atmosphere and the groundwater is mostly connected through evaporation and rainfall. As a result of the climate change, weather conditions including heavy rainfall in combination with a strongly heated surface will occur more often. Today these phenomena mostly occur in desert areas with very few rainfall events and high mean temperatures. In future the contrast of high temperature and heavy rainfall will also increase in temperate zones. The water exchange between the ground water and the atmosphere is controlled by the fluid flow process in the unsaturated zone. Modeling is mostly done with the assumption of a local thermal equilibrium between the fluid phases (water and air) and the solid matrix. The idea of this assumption is that in a defined limited area the temperatures for all phases are equal. This is an assumption which is valid for most processes in the subsurface.. One reason to use this assumption is the slow fluid motion which permits a longer exchange of energy between soil and fluid. The second reason is the even temperature distribution. But under certain conditions this assumption can be violated, for example in the previous discussed case of cold rain on a strongly heated surface. In these cases the assumption of a local thermal equilibrium might influence the fluid distribution and the fluid front shape. The distribution of the fluid in the unsaturated zone in turn is important for the coupling of the atmosphere and the ground water balance.

12 1.2 Purpose of this work Purpose of this work The purpose of this work is the evaluation of conditions under which the assumption of a local thermal equilibrium is valid and the use of multiple energy equations necessary. The first step is the modification of the existing model in DuMu X and the implementation of a model with two separate energy equations, one for the fluid phases and one for the solid phase. Additionally a third model is developed with separately calculated energy transfers in both fluid phases. Apart from the model implementation it is necessary to evaluate scenarios for which it is possible that the assumption of the local thermal equilibrium influences the fluid distribution, front shape and stability. The systematic comparison of the results will give the possibility to discuss the limitation and validity of the assumption of the local thermal equilibrium under different conditions. Therefore it is necessary to set up examples in which different physical processes are dominant.

13 Chapter 2 Basic concepts 2.1 Terms and definitions The first step to simulate physical processes is to develop a model concept. This approximates the real processes and is usually not able to provide the exact physical behavior of the examined system in its full complexity. However, it is the aim to develop a model as the best possible approximation of the real phenomena. Physical processes are described by mathematical formulations and new parameters have to be derived for the description of them. For the modeling of multiphase flow in porous media a comprehensive model concept already exists. This work starts with a two-phase, two-component non-isothermal model. Basic concepts, necessary assumptions and important parameters will be described in the following sections Scales Every description of physical processes depends on the used scale. Heat conduction, for example, can be described with molecular interaction on the microscale or with the help of a temperature gradient on the macroscale. Most physical processes are described and mathematically formulated on the microscale. On this scale a process is described on the basis of the molecular interaction. This concept requires an exact knowledge of all physical and geometrical properties. In the subsurface and in general in porous media this information is barely present. Experimental measurements inside a pore matrix are often a complex and expensive task. Therefore it can be appropriate to apply a more abstract model to describe the important processes on a larger scale, the macroscale. The averaging of the microscopic quantities results in new quantities which depend on the used scale for the averaging. On a small scale this new values

14 2.1 Terms and definitions 5 are fluctuating due to discontinuities on the micro scale. But from a certain size of the scale these fluctuations are not influencing the average any more. In the case that these macroscopic averaged values can describe the influence of the sum of microscopic values, it is not longer necessary to have the exact information of each value itself. One chooses an appropriate scale on which fluctuations on the microscale are no longer to be identified in the macroscale. On this scale new average quantities like saturation or relative permeability sufficiently describe the microscale information inside a representative elementary volume (REV) (see Helmig, 1997, p.24). An example for a composition of a soil on the micro scale and a illustration of the averaging process is given in figure 2.1. micro scale REV solid matrix liquid phase transition + gas phase averaging Figure 2.1: Schematic averaging of the microscale Fundamentals Phase and components A Phase (α) is a macroscopic system which is separated by a sharp interface from another phase. Phases do essentially not consist of one chemical species, but also of a mixture of components. Important is that a phase has got homogenous chemical and physical properties. In the described model there are three different phases. It consists of one immobile solid phase and two mobile phases, the wetting and non-wetting phase. Interfaces between fluids are able to move and change their shape, hence their mathematical description is quite complex. In contrast the interface of the solid phase is fixed. A phase itself can be composed of different components (c). In general a component is a single chemical species (e.g. water). One exception in the model is air. It is a pseudo-component with averaged properties. This simplification is possible because

15 6 the composition of air in the model domain is only changing in a range which is not of interest for the examined system. The composition of a phase is specified by the mass fractions of the different components. It is the ratio of the mass of the component c in the phase to the total mass in the phase. Obviously the sum of all mass fractions in one phase is unity. x c α = mc α m α with m α = c m c α (2.1) Different processes responsible for the interfacial mass transfer are considered in the model. They are illustrated in figure 2.2. Water is present in both phases and can switch between the wetting and non-wetting phase through evaporation and condensation. This mass transfer is always attended by an energy transfer (see 2.1.2), but only in the form of latent heat. Interfacial energy transfer is only accounted between the solid and the mixture phase (see chapter 3.2.3). It is called interphasial energy transfer. Figure 2.2: Transfer processes in multiphase flow Saturation and porosity The description of the mass distribution is essential for the balance equations. As it is not possible to represent the distribution of molecules on the microscale, it is necessary to introduce new quantities to represent the distribution on the macroscale. On this scale the volume filled by the different phases is described by the averaged quantities saturation and porosity. Porosity is the ratio of free pore volume, filled by the fluid phase and the total cell volume.

16 2.1 Terms and definitions 7 φ = V pore V tot (2.2) The pore volume is either filled by the wetting or non-wetting phase. The saturation is the ratio of the space filled by one phase and the entire pore volume. S w = V w V pore (2.3) Following this definition the sum of saturation of all phases is unity: n phases α S α = 1 (2.4) For the description of multiphase flow it is necessary to introduce the residual saturation S rα. When the saturation is below the residual saturation, no more Darcy flow takes place. Residual trapping and the absence of connected flow paths lead to this immobility at low saturations. The residual saturation exists for the wetting and non-wetting phase. In the range of saturation, in which the fluids are mobile, we can define a new effective saturation. It is defined as: S e = S w S rw 1 S rw S rn with S rw S w 1 S rn (2.5) Permeabilities The flow of any kind of fluid through porous media mainly depends on the interaction between the participating fluids and the soil matrix. On the macroscopic scale the influence of the soil matrix is specified by the intrinsic permeability K. It describes the adhesive influence which affects the fluid flow. In an isotropic medium, the intrinsic permeability is equal in all directions. To account for the fluid properties the hydraulic conductivity K f is used which includes the intrinsic permeability and additionally the fluid density ρ f, the gravity g and the viscosity µ: K f = K ρ fg µ (2.6) The hydraulic permeability has the dimensions of a velocity [m/s]. In the case of multiphase flow, we have to consider the mutual influence of the phases in addition. To account for this, the permeability is modified with a factor called relative permeability:

17 8 K fα = k rα K ρ fg µ with 0 α k rα 1 (2.7) The relative permeability k rα depends on the saturation of the phase respectively the volume filled by the phase itself. Typically the sum of the relative perm abilities is less than one that means the phases as a mixture are more inhibited than in single phase flow. Normally the relations found by Van Genuchten or Brooks-Corey are used for modeling (see figure 2.3). In this work the formulations of Brooks-Corey are used. 1.0 relative permeability [-] Van Genuchten: n=4.37 m=.77 =.37 Brooks-Corey: =2 P d =2 S wr =.1 Van Genuchten Brooks-Corey saturation [-] Figure 2.3: Relative permeability-saturation relationship after (Helmig, 1997) Viscosity Viscosity describes the internal flow resistance of a fluid due to fluid friction and links the angular strain γ with the shear stress τ: τ = µ γ t (2.8) The viscosity of water is decreasing with rising temperatures. In addition it also depends on the solute substances. The viscosity of pure air is in contrast increasing with rising temperatures. In case of saturated air, the additional concentration of water

18 2.1 Terms and definitions 9 caused by higher temperatures results in an overall decrease of the viscosity at rising temperatures (see Helmig, 1997). In displacement processes the viscosities of the involved fluid are very important, as they determine the character of the process. In case of the displacement of a fluid with a high viscosity by a low-viscous fluid, the interface remains as a sharp front. In the opposite case, the interface is not stable and forms fingering. Fingering is the effect of a displacement process forming an instable front; hence the displacement in some areas is faster along preferential flow paths. Capillarity and wettability At an interface between two phases molecular cohesion effects within a phase and adhesion effects between the phases occur. This phenomenon is called surface tension which depends on the physical properties of the two phases. Assuming a mechanical equilibrium at the interface connected to a solid wall (see figure 2.4), this equilibrium of forces is described by the Young s equation: or σ S2 = σ S1 + σ12cosα (2.9) ( ) σs2 σ S1 γ = arccos σ 12 (2.10) The angle γ is used to differ between wetting and non-wetting phases. A fluid forming a boundary angle 0 <γ <90 is called wetting phase, in this work it is water. Air is the non-wetting phase with a boundary angle 90 <γ <180. Figure 2.4: Interfacial tension and wetting angle after Helmig (1997) As the interface between both fluids is at equilibrium, a pressure drop occurs. The difference of the phase pressures is called capillary pressure. p c = p w p n (2.11)

19 10 For a capillary pore one can define the capillary pressure which depends on the surface tension and the pore diameter as follows: p c = 4σ 12cosγ d (2.12) With the help of this definition it is possible to evaluate the capillary pressure dependent on the saturation on the macroscale. At small saturations the wetting phase fills the smaller pores at first. The interface between wetting and non-wetting phase is thereby located in a pore with a small diameter d, which leads to a high capillary pressure, see equation Therefore the capillary pressure is rising with the reduction of the saturation S w. In the case that more water is entering the pore the wetting phase penetrates larger pores which leads to a decreasing of p c. The capillary pressure saturation-relation p c = p c (S w ) evaluated by Brooks and Corey is used. For a detailed description see Helmig (1997). The parameters in the equations for the capillary pressure are soil dependent parameters which characterize the grain size and distribution in the soil matrix. 10 capillary pressure [*10 5 Pa] Brooks-Corey Van Genuchten effective watersaturation [-] p d Figure 2.5: p c S w -relation after Brooks-Corey and Van Genuchten Helmig (1997)

