ARAB ACADEMY FOR SCIENCE & TECHNOLOGY AND MARITIME TRANSPORT (AASTMT) College of Engineering and Technology Department of Mechanical Engineering
Flow and Transport Problems in Porous Media Using CFD By NAGI ABDUSSALAM ABDUSSAMIE
[email protected]
A Thesis submitted to AASTMT in partial Fulfilment of the requirements for the award the degree of
MASTER of SCIENCE IN Mechanical Engineering
Supervisors Prof Dr Hassan El-Gamal
Assistant Prof Amr Aly
Mechanical Engineering Department Faculty of Engineering Alexandria University
Head of Marine Engineering Department College of Engineering AASTMT
2009
DECLARATION We hereby certify that we have read the present work and that in our opinion it is fully adequate in scope and quality as thesis towards the partial fulfillment of the Master Degree requirements in Mechanical Engineering from the Arab Academy for Science & Technology and Maritime Transport.
Supervisors: Name: Position:
Signature: Name: Position:
Hassan El-Gamal Professor in Mechanical Engineering Department at Alexandria University
Amr Aly Head of Marine Engineering Department at AASTMT-Alexandria
Signature:
Examiners: Name: Position:
Signature: Name: Position:
Signature:
Kamal Abdelaziz Professor in Mechanical Engineering Department at Elmonovia University
Raof Nasif Professor in Mechanical Engineering Department at Ain Shams University
ACKNOWLEDGEMENT First, from my deep heart I‟d like to thank God to make me able to finish this research. A lot of thanks for God‟s permission and for giving me the ability to overcome all problems which might make me fail. Second, it‟s a little to say „thanks‟ to my professor Dr Hassan ElGamal, his patience was continuous throughout this research and other courses which I studied under his observation, he always encourages me and his notes make this work beneficial. Due to his support and effort I followed the right way in my years‟ studies. Third, plenty of thanks to my dear supervisor Dr Amr Aly for his guidance and his support, surely without his help I‟ll be never at this level. He exerted a lot of effort for my studies. Out of his love, courage, and encouragement, I got all the right meanings. Owing to his intelligence I could set things to rights. At last, but not least, thanks for my friends, especially, Mr Hussein who often supports me, and advises me, many thanks for each member in my family, and also for my father‟s soul, I need to say I can never thank every body enough. I‟ll be remembering your help and support all my life.
Alexandria, January 2009 Nagi Abdussamie
ABSTRACT
Two-phase flow and transport processes in porous media are involved in a wide variety of engineering applications, such as oil recovery and groundwater remediation. Hence, in this work, the flow and transport problems have been studied using pore-scale modelling approach, which allowed investigating microscale processes and their effects on macroscale behaviour. The present pore-scale model takes one of the 2D micro-media images of realistic porous media and solves velocities and pressures of pore fluids using the Navier-Stokes equations in Cartesian coordinates. Because of the high complexity of the problem of dynamic immiscible two-phase flow in a system containing a large number of interconnected pores of variable shapes, original exact analytical solution is not feasible option. Therefore, the potential for using standard computational fluid-dynamics (CFD) methods has been investigated to analyze and predict the effects of several mechanisms on mobilization of trapped non-wetting fluid. The „Volume of Fluid‟ (VOF) method implemented in FLUENT has been used to model the immiscible two-phase flow specific to our problem. Eventually, it can be stated that it is possible, to specify the right conditions, to perform comparatively accurate simulations concerning twophase flow in porous media applications with the VOF model available in FLUENT.
TABLE OF CONTENTS ACKNOWLEDGEMENT………………………………………………
i
ABSTRACT…………………………………………………………….
ii
TABLE OF CONTENTS………………………………………………..
iii
LIST OF TABLES………………………………………………………
vi
LIST OF FIGURES…………………………………………………….
vii
LIST OF ABBREVIATIONS…………………………………………..
xi
NOMENCLATURE ……………………………………………………
xii
CHAPTER ONE: INTRODUCTION……………………………..........
1
1.1 Overview……………………………………………………………
1
1.2 Porous Media……………………………………………………….
1
1.3 Issues of scale………………………………………………………
4
CHAPTER TWO: LITERATURE REVIEW…………………………..
6
2.1 Pore-scale Properties………………………………………….........
6
2.1.1 Introduction to Network Models……………………………..
6
2.1.2 Interfacial Tension and Wettability…………………………..
8
2.1.3 Capillary Pressure…………………………………………….
12
2.2 Macro-scale Properties…………………………………..................
12
2.2.1 Rock Properties……………………………………………….
13
2.2.2 Darcy‟s Law…………………………………………………..
13
2.2.3 Oil Recovery Processes……………………………………….
15
2.3 Recent theoretical and experimental studies………………………..
17
2.4 Research objectives…………………………………………………
26
2.5 Thesis organization…………………………………………………
27
CHAPTER THREE: PORE-SCALE FLOW WITH CFD……………… 28 3.1 Introduction…………………………………………………………
28
3.2 Pore-scale modelling approach……………………………………..
29
3.3 Experimental imaging………………………………………………
30
3.4 Governing Equations of Fluid Flow………………………………...
32
3.4.1 Equation of Continuity…………………………………..........
32
3.4.2 Equation of Momentum………………………………………
33
3.5 Permeability calculations…………………………………………..
34
3.6 Modelling of Inertial body force…………………………………..
35
3.7 Computational Fluid Dynamics (CFD)……………………………
36
3.7.1 Pre-processor………………………………………………..
37
3.7.2 Solver…………………………………………………………
37
3.7.3 Post-processor………………………………………………..
37
3.8 The Finite Volume Method (FVM)……………………………….
38
3.8.1 Integration of the Equation of Transport over a Control
40
Volume………………………………………………………………… 3.8.2 Discretization………………………………………………..
40
3.8.3 Solution……………………………………………………..
42
3.9 Multiphase Flow Models in FLUENT……………………………
43
3.9.1 The Euler-Lagrange Approach……………………………..
43
3.9.2 The Euler-Euler Approach…………………………………
43
3.10 Volume of Fluid (VOF) Model…………………………………..
44
3.10.1 The Volume Fraction Equation……………………………
45
3.10.2 The Momentum Equation…………………………………
45
3.10.3 Interfacial Face Flux Calculations…………………………
46
3.10.4 Surface Tension……………………………………………
48
3.11 Summary…………………………………………………………..
49
CHAPTER FOUR: MODEL FORMULATION………………….......
50
4.1 Overview…………………………………………………………..
50
4.2 Geometry Description…………………………………………….
51
4.3 Initial and Boundary Conditions………………………………….
52
4.4 Material Properties………………………………………………..
52
4.5 Mesh Generation…………………………………………………..
53
CHAPTER FIVE: RESULTS AND DISCUSSION…………………..
59
5.1 Single Phase Flow Simulations………………………………........
59
5.1.1 Estimation of absolute permeability…………………….......
65
5.1.2 Effect of dynamic viscosity………………………………….
67
5.2 Two Phase Flow Simulations……………………………………..
68
5.2.1 Pressure-driven flow of interfaces in a pore body…………….
68
5.2.2 Prediction of the effect of vibration …………………………
76
CHAPTER SIX: CONCLUSIONS AND RECOMMENDATIONS` FOR FUTURE WORK…………………………………………………
85
6.1 Conclusions…………………………………………………………
85
6.2 Recommendations for future work …………………………………
86
REFERENCES………………………………………………………….. 88 APPENDIX A…………………………………………………………..
94
APPENDIX B…………………………………………………………..
96
APPENDIX C………………………………………………………….
97
LIST OF TABLES Table 2.1 Examples of Contact Angle………………………………
11
Table 4.1 Physical properties of oil-water flow system applied in the simulations…………………………………………………
53
Table 4.2 Results of the mesh resolution study………………………
54
Table B.1 Suggested values of time step size and under-relaxation factors………………………………………………………
96
LIST OF FIGURES Figure 1.1
Schematic drawing of a porous medium filled with one or two fluids……………………………………………….
2
Figure 1.2
Different scales in a porous medium………………………
4
Figure 2.1
An illustrative example of network model………………
7
Figure 2.2
Interfacial tensions between a solid surface, water and oil………………………………………………………......
Figure 2.3
8
The contact angle represents the balance between surface tension forces and adhesion forces between wall and water………………………………………………………
Figure 2.4
Different saturation states in a water-wet porous media system……………………………………………………
Figure 2.5
11
Diagram showing definitions and directions of Darcy‟s law………………………………………………………..
Figure 2.6
9
14
Geometry of an immiscible two-phase flow through a horizontal straight channel with a constriction under an external pressure gradient …………………………………
Figure 2.7
20
Composite image of the etched repeat pattern in the micromodel (approximately 509x509 µm)………………
23
Figure 3.1
General concept of pore-scale approach…………………
30
Figure 3.2
Infinitesimal fixed control volume………………………
32
Figure 3.3
Control volume around node P……………………………
40
Figure 3.4
Distribution of at different Pe numbers…………………
42
Figure 3.5
Actual interface shape……………………………………
47
Figure 3.6
Interface shape represented by the geometric Reconstruction scheme……………………………………
47
Figure 3.7
Interface shape represented by the donor-acceptor scheme………………………………………………. 47
Figure 4.1
Scanning electron microscope image of the repeat pattern in the silicon wafer……………………………………….
50
Figure 4.2
CAD geometry……………………………………………
51
Figure 4.3
The quadrilateral mesh (Level 1) ………………………..
55
Figure 4.4
Magnified view of the mesh showing the interior cells at Level 1…………………………………………………..
Figure 4.5
The quadrilateral mesh (Level 2) used in FLUENT simulations………………………………………………..
Figure 4.6
57
Illustration of the triangular mesh (interior between two grains) used in FLUENT simulations……………………...
Figure 4.10
57
Magnified view of the mesh showing the interior cells at Level 3………………………………………………….…
Figure 4.9
56
Default display of numerical mesh using quadrilateral elements at Level 3 used in FLUENT simulations………...
Figure 4.8
56
Magnified view of the mesh showing the interior cells at Level 2………………………………………………….…
Figure 4.7
55
58
Basic structure of CFD simulation procedure using FLUENT…………………………………………………...
59
Figure 5.1
Velocity vectors coloured by velocity magnitude (m/s)…..
60
Figure 5.2
Comparison of velocities along the outlet boundaries obtained by FLUENT (present study) to COMSOL (Keller‟s study) using Level 2 of grid..................................
Figure 5.3
61
Comparison of velocities along the outlet boundaries obtained by FLUENT (present study) to COMSOL
62
(Keller‟s study) using Level 3 of grid.................................. Figure 5.4
Contours of static pressure (Pa)……………………………
63
Figure 5.5
Stream function (kg/s) of water……………………………
63
Figure 5.6
Comparison of velocities along the outlet boundaries obtained by FLUENT (present study) to COMSOL (Keller‟s study) using triangular cells..................................
Figure 5.7
64
Comparison of velocities along the outlet boundaries obtained by FLUENT using both quadrilateral cells (Level 2 and 3) and triangular cells................................………….
Figure 5.8
The absolute permeability versus the velocity in x-direction along outlet boundaries……………………….
Figure 5.9
Figure 5.11
The effect of oil viscosity on the required external pressure to mobilize oil through the model………………………… Contours of initial volume fraction of Oil (shown in red colour)……………………………………………………..
Figure 5.12
66
The absolute permeability as a function in y-direction along outlet boundaries……………………………………
Figure 5.10
65
66 67
69
Results of the simulation of oil (shown in red colour) displaced by water. External pressure difference is 30 kPa…………………………………………………………
Figure 5.13
Relative saturation of Oil as a function of time. External pressure is 30 kPa………………………………………….
Figure 5.14
70
71
Results of the simulation of oil (shown in red colour) displaced by water. External pressure difference is 30 kPa and by using Level 3 of grid……………………………….
Figure 5.15
Relative saturation of Oil as a function of time. External pressure is 30 kPa and by using Level 3…………………...
Figure 5.16
73
74
Relative saturation of Oil as a function of time for several simulations with various values of external pressure 75
difference.............................................................................. Figure 5.17
Increasing of initial volume fraction of Oil (shown in red colour)……………………………………………………..
Figure 5.18
Results of the simulation of oil (shown in red colour) displaced by water. External pressure difference is 30 kPa…
Figure 5.19
75
76
Results of the simulation of oil (shown in red colour) displaced by water. External pressure difference is 30 kPa and the vibration amplitude is 10 m/s2……………
Figure 5.20
79
Relative saturation of Oil as a function of time for several simulations with the same initial conditions. External pressure is 30 kPa and vibration frequency is 10 Hz...........
Figure 5.21
80
Results of the simulation of oil (shown in red colour) displaced by water. External pressure difference is 30 kPa, the vibration amplitude is 10 m/s2 and using mesh at Level 3……………………………………………………
Figure 5.22
81
Relative saturation of Oil as a function of time. External pressure is 30 kPa and by using Level 3 of mesh , (f = 10 Hz and A = 10 m/s2 )……………………………………….
Figure 5.23
82
Relative saturation of Oil as a function of time at various values of frequencies ( 10 Hz and 100 Hz). External pressure is 30 kPa and the vibration amplitude is 10 m/s2…………………………………………………….
Figure 5.24
83
Relative saturation of Oil as a function of time at various values of frequencies ( 100 Hz and 500 Hz). External pressure is 30 kPa and the vibration amplitude is 10 m/s2…………………………………………………….
