Fereidoun Rezanezhad PhD Thesis 2007

Fereidoun Rezanezhad PhD Thesis 2007
Dissertation
submitted to the
Combined Faculties for the Natural Sciences and for Mathematics
of the Ruperto-Carola University of Heidelberg, Germany
for the degree of
Doctor of Natural Sciences
presented by
Diplom-Physicist: Fereidoun Rezanezhad
born in:
Ramsar, Iran
Oral examination: May 23, 2007
Experimental Study of Fingering Flow in
Porous Hele-Shaw Cells
Referees:
Prof. Dr. Kurt Roth
Prof. Dr. Bernd Jähne
Zusammenfassung
Mit dem Ziel, die physikalischen Prozesse der Bildung von Flussinstabilitäten
(fingering) in zwei Dimensionen zu untersuchen, wurden in Laborexperimenten
vertikale Infiltrationsexperimente in geschichtetem Sand mit Hilfe von HeleShaw Zellen durchgeführt. Es wurde eine Lichttransmissions-Methode entwickelt, um die Dynamik der Wassersättigung innerhalb der Fließfinger detailliert,
mit hoher räumlicher und zeitlicher Auflösung zu untersuchen. Die Methode wurde mit Hilfe von Röntgen-Absorptionsmessungen kalibriert. Die bei
der Lichttransmission auftretenden Streueffekte wurden über eine Dekonvolution mit Hilfe der Punktbildfunktion korrigiert. Dies ermöglicht quantitative,
räumliche hoch aufgelöste Messungen. Nach der vollständigen Entwicklung
der Finger wurde ein Farbstoff aufgegeben, um mobile und immobile Anteile
des Wassers unterscheiden zu können. Vollständig entwickelte Finger bestehen
aus einer Fingerspitze, einem Kern aus mobilem Wasser und einem Rumpf aus
immobilem Wasser. Es wurde die Dynamik der Wassersättigung innerhalb der
Fingerspitze, entlang des Kerns und im Bereich des Randes während seines radialen Wachstums untersucht. Dabei konnten vorausgegangene Untersuchungen
bestätigt werden, die ein Überschwingen der Sättigung im Bereich der Fingerspitze zeigten. Weiterhin wurde ein Sättigungsminimum direkt hinter der
Spitze als neues Phänomen gefunden. Die Entwicklung eines Fingers lässt sich
durch einen sukzessiven Anstieg des Wassergehaltes innerhalb des Fingerkerns
hinter dem Minimum sowie eine kontinuierliche Verbreiterung bis hin zu einem
quasi-stabilen Zustand charakterisieren. Dieser Zustand ist erst lange Zeit nach
der Entwicklung des Fingers erreicht. In diesem Stadium lässt sich ein Kern mit
einem schnellen konvektiven Fluss sowie ein Rand mit zunehmend langsamerem
Fluss feststellen. Sämtliche beobachteten Phänomene, außer des Überschwingens der Sättigung, konnten mit der hysteretischen Natur der Boden-WasserCharakteristik erklärt werden.
Abstract
With the aim of studying the physical process concerning the unstable fingering phenomena in two dimensions, experiments of vertical infiltration through
layered sand were carried out in the laboratory using Hele-Shaw cells. We developed a light transmission method to measure the dynamics of water saturation
within flow fingers in great detail with high spatial and temporal resolution.
The method was calibrated using X-ray absorption. We improved the measured
light transmission with correction for scattering effects through deconvolution
with a point spread function which allows us to obtain quantitative high spa-
tial resolution measurements. After fingers had fully developed, we added a
dye tracer in order to distinguish mobile and immobile water fractions. Fully
developed fingers consist of a tip, a core with mobile water, and a hull with
immobile water. We analyzed the dynamics of water saturation within the
finger tip, along the finger core behind the tip, and within the fringe of the fingers during radial growth. Our results confirm previous findings of saturation
overshoot in the finger tips and revealed a saturation minimum behind the tip
as a new feature. The finger development was characterized by a gradual increase in water content within the core of the finger behind this minimum and
a gradual widening of the fingers to a quasi-stable state which evolves at time
scales that are orders of magnitude longer than those of fingers’ evolution. In
this state, a sharp separation into a core with fast convective flow and a fringe
with exceedingly slow flow was detected. All observed phenomena, with the
exception of saturation overshoot, could be consistently explained based on the
hysteretic behavior of the soil-water characteristic.
Contents
1. Introduction
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2. Theoretical Background
2.1. Water Flow through Porous Media . . . . . . . . . . . . . . . .
2.1.1. Water flow through saturated porous media . . . . . . .
2.1.2. Water flow through unsaturated porous media . . . . .
2.1.3. Infiltration with wetting front through porous media . .
2.1.4. Hysteresis of soil water-retention function . . . . . . . .
2.1.5. Multiphase flow in porous media . . . . . . . . . . . . .
2.2. Physical Phenomena of Viscous, Gravity and Capillary Forces .
2.2.1. Stable and unstable fluid displacement . . . . . . . . . .
2.3. Visualization of Flow and Transport in Porous Media . . . . .
2.4. Preferential Flow Phenomena . . . . . . . . . . . . . . . . . . .
2.4.1. The history of preferential flow and why it is important
2.4.2. Macropore flow . . . . . . . . . . . . . . . . . . . . . . .
2.4.3. Funnel flow . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.4. Fingering flow . . . . . . . . . . . . . . . . . . . . . . .
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3. Experimental Methods and Materials
3.1. Laboratory Materials and Setup . . . . . . .
3.1.1. Materials and preparation of samples .
3.1.2. Saturated hydraulic conductivity . . .
3.1.3. Hele-Shaw cell . . . . . . . . . . . . .
3.1.4. Light source . . . . . . . . . . . . . . .
3.1.5. Camera setting . . . . . . . . . . . . .
3.1.6. Tensiometer construction . . . . . . .
3.1.7. Experimental setting . . . . . . . . . .
3.2. Methods . . . . . . . . . . . . . . . . . . . . .
3.2.1. Light Transmission Method (LTM) . .
3.2.2. X-ray absorption . . . . . . . . . . . .
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Contents
3.2.3. Limitations of techniques . . . . . . . . . . . . . . . . .
3.3. Water Content Calibration . . . . . . . . . . . . . . . . . . . .
3.3.1. Calibration error . . . . . . . . . . . . . . . . . . . . . .
4. Image Processing
4.1. Pre-processing of Images . . .
4.2. Deconvolution Processes . . .
4.2.1. Point Spread Function
4.2.2. 2D deconvolved image
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5. Results and Discussion
5.1. Experimental Evidences . . . . . . . . . . . . . . . . . . . . .
5.1.1. Observations . . . . . . . . . . . . . . . . . . . . . . .
5.1.2. Flow field structure using tracer experiment . . . . . .
5.1.3. Finger development in initially dry sand . . . . . . . .
5.1.4. Saturation overshoot . . . . . . . . . . . . . . . . . . .
5.2. Physical Explanation of the Finger Initiation . . . . . . . . .
5.2.1. Distribution of flow in the fine-textured toplayer . . .
5.2.2. Initiation of the finger in the coarse-textured sublayer
5.3. Why does the Saturation Overshoot Occur? . . . . . . . . . .
5.4. Dynamics of Water Saturation and Pressure . . . . . . . . . .
5.4.1. Dynamics and stabilization of the fingers . . . . . . .
5.5. Finger Width . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6. Finger Tip Velocity . . . . . . . . . . . . . . . . . . . . . . . .
5.7. Fingering Flow under Different Flux Rate Infiltration . . . .
5.8. Finger Persistence . . . . . . . . . . . . . . . . . . . . . . . .
5.9. Effects of High Initial Water Content . . . . . . . . . . . . . .
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6. Summary and Conclusions
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Bibliography
A. Appendix
ii
127
List of Figures
2.1. Schematic cross-section of the saturated and unsaturated zones
2.2. Moisture zones during infiltration with a sharp wetting front .
2.3. Hydraulic conductivity and soil-water characteristic curves . . .
2.4. Schematic diagram of flow processes in the mixing layer . . . .
2.5. Schematic diagram of water flow through macropores . . . . . .
2.6. Photographs of macropore flow paths in structured soil . . . . .
2.7. Schematic diagram of of water flow through funnel flow . . . .
2.8. Photographs of funnel flow paths in sandy soil . . . . . . . . .
2.9. Schematic diagram of water flow through fingering flow . . . .
2.10. Formation of water fingers in homogeneous sandy soils . . . . .
2.11. Three-dimensional form of fingering flow . . . . . . . . . . . . .
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3.1.
3.2.
3.3.
3.4.
3.5.
3.6.