20 2.1 Terms and definitions 11 Internal energy and enthalpy The internal energy U of a system is equal to the total energy of a non-moving system. One can regard the internal energy of a phase as the sum of energy of its molecules. The energy of a single molecule is consisting of the energy of translation, rotation and vibration, and the inter-molecular forces between the molecules. The energy of the molecules mainly depends on the temperature whereas the inter-molecular forces are influenced by the pressure. A detailed description can be found in Baehr and Kabelac (2006). In detail the internal energy can be classified in three groups. thermal internal energy, influenced by the change of pressure and temperature. chemical internal energy corresponds with the binding energy of electrons and changes through chemical reactions. nuclear internal energy, influenced through nuclear reaction. The enthalpy H is defined as the sum of internal energy u and volume changing work pv : H = U + pv (2.13) Both internal energy and enthalpy are extensive state variables. The corresponding specific variables are the specific internal energy u and the specific enthalpy h in [kj/kg]: u = u(t, υ) = U m (2.14) and h = h(t, υ) = H m = u + pv (2.15) The material law, which describes the relation of the specific internal energy u and the other two intensive variables T and υ, is called caloric state equation. In Figure 2.6 the functions u(t, p) and h(t, p) are illustrated. Evaporation enthalpy Transfer of the component water to the gaseous phase is called evaporation; the opposite way is called condensation. The difference of the specific enthalpy of saturated steam and the boiling fluid at the equal temperature and pressure is called specific evaporation enthalpy. It is the energy that has to be put into a boiling fluid for the complete isothermal-isobaric evaporation. The evaporation enthalpy depends on temperature, it is monotonically falling with increasing temperature (see figure 2.7). The

21 12 Figure 2.6: Specific internal energy and specific enthalpy of water grade increases up to the critical point where the state of gas and fluid is identical. At this point the evaporation enthalpy is zero. The same amount of energy is released during the condensation of steam. Figure 2.7: Evaporation enthalpy of different fluids (Gnielinski et al., 2006) For the calculation of the enthalpies of water and air the formulations proposed by the IAPWS (International Association for the Properties of Water and Steam) (IAPWS:, 1997) are used. The specific enthalpy of a phase is calculated by the mass fractions of the different components:

22 2.1 Terms and definitions 13. h α = h g αx a α + h w αx w α (2.16) Moist air Moist air is a mixture of gas and steam with one component that can condensate in the examined temperature and pressure range. In this case that component is water or water steam. The other component, dry air, is assumed as a mixture of ideal gases with a fixed composition consisting of mainly Nitrogen N 2 and in addition carbon dioxide CO 2, oxygen O 2 and argon Ar. Partial pressure and saturated steam pressure At a defined temperature and volume the mass m w of the steam in the gas phase is given by the Ideal gas law for a component: x w = p wv R W T (2.17) with p w as the water phase pressure and R W the individual gas constant. The partial pressure increases with bigger values of x w. With the assumption of an ideal gas the pressure of the gas component is equal to the one if the gas fills the volume on alone. p w g = x w g p g (2.18) The solubility of a component in a phase has a maximum value depending on the temperature. The maximum water content in air is reached when the partial pressure of the non-wetting phase is equal to the saturated steam pressure. At this point, surplus water steam condensates and forms a liquid phase. The saturated steam pressure only depends on the temperature and can be calculated using the Antoine equation: log p s = A B T C (2.19) The three contained coefficients are given with high accuracy in the literature for many chemical species and temperature ranges. Exact experimental measurements were used to evaluate these empirical parameters for water which are: A = , B = and C =

23 14 Figure 2.8: Saturated vapor pressure for water after equation Local thermodynamic equilibrium It is a basic assumption for the formulation of the mathematical equation of multiphase flow in porous media. It is applied for the processes inside a control volume. It contains three parts. Thermal equilibrium All phases within a REV have the same temperature ( T α = T s = T ).This assumption is valid as the energy exchange between the phases is significantly faster than the energy transport within a phase. This is valid for small grain sizes and their linked large specific soil surface area between the phases (see section 3.2.3) In the case that local thermal equilibrium is assumed there is no need for the calculation of energy transfer between the phases. The local thermal equilibrium implies several assumptions, like low fluid velocities, fast energy exchange between the phases and a smooth spatial temperature distribution.

24 2.1 Terms and definitions 15 Mechanical equilibrium The pressure forces of two phases are equal at the separating interface. For the description of the pressure at the interface in porous media a jump in the pressure is considered due to capillarity. Chemical equilibrium Mass transport between the phases is at equilibrium in the case that the chemical potentials of the components in each phase are equal. Furthermore the kinetics of all chemical reactions are neglected. The assumption of chemical equilibrium implies that all reactions are balanced and have reached their stationary value. For example, the kinetic of dissolution of air in water, which is a time dependent process, is not considered. It is always assumed that the dissolution is instantaneously at maximum Dimensionless numbers Dimensionless quantities are parameters in a mathematical model to describe a physical process. Mainly it is a dimensionless description of a set of processes with the aim to give a statement of the dominating one. As it is a dimensionless description it is not dependend on a certain problem or geometry but can give general information. Therefore dimensionless numbers are often used to transfer a found solution from one problem to a similar one. This section describes some relevant dimensionless numbers for the examination of the thermal non-equilibrium. Reynolds number The Reynolds number is an important quantity to characterize the flow behavior. It is defined as the ratio of the inertial to viscous forces. For small Reynolds numbers, the viscous forces are dominating the inertia forces. In that case it is possible to simplify the flow equations with the help of the Darcy s law. The Reynolds number is defined as: Re = vl ν (2.20) with the flow velocity v, the characteristic length L and the dynamic viscosity ν. The choice of the characteristic length is depending on the geometrical setup. In porous media either the pore diameter or the grain size can be used. For Reynolds number

25 16 smaller than one the use of the Darcy s law is accurate. For the range 1 to 2300 laminar flow occurs. Above 2300 a transition to turbulent flow is taking place. The interfacial energy exchange between the solid and the fluid phase depends, among other parameters, on the fluid flow characteristic and therefore on the Reynolds number. Prandtl number The Prandtl number is a dimensionless number containing only fluid properties and depends on the involved fluids. It is defined as the ratio of the dynamic viscosity µ, the heat capacity c p and the thermal conductivity λ: P r = µc p λ (2.21) The Prandtl number links the behavior of the velocity field to the temperature field. In energy transfer processes, the Prandtl number describes the relative thickness of the momentum and thermal boundary layers. A small Prandtl number states that the conductive energy flow is the dominant process compared to the convective energy flow. Nusselt number The Nusselt number characterizes convective energy transfer between a surface boundary and a moving fluid. It it describes the connection of the temperature boundary layer to the velocity boundary layer on the surface of the solid. The concept of the boundary layer is described in chapter 3. In general it is defined as: Nu = h exl (2.22) λ with the energy exchange coefficient h ex, the characteristic length L and the thermal conductivity of the fluid λ. It is equal to a dimensionless temperature gradient at the surface, and provides a measure of the convective energy transfer occurring at the surface (see Incorpera et al., 2007). It is derived with help of the boundary layer concept, which is described in section Nusselt numbers for different geometries and flow types can be found in the literature. The evaluation of the Nusselt number for the present case is used to evaluate the energy exchange coefficient.

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27 Chapter 3 Physical model This section provides a basic introduction to the conceptual model used for the simulation of mass and energy transfer. The conceptual models are the basis for the derivation of the mathematical and numerical model. The reduction of a physical process to a simplified concept is necessary for the mathematical formulation of almost every natural process. The focus in this work is on the extension of the energy balance equation, hence the description of the basic multiphase equations for mass transport are only described in short. In addition this chapter provides a detailed description of the different energy transfer concepts and the underlying physical processes. 3.1 Mass transfer Diffusion Mass diffusion is composed of different mechanisms. Most relevant is the molecular diffusion, which is the transport of molecules from high to low concentrations. It is caused by the random molecular motion, the Brownian motion. It is directed from higher concentrations to lower ones. This leads to an effective diffusive transport. It is described by the Fick s law, q D = D c (3.1) with the molecular diffusion coefficient D. The molecular diffusion coefficient is unidirectional, which means it acts in all directions in the same way. An additional diffusion effect, caused by the velocity distribution in the pores and the heterogeneities in the soil is called dispersion. Due to the higher velocities in the center of the pore, molecules are transported faster in the center. This leads to an additional

28 3.2 Types of energy transfer 19 smearing of a sharp front. Because of the assumption of uniform velocity in the porous media, this additional diffusion is not included in the numerical simulation. Therefore an additional diffusion coefficient for dispersion has to be considered in principle. In the case of numerical simulation, additional numerical diffusion occurs. This is an artificial effect, without connection to a physical effect. Numerical diffusion depends on the used grid and usually decreases with finer meshes. Due to the fact that the additional numerical diffusion occurs it is not necessary to account for the dispersion Advection The mass transport with the fluid flow, caused by a pressure gradient is called advective transport. For the fact that it is a directed quantity, which is acting along the pressure gradient the discretization is different compared to the diffusive transport. The advective flux is described by: q A = c v (3.2) For small Reynolds numbers the velocity is proportional to the hydraulic head. The proportionality factor is the hydraulic conductivity K f. with the hydraulic head defined as: v = K f h (3.3) h = p ρg + z (3.4) For multiphase flow additionally the mutual influence of the involved phases must be accounted for. This leads to a modified Darcy s law. It is described in detail in Helmig (1997) and is defined as: 3.2 Types of energy transfer Energy conduction v α = k rα µ α K( p α ρ α g) (3.5) Conduction is the transfer of energy between adjacent molecules. A higher energy amount leads to higher activity of the molecules and therefore to a higher energy