84
LIST OF ABBREVIATIONS AMMoC
Advection Meshfree Method of Characteristics
CAD
Computer Aided Drafting
CFD
Computational Fluid Dynamics
CMT
Computed Micro-Tomography
CPU
Central Processing Unit
CT DCA
Dynamic Contact Angle
DXF
Drawing Exchange Format
EOR
Enhanced Oil Recovery
Eq
Equation
FVM
Finite Volume Method
GUI
Graphical User Interface
IFT
Interfacial Tension
PDE
Partial Differential Equations
pH
potential Hydrogen
REM
Representative Elementary Volume
SC
Scanning Curve
SCA
Static Contact Angle
SEM
Scanning Electron Microscope
SL
Stereo Lithography
T/C
Throat Colloid diameter ratio
TDMA
Tri-Diagonal Matrix Algorithm
UDF
User Defined Function
VOF
Volume of Fluid
WP
Wetting Phase
NOMENCLATURE Symbols Roman Letters A
Vibration amplitude
[m/s2]
B
Inertial body force
[N/m3]
ED
Displacement efficiency
f
Angular frequency
G
Pressure gradient
[Pa/m]
g
Acceleration of gravity
[m/s2]
K
Permeability
[m2]
L
The length of the pore channel
[m]
M
Viscosity ratio
Pc
Capillary pressure
[Pa]
p
Pressure
[Pa]
Q
Volume flow rate
[m3/s]
R
Mean pore radius
[m]
r
Mean radius of curvature
[m]
sij
Strain-rate tensor
[1/s]
t
Time
u, v, w
Velocity components in x, y, z-directions
[m/s]
U
Superficial velocity
[m/s]
V
Velocity vector
[m/s]
[Hz]
-
[s]
Greek Letters δij
Kronecker delta
-
ε
Porosity
-
θ
Contact angle
κ
Curvature of the interface
μ
Dynamic viscosity
[Ns/m2]
ρ
Density
[kg/m3]
ζ
Surface tension
[N/m]
ηij
Stress tensor
[N/m2]
General flow variable
-
Under-relaxation factor
-
α
Volume fraction
-
Γ
Diffusion coefficient
-
Δt
Time step size
Grad, del, napla
Subscripts c
Capillary
crit
Critical
D
Displacement
nw
non-wetting
q
qth phase
w
Wetting
Superscripts T
Transposition
[Deg.] -
[s] -
Dimensionless Numbers Bo
Bond number, Bo
g w nw R 2
Ca
Capillary number, Ca
Re
Reynolds number, R e
U ww
uD
CHAPTER ONE INTRODUCTION
1.1 Overview Two phase flow and transport processes in porous media are involved in a wide variety of engineering applications, such as oil recovery and groundwater remediation. The inherent heterogeneity of subsurface porous media, as well as the complexity involved in the multiphase physics, result in a significant challenge to the fundamental understanding of multiphase flow and transport (Pan, 2003). In this chapter, the different physical aspects of flow in porous media will be briefly introduced to the reader. The beginning is with a definition of porous media, their types, and their most common length scales. A review of literature and theoretical background relevant to the research work is introduced in Chapter two.
1.2 Porous Media A porous medium is a body composed of a persistent solid part, called solid matrix, and the remaining is void space (or pore space) that can be filled with one or more fluids (e. g. water, oil and gas). Typical examples of a porous medium are soil, sand, cemented sandstone, karstic limestone, foam rubber, bread, lungs or kidneys. The flow in a porous medium takes place through connected pores in the rock. Regions on a larger scale that contain oil or gas are called reservoirs (Bastian, 1999). A phase is defined in (Bastian, 1999) as a chemically homogeneous portion of a system under consideration that is separated from other portions by a definite physical boundary. In the case of a single–phase system the void space of the porous medium is filled by a single fluid (e. g. water) or by
several fluids completely miscible with each other (e. g. fresh water and salt water). In a multiphase system the void space is filled by two or more fluids that are immiscible with each other, i. e. they maintain a distinct boundary between them (e. g. water and oil). There may only be one gaseous phase since gases are always completely miscible. Formally the solid matrix of the porous medium can also be considered as a phase called the solid phase. Figure 1.1 shows a two–dimensional cross section of a porous medium filled with water (single–phase system, left) or filled with water and oil (two–phase system, right).
Figure 1.1: Schematic drawing of a porous medium filled with one or two fluids.
Bastian (1999) defines a component to be part of a phase that is composed of an identifiable homogeneous chemical species or of an assembly of species (ions, molecules). The number of components needed to describe a phase is given by the conceptual model, i. e. it depends on the physical processes to be modelled. The example of fresh and salt water given above is described by a single–phase two component system. In this work the components are neglected and only the phases are considered. Adler (1992) classified porous media into three different types, which are artificial media, biological media and geological media. A regular array of spheres or fibers (or cylinders) is one of the most artificial media in that it
never occurs in the natural system. However, because of the relative simplicity of these artificial structures, many studies have been focused on artificial media, either experimentally, theoretically or numerically. These studies contribute to an important part of pore-scale modelling. Biological media (e.g. bones, tissues and membranes) are a special kind of porous media that are filled with a variety of fluid substances. The structures of biological media, in some cases, support the pore-scale models, such as a capillary network. For instance, pores and tubes form an important part of plant tissues. Geological media are of great interest because of their practical importance in numerous aspects of water resources and contaminant remediation (Pan, 2003). This work is related only to the geological porous media. In environmental engineering, it is conventional to distinguish between unconsolidated and consolidated media because groundwater is found in the pores of unconsolidated sediments, or in the fractures and openings of the consolidated rocks. A good example of unconsolidated media is a bed of sand particles. Sand is made of rough silica particles that are approximately spherical, so it can be idealized as a pack of different-sized spheres. Unconsolidated sands typically have porosities of between 30 and 45%. When sand is buried, chemical solutions in groundwater crystallize in some of the pore spaces between sand grains, which create hard cement that holds the sand grains together. The resulting material is sandstone, which is one of the consolidated sedimentary rocks. Sandstone makes up roughly 25% of the sedimentary rocks of the world. Other classes of consolidated porous materials includes shales, which have porosity from 10 to 25%, and carbonated rocks, i.e. rocks mostly made of calcite, which have porosity from 20% to more that 50%. As one may expect, these rocks exhibit chaotic shapes
in solid/pore interfaces, and the distribution of the pores in these rocks is much more spread out than that in unconsolidated sand and sandstones. If the geometry of artificial and biological media is relatively easy to obtain in most cases, the complexity involved in various types of geological porous media is confusing. How can one describe the pore morphology of geological material in three dimensions has been an immense topic in modern pore-scale studies (Pan, 2003).
1.3 Issues of scale The important feature in modeling porous media flow is the consideration of different length scales. Figure 1.2 shows a cross section through a porous medium containing different types of sands on three length scales (Bastian, 1999). In Figure 1.2a the cross section is on the order of 10 meters wide. This scale is called the macroscopic scale. There we can identify several types of sand with different average grain sizes. A larger scale than the macroscopic scale is often called regional scale but is not considered here, (Helmig, 1997).
(a) Macroscopic scale
(b) Microscopic scale
(c) Molecular scale
Figure 1.2: Different scales in a porous medium.
If we zoom in to a scale of about 10-3 m as shown in Figure 1.2b we reach the microscopic scale where individual sand grains and pore channels are visible. In the figure we see the transition zone from fine sand to coarser
sand. The void space is supposed to be filled with water. Our goal is basically to study the microscopic scale flow using computational fluid dynamics (CFD) to perform numerical simulations concerning single phase and twophase flows. Magnifying further into the water filled void space one would finally see individual water molecules as shown in Figure 1.2c. The larger black circles are oxygen atoms, the smaller white circles are the hydrogen atoms. This scale of about 10-9 m will be referred to as the molecular scale.
CHAPTER TWO LITERATURE REVIEW
The literature search was compiled from over fifty sources including websites, journals, conference proceedings, and books. Many papers and books have been consulted, but most of them are briefly mentioned and some of them are discussed along the thesis. The goal of this chapter is to focus on the study of multiphase flow processes as researchers manifested at various scales and on how the physical description at one scale can be used to obtain a physical description at a higher scale. Thus, some papers start at the pore scale and, mostly through pore-scale network modelling to obtain an average description of multiphase flow at the (laboratory or) core scale. Some other papers start at the core (macroscopic) scale.
2.1 Pore-scale Properties Single–phase flow is governed by pressure forces arising from pressure differences within the reservoir and the exterior gravitational force (Bastian, 1999). In multiphase flows the sharp interfaces between fluid phases on the microscopic level give rise to a capillary force that plays an important role in these flows. Even though flow in a single tube is given by simple equations, the network of the tubes is impossible to be known in detail. For instance, the characteristic length of a reservoir containing water and oil may be about 1 km, thus reservoir engineers are usually not interested in the behavior at pore scale.
2.1.1 Introduction to Network Models Pore-network models are effective tools used to investigate or predict macroscopic properties from fundamental pore scale behaviour of processes and phenomena based on geometric volume averaging. This is mainly due to the fact that explicit solutions to describe the behaviour of fluids in porous media system are impractical because of the complexity of these systems. Pore-network models are of two types: quasi-static and dynamic models (AlRaoush, 2002). Quasi-static models apply a capillary pressure on the network and calculate the positions of fluid-fluid interfaces ignoring the dynamics of pressure and the interfaces. Dynamic models apply inflow rate for each one of the components of the system and calculate the corresponding pressure and the interface positions. Pore-networks models are mechanistic models that idealize the complex pore space geometry of the porous media by representing the pore space by pore elements having simple geometric shapes. Figure 2.1 illustrates a simple configuration of network model. The most critical part in constructing a pore-network model is defining its structure and geometry, i.e., the locations of pore bodies (and hence pore-throat length distribution), porebody size distributions, throat-body size distribution, connectivity and the spatial correlation between pore bodies and pore throats (Al-Raoush, 2002).
Figure 2.1: An illustrative example of network model.
When the pore space of porous media is occupied by two or more fluids, two types of flow are possible, miscible displacement and immiscible displacement (Fanchi, 2006). In miscible displacement, the two fluids can completely dissolve in each other and there is no fluid-fluid interface. In the case where a fluid-fluid interface exists due to the interfacial tension between fluids, an immiscible displacement occurs as a form of simultaneous flow of the phases that exist in the system (e.g., oil, water, and gas). The second one will be involved in this work. The following sections provide discussion about single and multiphase processes and how they are related to the pore-network structure. Several basic concepts are needed to understand multiphase flow. They include interfacial tension, wettability and contact angle. These concepts lead naturally to a discussion of capillary pressure.
2.1.2 Interfacial Tension and Wettability Interfacial tension is known to be a major factor constraint on multiphase flow in porous media (Fanchi, 2006). Wettability is the ability of a fluid phase to preferentially wet a solid surface in the presence of a second immiscible phase. The wetting, or wettability, condition in a rock/fluid system depends on interfacial tension (IFT). Changing the type of rock or fluid can change IFT and, hence, wettability of the system. Adding a chemical such as surfactant, polymer, corrosion inhibitor, or scale inhibitor can alter wettability. Figure 2.2 shows two immiscible fluids (water and oil) in contact with a solid surface. Wettability is measured by a contact angle. The contact angle is always measured through the denser phase. Contact angle is related to interfacial tensions by (Fanchi, 2006), os ws ow cos
(2.1)
where os interfacial tension between oil and solid (N/m)
ws interfacial tension between water and solid (N/m)
ow interfacial tension between oil and water (N/m)
θ is called the contact angle and it determines which fluid (oil or water) will preferentially wet the solid (i.e., the wettability of a solid by a liquid). If θ is less than 90o, then water preferentially wets the surface and is called the wetting fluid. If θ is more than 90o, then oil preferentially wets the surface and water would be the non-wetting phase.
ow Oil
os
Water
ws
Solid
os
Figure 2.2: Interfacial tensions between a solid surface, water and oil.
When water is brought into contact with a solid surface, adhesion of the solid with the water and cohesion of the water become interacting forces (Figure 2.3). The contact angle is often seen as the result of the balance of these forces (Al-Raoush, 2002). The contact angle is the angle between the water and the solid at the contact line, the location where the water surface meets the solid surface. Whence the cohesion and adhesion forces are in equilibrium, the matching contact angle is static. Out of equilibrium, the water near the solid tends to move towards its equilibrium state. The velocity of the contact line depends monotonically on the deviation of the time-dependent (dynamic) contact angle (DCA) with the static contact angle (SCA).
Figure 2.3: The contact angle represents the balance between surface tension forces and adhesion forces between wall and water.
The DCA plays an important role in the fluid dynamics, especially if gravitational forces are not dominant, and surface tension forces are dominant (Al-Raoush, 2002). Constitutive relations for multiphase flow in porous media are dominated by capillary forces at the pore scale. Gravity and viscous forces are also contributed to multiphase flow behaviour. To evaluate the effect of these forces on the experimental system considered, three nondimensional variables, the capillary number Ca, which describes the ratio of viscous forces to capillary forces, the viscosity ratio M, and the Bond number Bo, which describes the ratio of gravity forces to capillary forces, are often used and defined as (Al-Raoush, 2002). Ca
U ww
M
Bo
nw w
g w nw R 2
where Uw is the aqueous phase (hereafter, “water”) Darcy velocity, g is the gravitational constant, and R is the mean pore radius.
Wettability is usually measured in laboratory. Laboratory fluids should also be at reservoir conditions to obtain the most reliable measurements of wettability. Based on laboratory tests, most known reservoirs have intermediate wettability and are preferentially water wet. Examples of contact angle are presented in Table 2.1 for different wetting conditions (Fanchi, 2006). Table 2.1: Examples of Contact Angle.
Wetting Condition
Contact Angle (Degrees)
Strongly water-wet
0-30
Moderately water-wet
30-75
Neutrally wet
75-105
Moderately oil-wet
105-150
Strongly oil-wet
150-180
Saturation ratio of a fluid, or simply saturation, is the fraction of the void space filled by that fluid. In a water-wet porous media system, three types of fluid saturation can be identified (Al-Raoush, 2002). At very low water saturation, water forms a thin film around the solid surface and isolated water-rings around the solid contact angle called pendular rings. These pendular rings become continuous as the wetting phase saturation increases. If the wetting phase saturation increases any further, the saturation is called funicular and the flow becomes possible due to the continuity of the wetting phase. As the saturation of the wetting phase increases, the nonwetting phase loses its continuity and form isolated droplets. This state is called insular saturation. Figure 2.4 shows these saturation states. The same concept can be applied to oil-wet porous media systems.
Pendular
Funicular
Insular
Figure 2.4: Different saturation states in a water-wet porous media system.