Sketch of falling head method . . . . . . . . . . . . . . . . .
Experimental Set-Up . . . . . . . . . . . . . . . . . . . . . .
Sketch and photo of the tensiometer installed over the cell .
Visualization of flow fingering using RGB and HSI formats
Sketch of the optical paths of light through Hele-Shaw cell .
Calibration results between two LTM and X-ray methods .
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4.1.
4.2.
4.3.
4.4.
4.5.
4.6.
4.7.
4.8.
4.9.
RGB and HSI color spaces for an example image . . .
Flat-field correction for an example image . . . . . . . .
Sequential steps of the image pre-processing . . . . . . .
Schematic of the method for measurement of PSF . . .
The process procedure to measure the PSF . . . . . . .
Flowchart of the procedure of the image deconvolution .
Example of deconvolution results for an observed image
Sequence of the image processing steps . . . . . . . . . .
Comparison of X-ray and LTM for a cross-section of two
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fingers
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5.1. Photographic sequence of images showing the water fingering .
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iii
List of Figures
5.2. Redistribution of water through two-layered sand . . . . . . . .
5.3. Infiltration of dye tracer into stabilized water fingers . . . . . .
5.4. Photographic sequence of images showing the dye fingers . . . .
5.5. Sequence illustrating the development of instability . . . . . . .
5.6. Fingered flow in a homogeneous initially dry sand . . . . . . . .
5.7. Photographs of the grain size and particle shapes . . . . . . . .
5.8. Evaluation of the fingering development into a two-layered sand
5.9. Sketch of saturation overshoot in flow fingers . . . . . . . . . .
5.10. Microscopic view of water-air interface and pressure . . . . . .
5.11. Measured pressure drop across the porous medium . . . . . . .
5.12. The temporal dynamics of saturation inside the finger core . .
5.13. Water saturation versus time for three areas within a finger . .
5.14. Plots of water saturations and water potential measurements .
5.15. Sketch of hydraulic states evolution during passage of the finger
5.16. Horizontal transverse intensity and saturation profiles . . . . .
5.17. Schematic diagrams of downward growth of a single finger . . .
5.18. Horizontal intensity profiles during 10 days infiltration . . . . .
5.19. Images illustrating the advancement of four fingers . . . . . . .
5.20. Finger tip velocity for two major fingers . . . . . . . . . . . . .
5.21. The number of the fingers as a function of flow rate . . . . . .
5.22. Plots of the finger properties as a function of flow rate . . . . .
5.23. Horizontal profiles for three fingers at three different flow rates
5.24. The width of six fingers as a function of time . . . . . . . . . .
5.25. Illustration of the persistence of fingered flow paths . . . . . . .
5.26. Sketch of hydraulic states evaluation showing the persistence .
5.27. Photographs of the effects of high initial water content . . . . .
5.28. The longitudinal and transverse dynamics of water saturation .
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A.1. Digital images of water fingering . . . . . . . . . . . . . . . . . 129
A.2. Digital images of dye tracer infiltration . . . . . . . . . . . . . . 131
A.3. Digital images of six experiments with different flux . . . . . . 133
iv
List of Tables
3.1. Grain sizes and saturated hydraulic conductivity of sands . . .
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5.1. The fingers number, average width and average tip velocity . .
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v
List of Symbols and Abbreviations
This list contains the most important symbols and notations used. The dimension is indicated in brackets. The mathematical structure of symbols is
indicated by their typographical appearance:
a scalar
b vector, unit length vector
a, a
A tensor
sin standard function
Subscripts usually refer to a component of a vector (x, y, z, or 1, 2, 3) or to
the phase (air, water, matrix).
The arguments of functions are suppressed if they are clear from the context,
e.g., ∂∂x f instead of ∂∂x f (x). They are written, however, if the dependence on
an argument is emphasized, e.g., A(θ(z)) for a vertical profile that depends on
the water content.
The pressure is sometimes given in units of [cm] or [cmWC] which means the
pressure of a hanging water column with the same height.
Below L stands for length, T for time, and M for mass.
Sign Convention
The vertical (z) axis points downward, in the direction of the acceleration of
gravity. Its origin is typically chosen at the soil surface. Accordingly, z is called
the depth.
Lowercase Latin Symbols
g
jw
h
p
pa
acceleration of gravity [L T−2 ]
volume flux of water [L T−1 ]
matric head [L]
pressure [M L−1 T−2 ]
pressure in air phase [M L−1 T−2 ]
Symbols
pw
pressure in water phase [M L−1 T−2 ]
pwt
threshold water pressure [M L−1 T−2 ]
r
pore radius [L]
v
velocity [L T−1 ]
x
position [L]
Uppercase Latin Symbols
A
light absorption moduli
Bo
bond number
Ca
capillary number
F
force [M L T−2 ]
I
light intensity [counts]
I0
Id
measured light intensity without any porous material
(input light) [counts]
measured intensity of the dark image [counts]
K
hydraulic conductivity tensor [M −1 L3 T]
Ks
hydraulic conductivity at water saturation [L 3 T−1 ]
M
viscosity ratio
n
refractive indices
S
saturation [–]
τ
light transmission factor
V
volume [L3 ]
Lowercase Greek Symbols
ψg
gravitional potential [M L−1 T−2 ]
ψw
water potential [M L−1 T−2 ]
ψm
matric potential [M L−1 T−2 ]
ρ
mass density of soil sample [M L−3 ]
ρw
mass density of water [M L−3 ]
θ
volumetric fraction [–]
θw
volumetric water content of water phase [–]
θs
volumetric water content at saturation [–]
viii
θr
residual volumetric water content [–]
φ
porosity of porous media [–]
η
viscosity [M T−1 L−1 ]
σ
surface tension [M T−2 ]
µ
X-ray absorption coefficient, the subscripts will
indicate the material [L−1 ]
Mathematical Notation
d
dt
∂
∂t
total derivative with respect to time [T −1 ]
∇
partial derivative with respect to space [L −1 ]
partial derivative with respect to time [T −1 ]
Abbreviations
HSI
LTM
system to specify the color space Hue, Saturation,
Intensity
Light Transmission Method
PSF
Point Spread Function
RGB
system to specify the color space Red, Green, Blue
ix
1. Introduction
The study of water flow and transport in porous media has applications in
many disciplines such as fate and transport of chemicals, and plant-root activity, dissolved contaminants and non-aqueous phase liquids in soils and it is
of fundamental importance in hydrologic science. Of particular interest is the
unsaturated zone, the so-called vadose zone, found between the ground surface
and the groundwater table. The most important processes in this zone include evaporation, plant-soil interactions, water infiltration, and contaminant
transport. These processes control the formation of groundwater, the transport of chemicals, and various remediation technologies. Soil physicists and
hydrologists have given an enormous attention to the problem of infiltration
and redistribution of water in soil. Redistribution of water in soil following
infiltration is important because it determines the amount of water held near
the surface for subsequent use by plants.
Due to different types of geological media in the unsaturated zone (ranging
from relatively homogeneous sand to heterogeneous fractured media.), it is
generally accepted that water and solutes may flow through the unsaturated
zones via preferential paths until they reach groundwater. Preferential flow in
the vadose zone is the focusing of flow into narrow channels. The flow through
preferential paths is extremely important in agriculture, in the hydrological
processes of infiltration and in the transport of agrochemicals through the soil
profile. The important aspect of preferential flow is that water, as well as
pollutants dissolved in it, can infiltrate downward through such pathways much
faster than predicted by homogeneous flow with plane wetting fronts.
The term preferential flow actually describes three different processes: macropore, funnel and fingered flow. Macropore flow refers to water passing through
some preferred path such as decayed roots and earthworm paths. Funnel flow
occurs when the downward water flow gets funneled or diverted toward a preferred direction because of impermeable layers. Fingered flow occurs in a perfect homogeneous sandy porous media, where the wetting front breaks up like
a flame front. It refers to an instability of the interface between two immiscible
fluids when one invades the other. Early examples were air invading water and
water invading oil where the instability results from a difference in the viscosity.
1
1. Introduction
More recently, fingered flow in porous media was studied, both in petroleum
industry to better understand oil recovery and in environmental sciences as
an instance of preferential flow of water through soil. The latter is of obvious
interest for the general understanding of various hydrological aspects including
the quantity and quality of groundwater recharge or the efficiency of irrigation.
Fingered flow is also of fundamental interest, however, since it challenges our
understanding of multiphase flow in porous media.
Unstable flow of water during infiltration was first reported by Hill (1952).
The fingering phenomenon has been the subject of numerous experimental
studies. Hill and Parlange (1972) examined instability of the wetting front in
layered soils by direct visual observation of the flow field through transparent panels. Diment and Watson (1985) described laboratory results for front
infiltration into layered soils with different uniform initial moisture contents.
