Download Coupling of Free Flow and Flow in Porous Media

Universität Stuttgart - Institut für Wasserbau
Lehrstuhl für Hydromechanik und Hydrosystemmodellierung
Prof. Dr.-Ing. Rainer Helmig
Master Thesis
Coupling of Free Flow and Flow in Porous
Media - Dimensional Analysis and
Numerical Investigation
Submitted by
Vinay Kumar
Matrikelnummer 2550493
Stuttgart, March 9, 2012
Examiner: Prof. Dr.-Ing. Rainer Helmig
Supervisors: Dipl.-Ing. Klaus Mosthaf, Dipl.-Ing. Katherina Baber
I, Vinay Kumar, hereby certify that I have prepared this Master Thesis
independently and that only the sources, aids and supervisors that are duly noted
herein have been used and/or consulted.
Signature:
Date: March, 9, 2012
Contents
1 Introduction
1.1 Objective and Structure of the Work . . . . . . . . . . . . . . . . . . .
2 Definitions
2.1 Representative Elementary Volume (REV) . . . . . .
2.2 Concepts Pertaining to Fluids . . . . . . . . . . . . .
2.2.1 Density . . . . . . . . . . . . . . . . . . . . .
2.2.2 Stresses and Deformations . . . . . . . . . . .
2.2.3 Newton’s Law of Viscosity . . . . . . . . . . .
2.2.4 Viscous Flow and The Navier-Stokes Equation
2.2.5 Stokes Flow . . . . . . . . . . . . . . . . . . .
2.3 Concepts Pertaining to Porous Media . . . . . . . . .
2.3.1 Porosity . . . . . . . . . . . . . . . . . . . . .
2.3.2 Hydraulic Conductivity . . . . . . . . . . . . .
2.3.3 Wettability . . . . . . . . . . . . . . . . . . .
2.3.4 Saturation . . . . . . . . . . . . . . . . . . . .
2.3.5 Relative Permeability . . . . . . . . . . . . . .
2.3.6 Capillary Pressure . . . . . . . . . . . . . . .
2.4 Thermodynamic Properties . . . . . . . . . . . . . .
2.4.1 Enthalpy and Internal Energy . . . . . . . . .
2.4.2 Specific heat . . . . . . . . . . . . . . . . . . .
2.5 Constitutive Relationships . . . . . . . . . . . . . . .
2.5.1 The pc − Sw relation . . . . . . . . . . . . . .
2.5.2 The kr − Sw relation . . . . . . . . . . . . . .
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3 Mathematical Model
3.1 Porous Medium Region . . . . . . . . . . . . . . . . . .
3.1.1 Assumptions . . . . . . . . . . . . . . . . . . . .
3.1.2 Compositional Multiphase Flow . . . . . . . . .
3.1.3 Non-Isothermal Compositional Multiphase Flow
3.2 Free-Flow Region . . . . . . . . . . . . . . . . . . . . .
3.2.1 Assumptions . . . . . . . . . . . . . . . . . . . .
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IV
3.3
3.2.2 Single-Phase Flow . . . . . . . . . . .
3.2.3 Compositional Single Phase Flow . .
3.2.4 Non-Isothermal Compositional Single
Interface Description and coupling . . . . . .
3.3.1 Mechanical Equilibrium . . . . . . .
3.3.2 Thermal Equilibrium . . . . . . . . .
3.3.3 Chemical Equilibirum . . . . . . . .
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Phase Flow
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4 Dimensional Analysis
4.1 Characteristic Values . . . . . . . . . . . . . . . . . . . .
4.2 Dimensionless System of Equations . . . . . . . . . . . .
4.3 Model Applications . . . . . . . . . . . . . . . . . . . . .
4.3.1 Capillary-Tissue Model . . . . . . . . . . . . . . .
4.3.2 Soil-Air Model . . . . . . . . . . . . . . . . . . .
4.4 Trends of Dimensionless Numbers . . . . . . . . . . . . .
4.4.1 Porous Medium Region . . . . . . . . . . . . . . .
4.4.2 Free-Flow Region . . . . . . . . . . . . . . . . . .
4.4.3 Dimensionless Numbers Common to Both Regions
4.4.4 Fourier Number . . . . . . . . . . . . . . . . . . .
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5 Numerical Model
5.1 Weighted Residuals and the Box Method (FV-FE Method)
5.2 Temporal Discretization of Equations . . . . . . . . . . . .
5.3 Discretised Equations of the Coupled Model . . . . . . . .
5.3.1 Free Flow Mass Balance . . . . . . . . . . . . . . .
5.3.2 Stokes Equation for Momentum Balance . . . . . .
5.4 The Structure in DuMux . . . . . . . . . . . . . . . . . . .
5.4.1 Sub Models . . . . . . . . . . . . . . . . . . . . . .
5.4.2 The Coupling Operators . . . . . . . . . . . . . . .
5.5 Implementation of the Coupling Concept . . . . . . . . . .
5.5.1 Boundary Flux in the Stokes Domain . . . . . . . .
5.5.2 Stabilization at the Boundary . . . . . . . . . . . .
5.6 The Capillary Tissue Model . . . . . . . . . . . . . . . . .
5.6.1 Motivation . . . . . . . . . . . . . . . . . . . . . . .
5.6.2 Model Set Up . . . . . . . . . . . . . . . . . . . . .
5.6.3 Boundary Conditions . . . . . . . . . . . . . . . . .
5.7 Choice of Characteristic values . . . . . . . . . . . . . . . .
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25
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6 Results and Discussion
59
7 Summary And Outlook
65
List of Figures
1.1
1.2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
3.1
3.2
4.1
4.2
5.1
5.2
5.3
5.4
5.5
5.6
5.7
Macro-scale example of the application of the coupled model: modelling
of evaporation in the unsaturated zone, after [17] . . . . . . . . . . . .
Micro-scale example application of the coupled model: modelling the
transfer of therapeutic agents between blood and tissue after [14] and [3]
Interface descriptions, after [17] . . . . . . . . . . . . . . . . . . . . . .
Dual-Domain concept of coupling after [17] . . . . . . . . . . . . . . . .
Example to explain the concept of REV after [4], source:[11] . . . . . .
Wetting phase fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Non-wetting phase fluid . . . . . . . . . . . . . . . . . . . . . . . . . .
pc − Sw curve for the Brooks-Corey and the Van Genuchten models,
(source [11]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Typical kr -sw relationship after Brooks-Corey and Van Genuchten
(source [11]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Normal component of the mechanical equilibrium coupling condition
after [17] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Tangential component of the mechanical equilibrium coupling condition
after [17] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Trends of the capillary number and the gravity number with increasing
characteristic velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Trends of the Euler number and the Reynolds number with increasing
characteristic velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Schematic diagram of the box method after [1] and [3] . . . .
Overview of the FE-FV grid for the coupling model . . . . . .
Pictorial description of the coupling concept . . . . . . . . . .
Illustration of the handling of pressure as a part of momentum
Illustration of the Capillary-Tissue model, after [23] . . . . . .
Dimensions of the domain . . . . . . . . . . . . . . . . . . . .
Boundary conditions of the domain . . . . . . . . . . . . . . .
V
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coupling
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57
VI
5.8
Dimensions of the domain with different characteristic lengths in the sub
domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.9 Model domain in dimensionless form, with lc = 0.2mm . . . . . . . . .
5.10 Model domain in dimensionless form, with lc = 0.05mm . . . . . . . . .
6.1
6.2
6.3
6.4
Dimesnionless pressure and velocity distribution in the free-flow region.
Dimensionless pressure and velocity distribution in the porous-medium
region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Transport in the free-flow region for t̂ = 0.005, 0.05, 0.07 and 0.3. . . .
Transport in the porous-medium region for t̂ = 10, 145, 800 and 3062. .
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60
61
62
63
List of Tables
3.1
Summary of phases and components in the two model applications . . .
22
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
Definition of characteristic quantities with possible choices
Dimensionless equations of the porous medium . . . . . . .
Dimensionless equations of the free flow region . . . . . . .
Dimensionless coupling conditions . . . . . . . . . . . . . .
Dimensionless numbers . . . . . . . . . . . . . . . . . . . .
Parameter values for two model applications . . . . . . . .
Process time scales for the capillary-tissue model . . . . .
Process time scales for the soil-air model . . . . . . . . . .
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43
5.1
5.2
Names of dimensionless models in DuMux . . . . . . . . . . . . . . . .
Parameter overview of the Capillary-Tissue model . . . . . . . . . . . .
50
55
VII
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Nomenclature
βp
isobaric thermal expansion coefficient
[1/P a]
βT
isothermal compressibility coefficient
[1/K]
λpm
thermal conductivity of the porous medium
g
gravity vector
K
intrinsic permeability
[m2 ]
Kf
hydraulic conductivity
[m/s]
µ
dynamic viscosity
[P a.s]
φ
porosity
σ
surface tension
τ
shear stress
K
intrinsic permeability tensor
θ
contact angle
%α
density of phase α
c
specific heat capacity
[J/kg.K]
cs
specific heat of the solid phase
[J/kg.K]
Dα,pm diffusion coefficient in porous medium
[def ine]
h
specific enthalpy
hα
enthalpy of phase α
krα
relative permeability of phase α
m
mass
[def ine]
[m/s2 ]
[-]
[N/m]
[N/m2 ]
[m2 ]
[◦ ]
[kg/m3 ]
[J/kg]
[W/(m.K)]
[-]
[kg]
VIII
LIST OF TABLES
1
mκα
mass of component κ in phase α
[kg]
pα
pressure of phase α
[P a]
pc
capillary pressure
[P a]
qακ
mass source or sinks of component κ in phase α
qT
energy source or sink
R
universal gas constant
Sα
saturation of phase α
[−]
T
temperature
[K]
t
time
u
specific internal energy tension
uα
internal energy of phase α
V
volume
vα
velocity of phase α
Xακ
mass fraction of component κ in phase α
Z
real gas factor
[kg/m3 .s]
[J/m3 .s]
[kJ/(kmolK)]
[s]
[J/kg]
[def ine]
[m3 ]
[m/s]
[-]
[N m/(kgK)]
Chapter 1
Introduction
Free fluid flow interacts with fluid flowing through a porous medium in numerous
instances which are of significant interest. Examples on the visible scale are the
process of evaporation or drying in soil (see fig 1.1), the infiltration of water and
pollutant from the surface into the ground during events of rainfall and pollutant
spills respectively. An example on the micro scale is the transfer of therapeutic agents
from the blood stream into the surrounding tissue (see fig 1.2).
The study of the phenomena mentioned above is of immense importance in today’s
world and can help to understand important natural processes or formulate a better
treatment strategy to cure diseases. This warrants a detailed study of how free flow
interacts with flow through a porous medium. But, the processes occur, as already
mentioned on different physical scales which are, by many orders of magnitude,
different from each other. The processes on each scale have their own set of conditions
and driving forces which influence the behaviour of the system in a unique way. These
driving forces, though of the same nature, affect different systems in different ways
based on the importance of specific forces at specific scales.
