Examination of the assumption of thermal equilibrium on the fluid

Transcription

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 2189945
Stuttgart, July 2009
Examiners: Prof. Dr.-Ing. Rainer Helmig, Dr.-Ing. Holger Class
Supervisor: Dipl.-Ing Klaus Mosthaf
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
Contents
1 Introduction
2
1.1
Climate modelling
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.2
Purpose of this work . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2 Basic concepts
2.1
Terms and definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2.1.1
Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2.1.2
Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.1.3
Moist air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
2.1.4
Local thermodynamic equilibrium . . . . . . . . . . . . . . . . .
14
2.1.5
Dimensionless numbers . . . . . . . . . . . . . . . . . . . . . . .
15
3 Physical model
3.1
3.2
4
18
Mass transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
3.1.1
Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
3.1.2
Advection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
Types of energy transfer . . . . . . . . . . . . . . . . . . . . . . . . . .
19
3.2.1
Energy conduction . . . . . . . . . . . . . . . . . . . . . . . . .
19
3.2.2
Convective energy transport . . . . . . . . . . . . . . . . . . . .
22
3.2.3
Energy exchange between phases . . . . . . . . . . . . . . . . .
23
4 Mathematical model
30
CONTENTS
I
4.1
Conservation of momentum . . . . . . . . . . . . . . . . . . . . . . . .
30
4.2
Conservation of mass . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
4.3
Conservation of energy . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
4.3.1
General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
4.3.2
Model 1 - thermal equilibrium . . . . . . . . . . . . . . . . . . .
33
4.3.3
Model 2 - thermal non-equilibrium . . . . . . . . . . . . . . . .
34
4.3.4
Model 3 - thermal non-equilibrium and separated soil conduction 36
5 Numerical Model
5.1
Box method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6 Model results
6.1
42
42
46
Parameter analysis of the effect on the local thermal equilibrium . . . .
46
6.1.1
Analytical analysis . . . . . . . . . . . . . . . . . . . . . . . . .
46
6.1.2
Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . .
49
6.2
Example: Infiltration in hot soil . . . . . . . . . . . . . . . . . . . . . .
54
6.3
Example: Evaporation in the subsurface . . . . . . . . . . . . . . . . .
58
6.4
Example: Evaporation affected by solar radiation . . . . . . . . . . . .
64
7 Discussion and outlook
72
List of Figures
2.1
Schematic averaging of the microscale . . . . . . . . . . . . . . . . . . .
5
2.2
Transfer processes in multiphase flow . . . . . . . . . . . . . . . . . . .
6
2.3
Relative permeability-saturation relationship . . . . . . . . . . . . . . .
8
2.4
Interfacial tension and wetting angle . . . . . . . . . . . . . . . . . . .
9
2.5
pc − Sw -relation after Brooks-Corey and Van Genuchten . . . . . . . .
10
2.6
Specific internal energy and specific enthalpy of water . . . . . . . . . .
12
2.7
Evaporation enthalpy of different fluids . . . . . . . . . . . . . . . . . .
12
2.8
Saturation vapor pressure for water after equation . . . . . . . . . . . .
14
3.1
Correlations for the effective conductivity for a water saturated soil . .
21
3.2
Schematic structure of porous contact areas at different saturations . .
22
3.3
Velocity and thermal boundary layer . . . . . . . . . . . . . . . . . . .
23
3.4
Packing types with different porosities . . . . . . . . . . . . . . . . . .
27
3.5
Specific interfacial area for different sphere diameters . . . . . . . . . .
29
4.1
Schematic energy transfer processes for model 1 . . . . . . . . . . . . .
33
4.2
35
4.3
37
4.4
Schematic energy transfer area for model 3 . . . . . . . . . . . . . . . .
37
4.5
Dependency of the correction factor C . . . . . . . . . . . . . . . . . .
40
5.1
Example of a box mesh . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
II
LIST OF FIGURES
III
6.1
Results of the analytical analysis . . . . . . . . . . . . . . . . . . . . .
48
6.2
Comparison of the fluid temperature in the standard case . . . . . . . .
51
6.3
Comparison of the water saturation in the standard case . . . . . . . .
52
6.4
Results of the numerical study . . . . . . . . . . . . . . . . . . . . . . .