Glass et al. (1989a) and Liu et al. (1993) found that if the moisture content was
spatially variable as a result of an earlier infiltration with fingered flow, the previous finger paths were preserved during subsequent infiltration cycles because
of hysteresis in the soil moisture characteristic curve. Selker et al. (1992a) reported a study of matric potential and finger development as functions of time,
using a homogeneous sand. The occurrence and types of fingering flow in homogeneous soils, however, is sensitive to many factors such as initial water content,
size and distribution of soil particles, rainfall intensity, water repellency and so
on. This was studied by Diment and Watson (1985), Baker and Hillel (1990),
Selker et al. (1992b), Dekker and Ritsema (1994), Yao and Hendrickx (1996)
and DiCarlo (2004). It has also been observed a phenomenon where the saturation and pressure profiles within the fingers are inverted for certain initial and
boundary conditions (Geiger and Durnford 2000, DiCarlo 2004). This is often
called “saturation overshoot” as there exists a high water saturation followed
by region with a low water saturation directly behind the wetting front.
The theoretical analysis of wetting front instability has been studied for many
years. Most of the studies were based on the stability analysis of the classic governing equation for unsaturated flow through porous media (Hill and Parlange
1972, Raats 1973, Philip 1975a, Diment et al. 1982, Glass et al. 1989a). Many
mathematical models have been developed to attempt to model the saturation overshoot and it has been shown that this phenomenon simply cannot be
described by the Richards’ equation with standard non-monotonic pressuresaturation curves (Eliassi and Glass 2001, Egorov et al. 2003, DiCarlo 2005).
This is because of the parabolic nature of the Richards’ equation, which requires the saturation to move continuously from the initial low saturation to
the final saturation. Implicit in the Richards’ equation is the fact that one can
2
define a length scale where properties such as porosity, conductivity, and saturation can be considered continuous. More exactly, the continuum description
of the phenomena only fails at the wetting front where there is a jump from the
initial low saturation to a high saturation. Behind this jump, Richards’ equation has been shown to describe the flow behavior well Selker et al. (1992b).
Although, it has been argued that additional continuum terms are necessary,
in particular for when the local saturation changes quickly (Eliassi and Glass
2002; 2003, DiCarlo 2005). Some simulation and conceptual models have been
used for the mechanisms of finger formation, propagation and persistence in
unsaturated porous media and calculating the size and speed of the fingered
preferential flow (Wang et al. 1998, Jury et al. 2003). Therefore, the theory of
the fingered flow is still under development and not understood well.
Laboratory measurements of soil moisture are of great interest in soil science
and are limited by the ability to measure dependent variables in heterogeneous
and/or transient systems. A number of tools for nondestructive measurement
of water saturation in the laboratory have been developed. Advanced methods
for visualization and imaging of flow and transport in porous media are Magnetic Resonance Imaging (MRI), X-ray Computed Tomography (XCT), Neutron Computed Tomography (NCT), and Gamma-ray Computed Tomography
(GCT). Each of these methods for measuring saturation within laboratory test
cells is limited in spatial or temporal resolution or in size of the sample and
requires very specialized and expensive equipment. In this study, we applied
the very simple visible light transmission laboratory techniques of Glass et al.
(1989c) for mapping the water content at high spatial and temporal resolutions in a thin porous Hele-Shaw cell. The 3 mm thin and thus translucent
sand sample was placed in front of a constant light source and images were
recorded with a digital camera. The Light Transmission Method (LTM) captures the spatial resolution of the water content and can provide new insights
into rapidly changing, two-phase and three-phase flow systems.
This research presents the development and application of the LTM for twophase flow, aimed at investigating unstable fingered flow in a sand-air-water
system. We established a Hele-Shaw cell where a layer of fine-textured sand
was placed on top of a coarse-textured sand and studied experimentally the
flow paths and instabilities of gravity driven fingers through an initially dry
porous medium. We used the intensity of transmitted light through the HeleShaw cell to measure water saturation, since water saturation is a function of
light intensity.
The overall objectives of the present study has been to advance: 1) infiltration experiments in a large Hele-Shaw cell to observe fingering flow patterns
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1. Introduction
during redistribution, 2) our understanding of the special and interesting case of
water infiltration through porous media (fingered flow), 3) realistically describing this type of flow using the relevant physical processes, 4) understanding of
two crucial phenomena for fingering flow (i) the observed saturation overshoot
which initializes a finger and (ii) the hysteresis of the soil-water characteristic which stabilizes it by dramatically reducing lateral flow, and 5) application of a visualization and monitoring method (LTM) with high spatial and
temporal resolution to characterize multiphase and transient flow in porous
media.
Outline
The thesis is divided in the following order. In chapter Theoretical Background
(2), first an overview of the theoretical background and fundamental principles
of water flow in porous media is illuminated (sec. 2.1). Section 2.2 describes the
physical phenomena of viscous, gravity and capillary forces. Different visualization and imaging methods to visualize the flow and transport are described
in sec. 2.3. In section 2.4, the different terms of preferential flow and a history
of this type of flow is reviewed.
A description of the experimental setup, used materials and methods together with a detailed description of the calibration process using X-ray absorption are presented in chapter 3.
Readers who are familiar with the physics of image analysis may continue
directly to the next chapter, Image Processing (4). The purpose of this chapter
is to provide an accurate and complete presentation of the image processing
applied in this study. In section 4.2 an advanced and improved process for the
LTM is presented, where a deconvolution is applied on images to correct for
the scattering effect occurring during the transmission of light through porous
materials.
The results of our experiments and analysis are presented in the fifth chapter, Results and Discussion (5). All essential results with a complete discussion
of the experimental evidences are given here. Section 5.2 and 5.3 describe the
physical explanation of the finger initiation and the saturation overshoot phenomenon. After the qualitative description, quantitative analysis of measurements of finger behavior, including dynamics of water saturation and pressure
in flow fingers, dynamics and stabilization of fingers, finger width, and finger tip
velocity under different conditions are presented. Further experiments follow
in section 5.7 and 5.8 and 5.9.
Finally, there is a short closing chapter, Summary and Conclusions (6) where
the whole work is summarized.
4
2. Theoretical Background
Over the years, there has been considerable interest in the multiphase fluid
flow and transport phenomena in the unsaturated and saturated zones of the
subsurface environment. Many of the current pressing problems of single-phase
flow and very complex processes of multiphase flow involve flow and transport
issues including preferential flow development, groundwater contamination and
subsurface storage of materials. Unsaturated hydrology has gained increased
attention within the last decades. Different geological material is found in the
unsaturated zone, ranging from relatively homogeneous sand to heterogeneous
fractured media. Within all these different types of media preferential flow and
transport have been observed. Preferential flow paths significantly increase the
vertical water and solute velocity which in turn may lead to the rapid movement
of harmful chemicals and microbial organisms. These flow paths may arise as a
result of fluid instabilities, created by density or viscosity differences, or because
of capillary and gravity effects.
The following chapter will briefly introduce the reader to the different physical aspects of flow in porous media. In the first section we will have an introduction and review on the fundamental principles and scientific progress in
the case of water flow through saturated and unsaturated porous media. The
second section presents the physical phenomena of viscous, gravity and capillary effects to generation of stable and unstable flow conditions. Section 2.3
introduces the different methods for visualization and imaging of transient flow
phenomena. Section 2.4 describes three different types of preferential flow in
unsaturated porous media.
2.1. Water Flow through Porous Media
A porous medium consists of a matrix with a large amount of microscopic pores
and throats. The pores are typically connected such that there are narrow tubes
through which one or more fluids (e.g. water, oil and gas) can pass through. The
flow process in porous media is very complex. The reasons for the complexity
are many, but a major factor is the intricate nature of the pore spaces between
matrix material. Such a matrix and pore space configuration can be found in a
5
2. Theoretical Background
lot of materials, like soils, rocks, cemented sandstones, foam rubbers and many
others. The pores tend to have irregular surfaces, which is one of the reasons
why fluid flow through them is so complex.