Therefore, although the process on different scales can be described by one
mathematical model, the study of these forces and the manner in which they influence
the system is of great importance since a major driving force on one scale may be
of little or no importance at the other scale. This information not only helps in
comparing the processes in the specific applications, but also helps in making further
simplifying assumptions to the model, therefore making it easier to find acceptable
solutions faster compared to a ”fully-generic” case of the model.
Dimensional analysis is a very helpful tool to aid such a study since it helps in
the resolution of forces, their relative dominance in the system (compared to other
forces) and how they help in speeding up or slowing down specific processes in the
system under consideration. So, dimensional analysis was chosen as the preferred tool
3
4
Figure 1.1: Macro-scale example of the application of the coupled model: modelling of
evaporation in the unsaturated zone, after [17]
Figure 1.2: Micro-scale example application of the coupled model: modelling the transfer of therapeutic agents between blood and tissue after [14] and [3]
1.1 Objective and Structure of the Work
5
to study the systems under consideration.
The model (formulated with certain assumptions) was examined with dimensional analysis and dimensionless equations were derived from the model [16]. The
dimensionless models were examined analytically for the parameter sets applicable.
These dimensionless equations were then implemented in DuMuX . A representative
test case was run to compare the qualitative behaviour of the dimensionless model to
the existing model.
1.1
Objective and Structure of the Work
In the following work, the dimensionless model of coupling of free flow and flow in
porous media is implemented in DuMuX . One of the dimensionless numerical model
has been run for a representative initial and boundary conditions to reproduce the
behaviour of the dimensional model. The results are analysed and discussed for the
simulation case. The report is structured in the following manner
− The basic definitions and concepts necessary to understand the processes have
been given in Chapter 2.
− The model concept along with the simplifying assumptions and the mathematical
model have been explained in Chapter 3.
− The dimensionless equations and an analytical overview of the dimensionless
numbers from the point of view of the respective applications are discussed in
Chapter 4
− A brief introduction to the numerical scheme used, the implementation of the
equations in the DuMuX framework and certain specific details of the implementation have been discussed in Chapter 5
− The model set up with the boundary conditions for the test case, results from
the model run has been discussed in Chapter 6
− The current work has been summarized and the scope for future work has been
laid down in Chapter 7
Chapter 2
Definitions
The model, free flow coupled with flow in porous media, has been previously investigated with two different approaches (see figure 2.1) after [13, 24]. They are:
− The single-domain approach,
− The two-domain approach.
In this work the two-domain approach has been chosen (see figure 2.2). This
necessitates dividing the model into two different regions, namely the free-flow and
the porous-medium region. Consequently they have to be investigated individually at
first, with different equations used to model flow and transport of mass, momentum
and energy. These two sub domains are then brought together at the interface where
suitable conditions have to be formulated such that the model correctly depicts the
processes occurring in reality.
The above mentioned modelling process has been applied to and investigated in two
different applications which are of importance in the scope of this work. They are:
− The modelling of transport of therapeutic agents from the blood into the interstitial fluid,
− The modelling of evaporation from the unsaturated zone.
These two applications bring with them their own set of constraints, assumptions
and modelling approaches. The first application has a single fluid phase in which the
transported property (therapeutic agent) propagates. The need to model transport of
heat or energy does not arise since the whole system will be at body temperature.
The second application has two fluid phases in the porous medium, air and water, which are soluble in each other thus giving rise to the concept of components.
These flows through the porous medium, under certain conditions, are transported into
the adjacent free-flow region which has a single phase. The explanation of the model
7
8
c)
Transition zone
a)
b)
Free Flow
Air
Interface
Sharp interface
Water Air Solid
Porous Medium
Figure 2.1: Interface descriptions, after [17]
Figure 2.2: Dual-Domain concept of coupling after [17]
2.1 Representative Elementary Volume (REV)
9
concept in subsequent chapters refers to the two-phase non-isothermal compositional
porous-media flow coupled with a single-phase non-isothermal compositional free flow
which is the more complex of the two models studied in this work. The conversion
to a single-phase isothermal compositional model to describe transport of therapeutic
agents between blood and tissue follows fairly straightforward simplifying assumptions.
2.1
Representative Elementary Volume (REV)
Due to the random nature of porous media, the task of computing solutions to
equations of flow and transport at the level of the pore space, where there exists a
clear distinction between the existing fluid phases and the solid phase, becomes highly
resource intensive. The fluid and the porous medium is instead considered at a certain
macroscopic scale by averaging the equations at the pore scale over a certain averaging
volume. At this scale the discontinues between phases appearing on the pore scale
are no longer visible and all phases coexist within this volume at the same time. The
transition of consideration of porous media from the pore scale to this macroscopic
scale by the process of volume averaging ([27]) gives rise to a new set of equations
(ex: Darcy’s law) and parameters (ex: permeability, porosity, saturation) which
hold at this scale and also correspond to the equations and parameters which are
relevant at the pore scale. At this scale the variations of properties due to molecular
effects, arising from the consideration of the porous media at smaller scales or due to
heterogeneities, which occur at larger scales, are completely absent [11]. A volume
element at this scale (over which the process of volume averaging is done) is called the
Representative Elementary Volume (REV) (see figure 2.3) [4].
From the consideration of fluid at the REV scale, the properties and parameters of
the fluid and of the porous medium relevant for the current study has been explained
in subsequent sections. From this brief overview of the applications, it is evident that
a few concepts need to be defined and explained in order to comprehend the model.
These concepts will be presented in the following sections.
2.2
2.2.1
Concepts Pertaining to Fluids
Density
The density of phase α is defined as the mass of the phase α in a unit volume occupied by
the phase α. It should be noted that the definition of density of a fluid already considers
the fluid at a scale where principles of continuum mechanics can be applied (i.e., the
fluid is not investigated by its individual molecules). The density of the phase α is
therefore a function of pressure and temperature of the system due the dependence of
the density on the compressibility of the fluid phase due to pressure, and the expansion
multiphase character on a macro scale. The average values result from the integration of
the microscopic balance equation [55] over an appropriate control volume, the representative elementary volume (REV) V0 . The macroscopic parameters to be averaged must
10
be independent of V0 (Fig. 2.1) and include continuous correlations. We will explain the
necessary properties of the REV with the example of the behavior of the average phase
density of phase α at a fixed point in space and time (see Fig. 2.2). If the volume,
consideration as porous medium is
average mass density
of the two-phase system
not possible
possible
inhomogeneous
medium
!a
V0 = REV
0
l
d
homogeneous
medium
L
volume
fracture
2.1: Definition
of the
Figure 2.3: ExampleFigure
to explain
the concept
ofREV
REV after [4], source:[11]
represented by a microscopic characteristic length d, is very small, the density of phase
of the fluid phase with the change in temperature. The compressibility of the fluid is
α is either finite or zero. An extension of the investigation volume causes fluctuations,
defined
under isothermal conditions and the expansion under isobaric conditions. For
depending
on the question
whether
large
phase
α areofincluded
the volume
a two phase system
containing
water
andparts
air, of
the
density
water isingiven
by theortotal
not. As an
ideal
derivative
after
[28]case,
as these fluctuations vanish when the investigation volume is large
enough. Then, within an interval, the average density and, in general, each averaged
parameter are independent from∂%
the=averaging
%βp dp + domain.
%βT dT This corresponds to the REV(2.1)
with its characteristic length l (see Fig. 2.1). A further extension of the averaging volume
where βp and βT , the expansion coefficients of water under isothermal and isobaric
leads to new
fluctuations
byby
macroscopic heterogeneities of the medium, which
conditions
respectively
arecaused
defined
are characterized by the length L. The averaging value is representative if the condition
1 ∂%
β
=
p
d!l!
L
% ∂p
(2.1)
(after Whitaker (1973) [241]) within the REV1is∂%
satisfied. The existence of a fluctuation–
β =
.
T
free scale in spatially variable porous media
%must
∂T be questioned particularly for multi-
The density of gas in a multi-component system is calculated by summing over the
partial density of each component κ, %κg . Therefore, the density of gas for a twocomponent system is defined as
X
%g =
%κg
κ∈w,a
where the density is a function of pressure and temperature
%κg =
pκg
Z κ Rκ T
2.2 Concepts Pertaining to Fluids
11
where Z κ and Rκ are the real gas factor and the gas constant of component κ. Under
the assumption that air is an ideal gas (Z κ = 1), the equation is simplified to the ideal
gas law
pg = %g RT
with %g =
2.2.2
n
,
V
where n is the number of moles and V the volume of the gas.
Stresses and Deformations
Fluids in motion develop stresses due to viscosity. For a fluid element in motion there
can be two types of forces which have to be taken into account, namely
− surface forces and
− body forces.
The surface forces are due to pressure and also due to the interaction of the fluid element
with other fluid elements (moving at different velocities), they act along the surface of
the fluid element. The body forces on the other hand are distributed throughout the
body of the element. The only body force which is of practical interest in the current
work is gravity. The normal and shearing stresses in the fluid are given by the stress
tensor τ


σxx τxy τxz
τ =  τyx σyy τyz  ,
(2.2)
τzx τzy σzz
where σii represents the normal stresses, and τij represents the shear stresses. Due to
the effect of these stresses and the definition of a fluid, i.e., to deform continuously with
stresses, there are instances when the fluid element undergoes linear and/or angular
deformation. From the law of conservation of mass inside each fluid element, it follows
that the density of the fluid should change related to the change in volume due to
this deformation. The rate of change of volume to the original volume of the fluid
element is called the volumetric dilatation rate. A detailed explanation of the volumetric
dilatation rate can be found in [26] and [19]. The components of the stress tensor are
then calculated as a function of the flow velocity using any of the standard relations for
various fluids. The relation applicable for the current work is explained in the following
section.
2.2.3
Newton’s Law of Viscosity
A fluid in motion can be resolved into layers moving at different velocities. The differences in friction between the fluid and boundary and between the layers of the fluid
set up a gradient of flow velocity perpendicular to the direction of flow. This creates
a shear between the layers of the fluid which is proportional to the velocity gradient
12
x
perpendicular to the direction of flow. Therefore, if ∇vx = ∂v
is the gradient of the x
∂y
velocity in the y direction, then the shear stress τ is given as
τ ∝ ∇vx .
The proportionality constant for the above equation is the fluid property called viscosity
or more precisely the dynamic viscosity (µ) which provides the measure of the internal
resistance of the fluid to flow.
τ = µ∇vx
(2.3)
The law is called the Newton’s law of viscosity. In general, the shear stress is expressed
as a function of the gradient of the i velocity in the j direction by the tensor equation
∂vj
∂vi
+
(2.4)
τij = µ
∂xj ∂xi
The Newton’s law of viscosity relates the stresses as a linear function of the gradient
of velocity. Fluids obeying such a law are called Newtonian fluids else they are called
non-Newtonian fluids.
