Universität Stuttgart - Institut für Wasser- und
Umweltsystemmodellierung
Lehrstuhl für Hydromechanik und Hydrosystemmodellierung
Prof. Dr.-Ing. Rainer Helmig
Master’s Thesis
Numerical Analysis of Modeling Concepts
for Salt Precipitation and Porosity Permeability Evolution during Brine
Evaporation
Submitted by
Tianyuan Zheng
Matriculation number 2709257
Stuttgart, May 10, 2014
Examiners: Prof. Dr.-Ing.Rainer Helmig; apl.Prof. Dr.-Ing.Holger Class
Supervisor: M.Sc. Eng.Vishal Arun Jambhekar
Contents
1 Introduction
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Objective and Structure of the Work . . . . . . . . . . . . . . . . . . .
2 Fundamentals
2.1 Evaporation . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Stages of Evaporation . . . . . . . . . . . . . . . . . . . .
2.2.1 Pure Water Evaporation . . . . . . . . . . . . . .
2.2.2 Brine Water Evaporation . . . . . . . . . . . . . .
2.3 Representative Elementary Volume (Macro-scale) . . . .
2.3.1 Porosity . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Saturation . . . . . . . . . . . . . . . . . . . . . .
2.3.3 Permeability and Darcy’s Law . . . . . . . . . . .
2.3.4 Capillary Pressure . . . . . . . . . . . . . . . . .
2.3.5 Hydraulic Conductivity . . . . . . . . . . . . . . .
2.3.6 Relative Permeability and Extended Darcy’s Law
2.3.7 Constitutive Relationships . . . . . . . . . . . . .
2.4 Fluid Properties . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Phase Composition . . . . . . . . . . . . . . . . .
2.4.2 Density . . . . . . . . . . . . . . . . . . . . . . .
2.4.3 Viscosity . . . . . . . . . . . . . . . . . . . . . . .
2.4.4 Phase Equilibrium . . . . . . . . . . . . . . . . .
2.4.5 Vapor Pressure . . . . . . . . . . . . . . . . . . .
2.4.6 Dalton’s Law . . . . . . . . . . . . . . . . . . . .
2.4.7 Raoult’s Law . . . . . . . . . . . . . . . . . . . .
2.4.8 Henry’s Law . . . . . . . . . . . . . . . . . . . . .
2.5 Properties of Aqueous Electrolyte Solutions . . . . . . .
2.5.1 Mole Fraction . . . . . . . . . . . . . . . . . . . .
2.5.2 Mass Fraction . . . . . . . . . . . . . . . . . . . .
2.5.3 Concentrations . . . . . . . . . . . . . . . . . . .
2.5.4 Molality . . . . . . . . . . . . . . . . . . . . . . .
2.5.5 Molarity . . . . . . . . . . . . . . . . . . . . . . .
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2
2
5
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CONTENTS
2.6
2.7
I
2.5.6 Activity . . . . . . . . . . . . . . . . .
2.5.7 Ionic Strength . . . . . . . . . . . . . .
2.5.8 Activity Coefficients for Individual Ions
2.5.9 Saturation Index . . . . . . . . . . . .
2.5.10 Solubility Product . . . . . . . . . . .
Thermodynamic Properties . . . . . . . . . .
2.6.1 Enthalpy and Internal Energy . . . . .
2.6.2 Heat Capacity . . . . . . . . . . . . . .
Transport Processes . . . . . . . . . . . . . . .
2.7.1 Advection . . . . . . . . . . . . . . . .
2.7.2 Di↵usion . . . . . . . . . . . . . . . . .
3 Model Concept and Implementation
3.1 Model Concept . . . . . . . . . . . . .
3.2 Mathematical Model . . . . . . . . . .
3.2.1 Assumptions . . . . . . . . . . .
3.2.2 Compositional Multiphase Flow
3.2.3 Primary Variables . . . . . . . .
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4 Numerical Model and DuMux
4.1 Space Discretization: Weighted Residual and
4.2 Time Discretization . . . . . . . . . . . . . .
4.3 The Structure in DuMuX . . . . . . . . . . .
4.3.1 The Numerical Simulator DuMuX . .
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the Box Method
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23
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27
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28
35
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37
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40
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5 Case Studies
5.1 General Problem Setup . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 Isothermal & non-isothermal case . . . . . . . . . . . . . . . . .
5.2 Layman’s Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Grid Convergence . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.2
- K Relationship . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.3 Isothermal and Non-isothermal Conditions (Layman’s Approach)
5.3 Chemistry Driven Approach . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Grid Convergence . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2
- K Relationship . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.3 Isothermal and Non-isothermal Conditions (Chemistry Driven
Approach) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Comparison of Layman’s Approach and Chemistry Driven Approach . .
5.5 Heterogeneous Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.1 Heterogeneous Scenario Setup . . . . . . . . . . . . . . . . . . .
5.5.2 Horizontal Heterogeneous Case . . . . . . . . . . . . . . . . . .
5.5.3 Patchy Heterogeneous Case . . . . . . . . . . . . . . . . . . . .
41
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49
52
55
55
56
57
CONTENTS
5.5.4
II
Random Heterogeneous Case
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59
6 Summary and Outlook
6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
63
64
List of Figures
1.1
1.2
1.3
Severe salinization problems in central Tunisia . . . . . . . . . . .
Saline and sodic soils in the world . . . . . . . . . . . . . . . . . .
Relevant interface processes for evaporation driven salt precipitation
soil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. .
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in
. .
3
3
Evaporation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conceptual idea of di↵erent stages of saline water evaporation from
porous media.Grains building the porous matrix are represented in gray
color, blue color represents saline water and white color represents the
dry matrix. Solid and dotted curved black arrows represent di↵usive
vapor transport through gas phase and salt crust respectively. Straight
black arrows represent capillary-driven fluid flow toward the evaporation front. Green clusters represent the precipitated salt within the pore
space or on top of the porous-medium. . . . . . . . . . . . . . . . . . .
2.3 Evaporation rate of Pure water and Saline water . . . . . . . . . . . . .
2.4 Relation between an averaged quantity and the size of the averaging
volume after [7] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Schematic illustration of transition from micro-scale to REV scale . . .
2.6 pc Sw curve for the Brooks-Corey ( = 1.29, pd = 619.45) and the Van
Genuchten relationships (↵ = 0.001, n = 3, m = 0.77) after [19] . . . . .
2.7 Typical kr Sw relationship based on Brooks-Corey and Van Genuchten
after [19] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8 Fluid volume deformed with angular velocity by shear stress . . . . . .
2.9 Applicability of Henry’s law and Raoult’s law for a binary gas-liquid
system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.10 Advection and di↵usion . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.1
2.2
3.1
4.1
4
9
9
10
12
16
17
19
22
26
Model concept with single phase in the free flow that interacts with two
fluid phases in the porous media, after [28] . . . . . . . . . . . . . . . .
28
Schematic diagram of the box method
39
III
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LIST OF FIGURES
4.2
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.9
5.10
5.10
5.12
5.13
5.13
5.15
5.16
5.16
5.17
5.18
5.19
5.20
5.21
5.22
5.23
5.24
5.25
5.26
5.27
5.28
solution technique. Left: Explicit solution. Right: Implicit solution.
Filled circles represent nodes with known primary variables, empty circles represent unknowns. . . . . . . . . . . . . . . . . . . . . . . . . . .
model setup for homogeneous case . . . . . . . . . . . . . . . . . . . . .
Temperatures along the domain in isothermal condition . . . . . . . . .
Temperatures along the domain in non-isothermal condition . . . . . .
Grid convergence test for non-isothermal layman’s approach . . . . . .
Permeability factor under di↵erent porosity permeability relationship
models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Grid convergence test for non-isothermal chemistry driven approach . .
Permeability factor under di↵erent porosity permeability relationship
models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Porosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Porosity and permeability setup for horizontal heterogeneous case . . .
Porosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Porosity and permeability setup for patch heterogeneous case . . . . . .
Porosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Porosity and permeability setup for stochastic heterogeneous case . . .
Porosity for the horizontal heterogeneous case . . . . . . . . . . . . . .
Gas phase saturation for horizontal heterogeneous case . . . . . . . . .
NaCl mass fraction for horizontal heterogeneous case . . . . . . . . . .
Solidity for horizontal heterogeneous case . . . . . . . . . . . . . . . . .
Porosity for the patch heterogeneous case . . . . . . . . . . . . . . . . .
Gas phase saturation for patch heterogeneous case . . . . . . . . . . . .
NaCl mass fraction for patch heterogeneous case . . . . . . . . . . . . .
Solidity for patch heterogeneous case . . . . . . . . . . . . . . . . . . .
Porosity for the random heterogeneous case . . . . . . . . . . . . . . .
Gas phase saturation for random heterogeneous case . . . . . . . . . .
NaCl mass fraction for random heterogeneous case . . . . . . . . . . . .
Solidity for random heterogeneous case . . . . . . . . . . . . . . . . . .
IV
40
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44
46
50
51
56
56
56
56
56
56
57
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57
58
58
58
58
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60
60
60
61
61
62
62
List of Tables
1.1
saline soils in the world . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1
3.2
3.3
Values of parameters for the Debye-Hückel and Truesdell-Jones equation 30
Parameters for the Truesdell-Jones equation . . . . . . . . . . . . . . . 31
Primary variables for di↵erent phase states and switch criteria . . . . . 36
5.1
5.2
Values of the reference case and considered parameter . . . . . . . . . .
Salt content varieties in non-isothermal and isothermal conditions under
layman’s approach after 20.83 days . . . . . . . . . . . . . . . . . . . .
Comparison of non-isothermal and isothermal model under chemistry
driven approach after 20.83 days . . . . . . . . . . . . . . . . . . . . . .
Comparison of the salinity calculated by layman’s approach and chemistry driven approach . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3
5.4
V
2
42
48
53
55
Nomenclature
[Cl ] Activity of Cl ion
[mol/lt]
[N a ] Activity of Na ion
[mol/lt]
↵
Specific phase in the medium
↵l
Longitudinal dispersivity
[m]
↵t
Transversal dispersivity
[m]
The pressure di↵erence
[Pa]
p
k
[-]
Activity coefficient
[-]
Brooks-Corey pore size distribution parameter
[-]
Heat conductivity through porous medium
[K/ms]
µ
Dynamic viscosity
[kg/ms]
rp
Pressure gradient
[Pa]
⌦
Saturation index
[-]
Porosity of the medium
[-]
pm
⌧pm,↵ tortuosity of a phase in porous medium
[-]
%
Density of a fluid phase
[kg/m3 ]
A
The cross sectional area
[m2 ]
C
Volumetric concentration
[kg/m3 ]
aCl
cN
Molar concentration
s
[mol/m3 ]
D↵
Molecular di↵usion coefficient of phase ↵
[m2 ]
F
Number of degrees of freedom
[m2 ]
VI
LIST OF TABLES
1
h↵
Specific enthalpy
I
Ionic strength
k
Specific component in porous medium
[-]
kr
Relative permeability
[-]
KN aCl Solubility procduct of NaCl
[J/kg]
[mol/kg]
[(mol/lt)2 ]
M
Molarity
[mol/m3 ]
m
Molality
[mol/kg]
n
Number of phase present
pc
Capillary pressure
[Pa]
pc
Pressure of the wetting phase
[Pa]
pd
Brooks Corey entry pressure
[Pa]
pn
Pressure of the nonwetting phase
[Pa]
Q
The discharge
q solidity Source/sink term of salt
[-]
[m2 /s]
[mol/s]
Se
E↵ective saturation
[-]
S↵
Saturation of a phase ↵
[-]
Snr
Residual non-wetting phase saturation
[-]
Swr
Residual saturation of the wetting phase
[-]
T
Temperature
[K]
u↵
Internal energy
[J]
Xi
Mass fraction of component i
[-]
xi
Mole fraction of component i
[-]
g
Gravitational vector
K
Intrinsic permeability
v
Darcy flow velocity
[m/s2 ]
[m2 ]
[m/s]
Chapter 1
Introduction
1.1
Motivation
Degradation of the soil productivity due to salinization is a major global issue. Soil
salinization is an old concern, but enhanced irrigation practices has been rapidly increasing its intensity & magnitude in the past century. The problem of salinization is
normally wide spread in arid, semi-arid and coastal regions where saline field is extensively used for agriculture. In some parts of the world soil salinization is so serious that
the agricultural lands have been abandoned. It has been already widely known that
the excess of the irrigated saline water contributes significantly to evaporation from
the bare soil surface and lead to the salinization. The salinity a↵ected areas around
the world are summarized in Table 1.1
Region
Africa
Near and Middle East
Asia and Far East
Latin America
Australia
North America
Europe
E↵ected Area (106 ha)
69.5
53.1
19.5
59.4
84.7
16.0
20.7
Table 1.1: saline soils in the world
Figure 1.1 gives an visual scene of the severe salinization in central Tunisia. Figure 1.2
describes the saline and sodic soils distribution worldwide. It is assumed that around
77 milion hectares of soil is salinised by human activities. The evaporative drying
of soil is normally influenced by the interaction between the free flow and the flow
and transport processes in the porous media (see Figure 1.3). For the free flow part,
wind velocity, temperature and radiation all have a strong impact on the evaporation
1.1 Motivation
3
Figure 1.1: Severe salinization problems in central Tunisia
Figure 1.2: Saline and sodic soils in the world
1.1 Motivation
4
process. Furthermore, turbulence in the free flow causes mixing of air, enhances
the vapor transport in free-flow and builds a boundary layer at the interface which
significantly a↵ects the vapor transport. For the porous media part, the transport
of the dissolved salt is influenced by viscous forces, capillary forces, gravitational
forces and advective and di↵usive fluxes. Continuous evaporation promotes salt
accumulation and precipitation resulting in soil salinization. Finally soil salinization
would influence the porous medium properites.