The fundamental equation governing motion of fluid substances such as liquids and gases in a medium at the pore-scale is the well-known Navier-Stokes
equations. These equations established using the Newton’s law into the conservation law of linear momentum (acceleration). The external forces F ext
acting
i
on a moving fluid element of arbitrary volume 4V are pressure (F p ), dissipativeR viscosity (Fv ) and gravity (Fg ). The momentum mv of the fluid element
is [ V ρ dV ]v, where v is the velocity vector and ρ is the density of fluid. The
rate of change of the momentum is m dv/dt and by definition the mass of a
fluid element remains constant. Thus, the Navier-Stokes equations are a dynamical statement of the balance of forces acting at any given region of the
fluid. hence
dv X ext
=
Fi
m
dt
i
Z
}|
{ Z
dv z
= Fg + F p + F v =
[ρg − ∇p + η∇2 v]dV ,
(2.1)
ρ dV
dt
V
V
where p is the pressure and η is the dynamic viscosity of the fluid. Since V is
an arbitrary volume, this implies
ρ
dv
= ρg − ∇p + η∇2 v ,
dt
(2.2)
which is called the Navier-Stokes equation. Substituting the total derivative
dv
∂v
dt = ∂t + v · ∇v leads to the Navier-Stokes equation for an incompressible
Newtonian fluid
ρ
∂v
+ ρ[v · ∇]v = ρg − ∇p + η∇2 v .
∂t
(2.3)
Dividing it by the constant fluid density ρ, yields the kinematic form
∂v
1
η
+ (v · ∇)v = g − ∇p + ∇2 v .
∂t
ρ
ρ
(2.4)
This equation describes the classical hydrodynamics of momentum conservation for an incompressible Newtonian fluid in the void space of a medium on the
microscopic level with the known geometry of the solid phase and the accompanying surface properties (Roth 2006). This equation is a non-linear partial
6
2.1. Water Flow through Porous Media
differential equation and, in theory, it is too difficult to solve it analytically
because of the non-linear term (v · ∇)v. During the last years much effort was
made to find a general solution for the incompressible Navier-Stokes equations
with spatial assumptions and boundary conditions.
Henry Darcy (1856), a French scientist seeking a means to design sand filters
for drinking water, innovated the theoretical basis for quantifying the process
of water flow through soil. He investigated the uniform flow of water through
packed sand columns and introduced the well known Darcy Equation, which describes the apparent water velocity based on the discharge per unit area.
For a saturated medium the only resistivity to the fluid flow is given by
the solid matrix. The so-called saturated hydraulic conductivity is influenced
by the pore geometry and the fluid mobility. Conductivity of an unsaturated
porous medium is additionally influenced by the presence of other fluids as
they share the same pore space. In general, water flow through porous media
can be considered into two regimes; the saturated and the unsaturated zones
(Fig. 2.1).
2.1.1. Water flow through saturated porous media
The saturated zone is the region below an underlying water table. It is recharged
by the percolation of water through the unsaturated zone that reaches the saturated zone. When soil is saturated, all the pores are filled with water and
the water phase becomes continuous (Fig. 2.1). The most common example
for a saturated zone is the groundwater region in an aquifer. When soils are
saturated with isotropic homogeneous or anisotropic heterogeneous media, the
flow of water in this zone is well described by Darcy’s law. The flux density
can then easily be obtained if the conductivity of the soil is known and the
total potential or matric potential at the ends of the soil are specified. Darcy
developed the following relationship for isotropic media where the volumetric
water flux is proportional to the pressure gradient as
jw = −Ks ∇p ,
(2.5)
where jw = (jw,x , jw,y , jw,z ) is the volumetric water flux passing through the soil,
Ks the saturated hydraulic conductivity and ∇p the applied pressure gradient
vector as the driving force. Equation 2.5 is the so-called Darcy’s law which
describes the water movement in saturated materials.
This equation for the one-dimensional vertical flow can be simplified as
jw,z = −Ks,z
∂p
,
∂z
(2.6)
7
2. Theoretical Background
pores filled with
air and water
z
land surface
unsaturated or
vadose zone
pores filled with
water
PSfrag replacements
capillary fringe
water table
θw
saturated zone
(groundwater)
θw,sat
Figure 2.1.: Schematic cross-section of the saturated and unsaturated zones
with a vertical distribution of soil moisture. The unsaturated zone
is the part of the subsurface between the ground water table and
the land surface where water content (θ w ) is typically small and
the pores are partially filled with water and air. The saturated
zone is below the water table and the pores are completely filled
with water.
where z is the vertical distance, positive upwards, and the saturated hydraulic
conductivity (Ks,z ) depends on the fluid and solid material property.
When there are other driving forces (e.g. gravitation force) in addition to
the negative pressure gradient, they can be included by replacing ∂p/∂z with
the sum of all driving forces per unit volume. Darcy’s law may be written as
jw,z = −Ks,z
∂p
− ρw g
∂z
,
(2.7)
where ρw is the density of water and g the gravitational acceleration constant.
8
2.1. Water Flow through Porous Media
2.1.2. Water flow through unsaturated porous media
The unsaturated zone, so-called vadose zone, is the portion of the subsurface
from above the groundwater table to the land surface. In contrast to the
saturated zone, the pores are partially filled with water (Fig. 2.1). Its thickness
can range from 0, when a lake or marsh is at the surface, to hundreds of
meters, as is common in arid regions. It is a zone that to a large degree
controls the transmission of water to other substances, as well as to the land
surface, to water on the surface, and to the atmosphere. The most important
processes in the unsaturated zone include evaporation, plant-soil interactions,
water infiltration, and transport. These processes control, e.g., the formation of
groundwater, the transport of chemicals, and various remediation technologies.
Water flow and chemical transport in unsaturated soils is a complex process
and difficult to describe quantitatively. Since the hydraulic conductivities in
unsaturated soils depend upon the heterogeneous nature of soil which often
entail changes in the state and content of soil water during flow.
The flux density of water movement through saturated soil can be easily
obtained if the saturated hydraulic conductivity of the soil is known and if
the total considered soil volume potential is specified (Eq. 2.5). However, the
problem becomes more difficult when the soil is unsaturated, i.e. when air
is also present in the pores, and water infiltrates through partially saturated
pores to the groundwater.
Edgar Buckingham (1907) extended Darcy’s law (Eq. 2.5) to a more general
form of flux law to describe flow through unsaturated soils yielding
jw = −K(θw )∇ψw ,
(2.8)
where K(θw ) is the unsaturated hydraulic conductivity, which is sensitive and
highly non-linear dependent on volumetric water content, θ w , in contrast to
the constant Ks in Eq. 2.5. The water potential ψw is the sum of the matric
and gravitational potentials. This equation is called the Buckingham-Darcy
flux law.
The more general case of unsteady or transient water flow (varies in time as
well as space) in unsaturated soils is a highly dynamic phenomenon. At the
macroscopic scale, the dynamics of water flow in unsaturated porous media is
usually described by the highly non-linear Richards’ equation (Richards 1931).
This equation is represented quantitatively by a combination of Darcy’s law
(Eq. 2.9) and the conservation of mass law,
∂
θw + ∇ · j w = 0 .
∂t
(2.9)
9
2. Theoretical Background
Inserting Eq. 2.8 into the Eq. 2.9 yields a formula which leads to the differential
equation
i
h
∂
(2.10)
θw − ∇ · K(θw )∇ψw = 0 ,
∂t
which includes two state variables. The water potential can be written as
ψw = ψm − ρw gz where ψm is the matric potential that represents the pressure
jump across the water-air interface and is negative for unsaturated soil. Hence,
basically Richards’ equation describes the one-dimensional uniform (equilibrium) form of unsaturated water movement through a partially saturated rigid
porous medium and is written as
h
i
∂
θw − ∇ · K(θw )[∇ψm − ρw g] = 0 .
(2.11)
∂t
This equation was deduced with the assumptions that the effect of displaced
air during infiltration can be neglected, and that air pressure and temperature
are constant. Inserting θ(ψm ), a typically strongly hysteretic water retention
function, into Eq. 2.11 yields a form with one state variable as
h
i
∂
Cw (ψm )
ψw − ∇ · K(ψm )[∇ψm − ρw g] = 0
(2.12)
∂t
∂θ
with Cw (ψm ) :=
,
∂ψm
where Cw (ψm ) is the soil water capacity function. This equation is called
the potential or ψ-form of Richards’ equation. And when matric potential is
expressed in terms of matric head h = ψ m /ρw g, Eq. 2.12 changes into
h
i
∂
h − ∇ · K(h)[∇h + 1] = 0
(2.13)
Cw∗ (h)
∂t
∂θ
with Cw∗ (h) :=
,
∂h
(Roth 2006). Since Richard’s equation is a non-linear partial differential equation, it generally can not be solved directly but only by some approximations
and numerical methods.
A mathematical description of water flow and transport in the unsaturated
zone by Richards’ equation requires knowledge of two functional relationships
characterizing the soil: (1) the unsaturated hydraulic conductivity function,
K(θw ), and (2) the soil-water retention function, θ(ψ m ). The functional relationships have been parameterized by several researchers, where the most common applied model is by Brooks and Corey (1966) and van Genuchten (1980).