29 20 Substance λ in W/ K m Water Air Solid (granite) 2.8 Solid (sand) 4.2 Solid (quartz sand) 8.5 Table 3.1: Thermal energy conductivities of selected soil types transfer due to collisions with adjacent molecules. As a consequence the molecules pass their energy to those nearby. The direction of the transport is always from high to low temperatures. It takes place in all continua - in solid, liquid and gaseous phases. It is described in the Fourier s law (3.6). The negative sign in the equation, given below, is due to the fact that the energy transport acts in the opposite direction to the temperature gradient. q = λ T (3.6) The dimension of the energy flux q is J/(m 2 s). The energy conductivity is a scalar property of the material. For fluids it depends on temperature and pressure, however in this work the values are assumed as constant in the examined temperature and pressure range. Table 3.1 shows the conductivity used in the model. The values are adopted from the work of Côté and Konrad (2009). The complexity of conduction, occurring simultaneously in three different phases, with large variation of conductivities, is not easy to describe mathematically. If an energy balance is set up for each phase (no local thermal equilibrium) on the microscale, the conductivities of the phases are material constants. On the macroscale all influences of the soil geometry and the fluid distribution have to be accounted for. The saturation of water is very important, because the energy conductibility of water is much higher than the one of air. Hence air can here be seen as a thermal isolator. In the case of an assumed local thermal equilibrium, only one energy balance is necessary, and so the conduction in all three phases is merged in one term as the effective conduction of all phases. This is done by weighting the conductivities by the volume ratios of the phases. On the macroscale, only two parameters are available for this weighting, the porosity and the saturation. These parameters are not sufficient to describe the composition of the unsaturated soil, as the location or distribution of the phases is not included. In the literature evaluations for the maximum and minimum possible effective conductivities are given, known as the Weiner bounds (see (see Côté and Konrad, 2009)). The upper Weiner bound is the theoretical possible maximum conductivity for a parallel configuration of the porous medium (Eq. 3.7).

30 3.2 Types of energy transfer 21 λ eff,max = (1 φ)λ s + φλ fl (3.7) The lower Weiner bound is the lowest possible conductivity for a serial configuration of the porous medium (Eq. 3.8). λ eff,min = λ fl λ s (1 φ)λ fl + φλ s (3.8) Krupiczka (1967) evaluated a relation for the thermal conductivity for single phase flow in a porous medium, composed of spherical grains: λ eff,α λ α = ( λs λ α ) logφ 0.057log( λs λα ) (3.9) This relation for the effective conductivity is obviously located between the upper and lower Weiner bound. Figure 3.1 shows that the given relation exceeds the upper Weiner bound for extreme values of the porosity. This is accepted to keep the mathematical formulation simple. In order to prevent an influence of this formulation the used porosities in the models have to stay in a range between 0.1 and 0.8, which is completely sufficient for the examined cases. 7 effective conductivity upper weiner bound lower weiner bound Correlation after Krupiczka porosity Figure 3.1: Correlations for the effective conductivity for a water saturated soil

31 22 Equation 3.9 for single phase flow is used to evaluate two conductivity coefficients: one for the porous medium completely filled with water and one for the medium filled with air. These effective conductivities for the single phase flow are now weighted by the saturation to get the effective conductivity for all three phases. For the weighting two methods are selected from literature. Possibility is a simple linear weighting: λ eff = λ eff,n + S w (λ eff,w λ eff,n ) (3.10) and the weighting with the square root of the saturation: λ eff = λ eff,n + S w (λ eff,w λ eff,n ) (3.11) The single phase conductivities λ eff,α uses in the formulas are evaluated with the help of equation 3.9. For low saturation the increase of water saturation has a big influence on the overall conductivity, as the contact areas between the soil grains are filled with water.(see figure3.2) The significant higher conductivity of water compared to air is therefore supporting the energy exchange in the new existing contact areas. This effect is illustrated in the first two pictures in figure 3.2. At higher saturations, the same increase of water saturation only creates a smaller increase of these contact areas, shown in the third picture of figure 3.2. This effect is illustrated in figure 3.2. It is illustration the way For this reason in this work the correlation 3.11, using the square root of the water saturation, as mentioned in Class (2007) is used: Figure 3.2: Schematic structure of porous contact areas at different saturations Convective energy transport In a moving fluid, energy is not only transported by energy conduction but also by the movement of the molecules itself. The energy of a phase, which is equal to the

32 3.2 Types of energy transfer 23 enthalpy, is transported with the mass flux. In this context enthalpy is treated as a component, similar to the transport of a solved conservative component. Therefore the same formulation is used as for the advective mass transport. q energy,α = T α ρ α c p,α q m,α (3.12) Energy exchange between phases In case of local thermal disequilibrium the exchange of energy between the phases inside one REV occurs. At a solid wall the superposition of thermal conduction and energy transport by the moving fluid is the energy exchange between the solid and the fluid phase. It is the energy flux normal to the boundary surface. The energy transport between two phases implies that the two phases are not in thermal equilibrium. The different temperatures of the phases yield a compensating energy transfer over the interface. The energy flux is thereby proportional to an energy exchange coefficient and the interfacial area. Both are evaluated in the following section. Boundary layer concept For the understanding of the energy exchange, which takes place between a solid surface and a fluid flowing past it, we introduce the concept of the boundary layer, evaluated by L. Prandtl. With the help of this concept the energy exchange between a fluid and a flat plate is described with the help of the dimensionless Nusselt number (see section 2.1.5). Based on this concept, empirical evaluations for the Nusselt number for complex flow fields and solid shapes are derived. y u free stream Τ u u velocity boundry layer δ (x) thermal boundary layer x Ts Figure 3.3: velocity and thermal boundary layer after Incorpera et al. (2007) In this section the velocity boundary layer, which is connected to the friction coefficient of the fluid, and the thermal boundary layer, which depends on the energy exchange

33 24 coefficient, are described. To illustrate this concept one considers a fluid flow over a flat plate (see Fig.3.3). In front of the plate, the flow is undisturbed and we assume a uniform velocity distribution, called u 0. Due to the adhesive forces the boundary condition for the flow velocity on the plate surface is assumed to be zero. With increasing distance to the plate the retardation of the fluid particles is decreasing. The retardation of the fluid depends on the shear stress acting parallel to the fluid motion. The thickness of the boundary layer is defined as the distance for which the velocity reaches 99% of the free stream velocity. The free stream velocity is the velocity away from the plate which is not influenced by the plate. As illustrated in Figure 3.3 the thickness of the boundary layer is increasing with x. In general the geometry of the surface, the property of the surface and the characteristic of the fluid flow determine the shape and thickness of the boundary layer. Later the dependency is described with the help of the Reynolds number. The thermal boundary layer also changes from T s at the plate surface to T f in some distance from the plate. The boundary layer thickness is again the range, in which the temperature is below 99% of the free stream temperature. The energy flux at the solid wall q s depends on the velocity of the fluid and the temperatures gradient. For the evaluation of the energy flux we set up the following equation, applying Fourier s law to the fluid at the wall, with y=0. q s = λ f T y (3.13) The thermal conductivity is the one of the fluid at wall temperature. Directly at the wall the exchange of energy takes only place by conduction. Here the fluid adheres to the solid wall and no convective transport takes place. The temperature gradient at the wall, perpendicular to the wall is the limiting factor for the exchange of energy. With the help of the Newton s law of cooling it is also possible to describe the energy flux q s as follows, q s = h(t s T F ) (3.14) with a new parameter, the local energy exchange coefficient h. By merging equation 3.13 and equation 3.14, it is possible to set up an equation to describe the local energy exchange coefficient only by the temperature gradient and the conduction coefficient: h ex = λ f T y T s T F (3.15) The energy exchange coefficient is also unknown for most problems but related to the temperature field in the fluid. With this definition it is easy to see that the energy exchange coefficient h ex is determined by the gradient of the temperature profile at the

34 3.2 Types of energy transfer 25 wall and the different temperatures of the wall and the fluid. The temperature field in turn is influenced by the velocity field, as the velocity is responsible for the advective energy transport in some distance of the wall. Hence the evaluation of the local energy exchange coefficient is an ambitious task and can only be handled theoretically in relative simple cases. One of these cases is the fully developed laminar flow over a flat plate. For more demanding cases, which is the case in this work the evaluation of the exchange coefficient still depends on experimental results and empirically found parameters. Since it is not reasonable to set up an experiment to evaluate all parameters for all possible cases, the usage of dimensionless quantities is required. With their help it is possible to transfer experimentally evaluated results to similar problems. A system is represented by dimensionless variables as dividing the position coordinates by a characteristic length, the velocity by a constant reference velocity and the temperature by a characteristic temperature. Using this technique, similar problems can be described with the help of their reference values. This gives the possibility to transfer known quantities to similar problems, only by transferring them to new reference values. To derive a dimensionless form of equation 3.15, which links the local energy exchange coefficient h to the temperature gradient; the dimensionless distance from the wall is defined as: y := y/l 0, (3.16) with L 0 as a characteristic length, for example the pore diameter or the grain size. In the same way, a dimensionless temperature is set up by dividing the temperatures by a characteristic temperature difference T 0, which is adequate to the problem. For the energy exchange this can be the maximally occurring temperature difference between solid and fluid phase. T := T T 0 T 0 (3.17) Including this modification in equation 3.15 the following formula is obtained: h = λ L 0 ( T y ) wall T s T F (3.18) The division by λ and the multiplication by the characteristic length L 0 lead to a dimensionless expression. This expression was previously introduced as the Nusselt number (see section 2.1.5) Nu := hl 0 λ = ( T y ) wall T s T F (3.19)