Soil salinity is the salt content in the soil and salinization is the process of increasing
the salt content. It can be caused by natural processes such as mineral weathering or
some artificial processes such as irrigation. Soil salinization most pronounced at the
soil surface.
Figure 1.3: Relevant interface processes for evaporation driven salt precipitation in soil
Soil Salinization can be dominant under following conditions:
The presence of soluble salts, such as sulfates, chlorides, sodium, calcium, and
magnesium in the soil
A high water table
A high rate of evaporation
Low annual rainfall
Soil salinization can a↵ect plant growth in several ways:
Decreased water uptake High salts concentration results in high osmotic potential
of the soil solution, so the plant has to use more energy to absorb water. Under
extreme salinity conditions, plants may be unable to absorb water and will wilt,
even when the surrounding soil is saturated.
1.2 Objective and Structure of the Work
5
Ion-specific toxicity When a plant absorbs water containing ions of harmful salts
(e.g. Sodium, Chloride, excess of Boron etc.), visual symptoms might appear,
such as stunted plant growth, small leaves, marginal necrosis of leaves or fruit
distortions.
Interference with uptake of essential nutrients An imbalance in the salts content
may result in a competition between elements. This condition is called ”antagonism”, i.e. an excess of one ion limits the uptake of another ion. For example,
excess of chloride reduces the uptake of nitrate, excess of phosphorus reduces the
uptake of manganese, and excess of potassium limits the uptake of calcium.
Sodium e↵ect on soil structure In saline soils, sodium replaces calcium and magnesium, which are adsorbed to the surface of clay particles in the soil. Thus,
aggregation of soil particles is reduced, and the soil will tend to disperse. When
wet, a sodic soil tends to seal, its permeability is dramatically reduced, and thus
water infiltration capacity is reduced as well. When dry, a sodic soil becomes
hard has the tendency to crack. This may result in damages to roots.
In this master thesis we implemented di↵erent model concepts to deal with salt
precipitation related to the evaporation of saline water from an unsaturated porous
medium in contact with atmosphere. Especially we will go into details about the
relationships between porosity and permeability and analyze di↵erent model concepts
under brine evaporation processes.
Numerical modeling is a valuable and widely used tool to analyze the flow and
transport in porous-media and transfer it to the application scale. However, numerical
modeling of saline water evaporation from a real heterogeneous porous media flow
system is a difficult task. Many scientists have devoted their e↵orts to develop
mathematical models and analyze saline water evaporation [33], [38], [40], [25], [8].
The experimental and numerical models of saline water evaporation have been studied
after [36], [31], [22] and [3]
Model concepts for a free flow - porous media coupling single phase system was discussed after [14], [26], [35]. A macro scale model concept for coupling two-phase
compositional porous media flow with single-phase compositional laminar free flow
is explained by [28], [2].
1.2
Objective and Structure of the Work
In this Master thesis we describe evaporation driven salinization by developing an
REV scale model concept for multi-phase multi-component porous-media flow. In the
following work, the model is implemented in DuMuX based on Vishal Jambhekar’s PhD
project [20] and the report is structured in the following manner.
1.2 Objective and Structure of the Work
6
The fundamental definitions and concepts are given in Chapter 2.
The conceptual model along with the simplifying assumptions and the mathematical model are described in Chapter 3.
A brief introduction to the numerical discretization, the implementation of the
equations in the DuMuX framework and certain specific details of the implementation are given in Chapter 4.
The problem set up, boundary conditions and the results are discussed in Chapter
5.
The summary of the current work and the scope for the future work are described
in Chapter 6.
Chapter 2
Fundamentals
This chapter contains the fundamental definitions required to understand related to
evaporation and multiphase flow processes in porous media.
2.1
Evaporation
Evaporation (see Figure 2.1) is vaporization of a liquid phase into gas phase. For
molecules of a liquid to evaporate, they must be located near the surface, be moving
in the proper direction, and have sufficient kinetic energy to overcome liquid-phase
intermolecular forces. And the proportion of the molecules which fulfill the condition
determines the rate of evaporation. When only a small proportion of the molecules meet
these criteria, the rate of evaporation is low. Since the kinetic energy of a molecule is
Figure 2.1: Evaporation
proportional to its temperature, evaporation proceeds quicker at higher temperatures.
2.2 Stages of Evaporation
2.2
Stages of Evaporation
2.2.1
Pure Water Evaporation
8
Based on the research before [13], [15], [29], [24], the classical approach for evaporation
of pure water from homogeneous porous media is considered into the following stages:
Stage 1 (S1):This stage is characterized by a relatively constant and high evaporation rate. Here, the evaporation process is capillary-driven and continues as
long as hydraulic connectivity exists between the receding drying front and the
free-flow-porous-media interface to sustain a high evaporation rate.
Stage 2 (S2):This stage starts when the hydraulic conductivity is disturbed and
evaporation is restricted by vapor di↵usion in the porous medium.
Stage 3 (S3):This stage is characterized by a low evaporation rate governed by
di↵usive vapor transport. It is established when the evaporation front is far below
the free-flow-porous-media interface.
2.2.2
Brine Water Evaporation
Under many natural conditions in real world (e.g. in arid and coastal fields) water
contains large amount of dissolved salts. Salt precipitation and its e↵ect on saline
water evaporation under homogeneous and heterogeneous conditions were examined
after [36], [30], [18] [29]. The saline water evaporation process can be described by
three stages as follows:
Stage 1 (SS1): This stage is characterized by high evaporation rate with minor
and gradual reduction, due to variations in the osmotic pressure.
Stage 2 (SS2): During this stage, the evaporation rate falls subsequently as a
result of salt precipitation and salt crust formation on top of the porous-medium.
Stage 3 (SS3): This stage exhibits a constant low evaporation rate governed by
di↵usion across the salt crust.
The stages SS1, SS2 and SS3 seems phenomenologically similar to the classical evaporation stages (S1, S2 and S3), in fact they have di↵erent mechanisms. For e✏orescent salt
precipitation, stages SS2 and portion of SS3 happen while the hydraulic conductivity is
sufficient to support the first stage of evaporation, [29] (see Figure 2.2). Furthermore,
Extended S1 stage was observed in heterogeneous case due to higher capillary forces
in the finer parts of porous media [24]. The comparison of evaporation rate between
pure water evaporation and brine water evpoartion is given in Figure 2.3
2.2 Stages of Evaporation
9
Figure 2.2: Conceptual idea of di↵erent stages of saline water evaporation from porous
media.Grains building the porous matrix are represented in gray color, blue color represents saline water and white color represents the dry matrix. Solid and dotted curved
black arrows represent di↵usive vapor transport through gas phase and salt crust respectively. Straight black arrows represent capillary-driven fluid flow toward the evaporation front. Green clusters represent the precipitated salt within the pore space or
on top of the porous-medium.
Figure 2.3: Evaporation rate of Pure water and Saline water
2.3 Representative Elementary Volume (Macro-scale)
2.3
10
Representative Elementary Volume (Macroscale)
Figure 2.4: Relation between an averaged quantity and the size of the averaging volume
after [7]
The model concepts in this work depends on the consideration of the porous medium
as a continuum consisting of a solid and one or more fluid phases being present at every
point in space. This means that the fluid flow is not considered in individual pores but
is averaged over a representative elementary volume (REV) and described by averaged
velocities. The reason for this average is that the microscale modeling for instance,
the modeling of the complex velocity field inside every pore is not practicable for
the most interesting questions since already a model domain of one cubic meter may
contain millions or billions of pores. Thus, neither the exploration of their topology nor
the calculation of microscale flow through every single pore lies within todays problems.
As basis for a correct averaging, a size of the REV has to be determined. A minimum
size is reached, when the average of the considered quantity keeps constant, i.e. that
slight extensions or reductions of the considered volume do not lead to significant
changes of the averaged quantity. Figure 2.4 describes the relation between the
size(radius r) of an averaging volume and the value of an averaged physical quantity
on the example of the porosity . For a disappearing radius, the porosity is either
zero or one as becomes clear from Figure 2.4. With increasing radius, the porosity
fluctuates until it reaches a constant value at rmin . If we consider a heterogeneous
medium, the porosity starts to change again when the volume is large enough to hit
heterogeneities as, e.g., sand lenses in a loamy soil. The same considerations as for the
porosity can be applied to other physical quantities giving the possibility to replace
the actual medium by a fictitious continuum in which we may assign values of any
property to any point in space [6]. That is, all physical quantities are now illustrated
by functions continuous in space, where it has to be mentioned that many quantities
2.3 Representative Elementary Volume (Macro-scale)
11
which are of importance for the following derivations such as porosity, permeability
and saturations only exist in this continuum approach, making the introduction of
the REV concept indispensable. Finally it has to be mentioned that minimum and
maximum REV sizes may vary between di↵erent parameters (such as porosity or
permeability) or processes (such as fluid flow or heat conduction).
In Figure 2.5 it can be seen that for the micro scale the local distribution of di↵erent phases is available but for the macro scale (REV) the local distribution of di↵erent
phases is not available. The transition from the micro-scale to the macro-scale consideration (see Figure 2.5) looks to momentum equations(e.g., Darcy’s law) with parameters
(e.g., saturation). In the following sections, the Darcy equations and parameters will
be talked.
2.3.1
Porosity
The porosity is defined as the ratio of the pore space to the total volume of the REV.
=
volume of pores in one REV
volume of the REV
(2.1)
In some applications, the porosity of the porous media is influenced by temperature
of the medium and pressure applied on it (for example , external pressure). But for
modelling flow through porous media, the porosity is considered constant (i.e. porousmedia is assumed to be rigid).
2.3.2
Saturation
As the porosity, the saturation S is a dimensionless number describing a volume ratio.
The saturation of phase ↵ is defined as the fraction of pore space, which is occupied by
the phase. The pore space is always filled by some fluid and the volume of all phases
must sum up to the pore space volume and therefore all saturations must sum up to
unity:
volume of phase ↵
S↵ =
(2.2)
volume of the pores
If the saturation of one phase reaches unity which means all the pore space is filled
with this phase, the porous media is commonly referred to as being saturated or fully
saturated by this phase.
If in real case, one phase is displaced by another, not all of the first phase can be
removed. On the opposite, there is a certain saturation which is held inside the medium.
This is due to capillary forces acting between the solid and the fluid phases. The
saturation of a phase which can not be removed through displacement by another
phase is called residual saturation Sr↵ It depends on the shape and size of the pores,
the temperature and the displacement itself.
2.3 Representative Elementary Volume (Macro-scale)
Figure 2.5: Schematic illustration of transition from micro-scale to REV scale
12
2.3 Representative Elementary Volume (Macro-scale)
2.3.3
13
Permeability and Darcy’s Law
Fluid flow in porous media is determined by pressure di↵erences. The relationship
between the pressure di↵erence p and the discharge Q of a fluid through a onedimensional column without considering gravity influences is given by the permeability
of the porous media and the viscosity of the fluid:
Q
1
p
= K
A
µ
l
(2.3)
where K represents the permeability and A the cross-section area of the column and l
its length . The ratio of discharge over area is defined as Darcy velocity. It is also often
referred to as filtration velocity or specific discharge and the is normally the quantity
when speaking of velocity of a fluid in this work. Equation 4.3 is a definition of a scalar
permeability in a one-dimensional fashion. However, in reality, the permeability can
be anisotropic, that is why it is normally written as a tensor K. Furthermore, not only
the pressure gradient but also gravity influences the fluid flow and thus Darcy’s law for
single phase flow is expressed in a vectorial notation as
v=
1
K(rp + g%)
µ
(2.4)
where the negative sign in front of the pressure gradient points out that flow is in the
direction of falling pressures. As stated above, the Darcy velocity is not the actual
velocity of the fluid inside the porous medium, because it is an area average. To
approximate the actual movement of the fluid in the flow direction, the Darcy velocity
is divided by the e↵ective porosity e to get the result of average velocity
va =
v
(2.5)
e
The e↵ective porosity considers the pore volume where flow actually occur which
means that dead-end pores are excluded. For the sake of simplicity, however, the
e↵ective porosity ( e ) is normally approximated by the known porosity .