2.2.4
Viscous Flow and The Navier-Stokes Equation
Newton’s second law of motion says the force in any direction causes the acceleration
of mass in that direction
Force = Mass × Acceleration
(2.5)
The forces considered are normal forces, shearing forces and gravity forces. The acceleration a is defined by
∂v
a=
+ (v · ∇) v,
(2.6)
∂t
the continuity equation is given by
∂%
+ ∇ · (%v) = 0.
∂t
(2.7)
The product of mass and acceleration is equal to the sum of forces acting on a fluid
element. These forces are divided into
− surface forces or stresses
− body forces, in this case, only gravity
The equation is formulated as follows
∂%v
+ % (v · ∇) v = ∇ · (τ ) + %g.
∂t
(2.8)
2.3 Concepts Pertaining to Porous Media
13
The stress tensor τ has been introduced already in equation 2.2. The Newton’s law of
viscosity (equation 2.4) is used to evaluate the components of the stress tensor τ . The
stresses have the contribution from the pressure and the shear stress from the relation
2.4. Expanding the expression for better understanding, we get

 



∂u
∂u
∂u
∂u
∂v
∂w 
1 0 0
∂x
∂y
∂z
∂x
∂x
∂x
 ∂v ∂v ∂v   ∂u
∂v
∂w 
τ = −p  0 1 0  + µ  ∂x
(2.9)

∂y
∂z  +
∂y
∂y
∂y
∂w
∂w
∂w
∂u
∂v
∂w
0 0 1
∂x
∂y
∂z
∂z
∂z
∂z
where u, v and w are velocity components in the x, y and z directions respectively. The
above equation can be expressed in tensor form as
τ = −pI + µ ∇v + (∇v)T
(2.10)
A comprehensive derivation of the Navier-Stokes equation can be found in [26] and
[19].
2.2.5
Stokes Flow
A special case of the Navier-Stokes equation occurs when the flow velocities in the fluid
are so low that the second-order inertial term of velocity, % (v · ∇) v, in the NavierStokes equation can be neglected. With this simplification, the Stokes equation is
obtained and is given below
∂%v
+ ∇ · (τ ) − %g = 0,
∂t
(2.11)
where τ is given by equation 2.10
2.3
2.3.1
Concepts Pertaining to Porous Media
Porosity
Porous media has pore spaces due to which fluid flow is possible. This property is
called porosity and is measured as the ratio of the pore spaces to the total volume in
one REV.
volume of pores in one REV
(2.12)
volume of the REV
On a more general consideration, the porosity of the medium is dependent on temperature of the medium and the pressure applied on it (for example, external loads).
But as a simplification for modelling flow through porous medium, the porosity may
be considered constant.
φ=
14
Figure 2.4: Wetting phase fluid
2.3.2
Hydraulic Conductivity
Hydraulic conductivity is a measure of the resistance of the porous medium to fluid
flow. Therefore it is a lumped factor containing properties of the porous medium (the
intrinsic permeability) and also of the fluid (the viscosity and density of the fluid). The
hydraulic conductivity Kf can be expressed as
Kf = K
%g
,
µ
(2.13)
where µ is the viscosity of the fluid, g is the acceleration due to gravity and K is the
intrinsic permeability tensor which for a three dimensional system is given by


Kxx Kxy Kxz
K =  Kyx Kyy Kyz  .
Kzx Kzy Kzz
(2.14)
The above case is, however, for a single fluid completely filling the porous medium.
Practically, there is usually more than one fluid filling the pore space which makes
only a part of the total pore volume available for the flow of one fluid. The specific
discharge (discharge per unit cross section) of a fluid is always lesser when there are
two fluids flowing simultaneously. In such a case, there is a decrease in permeability
for the fluid under consideration. This will be explained in section 2.3.5
2.3.3
Wettability
When two immiscible fluid phases fill a pore volume, the angle of contact θ between the
fluid phase and the solid phase determines the preferential wettability of the fluid phase.
The fluid with a contact angle less than 90◦ (see figure 2.4) is said to preferentially
wet the solid phase [4] and is called the wetting phase. The other fluid is then the
non-wetting phase (see figure 2.5). Wettability of a fluid to a solid is only relative to
another fluid.
2.3 Concepts Pertaining to Porous Media
15
Figure 2.5: Non-wetting phase fluid
2.3.4
Saturation
In a REV which has its pore space filled with two immiscible fluids, the saturation of
a phase α is the ratio of the volume that the phase occupies in the REV to the total
pore volume available in the REV.
Sα =
2.3.5
volume of pore spaces filled by phase α
total pore volume of REV
(2.15)
Relative Permeability
The flow of one fluid in a two-phase system is dependent on the extent to which
the other phase is occupying the pore space. When the second fluid is occupying
all the pores, the discharge of the first fluid cannot occur. In this case, the relative
permeability krα of the first fluid with respect to the second is 0 and that of the second
is 1. Considering the concept of relative permeability defined above, the equation of
the hydraulic conductivity(2.13) can be modified to include multiple fluid phases as
%α g
Kf = Kkr,α
(2.16)
µα
Relative permeability for each phase α varies from 0 to 1 and is strongly dependent on
the saturation (see sec. 2.5.2). The sum of relative permeabilities over all fluid phases
is 1.
X
krα = 1.
(2.17)
α
2.3.6
Capillary Pressure
At the area of contact of two fluid phases, there exist cohesive forces between the
molecules in one phase and adhesive forces between molecules of different phases. The
difference between the cohesion and adhesion causes free interfacial energy which shows
up as interfacial tension [4]. This interfacial tension is responsible for effects of capillarity which is the difference in pressure between the wetting phase and the non-wetting
16
phase. This can be derived by equilibrium considerations between the fluid-fluid interface and the solid medium. It is given by the relation
pc =
4σ cos θ
d
(2.18)
where σ is the surface tension of the fluid, θ is the angle of contact between the fluid
and the soil and d is the diameter of the capillary. Considering the equilibrium of forces
between the two phases, it is clear that the non-wetting phase pressure has to balance
not only the wetting phase pressure but also the capillary pressure, given by
pn − pw = pc = pc (Sw ) .
(2.19)
A detailed explanation and derivation can be found in [4].
2.4
2.4.1
Thermodynamic Properties
Enthalpy and Internal Energy
The enthalpy of the system is the property describing the thermodynamic potential of
the system [18]. It is defined mathematically as the sum of the internal energy u and
the volume changing work pV
H = U + pV
(2.20)
If the above equation is divided by the mass m of the medium contained in the system,
a specific enthalpy and specific internal energy is obtained
U
pV
H
=
+
m
m
m
h=u+
2.4.2
p
%
(2.21)
(2.22)
Specific heat
Specific heat of the system is defined as the amount of heat required to change the
temperature of a mass of 1 kg by 1 K. There can be two different definitions of specific
heat, one at constant volume cv and the other at constant pressure cp . The formulations
for cv and cp are given below.
∂u
cV =
(2.23)
∂T V
∂h
cp =
(2.24)
∂T p
2.5 Constitutive Relationships
2.5
2.5.1
17
Constitutive Relationships
The pc − Sw relation
On the REV scale, for instance when the saturation of the non-wetting phase
increases, the saturation of the wetting phase decreases (since the sum of saturations
is always unity). This can be accounted for by the fact that the non-wetting
phase first penetrates into the largest pores (because they have the lowest capillary
pressure according to equation 2.18 and hence are the easiest for the non-wetting
phase to enter) and pushes the wetting phase into smaller pores. This happens till
non-wetting phase has pushed the wetting phase into such pores where the pressure
of the non-wetting phase is no longer able to displace the wetting phase out of those
pore. The saturation of the wetting phase at this point is called the residual saturation.
From the above explanation and equation 2.18 it can be inferred that the capillary
pressure increases with decreasing pore diameter. This means that there is an increase
in capillary pressure with decrease in wetting phase saturation. Therefore the capillary
pressure in a multi-phase system is a function of the wetting-phase saturation. Furthermore, the capillary pressure accounts for the discontinuity of pressure at the interface
of the two fluid phases at the REV scale arising from the equilibrium of forces between
the two fluids in a pore space[4] (see equation 2.19). This phenomenon shows hysteresis.
The Brooks and Corey [6] model and the van Genuchten [10] model are two of the
commonly used models to find the pc − Sw relation. The capillary pressure - saturation
relations for the Brooks Corey model is give by
Sw − Swr
Seff (pc ) =
=
1 − Swr
pd
pc
λ
(2.25)
and for the Van Genuchten model is given by
Seff (pc ) =
Sw − Swr
= [1 + (α · pc )n ]m
1 − Swr
(2.26)
where λ is the fitting parameter of the Brooks-Corey model, with pd being the entry
pressure and α, n and m are the fitting parameters for the Van Genuchten model (see
figure 2.6 for an exemplary graph).
2.5.2
The kr − Sw relation
By extending the concept of permeability to multi-phase flows, the concept of relative
permeability has been briefly explained in section 2.3.5, it follows that for each saturation value of a particular phase, the phase occupies a distinct pathways through
the porous medium [4], thereby, influencing the flow of the other phase and hence
α
:
VG–parameter
[1/Pa]
λ
:
BC–parameter
[-]
pd
: BC–parameter, entry pressure
[Pa]
are based on parameters characterizing the pore space geometry. They are determined
by fitting to experimental data.
18
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 water saturation [-]
Figure
Definition
of the Brooks–Corey parameters λ and pd
Figure
2.6:2.19:
pc −S
models, (source
w curve for the Brooks-Corey and the Van Genuchten
[11])
The λ–parameter (BC) usually lies between 0.2 and 3.0. A very small λ–parameter
relative
permeability.
Thewhile
relative
permeability
is therefore
a function
of the wetdescribesits
a single
grain
size material,
a very
large λ–parameter
indicates
a highly
ting material.
phase saturation
Sw .pressure
Based on
capillary pressure
- saturation
relationship
by
non–uniform
The entry
pd the
is considered
as the capillary
pressure
reBrooks-Corey [6] and van Genuchten [10] explained in the previous section to calculate
quired to displace the wetting phase from the largest occuring pore (cf. Fig. 2.19 and
the effective saturation Se , the relative permeability of the wetting and non-wetting
2.20). The
influence
of the
pressure on multiphase
heterogeneous
phase
is given
forentry
the Brooks-Corey
model as processes
a functionin of
the effective media
saturation Se
will be discussed
in detail inBrooks-Corey
Chapter 5. parameter λ as
and the empirical
After Lenhard et al. (1989) [146] [145], we can derive
2+3λ the following correlations beλ
k
=
S
(2.27)
e
r,w
tween the Brooks–Corey and Van Genuchten form
parameters:
2+λ !
2 "
λ
m
1/m
k
=
(1
−
S
)
1
−
S
e
r,n
λ =
1 − Sewe
(2.36) (2.28)
1−m
4
The relative permeability
saturation
relationship
can also be given(2.37)
for the van
Sx = - 0.72
− 0.35e−n
Genuchten model as a function of the effective saturation Se and the van Genuchten
Sx
1/m
(Sx
− 1)1−m .