35 26 The relation of the Nusselt number and the thermal boundary layer is comparable with the relation of the friction coefficient and the velocity boundary layer. This local Nusselt number still depends on the local energy exchange coefficient h. The averaging over the surface of the body leads to an averaged Nusselt number Nu m, independent of the spatial variable x*. Finally the averaged Nusselt number Nu m only depends on the Reynolds number Re, the Prandtl number Pr and the geometry. Nu m := hl 0 λ = f(re L, P r, Geometry) (3.20) Averaged Nusselt number for packed spheres For many geometrical problems the literature offers empirical equations for the calculation of the averaged Nusselt number Nu m. As already mentioned, it is assumed that the soil matrix consists of circular spheres. For forced flow around a sphere Baehr and Stefan (1996) introduced the following equations for the calculation of Nu m : Nu m,sphere = 2 + Nu 2 m,lam + Nu2 m,turb Nu m,lam = 0.664Re 1/2 P r 1/3 Nu m,turb = 0, 037Re 0.8 P r Re 0.1 (P r 2/3 1) (3.21) The application of the equation in this form is only valid for a single sphere. The use of the derived Nusselt number in porous media requires additional modifications. In this work it is assumed that all spherical particles in the soil have the same diameter d p. Qualitatively, the values of the Nusselt number in a packing of spheres is larger than for flow around a single sphere. The frequent change of flow direction and the so caused changed miscibility is responsible for that. The relation between the Nusselt number of a single sphere and the one for a spherical packing is an arrangement factor which depends on the porosity. The following equation is used to calculate the Nusselt number for a packing of spheres. Nu m = f φ Nu m,sphere (3.22) According to Schluender (1972) the following equation for the arrangement factor f φ holds for a range of the porosity 0.4 < φ < 1 and only for creep flow. f φ = (1 φ) (3.23)

36 3.2 Types of energy transfer 27 Figure 3.4 show a simple example which illustrates that the packing of the spheres influences the porosity and thus the arrangement factor. Interfacial area of the solid phase Regarding a energy flux q s with the dimension J/sm 2, it is obvious that the energy transfer is proportional to the interfacial area between the two phases. The evaluation of the interfacial area between two moving fluids is a very complex task because the interfaces are moving and can form different shapes. In the scope of this work it was not possible to include the computation of these areas. Therefore additional simplifications are used. According to the work of Wang and Beckmann (1993)the fluid phase can be regarded as a mixture of water and air phase. Inside this bulk phase, equal temperatures are assumed and the two fluid phases have the same shared interface to the solid phase. This simplification is only used for the calculation of the interfacial energy transfer. Mass transport and convective energy transport are calculated separately for each phase. This leads to some new averaged quantities, which depend on the composition of the fluid mixture phase. With this simplification the only interfacial exchange takes place from the immobile phase (soil matrix s) and the introduced mixture phase (mixture phase m). By the means of energy transport the mixture fluid phase will hereafter be mentioned as the fluid phase, including the wetting and non-wetting phase. The size of interfacial area in this case only depends on the porosity and the shape of the single soil grains. It does not depend on time, because the soil matrix is assumed as immobile. To characterize the soil matrix, equally sized spheres of a diameter d p are assumed. Another important parameter is the porosity, which is changing for different types of packing. The influence of the packing order is illustrated in Fig. 3.4 Figure 3.4: Packing types with different porosities The porosity is a sufficient parameter to characterize the packing of the spheres. The volume of the solid phase V s is found from the number of particles and the volume of a single particle, V s = NV p. With the definition of the porosity (see section 2.1.2) and the volume of a single sphere V p it is possible to calculate, the number of particles per volume.

37 28 The number of particles per volume n V 1 φ = V s V tot = NV p V tot (3.24) is expressed as: n V = N V tot = 1 φ V p (3.25) The specific interfacial surface area between the particles and the mixture phase is than defined as a p = NA p V tot (3.26) with A p as the surface area of a single particle. The SI unit of the specific area is m 2 /m 3. It is the characteristic property for the packing of the spheres. Including equation 3.25 it follows that: a p = NA p V tot = A p V p (1 φ), (3.27) in which A p /V p is the specific surface area of a single particle. This surface-area-tovolume ratio can be calculated for every object and for a sphere this ratio is the smallest of all geometrie objects. For this reason one has to keep in mind that the interfacial area calculation used in this work is underestimating the real area of non-ideal grain particles: A p V p = πd2 p πd 3 p 6 = 6 d p (3.28) Inserting this in equation 3.27 it simplifies to: a p = 6 d p (1 φ) (3.29) The importance of the sphere diameter is illustrated in figure 3.5 where the specific interfacial area is plotted over the diameter with a fixed porosity of 0.3. It shows clearly that the area available for the energy exchange can become very large. For this reason the grain diameter is one of the most important parameters, which influences the applicability of the assumption of thermal equilibrium.

38 3.2 Types of energy transfer specific area specific interfacial area [m^2/m^3] Particle diameter [m] Figure 3.5: Specific interfacial area for different sphere diameters

39 Chapter 4 Mathematical model In this chapter the mathematical model is described. It describes the basic formulations for the conservation of mass, momentum and energy. To achieve this, balance equations for each component and the energy are set up. These include the model assumptions made in the previous chapters. 4.1 Conservation of momentum For flow in porous media with small Reynolds numbers, the momentum balance is not solved. Instead, as a simplification, Darcy s law is used for the calculation of the velocity distribution in the porous media. An adapted form of Darcy s law for multiphase flow can be found in Helmig (1997). It accounts for the mutual influence of the different phases and is defined as: with the parameters described in chapter 3. v α = k rα µ α K( p α ρ α g) (4.1) 4.2 Conservation of mass For the conservation of mass it is necessary to set up one balance equation for each component. The equations are solved for every single control volume. For the detailed description of the following multiphase balance equation (see Helmig (1997)) which is set up as follows:

40 4.3 Conservation of energy 31 α φ ρ αxα c div{ k r α ρ α X } {{ t } µ αk( p c α ρ α g)} div{deffρ c α Xα} c q c = 0 α α α storage } {{ } } {{ } advection diffusion (4.2) In equation 4.2 the momentum balance, in the form of the Darcy velocity is already included in the term for the advective transport. 4.3 Conservation of energy General For non-isothermal processes the temperature, as an additional primary variable, is necessary. Therefore an additional balance equation for the energy is set up. Therefore the first law of thermodynamics, which states that in a closed system there is no change of energy, is used. The total energy of a system is only changed by a flux over the system boundary. Hence a balance equation for the specific internal energy in a REV inside a porous medium is set up. The energy of a system is composed of the internal energy, the kinetic energy and the potential energy. For the modeling of processes inside the system E kin and E pot are no considered. Both are connected to the movement of the whole system itself. For a resting System we assume the total energy of the system to be equal to the internal energy of the molecules inside the system. E = U = G e dg (4.3) The processes considered in the model are energy storage, volume changing work and different energy transport mechanisms. Dissipation work and kinetic energy of the moving molecules are neglected due to the small velocities and viscous forces. This leads to the formulation of the first law of thermodynamics: G ( de dt dg = dwv G dt + dq ) dt dg (4.4) It contains three terms the change of the internal energy in the system ( de/dt), the volume changing energy (w v /dt) and the energy flow (dq/dt). In the first term, the specific energy e of a system is the sum of the kinetic energy and the internal energy of the fluid. For small velocities, usually found in porous media the

41 32 kinetic energy can be neglected. Therefore we assume e = u. The change of the specific internal energy ( u/ t) can be formulated with the Reynolds transport theorem which is described in detail in Helmig (1997) du dt = d G dt u dg + u(v n)dγ (4.5) Γ reformulated the third term with the Gauss s theorem: du dt = d G dt u dg + (uv)dg (4.6) G The second term of equation 4.4 is the volume change energy which is mainly important for compressible fluids. It is the amount of work which is needed/released by compression/expansion of a fluid. Density changes are considered for both fluids. For more details see Helmig (1997). The following equation is defining the volume change energy for the volume G: G dw v dt = dg 1 1 (vp)n dγ = (vp) dg (4.7) ρ G ρ The energy transport in the third term (dq/dt) of equation 4.4 is composed of conductive and radiative energy transport. Heat conduction is the diffusive transport of energy between adjacent molecules within one phase. The driving force of the energy conduction is the temperature gradient between two neighboring cells. The energy exchange within a material depends on its physical properties, for example the distance of the molecules and their bonding. On the macroscale this ability of energy exchange is quantified with a energy conduction coefficient for each material. Heat conduction takes place in both fluid phases and in the solid phase. Radiant energy transport is neglected in this work due to the normally low temperatures in the subsurface which leads to small amount of radiative energy transfer. Accounting for this, the energy transport can be described as follows: dq dt = 1 Γ ρ λ T dγ = 1 (λ T ) (4.8) G ρ Now equation 4.4 is reformulated with the three new terms and a term for the source/sinks r is added: G d dt udg+ 1 (uv)dg G G ρ (vp)dg+ 1 G ρ (λ T )dg+ 1 rdg = 0 (4.9) G ρ

42 4.3 Conservation of energy 33 Written in a differential form and with the use of the specific enthalpy h = u + p ρ (see 2.1.2), the following equation for single phase flow can be formulated: d dt u + (ρhv) + 1 ρ (λ T ) + 1 ρ r = 0 (4.10) Model 1 - thermal equilibrium In model 1 local thermal equilibrium is assumed. Only one energy equation is set up. Inside a REV it is assumed that temperatures for the wetting, the non-wetting and the solid phase are equal. Between the cells two energy transfer mechanisms are considered. Figure 4.1 illustrates the schematic model. Due to the local thermal equilibrium no exchange processes are considered inside one REV. Figure 4.1: Schematic energy transfer processes for model 1 To set up the final mass balance equation, equation 4.10 is modified for multiphase flow and the influence of the solid phase is included. Therefore an additional storage term for the soil is included and new energy conduction coefficient considering the presence of the solid phase is necessary