53
6.5
Conceptional setup of the infiltration examples . . . . . . . . . . . . . .
55
6.6
Fluid temperature distribution for the infiltration example . . . . . . .
57
6.7
Distribution of the water saturation for the infiltration example . . . .
57
6.8
Conceptional setup of the evaporation example . . . . . . . . . . . . . .
58
6.9
Overview of the simulation progress . . . . . . . . . . . . . . . . . . . .
61
6.10 Velocity distribution of the water phase for homogenous permeability .
62
6.11 Fluid temperature distribution in the evaporation example . . . . . . .
63
6.12 Evaporation rate of the evaporation case . . . . . . . . . . . . . . . . .
64
6.13 Conceptional setup of the time dependent evaporation problem . . . . .
65
6.14 Temperature oscillation of a representative node in the system . . . . .
67
6.15 Overview of the simulation progress . . . . . . . . . . . . . . . . . . . .
68
6.16 Evaporation rate of the three compared models . . . . . . . . . . . . .
69
6.17 Differences of the non-equilibrium model over time . . . . . . . . . . .
70
List of Tables
3.1
Thermal energy conductivities of selected soil types . . . . . . . . . . .
20
6.1
Varried parameters in the analytical analysis . . . . . . . . . . . . . . .
47
6.2
Characteristic values of the analytical analysis . . . . . . . . . . . . . .
49
6.3
Variation of the parameters . . . . . . . . . . . . . . . . . . . . . . . .
49
6.4
Boundary condition of the infiltration example . . . . . . . . . . . . . .
56
6.5
Soil parameter for the infiltration example . . . . . . . . . . . . . . . .
56
6.6
Boundary condition of evaporation example . . . . . . . . . . . . . . .
59
6.7
Soil parameter for the evaporation example . . . . . . . . . . . . . . . .
60
6.8
Boundary conditions of solar dependent evaporation . . . . . . . . . . .
66
6.9
Soil parameters for the solar dependent evaporation . . . . . . . . . . .
66
IV
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.
1.2 Purpose of this work
1.2
3
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 DuM uX 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.
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.
2.1.1
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
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.
REV
micro scale
solid matrix
liquid phase
transition +
averaging
gas phase
Figure 2.1: Schematic averaging of the microscale
2.1.2
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
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.
xcα
mcα
=
mα
with
mα =
∑
mcα
(2.1)
c
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.
7
φ=
Vpore
Vtot
(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.
Sw =
Vw
Vpore
(2.3)
Following this definition the sum of saturation of all phases is unity:
∑
nphases
Sα = 1
(2.4)
α
For the description of multiphase flow it is necessary to introduce the residual saturation Srα . 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:
Se =
Sw − Srw
1 − Srw − Srn
Srw ≤ Sw ≤ 1 − Srn
with
(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 ρf , the gravity g and the
viscosity µ:
Kf = K
ρf g
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:
8
K f α = krα K
ρf g
µ
0≤
with
∑
krα ≤ 1
(2.7)
α
The relative permeability krα 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
Van Genuchten:
n=4.37
m=.77
=.37
relative permeability [-]
0.8
0.6
Brooks-Corey:
=2
Pd=2
Swr=.1
0.4
Van Genuchten
Brooks-Corey
0.2
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
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
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:
σS2 = σS1 + σ12cosα
(2.9)
or
(
γ = arccos
σS2 − σS1
σ12
)
(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 = pw − p n
(2.11)
10
For a capillary pore one can define the capillary pressure which depends on the surface
tension and the pore diameter as follows:
pc =
4σ12 cosγ
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 2.12. Therefore the capillary pressure is rising with the reduction of the saturation Sw . In the case that more water is entering the pore the wetting
phase penetrates larger pores which leads to a decreasing of pc . The capillary pressure
saturation-relation pc = pc (Sw ) 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
5
capillary pressure [*10 Pa]
8
6
4
2
pd
Brooks-Corey
Van Genuchten
1
0.1
0.2
0.4
0.6
0.8
1.0
effective watersaturation [-]
Figure 2.5: pc − Sw -relation after Brooks-Corey and Van Genuchten Helmig (1997)
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
= u + pv
(2.15)
m
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.
h = h(T, υ) =
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
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:
13
w
hα = hgα xaα + hw
α xα
(2.16)
.