(2.38)
parameters , m and γα as=
pd
h
1 m i 2
m
kr,w = Se 1 − 1 − Se
(2.29)
h
1 i2m
kr,n = (1 − Se )γ 1 − Sem
A typical kr − Sw graph can be seen in fig 2.7.
(2.30)
2.7 Relative permeability
7
2.5 Constitutive Relationships
19
where λ is the empirical constant from the Brooks–Corey pc (S)–relationship (eqs. 2.3
and 2.39). The Van Genuchten model is applied in conjunction with the approach
Mualem
krw =
Se!
!
"
1
m
1 − 1 − Se
krn = (1 − Se )
γ
!
#m $2
1
m
1 − Se
$2m
(2.5
,
(2.5
where m comes from eqs. 2.35 and 2.40. The parameters " and γ are form paramete
which describe the connectivity of the pores [166]. Generally, " =
1
2
and γ =
1
3
[35].
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 2.7: Typical kr -sw relationship after Brooks-Corey and Van Genuchten (source
[11])
Figure 2.31: Relative permeability–saturation function after Brooks–Corey and Va
Genuchten
Figure 2.31 shows the kr (S)–relations after Brooks–Corey and Van Genuchten. Th
function krw (Sw ) for the wetting fluid (here: water — fluid with a higher affinity)
characterized by a gentle increase for low saturations and a very strong increase f
higher saturations. This is due to the fact that, in the case of low saturations, the flu
fills only very small pores, where flow movements are nearly impossible because of th
strong molecular attraction, whereas the largest pores are only filled in the case of a
almost fully saturated system.
The function krn (Sn ) for the non–wetting fluid (here: gas — fluid with a lower affi
ity), however, shows a considerably faster increase for low saturations. The reason
that, for lower saturations, the larger pores are filled first, whereas, in the case of almo
full saturation, only very small pores are filled which have no significant influence on th
Chapter 3
Mathematical Model
In this Chapter, the mathematical model and the coupling conditions are explained
considering the non-isothermal, two-phase, two-component porous media flow coupled
with the non-isothermal, single-phase, two-component free flow. A simplification to
isothermal and/or single phase systems in porous media is straightforward.
3.1
3.1.1
Porous Medium Region
Assumptions
In order to formulate a mathematical model for the porous medium from the twodomain model concept, certain simplifying assumptions have been made according to
[17]. The assumptions are
− The solid phase is rigid.
− Slow or creeping flow (Re 1), therefore the validity of the multiphase Darcy
law.
− Due to slow flow velocities and higher diffusion, dispersion caused due to different
flow velocities, is neglected and only binary diffusion is considered.
− Local thermodynamic equilibrium (mechanical, chemical and thermal) prevails
due to slow flow velocities.
− An ideal gas phase according to [12] and [21].
The phases and components relevant to the current work are summarized in table 3.1
3.1.2
Compositional Multiphase Flow
The Darcy Law for single-phase flow is extended to include multiple fluid phases. Then
the Darcy velocity of each phase α is given as a function of
21
22
Liquid (Interstitial Fluid)
Phases
Water and Air
Interstitial Fluid
and Therapeutic
Agent
Components
Gas
Liquid
(Blood)
Phases
Water and Air
Blood and Therapeutic Agent
Components
Free Flow Region
Liquid and Gas
Porous Medium Region
Table 3.1: Summary of phases and components in the two model applications
Application
Capillary-Tissue
Model
Soil-Air Model
3.1 Porous Medium Region
23
− the external driving forces, in this case it is assumed only to be the pressure
gradient ∇p and gravity g,
− the fluid properties such as density % and viscosity µ and
− the matrix properties such as the intrinsic permeability K and the relative permeability kr
vα = −
krα
K (∇ pα − %α g) ,
µα
α ∈ {l, g}.
(3.1)
But, real systems cannot be considered to be having completely immiscible fluids. The
fluids dissolve into each other and hence, apart from existing individually as phases,
they exist and are transported as components. To describe the transport of the
components the general equations of conservation of mass is extended to multiphase
flow.
Two mass balance equations can be formulated, one for each component summed over
the phases. Hence for κ ∈ {w, a},
X
α∈{l,g}
φ
X
∂ (%α Xακ Sα )
+ ∇ · Fκ −
qακ = 0,
∂t
(3.2)
α∈{l,g}
where the component flux Fκ is the sum of advective and diffusive flux
X
κ
Fκ =
%α v α Xακ − Dα,pm
%α ∇Xακ .
(3.3)
α∈{l,g}
The transported property is the mass fraction Xακ , which is the ratio of the mass of
component κ in phase α to the total mass of all components in phase α
Xακ =
mκ
P α
mκα
α ∈ {l, g}.
(3.4)
κ∈{w,a}
The total mass balance is then calculated by a summation of equation 3.2 along with
the constitutive relationship
X
Xακ = 1
α ∈ {l, g}.
(3.5)
κ∈w,a
The total mass balance of the porous-medium domain is given by
X
α∈{l,g}
φ
X
X
∂ (%α Sα )
+∇·
(%α v α ) −
qα = 0.
∂t
α∈{l,g}
α∈{l,g}
(3.6)
24
3.1.3
Non-Isothermal Compositional Multiphase Flow
The balance of energy is obtained by inserting the enthalpy of the system h into the
transport equation. While the flow and transport of mass occurs only in the pore space,
the transport of energy can occur both in the solid phase (by conduction) and in the
liquid phase. Hence, under the assumption of local thermal equilibrium, the energy
balance in the porous medium should account for both and is given by [8] as
X
φ
α∈{l,g}
∂ (%α uα Sα )
∂ (%s cs T )
+ (1 − φ)
+ ∇ · FT − qT = 0,
∂t
∂t
α ∈ {l, g}
(3.7)
where cs is the specific heat of the solid phase and FT is the energy flux defined by
FT =
X
α∈{l,g}
%α hα v α − λpm ∇ T,
(3.8)
where hα is the enthalpy of phase α and λpm is the effective thermal conductivity of
the porous medium expressed as a function of saturation of the fluid phases according
to [25]
p
λpm = λeff,g + Sl (λeff,l − λeff,g ) .
(3.9)
3.2
3.2.1
Free-Flow Region
Assumptions
The mathematical model for the free-flow region is formulated based on the following
assumptions made in [17].
− only one fluid phase is present
− two-component flow
− non-isothermal flow in the soil-air model
− Laminar flow and slow flow velocities which allows the use of the instationary
Stokes equation
− gravity as the only external body force
Additional to these assumptions certain other assumptions are made in the specific
application of the model.
3.3 Interface Description and coupling
3.2.2
25
Single-Phase Flow
The free-flow region is modelled with the transient Stokes equation. The velocity and
the pressure in the free-flow domain is given by the equation
∂ (%g v g )
+ ∇ · Fu − %g g = 0.
∂t
where Fu , the momentum flux term, is given by
Fu = τ
(3.10)
(3.11)
and the shear stress τ is given, for simplicity by Newton’s law of viscosity (see equation
2.10)
τ = pg I − µ ∇v + (∇v)T .
3.2.3
(3.12)
Compositional Single Phase Flow
The velocity field from the Stokes equation is used to compute component transport
by solving the transport equation
∂ %g Xgκ
+ ∇ · Fκ − qgκ = 0,
κ ∈ {w, a}.
(3.13)
∂t
where Fκ is the mass flux of component κ in the free-flow domain
Fκ = %g v g Xgκ − Dgκ %g ∇Xgκ ,
3.2.4
κ ∈ {w, a}.
(3.14)
Non-Isothermal Compositional Single Phase Flow
The energy fluxes are considered in the energy balance equation which is similar to the
energy balance equation 3.7 in the porous medium. It is given by
∂(%g ug )
+ ∇ · FT − qT = 0,
(3.15)
∂t
where the energy flux in the free-flow region FT is given, similar to equation 3.8, by
FT = %g hg v g − λg ∇T.
3.3
(3.16)
Interface Description and coupling
The two-domain approach is modelled with two different sets of equations, one set
each in the Darcy and the Stokes domain. At the interface, appropriate conditions
have to be applied to describe the transfer of mass, momentum and energy. These
conditions are derived from the individual mass, momentum and energy balance
26
equations by an assumption of thermodynamic equilibrium. Although the application
of these conditions at the interface results in a simple model yet having interface
conditions which are physically meaningful [17], rigorous thermodynamic equilibrium
cannot be achieved. In the following sections, the coupling conditions are described.
In general, the interface is approximated as a simple interface, meaning that it has no
thickness and cannot store mass, momentum or energy. The coupling conditions at
this interface are formulated based on the following assumptions as stated in [17]:
− mechanical equilibrium with the continuity of stresses in the normal and tangential directions
− chemical equilibrium given by the continuity of mole fractions
− thermal equilibrium given by the continuity of temperatures
− balance of the total mass fluxes, the component mass fluxes and the heat fluxes
3.3.1
Mechanical Equilibrium
The mechanical equilibrium consists of two components, the tangential and the normal
component. The normal component is given by equating the normal components of
forces in both domains. The normal component of stresses in the free-flow region given
by
σn = (−pg I + τ ) n
(3.17)
should balance the normal component of stresses in the porous medium region. Assuming that the solid phase is rigid, only conditions for the continuity of momentum
fluxes across the fluid-fluid interfaces suffice for the normal component of mechanical
equilibrium (see figure 3.1). The normal forces act along the area of contact of the gas
phase in the free-flow region and the liquid and gas phase in the porous medium region
individually. They can be expressed after [17] as
n · [Ag ((pg I − τ )n)]ff = [pg Ag ]pm
(3.18a) + (3.18b)
n · [Al ((pg I − τ )n)]ff = [(pl + pc ) Al ]pm
| {z }
ff
n · [((pg I − τ ) n)] = [pg ]
pg
pm
(3.18a)
(3.18b)
(3.18c)
where Al and Ag are the areas of contact of the liquid and gas phases respectively. The
equations 3.18a and 3.18b reduce to the equation 3.18c after the consideration that
for a two-phase system in the porous medium region, the gas phase is in balance with
the liquid phase and the capillary pressure given by the relation 2.19. The gas-phase
pressure in the free-flow region should be in balance with the liquid-phase pressure
3.3 Interface Description and coupling
27
gas phase
solid phase
gas phase
water phase
Figure 3.1: Normal component of the mechanical equilibrium coupling condition after
[17]
gas phase
solid phase
gas phase
water phase
Figure 3.2: Tangential component of the mechanical equilibrium coupling condition
after [17]
and the capillary pressure in the porous medium over the area Al and the gas-phase
pressure in the porous medium over the area Ag . But the liquid phase pressure and
the capillary pressure is related to the gas pressure by 2.19, therefore only equation 3.18c is sufficient to describe the normal component of the mechanical equilibrium.