2.1.3 Capillary Pressure According to Young-Laplace equation, capillary pressure is the pressure difference across the curved interface formed by two immiscible fluids in a small capillary tube (Fanchi, 2006): Pc Pnw Pw
(2.2)
where Pnw is the pressure of the nonwetting phase (hereafter, “oil”) and Pw is the pressure of the wetting phase (hereafter, “water”). The capillary pressure depends on the pore geometry (pore radius), on the interaction between fluids, ζ and on the amount and location of each phase in the pore space. Due to the complexity of natural porous media systems, it is difficult to obtain these parameters analytically. Therefore, the pore space is usually idealized (e.g., capillary tube or uniform spheres) to predict the capillary pressure as a function of the fluid saturation or determined experimentally. In addition, there is no unique relationship between capillary pressure and fluid saturation because of the hysteresis of the capillary pressure. Expressing capillary pressure in terms of force up per unit area gives: Pc
2r cos 2 cos r 2 r
(2.3)
where ζ is the interfacial tension between fluids and r is the mean radius of curvature.
2.2 Macro-scale Properties The porous medium is considered statistically homogeneous at large length scales but random on the pore scale due to the pore size distribution. Thus, the local randomness in the structures of the two fluids is caused by the capillary pressure while the macroscopic flow behavior is dominated by viscous forces (Das and Hassanizadeh, 2005; Iassonov, 2005).
2.2.1 Rock Properties Rock properties significantly influence the production of hydrocarbons from porous media (Fanchi, 2006). There are two important quantities describing the properties of a porous medium: the porosity and the permeability K (Bastian, 1999). The porosity of a porous medium is defined as:
pore volume matrix volume
(2.4)
where the pore volume denotes the total volume of the pore space in the matrix, and the matrix volume is the total volume of the matrix including the pore space. Thus, 0 1 . The porosity has to be averaged over many pores and the considered matrix volume must be larger than the pore size. Often the porosity can be chosen as constant for the whole medium. The permeability K describes the ability of the fluid to flow through the porous medium. K is often called the absolute permeability and is a quantity depending on the geometry of the medium only. There has been much effort to establish relations between the permeability and the porosity. However, a general formula seems impossible to find and the permeability was found to be proportional to m where m is constant in the range of 3 to 6 depending on the geometry of the porous medium (Fanchi, 2006).
2.2.2 Darcy’s Law In fluid dynamics and hydrology, Darcy's law is a phenomologically derived constitutive equation that describes the flow of a fluid through a porous medium. The law was formulated by Henry Darcy based on the results of experiments on the flow of water through beds of sand (Darcy, 1856). It also forms the scientific basis of fluid permeability used in the earth sciences. Figure (2.5) shows Darcy‟s experiment.
Figure 2.5: Diagram showing definitions and directions for Darcy's law.
Although Darcy's law (an expression of conservation of momentum) was determined experimentally by Darcy, it can be shown that under appropriate assumptions the momentum conservation of the Navier–Stokes equation reduces to the Darcy–Law on the macroscopic level. Consider a porous medium of absolute permeability K in a homogeneous gravitation field where one fluid of viscosity is injected through the medium by applying a pressure drop gradient P across the matrix. Then the superficial velocity U of the fluid through the medium is given by Darcy's equation: U
(2.5)
K
(P g )
where g denotes the acceleration due to the gravitational forces and is the density of the fluid. For fluid flow in a horizontal layer g can be neglected (our model). Calculation of the permeability using CFD is one of the goals of the present work and this will be shown in the next chapters. If we perform a dimensional analysis, we see that permeability has dimension of (m2) which is physical related to the cross-sectional area of pore throats in rock. One application of Darcy's law is to water flow through an aquifer. Darcy's law along with the equation of conservation of mass is equivalent to the groundwater flow equation, one of the basic relationships of hydrogeology. Darcy's law is also used to describe oil, water, and gas flows through petroleum reservoirs. Experimental tests have shown that for flow regimes with values of Reynolds number up to 10 may still be Darcian (Darcy, 1856). Darcy‟s Law correctly describes laminar flow, and may be used as an approximation of turbulent flow. Permeability calculated from Darcy‟s Law is less than true rock permeability at turbulent flow rates. The linearity of Darcy‟s Law is an approximation that is made by virtually all commercial simulators (Fanchi, 2006). So we must ensure that the flow in the channel is laminar at any time. This can be done by calculating the Reynolds number (Iassonov, 2005), Re
uD
(2.6)
where u is the maximum fluid velocity and D is the diameter of the channel. Assuming the parabolic (Poiseuille) velocity profile, u can be calculated for single channel model as twice the average fluid velocity in the narrowest part of the pore (Iassonov, 2005), u2
Q 2 rmin
(2.7)
2.2.3 Oil Recovery Processes Primary production is ordinarily the first stage of production. It relies entirely on natural energy sources. To remove petroleum from the pore space it occupies, the petroleum must be replaced by another fluid, such as water, natural gas, or air. Oil displacement is caused by the expansion of in situ fluids as pressure declines during primary reservoir depletion. This primary stage of oil production rarely exceeds 30% of the total original oil in place (Fanchi, 2006). To recover part of the remaining oil after the primary recovery, a fluid (usually water) is injected into some wells (injection wells) while oil is produced through other wells (production wells). This process serves to maintain high reservoir pressure and flow rates. It also displaces some of the oil and pushes it toward the production wells. This stage of oil recovery is called secondary recovery. In this recovery stage, there is two phase immiscible flow, one phase being water and the other being oil, without mass transfer between the phases (Chen, Huan and Ma, 2006). Both primary and secondary recovery processes are designed to produce oil using immiscible methods. Additional methods may be used to improve oil recovery efficiency. EOR processes are usually classified as one of the following processes: chemical, miscible, thermal, and microbial. Because water injection process rarely exceeds 50% recovery threshold; at the same time, associated costs and environmental impact may be significant. Power wave is considered one of EOR methods, recently, although the reason for the vibratory mobilization of the residual oil has remained unclear (Beresnev et al., 2005). To better understand the reason why the elastic waves can mobilize residual oil, we will attempt to explain the effects of vibration on oil recovery in this research. Vibratory stimulation might not only result in higher recovery efficiency, but also would be ecologically clean and economical to
implement, which further promotes the interest in developing such a technology (Iassonov, 2005). It is essential to determine the displacement efficiency by comparing initial and final volumes of fluid in place. Displacement efficiency accounts for the efficiency for recovering mobile hydrocarbon. To be specific, we define displacement efficiency for oil as the ratio of mobile oil to original oil in place (Fanchi, 2006), ED
oi or oi
(2.8)
where ED
displacement efficiency
oi
initial saturation (volume fraction) of oil
or
residual saturation (volume fraction) of oil Residual oil saturation is replaced by oil saturation at abandonment in
floods that do not reduce initial oil saturation to residual oil saturation during the life of the flood. One of the goals of enhanced oil recovery (EOR) processes such as water injection and vibratory stimulation is to reduce residual oil saturation and increase displacement efficiency.
2.3 Recent Theoretical and Experimental Studies The following is a review of some papers that are related to both micro and macro scales, most of them were done at laboratory and then studied numerically using VOF model which is the core of our study. A short description of the papers is given below.
Crandall, Ahmadi and Smith (2007) designed and created a new flowcell using stereo-lithography (SL). This porous media geometry was used for experimental and numerical studies of immiscible two-phase drainage. With SL construction both the throat heights and the throat widths have been
varied. This has created a cell with a wide variability of throat properties. Additionally, a Volume of Fluid (VOF) of the same flowcell geometry was studied. The experimental studies of air invading the flowcell saturated with water were shown a change from dendritic fingering structures to more stable displacements as the constant injection rate was reduced. This corresponds to a decreased fractal dimension and percent saturation of the invading air at low flow rates. The numerical model was first tested with the same fluids (air and water) and was shown to be in good agreement with the experiments. This VOF model was then extended to study different fluid-fluid pairings within the flowcell. As the defending fluid viscosity was increased the percent saturation of the invading fluid was shown to decrease. The SL construction was enabled the same geometry to be studied experimentally and numerically, thus allowing a wide variety of conditions to be analyzed.
Ahmadi (2006) used experimental and computational modelling methods for studying multiphase flows in porous and fractured media. Particular attention was given to the flows in a laboratory-scale flow cell model. It is shown that the gas-liquid flows generate fractal interfaces and the viscous and capillary fingering phenomena were discussed. Experimental data concerning the displacement of two immiscible fluids in the lattice-like flow cell were presented. The flow pattern and the residual saturation of the displaced fluid during the displacement were discussed. Numerical simulations results of the experimental flow cell were also presented. The numerical simulation results for single and multiphase flows through rock fractures were also presented. Fracture geometry studied was obtained from a series of CT scan of an actual fracture. Computational results show that the major losses occur in the regions with smallest apertures. An empirical expression for the fracture friction factor was also described. Applications to
CO2 sequestration in underground brine fields depleted oil reservoir stimulation were discussed.
Beresnev et al. (2005) studied numerically and experimentally a porescale model to predict the effects of vibration on residual oil that is entrapped as ganglia in pore constrictions because of resisting capillary forces. An external pressure gradient exceeding an „„unplugging‟‟ threshold is needed to carry the ganglia through. The vibration helps overcome this resistance by adding an oscillatory inertial forcing to the external gradient; when the vibratory forcing acts along the gradient and the threshold is exceeded, instant „„unplugging‟‟ occurs. They showed that mobilization effect is proportional to the amplitude and inversely proportional to the frequency of vibrations. They observed this dependence in a laboratory experiment, in which residual saturation was created in a glass micromodel, and mobilization of the dyed organic ganglia was monitored using digital photography. They also directly demonstrated the release of an entrapped ganglion by vibrations in a computational fluid-dynamics simulation.
Hai Huang and Paul Meakin (2005) successfully applied a popular multiphase flow simulation method in computational fluid dynamics, the volume of fluid (VOF) method, to model the liquid motion in partially saturated fracture apertures and fracture network under a variety of flow conditions. The effects of inertial forces, viscosity, and gravity acting on the fluid densities, fracture wall wetting dynamics, and the pressure drop across the fluid-fluid interfaces due to surface tension are systematically incorporated into the model. Complex dynamics of fluid-fluid interfaces and free surfaces in fracture apertures and pore spaces were handled very well by this model. Their simulation technique provided quantitative ways for better understanding of the fundamental physics governing unsaturated fluid motion
in fractures and porous medium. Their simulation results revealed that fluid motion in unsaturated fractures is complex, even for very simple fracture geometries under constant liquid injection conditions.
Iassonov (2005) presented new models to better understand the capillary trapping and vibration induced mobilization of a non-wetting phase ganglia in porous media, and to develop a method capable of estimating the effect of vibration on oil production in real-world applications. He examined and compared three techniques of modelling the effect of vibration on twophase immiscible displacement in porous media, differing mainly in the way a porous medium is represented in a computer model. These methods were pore-network modelling, a custom computer model for a flow in a single pore, and utilization of an existing commercial computational fluid-dynamics (CFD) software package FLUENT. Iassonov explained the main reason for the entrapment of non-wetting fluid which is varied in pore sizes, leading to variations in capillary pressures. This general idea can be illustrated with the simple case of a single blob (ganglion) of non-wetting fluid (hereafter, “oil”) in a long filled with wetting fluid (hereafter, “water”) with a constriction (“pore throat), schematically illustrated in Figure 2.6. Iassonov assumed that there is a constant external pressure gradient, driving the ganglion towards the constriction in the pore channel. As the leading (right) interface enters the pore throat (Figure 2.6a), its radius must decrease, resulting in an increase in the capillary pressure created by the meniscus. Since the radius of the leading interface is smaller than that of the trailing (left) meniscus, this will create imbalance within the oil blob, resisting the external pressure gradient (shown as an opposing arrow inside the blob in Figure 2.6b). If the external pressure is not sufficiently high, the leading
interface will not pass through the constriction, resulting in the entrapment of the oil ganglion.
(a)
(b) Figure 2.6: Geometry of an immiscible two-phase flow through a horizontal straight channel with a constriction under an external pressure gradient: (a) Ganglion in the open channel. (b) The ganglion is trapped.
Exact conditions for the entrapment will depend on many parameters. For simplicity, he ignored any dynamic effects (such as inertia, or the variation in the contact angle due to menisci movement). Using the Laplace formula capillary pressure and assuming an 180o contact angle for the oilwater-solid contact line, the entrapment criterion can be written as, 1 1 G 2 rmin rmax
/ L
(2.9)
where G is the absolute value of the external pressure gradient, is the oil/water surface tension, rmax and rmin are the maximum and the minimum radii of the pore (Figure 2.6), and L is the length of the pore channel (in case of a relatively long ganglion located in a series of interconnected pores, L can approximate the length of the oil ganglion itself). The right-hand side of
Formula (2.9) represents the critical pressure gradient (Gcrit) required for the mobilization of a particular blob. Formula (2.9) shows two important aspects of the capillary trapping. First, the higher variation in pore sizes leads to higher the potential for entrapment. Second, the shorter ganglia are more difficult to mobilize by a constant external pressure gradient, and thus are more likely to remain trapped.