2.1.3
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 N2 and in addition carbon dioxide
CO2 , oxygen O2 and argon Ar.
Partial pressure and saturated steam pressure
At a defined temperature and volume the mass mw of the steam in the gas phase is
given by the Ideal gas law for a component:
xw =
pw V
RW T
(2.17)
with pw as the water phase pressure and RW the individual gas constant. The partial
pressure increases with bigger values of xw . 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.
w
pw
g = xg 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 ps = 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 = 8.19621, B = 1730.63
and C = 233.436.
14
Figure 2.8: Saturated vapor pressure for water after equation 2.19.
2.1.4
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 ( Tα = Ts = 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.
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.
2.1.5
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
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 cp and the thermal conductivity λ:
Pr =
µcp
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:
hex L
(2.22)
λ
with the energy exchange coefficient hex , 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 3.2.3. 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.
Nu =
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
3.1.1
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,
qD = −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
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.
3.1.2
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:
qA = ∇c v
(3.2)
For small Reynolds numbers the velocity is proportional to the hydraulic head. The
proportionality factor is the hydraulic conductivity Kf .
v = −Kf ∇h
(3.3)
p
+z
ρg
(3.4)
with the hydraulic head defined as:
h=
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:
vα = −
3.2
3.2.1
krα
K(∇pα − ρα g)
µα
(3.5)
Types of energy transfer
Energy conduction
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
20
Substance
Water
Air
Solid (granite)
Solid (sand)
Solid (quartz sand)
λ in W/ K m
0.598
0.0257
2.8
4.2
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.
q̇ = −λ∇T
(3.6)
The dimension of the energy flux q̇ is J/(m2 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).
21
λef f,max = (1 − φ)λs + φλf l
(3.7)
The lower Weiner bound is the lowest possible conductivity for a serial configuration
of the porous medium (Eq. 3.8).
λef f,min =
λf l λs
(1 − φ)λf l + φλs
(3.8)
Krupiczka (1967) evaluated a relation for the thermal conductivity for single phase flow
in a porous medium, composed of spherical grains:
λef f,α
=
λα
(
λs
λα
)0.280−0.757logφ−0.057log( λλs )
α
(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
upper weiner bound
6
effective conductivity
lower weiner bound
5
Correlation after Krupiczka
4
3
2
1
0
0
0.2
0.4
0.6
0.8
1
porosity
Figure 3.1: Correlations for the effective conductivity for a water saturated soil
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:
λef f = λef f,n + Sw (λef f,w − λef f,n )
(3.10)
and the weighting with the square root of the saturation:
λef f = λef f,n +
√
Sw (λef f,w − λef f,n )
(3.11)
The single phase conductivities λef f,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
3.2.2
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
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.
qenergy,α = Tα ρα cp,α qm,α
3.2.3
(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.
free stream
u
Τ
y
u
velocity boundry
layer
δ (x)
u
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
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 u0 . 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 Ts at the plate surface to Tf 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 q̇s 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.
q̇s = −λf
∂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 qs as follows,
q̇s = h(Ts − TF )
(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:
hex =
−λf ∂T
∂y
Ts − TF
(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 hex is determined by the gradient of the temperature profile at the
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/L0 ,
(3.16)
with L0 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 4T0 , 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 − T0
4T0
(3.17)
Including this modification in equation 3.15 the following formula is obtained:
∗
∂T
λ ( ∂y∗ )wall
h=−
L0 Ts∗ − TF∗
(3.18)
The division by λ and the multiplication by the characteristic length L0 lead to a
dimensionless expression. This expression was previously introduced as the Nusselt
number (see section 2.1.5)
∗
( ∂T
)
hL0
∂y ∗ wall
=− ∗
N u :=
λ
Ts − TF∗
(3.19)
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 N um , independent
of the spatial variable x*. Finally the averaged Nusselt number N um only depends on
the Reynolds number Re, the Prandtl number Pr and the geometry.
N um :=
hL0
= f (ReL , P r, Geometry)
λ
(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 N um . 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 N um :
N um,Sphere = 2 +
√
N u2m,lam + N u2m,turb
N um,lam = 0.664Re1/2 P r1/3
N um,turb
(3.21)
0, 037Re0.8 P r
=
1 + 2.443Re−0.1 (P r2/3 − 1)
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
dp . 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.