43 34 t (φ ρ α u α S α ) + t ((1 φ)ρ sc s T s ) α } {{ } storage ( K rα ρ α h α K( p α ρ α g)) (λ eff T ) + q µ α α } {{ } α = 0 α } {{ } conduction } {{ } advection source (4.11) This is the final equation used in the existing model. It is a two-phase-two-component model, assuming local thermal equilibrium. It is used for a comparison with both extended model not using the assumption of local thermal equilibrium Model 2 - thermal non-equilibrium Without the assumption of a thermal equilibrium it is now necessary to compute different phase temperatures and the energy exchange between the different phases inside one REV. Therefore a balance equation for the energy of each phase is set up, which is considered separately. In the model the mixture of the both mobile fluids (water and air) is regarded as one bulk phase. This is only considered for the energy transport. According to the existing non-isothermal two-phase two-component model the mass transport of the two phases is still computed separately. For the computation of the energy flow and allocation, one sets up two energy balance equations and two temperatures, T s as the temperature of the soil grains and the bulk temperature T f water and air. It is now necessary to compute the energy exchange between the two phases inside one REV (see Crone et al., 2002). For both energy equations in model 2, a new interfasial energy transfer term is included. In both phases the amount of energy for this exchange is equal in an absolute value, but with contrary sign. This is obvious, as no energy can disappear within the exchange. In DuMu x the term was treated as a source/sink term, which was added to the already included source/sink term. Solid phase energy equation In the separately calculated solid phase three processes are considered. The storage term, the energy conduction and the energy exchange between the other phases. For the storage term we assume the temperature to be homogeneously distributed in the particle and the particle properties are independent for the temperature. This allows setting up the following energy balance equation for the solid phase:

44 4.3 Conservation of energy 35 Figure 4.2: Schematic energy transfer processes for model 2 t ((1 φ)ρ sc p,s T s ) = (h ex,ws a ws + h ex,as a as )(T f T s ) +q s (4.12) } {{ } } {{ } interfacial transfer storage In addition we need a term for the energy exchange between the two phases. This term depends on the fluid temperature T f and the solid temperature T s, the energy exchange coefficient h ex,αs between the two phases as well as on the specific surface area a αs. Regarding equation 4.12 it is easily noticeable that the term for the energy conduction in the soil matrix is missing. In the model described by Crone et al. (2002) this disadvantage is compensated by the modification of the conductivity of the fluid phase. In order to reproduce the physical behavior, the conductivity of soil is included in the fluid phase. With this modification it is possible to achieve physically appropriate overall energy conduction, but it is not possible to simulate energy propagation inside the solid phase without the transport in the fluid phase. Fluid phase energy equation The new energy equation for the fluid phase is quite similar to equation The storage term of the solid phase is detached and moved to the balance equation for the solid phase. In addition we need a term for the energy exchange between the two

45 36 phases. It depends on the fluid temperature T f and the solid temperature T s, the energy exchange coefficient h αs between the two phases as well as the specific surface area a αs. t (φ ρ α u α S α ) α } {{ } storage ( K rα ρ α h α K( p α ρ α g)) + (λ eff T ) + µ α α } {{ } } {{ } conduction advection (h ex,ws a ws + h ex,as a as )(T f T s ) + = 0 } {{ } α interphasial transfer } {{ } source q α (4.13) The energy exchange coefficients h αs and the interfacial areas a αs are defined in section and are strongly dependent on the grain size. Later this will be the main parameter for varying the energy exchange between the solid and the fluid phase. In respect of the conductivity, model 2 shows no differences compared to the equilibrium model. The temperature gradient in the soil matrix is not accounted for in the calculation of the conductive energy transport Model 3 - thermal non-equilibrium and separated soil conduction In model 3, the conduction in the solid phase is calculated separately from the conduction in the fluid phase. This compensates the disadvantages of model 2. Thereby the independent temperature gradients in the fluid and the solid phase are utilized to calculate the conductive energy fluxes. In the case of two separately computed conductive fluxes one has to choose an appropriate approximation for the energy conduction areas for each flux. Like proposed by Postelnicu (2008) we state that the available area is weighted with the help of the porosity: A s = φa scvf and A f = (1 φ)a scvf (4.14) This assumption is inconsistent with the assumption of a soil matrix composed of ideal spherical grains. But this assumption results in a direct contact area close to zero. Therefore we accept this inconsistency of the model.

46 4.3 Conservation of energy 37 Figure 4.3: Schematic energy transfer processes for model 3 Figure 4.4: Schematic energy transfer area for model 3 Solid phase energy equation Compared to both other models the energy balance is extended by a new term for the conduction in the soil matrix. This transport depends on the temperature gradient in the soil, the conductivity of the soil, and the transport area which is appropriated with the help of the porosity (see equation 4.14). t ((1 φ)ρ sc p,s T s ) = (1 φ) (λ s T ) + (h } {{ } } {{ } ws a ws + h as a as )(T f T s ) +q } {{ } s (4.15) conduction interphasial transfer storage

47 38 Fluid phase energy equation The fluid phase equation is the nearly the same as in model 2. The only difference states the weighting of the energy conduction with the porosity, which is caused by the smaller energy transfer areas. In addition one has to keep in mind that in this equation the energy conduction coefficient does not include any more the influence of the solid phase. d t (φ ρ α u α S α ) α } {{ } storage ( K rα ρ α h α K( p α ρ α g)) + φ (λ fl T ) + µ α α } {{ } } {{ } conduction advection (h ws a ws + h as a as )(T f T s ) + q } {{ } α = 0 α interphasial transfer } {{ } source (4.16) The conductivity of the fluid phase λ fl is now calculated with the formula: λ fl = λ air + S w (λ water λ air ) (4.17) Modification of the energy transfer coefficients The separated energy transfer, implemented in model 3 does not lead to an equal effective energy conductivity. Main reason for this is the use of the conductivities of the chemical species. These coefficients do not account for the fact, that the phases are no continuous. The averaging of the energy coefficient in model 1 (see section 3.2.3) is based on experimental results. It takes into account that the solid phase is not a continuous phase, but composed of single grains. As this correlation is often used in the literature, we accept this averaging as the best available averaging. In oder to make the two approaches comparable it is necessary to modify the energy conduction in model 3 with the help of a correction factor C. As derived in section the steady-state energy transport for local thermal equilibrium has to be the same in all three models. The model without separated energy transport does not differ in respect of the energy conduction. Therefore we only discuss the differences between separated and averaged energy conduction. For high energy exchange coefficients and an established local thermal equilibrium, the conductive energy transport must be the same in model 2 and 3:

48 4.3 Conservation of energy 39 Q 1 = Q 3 (4.18) Q 1 is the conductive energy flow in model 1 and Q 3 the one in model 3, composed of the two separated flows for the fluids and the soils. λ eff,i A T = C( λ fl φa T λ s (1 φ)a T ) (4.19) As we compare the same boundary conditions, the area A and temperature gradient T is the same. This reduction leads to: λ eff,i = C (φλ F l + (1 φ)λ s ) (4.20) Inserting the calculation of λ eff,i (see section 3.2.1) finally an equation is derived which can be used for the calculation of the correction factor C. C(φ λ F l + (1 φ)λ s ) = (1 ( ) logφ 0.057log( λs λs λn ) S w ) λ n λ n + S w λ w ( λs λ w ) logφ 0.057log( λs λw ) (4.21) After solving equation 4.21 for the correction factor C the equation represents a correction of the averaging method in model 3. This correction factor thereby depends on the saturation and the porosity. The dependency on the water saturation does clearly reveal that this equation is not the description of a physical effect. One can assert that this dependency is less meaningful, as the energy conduction of soil is a material constant. But in fact for a non-continuous solid phase it strongly dependent on the shape and distributions of the grains. For the purpose of this work, the comparability of the different models is essential. A more physical model, which is incomparable with respect to the thermal equilibrium, is not able to give conclusions in respect of the influence of the local thermal equilibrium. The dependency of C = C(S w, φ) is illustrated in figure 4.5.

49 porosity = 0.5 porosity = 0.4 porosity = 0.3 porosity = 0.2 porosity = 0.1 correction fator water sturation Figure 4.5: Dependency of the correction factor C

50

51 Chapter 5 Numerical Model The equations set up in chapter 4, including different partial differential and non-linear equations, can not be solved with analytical methods. The numerical computation is an approximation of the exact solution with the help of discretization and linearization. This transcription process requires special attention to ensure a mass conservative discretization and physical correct solutions, especially at heterogeneities. Beside this, the numerical scheme has to fulfill the following criteria: stability convergence consistency Stability of a numerical scheme ensures that small disturbances do not grow and leads to a divergent solution. Convergency is the criterion which is fulfilled, when the numerical solution is approaching the exact solution for discretization step sizes tend to zero. Consistency means, that the local error is approaching zero, for decreasing temporal and spatial discretization steps sizes (see Helmig (1997)). The standard Galerkin method is a central difference method for discretization and is used in various numerical schemes, like finite differences method (FD), finite element method (FE), and finite volume method (FV). For the time discretization the fully implicit Euler method is used. The spatial discretization is done with the help of the box method, which is described in the next section. 5.1 Box method The box method is a combination of the finite element and the finite volume method. The finite element method is based on the principle of weighted residuals. Historically