Iske and Käser (2005) proposed the application of a novel meshfree particle method to the Buckley-Leverett model. The utilized meshfree advection scheme, called AMMoC, is essentially a method of characteristics, which combines an adaptive semi-Lagrangian method with local meshfree interpolation by polyharmonic splines. The method AMMoC was applied to the five-spot problem, a well-established model problem in petroleum reservoir simulation. The numerical results and subsequent numerical comparisons with two leading commercial reservoir simulators, ECLIPSE and FrontSim, showed the good performance of their meshfree advection scheme. Since the present work mainly depends on Keller’s model, it will be discussed in some detail. Transport in groundwater of some contaminants, such as radionuclides, can be facilitated in the presence of colloidal particles. Therefore, Sirivithayapakorn and Keller (2003) presented experimental evidence of the effect of colloid exclusion from areas of small aperture sizes, using direct observations at the pore-scale using a realistic micromodel of porous media. Four sizes of hydrophobic latex spheres in aqueous suspension, from 0.05 to 3 μm, were introduced into the micromodel at three different pressure differences. They observed the frequency of occurrence of the size exclusion effect and the influence of relative size of pore throats and colloids (T/C ratio) and flow velocity. From their observations the smallest T/C ratio
entered by these different colloids, was 1.5. They also observed certain preferential pathways through the pore space for different colloid sizes, such that size exclusion eventually results in distinct pathways. These preferential paths become more important for larger colloids and for greater pressure gradients. Measured colloid velocities were 4.0 to 5.5 times greater than estimated pore water velocities. Acceleration factors (ratio of colloid to water velocity) increased for all colloid sizes with increasing pore-scale. Smaller particles appeared to travel along faster streamlines in pore throats, while larger particles travel along with a number of streamlines, thus at a slightly lower velocity than the small colloids. At larger scales the acceleration factor is decreased owing to Brownian motion, adsorption, colloid and straining filtration, and other factors, but these pore-scale results shed light on the size exclusion effect and its role in determining early colloid breakthrough. A thin slice of porous media (fine sand) was imaged through an optical microscope (approximately 600 by 600 µm) and then digitized. To improve the connectivity in the micromodel, the digitized image was modified slightly. The pattern was then repeated 100 x 100 times. The size of their pore space was chosen based on the physical limitations of their optimal viewing window. Although it is difficult to determine whether 100 units result in a scale-invariant model, it does represent a significant number of correlation lengths, leading toward more realistic behaviour. The digital image of the pore space was then transferred to a chrome plated glass mask, without magnification from the original porous media. Using technology similar to the manufacture of microchips, the image was photo-chemically etched onto a silicon wafer, at a constant etching thickness (for this particular study, micromodels have been constructed with etching depths of about 15 µm). Pore diameters range from 2.4 to 30 µm. Figure 2.7 is a scanning electron micrograph of a section of the repeat pattern, which
shows the shape of the grain walls. The porosity of the micromodel used has been experimentally determined to be 37%.
Figure 2.7: Composite image of the etched repeat pattern in the micromodel (approximately 509x509 µm), (Sirivithayapakorn and Keller, 2003).
The silicon surface was oxidized after etching, leaving a water-wetting surface, approximately 15 µm deep. The etched silicon wafer was then placed between two glass plates. The top glass plate was attached to the silicon wafer using anodic bonding. Ports were constructed on the top plate to allow independent injection and extraction from different directions. The bottom plate was attached with epoxy to the silicon wafer, and was mainly used to provide support for the micromodel. To ensure that the model remains waterwet, carbon dioxide was introduced to displace the air, and then water was injected, which eventually dissolves the carbon dioxide, allowing the wetting fluid to saturate the model completely. Flow rate was measured at the inlet, using digital flowmeters. Pressure was measured at the inlet and outlet lines using calibrated pressure transducers. The colloids used in that study were fluorescent latex spheres, with mean diameter of 3.0, 2.0, 1.0 and 0.05 µm. For each colloidal suspension, 3 samples of 1 mL of the suspension were taken. The
concentrations of colloid suspension were calculated in the range of 104 to 105 particles per mL. Their experiments were conducted under three pressure differences: 24, 52 and 210 kPa. The colloid solution was buffered at pH 7.8 to minimize colloid adhesion to each other. Their experiments were conducted under water–saturated conditions. The images and video streams were taken from selected regions in the micromodel, which include various pore bodies and pore throat sizes. From the unit pattern, 16 pore spaces in the micromodel were selected. The selected pore spaces have pore bodies with 3 to 9 throats. The throat size of the selected pores ranged from 2.4 to 14 µm. Their experiments were planned to observe the frequency of occurrences of size exclusion effect and the influence of relative size of pore throats to colloid (T/C ratio) and flow velocity on this phenomenon. Finally, to understand the velocity field in the porous space, they constructed a finite element numerical model of the pore space, using Femlab (COMSOL, Inc.). They solved the Navier–Stokes equation for the micromodel, using a 2–D finite element grid, to obtain the relative water velocity in the streamlines around the grains. They considered no–flow boundaries along the top and bottom of the flow direction and a constant pressure drop across the flow direction. This provided them as an estimation of the relative flow velocities in the pore space. The actual pore water velocity is estimated from the available physical data of the micromodel, the measured pressure gradient and the actual discharge.
Iassonov and Beresnev (2003) presented a model of enhanced fluid percolation based on two assumptions: the pore-filling fluid is viscoelastic, and the capillary entrapment is caused by contact-angle hysteresis. Vibration was modelled as an oscillating inertial body force acting on the fluid. This method did not rely on the resonant behaviour to explain the enhancing effect
of low-frequency vibration on the movement of the blob, it was plagued by an oversimplified geometry (a straight circular channel) and a certain assumptions utilized to calculate the net effect.
2.4 Research objectives To sum up, despite the large number of numerical studies of singlephase and multiphase systems that have been done, pore-scale study in porous media is still in its scientific infancy, since it is only over the last decade that relatively inexpensive high-performance computers have become available. Some models are undergoing theoretical development and improvement; mature, robust and efficient simulators are not available. Furthermore, current pore-scale applications are limited to relatively small domains and simple problems; for instance, predictive pore-scale modelling of three-phase properties is still at its early stage. The purpose of this thesis is to simulate single and two-phase flows in porous media using microscopic (pore) scale model with the help of CFD (Computational Fluid Dynamics) code called FLUENT 6.3. Among the pore-scale approaches applied in this work, Volume of Fluid (VOF) method modelling figures most prominently. The specific objectives of this research are: To construct a CFD model of the pore space for simulating single-phase and two-phase systems; To
determine
the
constitutive
relation
between
permeability and properties of a realistic porous medium using the single-phase simulator; and To present a new model for better understand the capillary trapping and vibration induced mobilization of a nonwetting phase ganglia in porous media, and to develop a
method capable of estimating the effect of vibration on oil production in real-world applications. 2.5 Thesis organization This thesis is divided into six chapters. Chapter three describes the pore-scale approach, and gives the mathematical model of our problem as well as an overview about computational fluid dynamics (CFD) techniques. Chapter four describes the micro-model of porous media and the numerical solution procedures including geometry description and mesh generation. Chapter five includes the result obtained using CFD software package FLUENT for two dimensional pore structures. Finally, Chapter six presents the general conclusions and provides recommendations for further research and development in this area.
CHAPTER THREE PORE-SCALE FLOW WITH CFD
3.1 Introduction In a porous media flow simulation, we are interested in modelling the displacement, within a porous media, of either oil or water. For the most part of this thesis, including all computations, the case of water-oil simulations will be shown. By porous medium, as mentioned in Chapter one it means a solid with many small voids, or pores, potentially connected, through which fluid may flow. The volume fraction of the pores as a total of the whole volume is known as the porosity (presented in Chapter 2). Since it is typical to view the pores as a microscale feature, this porosity is a macroscale feature, given point wise.
It is usually considered that one of the fluids to be
displacing the other, as in the case of an oil-water flow where the water is pumped in so as to displace the oil. While the displacing fluid may be immiscible with the fluid being displaced, the displacement does not take place as a piston like process with a sharp interface between the two-fluids. Rather, simultaneous flow of the immiscible fluids takes place within the porous media (Westhead, 2005). In considering this simultaneous flow we assume, for the present, no mass transfer between the fluids. Mass transfer could potentially occur if there was a chemical reaction taking place between the fluids (Westhead, 2005). Typically, one of the fluids wets the porous media more than the other; we refer to this as the wetting phase fluid, and we refer to the other as the nonwetting phase fluid. Wettability describes the relative preference of a rock (from which the porous media is formed) to be covered by a certain phase. In a water-oil system, water is most often the wetting phase (discussed in Chapter 2). This chapter presents the methodology which has been selected to
solve the problem of flow and transport through pore-scale model. The following sections discuss a lot of issues including pore-scale modelling approach, the mathematical model, as well as an introduction to CFD.
3.2 Pore-scale modelling approach In this section, the flow and transport problems will be studied by using pore-scale modelling approach, which allows us to investigate microscale processes and their effects on macroscale behaviour. Pore-scale modelling provides opportunities to study transport phenomena in fundamental ways because detailed information is available at the microscopic pore scale (Pan, 2003). Pore-scale modelling offers an important tool to develop constitutive relations that are difficult and even impossible to be obtained by lab experiments. The basic strategy is to perform numerical experiments analogous to those performed in the laboratory. However, the pore-scale simulation provides more versatility in choice of parameters, a greater variety of quantitative data, and more importantly, easier design of numerical experiments. In addition, until new experimental methods are developed to measure processes at highly resolved scale, pore-scale modelling provides unique opportunities to understand the fundamental pore scale processes in multiphase porous medium systems. The effect and significance of these pore-scale processes are then able to be incorporated into constitutive theories to achieve an accurate description of larger-scale phenomena of interest. The dramatic evolution of computational capabilities offers to us new opportunities for simulating larger domains and modelling a wider range of processes. This makes pore-scale approaches potentially attractive for industrial or field applications as measurement tools to compute transport properties, such as permeability of a particular subsurface system.
The principle of pore-scale modelling in general is composed of two major steps. The first involves the detailed identification and specification of the porous medium morphology. There are several approaches to this difficult step. An experimental imaging is one of them. Once the pore morphology is obtained, in the second step, flow and transport problems can be solved by using various methods, the decision was using computational fluid dynamics techniques to solve these problems. In the following sections CFD will be discussed in details. Figure 3.1 represents our concept of the pore-scale approach.
Pore-scale modelling approach
How to obtain the morphology information of the porous media?
Experimental imaging
+
How to solve the flow and transport in porous media?
CFD/FEM
Figure 3.1: General concept of pore-scale approach.
3.3 Experimental imaging In many situations, a heterogeneous porous medium system is both inert and nondeformable (Pan, 2003). However, even with this simplification, the description of pore morphology in natural porous media still remains a formidable problem. One of the major problems which had severely slowed
the pace of research in porous media is the fact that the geometrical influence is crucial and that it was very hard to get detailed information about it. In the last fifty years, there have been intensive researches on realistic porous medium morphology. The morphology of a porous medium consists of its geometrical properties: the shape, volume of its pores, and its topological properties, i.e. the way in which the pores are connected to each other. Recent advances in micro-model experiments and high-resolution tomographic imaging, which allows for accurate representation of pore morphology, have spurred an explosion of interest in pore-scale modelling (Pan, 2003). Experimental determination of detailed pore morphology began in the early 1980's, using a classic method that called “serial sectioning”, i.e. cutting the samples and examining the thin sections. The classic cut-and-examine method is extremely tedious though it may yield interesting two-dimensional information about thin sections. Further, obtaining three-dimensional pore structure by serial sectioning is much more difficult. Recently non-destructive imaging techniques have been developed and applied to examine porous medium morphology. For example, a synchrotron computed microtomography (CMT) method that employs an X-ray synchrotron source and produces images based upon fully three-dimensional reconstruction with higher accuracy and higher resolution than the juxtaposition of many two-dimensional planes. Despite being extremely useful, the non-destructive imaging techniques, such as CMT, have been utilized only quite recently. Widespread use of these techniques is not readily available. The scale of the measurable medium using these techniques is typically limited to few centimeters or less at present. However, the results so far, although derived from small systems, have implications for many practical problems. In addition, the microtomography techniques are also undergoing continuous development and refinement, so they offer the
possibility of the application to larger systems and use for validating numerical models in greater detail. The scanning electron microscope (SEM) is a type of electron microscope that images the sample surface by scanning it with a high-energy beam of electrons in a raster scan pattern. The electrons interact with the atoms that make up the sample producing signals that contain information about the sample's surface topography, composition and other properties such as electrical conductivity. The present geometry of this work takes one of the micro-image of a realistic porous medium and this was created by SEM.
3.4 Governing Equations of Fluid Flow In this thesis FLUENT was used to perform calculations of the fluid flow field of concern. This section will state the differential equations, which describe any fluid in any general motion. These equations will be presented in Cartesian tensorial notation without any derivation.
3.4.1 Equation of Continuity Consider the infinitesimal fixed control volume shown in Figure 3.2.
Figure 3.2: Infinitesimal fixed control volume.
The following equation can be derived if a mass conservation is applied on the above control volume: ui 0 xi
(3.1)
where i = 1, 2, 3 is referred to x, y, z axes. This is the equation of continuity, which, in its present shape, is valid for situations when the flow is incompressible and no source or sink singularities exist within the volume (Versteeg and Malalasekera, 1995).