The validity of Darcy’s law is limited to creeping flow conditions, which are in general
defined by Reynolds numbers below one. The dimensionless Reynolds number is defined
as
|Va | · d
Re =
(2.6)
⌫
where d denotes characteristic length and ⌫ the kinematic viscosity. The characteristic
length would be a pore diameter. Since this is not easily measured, In real system, the
median grain diameter d50 is widely used. Although creeping flow is limited to Re < 1,
the influence of inertia and turbulent e↵ects is negligible for Reynolds numbers up to
10 after [17]
2.3 Representative Elementary Volume (Macro-scale)
2.3.4
14
Capillary Pressure
At the interface of two fluid phases, there exist cohesive forces between the molecules
in one phase and adhesive forces between molecules of di↵erent phases. The di↵erent behavior between the cohesion and adhesion causes free interfacial energy which
appears as interfacial tension after [7]. This interfacial tension determines capillarity
which is the di↵erence of the wetting phase pressure and non-wetting phase pressure.
It can be derived from equilibrium considerations between the fluid-fluid interface and
the solid medium.
4 cos✓
pc =
(2.7)
d
where denotes the surface tension of the fluid, ✓ contact angle between the fluid and
the soil and d the diameter of the pore. Considering the equilibrium of forces between
the two phases, it is clear that the non-wetting phase pressure has to be balanced with
both the wetting phase pressure and the capillary pressure.
pn
2.3.5
pw = pc = pc (Sw )
(2.8)
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 the fluid (the viscosity and density of the fluid). The
hydraulic conductivity Kf can be expressed as
Kf = K
%g
µ
(2.9)
where µ is the viscosity of the fluid, g is the gravitational acceleration and K is the
intrinsic permeability tensor in which a three dimensional system is given below
2
3
Kxx Kxy Kxz
K = 4Kyx Kyy Kyz 5
(2.10)
Kzx Kzy Kzz
The above case is for a single phase flow. Practically, there is usually more than one
fluid phase in the pore space which makes only a certain ratio 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 for multiple phases . In such condition, the e↵ect of
existance of multiple phases’ permeability is taken into account as below.
2.3.6
Relative Permeability and Extended Darcy’s Law
In the case of multiphase flow, the di↵erent phases influence each other in their flow
properties. This is taken into account by introducing a dimensionless factor kr , the
2.3 Representative Elementary Volume (Macro-scale)
15
relative permeability. A basic consideration for the determination of this factor is
that one phase will occupy parts of the flow paths which are therefore blocked by the
second phase and vice versa. If the phases are assumed to be randomly distributed
over all the flow paths, it turns out reasonable to linearly scale the permeability of
a phase with its saturation and therefore set the relative permeability to kr↵ = S↵ .
However due to capillary e↵ects the distribution of the phases will not be random but
more specific. Capillary pressures are low in wide pores which allows a non-wetting
phase to penetrate these at first, however a wetting phase will stay inside narrow pores
when the saturation drops. The narrower a pore, the higher viscos friction.
The relative permeability concept is incorporated into Darcy’s law by multiplication of
the velocity of phase ↵ by its relative permeability kr↵
v↵ =
↵ K(
rp↵ + %↵ g)
(2.11)
where ↵ = kµr↵
is the mobility of phase ↵ and p↵ points out that the pressure of
↵
di↵erent phases may not be equal (caused by capillary pressure).
2.3.7
Constitutive Relationships
2.3.7.1
The pc - Sw Relation
On the REV scale, when the saturation of the non-wetting phase increases, the saturation of the wetting phase decreases. This can be accounted for that the non-wetting
phase first penetrates into the largest pores (since they have the lowest capillary
pressure according to Equation 2.7 and therefore have the highest probability for the
non-wetting phase to enter) and pushes the wetting phase into smaller pores. This
phenomenon happens until non-wetting phase has pushed the wetting phase into the
pores where the pressure of the non-wetting phase can not displace the wetting phase
out of the pores. This saturation of wetting phase is called the residual saturation.
From equation 2.7 it can be inferred that the capillary pressure increase with
decreasing pore diameter. This means that there is an increase in capillary pressure
due to the decrease in wetting phase saturation. Thus the capillary pressure in a
multi-phase system is a function of the wetting phase saturation. Furthermore, the
capillary 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 after [7] and equation 2.8
The Brooks and Corey model [9] and Van Genuchten model [39] are the two of the
commonly used models to find the pc Sw relation. The capillary pressure - saturation
relations for the Brooks Corey model is given by
✓ ◆
Sw Swr
pd
Sef f (pc ) =
=
(2.12)
1 Swr
pc
2.3 Representative Elementary Volume (Macro-scale)
16
and for the Van Genuchten model is given by
Sef f (pc ) =
Sw Swr
= [1 + (↵ · pc )n ]m
1 Swr
(2.13)
where is the fitting parameter of the Brooks-Corey model, with pd being the entry
pressure and ↵, n and m are the fitting parameters for the Van Genuchten model (see
Figure 2.6).
Figure 2.6: pc Sw curve for the Brooks-Corey ( = 1.29, pd = 619.45) and the Van
Genuchten relationships (↵ = 0.001, n = 3, m = 0.77) after [19]
2.3.7.2
The kr - Sw Relation
For extending the concept of permeability to multi-phase flows, the relative permeability has been briefly explained. it follows that for each saturation value of a certain
phase, the phase occupies a distinct pathway through the porous media, therefore,
influencing the flow of the other phase and its relative permeability. The relative permeability is thus a function of the wetting phase saturation Sw . Based on the capillary
pressure - saturation by Brooks and Corey [9] and van Genuchten [39] explained in the
previous section to get the e↵ective saturation Se , the relative permeability of the wetting and non-wetting phase is determined by the Brooks - Corey model as a function
of the e↵ective saturation Se and the empirical Brooks - Corey parameter as:
2+3
kr,w = Se
kr,n = (1
⇣
Se ) 2 1
(2.14)
2+
Se
⌘
(2.15)
The relative permeability - saturation relationship can also be created by the Van
Genuchten model as a function of the e↵ective saturation Se and the Van Genuchten
2.4 Fluid Properties
17
parameters ✏, m and gamma as:
h
kr,w = Se✏ 1
kr,n = (1
A typical kr
Se )
1
1
h
Sem
⌘m i2
1
1
Sem
i2m
(2.16)
(2.17)
Sw relationship model is given in Figure 2.7
Figure 2.7: Typical kr
after [19]
2.4
⇣
Sw relationship based on Brooks-Corey and Van Genuchten
Fluid Properties
The first step in the development of a model for multiphase flow is to distinguish
the phases. A phase is a continuous substance with homogeneous properties which is
separated from other phases(with other properties) by a sharp interface.
2.4.1
Phase Composition
Each phase is made up of one or more components, where a component can be any
chemical substance or group of substances such as air (which is then called pseudo
component). Although di↵erent phases may not be miscible, there is mass transfer
between them at the interface. Mass transfer from a solid, liquid or gaseous phase
to a liquid phase is normally called dissolution. Mass transfer from a liquid phase
to gaseous phase is called vaporization and if a gaseous component is transferred
it is called degassing. The mass transfer from a liquid or gaseous phase to a solid
phase and vice versa is named adsorption and desorption. If a usual solid component
is transferred from a fluid to a solid phase, it is addressed by precipitation. The
2.4 Fluid Properties
18
composition of a phase can have strong impacts on its physical and chemical properties
and behavior and make it necessary to introduce the measurement for the amount of
contained components. In general, the index ↵ represents the phase and the index 
denotes the component.
The mole fraction (dimensionless) relates the number of molecules n↵ of one
component in a phase to the overall number of molecules in the phase n↵ :
x↵ =
n↵
n↵
(2.18)
The mass fraction(dimensionless) relates the mass of a single component  to the
total mass of the phase ↵ and is related to the mole fraction via the molar masses
M  by
x M 
X↵ = P ↵  
(2.19)
 x↵ M
Concentrations (in units of density) are defined as mass of a component per unit
volume and are related to the mass fractions via the phase density %↵ by
C↵ = X↵ %↵
By definition, mass fractions as well as mole fractions sum up to unity
X
X
X↵ =
x↵ = 1

2.4.2
(2.20)
(2.21)

Density
The density of phase ↵ is defined as the mass of the phase ↵ in a unit volume occupied
by the phase ↵. It should be known 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
↵ is therefore a function of pressure and temperature of the system for the dependence
of the density on the compressibility of the fluid phase due to pressure, and the
expansion of the fluid phase with the change of temperature. The compressibility
of the fluid is determined under isothermal conditions and the expansion is under
isobaric terms. For a two phase system containing water and air, the density of water
is given by the total derivative after [41].
@% = % p dp + %
T dT
(2.22)
where p and T are the expansion coefficients of water under isothermal and isobaric
conditions. They are respectively defined by
2.4 Fluid Properties
19
p
=
1 @%
% @p
(2.23)
T
=
1 @%
% @T
(2.24)
The density of gas in a multi-component system equals to the sum of the partial density
of each component , %g . Then the density of gas for a two-component system is defined
as
X
%g =
%g
(2.25)
2w,a
where the density is determined by pressure and temperature
%g =
Pg
Z  R T
(2.26)
in which Z  and R are the real gas factor and the gas constant of component . Under
the assumption that air is an ideal gas (Z  = 1), the equation is converted to the ideal
gas law
Pg = %g RT
(2.27)
with %g =
2.4.3
n
,
V
where n is the number of moles and V is the volume of the gas.
Viscosity
Fluids may be defined as materials that continue to deform in presence of any shear
stress [7]. Viscosity is a measure that relates the extent of this deformation to the
applied shear stress. Think about an infinitesimal fluid volume shown in Figure 2.8.
Shear stress ⌧yx in longitudinal direction applies to the top and bottom edge of the
volume and therefore it is deformed with an angular velocity @'
. Newtonian fluids
@t
have a linear ratio between shear stress and deformation velocity which is described by
the dynamic viscosity µ with the unit [P a · s] :
Figure 2.8: Fluid volume deformed with angular velocity by shear stress
2.4 Fluid Properties
20
@'
(2.28)
@t
Another common parameter in hydromechanics is the kinematic velocity with the unit
2
[ ms ], obtained through the dynamic viscosity divided by the fluids density
⌧ =µ
µ
%
⌫=
(2.29)
For liquids, viscosity will not change significantly with varying pressures, but it decreases with increasing temperature. Gases, on the opposite, the viscosity
will increase
p
with the increasing temperature, where the proportionality µ / T can be used for
explaining ideal gases after [1]. The viscosity of ideal gases is independent of pressure
and also in real gases, the viscosity is not influenced by pressure through wide ranges.
2.4.4
Phase Equilibrium
A basic assumption in this subsection is that the compositions of phases are in thermodynamic equilibrium. In this case, the phases have the same temperature, partial
pressure for components and no more mass transfer between the phases takes place.
The basic thermodynamic laws for the description of the phase equilibrium are introduced in the following.
2.4.5
Vapor Pressure
Consider a pure substance which is present as gas and liquid phase. The two phases are
in contact and in equilibrium, i.e. there is no mass exchange of the substance between
the phases. Then the pressure of the gas phase is called vapor pressure. Its non-linear
dependence on the temperature T can be described with Antoine’s equation.
Pvap = 10[a
b/(T c)]
,
(2.30)
where the parameters a,b and c are fluid specific constants.
2.4.6
Dalton’s Law
Dalton’s Law states that the pressure of a mixture of gases equals the sum of the
pressures each component would have if it filled the whole volume alone
X
Pg =
pg ,
(2.31)

where the pg is called partial pressure of component . This definition, however, is
only valid for ideal gases where the pressure is a linear function of the molar density.
2.4 Fluid Properties
21
To make 3.21 valid for real gases, the partial pressure pg is defined to be proportional
to the mole fraction of component  in the gas.
Pg = xg pg
(2.32)
If the definition is combined with the vapor pressure, the mole fraction of a component
 inside a gas over a liquid can be evaluated as the ratio of the vapor pressure over the
gas pressure.