10
2.1. Water Flow through Porous Media
Simulation of water dynamics in this zone requires input data including the
model parameters, the geometry of the system, the boundary conditions and,
when simulating transient flow, initial conditions. A complete theoretical details can be found in several textbooks, e.g. (Bear 1972, Jury et al. 1991,
Kutı̀lek and Nielsen 1994, Roth 2006).
2.1.3. Infiltration with wetting front through porous media
Water infiltrating downward into dry unsaturated soils forms the air-water
interface which, in soil physics, is called the wetting front. A knowledge of the
structure of the wetting front is very important for predicting wetting front
propagation through soil, which in turn makes it possible to predict both deepsoil and ground water pollution. When water is poured in excess at the top of a
soil column, the water will flow into the soil due to the gravitational forces at a
rate depending on the water flux, water content in the column and the physical
properties of the soil. In the first stage the moisture profile gradually changes,
but later, in a column of uniform soil that is initially at a constant moisture
content, it maintains a fixed profile which moves downwards at constant speed.
Infiltration is defined as the initial process of water entering the soil resulting
from application at the soil surface. Downward infiltration into an initially
unsaturated soil generally occurs under the combined influence of capillary
and gravity gradients where capillary forces are dominated by gravity forces.
Three different zones can be assumed during infiltration into a soil column: a
saturated zone with constant water content θ s (except for entrapped air), a
transition zone where moisture content decreases rapidly from θ s to θi (initial
water content) with depth and the wetting front zone which is a zone of a very
steep moisture gradient (Fig. 2.2).
Green and Ampt (1911) neglected the evolution of the infiltration front during the initial stage of the infiltration. They quantitatively reproduced the
temporal change of the infiltration flux and of the depth profile of the water.
Green and Ampt assumed a homogeneous soil and an infiltration wetting front
that is sharp. This model is also called ”plug flow” or ”piston flow” model.
The water content profile is piston-type with a well-defined wetting front. The
piston-type model assumes the soil is saturated at a volumetric water content
of θs down to the wetting front. At the wetting front, the water content drops
abruptly to an antecedent value of θ i .
Raats (1973) studied the infiltration process with wetting front instability
for Green and Ampt flow, i.e. infiltration with a sharp front separating the
dry from the wetted soil. He noted that the wetting front would break up into
11
2. Theoretical Background
infiltrating water
soil
surface
θi
depth
depth
θ
jw
saturated zone
transition zone
θs
PSfrag replacements
wetting frot
moves down
into dry soil
dry soil
z
z
Figure 2.2.: Left: Moisture zones during infiltration and Right: Illustration
of transient one-dimensional infiltration following the Green-Ampt
approach with conceptualized water content profile, which demonstrated the sharp wetting front during water infiltration through
soils.
fingers if the velocity of the front increased with depth. He also found that
infiltration of water into a soil without the possibility for air to escape would
lead to an unstable front with downward moving fingers of water inter-spaced
with fingers of air moving up. For different cases he derived criteria that must
be satisfied for the wetting front to become unstable.
2.1.4. Hysteresis of soil water-retention function
The most basic information on water in an unsaturated medium is water content or wetness. It is defined as the volume of water per bulk volume of the
medium. Water is held in an unsaturated medium by forces whose effect is
expressed in terms of the energy state or pressure of the water, the so-called
matric pressure or matric potential. It is the pressure of water in a pore of
an unsaturated medium relative to the air pressure (ψ m = pw -pa ), where pw
is the water pressure and pa is the air pressure. In an unsaturated medium,
the water is generally at lower pressure than the air, so the matric potential is negative. Higher water content is related with higher matric poten-
12
−15
0
1
2
il
so
−10
d
re
tu
il
ed so
xtur
se te
ex
et
−5
3
4
matric potential log(−ψm [cmWC])
fin
co a r
conductivity log(K [cm h−1 ])
0
4
fine
3
tex
tur
coar
2
se te
xtur
ed
1
ed
soi
l
soil
0
0
0.20
0.40
matric potential ψm [-cm]
2.1. Water Flow through Porous Media
PSfrag replacements
m
ain
sca
ma
in
ing
dr
yi
nn
ng
cu
cu
rv
rve
s
we
e
ttin
gc
urv
e
θs
θr
matric potential log(−ψm [cmWC])
water content θw [–]
water content θw [–]
a
b
c
Figure 2.3.: a,b) Typical hydraulic conductivity function and soil-water characteristic curves for two different classes of the fine-textured soil,
e.g. loamy soil, (solid line) and a coarse-textured soil, e.g. sandy
soil, (dashed line). The thin dashed line represents a sand with
larger grains and thus, larger pores. c) The main soil-water characteristic curves for wetting and drying (solid lines), and three sets
of scanning curves (dashed lines) in hysteresis loop.
tial and as matric potential decreases the water content decreases, but in a
way that is non-linear and hysteretic. Figure 2.3a,b shows typical shapes of
conductivity-pressure and pressure-saturation relationships of a coarse textured
porous medium , e.g. sandy soil, and a fine textured porous medium, e.g. loamy
soil, for a drainage cycle. As the soil is finer dispersed the soil-water interaction
forces are greater. This means that in the same humidity conditions, clays have
considerably higher values of soil suction in comparison to sands. The curve
showing the relationship between water content and matric potential, θ(ψ m ),
for a soil is called soil-water characteristic curve or retention curve, which is
characteristic for a porous medium that depends on the nature of its pores.
This relation strongly influences the movement of water and other substances
in unsaturated media.
There are two branches defining the soil-water characteristic curve. One
is related to wetting whereas the other to drying. Spatial variations in soil
properties dominate over hysteretic effects. In order to describe the hysteretic
behavior of a particular soil, many wetting and drying experiments have been
performed during the last years (Topp and Miller 1966, Parlange 1976b, Jaynes
1992, Lehmann et al. 1998), because the water-retention function may change
with each drying and wetting process. Thus, a theory is needed to estimate
13
2. Theoretical Background
the water-retention function for any drying and wetting loop based on the envelope of main drying and wetting curves. The main drying curve (drainage
equilibrium curve) describes the drying from the highest reproducible saturation degree (θs ), which is usually not complete due to entrapped air, to the
residual water saturation (θr ). The main wetting curve (imbibition equilibrium
curve) describes the wetting from the residual water content to the highest saturation degree. Starting from a boundary wetting or drying curve, a sequence
of wetting and drying cycles can be expressed by scanning curves in primary,
secondary or higher order. According to the hysteresis function, the drying
scanning curves are scaled from the main drying, and wetting scanning curves
from the main wetting curves (Fig. 2.3c). The hysteresis effect is caused by
different radii of the controlling pores and different contact angles for wetting
and drying processes.
2.1.5. Multiphase flow in porous media
Darcy’s equation is an empirical macroscopic equation based on average quantities and derived for one-dimensional single-phase flow. Oftentimes, fluid flow
in porous media involves more than one fluid and the difficulties arise when
describing two or multi-phase flow in porous media. In case of a single-phase
system the pore space of the porous medium is filled by a single fluid (e.g. water)
and in a multiphase system the pore space is filled by two or more fluids (e.g. air,
water, oil, etc.). Multiphase flow is a very complex physical phenomenon,
where many flow types can occur (e.g. gas/solid, gas/liquid, liquid/liquid and
solid/liquid) and with each flow type, several possible flow regimes can exist.
Two general types of fluid displacement are possible when two or more fluids
in motion occupy a porous medium (Bear 1972):
Miscible displacement: In this process, a fluid is displaced in a porous
medium by another fluid that is miscible with the first one. The two fluids
are completely soluble in each other. The interfacial tension between the
two fluids is zero and the fluids dissolve in each other. Therefore, in
this type of displacement, there is no capillary effect, but instead there
is dispersive mixing between the two fluids which can play an analogous
role. In miscible displacement, when two fluids are in contact with each
other, a transition zone due to hydrodynamic dispersion is immediately
created. The composition of the fluids varies from one to the other fluid
across the zone. This makes the miscible displacement a very efficient
14
2.1. Water Flow through Porous Media
recovery procedure, where the elimination of capillary forces might lead
to a total recovery of the displaced phase.
Immiscible displacement: This process is a transient process where
one fluid displaces another fluid from a porous medium with no mixing at
the interface. It is a simultaneous flow of two or more immiscible fluids
or phases in the porous medium. The interfacial tension between the two
fluids is non-zero and a distinct fluid-fluid interface separates the fluids
within each pore. The flow of immiscible fluids in a porous medium can
be conveniently subdivided into two types: steady-state, where all the
macroscopic properties of the system are time independent at all points,
or unsteady-state where the fluid and flow properties change with time.