3.4.2 Equation of Momentum The differential momentum equation valid for any fluid in any general motion is obtained by applying Newton's second law of motion on the control volume above:
u i ui p ij u j g i t x j xi x j
(3.2)
Because the fluids concerned in this thesis (i.e., water and oil) both are classified as Newtonian fluids, the viscous stress tensor in equation (3.2) is proportional to the velocity gradients and the viscosity : ij 2sij
(3.3)
where sij is the strain-rate tensor defined as 1 u u sij i j 2 x j xi
(3.4)
Equation (3.3) can now be rewritten as the famous Navier-Stokes equation:
u i u j ui g i p 2sij t x j xi x j
(3.5)
3.5 Permeability calculations Symmetric tensor of absolute permeability with dimensions (m2) is a parameter of the solid matrix only and may depend on position in the case of a heterogeneous porous medium. Furthermore permeability may be anisotropic if the porous medium has a preferred flow direction as explained in subsection. According to Darcy‟s Law (presented in Chapter 2) for flow in a porous medium, the absolute permeability of the present model can be numerically calculated. The seepage or superficial velocity in horizontal direction (our model) is, U
K
P
(3.6)
From (3.6) K can be easily estimated as K
U p
(3.7)
where K is the absolute permeability (m2). Darcy‟s Law is valid for the slow flow (inertial effects can be neglected) of a Newtonian fluid through a porous medium with rigid solid matrix (Heinrich, Proirier and Nagelhout, 1996). However, the Reynolds number for flow in porous media has been obtained based on the ranges of pore diameters of our model and we have found that the flow is still Darcian at given parameters ( Re 8 ). We now wish to solve the problem (3.1) and (3.5) in two-dimensional system (2-D Navier-stokes equations). Therefore, the system of equations will have to be solved numerically. Then the absolute permeability can be easily determined from (3.7) based on the approximation of (U u). 3.6 Modelling of Inertial body force Since the elastic-wave propagation speed is significantly faster in the solid “skeleton” than in the pore-filling fluids, it is reasonable to assume that
the fluid reacts to the passing wave primarily through the oscillation of the solid walls of the pore (Iassonov, 2005). If we consider the problem from the moving wall‟s frame of reference, we find that movement of the wall will result in an inertial body force acting on the fluid (Biot, 1956). For most cases of vibratory stimulation of naturally occurring porous media, components of the body force dependent on local fluid-velocity gradients can be safely ignored (Beresnev et al., 2005). Thus, for harmonic oscillation of the porous medium, the vibration-induced inertial force acting on the fluid at any given time (t) can be written simply as, B A sin 2ft
(3.8)
where A is the vector defining the acceleration amplitude (m/s2) and direction of vibration, is the density of the fluid (kg/m3), and f is the wave frequency (Hz). Equation (3.8) can be added to (3.5) in the right-hand side as external force due to vibration. It is, effectively, the inertial force acting on the fluid because of the pore wall (solid skeleton) oscillations. This term, along with the no-slip boundary condition, is what determines the coupling between the fluid and the solid matrix in our model when vibrations are present. Above only a generic illustration of the vibratory mobilization mechanism was presented, which cannot provide a ready-to-use formula for any situation. While we can make certain predictions (e.g., increase in the mobilization effect with increase in vibration amplitude), the exact parameters of the mobilization effect will depend on a multitude of factors, including pore geometry, viscosities of pore-filling fluids, vibration amplitude and frequency, and the external pressure gradient. Thus, the problem needs to be considered in a detailed quantitative manner, looking into the effect of some or all of the aforementioned factors in mobilization of the oil. We can now consider the effect of longitudinal vibration (x-direction) of the pore wall to the flow of the fluid. Since the vibration-induced body force oscillates harmonically, Equation (3.8) can be applied as a momentum source in x-
direction via User-Defined Function (UDF) with help of C Language. However, most researchers are interested in studying the effect of vibration with Low-frequencies range from 1 to 500 Hz and then the vibration amplitude can be predicted (Beresnev et al., 2005).
3.7 Computational Fluid Dynamics (CFD) Because of the high complexity of the problem of dynamic immiscible two-phase flow in a system containing a large number of interconnected pores of variable shapes, original exact analytical solution is not feasible option. The next best alternative to an analytical solution would be a finite-element or finite-volume model, explicitly accounting for all the physical mechanisms involved. Therefore, the potential for using standard computational fluiddynamics (CFD) methods has been investigated to analyze and predict the effects of the different mechanisms on mobilization of trapped non-wetting fluid. Due to the fast development of computer power during the past decade, the usage of CFD has increased enormously (Versteeg and Malalasekera, 1995). The purpose with CFD is to analyse systems involving fluid dynamics, heat transfer and other associated phenomena such as chemical reactions and phase changes by numerical calculations. CFD codes are all structured around the numerical algorithms that will solve the fluid flow problems (Versteeg and Malalasekera, 1995). The CFD code consists of the following three fundamental features: 1. Pre-processor 2. Solver 3. Post-processor
3.7.1 Pre-processor The task for the pre-processor is to gather the essential information needed for the solver to tackle the problem, i.e., the indata. This involves: Generation of the grid that defines the geometry of interest, i.e., the computational domain. Specifications of the physical and chemical phenomena that need to be modelled. Definition of the fluid properties. Specification of appropriate boundary conditions.
3.7.2 Solver The aim for the solver is to carry out the numerical calculations necessary to produce satisfactory simulations of the flow problem. The solver can be based on the following techniques: finite difference, finite element, finite volume and spectral methods. The main differences between these techniques are based on how the variables of flow are approximated and on the discretisation process. In this research, a decision was made to use the commercial CFD software package FLUENT, well-known for its extensive capabilities in solving a wide variety of fluid-flow problems (including multiphase flow) using the finite-volume approach. This method was originally developed as a special finite difference formulation and will be discussed in the following section.
3.7.3 Post-processor The post-processor returns the results of the simulation calculated by the solver. Today, most of the available CFD programs have developed graphical tools, which make it possible to receive a visualization of the calculated data. Examples of this follow below: Vector plots of the velocity field.
The ability to track the path of a particle through the domain. Contour plots Animations of the fluid flow. View manipulation (translation, rotation, scaling etc.)
3.8 The Finite Volume Method (FVM) Due to the fact that FLUENT attacks physical or engineering problems by using the finite volume method, a short description of this method will be given. The numerical algorithm of FVM has the following structure (Versteeg and Malalasekera, 1995): 1. Exact integration of the governing equations of fluid flow over all the control volumes within the solution domain. 2. Discretisation involving substitution of a variety of finite difference-type approximations for the terms in the integrated equation representing the flow processes. The terms in equation are convection, diffusion and source terms. And then this system of integral equations is transformed into a system of algebraic equations that can be solved numerically. 3. Solution of the algebraic equations by an iterative method. It is the first step that the finite volume method differs from the other numerical techniques. The finite volume method expresses the conservation of relevant properties for each finite size cell. It is clear that relationship between the numerical algorithm and the physical principal of conservation that makes the finite volume method easier to apply and understand than finite element and spectral methods.
With the finite volume method the conservation of a general flow variable, , can be described as a balance between different processes which tend to increase or decrease . In other words: Rate of change of in the control volume with respect to time
= +
Net flux of into the control volume due to convection Net flux of into the control volume due to diffusion
+
Net rate of creation of inside the control volume due to diffusion
Below, a brief run-through of the three steps from which the finite volume method is build follows: The general equation of transport, which serves as a starting point for computational procedures in the finite volume method, is written in the following form: ui S t xi xi xi
(3.9)
where Γ is a diffusion coefficient, e.g., thermal conductivity, ρ is the density (kg/m3) and S is a source term. For a more straightforward and easy explanation a steady, onedimensional convection and diffusion problem will be considered. To make it even more basic there will be no production of inside the control volume (i.e. S =0). These simplifications make it possible to write equation (3.9) in the following manner: d u d d dx dx dx
(3.10)
Mass conservation yields: d u 0 dx
(3.11)
3.8.1 Integration of the Equation of Transport over a Control Volume Consider a one-dimensional control volume shown in Figure 3.3 below. The node of interest is P and the neighbouring nodes are W and E. The control volume for which P represent the centre is restricted by faces w and e.
Figure 3.3: Control volume around node P.
Integration of equations (3.10) and (3.11) over the control volume gives:
uA e uA w A
(3.12)
uAe uAw 0
(3.13)
A x e x w
where Ai is the face area for w and e which are assumed to be equal, i.e., Ae = Aw = A. 3.8.2 Discretization The terms in equation (3.12) must be approximated in order to achieve discretised equations. It has been found convenient to define two variables F and D to represent the convective mass flux per unit area and diffusion conductance at cell faces, respectively: F u
(3.14)
D
x
(3.15) The variables F and D at the different cell faces can be expressed as, Fw u w , Dw
w , xWP
Fe u e De
e xPE
Using the central differencing scheme, the following expressions are achieved: P De AE P A e Ae E x e xPE
(3.16)
A w Ae P W Dw AP W x w xWP
(3.17)
uAe uAe E P
(3.18)
2
uAw uAw P W 2
Fe AE P 2
Fw AP W 2
(3.19) The integrated equation of mass conservation can be written as: Fe Fw 0
(3.20)
By inserting equations (3.16) to (3.19) into equation (3.12) the following expression is achieved: Fw Fe Fw Fe D w 2 De 2 Fe Fw P Dw 2 W De 2 E
(3.21)
this can be rewritten in the following fashion: aPP aWW aEE
(3.22)
where aW Dw
Fw , 2
aE De
Fe , 2
aP aW aE Fe Fw
Due to this the variable in the node P can now be described with the values in the neighboring nodes W and E.
The central differencing possesses one major weakness, which is revealed for flows where convection is not negligible compared to diffusion. The relationship between convection to diffusion is often measured with the Peclet number, Pe, which is defined as Pe
F u D / x
(3.23)
Figure 3.4 shows the distribution of depending on different values of Pe.
Figure 3.4: Distribution of at different Pe numbers.
For the case where Pe = 0 (i.e., pure diffusion) conditions at node E will be influenced by those upstream at node P but also by those further downstream. As Pe increases the conditions at E will be more influenced by those at P and less by those further downstream. In the case of pure convection (Pe → ∞) the conditions at E will be equal to those at P. The central differencing scheme assumes equal influence from all neighboring nodes when calculating the value at P. This will cause some serious problems for flows where the value of Pe is not negligible. Therefore, CFD codes utilize the central differencing scheme only for the discretisation of the diffusion terms, i.e., the terms on the right hand in equation (3.12). The convective terms are discretised by so-called upwind schemes (Versteeg and Malalasekera, 1995). FLUENT supports four different upwind schemes for
the user to utilize: First Order Upwind, Power Law, Second Order Upwind and QUICK (Fluent Inc., 2005).
3.8.3 Solution When the equations of interest have been discretised, the problem is solved using iterative methods. An example is the TDMA method, which stands for Tri-Diagonal Matrix Algorithm (Versteeg and Malalasekera, 1995). It is considered to be computationally inexpensive and to require a minimum amount of storage. This has made it very popular and is thus widely used in CFD programs, FLUENT included (Fluent Inc., 2005).
3.9 Multiphase Flow Models in FLUENT Today there exist two different approaches for modelling multiphase flows by the way of numerical calculations: The Euler-Lagrange approach and the Euler-Euler approach. Short introductions to these approaches are given below (Fluent Inc., 2005).
3.9.1 The Euler-Lagrange Approach The Euler-Lagrange approach treats the fluid phase as a continuum by solving the time-averaged Navier-Stokes equations (discussed in section 3.4) while the dispersed phase is solved by tracking a large number of particles, bubbles or droplets through the calculated flow field. The dispersed phase can interact with the fluid phase by exchanging momentum, mass and energy. The approach computes the trajectories of the particles or droplets at specified intervals during the fluid phase calculation. This makes it ideal for flow problems involving spray dryers, coal and liquid fuel combustion and some particle-laden flows. A fundamental assumption made by the EulerLagrange approach is that the dispersed second phase occupies a low volume fraction making it inappropriate to model flow problems involving liquid-
liquid mixtures, fluidised beds or any other application where the second phase is not negligible. FLUENT possesses one model that adopts the EulerLagrange approach (the Lagrangian discrete model). 3.9.2 The Euler-Euler Approach The Euler-Euler approach treats the different phases as interpenetrating continua. The concept of phasic volume fraction is introduced in this model because the volume of a phase cannot be occupied by other phases. The sum of these volume fractions is equal to unity in every cell and they are assumed to be continuous functions of space and time. FLUENT possesses three models that use the Euler-Euler approach: • The volume of fluid model (VOF) • The mixture model • The Eulerian model Because we used the „Volume of Fluid‟ (VOF) method implemented in FLUENT to model the immiscible multiphase flow specific to our problem, the VOF model will be discussed more thoroughly in next section.
3.10 Volume of Fluid (VOF) Model The volume of fluid model assumes that the phases are immiscible and are not interpenetrating each other (Ferziger and Peric, 2002). This means that it does not completely follow the Euler-Euler approach, which states that the phases are considered as interpenetrating continua. This model is used whenever the interface between different phases is of interest. The model solves the continuity equation for one or more of the phases in order to track the interface(s) between the phases. The model adopts homogeneous multiphase theory, which assumes that the velocities of the different phases are equal. This requires that only a single momentum equation is solved throughout the domain and the resulting velocity field is then shared by all the phases (Fluent Inc., 2005).
The model also solves volume-fraction equations; in our research the energy transfer is neglected. Incompressible flow is assumed, and gravitational forces are ignored. The convergence criteria are the scaled residuals for the continuity equation and each component of the fluid velocity, which are defined by the ratios of the total residuals summed over all computational cells and corresponding total absolute values. Because the VOF model is based on the fact that two or more phases are not interpenetrating and that the sum of the volume fractions of the phases equals unity in every volume cell the following three conditions are possible (Hirt and Nichols, 1981).: αq = 0: the cell is empty (of the qth phase). αq = 1: the cell is full (of the qth phase). 0 < αq < 1: the cell contains the interface between the qth and one or more phases.
3.10.1 The Volume Fraction Equation The continuity equation, discussed earlier, is solved for the volume fraction of one (or many) of the phases in order to track the interface(s) between the phases. It is defined in the following fashion for the qth phase: q t
S V . q q
q
(3.24)
where S is a source term, which by default is set to zero. q
Equation (3.24) is not solved for the primary phase but instead its volume fraction is computed by the following equation: n
q 1
q
1
(3.25)
This means that for a multiphase flow problem involving n different phases, the primary phase will be calculated as
1 1 2 3 4 ... n
3.10.2 The Momentum Equation The single momentum equation that is shared by all the phases is defined in FLUENT as
T V V V p V V g F t
(3.26)
The momentum equation is dependent on the volume fractions of all phases through the properties ρ and μ which, like all other properties, are calculated in the following fashion: n
q q
(3.27)
q 1 n
q q
(3.28)
q 1
For a problem involving, for example, a two-phase flow in which the volume fraction of the second phase is being tracked, the density in each cell is given by 2 2 1 2 1
(3.29)
The indices 1 and 2 denote phase one and phase two, respectively.
3.10.3 Interfacial Face Flux Calculations The VOF model offers four different schemes for calculation of the face fluxes through the cells: The geometric reconstruction scheme The donor-acceptor scheme The Euler explicit scheme The implicit scheme The two first schemes use special interpolation treatments to calculate the face fluxes for the cells situated near the interface and standard
interpolation schemes (i.e., standard upwind, second-order, or QUICK scheme) for the cells that are completely filled with one phase or another. The results based on the different techniques of these schemes are shown in the figures down below.
Figure 3.5 shows an actual interface shape whereas Figures 3.6 and 3.7 show the interface shape represented by the geometric reconstruction scheme and the donor-acceptor scheme, respectively.
Figure 3.5: Actual interface shape.
Figure 3.6: Interface shape represented by the geometric Reconstruction scheme.