2.4.7
Raoult’s Law
In a liquid which does not consist of only one component, the vapor pressure of each
component is lowered. This lowering is described by Raoult’s law to be proportional
to the mole fraction of the component in the liquid phase
Pg = Xw pvap ,
(2.33)
where pvap is the vapor pressure of the pure liquid. Mixtures in which Raoult’s law is
valid for all mole fractions are called ideal. Although this behavior is seldom observed
in reality, Raoult’s law sufficiently describes the case of rather large mole fractions and
is true for the vapor pressure of solvent in an ideally diluted solution.
2.4.8
Henry’s Law
Since Raoults law is in reality only valid for solvents, another relation between partial
gas pressure and liquid phase mole fraction is needed for the solute. This relation is
given by Henrys law.
(2.34)
Pg = Hw xw ,
where Hw is the Henry coefficient. It has a strong temperature dependence which may
be described according to [34]
✓
✓
◆◆
4hsol 1
1
✓
H(T ) = H · · · exp
(2.35)
R
T
T✓
The superscript ✓ denotes values at standard temperature 298.15K and 4hsol is the
solution enthalpy.
In Figure 2.9 shows the range of applicability of both Henry’s law and Raoult’s law
for a binary system, where component 1 is component forming a liquid phase, e.g.
water, and component 2 is component forming a liquid gaseous phase, e.g. air. It is
obvious that for low mole fractions of component 2 in the system (small amounts of
dissolved air in the liquid phase), Henry’s law can be applied whereas for mole fractions
of component 1 close to 1 (small amounts of vapor in the gas phase), Raoult’s law is
the appropriate description. In general, the solvent follows Raoult’s law as it is present
in excess, whereas the dissolved substance follows Henry’s law as it is highly diluted.
2.5 Properties of Aqueous Electrolyte Solutions
22
Figure 2.9: Applicability of Henry’s law and Raoult’s law for a binary gas-liquid system
2.5
2.5.1
Properties of Aqueous Electrolyte Solutions
Mole Fraction
The mole fraction xi is defined as the ratio of number of moles of a specific component
to that of the total number of moles of the entire solution.
xi =
2.5.2
moles of a component
[ ]
moles of total components in the solution
(2.36)
Mass Fraction
The mass fraction is defined as the ratio of the mass of the specific component to that
of the specific component to that of the total mass of the substance.
eq : 31X =
2.5.3
mass of the specfic substance
[ ]
total mass of the substance
(2.37)
Concentrations
The molar concentration c is defined as the ratio of the number of moles of a specie to
that of the total volume.
number of moles mol
c=
[ 3]
(2.38)
total volume
m
Similarly, the mass concentration C is defined as the ratio of the mass of the solute to
that of the total volume of the solution.
c=
mass of the substance kg
[ 3]
total volume
m
(2.39)
2.5 Properties of Aqueous Electrolyte Solutions
2.5.4
23
Molality
The Molality is defined as the ratio of the number of moles of a specific component to
the mass of the solvent.
number of the moles of a component mol
c=
[
]
(2.40)
total mass of the solvent
kg
2.5.5
Molarity
Molarity is defined as the ratio of the number of moles of teh specific component to
the volume of the solution.
number of the moles of a component mol
c=
[ 3]
(2.41)
total volume of the solvent
m
2.5.6
Activity
Activity measures the e↵ective concentration of a species in a mixture under non-ideal
conditions which determines the real chemical potential for a real solution rather than
an ideal one. The activity of a species i, denoted ai is defined as :
✓
◆
µi (P, T ) µ0i (P, T ))
ai = exp
(2.42)
RT
where µi (P, T ) is the chemical potential of the species, µ0i (P, T ) is the chemical potential of that species in the chosen standard state, R is the gas constant and T is
the thermodynamic temperature. This definition can also be explained in terms of the
chemical potential:
µi (P, T ) = µ0i (P, T ) + RT lnai
(2.43)
2.5.7
Ionic Strength
The eletrostatic forces between the charged solute species in an aqueous electrolyte
solution depend on the charges of the species (the higher the charges,the greater the
electrostatic forces) and the total concentration of the species (because an increase in
the total solute concentration decreases the mean distance between the ions and, thus,
the electrostatic forces). Both these factors are included in the ionic strength (I) of an
aqueous solution, a concept introduced after [27], which is used for calculating activity
coefficients. The ionic strength is calculated by the following formula:
1X
I=
mi zi2
(2.44)
2 i
where mi is the molality of the ith species (which is essentially the same as molarity, except in concentrated solutions with total dissolved solids in excess of about
7000mg I 1 ) and zi the charge (positive or negative) of the ith species.This formulation of I emphasizes the e↵ect of higher charges of multivalent ions (positive and
negative) and does not include any contribution from neutral molecules.
2.6 Thermodynamic Properties
2.5.8
24
Activity Coefficients for Individual Ions
The activity coefficient measures how much an actual system deviates from a reference
system. Mathematically, the activity coefficient is defined as the limit of the equation
derived for activity and has no units. In a non-ideal solution
µi (P, T ) = µ0i (P, T ) + RT ln i ai
(2.45)
where µ0i (P, T ) is the standard state chemical potential and i the rational activity
coefficient. In the case of aqueous solutions, concentrations are commonly measured
as molalities rather than mole fractions, so that the expression for ai is modified to
ai =
i mi
(2.46)
where i is the practical activity coefficient and mi the molality. As i ! 1, the solution
approaches ideal behavior. For a very dilute solution, i ⇡ 1 and ai ⇡ mi .
2.5.9
Saturation Index
The saturation index (⌦) is a useful quantity to determine whether the water is saturated, undersaturated, or supersaturated with respect to the given mineral.
⌦=
2.5.10
[N a+ ][Cl ]
KN aCl
(2.47)
Solubility Product
Solubility product constants (K) are used to describe saturated solutions of ionic compounds of relatively low solubility. A saturated is in a state of dynamic equilibrium
between the dissolved, dissociated, ionic compound and undissolved solid.
Mx Ay (s) ! xM y+ (aq) + yAx (aq)
(2.48)
The general equilibrium constant for such processes can be written as:
Kc = [M y+ ]x [Ax ]y
(2.49)
Since the equilibrium constant refers to the product of the concentration of the ions
that are present in a saturated solution of an ionic compound, it is given the name
solubility product, and given the symbol Ksp .
2.6
2.6.1
Thermodynamic Properties
Enthalpy and Internal Energy
Enthalpy measures the total energy of a thermodynamic system. It contains the system’s internal energy, thermodynamic potential, volume and pressure. The unit for
2.7 Transport Processes
25
enthalpy is joule. The enthalpy of a homogeneous system is defined as:
H = U + pV
(2.50)
where H is the enthalpy of the system, U the internal energy of the system, p the
pressure of the system and V the volume of the system.
2.6.2
Heat Capacity
Heat capacity, also thermal capacity, is a measurable physical property that specifies
the amount of heat energy demanded to change the temperature of a certain object by
a given amount. The unit of the heat capacity is joule per kelvin ( KJ )
2.7
2.7.1
Transport Processes
Advection
A moving fluid carries the fluid’s constituents with it. To this transport mechanism
we refer as advection. The advective mass flux density Ja is:
Ja = qc,
(2.51)
where q[LT 1 ] is the specific discharge (discharge per cross-sectional area) and c[M L 3 ]
is the volumetric concentration
2.7.2
Di↵usion
Di↵usion is a mass transfer process caused by the random Brownian motion of solute
particles in fluids. The mass flux density is proportional to the concentration gradient.
The flux is oriented from the region of high to that of low concentration. The mass
flux density is described by Fick’s law:
Jd = Dm rc,
in which Dm [L2 T
1
] is the molecular di↵usion coefficient.
(2.52)
2.7 Transport Processes
Figure 2.10: Advection and di↵usion
26
Chapter 3
Model Concept and
Implementation
In this chapter, the porosity permeability relationship and the component specific transport equations are described. Furthermore, the mathematical model is also discussed.
The model used in this study is developed by Vishal Jambhekar (discussed in [20]).
We implemented permeability and porosity relationship, chemistry driven approach for
salt precipitation, comparison with layman’s or engineering driven approach (discussed
in [20]), non-isothermal and isothermal model. Heterogeneities are also performed.
3.1
Model Concept
Foe the considered system we use the compositional two-phase three-components
porous medium flow model under both isothermal and non-isothermal conditions The
two fluid phases are:
Liquid phase: This phase is composed of water(w ) as its main component and
dissolved salt(NaCl ) and air(a),
Gas phase: This phase is composed of two components air(a) and vapor(w ).
3.2
Mathematical Model
3.2.1
Assumptions
In order to create a mathematical model for the porous media, several simplified assumptions are made. The assumptions are:
The solid matrix (subscript s) is rigid (the porosity does not change) and inert.
Two-phase flow is considered, consisting of a liquid phase (subscript l) and gas
phase (subscript g), both of them are assumed to be Newtonian fluids.
3.2 Mathematical Model
28
Slow or creeping flow (Re ⌧ 1), therefore the multiphase Darcy’s law is valid.
Due to slow flow velocities and high di↵usion rate, dispersion caused by di↵erent
flow velocities, is ignored and only binary di↵usion is considered.
Local thermodynamic equilibrium (mechanical, chemical and thermal) fulfills due
to slow flow velocities.
An ideal gas phase based on [32].
Each fluid phase is composed of two components: water (superscript w) and air
(superscript a).
A compositional model is employed, facilitating a phase transition of components.
A static capillary-pressure / saturation relationship without hysteresis is considered.
Precipitated salt is the solid phase apart from the solid matrix and is considered
to be immobile.
3.2.2
Compositional Multiphase Flow
Figure 3.1: Model concept with single phase in the free flow that interacts with two
fluid phases in the porous media, after [28]
In the porous media domain ⌦pm , two fluid phases ↵ 2 {l, g} can be presented, liquid
phase and gas phase as shown in Figure 3.1. The liquid phase consists of dissolved
3.2 Mathematical Model
29
air and Halite and the gas phase consists of water vapor. With the mole fractions x↵
of component  2 {w, a, N aCl} in a phase ↵ and the phase saturation S↵ , the mass
conservation for each component is given as:
X @( %mol,↵ x S↵ )
↵
+ r · F
@t
↵2{l,g}
X
q↵ = 0
↵2{l,g}
8 2 {w, a, s},
where the mass flux of a component is given by
X

F =
(%mol,↵ v↵ x↵ D↵,pm
%mol,↵ rx↵ ).
(3.1)
(3.2)
↵2{l,g}

Here, represents the porosity and q↵ represents the source/sink term. D↵,pm
is the
macroscopic di↵usion constant of component  in the porous media for a multiphase
system. The di↵usion coefficient is a function based on the soil properties (e.g. porosity
and tortuosity) and the fluid saturation. The phase velocities v↵ are described with
the extended Darcy’s law:
v↵ =
kr↵
K(rP↵
µ↵
%↵ g),
↵ 2 {l, g},
(3.3)
where µ↵ , %↵ and kr↵ are the dynamic viscosity, mass density and relative permeability
respectively, g , K and p↵ are the gravity vector, the intrinsic permeability tensor and
phase pressure respectively.
Inserting 3.3 and 3.2 into 3.1 and assuming precipitation of dissolved NaCl at its
saturation concentration, the mass balance for each component  2 {w, a, N aCl} is
given:
⇢
X @ ( %mol,↵ x S↵ )
X
kr↵
↵
r·
%mol,↵ x↵ K (rP↵ %↵ g)
@t
µ↵
↵2{l,g}
↵2{l,g}
|
{z
} |
{z
}
StorageT erm
AdvectiveF lux
(3.4)
X
X

r · Dpm,↵
%mol.↵ rx↵
q↵ = 0
8 2 {w, a, s},
↵2{l.g}
3.2.2.1
|
{z
Dif f usiveF lux
Precipitation approaches
}
↵2{l,g}
| {z }
SourceT erm
The first method we use is based on the solubility limit of salt. We can call it layman’s
approach or engineering driven approach (equation 3.5).
⇢
aCl
aCl
@( %mol,l Sl (xN
xN

l
l,max )) for  = N aCl, ↵ = l
q↵ =
(3.5)
0
else
3.2 Mathematical Model
30
The second method consider the reactive precipitation or we can call it chemistry
approach. The Ionic strength (the definition of Ionic strength can be found in Chapter
2) in this case is given as:
1
1
IN a = µN a , ICl = µCl .