N um = fφ N um,Sphere
(3.22)
According to Schluender (1972) the following equation for the arrangement factor fφ
holds for a range of the porosity 0.4 < φ < 1 and only for creep flow.
fφ = 1 + 1.5(1 − φ)
(3.23)
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 q̇s with the dimension J/sm2 , 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 dp 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 Vs is found from the number of particles and the volume of
a single particle, Vs = N Vp . With the definition of the porosity (see section 2.1.2) and
the volume of a single sphere Vp it is possible to calculate, the number of particles per
volume.
28
Vs
N Vp
=
Vtot
Vtot
The number of particles per volume nV is expressed as:
1−φ=
nV =
N
1−φ
=
Vtot
Vp
(3.24)
(3.25)
The specific interfacial surface area between the particles and the mixture phase is
than defined as
N Ap
ap =
(3.26)
Vtot
with Ap as the surface area of a single particle. The SI unit of the specific area is
m2 /m3 . It is the characteristic property for the packing of the spheres. Including
equation 3.25 it follows that:
ap =
N Ap
Ap
=
(1 − φ),
Vtot
Vp
(3.27)
in which Ap /Vp 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:
πd2p
Ap
6
= πd3 =
p
Vp
dp
(3.28)
6
Inserting this in equation 3.27 it simplifies to:
ap =
6
(1 − φ)
dp
(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.
29
4500
specific area
4000
specific interfacial area [m^2/m^3]
3500
3000
2500
2000
1500
1000
500
0
0.01
0.02
0.03
0.04
0.05
0.06
Particle diameter [m]
0.07
0.08
0.09
0.1
Figure 3.5: Specific interfacial area for different sphere diameters
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:
vα = −
krα
K(∇pα − ρα g)
µα
(4.1)
with the parameters described in chapter 3.
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:
4.3 Conservation of energy
31
∑
φ
|
∑
ρα Xαc ∑
k r ρα
c
c
c
−
div{ α Xαc K(∇pα − ρα g)} −
div{Def
f ρα ∇Xα } −q = 0
∂t
µ
α
{z
}
{z
} |α
{z
}
|α
storage
α
advection
dif f usion
(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
4.3.1
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 =
e dG
(4.3)
G
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
dG =
dt
∫ (
G
dwv dq
+
dt
dt
)
dG
(4.4)
∫
It contains three terms the change of the internal energy in the system ( de/dt), the
volume changing energy (wv /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
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
∫
G
∫
d
u dG +
dt
u(v · n)dΓ
(4.5)
Γ
reformulated the third term with the Gauss’s theorem:
du
=
dt
∫
G
d
u dG +
dt
∫
∇ · (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
dwv
=
dt
I
dG
1
(vp)n dΓ =
ρ
∫
G
1
∇ · (vp) dG
ρ
(4.7)
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
∫
Γ
1
− λ∇T dΓ =
ρ
∫
1
− ∇ · (λ∇T )
ρ
G
(4.8)
Now equation 4.4 is reformulated with the three new terms and a term for the
source/sinks r is added:
∫
G
d
udG+
dt
∫
∫
∇·(uv)dG−
G
G
1
∇·(vp)dG+
ρ
∫
1
− ∇·(λ∇T )dG+
ρ
G
∫
G
1
rdG = 0 (4.9)
ρ
Written in a differential form and with the use of the specific enthalpy h = u +
2.1.2), the following equation for single phase flow can be formulated:
d
1
1
u + ∇ · (ρhv) + ∇ · (λ∇T ) + r = 0
dt
ρ
ρ
4.3.2
33
p
ρ
(see
(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
34
∂ ∑
∂
(φ
ρα uα Sα ) + ((1 − φ)ρs cs Ts )
∂t
∂t
α
|
{z
}
storage
∑ Krα
∑
−
(
ρα hα K(∇pα − ρα g)) − ∇ · (λef f ∇T ) +
qα = 0
{z
}
|
µ
α
conduction
{z
}
|α
| α{z }
advection
(4.11)
source
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.