To derive the tangential component of the mechanical equilibirum, the Beavers and
Joseph condition [5] is utilised. It states that a slip velocity is induced on the porous
medium side at the interface which is greater than the Darcy velocity existing inside the
porous medium. This velocity is proportional to the shear stress along the interface [5].
By making a simplifying assumption after [22], the velocity in the porous medium can
be completely neglected, thus arriving at the condition for the tangential component
(see figure 3.2).
√
ff
ki
τ n · ti = 0,
vg +
αBJ µg
i ∈ {1, . . . , d−1}
(3.19)
where the empirical dimensionless Beavers and Joseph coefficient αBJ is to be found
28
out experimentally or numerically.
3.3.2
Thermal Equilibrium
The thermal equlibirum condition can be formulated based on the assumption stated
earlier of slow flow velocities and local thermodynamic equilibrium. The continuity of
temperature
[T ]ff = [T ]pm ,
(3.20)
and the continuity of heat fluxes across the interface are the coupling conditions. The
heat flux in the free-flow domain must balance the combined heat fluxes of the porous
medium domain[17].
[(%g hg v g − λg ∇T ) · n]ff = − [(%g hg v g + %l hl v l − λpm ∇T ) · n]pm .
3.3.3
(3.21)
Chemical Equilibirum
The continuity of chemical potential would make the ideal coupling condition for the
model. However, the from equation 3.18c it is apparent that the continuity of normal
forces results in a discontinuity of gas phase pressure at the interface. Hence, the
continuity of chemical potential cannot be completely fulfilled. It is assumed that the
coupling condition given by the continuity of mass fractions is valid [17].
[Xgκ ]ff = [Xgκ ]pm ,
κ ∈ {a, w}.
(3.22)
This is based on the assumption that the discontinuity is pressure has little influence
on the chemical equilibrium. The continuity of component fluxes given by
ff
%g v g Xgκ − Dg %g ∇Xgκ · n =
pm
− %g v g Xgκ − Dg,pm %g ∇Xgκ + %l v l Xlκ − Dl,pm %l ∇Xlκ · n
(3.23)
and the total mass balance equation given by
[%g v g · n]ff = − [(%g v g + %l v l ) · n]pm
form the additional coupling conditions.
(3.24)
Chapter 4
Dimensional Analysis
Due to the complexity of the system of equations, namely, different equations in the
two sub domains and the coupling conditions, it proves helpful to be able to identify the
driving forces and major processes in the two sub domains and also the driving forces
for transfer of mass, momentum and energy across the interface. To aid this study, a
dimensional analysis of the governing differential equations has been performed. Characteristic values for primary variables are chosen based on typical values observed in the
application under consideration. The governing equations have then been converted
to dimensionless equations, with the substitutions given below, such that they contain
only dimensionless numbers and variables. With suitable choices for the characteristic
values, the dominating forces in the system can be identified.
l
lc
t
time t̂ =
tc
ˆ = ∇lc
gradient, divergence operator ∇
T
temperature T̂ =
Tc
p
pressure p̂ =
pc
u
internal energy û =
uc
%
density %̂ =
%c
length ˆl =
29
30
4.1
Characteristic Values
In order to convert the equation to dimensionless equations, the characteristic
quantities were defined and chosen for the variables as described in table 4.1. As
a rule, only independent variables are chosen to set the characteristic quantities.
However, there can be dependent variables expressed as a function of one or more
independent variables, but, choosing as few dependent variables as possible is advised.
The variables of the governing equations are then substituted as a product of a
characteristic value of a variable and a dimensionless variable. These characteristic
values are grouped together to form dimensionless groups commonly identified by
dimensionless numbers.
The choice of these characteristic values were discussed before in [16]. In general the
choice should be made not only based on the dominant process of the sub-domains
but also based on the scale at which these processes are intended to be studied. Then
the dimensionless numbers calculated from these quantities indicate the dominance of
forces at this scale. For example, if the influence of advection or diffusion is to be
studied at the pore scale, then the pore diameter is to be taken for the characteristic
length. However the aim of the dimensionless analysis is to study the system and
eventually to have an understanding of the processes at the interface through the
dimensional analysis. To aid this, the system length has been chosen for the analysis
in this Chapter. Other possible choices of the characteristic values are discussed in
[16].
4.2
Dimensionless System of Equations
The dimensionless equations and numbers have been summarised in tables 4.2, 4.3
and 4.5. These equations have dimensionless primary variables and more importantly,
dimensionless numbers, whose values provide insights into the forces which drive the
system.
4.3
Model Applications
The dimensional analysis has been done to study two specific model applications which
are of interest in the current work. They are briefly described in subsequent sections.
4.3.1
Capillary-Tissue Model
The Capillary-Tissue model is an application of the coupled model at the Micrometre
to millimetre scale to describe the transport of therapeutic agent, which is injected
into the blood stream, in the tissue. The capillaries are of the order of a few millimetre
4.3 Model Applications
31
Table 4.1: Definition of characteristic quantities with possible choices
Name
Notation
Remarks
Length
lc
independent variable, system length, front
width, pore diameter
Time
tc
dependent variable, chosen as
Velocity
vc
independent variable, relevant velocity observed of the application
Temperature
Tc
independent variable, taken as a temperature
difference
Pressure
pc
independent variable, taken as a pressure
difference or the capillary pressure for twophase flow.
Density
%c
described by the ideal gas law, as a function
of a reference temperature and pressure.
lc
vc
in length and in the order of a few micrometre in diameter. The therapeutic agent
injected into the capillary enters the surrounding tissue across the capillary wall. The
capillary wall and tissue have different permeabilities and porosities due to which
the transport characteristics vary in the two regions. The free-flow velocities in the
capillary region is in the order of millimetre per second.
4.3.2
Soil-Air Model
The Soil-Air model is an application of the coupled model on the macro-scale to describe
the process of evaporation due to radiation and wind. The model is done in accordance
to a laboratory experiment where a closed box of soil with a certain water content is
kept in a wind tunnel and based on the temperature, moisture content and velocity
of the wind, evaporation takes place from the soil. The dimensions of the experiment
is then the input to the modelling and therefore the system length is 0.25 m. The
free-flow velocities for this application ranges from 1 to 10 meters per second.
From the descriptions of the models in the previous sections, it is evident that the
32
α∈{l,g}
=0
κt
qα
c
%c
=0
T̂
p̂g
Sl or Xακ
Primary variables
Table 4.2: Dimensionless equations of the porous medium
Balance equations
− qT %ctuccα = 0
(1−φ) Tc ∂ (%s cs T̂ )
+
%c ucα
∂ t̂
Tc λpm tc ˆ
∇T̂
%c ucα lc2
qα tc
%c
Mass balance for component
κ ∈ {w, a}: P
κ
α Sα )
ˆ · %̂α v̂ α Xα − 1 ∇X
ˆ α −
φ ∂(%̂α X
+∇
Pe
∂ t̂
α∈{l,g}
−
α∈{l,g}
Total Mass balance:
P
P
P
ˆ ·
φ ∂(%̂∂αt̂Sα ) + ∇
(%̂α v̂ α ) −
α∈{l,g}
hα
%̂ v̂
ucα α α
φ ∂(%̂α∂ût̂α Sα ) +
Energy balance:
P
α∈{l,g}
ˆ ·
∇
∂ t̂
uc
Energy balance:
∂(%̂g ûg )
ˆ
+ ∇ · hg %̂v̂ g −
∂ t̂
−
1 ˆ
∇v̂ g
Re
Tc λg tc ˆ
∇T̂
%c uc lc2
Momentum Balance:
∂(%̂g v̂ g )
ˆ · (p̂g I) − ∇
ˆ ·
+ Eu∇
Total Mass balance:
∂(%̂g )
ˆ · (%̂g v̂ g ) − qg tc = 0
+∇
%c
∂ t̂
=0
− %̂g F1r2 = 0
qT tc
%c uc
Mass balance for component κ ∈ {w, a}:
κ
∂ (%̂Xgκ )
ˆ gκ − qg tc = 0
ˆ · %̂g Xgκ v̂ g − 1 %̂g ∇X
+
∇
Pe
%c
∂ t̂
Balance equations
T̂
v̂ g
p̂g
Xακ
Primary variables
Table 4.3: Dimensionless equations of the free flow region
4.3 Model Applications
33
34
Condition
h
−
h
−p̂g I +
h
= [p̂]pm
pcpm
pcff
= 0 (i ∈ {1, . . . , d−1})
i
Equation
iff
1 1 ˆ
∇v̂ g
Eu Re
· ti
λg Tc tc ˆ
∇T̂
%c hc lc2
[Xgκ ]ff = [Xgκ ]pm
%̂g ĥv̂ g −
λpm Tc ˆ
∇T̂
%c hc lc2
= [T̂ ]pm
iff
=
ipm
·n
·n
[(%̂g v̂ g ) · n]ff = − [(%̂g v̂ g + %̂l v̂ l ) · n]pm
√
ki ˆ
∇v̂ g
αBJ lc
h
[Tc ]ff
[T̂ ]ff
[Tc ]pm
%̂g ĥg v̂ g + %̂l ĥl v̂ l −
(%c hc vc )ff
(%c hc vc )pm
(%c vc )ff
(%c vc )pm
v̂ g +
Table 4.4: Dimensionless coupling conditions
Mechanical Equilibrium:
Normal Component
Mechanical Equilibrium:
Tangential Component
Continuity of Mass Fluxes:
Continuity of Mass Fractions:
Continuity of Heat Fluxes:
Continuity of Temperature:
4.4 Trends of Dimensionless Numbers
35
Table 4.5: Dimensionless numbers
Name
Symbol
Definition
Forces Involved
Capillary Number
Ca
Kpc
lc vc µα
capillary force
viscous force
Euler Number
Eu
pc
%vc2
pressure force
inertia force
Fourier Number
Fo
αT tc
lc2
conduction
storage
Froude Number
Fr
√vc
glc
inertia force
gravity force
Gravity Number
Gr
K%α g
vc µα
gravity force
viscous force
Peclet Number
Pe
lc vc
κ
Dα
advection
diffusion
Reynolds Number
Re
%c vc lc
µg
inertia force
viscous force
scales and the parameter values are different. Therefore the dimensionless numbers for
the two model applications are compared in the following sections.
4.4
Trends of Dimensionless Numbers
The dimensionless numbers described in table 4.5 provide an insight into the driving forces of the system. From the dimensional analysis of the coupled model seven
dimensionless numbers have been obtained. Out of the seven, two, namely
− capillary number and
− gravity number
describe processes in exclusively in the porous medium, while three, namely
− Euler number
− Reynolds number and
− Froude number
36
describe processes exclusively in the free-flow region and lastly, two dimensionless numbers, namely
− Peclet number and
− Fourier number
describe the component transport processes in both the free-flow and the porousmedium region. It is to be noted that the definition of the capillary number varies
with the model application considered. For a two-phase system, the capillary number
is formulated by defining the characteristic pressure pc as the characteristic capillary
pressure. This is usually taken as the change in capillary pressure over the system
length or the front length [15]. This definition holds good only for models which have
a concept of capillary pressure. Therefore for a single phase system, a characteristic
drop in pressure is chosen for pc .