(3.6)
2
2
Based on the individual Ionic strength, activity coefficient (the definition of activity
coefficient can be found in Chapter 2) is accessible. Three equations are available:
p
log µi = Azi2 I,
(3.7)
This is called the Debye-Hückel limiting law because it applies to the limiting case
of very dilute solutions when I  0.001molkg 1 . Where A refers to a temperature
dependent parameter, and zi refers to the charge of the ith species. When the solutions
Parameters at 1
T emp( C)
A
0
0.4883
5
0.4921
10
0.4960
15
0.5000
20
0.5042
25
0.5085
30
0.5130
35
0.5175
40
0.5221
45
0.5271
50
0.5319
55
0.5371
60
0.5425
bar
B ⇥ 108
0.3241
0.3249
0.3258
0.3262
0.3273
0.3281
0.3290
0.3297
0.3305
0.3314
0.3321
0.3329
0.3338
Table 3.1: Values of parameters for the Debye-Hückel and Truesdell-Jones equation
with I  0.1[molkg 1 ], the extended Debye-Hückel equation is most useful and provides
adequate approximations for ionic strength up to about 1 [molkg 1 ].
p
Azi2 I
p
log µi =
(3.8)
1 + ai B i
where A and B are constant characteristic of the solvent at specified P (Pressure) and
T (Temperature), and ai is the hydrated ionic radius or ”e↵ective” ionic radius, of the
ith ion.
The form of the Debye-Hückel equation is such that the calculated activity coefficient
of an individual aqueous ion continuously decreases with increasing ionic strength of
3.2 Mathematical Model
31
the solution. Experimental data, however, show that in solutions of high ionic strength
(I is in a range of 0.8 and 1.0 for most ions) activity coefficient actually increase with
increasing ionic strength. To accommodate this behavior, Truesdell and Jones (1974)
proposed a modification that simply added a term bI to the D H equation:
p
Azi2 I
p + bI
log µi =
(3.9)
1 + ai B i
where b is a constant specific to the individual ion. The additional term compensates
for the lowering of the dielectric constant of water and increased ion pairing caused
by the increased concentration of solutes. Values of A and B are the same as given
in Table 3.1; values of a and b for selected ions are listed in Table 3.2. N a+ and
Cl are the only ions relevent to this work. Equation 3.9 is applicable to solutions
with ionic strength ranging from 0 to 2 [molkg 1 ]. With the Truesdell-Jones equation,
a ⇥ 10
4.78
4.32
3.71
5.46
10.65
3.71
5.31
Ion
H+
N A+
K+
M g 2+
OH
Cl
SO42
8
b
0.24
0.06
0.01
0.22
0.21
0.01
-0.07
Table 3.2: Parameters for the Truesdell-Jones equation
required parameters and molality of individual ions we get the activity of sodium ions
and chloride ions.
[N a+ ] = µN a · MN a , [Cl ] = µCl · MCl
(3.10)
The saturation index ⌦ (the definition can be found in Chapter 2) is calculated with
the following equation:
[N a+ ][Cl ]
(3.11)
⌦=
KN aCl
3.2.2.2
Mass Conservation of Precipitated salt
The assumption of salt precipitation also demands to account for the conservation of
the precipitated salt. The conservation of the precipitated salt is given as follows:
@(
N aCl N aCl
%mol,s )
s
@t
+ qlN aCl = 0,
(3.12)
Change in porosity is given as:
=
0
N aCl
,
s
(3.13)
3.2 Mathematical Model
3.2.2.3
32
Porosity and Permeability Relationships
A great amount of work is in the attempt to grasp the complexity of permeability
function into a model with general applicability [3]. These are empirical, statistical,
and the recently introduced ”virtual measurement” methods. They respectively make
use of empirically determined models, multiple variable regression, and artificial neural
networks. All these studies give a much better understanding of the factors influencing
permeability, but it also turns out that it is an illusion to have a ”universal” relation between permeability and other variables. Di↵erent relationships may fit various
kinds of soils related to di↵erent applications and these relations are investigated for
permeability determination. Originally Kozeny Carman formula was implemented to
describe the relationship between porosity and permeability in the model. In this study
a comparison is made with another five functions.
1. Kozeny Carman relation
The most used approach that connects the permeability to porosity was proposed
by Kozeny (1927) and later modified by Carman (1956). KC model considered
the porous medium as a bundle of cylindrical pores. The KC equation has the
form as:
✓ ◆3 ✓
◆3
K
1
0
=
(3.14)
K0
1
0
where is the porosity, 0 is the initial porosity, K is the permeability and K0
is the initial permeability.
2. Verma Pruess relation
Verma and Pruess (1988) derived a permeability-porosity relationship from a
pore-body-and-throat model in which permeability can be reduced to zero with
a finite porosity remaining. Verma Pruess model gives the following expression:
✓
◆2
K
c
=
(3.15)
K0
0
c
where is the porosity, 0 is the initial porosity, K is the permeability, K0 is
the initial permeability and c = 0.9 0 is the value of ’critical’ porosity at which
permeability goes to zero.
3. Modified fair-Harch relation
Fair-Hatch model was derived from dimension analysis and verified experimentally after [7]. A modified form of the Fair-Hatch relation was given by [10]
as:
#2
✓ ◆3 "
2/3
2/3
K
(1
)
+
(
)
0
f
0
=
,
(3.16)
2/3
K0
0
(1
) +( f
)2/3
3.2 Mathematical Model
33
where is the porosity, 0 is the initial porosity, f is the final porosity that is
calculated based on the initial porosity and volume of precipitated salt, K is the
permeability and K0 is the initial value of permeability
4. Timur model
Based on the work of Kozeny and Wyllie & Rose, Timur proposed a generalized
equation in the form:
B
K=A
.
(3.17)
C
Swi
That can be evaluated in terms of the statistically determined parameters A,
B and C. He applied a reduced major axis (RMA) method of analysis to data
obtained by laboratory measurements conducted on 155 sandstone samples from
three di↵erent oil fields from North America. Based both on the highest correlation coefficient and on the lowest standard deviation, Timur has chosen from
five alternative relationships the following formula for permeability.
K 1/2 = 0.136
4.4
Swi 2 ,
(3.18)
where is the porosity, K is the permeability and Swi is the residual saturation.
And for residual water saturation:
1.26
Swi = 3.5
2
Swi
,
(3.19)
with a standard error of 13% pore volume. This model is applicable when residual water saturation exists. Timur also assumed that a value of 1.5 for the
cementation factor, m, holds in all cases.
5. Coates model Coates and Denoo proposed the following formula for permeability
determination:
2
(1 Swi )
K 1/2 = 100
,
(3.20)
Swi
where K is in milidarcies. And this formula also satisfies the condition of zero
permeability at zero porosity and when Swi = 100%. The formation must be at
irreducible water saturation.
6. Tsypkin and Woods model In [38], G.Tsypkin & W.Woods derived a permeability porosity model. They use a parametric relation to describe the change in
permeability with the mass of precipitate given by:
K
1
=
K0
N aCl
exp(# (1
))
s
1 exp(# )
(3.21)
where # is a coefficient which accounts for di↵erent changes in the permeability
aCl
is
as a result of precipitation which we normally choose 10, 20 or 30 and N
s
the total precipitation.
3.2 Mathematical Model
3.2.2.4
34
Thermal Equilibrium
Based on the assumption of local thermal equilibrium (Tl = Tg = Ts = T ), one energy
balance equation accounts for the convective and conductive energy fluxes, energy
sources/sinks and energy storage in the liquid and solid phases [19] and [12]:
X @( %↵ u↵ S↵ ) @
+
@t
↵2l,g
N aCl N aCl N aCl
%s c s T
s
@t
+ (1
0)
@(%s cs T )
+ r · FT = qT (3.22)
@t
aCl
where u↵ represents the internal energy, N
the volume fraction of the precipitated
s
N aCl
salt, cs
the heat capacity of the precipitated salt, 0 the initial porosity and cs the
heat capacity of the porous medium. The total heat flux is given as:
X
FT =
%↵ v↵ h↵
(3.23)
pm rT,
↵2l,g
with the specific enthalpy h↵ . The e↵ective heat conductivity pm (Sl ) as a function of
phase saturation Sl represents the combined heat conduction of the fluid and the solid
phases. Approximately it can be calculated as the weighted sum of the e↵ective heat
conductivities of water-saturated soil ef f,l and air-saturated soil ef f,g based on [37],
[12] and [28]:
p
=
+
Sl ( ef f,l
(3.24)
pm
ef f,g
ef f,g ).
In which the e↵ective heat conductivities ef f,↵ of a saturated porous medium depend
on the conductivities of the pure fluids ↵ and of the solid s as below after [21]:
ef f,↵
↵
=
s
0.28 0.757 log
0.057 log(
s/ ↵)
.
(3.25)
↵
In order to add the above mentioned conservation equations for the model description, supplementary constraints and constitutive relationships are required for a closed
system of partial di↵erential equations (PDEs) for the porous medium flow domain.
3.2.2.5
Supplementary Constraints
1. Total void space within the porous medium is occupied by liquid and gas phases:
S g = 1 Sl .
2. The secondary phase pressure is determined by the capillary pressure: pc (Sl ) =
pg pl . The constitutive relationship for capillary pressure-saturation has been
discussed in Chapter 2.
3. It can be recognized that the given supplementary constraints hold for the mass
and mole fractions of all components  2 w, a, s in each phase ↵ 2 l, g: (e.g.Xlw +
a
s
Xla + Xls = xw
l + xl + xl = 1).
3.2 Mathematical Model
3.2.2.6
35
Constitutive Relationships
1. The capillary pressure-saturation relationship pc = pc (Sl ) is described by Brooks
Corey[9] or Van Genuchten[39] model.
2. The relative permeability kr↵ for each phase ↵ can be explained as a function
kr↵ (S↵ ) of the phase saturation S↵ using [9] or [39] model.
3. As mentioned by [5], the density and viscosity of the liquid phase are evaluated
as a function of its temperature, pressure, and salinity as follows:
%l = %w + 1000Xls {0.668 + 0.44Xls + [300p
+T (80 + 3T
3300Xls
2400pXls
13p + 47pXls )] ⇥ 10 6 }
3
µl = 0.1 + 0.333Xls + (1.65 + 91.9Xls )exp/ [0.42(Xls
(3.26)
0 .8
0.17)2 + 0.0045]T 0.8
(3.27)
The e↵ect of pressure variation on saline water viscosity is negligible after [5].
The density and viscosity of the gas phase are assumed to be same as that of the
air: %g = %a (p, T ), µg = µa (p.T ).
4. As a part of thermodynamic equilibrium, chemical equilibrium is assumed,The
phases are in equilibrium with respect to the exchange of components. In the
current work, the chemical equilibrium is ensured by assuming equal partial pressure of each component  2 {w, a, s} in each fluid phase ↵ 2 {l, g} as discussed
in [12] and [23].
5. The specific enthalpy is given as a function of temperature T and phase pressure
P↵ . For the gas and the liquid phases, enthalpy functions hg and hl are given
below:
hg (pg , T ) = Xgw hw + Xga ha ,
(3.28)
and
hl (pl , T ) = Xlw hw + Xla ha + Xls (hs +
where
hs ),
(3.29)
hs is the heat of dissolution of halite.
6. Same with specific enthalpies, specific internal energies can also be described as
a function of temperature T and phase pressure p↵ as follows:
u↵ (p↵ , T ) = h↵ (T )
3.2.3
p↵ /%↵
(3.30)
Primary Variables
For modeling, suitable primary variables must be chosen. Primary variables are computed directly within the iteration steps of the numerical model, while the secondary
3.2 Mathematical Model
36
variables are determined from the available primary variables. The number of primary variables is determined by the Gibbs phase rule. From the Gibbs phase rule, the
number of degrees of freedom indicate required number of primary variables.
F = K + 1,
(3.31)
where K represents the number of components present in the model. the primary
variables change based on the phase presence and it is required to specify the switch
criteria in the model.
1. When there is only the wetting phase present (condition: Sw = 1), the concentration of the gas (non-wetting phase) in wetting phase is smaller than the maximum
solvable concentration. The variable represents the mass fraction of gas dissolved
into the brine.
2. When both the phases are present (condition: 0 < Sw < 1), the concentration of
the wetting and the non-wetting phase is more than that of the minimum solvable
concentrations. The variable under consideration is S↵ .
3. When there is only the non-wetting phase present (condition: Sw = 0), the
concentration of water (wetting phase) in non-wetting phase is smaller than the
maximum solvable concentration. The variable represents the mass fraction of
water dissolved into the gas.