52 5.1 Box method 43 used in the field of structural mechanics, finite elements is today more and more used in the field of fluid mechanics. The general idea of the weighted residual method is the approximation of a unknown function with the help of appropriate base functions N 0, N 1,..., N n. For the description we are looking for the solution of a boundary value problem of a unspecific kind. L(u) = div F(u) r = 0 (5.1) The base Functions N 0, N 1,..., N n are chosen in a way that the inhomogeneous and homogeneous boundary conditions are satisfied. ũ = N 0 + n u, in i (5.2) The replacement of the function u by its approximation ũ, only defined at the nodes, leads to a numerical error, as L(ũ) cannot be the exact solution over the entire domain. This defect is denoted by ε. i=1 L(ũ) = divf(ũ) r = ε (5.3) The aim is now to minimize this defect within the domain. Therefore the expression is multiplied with an appropriate norm, to find the coefficients that the integrals over the hole domain G W i εdg =! 0 i = 1, 2,...n (5.4) become zero. The finite element method includes an isoparametric concept (see Helmig (1997))for the generation of the mesh. This is mainly important for the modeling of complex structures in the subsurface like fractures or flow channels. Unfortunately this concept does not provide a local mass conservation, as the defect is only minimized for the whole domain. To compensate this disadvantage the finite element method was combined whit a finite volume method, which provides a local mass balance. A second mesh is integrated in the FE mesh which forms the control volumes for the FV method (see figure 5.1). Inside this control volumes the conservation of all mass and energy components is assured. The use of two different meshes in one simulation lead to the fact that the information is available at different locations. The primary variable like saturation and pressures are defined at the nodes of the FE mesh. Therefore the FE-gradient and the gradients of the primary variables are defined along the edges which connects the nodes. Each edge is separating two elements and is therefore connected with two elements. In addition

53 44 an edge is connected to two sub control volume faces (SCVF) which are separating the control volumes. The integration points are located in the center of this SCVF. There the phase velocities are evaluated and defined. By the help of the Ansatz functions all primary variables are defined in the whole domain and so are the gradients. However some properties like the intrinsic permeability K are only defined at the nodes of the FE-grid. Therefore for the evaluation of the Darcy velocity the intrinsic permeability is calculated for the integration points by harmonic averaging of two adjacent nodes. The output of the phase velocities is also defined for the element nodes. B i portion of element e in Box B i (sub control volume associated with node i) i i e j e j k barycenter of element e k integration points Figure 5.1: Example of a box mesh Class (2007)

54

55 Chapter 6 Model results 6.1 Parameter analysis of the effect on the local thermal equilibrium In order to develop a better aunderstanding of the processes influencing the local thermal equilibrium this section provides an analysis of the influence of different parameters. For the first approach an analytical study is used to evaluate the governing parameters. Afterwards a numerical analysis is done to compare the results and to examine further parameter Analytical analysis For an analytical examination of the assumption of local thermal equilibrium the first step is to derive a characteristic value. For this it is important to derive a value that gives a statement on the quality of the applicability of the assumption of the local thermal equilibrium The important competitive processes are the advective energy transfer and the interfacial energy transfer. The advective transfer is mainly responsible for the disturbance of the thermal equilibrium inside a REV. The energy exchange between the phases equalizes temperature differences and is therefore working towards the local thermal equilibrium. At a given difference of the fluid and the solid phase temperatures, it is possible to derive a characteristic time, which is needed for the system to reach thermal equilibrium. The fluid temperature is therefore assumed to be constant. The temperature differences will generate an energy flux between the phases until the solid phase has reached the temperature of the fluid phase. The kinetic of the process is not considered and the energy transfer is assumed to be constant over the whole balancing process. The characteristic time is therefore the ratio of energy transferred in

56 6.1 Parameter analysis of the effect on the local thermal equilibrium 47 the whole process E s and the energy flow per second between the phases q ex. Strong simplifications are included in the derivation of this characteristic time. Especially since the temperature difference is decreasing during the balancing process, the real time for the balancing is certainly longer. In addition the presence of the non-wetting phase is neglected because the same behavior is expected for a mixture phase. t ch = E s = (1 φ)cs pρ s V T q ex h ex A p V T = (1 φ)cs pρ s h ex a p (6.1) Both processes, the advective energy transport and the interfacial energy transport, take place at the same time. The characteristic time t ch is now multiplied with the seepage velocity v a = q/φ to get a characteristic width of the front in which the thermal non-equilibrium occurs. b ch = v a t ch (6.2) Both the characteristic time t ch and the characteristic width b ch are no approximations of real values, but only qualitative properties to compare the influence of both processes mentioned above. Additionally the influence of the thermal conduction is not included as the convection is the dominating process. Apart from these simplifications the characteristic width is an indicator for the size of the area which is in thermal non-equilibrium. Therefore it is used to quantify the influence of the thermal non-equilibrium. The varied parameters are prepared in table 6.1. parameter standard case reduced case increased case permeability K porosity φ particle diameter d p 5 mm 1 mm 10 mm Table 6.1: Varried parameters in the analytical analysis To evaluate the influence of the parameters a standard case is computed first. For the following case only one parameter is changed at once. The results of the analysis are presented in figure 6.1. The characteristic width t ch is shown in the first row to allow the comparison of the different regarded cases.

57 48 Figure 6.1: Results of the analytical analysis Obviously the intrinsic permeability is mainly influencing the convective flow velocity. Therefore lower K values lead to lower convective flow, and so to a reduced characteristic width. This means that the thermal non-equilibrium is limited to a smaller area. The results of the variation of K is shown in the second and third row in figure 6.1. The next two rows show the results for the variation of the porosity φ. The porosity directly influences all energy transport processes. For a lower porosity a higher seepage velocity is observed. In addition the specific interfacial area between the fluid and the solid phase is increased. At least the higher seepage velocity causes an increase of the average energy exchange coefficient, by means of higher Reynolds numbers. Summarized, all process are increased by a decreased porosity, but the increase of the characteristic width indicates that the influence of the higher advective transfer is dominating. The highest influence is found for the variation of the particle diameter d p. It directly influences the energy exchange coefficient, the interfacial area between the phases and also the Reynolds number. A reduced diameter of the soil particles leads to higher Reynolds number and therefore higher exchange coefficients. In addition the interfacial area increases with a smaller diameter. Both influences lead to an increased energy exchange and with that to a smaller characteristic width. The influence of the particle diameter is illustrated in the last two rows of figure 6.1. The values of the characteristic time presented in table 6.2 all exceed one second except in the case of case of the small grain diameter. There the balancing process is significantly increased. The slow velocities, which are used for the calculation of the characteristic width of the disturbed area, show that the water front is not moving a

58 6.1 Parameter analysis of the effect on the local thermal equilibrium 49 parameter characteristic time characteristic width seepage velocity [s] [m] [m/s] standard case K = K = φ = φ = d p = 1mm d p = 10mm Table 6.2: Characteristic values of the analytical analysis significant distance in this characteristic time. The time values need to be seen in the context of the very slow propagation velocities Numerical analysis In addition to the analytical analysis the numerical model also give the opportunity to analyze the effects of different parameter variations. This method is not limited to the parameters which are included in the examined formulas used in the analytical method. Besides the assumptions for the derivation of the formulas are not necessary. Additionally it is possible to quantify the influence of numerical factors like the spatial and temporal discretization. All varied parameters are presented in table 6.3. parameter standard case reduced case increased case permeability K porosity φ energy conductivity soil λ partical diameter d p 5 mm 1 mm 10 mm vertical spacial discretization x 0.05 m 0.1 m m maximum time step size 100 s 10 s Table 6.3: Variation of the parameters In contrast to the analytical analysis the numerical analysis is strongly dependent on the computed example. The examined process for this analysis is the infiltration of cold water into hot soil. The soil parameter in this case is homogenous in the whole domain. The initial temperature of the soil is 50 K above the temperature of the infiltrating water. On top water is infiltrating the domain at a Dirichlet boundary, on the

59 50 bottom at the other Dirichlet boundary the fluid flow is leaving the domain. On both sides the domain is limited by Neumann no flow conditions. All processes are acting in vertical direction; therefore the spatial discretization in horizontal direction is not important and is not varied in this study. For the analysis, the differences in the results of both non-equilibrium models compared to the equilibrium model, are used as the indicator for the influence of the local thermal equilibrium. This is a valid approach, as the only difference is the dealing with the equilibrium assumption, which is used in the equilibrium model. In both other models, the temperature of the phases is explicitly calculated. Therefore the differences between the model can be lead back to the thermal equilibrium. Two primary variables of the simulation are selected to illustrate the differences between the models. The first one is the fluid temperature as it is the primary variable which is directly influenced by the thermal equilibrium. In addition the saturation is chosen because it is indirectly influenced. The reason for that are the temperature dependent properties of water like the viscosity and the density. In this mainly advective influenced problem, the main differences occur in the area of the saturation front. Ahead of the front the differences in the models are zero, as these cells are not yet affected, by the water propagation. This behavior is clearly visible in figure 6.3 which is illustrating the model differences for the standard case. The color bar is modified in the manner that the area with no differences is white. Blue areas are showing positive differences which mean that the regarded value is higher compared to the equilibrium model. Red areas indicate lower values of the compared model to the equilibrium model. For the aim of illustrating shortly the processes and the occurring differences the result of standard case is discussed. For the non-equilibrium case the energy exchange between the phases is retarded. In the case of cold water infiltrating the hot soil, the warming of the fluid phase is slightly slower. Therefore the fluid in figure 6.2 is showing that the fluid in the red area remains colder than in the equilibrium case. On contrary the soil material remains hot for a longer period. For this fact the temperature behind the temperature front is increased. The reason for this is the fact that the soil in this stage has more energy stored and is therefore able to transfer more to the fluid phase. The indirect influence of this slight temperature differences on the water saturation is shown in figure 6.3 which illustrates the distribution of the water saturation. First obvious fact is that the front propagation is slightly slower. This results in a very sharp area where the saturation has not yet reached the value of the equilibrium case in model 1. It is followed by a slim area, where the saturation is higher in the non-equilibrium case. These are the two areas where the maximal differences occur inside the domain (see figure 6.3). Behind the front another smooth jump is found. It is at the same location as the temperature front (compare figure 6.2). Upstream of the area the fluid phase has