In equilibrium (steady-state) flow of immiscible fluids, the saturation of
the medium with respect to all fluids contained in the system is constant
at all points. Therefore, in steady-state flow there is no displacement of
any fluid by any of the other fluids in the pores. This means each fluid
flows through its own path without affecting the flow of the other fluids.
However, in unsteady-state flow, the saturation at a given point in the
system is changing with time.
On the microscopic level the sharp interfaces between fluid phases give rise to
a capillary force that plays an important role in multiphase flows. When two
immiscible fluids are in contact within the interstices of a porous medium, a
discontinuity in pressure across the interface separates them. Its magnitude
depends on the interface curvature at that point. The difference between the
phase pressure of the two fluids is called capillary pressure. It is an important
parameter to quantify multiphase flow in porous media. At equilibrium, this
pressure (peq
c ) is the difference between the pressure in the non-wetting and the
wetting phase:
peq
(2.14)
c = pnw − pw ,
where nw and w label the non-wetting and wetting phases, respectively. These
differences occur because, in a confined geometry, the contact angles cause a
curvature of the fluid interface and thus a capillary pressure difference between
the two phases. At equilibrium, this pressure difference can be given through
the Young-Laplace equation and it depends on the interfacial tension between
the fluids (σ12 ) and the principal radii of curvature of the surface, r 1 and r2 ,
as shown:
1
2σ12
1
+
(2.15)
= ∗ ,
4p12 := p2 − p1 = σ12
r1 r2
r
15
2. Theoretical Background
where r ∗ is the mean radius of curvature (2/r ∗ =1/r1 +1/r2 ). The capillary
pressure is thus a measure of the tendency of a porous medium to suck in
the wetting fluid phase or to repel the non-wetting phase. The direction and
magnitude of this pressure depends on the interface geometry at any particular
instant in time. The interface is dynamically deforming or has arrived at a
stable equilibrium shape.
When two fluids are simultaneously present, the ability of one fluid to flow
depends on the local configuration of the other. Depending on the wetting
properties of the fluids, there are two basic terms of displacement in two-phase
flow in porous media to describe the fluid displacements: drainage and imbibition. The two processes are due to different mechanisms and usually result
in completely different pressure and saturation profiles. In the imbibition (displacement of non-wetting fluid by the wetting fluid or increase in wetting phase
saturation) the wetting fluid invades. The capillary pressure difference creates
a driving force such that the invading fluid spontaneously imbibes until it is
balanced by hydrostatic pressure. For drainage (displacement of wetting fluid
by the non-wetting invading fluid or reduction of wetting phase saturation),
the pressure at an invading interface must rise above the pressure in the adjacent displaced fluid by an amount that allows the interface to deform to the
curvature of the local pore.
The dynamic saturation during a displacement process and the final average
saturation that is left behind after displacement, are important parameters. In
many processes, they are important either directly or because of their effect on
other quantities. Saturation is affected by how uniformly a displacement front
moves through the medium and the amount of displaced fluid that is trapped
behind the front. The displacement pattern depends on the morphology of the
medium, mobility ratio, wettability, and the balance between gravity, capillary
and viscous forces. For any medium and specific fluids, the gravity, capillary
and viscous forces alone dictate the displacement.
2.2. Physical Phenomena of Viscous, Gravity and
Capillary Forces
Depending on the conditions of drainage displacement, the process where a
non-wetting fluid displaces a wetting fluid, different flow regimes may be observed. This process provides pattern formations between the interface of the
two fluids and the different structures obtained can be divided into two main
flow regimes of stable and unstable displacement. Before discussion on the
16
2.2. Physical Phenomena of Viscous, Gravity and Capillary Forces
properties of these two regimes, to quantify the relative importance of various
driving forces in flow domain, we employ three dimensionless numbers using
three type of forces in two-phase fluid displacements: viscous forces in the displacing fluid (Fv = η2 [v/r]r 2 ), capillary forces due to the interface (F c = σr)
between them and gravity forces (Fg = ρgr 3 ), where η1 and η2 refer to the
viscosity of the displaced and displacing fluid, respectively, v is the velocity, σ
is the surface tension, ρ is the fluid density, r is the microscopic length scale
of grain, and g is the gravitational acceleration constant. This leads to three
dimensionless numbers that characterize the different properties of immiscible
displacement in porous media: the capillary number (Ca), the bond number
(Bo), and the viscosity ratio (M). The capillary number describes the relative
magnitude of viscosity over capillary forces, the bond number describes the
relative magnitude of gravity over capillary forces and the viscosity ratio gives
the ratio of the two viscosities, as
Ca =
Bo =
M =
η2 v
Fv (viscous forces)
=
Fc (capillary forces)
σ
ρ g r2
Fg (gravity forces)
=
Fc (capillary forces)
σ
η2
,
η1
(2.16)
(2.17)
(2.18)
Friedman (1999) evaluated the magnitude of the various forces in soils for
a representative case of a water-air interface moving. Inserting the values,
when water is displacing air from a representative pore of radius 10 µm at
a high velocity of 0.1 cm s−1 (using the rounded values of g = 103 cm s−2 ,
ρ = 1 g cm−3 , σ=102 dynea cm−1 and η = 10−2 g cm−1 s−1 ), into Eq. 2.16 and
2.17 yield Ca ≈ 10−5 and Bo ≈ 10−5 . Hence, the interfacial forces are some
five orders of magnitude larger than the viscous forces.
2.2.1. Stable and unstable fluid displacement
The stability of an invading front is controlled by the interplay between viscous,
gravity and capillary forces that can act as either stabilizing or destabilizing
displacement.
Stable displacement occurs when the front between the two fluids is flat,
or when the front width (distance between the most and least advanced portion
a
dyne is the unit of force in the cgs system and equals 10−5 N.
17
2. Theoretical Background
of the front) is constant with time. Stable displacement takes place if viscosity
forces are dominated, and capillary effects and pressure drop in the displaced
fluid are negligible (M < 1 and Bo > 1). Here a flat front develops with only
a few irregularities of a few pores. Now, the question is that what conditions
give rise to transitions from flat to unstable displacement fronts within a porous
media?
Unstable displacement can arise as a result of fluid instabilities created by
density or viscosity differences between the two immiscible fluids. Destabilizing influences such as large viscosity ratio, action of gravity, or rapid drainage
velocities can cause the front width to grow without bound, resulting in instability and onset of fingering. Phase instability displacement or ”fingering”
separate from the single phase concept of flow channelization, occurs where
any of capillary, gravitational and viscous forces are dominated. For unstable
displacement, three major flow regimes have been identified: viscous fingering,
capillary fingering and gravity-driven instability. The properties of the different
regimes are briefly discussed below.
Viscous fingering occurs when a less viscous fluid displaces a more viscous one; a planar boundary between the fluids is unstable against small
perturbations, and in the course of time the interface adopts a fingered
configuration. The process is obtained by injecting a low viscosity fluid
into a medium of high viscosity fluid with a high injection rate. The
instability arises from a pressure gradient advancing the less viscous fluid
against the more viscous one. The action of such driving-force mechanisms leads to the viscous fingering patterns. Viscous fingering results
when viscous forces are significant relative to capillary forces (Ca 1).
Capillary fingering is executed by injecting the invading fluid at very
low injection rate. The low injection rate causes the viscous forces to
vanish and capillary forces dominate the viscosity (Ca 1 and Bo 1)
and local variations in pore throat size govern the flow path. In this case,
the non-wetting phase advancing on several pore throats will invade the
largest pore first where the critical pressure is lowest. Once that pore
has been drained, a new set of pore throats presents itself to the invading
front, and the process repeats. Consequently, the principal force is due to
capillary forces of the interface between the displacing and the displaced
phases.
18
2.3. Visualization of Flow and Transport in Porous Media
Gravity-driven instability occurs when a denser fluid displaces a lighter
one from above with the principal forces due to gravity, and viscous forces
do not act to fully stabilize the displacement front. With Bo 1, the
instability arises by gravity acts on the fluid phase and serves as a stabilizing force by flattening the interface and eliminating height differences
caused by viscous instability or capillary fluctuations. In this case, gravity
fingers propagate vertically in the medium and continue to grow rapidly
because of the gravitational instability of the process. Thus, fingering
due to gravity may take place when water infiltrates into unsaturated
porous media.
In the present study, we used experiments to explore the formation of gravitydriven fingers during the infiltration of water into non-horizontal and initially
dry porous sand column by influence of gravity and capillarity.