Figure 3.7: Interface shape represented by the donor-acceptor scheme.
The Euler explicit scheme and the implicit scheme do not use any special interpolation treatment for the cells near the interface. These schemes treat every cell as if it is completely filled with one phase or the other thus using the standard interpolation schemes mentioned above. The geometric reconstruction scheme (Figure 3.6) is the most accurate scheme for resolving the interface shape and was therefore used for the simulations in this thesis. The drawback lies in the fact that it is the most expensive scheme for interpolation near the interface in terms of CPU time for a given mesh. The interface between the fluids is then reconstructed at each time step using geometric reconstruction, where an interface in each cell is represented by a segment of a straight line (or a plane in case of a three-dimensional mesh), based on the known volume fraction within the cell and its neighbours. The VOF method is appropriate for modelling the effects of surface tension in problems requiring the interfaces to be sharply defined at any time during the simulation.
3.10.4 Surface Tension As mentioned in chapter 2, surface tension effects are of large importance to the hydrodynamic behaviour of multi-phase flows. The different flow patterns that may occur are all governed by surface tension
(Wallis, 1969). The VOF model allows the user to include the effects of surface tension, which will act along the interface between each pair of phases. This effect will be implemented in the momentum equation (3.26) as a source term, which for a two-phase flow is defined as F 12
1 1 1 2 2
(3.30)
where κ is the curvature of the interface and ρ is the volume-averaged density calculated in equation (3.29). 3.11 Summary In this chapter, we presented the most important aspects concerning the flow through pore-scale model. These aspects are to perform numerical simulations for this model, since a new approach has been introduced, porescale modelling approach which consisted of two steps, experimental imaging to collect data of the porous medium. This step was done previously using SEM. After that the second step started, CFD methods have been chosen to solve the system of PDEs with the help of FVM solver. The VOF model available in FLUENT has been discussed in some detail because it will be used to solve the two-phase flow problem. In the next chapter, the procedures of numerical solutions will be introduced starting from the geometry modelling to the generation of numerical grid.
CHAPTER FOUR MODEL FORMULATION 4.1 Overview This chapter describes a non-conventional model of porous media flow that will be using the Navier-Stokes equations in the interstices of porous media and discusses the procedures which have been carried out for the geometry modeler and mesh generator. The present micro-model is based on pore-scale flow experiments conducted by Sirivithayapakorn and Keller (2003). They designed their lab experiments on the basis of scanning electron microscope (SEM) images of thinly sliced rock (presented in Chapter 2). The scale at bottom in Figure 4.1 indicates that pore throat and body dimensions are on the order of 1-100 μm (Sirivithayapakorn and Keller, 2003).They etched the geometric patterns from the images onto solid with an elaborate process similar to the etching of silicon wafers. After that, they transferred these images to DXF files, which can be easily imported into AutoCAD software, and then it can be studied by using CFD tools.
Figure 4.1: Scanning electron microscope (SEM) image of the repeat pattern in the silicon wafer (typical pore size 10–100 μm, pore throats 3–20 μm).
4.2 Geometry Description Figure 4.2 shows the detailed geometry of Keller‟s model in AutoCAD environment. This model takes one of the 2D micromedia images and covers 640 x 320 μm. It consists of many solid elements, pore bodies and pore throats. Flow moves from right side to the left.
Flow Direction
Figure 4.2: CAD geometry.
It is typical to represent fluid flow in the subsurface as a continuum process using average or “continuous” properties for the bulk rather than detailing the shape and orientation of each solid particle within a porous medium. Inserting the bulk properties into an equation such as Darcy‟s law gives an average flow rate for the total volume (presented in Chapter 2). While bulk approximations typically produce excellent estimates sufficient for considering flow over large areas that a close-up Navier-Stokes analysis describes.
Accordingly, the flow of a single Newtonian fluid in the void space of a porous medium is described on the microscopic level by the Navier–Stokes system of equations with appropriate boundary conditions. However, the void space configuration is usually not known in such detail to make this description feasible. Moreover, a numerical simulation on that level is beyond the capabilities of today‟s computers and methods (Pan, 2003).
4.3 Initial and Boundary Conditions Flow is laminar in the pores and does not enter the grains. The inlet and outlet fluid pressures are known. Assume flow is symmetric about the top and bottom boundaries. The primary zone of interest is the rectangular region with an upper left corner at (0, 0) μm and lower right coordinates at (640, -265.0) μm. Owing to the problem‟s small scale, the model does not include a body force that accounts for gravity. Specific boundary conditions are applied to complete the formulation of the problem. Flow Boundary Conditions are as follows: p p0 , p p0 ,
n. u u 0
n. u u 0
at inlet
T
at outlet
T
at grains
u=0 n.u 0,
tn pI u u n 0 T
at sides
In these equations, n is the unit normal vector, tn is the unit tangential vector, p0 is a specified pressure, and (I) is the identity matrix. Initial conditions will be included in case of two-phase flow with VOF model.
4.4 Material Properties Computations and the conditions are classified based on character of the medium used, water or oil. The contact angle of water is 0.1 degree for all
simulations. This means we assume strongly water-wet condition (discussed in Chapter 2). Usually, Beresnev et al. (2005) tend to use the medium viscosity of the oil (5 mPa.s). The effect of dynamic viscosity will be shown in Chapter 5. Table 4.1 summarizes the main physical properties required by solver. Interfacial tension between water and oil is defined as 0.03 N/m. Table 4.1: Physical properties of oil-water flow system applied in the simulations.
Physical property
Water
Oil
Density (kg/m3)
1000
800
Dynamic viscosity (Pa.s)
0.001
0.005
4.5 Mesh Generation Because the Navier-Stokes equations are computationally difficult, it is important to use an appropriate mesh. If the mesh is too coarse, the solution might not converge at all or errors might be large. Conversely, if the mesh is too fine, the solution time for the nonlinear system of equations might be unnecessarily long. As mentioned above, we chose to model the porous medium as two-dimensional structure, with pore bodies connected with different throats. To create the finite-volume mesh for the FLUENT simulations, we utilized GAMBIT, an advanced CFD pre-processor designed for geometry description and meshing, all within a single user interface. GAMBIT imports the geometry (Figure 4.2) and does not mesh any unneeded regions. GAMBIT has two types of cells in 2D (triangular and quadrilateral). In case of single phase flow problem FLUENT can utilize both of them. While
the VOF model, with geometric reconstruction of the interfaces for a problem where interfacial tension is important, works best on quadrilateral (for twodimensional problems) meshes because of the higher accuracy of the interfaces representation (Fluent Inc., 2005), only this type of grid has been used. Because grid type and resolution can significantly affect on both the quality of the results and the calculations speed, a short series of tests with various grid resolutions was performed for each of the simulation presented in this chapter. The general mesh structure we used was generated using an automatic mesh formation in GAMBIT, with an unstructured mesh in the pore body. Three levels of grid were tested to ensure good results. Table 4.2 presents these levels. Each level has a particular number of quadrilateral cells.
Table 4.2: Results of the mesh resolution study.
Level No.
Total number of quadrilateral cells
1
6788
2
44982
3
114944
A coarser mesh (Level 1) may not be appropriate, since some of the smaller pore throats would have only one mesh element (cell) across. It‟s shown in Figure 4.3. By zooming in Figure 4.3 the interior elements can be visible clearly. This is illustrated in Figure 4.4, where unstructured mesh is shown with a poor quality causing a high error in the calculations.
Figure 4.3: The quadrilateral mesh (Level 1).
Figure 4.4: Magnified view of the mesh showing the interior cells at Level 1.
Level 2 might be classified as a medium mesh as shown in Figure 4.5. It‟s clearly shown in Figure 4.6 that this level is finer than Level 1, where the number of cells was increased to 44982 as well as the quality of mesh was enhanced. This leads VOF model available in FLUENT treats Level 2 well and gives reasonable results on behalf of Level 1. Level 2 can be created by
reducing the spacing between cells and then GAMBIT will generate more cells automatically.
Figure 4.5: The quadrilateral mesh (Level 2) used in FLUENT simulation.
Figure 4.6: Magnified view of the mesh showing the interior cells at Level 2.
The last testing level is to create Level 3. If the number of cells is more than Level 2, we will reach Level 3 which is shown in Figure 4.7 as a default display of the model grid, by zooming in, we can see that the interior of the
model is composed of a fine mesh of quadrilateral cells that are displayed in Figure 4.8.
Figure 4.7: Default display of numerical mesh using quadrilateral elements at Level 3 used in FLUENT simulations.
Figure 4.8: Magnified view of the mesh showing the interior cells at Level 3.
Finally, the potential problem was low resolution of the mesh in some of the more narrow throats. An increase in resolution or even an addition a boundary layer (providing higher mesh resolution near the pore walls) resulted in a significant degradation in speed performance.
Since FLUENT can also solve the problems with triangular cells in case of single phase flow, the model has been tested by using this type of cells (65632 cells) where two grains are shown in Figure 4.9 as a partial display, because the dimensions are very small only pore throat is displayed.
Figure 4.9: Illustration of the triangular mesh (interior between two grains) used in FLUENT simulations.
Once the numerical mesh has been created, it will be imported by FLUENT and then the solver functions will start to solve the problem and post the results to be analyzed. Figure 4.10 illustrates a flow chart that shows the structure of CFD needed to model our problem. In general, FLUENT is able to solve all the problems discussed in this work on unstructured meshes (Fluent Inc., 2005). The discussion of solving steps will be presented in next chapter.
Start
Create geometry ACAD or
Gambit
Surface generation
(Gambit)
Volume blocks generation (Gambit)
Grid generation and grid smoothing/check
(Gambit)
Import mesh files into FLUENT
Set case file: Grid scale, model, physical properties, boundary conditions
(FLUENT)
Initialize the flow field/ calculate a solution (FLUENT)
No
Is the
No
solution converging
? Run more iterations until convergence
Save case and data file Display and print simulation results (FLUENT)
Figure 4.10: Basic structure of CFD simulation procedure using FLUENT
CHAPTER FIVE RESULTS AND DISCUSSION
5.1 Single Phase Flow Simulations After an appropriate mesh is provided, FLUENT is used for the simulation setup, the solving process and the post-processing of the results. During the simulation setup all GUI-commands are scripted by the program but one has to be aware of the fact that not all possible settings are accessible by the GUI. Appendix A summarizes the major settings of simulation. Some special functions are only available if they are entered on the command line level leading sometimes to difficulties in finding the appropriate settings. The studies of water flow are conducted at the pore scale using a physical micromodel, based on the experimental setup and numerical results using finite element solver called COMSOL software (COMSOL Inc.,). Those have been previously developed and applied by Sirivithayapakorn and Keller (2003), so our strategy is to get comparatively acceptable results using FLUENT software. Specifying a set of suitable boundary conditions is similar to that presented in Chapter four. It is possible to solve the Navier-Stokes equations on a grid using the standard CFD techniques. Water generally moves from right to left across the model. Constant pressure difference of 715 Pa is assumed. However, the current applications of the finite volume method to flow through microscopic porous media have been limited to solving timeindependent Navier-Stokes equations in single-phase flow systems. Applying the finite volume method starts by discretizing the gradients of Equations (3.1) and (3.5) on the unstructured mesh. One has to apply the no-slip (u=0) boundary condition at solid-fluid interface. Therefore, some care must be taken to appropriately implement the boundary condition for each possible
shape of a solid-fluid interface. The lack of versatility on implementing the boundary conditions for arbitrary grain shapes precludes the broader applications of the CFD methods in porous medium systems. Figure 5.1 shows FLUENT solution predicted by a Navier-Stokes analysis for the relative velocities in the pore spaces of a micro-scale porous medium. Velocities are higher in the narrowest pores. The fluid velocities tend to decrease in stretches where the cross-sectional area for the flow increases.
Figure 5.1: Velocity vectors coloured by velocity magnitude (m/s).
The domain plot in Figure 5.2 shows the x-velocity at the outlet using mesh at Level 2. The velocities are negative because the flow is moving in the negative x-direction. The numerical model was first tested with the same fluid (water) and was shown to be in a good agreement to the experiments done by Keller using COMSOL software. There is a difference in the results obtained by using finite volume method through FLUENT as a result of the difference in number of cells.
50 Keller Present (Level 2)
0
-50
y, m
-100
-150
-200
-250
-300 -40
-35
-30
-25
-20
-15
-10
-5
0
x-velocity, mm/s
Figure 5.2: Comparison of velocities along the outlet boundaries obtained by FLUENT (present study) to COMSOL (Keller’s study) using Level 2 of grid.
Figure 5.2 reveals the highest velocities tend to occur in narrow pores with high pressure drops, as we might expect. This close-up view near the exit reveals that high velocities also develop in wide channels where pressure gradients are relatively shallow but multiple “tributaries” combine. Figure 5.3 shows the same comparison but using Level 3 of mesh. The next step is choosing the optimal mesh utilizing the quadrilateral cells for two-phase flow problem which will be solved by VOF model. This will be done based on single phase results. It‟s notable in Figure 5.2 and 5.3 that both results obtained by using Levels (2 and 3) of mesh are evenly matched along the curve of the previous study. It means that even though Level 2 contains only 44982 cells, it gives nearly the same result of which Level 3 (114944
cells) does. This decreasing in the total number of quadrilateral cells will be useful with VOF model because of reduction in the speed of calculations. 50
Keller Present (Level 3)
0
y, m
-50
-100
-150
-200
-250
-300 -40
-35
-30
-25
-20
-15
-10
-5
0
x-velocity, mm/s Figure 5.3: Comparison of velocities along the outlet boundaries obtained by FLUENT (present study) to COMSOL (Keller’s study) using Level 3 of grid.
Figure 5.4 shows filled contours of static pressure in the pore-scale model where the pressure decreases gradually from the inlet pressure to the atmospheric pressure at the outlet boundaries. FLUENT has the ability to present streamlines of water flow as contours of stream function. Figure 5.5 displays the streamlines through the pore spaces. Both of simulations shown in Figures 5.4 and 5.5 are obtained utilizing Level 2 of mesh.
Figure 5.4: Contours of static pressure (Pa).