Based on the parameter ranges which are feasible for the two model applications, a
preliminary analysis of the trends of forces in the two domains is done. A typical set
of parameters for the two model applications is given in table 4.6. From the table of
parameters it is evident that the models lie in very different physical scales. Due to
varying scales of the model application there are different forces at play for the flow
and transport processes. They are driven by different forces at different scales and the
forces are also influenced by the parameter ranges which are typical for each model
application. To give an example, the influence of gravity is more important for bigger
problem scales. For the same system length however, the influence of gravity may
become prominent as the flow velocity is slower.
The resulting numbers are plotted on the axes of a graph showing the variation of
numbers with change in a characteristic value while keeping the other parameters constant. In the following study, the numbers are plotted as a function of the characteristic
velocity while varying it in a meaningful range for both applications. The ranges for
velocities are given in table 4.6. The resulting plot of dimensionless numbers of the
porous-medium region is given in figure 4.1 and for the dimensionless numbers in the
free-flow region in figure 4.2.
4.4.1
Porous Medium Region
The graph in figure 4.1 shows the variation of the capillary number and the gravity
number with varying characteristic velocity for the porous medium region. The four
lines on the graph indicate the following
− For the capillary-tissue model
4.4 Trends of Dimensionless Numbers
37
Figure 4.1: Trends of the capillary number and the gravity number with increasing
characteristic velocity
38
Figure 4.2: Trends of the Euler number and the Reynolds number with increasing
characteristic velocity
4.4 Trends of Dimensionless Numbers
39
– The trend in the capillary wall which has a lower permeability than the rest
of the region in the porous medium
– The trend in the tissue region
− For the soil-air model describing evaporation
– The trend for the liquid phase
– The trend for the gas phase
With the given definition of the dimensionless numbers and the graph, it can be
inferred that the lines along which the dimensionless numbers are unity represent
the balance of forces involved and a trend towards one direction represents the
increase of one particular force over the other. The numbers, which are plotted for a
representative range of characteristic velocities lie in different quadrants of the graph.
This is because, to analyse the forces at the system level, the system length is chosen
for the calculation of the dimensionless numbers. The two applications have two
different system lengths and hence different forces are important at these lengths.
From the graphs, in general the following conclusions are made about the porousmedium part of the two models
− For the same velocity range, the gravity force is more important in the soilair model, because of the bigger length scale, than in the capillary-tissue model.
From this, a simplifying assumption can be made to the latter model by neglecting
the influence of gravity.
− Within the soil-air model, the gravity force is more important to the liquid than
to the gas owing to the density differences between the two fluids.
− Increasing permeability decreases the effect of viscous forces and increases the
effect of gravity and capillary forces.
4.4.2
Free-Flow Region
In the free-flow region the Reynolds and the Euler number are plotted against
each other. From this graph it is observed that for the same velocity range, the
capillary-tissue model is partly dominated by pressure forces and partly by inertia.
But this statement is clearly specific for one particular parameter set and boundary
conditions of one application. For a different parameter set which would increase the
influence of inertial forces in the system, the usage of the Stokes equation should be
done with caution as the Stokes equation may not be the proper equation to model
such a system with. The soil-air model on the other hand, appears to be dominated by
inertia compared to any other force. This can be explained considering the individual
applications. To achieve a certain velocity, there should be a higher pressure difference
40
in the capillary-tissue model than in the soil air model, owing to the density and the
viscosities of the fluids in the respective applications.
For the soil-air model, when the influence of gravity is considered along with the velocity, the Reynolds numbers tend to transitional flow (between laminar and turbulent)
or even to turbulent flow very quickly for all but a few very-slow velocity cases.
4.4.3
Dimensionless Numbers Common to Both Regions
The Peclet number and the Fourier number are the most important dimensionless
numbers as they provide insights into mass and heat transport processes in the system.
These numbers appear in the transport equation for mass and energy in both the freeflow and the porous medium regions. They are explained in the following sections.
4.4.3.1
The Peclet Number
The Peclet number arises from the general advection-diffusion equation and links the
time scales of diffusion and advection. Therefore if the time required to cover a distance
of lc by pure advection is
lc
tadvection =
(4.1)
vc
and the time required to cover a distance of lc by pure diffusion is
tdiffusion =
lc2
D
(4.2)
the Peclet number defines the ratio of the diffusive time scale to the advective time
scale. These time scales provide information on what process is relevant over a
distance lc . If the diffusion time is far less than the advection time, then the diffusion
process happens faster than the advection process and hence the system is said to be
diffusion dominated. Similarly, if the advection time is very small compared to the
diffusion time, then advection happens faster than diffusion and the system is then
advection dominated [20]. A detailed description of transport phenomenon can be
found in [7].
In this study, the processes are analysed at the system level, and hence the system
length is used in calculating the Peclet number. For the capillary-tissue model, using
the typical velocity ranges observed, the process time scales are summarised in table
4.7 and similarly a summary of the process times in the soil air model is given in table
4.8.
From the tables it is evident that the time for advection is generally much shorter over
the system length than diffusion especially in the capillary-tissue model. Hence the
4.4 Trends of Dimensionless Numbers
41
system can be considered advection dominated. From equations 4.1 and 4.2 the Peclet
number is written as
vc lc2
Pe =
(4.3)
Dlc
This formulation of the Peclet number makes it easier to visualise the effect of the
length scale on the speed of the processes. The diffusion time scale is quadratic with
length meaning that the time for diffusion increases four times when the length doubles
while the advection time is linear with length. To formulate the statement in another
way by solving for the length travelled by advection or diffusion in equations 4.1 and
4.2, the length travelled by advection is linearly proportional with time, but the length
travelled by diffusion is proportional to the square root of time. This means that
as time progresses or for very large systems, the effect of diffusion will be much less
observable than advection.
4.4.4
Fourier Number
The Fourier number arises from the non-dimensional form of the general transient
heat-conduction equation
∂ T̂
ˆ 2 T̂ .
= Fo∇
(4.4)
∂ t̂
In this equation the Fourier number (F o) is the ratio of thermal conduction and
thermal storage. This ratio indicates how a change in temperature propagates in the
system. For a very high thermal storage in the system (which would mean a small
Fourier number), a change in temperature at one end of the system may not have
propagated to the other end in the observation time. For large Fourier numbers, the
system can be assumed to have reached steady state. In general it can be stated
that the Fourier number gives the importance of the conduction to storage at the
observation time over the observation length.
The importance of the Fourier number in the soil-air model has to be further
investigated. Such an investigation is not in the scope of the current work.
From this analytical overview of the dimensionless numbers, the next step of implementing the dimensionless equations and the coupling conditions in a numerical framework
is discussed in the following Chapter.
42
Table 4.6: Parameter values for two model applications
Parameter Name
Capillary-Tissue
Model
Soil-Air Model
Permeability [m2 ]
10−18 (tissue),
10−24 (capillary
wal)
10−10
kg
Density [ m
3]
1030(interstitial
fluid) 1050(blood)
1000(water),
1.189(air)
Viscosity [P a.s]
0.0012(interstitial
fluid)
0.0021(blood)
0.0013(water),
1.71 × 10−5 (air)
Characteristic length [m]
3 × 10−4
0.25
Characteristic
velocity
m
(porous medium) [ s ]
1×10−10 to 1×10−5
1×10−10 to 1×10−5
Characteristic velocity (free
flow) [ ms ]
1 × 10−3 to 1 × 10−2
1 × 100 to 1 × 101
Diffusion coefficient (porous
2
medium) [ ms ]
5.12 × 10−14
1.9 × 10−9
Diffusion coefficient (free
2
flow) [ ms ]
2.93 × 10−14
2.6 × 10−5
4.4 Trends of Dimensionless Numbers
43
Table 4.7: Process time scales for the capillary-tissue model
Time
Free Flow
Porous Medium
tadvection (s)
10−3 to 101
100 to 106
tdiffusion (s)
3 × 106
1.7 × 106
Table 4.8: Process time scales for the soil-air model
Time
Free Flow
Porous Medium
tadvection (s)
0.25 to 0.05
103 to 109
tdiffusion (s)
2 × 103
106 (liquid)
103 (gas)
Chapter 5
Numerical Model
Due to the complexity of the equations of the coupled model and partial differential
equations in general, analytical solutions are not available for all but a few very simplified cases. Therefore they are always solved numerically by discretizing them in space
and in time. In this case the equations were solved using DuMux , which is a framework
based on continuum mechanical concepts to simulate multiphase flow and transport
processes in porous media [1]. DuMux is built on top of DUNE or Distributed and
Unified Numerics Environment and makes use of the object-oriented programming
language C + + along with extensive use of template programming. In this Chapter,
the concept of implementation of the mathematical model into a numerical model is
discussed.
5.1
Weighted Residuals and the Box Method (FVFE Method)
The partial differential equations intended to be solved are discretised in time and
in space, which means that the solution for the equations is not calculated for every
mathematical point in the entire domain, but for certain discrete points in space and
in time. To achieve this, the exact solution f (u) of a problem, defined as
f (u) =
∂
u + ∇ · F (u) − q = 0
∂t
in the integral form over a domain G
Z
Z
Z
∂
u dG +
∇ · F (u) dG −
q dG = 0
G ∂t
G
G
(5.1)
(5.2)
is approximated to f (ũ), where f (ũ) represents the approximate solution of the equation at discrete points in space, in this case at the nodes of a Finite-Element mesh.
The resulting approximation means that the equation 5.1 is no longer equal to zero,
but to an value ε which is the error obtained by approximating f (u) to f (ũ).
45
46
f (ũ) = ε
with
ũ =
X
Ni ũ∗i
i ∈ nodes of the element
i
(5.3)
(5.4)
where ũ∗i is the approximated value of u at node i of the Finite Element mesh and Ni
is a linear basis function. The error ε is then weighted by a weighting function W such
that the integral of the product over the whole domain G should be equal to zero. So
Z
!
Wj · ε = 0
(5.5)
G
and
X
Wj =1.
(5.6)
j
where j is the nodes of the elements. substituting it into equation 5.2 we get
Z
Z
Z
∂ X
∗
Ni ũi dG +
Wj · [∇ · F (ũ)] dG −
Wj q dG = 0.
Wj
∂t i
G
G
G
(5.7)
By using the Gauss Divergence Theorem and the chain rule the above equation can be
simplified to
P
Z
Z
Z
Z
∂ Ni ũ∗i
i
Wj
dG +
Wj · [∇ · F (ũ)] · n dΓ +
∇Wj · F (ũ) G −
Wj q dG = 0.
∂t
G
∂G
G
G
(5.8)
The storage properties of the grid are limited only to the nodal points using the mass
lumping defined by
(R
R
W
dG
=
N dG = Vi i = j
j
lump
G
G i
Mi,j
=
(5.9)
0
i 6= j
where Vi is the volume of the Finite-Volume box Bi around the node i (see figure 5.1).