60 6.1 Parameter analysis of the effect on the local thermal equilibrium 51 Figure 6.2: Comparison of the fluid temperature in the standard case reached the same temperature as the solid phase. No thermal exchange takes place any more. Downstream (upper area of the temperature front) the soil matrix is cooled to the temperature of the infiltrating fluid. Again the exchange process is balanced. The area between is the place where the model differences caused by the local thermal equilibrium arise. In the areas of cold water the reduced viscosity and increased density cause a higher water saturation which is balancing the mobility of water to ensure the same water transport. This transition in the temperature also effects the saturation transition which can be seen in the upper part of figure 6.3. Behind the front the compensation processes reduce the differences with increasing distance to the front. For this reason an average difference over the whole domain is always dependent on the location of the front. For the aim not only to compare timesteps with the same location of the front, the mean differences are not regarded in this work. In contrast the maximum absolute difference is independent of the front location, which allows the accounting of all timesteps and front locations. To compare the simulation cases to the standard case an average of the occurring maximal difference of all timesteps is evaluated. For the presented standard case, this difference is the reference value ε norm. To evaluate the influence of the single parameter variations of the standard case are simulated. In each case only one parameter is changed and the differences between the models are compared to the standard case. The differences ε are standardized with the following formula:

61 52 Figure 6.3: Comparison of the water saturation in the standard case ε rel = ε norm ε ε norm 100% (6.3) The results of the parameter analysis is presented in figure 6.4. The parameters can be divided in two classes, physical input parameters and numerical parameters. Thereby numerical parameter, as time step size and spatial discretization, are not changing the simulated condition, but only the accuracy of the calculation. The variation of the intrinsic permeability K is showing the expected influences known from the analytical analysis. For a higher permeability, the fluid velocities are increased, and therefore is the convective transport. As the convective is the major process which is violating the local thermal equilibrium, higher velocities lead to increased differences for the non-equilibrium models. The porosity φ is influencing all energy transfer processes. For higher porosities, the decreased pore space is leading to faster seepage velocities and therefore to a higher advective energy flux. A higher value of the porosity is also leading to an increased conductive energy flux by the soil matrix. At last the porosity is also influencing the energy exchange between the phases trough the energy transfer coefficient and the interfacial area, both are depending on the porosity. All these different factors complicate a prediction of the influence of the parameter. First the reduction of the porosity effects a higher difference in the saturation. The temperatures in contrast show a reduction of the difference. This unexpected behavior is discussed later. For a higher porosity the differences for both variables drop. This is expected, as the velocity

62 6.1 Parameter analysis of the effect on the local thermal equilibrium 53 is reduced due to the increased void space. In addition the energy storage capacity is reduced which causes a faster balancing. The conductivity λ s of the soil matrix is also varied in the range of physical properties of different sand species. In contrast to the permeability, the conductivity is quantifying the diffusive like process of energy transport in the soil matrix, which is compared to the other energy fluxes of small influence. Therefore the influence of the conductivity on the validity of the local thermal equilibrium is negligible. For this reason the graph 6.4 shows negligible differences to the standard case. Figure 6.4: Results of the numerical study The diameter d p of the soil grains is the parameter with the biggest influence on the interfacial energy exchange. First it is mainly influencing the interfacial area between the phases and it is also used for the calculation of the energy exchange coefficient h (see section 3.2.1). A small particle diameter leads to a decrease in the model differences, mainly because of the increased interfacial area. An increased area between the phases leads to a higher energy exchange between the phases. Therefore the temperatures differences are reduced. In contrast a bigger diameter leads to larger differences. The results indicate that the particle diameter has the biggest influence compared to all other examined parameters. At last the influence of two numerical parameters is examined. The better temporal

63 54 discretization is leading to smaller differences between the models. The main reason for this is the better temporal approximation of the energy exchange process, once a REV is disturbed. A variation including bigger timesteps as in the standard case was not possible as in the standard case the simulation already used the maximal possible timestep. In addition the observation timesteps in which the results are saved needed to be the same in the whole analysis to be able to compare the results and can not be enlarged. In the case of bigger spatial discretization the differences in the models are reduced. Most likely the reason for this is that the sharp area of large differences is simpley averaged over a bigger REV and therefore the value is smaller. The finer discretization is not significantly affecting the differences in the temperature. But in the saturation an unexpected increase of the difference is found. The unexpected behavior, for a small porosity and a fine spatial discretization, need to be discussed separately. The water saturation in all examples is only dependent on the local thermal equilibrium in an indirect manner by temperature dependent parameters. For this reason it is expected that large differences are direct consequences of large temperature discrepancies. This behavior was found in general for the parameter variation. Two exceptions are observed, one for a porosity of 0.1 and another for the fine spatial discretization. In both cases the differences in the saturation is increased, whereas the temperature does not show a similar behavior. In order to describe this unexpected behavior of the saturation the different kind of both parameters has to be accounted for. The temperature is a intensive property. It does not depend on the size of the REV. In contrast the saturation is a volume averaged quantity connected with the phase volume. In both cases, for a reduced pore space and a finer discretization, the total void volume of a single REV is reduced. For this reason an assumed change in the mass is causing a bigger effect on the saturation in this cases, whereas the temperature is unaffected. 6.2 Example: Infiltration in hot soil The first examined process, relating to the influence of the local thermal equilibrium is the water infiltration into a hot subsurface. In nature an example for this is a heavy rainfall event in desert regions. The poor heat conductivity of the subsurface, composed of sand grains and air filled void space, cause a high temperature in the top soil layer due to the solar radiation. The soil in dessert regions can therefore reach a temperature of up to 60 C and more. For a rainfall event the water is infiltrating the hot soil. In this case the thermal non-equilibrium and the separated calculation of the phase temperature can cause differences in the front propagation and front shape. The retarded balancing of the temperatures cause that a cold water front entering the hot surface remains colder.

64 6.2 Example: Infiltration in hot soil 55 For this example a rectangular model domain is created. Initially a water saturation of 0.01 is set for the whole domain. This is necessary for numerical reasons. This initial water saturation prevents the need of the change of the primary variables, which can cause additional numerical influence on the results. As the occurring differences between the models are anyway small, this additional influence is as far as possible avoided. The initial temperature of the soil was set to 60 C. Figure 6.5: Conceptional setup of the infiltration examples Boundary conditions The boundary conditions in this problem are shown in table 6.4. On top of the domain a Dirichlet condition was set for the water to infiltrate. A second Dirichlet boundary is set on the bottom. On the right and the left side Neumann no flow boundary conditions prevent any flow. The domain, as shown in figure 6.5, consists of two soil types, which the soil parameters described in table 6.5. The right side consists of the more permeable soil B. The higher porosity of soil B is indicating bigger pores in the soil, so a different dp c /ds e -relation is implemented. Thereby the entry pressure is equal in both areas but the slope dp c /ds e is different. The higher soil volume also implies a higher energy conductivity and energy storage capacity for soil B. Results of the infiltration example This infiltration example is a more complex model of the already discussed model in the parameter study. Therefore the occurring influences of the non-equilibrium similar.

65 56 Orientation Neumann Dirichlet top - S w = 0.5 p n = 10000P a T = K T s = K left side no flow - right side no flow - top - S w = 0.01 p n = 10000P a T = K T s = K Table 6.4: Boundary condition of the infiltration example type permeability porosity capillary pressure S wr S nr Soil A p d = 10000, α = Soil B p d = 10000, α = Table 6.5: Soil parameter for the infiltration example The retardation of the thermal balancing cause a slower heating of the infiltrating water, which is illustrated in figure 6.6. The temperature distribution in the equilibrium case is shown in the left column. The other two columns show both non-equilibrium cases. The red areas show the regions where the water is cooler in both non-equilibrium cases. This leads to an lower viscosity and higher density. However the differences in the non-equilibrium models of a maximum of 0.2 K are not significant and therefore the influence on the fluid properties is excpected to be small. Due to the higher density the occupied volume is smaller and in addition the lower viscosity leads to a slower fluid velocity and a slower propagation of the wetting phase at the front. This effect can be seen in the distribution of the water saturation which is illustrated in figure 6.7. Yet the saturation differences are limited on the area of the saturation front. Upstream the front no influence of the non-equilibrium is found. At the front, the difference is slowly increasing over the simulation time and reaches a maximum of This is a difference of less the one percent of the water saturation of the infiltrating water.

66 6.2 Example: Infiltration in hot soil 57 Figure 6.6: Fluid temperature distribution for the infiltration example Figure 6.7: Distribution of the water saturation for the infiltration example

67 Example: Evaporation in the subsurface The second examined process in the unsaturated zone is the direct evaporation of water from the subsurface to the atmosphere. As mentioned in dessert zones, high solar radiation can cause high temperatures in the top layer of the soil. Therefore it is possible that the local thermal equilibrium also affects the evaporation of water and the thereby implied water transport in the unsaturated zone. Especially the effect of an inhomogeneous soil with large variations in the permeability, energy conductivity and energy capacity may cause differences for the non-equilibrium case. The aim is to get a better understanding of the dominating processes in the unsaturated zone and the potential influence of the local thermal equilibrium. Therefore the simulation is done on a small scale which gives the opportunity for better adjustment of the different parameters and a better analysis of the occurring effects. On top of the domain a small part of the atmosphere is included. This is done as an approximation of the influence of the connection between the unsaturated zone and the atmosphere (see figure 6.8). The flow of air in this small atmospheric layer is also calculated with the Darcy approach. Therefore also all soil parameters need to be defined in this area. These parameters are set in a way to mainly prevent an influence of these parameters. For this reason the calculation of the free flow is not exact and is only a limited approach. However it is not the aim to model the processes in the air layer, but in the subsurface below. Primary the air channel is included to prevent that the upper boundary condition causes a strong influence on the examined system. Figure 6.8: Conceptional setup of the evaporation example

68 6.3 Example: Evaporation in the subsurface 59 orientation Neumann Dirichlet atmosphere layer left side - p n = Pa X wn = T = K T s = K left side - p n = Pa X wn = T = K T s = K top no flow - subsurface left no flow - right no flow - bottom no flow - Table 6.6: Boundary condition of evaporation example Boundary conditions In the subsurface no flow boundary conditions for all phases and energy are set at all borders. The upper part of the soil is coupled to the atmosphere layer. For the border on top of the atmosphere, no flow conditions for all variables are set. On the left and the right side of the layer Dirichlet boundary conditions are set to create a constant air flow along the surface. The values are prepared in table 6.6. This setup leads to an air flow in the channel which is important for the transport of the evaporated water out of the model domain. The conceptional setup of the soil composition was adopted from an experiment. The exact geometry of the experiment cause problems in the calculation and is not used for the calculation. However the used geometry is similar to the experiment. Unfortunately no published results of the experiment are available during this work. The soil consists of two areas filled with spherical grains of different diameter. In the tortuous area A grains with a small diameter are used. This results in a lower permeability and porosity and in higher capillary pressures in area A. The area around is filled with spheres of a bigger diameter. This arrangement is illustrated in figure 6.8. The results of the experiment indicate the existence of a capillary drag in the fine material due to the high capillary forces. There are several aims for this example: First of all and mainly related to this work, the influence of the assumed local thermal equilibrium is examined. For the second it is tried to simulate the upward flow of water to the surface in the finer material as indicated in the experiment.