Figure 5.5: Stream function (kg/s) of water.
To ensure a good comparison we tested the micromodel with help of 2D triangular cells and compared to Keller’s results, Figure 5.6 indicates water velocity in x-direction at outlet boundaries by using 65632 triangular cells of grid. The comparison showed approximately the same result obtained in case of using quadrilateral cells. There is a difference in this comparison as
a result of mesh quality as well as the number of cells. This type of cells is used only with single-phase flow. 50 Keller Present (Tri.)
0
y, m
-50
-100
-150
-200
-250
-300 -40
-35
-30
-25
-20
-15
-10
-5
0
x-velocity, mm/s
Figure 5.6: Comparison of velocities along the outlet boundaries using triangular elements (present study) to COMSOL (Keller’s study) using triangular cells.
Finally, the three last results obtained by the FLUENT simulations are compared together. It is concluded from Figure 5.7 that Level 2 almost matches with the others. Therefore, the decision is made to perform the most simulations concerning two-phase flow with using Level 2 of grid. Figure 5.7 proves that Level 2 of mesh is enough to use in case of twophase flow through VOF model, this is an important advantage because of limitation of flow time. This obtained conclusion is based on single phase simulations as well as the previous results.
50
0
Level 2 Level 3 Tri. Cells
-50
y, m
-100
-150
-200
-250
-300 -40
-35
-30
-25
-20
-15
-10
-5
0
x-velocity, mm/s
Figure 5.7: Comparison of velocities along the outlet boundaries obtained by FLUENT using both the quadrilateral cells (Level 2 and 3) and triangular cells.
5.1.1 Estimation of absolute permeability After the velocity field and pressure distribution have been obtained, the most important property of our model is the permeability, which has a significant effect on the macro-scale behaviour. Based on the x-velocity along outlet boundary obtained before, we can estimate the absolute permeability for our model by using Darcy‟s formula (3.6). From the inflow/outflow boundary conditions we have evaluated the pressures. It‟s obvious that the relation between the velocity in x-direction at outlet and the permeability is linear (Figure 5.8), where the maximum permeability at the maximum velocity (38.79 mm/s) is 3.47x10-11 m2 and the average absolute permeability is about 9.1x10-12 m2, which is almost similar to the typical value obtained experimentally for unconsolidated rocks where the petroleum existed.
Figure 5.9 shows the absolute permeability of the pore-scale model as a function in y-coordinate. -11
x 10
3.5
Absolute permeability, m2
3
2.5
2
1.5
1
0.5
0
0
5
10
15
20
25
30
35
40
x-velocity, mm/s
Figure 5.8: The absolute permeability versus the velocity in x-direction along outlet boundaries. 50
0
y, m
-50
-100
-150
-200
-250
-300
0
0.5
1
1.5
2
2.5
Absolute permeability, m
2
3
3.5 -11
x 10
Figure 5.9: The absolute permeability as a function in y-direction along outlet boundaries.
5.1.2 Effect of dynamic viscosity The viscosity of the pore-filling fluid (oil) should be an important factor controlling the mobilization of ganglia. Oil can be classified according to its viscosity. A natural petroleum reservoir may contain oil which has a dynamic viscosity ranges from 0.3 mPa.s (light oil) to 1 Pa.s or above (heavy oil). We expect almost linear increase in the external pressure gradient required for mobilization as well as the vibration amplitude, when the vibration is applied, with the increase in fluid viscosity. The numerical experiments
were
conducted
under
three
pressure
differences:
0.715, 2 and 6 kPa. Figure 5.10 shows the volumetric flow rate of oil plotted against the dynamic viscosity of oil. As expected, we see an almost linear dependence between the external pressure difference and the flow rate at all cases. 10 P=0.715kPa
9
P=2kPa P=6kPa
Oil viscosity, mPa.s
8 7 6 5 4 3 2 1
0
2
4
6
8
10
12
14
3
Volume flow rate, cm /s
Figure 5.10: The effect of oil viscosity on the required external pressure to mobilize oil through the model.
5.2 Two Phase Flow Simulations 5.2.1 Pressure-driven flow of interfaces in a pore body In order to remedy the limitations of pore-scale model, we decided to utilize FLUENT to simulate the variety of fluid-fluid interface configurations and their evolution in multiple pores. A two-dimensional geometry was originally chosen with the hope of improving the algorithm responsible for handling the menisci of the pore bodies. Since the general form of the pore geometry and fluid parameters (viscosity, density, surface tension, and contact angle) have already been defined in Chapter 4, all we had to do was choosing the proper set of methods and parameters defining the numerical algorithms in FLUENT. However, 0.1o contact angle (complete water wetting of the wall with oil being the nonwetting fluid) is used for all the calculations hereafter. The proper level of refinement was to be determined. It‟s shown in Chapter 4 that several levels of grid refinement considered, starting with as low as 1 mesh element across the pore throat (Level 1). A number of FLUENT simulations were then run to determine the lowest level of grid refinement that would result in an acceptable representation of the fluid flow and, more importantly, fluid-fluid interface behaviour. Running several simulations with progressively more refined meshes, we found the optimal mesh resolution should be at least Level 2 (44982 cells) which is sufficient to resolve fine details of the interfaces, yet the coarse mesh is enough to allow running a high number of simulations in a limited. Another important parameter of our numerical simulations is the time step. Although we conducted a time-step sensitivity study for various mesh resolution, we were most interested in the results for the coarsest acceptable mesh. It has been found that optimal time step for the selected mesh (Level 2) was 0.6 µs. Appendix B presents the default and recent values of time step used in this work. Based on these tests, as well as on a number of other
simulations, it is concluded that for the time steps below 0.6 µs all simulations were practically identical. For the values of time step between 0.6 and 0.8 µs, certain differences in the results could be observed. As time step approached 1 µs, the modelled fluid interfaces lost stability, showing oscillations and breakup without apparent cause. Following the same principles outlined above, we argue that a value of the time step of 0.6 µs is adequate for our modelling goals when Level 2 of mesh is used. Our model can be tested with pressure difference as the only driving force. We started with the invasion of a wetting fluid (water) from the right side (inlet) into the pore body, under various boundary conditions (defined as pressures in the inlet and outlet pore body). To do so, the model was originally partially filled with the non-wetting phase, and then the wetting phase was injected from one side of the model under constant predefined pressure as 30 kPa. Figure 5.11 displays the initial conditions of water/oil flow system, where the initial volume fraction of oil (shown in red colour) is about 17 % while the remaining volume is water (shown in blue colour). The dynamic testing is considered of comparing the fluid-phase distribution at various times. Several „scenarios‟ in our model geometry were simulated and then compared. We chose to calculate the residual of non-wetting (oil) phase.
Figure 5.11: Contours of initial volume fraction of Oil (shown in red colour), (Time = 0 s).
Figure 5.12 shows the results of six simulations of the water invasion into a pore body containing some oil located in the middle of the model (as
snapshots of the fluid-phase distributions taken at different times). The higher tortuosity of the pore space resulted in a higher residual saturation and formation of several isolated oil blobs. Time = 0.12 ms
Time = 0.84 ms
Time = 1.2 ms
Time = 1.74 ms
Time = 2.16 ms
Time = 3 ms
Figure 5.12: Results of the simulation of oil (shown in red colour) displaced by water. External pressure difference is 30 kPa and by using Level 2 of mesh.
The volume of the non-wetting phase remaining in the system was plotted against time. Figure 5.13 shows the volume of the non-wetting fluid (normalized to the initial, 17 % saturation volume) as a function of time at 30 kPa of the external pressure difference in 44982 elements (Level 2) pore model. It should be noted that the maximum residual concentration of the
non-wetting fluid in our model did not exceed 12.3 %. The discrepancy can be explained by the lack of film and crevice flow implementation in the numerical simulation. It is better to determine the displacement efficiency (presented in Chapter 2) by comparing initial and final volumes of fluid in place and it is in this case about 27.7%. 17
Relative saturation of Oil, %
16.5 16 15.5 15 14.5 14 13.5 13 12.5 12
0
0.5
1
1.5
2
2.5
3
Time, milisecond Figure 5.13: Relative saturation of Oil as a function of time. External pressure is 30 kPa and by using Level 2 of mesh.
However, when it came to modelling the „retreat‟ of the oil from a pore body, we knew that our pore-scale model was not handling such processes well. Thus, attempting to find some sort of a general classification scheme for the variety of „retreat‟ scenarios arising from the FLUENT simulations was our main priority. We soon realized that, though there were at least two reasons that would make such classifications practically implausible. First, the number of potential „scenarios‟ that could arise in the pore-scale model would be directly
proportional to the number of pores and the time of the simulation, and using a set number of pre-set „scenarios‟ would likely be inadequate (it would only extend the average simulated time until problems appeared in the model; it would not resolve the problem completely). Second, the configurations of the interfaces during even rather simple „retreat‟ scenarios simulated in FLUENT were so complex that they could not be translated into the model algorithm while keeping the generally simple description of the menisci. A variety of comparison methods could be used: for example, calculating the fraction of the model volume (area) occupied by the nonwetting fluid (oil) would provide a measure of the dissimilarity between two simulations with different mesh sizes (Level 2 & 3). In most of the tests conducted, this value varied in time but never exceeded 10 %, and it did not always correlate with important differences in interface configurations. Thus, a simple visual comparison of the fluid distributions in the models was conducted. The decrease in the linear scale resulted in a corresponding decrease in the time scale (time step), thus slowing the calculation speed even further. Since direct reduction in the resolution of the unstructured mesh was not possible. The time step chosen for all these simulations in this case (Level 3) was 0.1 µs, sufficiently small for the finest grid resolution considered. The interface curvature for the lowest grid resolution tested appears to be adequate. Because animation sequences of the simulations cannot be presented in this text, snapshots of fluid configurations at certain times will be shown in Figure 5.14 to illustrate the differences between the simulation results with mesh Level 3 (114944 cells), where „water invasion‟ scenario is illustrated. Several other scenarios have also been simulated, and in all cases the results obtained with the lowest mesh resolution considered (Level 3) were remarkably similar to those obtained with higher mesh resolution (Level 2), with very few exceptions creating the same general
interface configuration. Based on the results of this extensive testing, we can make an educated decision on the grid resolution optimized for speed. Time = 0 s
Time = 1 ms
Time = 1.3 ms
Time = 1.6 ms
Figure 5.14: Results of the simulation of oil (shown in red colour) displaced by water. External pressure difference is 30 kPa and by using Level 3 of grid.
Figure 5.15 represents the above results but as x-y plot where the relative saturation (volume fraction) of oil is plotted against the flow time at external pressure difference of 30 kPa. Owing to the high number of cells the flow time is limited to 2 ms.
17 16.5
Relative saturation of Oil, %
16 15.5 15 14.5 14 13.5 13 12.5 12
0
0.5
1
1.5
2
Time, millisecond
Figure 5.15: Relative saturation of Oil as a function of time. External pressure is 30 kPa and by using Level 3 of grid.
The FLUENT simulation showed that, in general, VOF available for the pore-scale model performed very well in simulating the behaviour of the menisci during the events of „water invasion‟ (including more complex cases of water entering the pore body various values of external pressure). Figure 5.16 the relative saturation (volume fraction) of oil is plotted against the flow time with three different external pressures 25, 30 and 35 kPa, where the final residual saturation of oil decreases at the higher pressure, therefore the displacement efficiency increases.
17 P=25kPa P=30kPa
Relative saturation of Oil, %
16
P=35kPa
15
14
13
12
11
0
0.5
1
1.5
2
2.5
3
3.5
Time, millisecond Figure 5.16: Relative saturation of Oil as a function of time for several numerical experiments with various values of external pressure difference.
The final testing stage was meant to investigate the model‟s capability to simulate relatively large initial volume fraction of oil which was increased to 32 %, with a large number of interfaces in complex configurations and the mesh is used at Level 2 as shown in Figure 5.17.
Figure 5.17: Increasing of initial volume fraction of Oil (shown in red colour), (Time = 0 s).
After running several simulations the oil was displaced out of the domain by water, the selected snapshots are shown in Figure 5.18.
Time = 0.18 ms
Time = 0.84 ms
Time = 1.2 ms
Time = 3 ms
Figure 5.18: Results of the simulation of oil (shown in red colour) displaced by water. External pressure difference is 30 kPa and by using Level 2 of mesh. .