4.3.3
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, Ts as the temperature of the soil grains and the bulk
temperature Tf 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 DuM ux 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:
35
∂
((1 − φ)ρs cp,s Ts ) = (hex,ws aws + hex,as aas )(Tf − Ts ) +qs
{z
}
∂t
|
{z
} |
storage
(4.12)
interf acial transf er
In addition we need a term for the energy exchange between the two phases. This
term depends on the fluid temperature Tf and the solid temperature Ts , the energy
exchange coefficient hex,αs between the two phases as well as on the specific surface
area aαs . 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 4.11. 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
36
phases. It depends on the fluid temperature Tf and the solid temperature Ts , the
energy exchange coefficient hαs between the two phases as well as the specific surface
area aαs .
∂ ∑
(φ
ρ α uα S α ) −
∂t
α
|
{z
}
storage
∑ Krα
ρα hα K(∇pα − ρα g)) + ∇ · (λef f ∇T ) +
(
|
{z
}
µ
α
conduction
|α
{z
}
advection
∑
qα = 0
(hex,ws aws + hex,as aas )(Tf − Ts ) +
|
{z
}
interphasial transf er
| α{z }
(4.13)
source
The energy exchange coefficients hαs and the interfacial areas aαs are defined in section 3.2.3 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.
4.3.4
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:
As = φAscvf
and
Af = (1 − φ)Ascvf
(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.
37
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).
∂
((1 − φ)ρs cp,s Ts ) = (1 − φ)∇ · (λs ∇T ) + (hws aws + has aas )(Tf − Ts ) +qs
{z
} |
{z
}
|∂t
{z
} |
storage
conduction
(4.15)
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 ∑
(φ
ρ α uα S α ) −
∂t
α
|
{z
}
storage
∑ Krα
(
ρα hα K(∇pα − ρα g)) + φ∇ · (λf l ∇T ) +
|
{z
}
µ
α
conduction
|α
{z
}
advection
∑
(hws aws + has aas )(Tf − Ts ) +
qα = 0
|
{z
}
| α{z }
(4.16)
source
The conductivity of the fluid phase λf l is now calculated with the formula:
λf l = λair +
√
Sw (λwater − λair )
(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 3.2.3 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:
39
Q1 = Q3
(4.18)
Q1 is the conductive energy flow in model 1 and Q3 the one in model 3, composed of
the two separated flows for the fluids and the soils.
− λef f,I A∆T = C(−λf l φA∆T − λs (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:
λef f,I = C (φλF l + (1 − φ)λs )
(4.20)
Inserting the calculation of λef f,I (see section 3.2.1) finally an equation is derived which
can be used for the calculation of the correction factor C.
( )0.280−0.757logφ−0.057log( λ s )
√
n
λs
C(φ λF l + (1 − φ)λs ) = (1− Sw ) λn
λn
)0.280−0.757logφ−0.057log( λλs )
(
√
w
λs
+ Sw λw
λw
λ
(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(Sw , φ) is
illustrated in figure 4.5.
40
1.4
porosity = 0.5
porosity = 0.4
1.2
porosity = 0.3
porosity = 0.2
correction fator
1
porosity = 0.1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
water sturation
0.8
Figure 4.5: Dependency of the correction factor C
1
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
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
N0 , N1 , ..., Nn . 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 N0 , N1 , ..., Nn are chosen in a way that the inhomogeneous and
homogeneous boundary conditions are satisfied.
ũ = N0 +
n
∑
u, iNi
(5.2)
i=1
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.
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
∫
Wi εdG =! 0
i = 1, 2, ...n
(5.4)
G
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
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.
portion of element e
in Box B i
(sub−control volume
associated with node i)
Bi
i
i
e
k
j
j
e
barycenter of
element e
k
integration points
Figure 5.1: Example of a box mesh Class (2007)
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.
6.1.1
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
6.1 Parameter analysis of the effect on the local thermal equilibrium
47
the whole process Es and the energy flow per second between the phases qex . 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.
tch =
Es (1 − φ)csp ρs V ∇T
=
qex
hex Ap V ∇T
(1 − φ)csp ρs
=
hex ap
(6.1)
Both processes, the advective energy transport and the interfacial energy transport,
take place at the same time. The characteristic time tch is now multiplied with the
seepage velocity va = q/φ to get a characteristic width of the front in which the thermal
non-equilibrium occurs.
bch = va tch
(6.2)
Both the characteristic time tch and the characteristic width bch 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
permeability K
porosity φ
particle diameter dp
standard case
5 · 10−11
0.3
5 mm
reduced case
2.5 · 10−11
0.1
1 mm
increased case
1 · 10−10
0.5
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 tch is shown in the first row to allow
the comparison of the different regarded cases.