In a recent study, Méheust et al. (2002) used 2D glass bead packs to study
drainage morphology for a range of Ca and Bo values. They defined a generalized bond number Bo∗ = Bo − Ca and showed that experimental drainage
front morphology was a function of Bo ∗ . For Bo∗ > 0, indicating Bo > Ca, the
predominance of gravitational forces resulted in stable displacement (flattened
fronts). For Bo∗ < 0 or Ca > Bo, the diminishing stabilizing force produced
fronts that became progressively more unstable, indicated by increasing front
width as Bo∗ became more negative. At slightly negative Bo ∗ values (in the
range of 0 < Bo∗ <- 0.05), capillary forces became significant resulting in behavior characteristic of capillary fingering. As Bo ∗ decreased further, viscous forces
became significant and viscous fingering was observed for values of Bo ∗ <- 0.08
(approximate range).
2.3. Visualization of Flow and Transport in Porous
Media
The spatial and temporal variability of water and solute fluxes in soil makes it
difficult to obtain a quantitative understanding of the dynamics of water and
solute inside the sample without destroying it. During the last half century ,
soil science methods to study the hydraulic behavior within a sample of porous
media without destruction of the sample became more attractive. This section
reflects the state of art of non-destructive measurements for visualization and
imaging as a means for quantitative analysis of flow and transport phenomena
in porous media. The analysis of the sequential dynamic events in two and
19
2. Theoretical Background
three-dimensional space involving one or more temporal variables in single or
multiphase systems is improving by outstanding progress in computer science,
technology and image processing. Attention is focused on visualization and
imaging of dynamic transport phenomena in animate and inanimate systems
(Sideman 2002). Of particular interest is the interplay between reality and
imaging as well as the application of the quantitative information contained
within the visual images of real systems. Visualization is the art of transforming
normally invisible phenomena into visible measurable events. Flow dynamic
visualization is still attracting the scientific world, probably since it can be
tackled both experimentally and theoretically. The need for better methods
to image transient flows in porous media has been discussed by a number of
authors (Darnault 2001, DiCarlo et al. 1997).
At present, very few methods exist for rapid, non-destructive and accurate
measurements of fluid contents in three-phase, sand or Non-Aqueous Phase
Liquid (NAPL)-air-water systems in transient flow fields. These methods are
time consuming and can be used only near steady state flow conditions. These
non-invasive imaging techniques that have the potential to provide the required information on the internal structure and/or moisture distribution in
soils are:
Neutron Computed Tomography (NCT) is a real-time, non-invasive
imaging method that can be used for quantitative imaging of hydrologic phenomena at video frame rates, with great sensitivity to variations
in moisture content, sub-millimeter resolution and image fields greater
than a few hundred square centimeters. This imaging technique can
provide two-dimensional images in transmission. The measurements of
moisture content within a flow field can be obtained by measuring the
intensity change of a thermal neutron beam as it passed through the experiment chamber and interacts with the nuclei of atoms (Deinert et al.
2004, Menon et al. 2006).
X-ray Computed Tomography (XCT) or synchrotron X-rays absorption (film-based radiography) imaging technique allows accurate and fast
measurements of fluid contents in transient flow fields, in any soil type.
However, they are limited to measurements of a few square millimeters
of the flow field at one time (Liu et al. 1993, McBride and Miller 1994,
DiCarlo et al. 1997, Bayer et al. 2004). The technique is based on the
attenuation of an X-ray beam by a substance. The attenuation pattern
of the beam is recorded as two-dimensional image in transmission, such
20
2.3. Visualization of Flow and Transport in Porous Media
as in traditional medical X-ray imaging. This technique is also used to
perform three-dimensional tomography images.
Magnetic Resonance Imaging (MRI) has also found a well established tool applied in studying fluids in porous media. It also provide
information on water distribution and also on other pore scale characteristics such as the specific surface, diffusion coefficient, or motion velocity.
In this process, the volumetric water content at one voxel is directly
proportional to the number of resonating spinning or magnetic properties of the water protons in the voxel. This magnetization at each voxel
of matrix material is measured with nuclear magnetic resonance. This
technique can also provide two and three-dimensional images to obtain
information on soil properties and flow at high spatial resolution. Using
this method, a quantitative measurement is not always possible, especially at low saturation levels (Johns and Gladden 1998, Deurer et al.
2002, Votrubovà et al. 2003).
Gamma-ray Computed Tomography (GCT) can also be used as
a tool to investigate and evaluate possible modifications in soil structure and to analyze the variations of soil physical properties such as
soil density, porosity, pore size distribution and bulk density within the
soil sample volume. This gamma radiation technique uses a radioactive
gamma ray source, e.g. 241 Am (Ferrand et al. 1986, Oostrom et al. 1995,
Pires et al. 2005).
Disadvantages of these methods are that they cannot measure transient flow
phenomena with high temporal resolution and that their use involve radiation
hazard.
Light Transmission Method (LTM) is the only method that does
not use radiation and is the best available method for rapid and accurate
measurements of transient flow in porous media on a sub-second time
scale. It has been applied since 50 years in Hele-Shaw cells with smooth
walls (Saffman and Taylor 1958, Chouke et al. 1959). The LTM is a nondestructive method that allows visualization and measurement of fluid
content in transient water flow occurring in sandy porous media. The
advantages of the LTM are that it does not involve radiation and that it
is able to visualize fluid content changes over the whole flow field with
a time resolution of tenths of second (Tidwell and Glass 1994, Darnault
2001, Mortensen et al. 2001). This method is useful for transparent and
21
2. Theoretical Background
thin slices of porous materials, e.g. coarse sand and crushed glass. The
LTM uses the hue or intensity of light transmitted through a slab chamber to measure fluid content, since total liquid content is a function of hue
or light intensity. The LTM captures the spatial resolution of the fluid
contents and can provide new insights into rapidly changing, two-phase
and three-phase flow systems. Application of the LTM as a visualization
technique for environmental and physical phenomena, groundwater remediation by surfactants as well as visualization of model cluster growth
and fractal dimensions is noted by Darnault et al. (2002). Stöhr (2003)
used the LTM methods to study flow and transport in porous media
particularly solids, liquids and fluorescent dyes and applied this optical
method for the highly precise matching of refractive indices using planar
laser-induced fluorescence with high temporal and spatial resolution.
Therefore, there is a method, LTM, that allows full field moisture content
visualization in two-phase systems with high spatial and temporal resolution.
In this research we adapt the LTM to measure the dynamics of water saturation
in a sand-air-water system.
2.4. Preferential Flow Phenomena
Two different types of immiscible flow are found in porous media: (1) transient
displacement classified as either drainage or imbibition, and (2) steady state
flow (Dullien 1992). During transient displacement complex patterns can be
found where preferential flow paths develop. From a phenomenological perspective, preferential flow refers to nonuniform and rapid movement of water
and solutes into narrow channels of some subregions of the soil. There, water is
gated by less permeable layers and funneled through more permeable regions.
For inclined and cross-bedded layers this may lead to a strong focusing of the
flow to narrow channels (Stagnitti et al. 1994, Roth 2006). The soil profile can
be distinguished in a distribution layer near the soil surface and a conveyance
(preferential) zone below it (Steenhuis et al. 1994, Ritsema and Dekker 1995).
The distribution zone acts as a linear reservoir resulting in an exponential loss
of solutes from this zone and it funnels water and solutes in flow paths within
the conveyance zone. By Lawes et al. (1882) observations, in the conveyance
zone the transport of water and solute collected in drains can be separated into
two constituents: preferential flow (direct drainage) and matrix flow (general
drainage). As shown schematically in Figure. 2.4, rainfall enters a mixing zone
where it mixes with the water and solutes in this layer. Water and solutes from
22
2.4. Preferential Flow Phenomena
distribution layer
matrix with
preferential paths
(a)
distribution layer
matrix without
preferential paths
(b)
Figure 2.4.: Schematic diagram of flow processes in the mixing layer: wetting
front advances in matrix with preferential flow paths (a) and matrix without preferential flow paths (b). The arrows represent the
infiltration flux. (Redrawn after Cornell-Website)
the mixing layer are transported downwards as matrix or preferential flow and
the latter passes with little modification through the soil channels to the deeper
soil or groundwater (Steenhuis et al. 1994).
Preferential flow, the rapid non-equilibrium transport of soil solution is more
often the rule than the exception. Due to its rapid movement, preferential flow
allows much faster contaminant transport. This can provide significant consequences for groundwater quality and has direct impacts on drinking water and
human health, animal waste management, nutrient and pesticide management,
as well as watershed management.
Matrix flow is a relatively slow and even movement of water and solutes
through the soil while sampling all pore spaces, obeying the convective-dispersion
theory which assumes that water follows an average flow path through soil.