Phase State
liquid phase only
gas phase only
both phases
Phase Present
liquid
gas
liquid and gas
Primary Variables
aCl
pg , xal , xsl , N
,T
s
w
s
N aCl
p g , xg , x g , s , T
aCl
pg , Sl , xsl , N
,T
s
Switch Criterion
xal
xal,max
xw
xw
g
g,max
S↵  0
Table 3.3: Primary variables for di↵erent phase states and switch criteria
Chapter 4
Numerical Model and DuMux
For the explanation of the system of equations, analytical solutions are not available
for all cases but for a few very simplified situations. Therefore they are normally
solved numerically by discretizing them in space and in time. The equations were
solved using DuMux , which is a framework based on continuum mechanical concepts
to simulate multiphase flow and transport in porous media. DuMux in turn is based on
the software DUNE (Distributed and Unified Numerics Environment) that contains a
library of di↵erent grid-based solutions techniques in order to solve partial di↵erential
equations and makes use of the object-oriented programming language C + + along
with extensive use of template programming. The concept of implementation of the
mathematical model into a numerical model will be explained in this chapter.
As discussed in Chapter 3, after creating conservation equations, supplementary
constraints and constitutive relationships, a closed non-linear system of PDEs can
be obtained for the complete description of flow and transport processes in a porous
domain. Here, we focus on the numerical treatment of the coupled system of
advection-di↵usion PDEs, especially the discretization in time and space.
The hyperbolic or elliptic character of a problem is considered by the global pressuresaturation formulation. Other formulations for a multiphase flow system have been
studied in [19] and [11].
4.1
Space Discretization: Weighted Residual and
the Box Method
The partial di↵erential equations intended to be solved are discretized in both time
and space, which means that the solution of the equations will not be calculated for
every continuous point in the entire domain, but for certain discrete points in space
4.1 Space Discretization: Weighted Residual and the Box Method
38
and time. For achieving this, the exact solution f (u) of a problem, defined as
f (u) =
@
u + r · F (u)
@t
is integrated over a domain G
Z
Z
@
udG +
r · F (u)dG
G @t
G
q = 0,
(4.1)
Z
(4.2)
qdG = 0
G
is approximately equals to f (ũ), where f (ũ) refers to the approximate solution of the
equation at discrete points in space, in this case at the nodes of a Finite Element mesh.
It is an approximated result so that the equation 4.1 is no longer equal to zero, but to
an value " which denotes as the error obtained by approximating f (u) to f (ũ).
f (ũ) = "
with
ũ =
X
Ni ũ⇤i
i 2 nodes of the element
i
ũ⇤i
(4.3)
(4.4)
where
is the approximated value of u at node i of the finite element mesh and Ni
is linear basis function. The error " 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.
Z
Wj · " = 0,
(4.5)
G
and
X
Wj = 1
(4.6)
j
where j is the nodes of the elements, substituting it into 4.2
Z
Z
Z
@ X
⇤
Wj
Ni ũi dG +
Wj · [r · F (ũ)]dG
Wj qdG = 0
@t i
G
G
G
(4.7)
By using the Gauss Divergence Theorem and the chain rule, 4.7 can be rewritten into:
P
Z
Z
Z
Z
@ i Ni ũ⇤i
Wj
dG +
Wj · [r · F (ũ)] · d + rWj · F (ũ)G
Wj qdG = 0 (4.8)
@t
G
@G
G
G
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 =
(4.9)
0
i 6= j
4.1 Space Discretization: Weighted Residual and the Box Method
39
where Vi is the volume of Finite Volume box Bi around the node i (see figure 4.1).
Substituting the mass lumping term into the equation 4.8, the new balance equation
is obtained after the discretization of equation 4.7. This is given as
Z
@ ũ⇤i
Vi
+
[Wj · F (ũ)]dG Vi · q = 0
(4.10)
@t
@G
The weighting function Wj is defined to be piecewise constant over the Finite Volume
box Bi
⇢
1 x 2 Bi
lump
Mi,j =
(4.11)
0 i2
/ Bi
so that the gradient of the weighting function is zero
rWj = 0
This gives the last form of the discretized equation
Z
@ ũ⇤i
Vi
+
[Wj · F (ũ)] · d
@t
@G
(4.12)
Vi · q = 0
(4.13)
Hence now for ũ 2 {v, p, X  }, ũ can be substituted by the approximating function for
the primary variables.
Figure 4.1: Schematic diagram of the box method
4.2 Time Discretization
4.2
40
Time Discretization
Normally, a multiphase multicomponent flow system is transient and is represented by
a system of parabolic equations in time. A first order finite di↵erence scheme or the
implicit Euler scheme is utilized. The fully implicit method is unconditionally stable,
but leads to a series of coupled equations which must be solved iteratively. The coupled
system of equations is denoted as:
xt+
t xt
t
= At+ t xt+ t ,
(4.14)
where x represents the vector containing unknown primary variables at time t+ t and
Matrix A contains coefficient functions that depend on the known primary variables in
x at time t + t
Figure 4.2: solution technique. Left: Explicit solution. Right: Implicit solution.
Filled circles represent nodes with known primary variables, empty circles represent
unknowns.
4.3
4.3.1
The Structure in DuMuX
The Numerical Simulator DuMuX
The modeling toolbox DuMuX was chosen as framework for the implementation of the
numerical scheme. DuMuX is a free and open-source simulator for flow and transport processes in porous media. It is based on the Distributed and Unified Numerical
Environment DU N E [4], which provides grids, solvers, discretization, etc. The main
intention of DuMuX is to provide a sustainable and consistent framework for the implementation and application of model concepts, constitutive relations and discretization
[16] with a focus on porous-medium applications.
Chapter 5
Case Studies
In this chapter the implemented 2pncmin model is applied to di↵erent scenarios. The
Non-isothermal condition is compared with isothermal condition and chemistry driven
approach for salinization is compared with layman’s approach to study the influence
of salt precipitation.
Layman’s approach and Chemistry driven approach are applied and studied under
isothermal and non-isothermal conditions for salt precipitation. Grid convergence
test is also carried out under homogeneous conditions.
The layman’s approach is also used to analyze the precipitation and porosity
evolution under heterogeneous conditions.
5.1
General Problem Setup
The problem setup for analysis is shown in Figure 5.1 with Neumann no flow boundary
at the bottom and the left and right sides of the domain. A far field Dirichlet boundary
is set at the top of the extended artificial zone (the white district in 5.1). The artificial
zone is set because in order to simulate evaporation process, we need an extended zone
in the middle of the Drichlet boudary and the initial water level. In the extended zone,
the porosity is set to be 0.4 and the permeability is set to be 2⇥10 10 m2 . The entire region is divided into 10 ⇥ 50 cell based on the grid convergence test which is shown later.
The initial condition are also discussed in 5.1. With the given primary variables and
boundary conditions we can get the variables through all the cells in the domain in
every time step by using the box scheme (discussed in Chapter 4).
In the following cases, porosity and permeability may change to analyze the e↵ect of
the heterogeneities, while all other parameters equals to the reference case shown above
in 5.1.
5.1 General Problem Setup
42
Figure 5.1: model setup for homogeneous case
5.1.1
Isothermal & non-isothermal case
Energy is a key factor influencing evaporation process, To simulate the evaporative
salt precipitation in real world, Non-isothermal and isothermal conditions are put
into e↵ect. Due to the radiation, the surface of the soil has the higher temperature
Table 5.1: Values of the reference case and considered parameter
Parameter
Porous medium
porosity
permeability
vertical layer permeability
Brooks corey Pe
Brooks corey Lambda
Swr
Snr
solubility limit
Reference
Unit
0.364
1 ⇥ 10 10
1 ⇥ 10 10
120
2
0
0
0.26
m2
m2
Pa
-
and the temperature slowly decreases with the depth. Thus also for the Isothermal
condition we consider linear variance of temperature from 313K on the top to 308K at
the bottom which does not change with time (see Figure 5.2).
5.2 Layman’s Approach
43
In order to account heat tranfer we introduce the non-isothermal scenario. For the
non-isothermal condition we set the temperature of the upper Dirichlet boundary to
308K and the initial temperature of the domain to 298K. The temperatures at the
beginning and end time of the simulation are shown in Figure 5.3.
Figure 5.2: Temperatures along the domain in isothermal condition
(a) Initial Condition
(b) Ending Condition
Figure 5.3: Temperatures along the domain in non-isothermal condition
5.2
5.2.1
Layman’s Approach
Grid Convergence
For the grid convergence test, four grid discretizations are used. Normally coarse grids
increase the simulation speed, but the accuracy is decreased. Vice versa, fine grids
have better performance but higher computational cost. Therefore, we need to find a
compromise of accuracy and cost.
The grid sizes of 30 ⇥ 6, 40 ⇥ 8, 50 ⇥ 10 and 60 ⇥ 12 and layman’s approach was used
to calculate salt precipitation under non-isothermal condition. (see Figure 5.4). From
di↵erent kinds of grid discretization, it is observed that the 60 ⇥ 12 grid discretization
does not have large di↵erence compared to the 50 ⇥ 10 grid. At the same time the
calculation cost of 50⇥10 is less than 60⇥12 grid. Thus, the 50 ⇥10 grid discretization
is the best choice for this approach and will be used for future simulations.
Figure 5.4: Grid convergence test for non-isothermal layman’s approach
5.2 Layman’s Approach
44
5.2 Layman’s Approach
5.2.2
45
- K Relationship
To study the influence of evaporative salinization on permeability. Permeability
factor ( KK0 [ ]) is used. Six relationship between porosity and permeability were
introduced into the model (see section 3.2.2.3 for relations). The precipitation is
mainly accumulated on the near surface in a homogenenous manner. The mid point
of the surface is chosen for observation. Here layman’s approach is used to calculate
salt precipitation (see Figure 5.5 ).
In Figure 5.5, the permeability factor calculated by verma pruss relation decreases
rapidly and reaches zero only after 6 days of evaporation, which means the permeability also reduces to 0 at that time. This kind of extreme phenomenon is unphysical. For
the Timur’s, Modified Fair-Hatch’s, Coates’ and Kozeny Carman relations, the calculated permeability factors have similar bahavior. After about 20 days, the permeability
factors are in a range of 0.2 and 0.3, which means the permeability has been reduced by
70% to 80%. These results confirm each other. The permeability factor calculated by
Tsypkin relation stays at around 69% and this value is much higher than all other relationships shown in Figure 5.5. Therefore, as a preliminary estimation, the four relation
which behave similar are best fitted for modeling salinization. It has to be mentioned
that the Coates’ and Timur’s formulations are also related to residual saturation (see
Equation 3.17 and Equation 3.20) and in order to make the initial permeability factor
to 1 we had to set the current residual saturation which may not be satisfied in the real
scenario. Therefore, Modified Fair-Hatch’s and Kozeny Carman relations are better
than the other relations. Here we would like to choose Kozeny Carman relation for the
following simulation. For more certain choice, experimental control is also required.
5.2.3
Isothermal and Non-isothermal Conditions (Layman’s
Approach)
In this section, the layman’s approach is used for salt precipitation (see Equation
3.5). A 2pncmin model for the isothermal condition and a 2pncminni model for the
non-isothermal condition are implemented. The aim of this comparison is to show
the influence of energy transport on the salt precipitation. For the homogeneous
setup and parameters given above, gas phase saturation, salt precipitation and NaCl
mass fraction are calculated under both isothermal and non-isothermal conditions. In
Figure 5.2, Sg represents gas phase saturation, solidity represents the volume fraction
aCl
of salt precipitation to total volume ( N
), and here NaCl represents the mass
s
N aCl
fraction of NaCl in wetting phase (Xw ). Qualitative results are shown on left and
quantative result are on right. In the quantative analysis, the data is plotted along the
center line. Both the isothermal and non-isothermal conditions are chosen at 20.83
days for observation.
Figure 5.5: Permeability factor under di↵erent porosity permeability relationship models
5.2 Layman’s Approach
46
5.2 Layman’s Approach
47
For the non-isothermal condition, gas phase saturation is 1 at the surface, phase
saturation decreases with the increase of depth and reaches 0 at the depth of 0.06 m.
Meanwhile the salt precipitation has formed in a range of 0.01 m under the surface
and the maximum solidity is about 14% on the surface. The highest mass fraction of
NaCl is approximately 0.46 on the surface and it continuously decreases to its initial
value at the depth of 0.03 m.