5.2.2 Prediction of the effect of vibration Since so far we have been unable to effectively model the effect of vibration on a large scale, explicitly accounting for the pore interconnectivity, and using only abstract representations of the fluid flow and the fluid-fluid interface dynamics, an application of the CFD methods to a large number of pores is in order. Even though in the time frame of this study the calculation speed of personal computers increased several-fold, detailed modelling of the dynamic multi-phase flow in a large number of pores is still unacceptably slow. However, we see that we are already sacrificing many small details, for example, we have to consider a complex pore geometry that does not
explicitly account for film flow, we approximate the flow rate in the pore throats, and we ignore pressure gradients within pore bodies. Thus, we can argue that a similarly low-detailed CFD model might be acceptable for simulating the general behaviour of ganglia in geometries containing large numbers of pores. To determine if such low-detailed stimulation can be effectively run through FLUENT, we conducted the following study. As described in chapter 4, the effect of vibration can be modelled by simply introducing an oscillatory body force. This force can be simulated in FLUENT by defining the source term (terms added to the right-hand side of the momentum equation). Even though only constant source terms can be defined directly, FLUENT supports a powerful mechanism of plug-ins called “user-defined functions” (UDF), allowing execution of external code (either interpreted or compiled from a C language source), with extensive access to the inner data of the model and the solver. Using this mechanism, we defined the time-dependent source terms for each component of the momentum equation. Appendix C includes an example of user-defined function used in this section, the parameters of vibration can be changed to predict the effect of vibration on the residual saturation of oil. Since the utilization of UDFs has a noticeable performance hit (decrease in calculation speed), we attempted to limit the number of external functions used. This was done by using the simplified form of vibration-induced force. Due to the time-consuming calculations, only 9 numerical tests were simulated. In the case of using a medium mesh (Level 2), the maximum time step that did not result in the lack of convergence or obvious differences in the simulation results was similar to the first selection (0.6 µs) in pressure-driven flow only. Even with these settings, FLUENT demonstrated a rather slowspeed performance. This was sufficient to demonstrate the ganglia formation and entrapment in our relatively small model, but not very useful for studying the long-term effects of vibration. Due to the slow performance associated
with using UDF, only short mobilization tests could be performed in a reasonable amount of time. A scenario was developed to demonstrate the effect of vibrations on the residual saturation of the non-wetting fluid. In the model, originally saturated with water, large oil ganglia (17 %) were manually inserted, and an external pressure difference of 30 kPa applied by setting the appropriate boundary conditions. These initial and boundary conditions were then used in several simulations runs with addition of an oscillatory body force. For the comprehensive comparison to study the effect of vibration parameters, a series of experiments was designed, sampling the frequency range from 10 to 500 Hz. The parameter for the comparison was the amplitude required for increasing the mobilization of oil under a given pressure difference applied between the inlet and outlet of the model. In FLUENT, a series of simulations was run, varying the acceleration amplitude of the applied body force (1 to 20 m/s2) until the mobilization threshold was bracketed within an acceptably small interval. The results of a series of simulations with the external pressure difference (30 kPa) and 10 m/s2 of the amplitude at 10 Hz of the frequency are shown in Figure 5.19 with Level 2 of grid. In this case, the results of the FLUENT simulations match almost perfectly in the lower frequency range. Although the optimization of the numerical mesh and the time step allowed us to simulate noticeably larger numbers of pores in a shorter timeframe, the resulting calculation speed is still insufficient to obtain statistically viable results for variations in the initial volume fraction of oil, pore geometry, or even an estimation of the required vibration amplitude. While shorter tests for the stability of a given ganglion configuration can be performed in as little as 3.5 ms, simulations of ganglion movement across the model space may take from 10 hours to several days.
Time = 0 s
Time = 1.74 ms
Time = 1.38 ms
Time = 2.7 ms
Figure 5.19: Results of the simulation of oil (shown in red colour) displaced by water. External pressure difference is 30 kPa and by using Level 2 of mesh, (A = 10 m/s2).
A comparison of relative saturation of oil at various values of the amplitude of vibration (1 to 20 m/s2) in a certain time limit can be seen in Figure 5.20, where the relative oil saturation (the ratio of the volume occupied by oil to the total volume of the model) is plotted versus time for several simulations with various parameters of vibratory stimulation. The effect of vibration can be clearly seen in the faster removal of oil from the model, as well as in the lower final residual saturation. Based on the estimates in section 5.2.1 for the lowest acceptable mesh resolution obtained (Level 3), that showed the results of the simulation of oil displacement by water in the given geometry, the higher tortuosity of the pore space resulted in a higher residual saturation and formation of several isolated oil blobs. To test the stability of these residual blobs, a series of simulations that included an oscillating body force (applied in the direction of the flow in the model, right-to-left) were conducted. The vibration amplitude tested for
the mobilization of oil was between 0.01 m/s2 and 30 m/s2 because of the relatively small pore throat diameters and low external pressure gradient, and the frequency is defined as 10 Hz. Obtaining a precise estimate of the mobilization amplitude for oil would require too many additional „iterations‟ (simulations runs with different vibration amplitudes), two different amplitudes are applied (10 and 20 m/s2).
17 No vibration 2
A=1 m/s (started at 0.0 s) 2
Relative saturation of Oil, %
16
A=10 m/s (started at 0.0 s) 2
A=20 m/s (started at 0.0 s) 15
14
13
12
11
0
0.5
1
1.5
2
2.5
3
3.5
Time, millisecond Figure 5.20: Relative saturation of Oil as a function of time for several simulations with the same initial conditions. External pressure is 30 kPa and by using Level 2 of mesh, (f = 10 Hz).
An example of a comparison in case of Level 3 can be seen in Figure 5.21, where „water invasion‟ scenario is illustrated. Several other scenarios have also been simulated, and in all cases the results obtained with the lowest mesh resolution considered (Level 3) were remarkably similar to those
obtained with higher mesh resolution, with very few exceptions creating the same general interface configuration but minor (under 10 %) variation in the flow rates. Based on the results of this extensive testing, we can make an educated decision on the grid resolution optimized for speed. Time = 0 s
Time = 0.9 ms
Time = 1.3 ms
Time = 1.7 ms
Figure 5.21: Results of the simulation of oil (shown in red colour) displaced by water. External pressure difference is 30 kPa, the vibration amplitude is 10 m/s2 and using mesh at Level 3.
A comparison of relative saturation of oil at the amplitude of vibration of 10 m/s2 in a certain time limit (2 ms) can be seen in Figure 5.22, where the relative oil saturation is plotted versus time for several simulations with constant parameters of vibratory stimulation (f = 10 Hz and A = 10 m/s2). The effect of vibration can be clearly seen in the faster removal of oil from the model, as well as in the lower final residual saturation.
17 No vibration 2
A=10 m/s (started at 0.0s)
Relative saturation of Oil, %
16
15
14
13
12
11
0
0.5
1
1.5
2
Time, millisecond
Figure 5.22: Relative saturation of Oil as a function of time. External pressure is 30 kPa and by using Level 3 of mesh, (f = 10 Hz and A = 10 m/s2).
Because our model is two-dimensional, it will not exhibit a noticeable film flow, nor will it show the snap-off of the non-wetting phase in pore throats (the interface curvature will always correspond to the wall curvature, leading to the spreading of the wetting fluid thinner along the wall). Thus, we can conclude that using the mesh-resolution Level 2 is sufficient for our purposes. Another important parameter that plays a great role in the vibration effect is the frequency. The model is tested under various values of the frequency, in all the above simulations the frequency was constant (10 Hz). The results obtained with the vibratory stimulation enabled are illustrated in Figure 5.23, where the initial condition was 17 % of the volume fraction assigned to the oil, and the external pressure is 30 kPa, besides the amplitude of the acceleration is defined as a constant value of 10 m/s 2 in all cases of
studying the effect of frequency. In Figure 5.23 the time limit is expanded to almost 4.6 ms in order to obtain clear comparison between two different frequencies (10 and 100 Hz). The FLUENT simulations are based on the same boundary conditions applied before in this section but for changing of the frequency via UDF.
Relative saturation of Oil, %
17 f = 10 Hz f = 100 Hz
16
15
14
13
12
11
0
1
2
3
4
Time, millisecond Figure 5.23: Relative saturation of Oil as a function of time at various values of frequencies (10 and 100 Hz). External pressure is 30 kPa and by using Level 2 of mesh, (A = 10 m/s2).
To ensure the understanding of the oil flow behaviour under different frequencies, the initial volume fraction of oil is raised to 32 % with the same boundary conditions. In Figure 5.24 the relative saturation of oil is plotted versus time flow of the model at two different frequencies (100 and 500 Hz), it is relatively clear that at the lower frequency we have a good recovery. But the model should be run more than 3.5 ms to predict the effect of the
frequency well. The relative saturation decreases rapidly in the case of higher frequency but the residual saturation of oil is low with the lower frequency. At the higher frequencies, FLUENT shows noticeably larger required vibration amplitudes (more than 10 m/s2).
34
Relative saturation of Oil, %
32
f = 100 Hz f = 500 Hz
30 28 26 24 22 20 18 16
0
0.5
1
1.5
2
2.5
3
Time, millisecond Figure 5.24: Relative saturation of Oil as a function of time at various values of frequencies (100 and 500 Hz). External pressure is 30 kPa and by using Level 2 of mesh, (A = 10 m/s2).
CHAPTER SIX CONCLUSIONS AND RECOMMENDATIONS
The focus of this research is on gaining a better understanding of the flow and transport problem in porous media by using pore-scale modelling approach. In order to meet the objectives of present work, this chapter introduces
two
sections,
where
the
thesis
is
summarized
and
recommendations for further studies are addressed.
6.1 Conclusions The motion of the fluid(s) based on the Navier-Stokes equations has been studied by implementing the model with the finite-volume technique and numerical analysis. Some predictions such as the absolute permeability has been found based on single-phase flow simulations. This obtained property plays a great role in the macroscopic flow behaviour as well as in real-world problems. Regarding to the tracking of interfaces in case of two-phase flow, we conclude that multiple-pores CFD modelling appears to be possible for the purposes of tabulating the dynamic interface configurations needed for improving the interface description in the pore-scale approach. The use of FLUENT to validate the processes modelled in a realistic porous medium proved that the choice was in order. But the calculation time is still relatively long which amounts to hours and days for a single simulation run. We demonstrated that careful optimization of the mesh resolution and the time step allows using FLUENT for simulating a reasonable number of interconnected pores in the timeframe of less than a day on an up-to-date personal computer. This type of model can be useful in
determining certain features of the effect of vibration on isolated entrapped ganglia in particular pore geometry (e.g., the mobilization threshold, or an effective duration of the vibratory simulation), provided someone is willing to invest a significant amount of computer time is these simulations. The usefulness of these results for the potential applications of the vibratory-stimulation technology will likely be limited because of the simplified model geometry, the small scale of the model, and two-dimensional representation of the pore space. Overall, it‟s concluded that, the direct use of CFD software for modelling the effect of vibration on fluid flow in large number of pores is unlikely to provide answers and estimates applicable to real-world utilization of the technology, primarily due to the flow calculation speed for even two-dimensional models of a relatively small number of pores. This conclusion is based on the assumption of using only a single personal computer for simulations. Thus, we can conclude that an abstract representation of the interfaces in a pore-scale model will not allow for adequate modelling of highly dynamics processes (such as vibration) in a system involving a large number of pores and/or complex configurations of the interfaces.
6.2 Recommendations for Future Studies After considering the results obtained in the thesis regarding the porescale modelling of realistic porous media flow and transport, a few recommendations for further studies are as follows: For the numerical modelling work, the challenge lies in the simulation of a sufficiently large porous domain be able to supply more precise predictions than those presented in this thesis.
Another interesting direction of investigation is the study of the effect of minor features of pore geometry (e.g., rough walls, and angular geometry of the model) on the mobilization potential of vibration on an interface in the pore constriction. This modelling can be performed using the existing CFD software. As the usefulness of these results concerning two-phase flow in realistic porous media at microscopic scale, macroscopic scale studies should be done to predict and understand the effects of vibration-induced mobilization of trapped oil in porous media. For experimental research on a laboratory scale involving realistic porous media (e.g, sandstone core samples), better documentation and elaboration of the conditions for effective mobilization is required, primarily in terms of the parameters of the vibrations for given external pressures, in order to directly correlate the experimental results with field data. Description of the pore morphology of geological material in three dimensions will be an immense topic in modern pore-scale studies. This is one of the studies that deal with multiphase flow through porous media. Hopefully more intense research will be done in this regard and makes porous media flow understandable.
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Appendix A Simulation Settings To set up a simulation of interest in FLUENT, the following steps were carried out: 1. The meshed geometry of interest was imported from GAMBIT into FLUENT. 2. Because the dimensions are small, the double-precision version of FLUENT was used. 3. We employed the unsteady, pressure-based (segregated) solver. 4. For two-phase flow system, the VOF model was selected and the number of phases was specified. Water was designated as the primary phase, and oil was designated as the secondary phase and the interaction between them, i.e., the surface tension, was specified. 5. The inlet conditions concerning velocity, pressure and volume fraction of oil were specified as boundary conditions. 6. The appropriate model-specific solution parameters were set. Appendix B gives an explanation on how these values were calculated. 7. The starting values of velocities, pressure and volume fraction of oil were initialized. 8. Appropriate residual conditions were defined. 9. The transient calculation with proper size of the time step was calculated and the results examined. 10. Although adaptive-time step evaluation methods were available, a constant time step was used in every simulation, because this approach provides more consistent results and therefore simpler post-processing for creating dynamics animations.
Appendix B Appropriate Values of Time Step Size and Under-Relaxation Factors Table B.1 shows the values of time step size and under-relaxation factors that were used for the simulations. These were selected to meet the appropriate number of iterations per time step suggested in the FLUENT manual (FLUENT Inc., 2005), i.e., 10 to 20 iterations. A reasonable courant number of 0.25 was used in the analyses. The Courant number is a dimensionless number that compares the time step in a calculation to the characteristic time of transit of a fluid element across a control volume given by the equation: t xcell / u fluid Table B.1: Suggested values of time step size and under-relaxation factors. Time step size [s] Under-relaxation factor Pressure Density Body forces Momentum Volume fraction
Default 1
44982 elements 6x10-7
114944 elements 1x10-7
0.3 1 1 0.7 0.2
1 1 1 1 1
1 1 1 1 1
Due to the fact that the equations solved by FLUENT are of non-linear and complex nature, it is necessary to be able to control the change of (FLUENT 6.2 User's Guide, 2005). This is often achieved with the help of under-relaxation, which is a process that reduces the change of during each iteration. The new value n 1 is achieved by the sum of the former value n and the computed change in n , multiplied by the under-relaxation factor : n 1 n
(B.1)
Appendix C User Defined Sub-routine in ‘C’ language to include the time varying acceleration
FLUENT does not have a provision to input a time-varying acceleration required in the case of vibration. A user defined sub routine, shown below, was written to apply the time-varying acceleration in the FLUENT software (FLUENT 6.3 UDF Manual, 2006). The user defined functions (UDF) update interval was set to 1 so that this program was called after every time step, and the acceleration was updated. Thus, the effect of vibration was simulated by applying the appropriate accelerations at different times during the analysis. The frequency (f) is set here as 10 Hz, and the amplitude (A) as 10 m/s2, other values of these parameters are not presented in this code but in another file similar to the presented file. #include "udf.h" #define f 10.0 /*vibration frequency in Hz*****/ #define A 10.0 /*vibration amplitude in m/s^2*****/ #define pi 3.14 DEFINE_SOURCE (x_mom,c,thread,dS,eqn) { real source,t,xaccel; t=CURRENT_TIME; xaccel=A*sin(2*pi*f*t); source=xaccel*C_R(c,thread); /*C_R is fluid density within a cell in kg/m^3*****/ dS[eqn]=xaccel; return source; }