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 dp . 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
parameter
standard case
K = 2.5 · 10−11
K = 5.0 · 10−10
φ = 0.1
φ = 0.5
dp = 1mm
dp = 10mm
characteristic time
[s]
3.61
4.07
3.11
2.46
4.63
0.18
12.45
characteristic width
[m]
6.75 · 10−4
3.81 · 10−4
1.16 · 10−3
1.38 · 10−3
5.20 · 10−4
3.43 · 10−5
2.33 · 10−3
49
seepage velocity
[m/s]
1.87 · 10−4
9.35 · 10−5
3.74 · 10−3
5.61 · 10−4
1.12 · 10−4
1.87 · 10−4
1.87 · 10−3
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.
6.1.2
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
permeability K
porosity φ
energy conductivity soil λ
partical diameter dp
vertical spacial discretization x
maximum time step size
standard case
5 · 10−11
0.3
4.2
5 mm
0.05 m
100 s
reduced case
2.5 · 10−11
0.1
2.8
1 mm
0.1 m
10 s
increased case
1 · 10−10
0.5
8.4
10 mm
0.025 m
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
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
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 εnorm .
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:
52
Figure 6.3: Comparison of the water saturation in the standard case
εrel =
εnorm − ε
· 100%
εnorm
(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
53
is reduced due to the increased void space. In addition the energy storage capacity is
reduced which causes a faster balancing.
The conductivity λs 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 dp 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
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.
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 dpc /dSe -relation is
implemented. Thereby the entry pressure is equal in both areas but the slope dpc /dSe 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.
56
Orientation
Neumann
top
-
left side
right side
top
no flow
no flow
-
Dirichlet
Sw = 0.5
pn = 10000P a
T = 283.15 ◦ K
Ts = 283.15 ◦ K
Sw = 0.01
pn = 10000P a
T = 333.15 ◦ K
Ts = 333.15 ◦ K
Table 6.4: Boundary condition of the infiltration example
type
Soil A
Soil B
permeability
1 · 10−11
5 · 10−11
porosity
0.1
0.3
capillary pressure
pd = 10000, α = 1.2
pd = 10000, α = 2
Swr
0.1
0.1
Snr
0.1
0.1
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 0.044. This is a
difference of less the one percent of the water saturation of the infiltrating water.
6.2 Example: Infiltration in hot soil
Figure 6.6: Fluid temperature distribution for the infiltration example
Figure 6.7: Distribution of the water saturation for the infiltration example
57
58
6.3
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
6.3 Example: Evaporation in the subsurface
orientation
Neumann
atmosphere layer
left side
-
left side
-
59
Dirichlet
top
no flow
pn = 10 1000 Pa
Xwn = 0.005
T = 303.15 ◦ K
Ts = 303.15 ◦ K
pn = 10 0000 Pa
Xwn = 0.005
T = 303.15 ◦ K
Ts = 303.15 ◦ K
-
subsurface
left
right
bottom
no flow
no flow
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.
60
type
permeability
[m2 ]
porosity
[-]
capillary pressure
[-]
Swr
[-]
Snr
[-]
atm. layer
soil A
soil B
1 · 10−3
1 · 10−9
1 · 10−12
0.9
0.3
0.1
pd = 0
, α = 0.0
pd = 15000, α = 1.2
pd = 10000, α = 2.0
0.0
0.1
0.1
0.0
0.0
0.0
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.
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 6.10. 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.
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 pn . 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
dpc /dSe 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
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 6.12. 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.
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/(m2 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.
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.
66
orientation
Neumann
atmosphere layer
left side
-
left side
-
Dirichlet
top
no flow
pn = 10 1000 Pa
Xwn = 0.005
T = 303.15 ◦ K
Ts = 303.15 ◦ K
pn = 10 0000 Pa
Xwn = 0.005
T = 303.15 ◦ K
Ts = 303.15 ◦ K
-
subsurface
left
right
bottom
no flow
no flow
no flow
-
Table 6.8: Boundary conditions of solar dependent evaporation
type
atm. layer
soil A
soil B
permeability
10−3
10−11
10−8
porosity
0.9
0.3
0.1
capillary pressure
pd = 0
,α = 0
pd = 25000, α = 1.2
pd = 25000, α = 2
Swr
0.0
0.0
0.0
Table 6.9: Soil parameters for the solar dependent evaporation
Snr
0.0
0.0
0.0
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 6.14. Around 3 · 105 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 6.15. 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.