For the isothermal condition, the water evaporation and salt precipitation also
happen on the surface layer like the non-isothermal condition. While the gas phase
saturation decreases and reaches 0 at the depth 0.065 m, the range of salt precipitation
accumulation is from 0 m to 0.018 m under the surface and the highest NaCl mass
fraction is almost same with that in the non-isothermal condition. The main reason
for the similar behavior of evaporation and precipitation under this two conditions
can be the relative independence of salt solubility limit on temperature. On the other
hand, due to the higher background temperature in the isothermal condition, more
evaporation and precipitation is observed.
it is important to point out, the higher mass fraction in the liquid phase at very low
saturation are the numerical e↵ects related to the equilibrium calculation and has
no physical significance. At this time it is dry at the surface, thus there is no salt
to precipitate. As a conclusion, under layman’s approach higher temperature lead to
higher evaporation rate. But when the temperature range is not large. The behaviour of
isothermal and non-isothermal cases are similar. Iso-thermal condition is good enough
when we do not look for significant result.
5.3 Chemistry Driven Approach
48
Table 5.2: Salt content varieties in non-isothermal and isothermal conditions under
layman’s approach after 20.83 days
Non-isothermal
5.3
Isothermal
Chemistry Driven Approach
In chemistry driven model, the salt precipitation is calculated based on the solubility
product (⌦) for dissolved species. More specific information about chemistry driven
approach can be found in Chapter 3.
5.3.1
Grid Convergence
In order to test the grid convergence, four kinds of grid discretization are implemented
(30 ⇥ 6, 40 ⇥ 8, 50 ⇥ 10 and 60 ⇥ 12). Chemistry driven approach was applied to
compute their salt precipitation under non-isothermal condition. The results can be
seen in Figure 5.6. The 60 ⇥ 12 grids discretization does not have large di↵erence
5.3 Chemistry Driven Approach
49
compared to the 50 ⇥ 10 grids. At the same time the calculation speed (less than
8000s) of 50 ⇥ 10 is also acceptable. Therefore, 50 ⇥ 10 grid discretization is the best
choice for this chemistry driven approach. Hence, in the following work with chemistry
driven approach will be simulated with this discretization.
5.3.2
- K Relationship
As discussed in chapter 3, six relationships between porosity and permeability were
used. The precipitation normally formed on the near surface, so the mid point of the
surface is chosen for observation. Hence, the chemistry driven approach was applied
to account for precipitation under non-isothermal condition (see Figure 5.7 ).
In Figure 5.7, the permeability factor given by verma pruss relation decreases sharply
and reaches 0 after 10 days of evaporation, which implies the permeability reduces
to 0. This kind of extreme phenomenon is unrealistic in natural system. For the
Timur’s, Modified Fair-Hatch’s, Coates’ and Kozeny Carman relations, the calculated
permeability factors behave similar. After about 25 days, the permeability factors are
in a range of 0.2 and 0.3, which means the permeability has been reduced by 70 to
80 percentage. These results verify each other. The permeability factor calculated by
Tsypkin relation stays at around 0.7 and this value is much higher than all the others.
Therefore, as a preliminary estimation, the four relation which behave similar are best
fitted for modeling salinization. Moreover, the Coates’ and Timur’s formulations are
also related to residual saturation and we have to set the residual saturation in order to
map the initial permeability factor to 1, which may not be valid in the real salinization
scenario. By contrast, Modified Fair-Hatch’s and Kozeny Carman relations should be
better than others. Similar to layman’s approach we choose Kozeny Carman relation
for simulations with chemistry driven approach. For more reliable decision, comparison
against experiments would be useful.
5.3.3
Isothermal and Non-isothermal Conditions (Chemistry
Driven Approach)
A 2pncmin model is implemented for the isothermal condition and a 2pncminni model
is used for the non-isothermal condition. The aim of this comparison is to find the
influence of temperature on the salt precipitation.
On the basis of the homogeneous scenario setup (see Figure 5.1) and parameters given
above, gas phase saturation, precipitation and NaCl mass fraction are computed under
both isothermal and non-isothermal conditions. In Figure 5.3, Sg means gas phase
saturation, solidity is the fraction of salt precipitation volume to porosity volume due
aCl
to evaporation ( N
), and NaCl represents the mass fraction of NaCl in the wetting
s
N aCl
phase (Xw ). For both the isothermal and non-isothermal conditions are oberved
Figure 5.6: Grid convergence test for non-isothermal chemistry driven approach
5.3 Chemistry Driven Approach
50
Figure 5.7: Permeability factor under di↵erent porosity permeability relationship models
5.3 Chemistry Driven Approach
51
5.4 Comparison of Layman’s Approach and Chemistry Driven Approach
52
after 20.83 days.
For the non-isothermal condition, gas phase saturation is 1 on the near surface,
decreases with the increase of depth and turns into 0 at the depth of 0.05 m. At the
same time the salt precipitation has developed within 0.005 m under the surface and
the maximum solidity is about 0.12 at the surface. The highest mass fraction of NaCl
is approximately 0.52 on the surface and it decreases to the initial mass fraction of
NaCl at the depth of 0.02 m.
For the isothermal condition, similar to non-isothermal, the water evaporation and
salt precipitation also takes place on the surface. However, the depth of gas phase
saturation is 0 is 0.06 m, the range of salt precipitation happening is from 0 m to
0.018 m and the NaCl mass fraction is also higher than that in the non-isothermal
condition. The main reason for the quantitative similar results of evaporation and
precipitation under isothermal and non-isothermal conditions can be the independence
of salt solubility limit on temperature which means the solubility limit does not vary
much with the change of temperature. Related to the higher temperature in the
isothermal condition, stronger evaporation and precipitation occur.
It important to be mention that, similar with layman’s approach, the higher mass
fraction in the liquid phase at very low saturation are the numerical e↵ects related to
the equilibrium calculation. At this time it is dry at the surface, thus there is no salt
to precipitate. As a conclusion, under chemistry driven approach higher temperature
lead to higher evaporation rate. But when the temperature range is not large. The
behaviour of isothermal and non-isothermal cases are similar. Iso-thermal condition is
good enough when we do not look for significant result.
5.4
Comparison of Layman’s Approach and Chemistry Driven Approach
The two approaches of precipitation have comparable consequence. While layman’s
approach has a solubility limit (see Equation 3.5), chemistry driven approach uses the
product solubility, saturation index (⌦) and ions strength..
For the chemistry driven approach, gas phase saturation is 1 of the surface, decreases
with the increase of depth and turns into 0 at the depth of 0.05 m. At the same
time the salt precipitation has developed within 0.005 m under the surface and
the maximum solidity is about 0.12 on the surface. The mass fraction of NaCl is
approximately 0.52 on the surface and it dramatically decreases to the initial mass
fraction of NaCl at the depth of 0.02 m.
5.4 Comparison of Layman’s Approach and Chemistry Driven Approach
53
Table 5.3: Comparison of non-isothermal and isothermal model under chemistry driven
approach after 20.83 days
Non-isothermal
Isothermal
5.4 Comparison of Layman’s Approach and Chemistry Driven Approach
54
For the layman’s approach, the evaporation and salt precipitation also take place at
the surface layer similar to the chemistry driven approach. The gas phase saturation
equals 0 at the depth of 0.06 m, the salt precipitation happens to the depth of 0.01m
from surface with the maximum solidity of 0.14. The highest NaCl mass fraction is
about 0.46 and is lower than that in the chemistry driven approach. It can be clear
seen that the area of high mass fraction of the chemistry driven approach is much
larger than that of the layman’s approach.
The main reason leading to this phenomenon is a explicit solubility limit (0.26) of
NaCl has been set for layman’s approach. On the other hand, the limit of chemistry
driven approach is implicitly decided by the solubility product. Therefore the salt mass
fraction for layman’s approach has to be close to the solubility limit (0.26). But for the
chemistry driven approach, higher dissolved salt concentration are observed. In order
to solve this problem, better fitting parameters for the chemistry driven approach is
necessary. That is to say, so far layman’s approach is more reliable to simulate the
precipitation process and the following calculation in heterogeneous cases are performed
using layman’s approach.
5.5 Heterogeneous Case
55
Table 5.4: Comparison of the salinity calculated by layman’s approach and chemistry
driven approach
Non-isothermal chemistry driven approach
5.5
Non-isothermal layman’s approach
Heterogeneous Case
5.5.1
Heterogeneous Scenario Setup
In this section, the domain size, initial and boundary conditions of the basic model
setup are same with the homogeneous case (see Figure 5.1). The porous medium
properties are given in table (see table 5.1 ). But the porous media spatial parameters
in the domain are di↵erent. In addition, non-isothermal condition is chosen to idealize
a normal system (see Figure 5.3). Layman’s approach is applied for computing the
precipitation. Here three di↵erent heterogeneities are used.
Horizontal Heterogeneity
As shown in Figure 5.10), the left red part is the coarse sand with the porosity
5.5 Heterogeneous Case
56
of 0.4 and permeability of 2e-10 and the right blue part represents fine sand with
the porosity of 0.364 and permeability of 1e-10.
Figure 5.10: Porosity and permeability setup for horizontal heterogeneous case
Patchy Heterogeneity
Patches with di↵erent porosity and permeability are indroduced along the domain
(see Figure 5.13). In Figure 5.13, the porosity of the media is 0.364 in blue part
and is 0.4 in red part. The media permeability in the blue part is 1e-10 and is
2e-10 in the red part.
Figure 5.13: Porosity and permeability setup for patch heterogeneous case
Random Heterogeneity
For the random distribution heterogeneous case, the permeability field is generated randomly in the whole domain based on gaussian distribution in a range of
5.83e-11 to 1.71e-10 (see Figure 5.16). The porosity of the soil is set to be 0.364.
5.5.2
Horizontal Heterogeneous Case
Horizontal heterogeneity is applied into the model in order to see the change of
porosity, gas phase saturation, NaCl mass fraction and solidity (see Figure 5.17, 5.18,
5.5 Heterogeneous Case
57
Figure 5.16: Porosity and permeability setup for stochastic heterogeneous case
5.19 and 5.20).
In Figure 5.17, 5.18, 5.19 and 5.20, two lines are chosen for plotting the data in
order to analyze the influence of heterogeneity. The red line represents the calculated
value on the coarse sand and the blue line reflects the calculated value on the fine sand.
After 16.2 days of evaporation, on the top layer (0-0.01 m) of the left side the porosity
gradually decreased from 0.4 to 0.28 and on the top layer (0-0.004 m) of the right side
decreased from 0.364 to 0.27. At the same time, for the gas phase saturation, the
saturation of the left side comes down from 1 at the surface to 0 at the depth of 0.045
m and that of the right side comes down from 0.98 at the surface to 0 at 0.04 m from
the surface. For the NaCl mass fraction, the fraction on the left side goes up from
0.15 to 0.44 on the near surface, while the fraction on the right side increases from
0.15 to 0.24. For the solidity, on the left side the solidity appears in a range of 0-0.01
m and accumulates from 0 to 0.125 and on the right side it appears from 0 m to 0.005
m and the maximum value is 0.09.
The reason of the di↵erent salt precipitation behavior between left side and right side
is caused by the di↵erence of capillary pressure and porosity. On the coarse sand part,
higher porosity gives larger interface of brine water and air. Thus the coarse sand
region has higher evaporation rate than the fine sand and has more salt precipitation.
While fine sand with smaller porosity has larger capillary pressure and can hold higher
water level.
5.5.3
Patchy Heterogeneous Case
Patchy heterogeneity is applied in order to see the change of porosity, liquid phase
saturation, NaCl mass fraction and solidity (see Fig 5.21, 5.22, 5.23 and 5.24).
5.5 Heterogeneous Case
Figure 5.17: Porosity for the horizontal heterogeneous case
Figure 5.18: Gas phase saturation for horizontal heterogeneous case
Figure 5.19: NaCl mass fraction for horizontal heterogeneous case
Figure 5.20: Solidity for horizontal heterogeneous case
58
5.5 Heterogeneous Case
59
Two lines are chosen for plotting the graphs in order to analyze salinization and
change in porosity and permeability at di↵erent locations. The red line represents the
calculated value on the left part and the blue line reflects the calculated value on the
right part.
After 16.2 days of evaporation, on the top layer (0-0.01 m) of the left side the porosity
gradually decreased from 0.4 to 0.26 and on the top layer (0-0.005 m) of the right
side decreased from 0.364 to 0.275. At the same time, the gas phase saturation on
the left side decreases from 1 at the surface to 0 at the depth of 0.04 m and that of
the right side comes down from 0.98 at the surface to 0 at 0.04 m from the surface.
For the NaCl mass fraction, the fraction on the left side goes up from 0.15 to 0.43 on
the near surface, while the fraction on the right side increases from 0.15 to 0.25. For
the solidity, on the left side the solidity appears in a range of 0-0.01 m and with a
maximum value of 11% and on the right side it appears from 0 m to 0.005 m and the
maximum value is 0.085.