Substituting the mass lumping term into the equation 5.8, the discretised form of the
balance equation 5.7 is obtained. This is given as
Z
Z
∂ ũ∗i
Vi
+
[Wj · F (ũ)] · n dΓ +
∇Wj · F (ũ) dG − Vi · q = 0
(5.10)
∂t
∂G
G
The weighting function Wj is defined to be piecewise constant over the Finite-Volume
box Bi
(
1 x ∈ Bi
Wj (x) =
(5.11)
0 x∈
/ Bi
5.1 Weighted Residuals and the Box Method (FV-FE Method)
a)
47
b)
secondary FV mesh
FE mesh
Bi
i
k
scv i
Bi
node i
c)
Ek
i
Ek
k
k
bi
eij
k
k
nij
xij
j
Figure 5.1: Schematic diagram of the box method after [1] and [3]
so that the gradient of the weighting function is zero
∇Wj = 0.
This gives the final form of the discretised equation
Z
∂ ũ∗i
+
[Wj · F (ũ)] · n dΓ − Vi · q = 0.
Vi
∂t
∂G
(5.12)
Now for ũ ∈ {v, p, X κ }, ũ can be substituted by the approximating function for the
primary variables.
X
p̃ =
Ni p̃i ∗
(5.13)
i
ṽ =
X
Ni ṽ∗
(5.14)
Ni X̃ κ∗
(5.15)
∇Ni p̃i ∗
(5.16)
∇Ni ṽ∗
(5.17)
∇Ni X̂ κ∗ .
(5.18)
i
X̃ κ =
X
i
∇p̃ =
∇ṽ =
κ
∇X̃ =
X
i
X
i
X
i
A detailed derivation can be found in [11].
48
5.2
Temporal Discretization of Equations
To discretise the equations in time, a first order finite difference scheme or the implicit
Euler scheme is utilized. With this the time derivative of the variable u is directly
calculated by the difference of u between two times over the time interval. If over the
time interval
∆t = tn+1 − tn
the function f (u) changes then the derivative according to the implicit Euler method
∂u
un+1 − un
=
∂t
∆t
(5.19)
gives the value of f (u) at the time n + 1.
5.3
Discretised Equations of the Coupled Model
Utilizing the concepts and techniques for spatial and temporal discretisation described
above, the system of dimensionless equations of the coupled model are discretized and 2
exemplary equations are presented in the proceeding sections. The rest of the equations
can be discretized analogously.
5.3.1
Free Flow Mass Balance
The discretized mass balance equation has the form
!n+1
Z
X
(%̂g )n+1 − (%̂g )n
Vi
+
%̂g
Ni v̂ ∗i
· n dΓBi −
∆t
∂Bi
i
!n+1
Z
X
p
tc
c
ˆ i p̂∗i
αh2
∇N
· n dΓBi − Vi αh2
q · n = 0, (5.20)
%c vc lc
%
v
l
c
c
c
∂Bi
i
where there are extra terms compared to the mass balance equation found in table 4.3.
This is because, for stability reasons, the mass balance is stabilized with the divergence
of the momentum balance equation and a stabilization factor [9]
∇ · (∇ · [p − µ∇v]) = ∇ · ∇p − µ∇2 v ,
(5.21)
The above equation is in the dimensional form. After simplification with the Divergence
Theorem and an additional stabilization factor αh2 given by
1
αh2 = α bki + bkj ,
2
(5.22)
5.4 The Structure in DuMux
49
where bki and bkj are the volumes of the sub control volumes belonging to nodes i and
j of the element, the stabilization term to be added to the mass balance equation
simplifies to
!
Z
X
p
c
ˆ i p̂∗i · n dΓBi
∇N
(5.23)
αh2
%c vc lc
∂Bi
i
in the non-dimensional form which has been implemented. This is done for stability
reasons to connect the velocity in the mass balance equation to the pressure. A detailed
explanation is found in the work by [9].
5.3.2
Stokes Equation for Momentum Balance
The momentum balance equation in the free-flow region is given below in the discretized
form
!
P
P
X
(%̂g i Ni v̂ ∗i )n+1 − (%̂g i Ni v̂ ∗i )n
ˆ i p̂∗i −
Vi
+ Vi Eu
∇N
∆t
i
!
Z
X
1
ˆ i v̂ ∗i dΓBi − Vi 1 %̂ = 0. (5.24)
%̂
∇N
F r2
∂Bi Re
i
The special aspect of this implementation which is to be noted is the term of the
momentum fluxes which is split into pressure and shear stress contributions
1 ˆ
1 ˆ
ˆ
ˆ
ˆ
∇ · Eu p̂I −
∇v̂ g = Eu ∇p̂ − ∇ ·
∇v̂ g .
(5.25)
Re
Re
In this form, the pressure part is handled as a volume term and the shear stresses
are reduced by the Gauss divergence theorem and handled as a flux term. This is
then implemented and hence the equation 5.24 is obtained. Further details about the
implementation is found in [2].
5.4
5.4.1
The Structure in DuMux
Sub Models
The previously described equations and the other discretized equations have been implemented in DuMux in various sub models for the free flow and for the porous medium.
The sub models which were involved in the implementation are given in table 5.1. The
sub models mentioned were adapted to the dimensionless form from the dimensional
form for this study.
The general naming convention followed is that ’p’ indicates phases, ’c’ indicates components, the numbers indicate number of phases and components, the suffix ’ni’ indicates the model is non isothermal and and the suffix ’dl’ indicates that the model uses
dimensionless equations.
50
Table 5.1: Names of dimensionless models in DuMux
Remark
Porous Medium
Free Flow
Single
Phase
Multi
Phase
Flow and Transport Model
1p2cdl
2p2cdl
stokesdl
Transport Model
included in
flow model
included in
flow model
stokes2cdl
Energy Model
not needed
2p2cnidl
stokes2cnidl
5.4.2
The Coupling Operators
The sub models in table 5.1 describe flow and transport processes in individual sub
domains. These models then interact with each other due to the coupling conditions
which are implemented in the following operators
− coupling stokes2cdl to 1p2cdl by the operator 2cstokes1p2cdl, to model the
Capillary-Tissue problem,
− coupling stokes2cnidl to 2p2cnidl by the operator 2cnistokes2p2cndl to model the
Soil-Air problem.
A pictorial overview of the coupling domain is given in figure 5.2. In the figure it can
be observed that there are certain nodes which are belonging exclusively to the sub
domains, in these regions, they are modelled with the sub models described in table 5.1.
At the interface the fluxes and the coupling conditions are modelled by the operators
described in this section.
5.5
Implementation of the Coupling Concept
The equations of the coupling model are described in Chapter 3. The dimensionless
equations and coupling conditions are given respectively in tables 4.2, 4.3 and 4.4.
These conditions are implemented in a Dirichlet-Neumann like setup. In such a set
up of a coupled problem having two sub domains, one of the sub domains is modelled
as a Neumann boundary-value problem at the coupling interface and the other sub
domain is modelled as a Dirichlet boundary-value problem at the coupling interface.
The individual sub domain problems and the coupling conditions are assembled in one
5.5 Implementation of the Coupling Concept
51
Figure 5.2: Overview of the FE-FV grid for the coupling model
global stiffness matrix and the whole system is solved at each time step. Such a set
up influences the dimensionless models as will be explained in later sections.
The model studied is set up in such a way that the Stokes domain provides fluxes to
the Darcy domain though the integration points on the interface, and these fluxes are
then resolved in the Darcy domain and reflect changes in the primary variables. These
primary variables are then set as Dirichlet-like conditions at the coupling nodes to the
Stokes domain (refer figure 5.3). Due to this coupling concept there are certain issues
which have to be addressed in the sub-domain models, especially in the Stokes models.
5.5.1
Boundary Flux in the Stokes Domain
The simplified, stationary, Stokes equation for the momentum fluxes (without gravity
and source or sinks) is of the form
Z
Bi
ˆ g dBi −
Eu ∇p̂
Z
∂Bi
1 ˆ
∇v̂ · n dΓBi = 0
Re
(5.26)
52
Figure 5.3: Pictorial description of the coupling concept
which is further divided into contributions from the internal and the boundary regions
Z
| Bi
ˆ g dBi +
Eu ∇p̂
1 ˆ
− ∇v̂ · n dΓBinternal +
Re
∂Binternal
{z
}
internal
Z
1 ˆ
− ∇v̂ · n dΓBboundary = 0. (5.27)
Re
∂Bboundary
|
{z
}
Z
boundary flux q N
Here it is to be noted that the pressure part is implemented as a volume term due
to stability reasons. At the coupling interface, the fluxes which are to be given to
the Darcy domain are taken from the boundary functions. These boundary functions
calculate the residual to be given to the Darcy domain as the sum of all internal fluxes
in the Stokes domain. These internal fluxes do no contain the pressure (only the
pressure gradient has been implemented). But the coupling condition for the normal
component of the momentum balance reads
ppm
1 1 ˆ
∇v̂ g = [p̂]pm cff .
−p̂g I +
Eu Re
pc
(5.28)
Comparing the free-flow side of the above coupling condition to the boundary flux term
in equation 5.26, it can be seen that the pressure part is missing. Therefore the pressure
term is added as a correction to the boundary flux part. This pressure correction is
done at the coupling interface and at Neumann flux boundaries (refer figure 5.4). The
pressure-corrected boundary flux term is then implemented in a way that the coupling
5.6 The Capillary Tissue Model
53
Figure 5.4: Illustration of the handling of pressure as a part of momentum coupling
condition is mathematically fulfilled. This reads
Z
1 ˆ
− ∇v̂ · n dΓBboundary +
Re
∂Bboundary
{z
}
|
boundary f lux
Z
Z
ppm
[p̂]pm cff Eu dBi (5.29)
Eu p̂g n dBi =
pc
Bi
| Bi
{z
}
pressure correction
This is done as explained in the work by [2].
5.5.2
Stabilization at the Boundary
The stabilization term found in the mass balance equation of the free-flow region (equation 5.20) has been removed at the coupling interface prior to coupling.
5.6
The Capillary Tissue Model
In the following sections the set up of the coupled model for the Capillary-Tissue
problem is explained.
5.6.1
Motivation
The motivation for the Capillary-Tissue model comes from the necessity to better
understand the phenomenon of drug delivery in the human body. For any therapy
to be successful and efficient, the drug has to be delivered to the target (in this
54
Figure 5.5: Illustration of the Capillary-Tissue model, after [23]
case a tumor) and, as far as possible, should not be delivered to other healthy
regions of the body to minimize side effects. The therapeutic agent enters the body
through the blood stream and then penetrates into the tissue across a microvascular
wall. A more detailed physiological background and motivation is found in the work [3].
To describe the physical system briefly, the blood flow is driven by the heart and
the oxygenated blood coming from the heart is transported in arteries. The smallest
branching of these arteries before the veins, which carry the de-oxygenated blood back
to the heart, are the capillaries.