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 6.16. 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.
69
Figure 6.16: Evaporation rate of the three compared models
Differences of the non-equilibrium models
In figure 6.16 the evaporation rates of all three models are shown. Both non-equilibrium
models show increased fluctuations mainly in the first forty-eight hours of the simulation. An increased fluctuation was expected in the period of the day and night change.
But the observed oscillation occurs within a period. One possible reason is that the
time step size is that big that a proper reproduction of the heat exchange is hindered.
In general the average evaporation rate is slightly increased in both non-equilibrium
models. Therefore in the non-equilibrium models the water mass is decreasing faster
and the top layer starts to dry out earlier. The initial reason for the increased evaporation rate is the reduction of the energy conductivity in the soil. This is the reason that
in the non-equilibrium the solid phase is heated slower and thus the full conductive
energy transport takes place later. Due to this reduced conduction the energy is not
transported to the deeper surface layers in the same amount as in the equilibrium case.
More energy remains in the top layer and cause higher fluid temperatures and so a
higher evaporation rate. The higher evaporation rate creates two additional effects.
First the temperatures in the fluid phase are reduced due to the latent energy which is
needed for the phase change of the water. In addition the higher evaporation leads to
lower water saturation in the upper layer of the subsurface. This leads to an increase of
the mobility of the non-wetting phase. Due to this lower water saturation and therfore
fewer water occupied pores the air flow in the subsurface is increase and with it the
transport of water steam out of the domain. This is a self increasing effect that causes
70
the fact that higher evaporation rates additionally increase themselves. Needless to
say that this effect is only arise as long as the energy for the phase change is available.
For that reason is is clear that the higher evaporation rate is not only an effect of
the non-equilibrium but also caused by the increase of the mobility of the non-wetting
phase.
This increased evaporation lead to a rise of the differences in the water saturation over
the simulation period. This increase is observable untill the start of the drying up of
the top soil layer. The results indicate that the start of the drying out is connected
with the certain amount of water mass remaining in the system. This point is the same
in all models but is not reached at the same time. For that reason the evaporation rate
of the equilibrium system remains on the high level for a longer period of time until
the system reaches the same water content. Figure 6.17 show the percentage difference
of the water mass in the system. It reaches its maximum when the non-equilibrium
models reach their maximum evaporation rates. Afterwards the rate drops in the nonequilibrium case, while it is still high in the equilibrium case. This leads to a reduction
of the differences.
Figure 6.17: Differences of the non-equilibrium model over time
In this example the local thermal non-equilibrium at least is initiating the higher evaporation rate. To really account only for influence of the non-equilibrium it may be
reasonable to implement a constant supply of water from the bottom layer to compensate the effect of the different water saturations. In that case the system will reach a
steady state, where the evaporation rates can be compared in a better way.
Chapter 7
Discussion and outlook
Discussion
This work covers a range of natural processes. The selected processes are chosen for
the reason that they are expected to show the biggest influence of local thermal equilibrium. For this processes including high temperature gradients and highest realistic
flow velocities are selected. However these velocities are still small and the moderate
temperature gradients in all natural processes in the subsurface make sure that the balancing processes always dominate. Maximum saturation differences in the magnitude
of 10−3 does not justify the significantly increased computation costs. Approximately
the calculation in the non-equilibrium case requires twice as much computer power as
in the equilibrium case.
Important influencing parameter
The analysis of different parameter showed that the validity of the assumption of local thermal equilibrium mainly depends on a few dominant factors. One of it is solid
surface area between the soil and the fluid phase. The second is the convective energy
transport in the fluid phase. The interfacial area is mainly influenced by the diameter
at a given geometry. In this study the diameter of 5 mmm was chosen as big as realistically possible. For smaller diameters in the range of 1mm, the differences between
the models were found in the range of the numerical errors. Therefore for this small
diameter the results need to be accepted as equal as no other statement can be done.
The second important parameter is the permeability of the soil as it is directly connected with the convective energy transport. In addition other parameter that effects
the advective transport are important, like the pressure gradient or the porosity.