69 60 type permeability porosity capillary pressure S wr S nr [m 2 ] [-] [-] [-] [-] atm. layer p d = 0, α = soil A p d = 15000, α = soil B p d = 10000, α = Table 6.7: Soil parameter for the evaporation example Results of the evaporation example The simulation period for this example is twelve days. Figure 6.9 shows an overview of the drying of the model domain due to the evaporation. It is clearly visible that in the simulation the evaporation mainly occurs in the left area of the surface, where the air flow is entering the system. This is mainly influenced by the boundary conditions. In addition the turbulent behavior can not be reproduced by this model as in the whole domain Darcy s law is used. The majority of the streamlines enter the subsurface at the left side which cause the high evaporation of water in this area. The simulation results in figure 6.9 are extracted of the equilibrium model.

70 6.3 Example: Evaporation in the subsurface 61 Figure 6.9: Overview of the simulation progress The simulation of the upward water flow in the fine material, indicated by the experimental results, was not completely possible for the setup of parameters described above. Main reason for this is three magnitudes higher intrinsic permeability in the fine material. It mainly prevents any flow in this area. Only a very slow balance process from the right to the left side is found. This is the case because the main evaporation occurs on the left side and lead to a saturation gradient from the right to the left. For this reason the simulation is also done with a homogenous intrinsic permeability in the whole subsurface. For this setup the capillary upward drag occurs. The velocity distribution is illustrated in figure In the lower area water moves in the fine material, and then it is transported upwards. In the horizontal segments the water flow again follows the flow path in the fine material. In the top segment of the fine material the water leaves to the surrounding area, mainly in the direction to the left where the main evaporation takes place. The result of this simulation indicates that the capillary drag also takes place in the model with the low permeability, but due to the drastically reduced velocity this effect is completely overlapped by the flow between the two high permeable areas and is not significant.

71 62 Figure 6.10: Velocity distribution of the water phase for homogenous permeability Another possible reason for the absence of the capillary drag is the implementation of the capillary pressure. Unfortunately no data for the capillary pressure is known. Therefore it is necessary to assume a proper set of parameter for the capillary pressuresaturation relation. In addition the implementation of the capillary pressure is not able to exceed the phase pressure of the primary variable. Reason for this is the use of the capillary pressure as a constitutive relation to calculate the second phase pressure. In contrast, the microscopic capillary pressure is able to exceed the atmospheric pressure. These high capillary pressures can not be reproduced in the simulation. To prevent negative pressure values, the capillary pressure is limited to values below the primary pressure variable in this case the pressure of the non-wetting phase p n. At low saturations when the limitation is reached, a difference in the saturation does not longer lead to differences in the capillary pressure. In this range the value of the expression dp c /ds e is zero and so no capillary influence is found. Differences of the non-equilibrium models The main influence of the non-equilibrium is found in the temperatures. Temperature dependent values, especially in this case, the maximum water content in the nonwetting phase are influenced indirectly. Figure 6.11 show the temperature distribution

72 6.3 Example: Evaporation in the subsurface 63 in the equilibrium model and in addition the differences found in the non-equilibrium models. Thereby blue areas show of lower temperatures in the non-equilibrium case. In the red areas the non-equilibrium models reach higher temperatures. Figure 6.11: Fluid temperature distribution in the evaporation example For a better examination of the differences between the three model concepts the evaporation rate during the simulation is presented in figure Regarding the evaporation rate, different stages of the simulation can be distinguished. In the first stage the hot air stream is heating the subsurface and the included water. This causes a fast rise of the evaporation rate. After twelve hours the upper layer is heated. During the next stage the water saturation slowly reduces. This causes a higher mobility of the non-wetting phase and therefore an increased air flow in the subsurface. This causes a steady increase of the evaporation rate. The evaluation rate reaches it maximum around hour eighty-five. From this point in time the rate is decreasing due to the drying of the upper layer of the subsurface. The more layers are falling dry the more air streamlines do not cross an area containing water anymore. For this fact along these streamlines no evaporation occur anymore. The only water transfer from one to another streamline is the diffusive transport, which is slow compared to the advective flow. Around simulation hour 240 the last area above the low-permeable area falls dry. By the time the evaporation rate reaches a very low level because the majority of the flow area is blocked by the low permeable soil. After a simulation time of twelve days, the simulation aborts. Main reason for this is that the pressure gradients get that small, that the Newton solver is not able to find a solution.

73 64 Figure 6.12: Evaporation rate of the evaporation case The evaporation rates for all models are nearly equal. By trend the evaporation rate in the non-equilibrium case is increased, due to the slightly higher fluid temperatures in the top soil layer. However in a period of five days, the differences in the total water mass differ in a range lower than 0.3%. Therefore the results indicate that for this example the use of the thermal non-equilibrium is not necessary. 6.4 Example: Evaporation affected by solar radiation The example for a evaporation process presented in section 6.3 is set up with respect of an experimental study. Now a more realistic model is regarded. In nature constant conditions like in the previous example do not occur. This was the reason to set up a similar evaporation example which is closer connected with the natural occurring directly evaporation from the ground water to the atmosphere. Therefore the solar radiation is included in the model that heats up the top layer of the subsurface in the daytime. This is implemented as a time controlled energy source term for the solid phase. The heat flux in the day period is set to 1000 J/(m 2 s). The soil consists, similar to the previous example, of two soil types, a finer material A and a coarse material B. Again differences in the non-equilibrium models are examined.

74 6.4 Example: Evaporation affected by solar radiation 65 Figure 6.13 is illustrating the setup in principle. Figure 6.13: Conceptional setup of the time dependent evaporation problem Boundary conditions The boundary conditions are similar to the first evaporation example. In the subsurface no flow Neumann conditions are implemented. The upper boundary is coupled to the atmosphere layer. On top of this layer, no flow conditions are set. On the left and the right side of the layer Dirichlet boundary conditions are set. The values are prepared in table 6.8. In addition a time controlled energy source is implemented in the top layer of the soil. The day-night cycle is simplified by two episodes during twenty-four hours. The day period is expressed with a constant energy source. During the night episode no energy is put in the system. As mentioned the soil consists of two different materials. A simplified setup was used for this example: in the center of the soil a column of fine material A is separating two higher permeable areas B. The soil parameters are listed in the following table 6.9.

75 66 orientation Neumann Dirichlet atmosphere layer left side - p n = Pa X wn = T = K T s = K left side - p n = Pa X wn = T = K T s = K top no flow - subsurface left no flow - right no flow - bottom no flow - Table 6.8: Boundary conditions of solar dependent evaporation type permeability porosity capillary pressure S wr S nr atm. layer p d = 0, α = soil A p d = 25000, α = soil B p d = 25000, α = Table 6.9: Soil parameters for the solar dependent evaporation

76 6.4 Example: Evaporation affected by solar radiation 67 Results of the solar dependent evaporation example This example is simulated for a period of six days. The Output of the interested variables is generated every hour. The simulation starts with a 12 h episode with a energy injection. The change of the episodes leads to an oscillation in the temperature in the upper area of the soil. This temperature oscillation is plotted for one node located in the fine area right under the surface (for the location see 6.13). The curve for the node is illustrated in figure Around s the water saturation of the node reaches zero. From this moment on the cooling effect of the evaporation does not take place any more in this cell. This leads to the increase of the temperature. This curve is characteristic for all nodes in the system, whereas the temperature jump is dependent on the location of the node. With increasing distance to the energy source the oscillation become smoother. Figure 6.14: Temperature oscillation of a representative node in the system The different stages of the simulation are illustrated in figure During the first eighty-four hours the evaporation rate increases. Reason for this is the decrease of the water saturation in the whole domain and the heating of the soil. Due to the capillary drag the water is transported upwards so that the top layer of the soil still contains water. The water content in the top layer keeps the evaporation rate on a high level.

77 68 From hour eighty-four the top layer dries up. Due to the lower permeability the finer soil material longer contains water in the pores. Figure 6.15: Overview of the simulation progress In addition to the output of the important parameters also the evaporation rate for every timestep is calculated. The change of the water mass in the system is equal to the evaporation in the system, because the water is only transported by the gas flow out of the domain. The mass difference between two timesteps is calculated and divided by the size of the timestep to obtain the average evaporation rate. The temporal sequence of the evaporation rate is shown in figure In this figure the different stages of the simulation can be described. Up to hour eighty-four the evaporation rate reaches its maximum. The reason for this is the decrease of the saturation and therefore the increase of the mobility of the non-wetting phase in the unsaturated zone. For this reason the flow of the non-wetting phase in the subsurface is increasing and with it the transport of water steam. After this period the drying of the top layers of the subsurface cause a radical reduction of the evaporation rate. This indicates that the main evaporation takes place in the top layer. The reason for this is that the air flow is faster in the top of the domain. With increasing depth of the subsurface the velocity of the non-wetting phase decreases.