After 16.2 days’ evaporation, the water level decreases by 0.04 m. So the di↵erence of
evaporation on both side is mainly influenced by the capillary pressure and permeability
on the top layer. The distribution of the media on the top layer can be considered as
the horizontal heterogeneity. In addition, the draw down of water level caused by
evaporation is much smaller than the capillary heights (the height of the water column
supported by the capillary pressure) of both coarse sand and fine sand. Therefore, the
evaporation is continuous and depends on the water flow rate in the porous medium.
Hence more water will evaporate through the fine sand on left side to the surface on
the top layer, which results in higher evaporation rate.
Figure 5.21: Porosity for the patch heterogeneous case
5.5.4
Random Heterogeneous Case
Random heterogeneity is applied into the model in order to see the change of porosity,
gas phase saturation, NaCl mass fraction and solidity (see 5.25, 5.26, 5.27 and 5.28).
The middle line for the domain from the top to the bottom is chosen for plotting the
5.5 Heterogeneous Case
Figure 5.22: Gas phase saturation for patch heterogeneous case
Figure 5.23: NaCl mass fraction for patch heterogeneous case
Figure 5.24: Solidity for patch heterogeneous case
60
5.5 Heterogeneous Case
61
graphs in order to analyze the porosity and salinity under random heterogeneity.
After 16.2 days of evaporation, on the top layer (0-0.02 m) the porosity gradually
decreased from 0.364 to 0.28. At the same time, for the gas phase saturation, the
saturation comes down from 0.95 at the surface to 0 at the depth of 0.05 m. For the
NaCl mass fraction, it goes up from 0.15 to 0.26 on the near surface. For the solidity, on
the left side the solidity appears in a range of 0-0.01 m with the maximum value of 0.08.
Larger permeability makes water easier to come through and hard to storage. Due to
the larger permeability on the right side near the lower evaporation zone (see Figure
5.16), the gas phase saturation on the right side is larger than that on the left side. In
other words, the water level on the right side is lower than that on the left side.
Figure 5.25: Porosity for the random heterogeneous case
Figure 5.26: Gas phase saturation for random heterogeneous case
5.5 Heterogeneous Case
Figure 5.27: NaCl mass fraction for random heterogeneous case
Figure 5.28: Solidity for random heterogeneous case
62
Chapter 6
Summary and Outlook
6.1
Summary
During this master thesis, we used the implementation a 2pncmin framework in
DuMuX in order to describe the salt precipitation of brine water in porous medium.
This research has been highly motivated by the global salinization of soil due to the
salt precipitation in porous medium. The two phases used in the model are liquid
and gas phase. The three components implemented in the model are water (w),
air (a), and Sodium Chloride (NaCl). The relationship between permeability and
changing porosity caused by salt precipitation has been studied. Non-isothermal
model and isothermal model, Chemistry driven approach and Layman’s approach for
precipitation and di↵erent kinds of heterogeneous cases are compared. The emphasis
of this thesis is to quantitatively recognize the evaporation and precipitation process
under di↵erent conditions and how di↵erent porosity permeability relationships behave
during salt precipitation.
The BOX scheme has been implemented for the spatial discretization and implicit
Euler method is used for the time discretization. The e↵ect of dissolution is neglected.
For chemistry driven approach, the main factor that is responsible for salt precipitation is the saturation index and solubility product as discussed in section 3.2.3. The
solubility product values for NaCl salt, are approximated by trial and error method.
For the layman’s approach, the key factor for salt precipitation is the solubility limit
which is also discussed in section 3.2.3.
Di↵erent kinds of scenario are implemented into the discussion in Chapter 5. The grid
convergence test shows that the 50⇥10 is the best discretization of the cells. The permeability factor (see section 5.2.2) was introduced to determine the best relationship
for the porosity and permeability on the salinization process and it proves that the
Kozeny Carman and Modified Fair-Hatch’s relation are the best for modeling salinization. Non-isothermal and isothermal cases are used to analyze the e↵ect of temperature
6.2 Future Work
64
on saturation, salt precipitation. It turns out the behavior of the two cases is similar.
Thus isothermal case is good enough if we are not looking for significant result. The
layman’s approach and chemistry driven approach are compared to see the influence of
di↵erent approaches on precipitation. Chemistry drvien approach describes the precipitation in a more realistic way and is suitable for mixed ions. Layman’s approach has
the explicit solubility limit. Three kinds of heterogeneities (Patch heterogeneity, Horizontal heterogeneity and Random heterogeneity) are implemented to see the impacts
of di↵erent heterogeneities on precipitation. Di↵erent porosity gives di↵erent capillary
pressure. The water level is determined by capillary pressure and permeability. The
evaporation rate and precipitation are determined by the porosity.
6.2
Future Work
1. To find more precise constant parameters for the chemistry driven approach to
get a better fit.
2. To add multiple minerals to investigate the influnece on precipitation.
3. To make the temperature and remediation periodically changing with time to
imitate day and night.
4. To couple the porous medium model with free flow to oberve the impact of wind
on evaporation.
5. To modify the water content in the free flow so we can find out the influence of
di↵erent humidity on evaporation.
6. Implement experimental control to verify the simulation result and determine the
best porosity and permeability relationship.
Bibliography
[1] Atkins, P. and De Paula, J. Atkins’ physical chemistry. Eight Ed, 2006.
[2] Baber, K., Mosthaf, K., Flemisch, B., Helmig, R., Müthing, S., and Wohlmuth,
B. Numerical scheme for coupling two-phase compositional porous-media flow and
one-phase compositional free flow. IMA journal of applied mathematics, 77(6):887–
909, 2012.
[3] Balan, B., Mohaghegh, S., and Ameri, S. State-of-the-art in permeability determination from well log data: part 1-a comparative study, model development. paper
SPE, 30978:17–21, 1995.
[4] Bastian, P., Blatt, M., Dedner, A., Engwer, C., Klöfkorn, R., Kornhuber, R.,
Ohlberger, M., and Sander, O. On the generic parallelisation of iterative solvers
for the finite element method. Computing, 4(1):121–138, 2008.
[5] Batzle, M. and Wang, Z.
57(11):1396–1408, 1992.
Seismic properties of pore fluids.
Geophysics,
[6] Bear, J. et al. Groundwater hydraulics. McGraw, New York, 1979.
[7] Bear, J. Dynamics of fluids in porous media. American Elseuier, 1972.
[8] Bechtold, M., Haber-Pohlmeier, S., Vanderborght, J., Pohlmeier, A., Ferré, T.,
and Vereecken, H. Near-surface solute redistribution during evaporation. Geophysical Research Letters, 38(17), 2011.
[9] Brooks, R. H. and Corey, A. T. Hydraulic properties of porous media. Hydrology
Papers, Colorado State University, (March), 1964.
[10] Chadam, J., Ortoleva, P., and Sen, A. A weakly nonlinear stability analysis of the
reactive infiltration interface. SIAM Journal on Applied Mathematics, 48(6):1362–
1378, 1988.
[11] Class, H. Models for non-isothermal compositional gas-liquid flow and transport
in porous media. 2008.
65
BIBLIOGRAPHY
66
[12] Class, H., Helmig, R., and Bastian, P. Numerical simulation of non-isothermal
multiphase multicomponent processes in porous media.: 1. an efficient solution
technique. Advances in Water Resources, 25(5):533–550, 2002.
[13] Coussot, P. Scaling approach of the convective drying of a porous medium. The
European physical journal B-condensed matter and complex systems, 15(3):557–
566, 2000.
[14] De Lemos, M. J. Turbulent flow around fluid–porous interfaces computed with a
di↵usion-jump model for k and " transport equations. Transport in porous media,
78(3):331–346, 2009.
[15] Fisher, E. Some factors a↵ecting the evaporation of water from soil. The Journal
of Agricultural Science, 13(02):121–143, 1923.
[16] Flemisch, B., Darcis, M., Erbertseder, K., Faigle, B., Lauser, A., Mosthaf, K.,
Müthing, S., Nuske, P., Tatomir, A., Wol↵, M., et al. Dumu¡ sup¿ x¡/sup¿: Dune
for multi-{phase, component, scale, physics,} flow and transport in porous media.
Advances in Water Resources, 34(9):1102–1112, 2011.
[17] Freeze, R. A. and Cherry, J. Groundwater, 604 pp, 1979.
[18] Fujimaki, H., Shimano, T., Inoue, M., and Nakane, K. E↵ect of a salt crust on
evaporation from a bare saline soil. Vadose Zone Journal, 5(4):1246–1256, 2006.
[19] Helmig, R. et al. Multiphase flow and transport processes in the subsurface: a
contribution to the modeling of hydrosystems. Springer-Verlag, 1997.
[20] Jambhekar, V. A. Evaporation-driven transport and precipitation of nacl in porous
media.
[21] Krupiczka, R. Analysis of thermal conductivity in granular materials. Int. Chem.
Eng, 7(1):122–144, 1967.
[22] Laurindo, J. B. and Prat, M. Numerical and experimental network study of
evaporation in capillary porous media. drying rates. Chemical engineering science,
53(12):2257–2269, 1998.
[23] Lauser, A., Hager, C., Helmig, R., and Wohlmuth, B. A new approach for phase
transitions in miscible multi-phase flow in porous media. Advances in Water
Resources, 34(8):957–966, 2011.
[24] Lehmann, P., Assouline, S., and Or, D. Characteristic lengths a↵ecting evaporative
drying of porous media. Physical Review E, 77(5):056309, 2008.
[25] Lehmann, P. and Or, D. Evaporation and capillary coupling across vertical textural contrasts in porous media. Physical Review E, 80(4):046318, 2009.
BIBLIOGRAPHY
67
[26] de Lemos, M. J. and Silva, R. A. Turbulent flow over a layer of a highly permeable
medium simulated with a di↵usion-jump model for the interface. International
journal of heat and mass transfer, 49(3):546–556, 2006.
[27] Lewis, G. N. and Randall, M. The activity coefficient of strong electrolytes. 1.
Journal of the American Chemical Society, 43(5):1112–1154, 1921.
[28] Mosthaf, K., Baber, K., Flemisch, B., Helmig, R., Leijnse, A., Rybak, I., and
Wohlmuth, B. A coupling concept for two-phase compositional porous-medium
and single-phase compositional free flow. Water Resources Research, 47(10), 2011.
[29] Nachshon, U., Shahraeeni, E., Or, D., Dragila, M., and Weisbrod, N. Infrared
thermography of evaporative fluxes and dynamics of salt deposition on heterogeneous porous surfaces. Water Resources Research, 47(12), 2011.
[30] Nassar, I. and Horton, R. Salinity and compaction e↵ects on soil water evaporation and water and solute distributions. Soil Science Society of America Journal,
63(4):752–758, 1999.
[31] Pel, L., Huinink, H., and Kopinga, K. Salt transport and crystallization in porous
building materials. Magnetic resonance imaging, 21(3):317–320, 2003.
[32] Reid, R. C., Prausnitz, J. M., and Poling, B. E. The properties of gases and
liquids. 1987.
[33] Risacher, F. and Clement, A. A computer program for the simulation of evaporation of natural waters to high concentration. Computers & Geosciences, 27(2):191–
201, 2001.
[34] Sander, R. Compilation of henry’s law constants for inorganic and organic species
of potential importance in environmental chemistry, 1999.
[35] Shavit, U. Special issue on “transport phenomena at the interface between fluid
and porous domains”. Transport in Porous Media, 78(3):327–330, 2009.
[36] Shimojimaa, E., Yoshioka, R., and Tamagawa, I. Salinization owing to evaporation
from bare-soil surfaces and its influences on the evaporation. Journal of Hydrology,
178(1):109–136, 1996.
[37] Somerton, W., El-Shaarani, A., and Mobarak, S. High temperature behavior of
rocks associated with geothermal type reservoirs. In SPE California Regional
Meeting, 1974.
[38] Tsypkin, G. G. and Woods, A. W. Precipitate formation in a porous rock through
evaporation of saline water. Journal of Fluid Mechanics, 537:35–53, 2005.
BIBLIOGRAPHY
68
[39] Van Genuchten, M. T. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Science Society of America Journal, 44(5):892–898,
1980.
[40] Zeidouni, M., Pooladi-Darvish, M., and Keith, D. Analytical solution to evaluate
salt precipitation during co¡ sub¿ 2¡/sub¿ injection in saline aquifers. International
Journal of Greenhouse Gas Control, 3(5):600–611, 2009.
[41] Zielke, W. and Helmig, R. Grundwasserströmung und schadsto↵transport. femethoden für klüftige gesteine. Wissenschaftliche Tagung Finite Elemente Anwendung in der Baupraxis. Universität Fridericiana Karlsruhe, Berlin: Verlag
Ernst u. Sohn, 1991.