Stability
For both non-equilibrium models the results of the primary variables show an additional
fluctuation compared to the normal case. This effect can be seen best in figure 6.16 in
72
73
which the evaporation rates in model 2 and 3 are oscillating and the equilibrium model
does not show this behavior. One possible reason for this is the coupling of the two
temperatures. The fluid temperature influences several constitutive relations like the
fluid properties of water and air. Especially the maximum water content is connected
with the fluid temperature; therefore a temperature change also causes a phase change
of water. Thereby the density difference of water and water steam strongly influences
the pressure and the volume that the fluid occupies. For example the evaporated water
steam has a thousand times lower density and occupies a larger volume. Therefore the
pressure is rising which again influences the maximum water steam content. In addition
the condensation and evaporation goes along with a change of the energy in the fluid
phase. Therefore a phase change of water lead to a change of the fluid temperature.
In the non-equilibrium models the fluid temperature is more decoupled from the soil
temperature and therefore fluctuations only occur in the fluid phase temperature not
in the solid phase. In contrast in the equilibrium model the influence of the solid phase
on the fluid temperature compensates fast changes in the temperature by reason of a
higher heat capacity. In this case the coupling of the phase temperatures lead to an
smoother fluctuation of the temperature and a higher stability
Differences between the two non-equilibrium models
During all stages of this study, both non-equilibrium models showed very similar result.
In the numerical study for the infiltration of cold water (see section 6.1.2) both models
showed different behavior only in one case. Also in all calculated examples model
3 does not show any significant differences to model 2. Main reason for this is most
likely the fact that in general the differences in the phase temperature are less than one
percent so that the calculation of two different temperature gradients for the separated
conductivities does not lead to significantly different fluxes. The second reason is the
domination of the energy convection compared to the conductivity. Hence the increased
complexity of model 3 is not producing significantly better results. Therefore the easier
model 2 can be preferred.
Validity of the assumption of the thermal equilibrium
The temperature which directly depends on the thermal equilibrium is more affected
by the assumption. The influence on the temperature dependent fluid properties like
the density and the viscosity is therefore indirectly influencing the fluid motion and
distribution. But this indirect influence on the water saturation is not significant.
Especially in the examples including evaporation the temperature is also influencing
the water steam content in the gas phase.
Finally the assumption of the local thermal equilibrium was found to be valid in all
examined natural processes. In these cases the more complex calculation of the nonequilibrium does not lead to a significant improvement of the results.
74
Outlook
As mentioned the low flow velocities in the subsurface lead to a domination of the
balancing processes inside a control volume. For this reason the examined cases can be
extended to artificially caused processes in future works. These processes can be the
remediation of a decontaminated soil with the help of steam injection or the simulation
of a heat-pipe. In both cases higher velocities and higher temperature gradients than
in natural processes occur. Especially the high amount of transported energy in water
steam probably leads to an increase of the influence of the thermal non-equilibrium.
Thereby in the non-equilibrium case the hot steam can infiltrate the cold material
deeper before it condensates. Therefore the proceeding of the stream front may also
be influenced.
To improve the model concepts the implementation of the chemical and mechanical
non-equilibrium is reasonable. Especially chemical processes can show significant dependency on the temperature. For the examined evaporation process the kinetics of
the evaporation can be included. This may lead to an increased influence of the thermal non-equilibrium on the evaporation rate. This includes all temperature dependent
processes like biological processes or solution processes.
At last the model concept in this thesis is limited on soil which consists of grains, like
sand. Even in the model the real specific surface areas are underestimated. Compared
to other soil types the specific soil durface area is huge. As the balancing processes
between the phases are proportional to the specific area it causes an amplified temperature balancing. Other soil type, like fractured rock, does not show similarly large
specific areas. Therefore the balancing processes are retarded and the thermal equilibrium assumption may have a bigger influence. In addition higher velocities are found
in these fractures which may also influence the local thermal equilibrium.
At last it is to mention that it was not possible to include a instable front behavior, like
fingering. Therefore it was not possible to examine the influence on the front stability.
The influence on this can examined in a model which is calculating the location and
the shape of the interface between the wetting and non-wetting phase.
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Examination of the assumption of thermal equilibrium on the fluid

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