Within these capillaries, the injected therapeutic agent filtrates into the tissue. The
filtration can be expected to be non-uniform across the length of the capillary since
the pressure gradually decreases from the arterial to the venal end. Along the length
of the capillary if the pressure in the capillary goes below the pressure in the tissue,
reabsorption occurs. A schematic diagram is given in figure 5.5.
From this brief explanation a model can already be conceptualised in such a way that
the capillary region is modelled as the free-flow region, the tissue and the capillary wall
as the porous medium
5.6 The Capillary Tissue Model
55
Figure 5.6: Dimensions of the domain
5.6.2
Model Set Up
The model domain of the Capillary-Tissue model is given in figure 5.6. The model has
a length of 0.3 mm and a total height of 57 µm. The porous medium is 52 µm wide and
the free flow region is 5 µm wide. The capillary wall across which the transfer takes
place is resolved in the porous medium region by assigning a different set of parameters
to this region. A summary of the parameters used for the Capillary-Tissue is given in
table 5.2. Additional to these, the other parameters which are used in the model are
listed below
− intrinsic permeability K of the tissue is 4.43 × 10−18 m2 in all directions
− intrinsic permeability of the wall: Kxx = 6.5 × 10−24 m2 and Kyy = 6.5 × 10−21 m2
Table 5.2: Parameter overview of the Capillary-Tissue model
5.6.3
Parameter
Porous Medium
Free Flow
Density [kg/m3 ]
1030
1050
Viscosity [P a.s]
0.0012
0.0021
Diffusion coefficient
[m2 /s]
5.12 × 10−14
2.93 × 10−14
Boundary Conditions
The capillary region has the following boundary conditions
56
− Neumann no flow for the mass fraction on the top boundary
− Outflow condition for the mass fraction and Dirichlet condition for the pressure
on the right hand boundary
− Dirichlet condition for the mass fraction and velocity on the left boundary
− Coupling outflow condition on the bottom boundary
Additional to these boundary conditions, the transient term in the Stokes equation has
been neglected. It is assumed that gravity has no influence (inferred from the analytic
analysis done in Chapter 4) and is hence negligible in the equations. The tangential slip velocity at the interface from the Beavers-Joseph-Saffman condition set to zero.
The Beavers-Joseph-Saffman condition provides an expression for the slip velocity at
the interface. In the model considered, it is assumed based on the physiology of the
capillaries that the transfer of the therapeutic agent happens across the capillary wall
through discrete pathways, modelled in this case by the one-dimensional Poiseuille
flow. Resulting from this assumption, the only component of velocity in the Poiseuille
channels is in the y-direction. Therefore the x-component of the velocity, in this case
the slip velocity at the interface is taken to be zero and implemented as a Dirichlet
condition at the interface.
The tissue region has the following boundary conditions
− Outflow conditions for pressure and mass fraction on the left and the right boundary below the capillary wall
− Neumann no-flow conditions on the left and right boundary of the capillary wall
− Outflow conditions on the bottom boundary
− Coupling inflow conditions on the top boundary
A summary of the boundary conditions can be found in figure 5.7
5.7
Choice of Characteristic values
The choice of the characteristic values to be used for the model run is the most crucial
choice which has to be done. These choices are based on the scales of the processes of
interest of the study. In the considered model, there is a fast process in the free-flow
region and a slow process in the porous-medium region. Therefore, the first intuition
would be to choose two different characteristic length scales for the processes and two
different characteristic times. However, since the coupled system is solved as a whole
at each time step, this possibility is ruled out since different length scaling would give
5.7 Choice of Characteristic values
57
Figure 5.7: Boundary conditions of the domain
Figure 5.8: Dimensions of the domain with different characteristic lengths in the sub
domains
a distorted system (see figure 5.8).
Choosing the characteristic time differently based on the time scales of the different
process time scales in the sub domains is also eliminated as the implementation of
the model is such that the whole system, i.e., both the sub domains are solved at
every time step. Therefore choosing different characteristic times would mean that
individual sub domains would move through different times and hence the coupled
model would not calculate a physically meaningful solution.
Due to the limitations elaborated in the previous paragraphs, the characteristic quantities for one simulation run have been chosen to be consistent with one sub-model at
a time. Therefore, with this system, the output has to be investigated only for the
sub model for which the chosen characteristic quantities fit. For this model run, the
length of the system (0.2 mm) is chosen as the characteristic length and hence the
58
Figure 5.9: Model domain in dimensionless form, with lc = 0.2mm
Figure 5.10: Model domain in dimensionless form, with lc = 0.05mm
whole system is scaled by this characteristic length. The new dimensions after rescaling are illustrated in figure 5.9. This rescaling enables the calculation of dimensionless
gradients required for the flow, transport and coupling equations.
The characteristic time was chosen to fit the fast process happening in the free flow
domain and therefore the characteristic time is take to be very short, in this case the
time required for the mass fraction to travel from the left-hand Dirichlet boundary to
the right-hand outflow boundary (0.05 s).
For the second model run, the characteristic quantities were chosen to fit the porous
medium region. Therefore the system was scaled to the characteristic length of 0.05 mm
which is approximately the dimension of the model breadth-wise. The characteristic
time was chosen to be 1000 seconds. The resulting dimensions are given in figure 5.10.
The output of the model runs and the discussion is presented in the following Chapter.
Chapter 6
Results and Discussion
The Capillary-Tissue model was run with the boundary conditions and the model set
up described in the previous Chapter. As stated in Chapter 1, the objective of this
work is to qualitatively reproduce the behaviour of the dimensional Capillary-Tissue
model. This has been achieved and the resulting pressure and velocity distribution
and the transport of mass fraction at various simulation times in the free-flow and the
porous-medium regions are presented in figures 6.1 to 6.4
It is observed in the figures that there is an uniform gradient of pressure from left
to the right, with a dimensionless pressure drop of 1.0 over the system length. The
pressure is coupled to the pressure in the porous medium region across the capillary
wall which has a much lower permeability than the rest of the tissue. Therefore
there is a high pressure drop across the capillary wall in the normal direction and
a related y-velocity in the free-flow region. This value of the pressure drop and the
y-velocity changes along the coupling interface length wise according to the gradient
in the free-flow region with the highest pressure drop and y-velocity being at near
the left-hand boundary. The therapeutic agent introduced into the flow field at the
left boundary saturates the free flow region within a very short time. Then, due to
the pressure difference across the interface, the therapeutic agent enters the capillary
wall, where the transport is slowed down due to the lower permeability and porosity
of the capillary wall. Once past the wall, the therapeutic agent migrates downwards
and finally towards the right boundary at later time steps.
The pressure drop is the main driving force for the porous-medium region. The
diffusion is also an important process in the model, but from the calculation of
time scales of advection and diffusion relevant to the model, the effect of diffusion
is far lesser than the effect of advection in both regions. However, if diffusion is to
be compared relative to the sub domains, is should be considered along the major
process directions. The Peclet number obtained by the characteristic quantities
of the free flow region shows that, for longitudinal transport, the diffusion can be
59
60
Figure 6.1: Dimesnionless pressure and velocity distribution in the free-flow region.
61
Figure 6.2: Dimensionless pressure and velocity distribution in the porous-medium
region.
62
Figure 6.3: Transport in the free-flow region for t̂ = 0.005, 0.05, 0.07 and 0.3.
63
Figure 6.4: Transport in the porous-medium region for t̂ = 10, 145, 800 and 3062.
64
considered irrelevant to the problem in this direction. This is due to the relatively
high velocities and low diffusion in the free-flow region. The Peclet number obtained
by the characteristic quantities of the porous medium region shows that the Peclet
number for the transverse transport in the porous medium is by orders of magnitude
lower than the Peclet number for longitudinal transport in the free flow. Here, the
longitudinal Peclet number in the free flow is compared to the transverse Peclet
number in the porous medium. As long as the capillary is not fully saturated with the
therapeutic agent, the diffusion is very strong due to the very small capillary diameter.
But, the injection of the therapeutic agent is along the entire left boundary and the
capillary region is saturated so quickly that the transverse diffusion in the capillary
region has little contribution to the transport.
The second case which is not considered for a detailed study is the longitudinal
diffusion in the porous medium. This is left out since the therapeutic agent enters
the porous medium region along the top boundary and hence the gradient of the
mass fraction is stronger in the y direction compared to the x direction. Therefore,
diffusion, happening due to the gradient of the mass fraction is more prominent in
the y direction in the initial times. Due to constantly more infiltration into the tissue
on the left compared to the right of the domain (and the free flow begin already fully
saturated), there is a gradient set up in the porous medium along the x direction
eventually at later times. But this gradient is presumably very small compared to the
advective transport, supported by the high Peclet numbers and therefore, the diffusive
transport is not considered to be a prominent process in the longitudinal direction of
the porous medium.
A more interesting choice of characteristic quantities would be a set which characterises
the interface. This seems challenging at the moment for certain reasons. The interface
in this model is approximated as a simple interface and it does not have a thickness
and cannot store mass, momentum or energy [17]. Although the capillary model has
a capillary wall which acts as an interface, it is exclusively resolved in the porous
medium model. Therefore, at the interface, there is a discontinuity of the variables
and the dimensionless numbers. As mentioned before in [16], dimensional analysis has
a limitation to strong jumps or regime changes.
Chapter 7
Summary And Outlook
During the course of this work, the coupled model was theoretically analysed based on
representative parameter values. One of the models was implemented in the numerical
framework DuMux . The model was run with boundary and initial conditions such
that it qualitatively reproduces the output for a previously-modelled test case. The
challenges of choosing the characteristic values, such as length and time, for a
numerical simulation were addressed and the resulting limitations were outlined. The
output of the model was assessed considering the limitations.
The current work proves as a first step in understanding the processes at the interface
of the coupled model. Although this is the ultimate goal of such a dimensional
analysis, the preliminary information of the processes in the sub domains provide
already a better insight into the driving forces of the system. The dimensional analysis
also provides justifications for simplifying assumptions which have been made or
provides a hint to include other aspects which have been previously not considered.
The dimensional analysis carried out has the following scope for future work to be
carried out
− Extension to two-phase non-isothermal flow,
− A more complete consideration of the stress terms in the Stokes equation. For
now the Stokes equation has been modelled in with Newton’s law of viscosity,
but in highly viscous flows, terms accounting for volumetric dilatation and fluidelement deformation have to be considered.
− A detailed parameter study has to be carried out to understand how the model
parameters, not just the choices of characteristic quantities, change the behaviour
of the system. A dimensionless system of equations would help in such a situation
since it is possible to change dimensionless numbers and still get the desired
behaviour instead of changing individual parameters.
65
66
− A study of how individual dimensionless numbers change the behaviour of flow
and transport can be performed by scaling all other dimensionless numbers and
terms in a dimensionless equation to one except for the term which need to be
analysed. Then the influence of the investigated number or term on the general
flow or transport behaviour can be studied in detail.
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