The chemical composition of preferential flow regions reflects the concentration of water near the surface, while matrix flow represents the concentration
23
2. Theoretical Background
of water around the drain. Thus, when a salt is equally distributed throughout
the soil, the salt content of matrix flow is higher than that of the water in
preferential flow paths, which is characterized by the rainfall composition. The
opposite is true (preferential flow has a higher solute content than the matrix
flow), when a fertilizer or tracer were recently applied surface. Consequently,
the solute concentration of drainage water depends on the ratio of water in
preferential and matrix flow paths (Steenhuis et al. 1994; 2001).
The relative importance of the two forms of percolation, preferential and
matrix flow depends on the soil type and rainfall intensity. For example, wellstructured soils consisting of clay and loam mixes typically experience low
permeability rates. In such soils, less than 1% of the pore-volume consists of
cracks and subsurface channels. However, during rain events, water infiltrating
from the soil surface, often flows through these channels in reference to the surrounding soil-matrix, whose small pores are penetrated comparatively slowly.
Even through these channels make up a relatively small percentage of the total pore volume, they may be responsible for the distribution of moisture and
solute transport after an infiltration event. Preferential flow may be initiated
well below soil-water saturation.
2.4.1. The history of preferential flow and why it is important
Natural soils are mostly non-uniform and often strongly heterogenous. Soil
properties are strongly dependent on structure, which determines not only
workability and water retention, but also strongly influences soil/plant interaction like root growth, transport of water, air and chemicals and thus can be
considered as important parameter for soil quality. Furthermore, soil structure
is a main factor determining preferential water and chemical fluxes in soils. The
term preferential flow refers to several phenomena which have in common the
non-uniform and often rapid movement of water through soils which does not
tend to move as a horizontal wetting front. This rapid movement bypasses the
bulk of the soil matrix, reducing the potential for pollutant adsorption and/or
degradation and increasing the threat of groundwater and surface water contamination.
The concept of bypass flow through macropores and preferred routes and
the fact that they will permit rapid movement of water and chemicals dates
back to Schumacher (1864) who stated that, the permeability of a soil during
infiltration is mainly controlled by big pores, in which water is not held under
the influence of capillary forces. Intense studies in the last few decades have
also indicated that the preferential flow mechanism is the most significant flow
24
2.4. Preferential Flow Phenomena
process influencing the pollution potential and transport of a given chemical
in the porous media. Scientists and engineers have devised measurement and
modeling strategies in order to characterize and quantify the role of preferential
flow processes in water and pollutant transport.
Preferential flow was investigated by Lawes et al. (1882) in field drainage
experiments. They found that a large portion of water applied to a soil only
slightly interacted with that already present in the root zone. As scientists were
forced to explain groundwater contamination problems, identified during the
mid 20th century, the long-neglected observation of preferential flow phenomena was eventually brought up again. Various pesticides, which were believed
to be quickly degraded or strongly sorbed near to where they were applied,
were unexpectedly found in the groundwater much more quickly and in higher
concentrations than experts would have predicted.
During the last two decades, high concentrations of pesticides in tile lines
or shallow groundwater were found in many studies shortly after application.
Contamination of groundwater by agricultural chemicals is becoming a serious
threat. Modern agriculture is based on a broad range of fertilizers and pesticides to assure reliable crop yields. A long-term hazard is involved that the
chemicals get into groundwater, they may remain there for hundreds of years.
Although integrated pest management may reduce the amount of chemicals
needed, a total ban is currently not feasible.
Preferential flow in soil may occur by three possible mechanisms: flow through
large and continuous voids, so-called macropore flow, sub-surface layering in
textural interfaces (funnel flow) and instabilities that lead to fingered flow or
wetting front instability. All of these terms refer to the fact that water tends
to flow only through a portion (sometimes a very small portion) of the total
soil volume. Bowman et al. (1994) found in one experiment that as much as
80% of the water percolating through the soil profile actually passes through as
little as 20% of the cross-sectional area of the profile. The three types of flow
are described briefly below, along with the respective soil textures.
2.4.2. Macropore flow
Macropore flow is the result of a variety of soil-forming factors such
as flow through non-capillary cracks or channels (Cornell-Website).
Macropores and sub-surface channels can result from either biological activity (e.g., root channels, worm-holes, etc.), geological forces (e.g., subsurface
erosion, desiccation and synaeresis cracks and fractures) or agrotechnical prac-
25
2. Theoretical Background
Macropore Flow
open macropore
ponded water
soil surface
blocked
macropore
Figure 2.5.: Schematic diagram of water flow through macropores. Water can
flow only through those macropores that are open to the soil surface. Blocked macropores remain dry.
tices (e.g., plowing, bores and wells). Surface cracks and channels that lead
to a bypass the root zone are also responsible for rapid transport of moisture and chemicals through the unsaturated zone (Beven and Germann 1982,
Larsson and Jarvish 1999, Langmaack et al. 1999).
Macropores do not begin to conduct water until the soil near the macropore
becomes saturated, and there is ponded water on the soil surface that can
flow into open or ”surface vented” macropores. However, macropores that
are blocked by debris or cut off by tillage tend to remain dry and non waterconducting (Fig. 2.5).
These macropores are the result of a variety of soil forming factors such as
flow through non-capillary cracks or channels within a profile. In clay and loam
soils, for example, areas of relatively low permeability are riddled with channels
consisting of cracks partially filled with sand and small stones, as well as passages formed by roots and earthworms as shown in Figure 2.6. When it rains,
water infiltrating the ground is more follow these channels than the surrounding
matrix, whose small pores are penetrated comparatively slowly. Flow through
the uppermost soil horizon (typically the plow layer for current or former agricultural soils) is often fairly uniform. This layer acts as a ”distribution layer”,
distributing the flow to macropores in the subsoil conveyance zone. Water and
chemicals that travel in macropores often bypass the bulk of the soil matrix.
26
2.4. Preferential Flow Phenomena
Macropore Flow
Distribution layer
Conveyance zone
Figure 2.6.: Photographs of macropore flow paths in structured soil dyed by
a blue tracer applied at the surface. Below the distribution layer,
the dye brached into many fine channels, following structural cracks
and continuous earthworm channels, indicating the presence of significant preferential flow paths through macropores. The picture
on the right shows an earth worm as pore generator in structured
soil (photographs from Cornell-Website).
Flow in macropores and channels can occur with little or no interaction with
the surrounding soil-matrix (Pivetz and Steenhuis 1995).
Macropore flow has been identified to be the culprit of contaminations of
groundwater with pesticides and other agrochemicals which are typically decomposed or at least retained in the biologically active top soil layers. Corresponding observations in drainage water, the first indirect reports on macropore
flow published by Lawes et al. (1882). Despite this outstanding study, the understanding of the phenomena remain qualitative and heuristic representations
mark the state of the art.
2.4.3. Funnel flow
Funnel flow arises when sloping geological layers cause pore water to
flow laterally, accumulating at a low region. If the underlying region
is coarser, finger flow may also occur (Cornell-Website).
27
2. Theoretical Background
Funnel Flow
coarse la fine
layer
yer
fine layer
Figure 2.7.: Schematic diagram of the funnel effect in sandy soil. A fast-moving
spout forms beneath the lower regions of an inclined layer.
The way soil is layered makes a big difference in how water and solutes find
paths down to the groundwater. The way these flow paths merge depends on
inhomogeneities in the soil. Sloping structural interfaces have a considerable
effect on the degree of merging and rate of flow (Walter et al. 2000).
Funneled flow is a unique category of flow phenomena referring to the situation in which a capillary barrier develops above a coarse layer which underlies
a relatively fine soil (Walter et al. 2000, Kung 1990b). At low flow rates, when
the matric potential at the textural interface is so low that water cannot enter
into the coarse, underlying soil, the capillary barrier effectively restricts vertical water flux, forcing the water to move laterally along the bedding interface
(Fig. 2.7). Since moisture is forced to take this narrow route, the concentration
of moisture and solutes may once again bypass much of the soil matrix and be
directed to the groundwater.
The diversion of flow caused by layering is significant because even in areas
where groundwater monitoring is conducted routinely; pollution may be missed
entirely if the detecting is placed in the wrong location. Therefore, in order to
prevent groundwater contamination, it is desirable to know where the layers
and funnels are.
Sloping interfaces of texturally different layers may act as funnels and concentrate flow. This phenomenon can also be observed in field experiments
using dye tracers (Fig. 2.8). Textural interfaces and subsurface layers may
28
2.4. Preferential Flow Phenomena
Funnel Flow
a
b
c
Figure 2.8.: Photographs of funnel flow at field site runs with dye tracers to
observe the funnel effect in sandy soil: a) Finger funnel flow with
dye penetrating into the fine sand appears to have moved freely
and directly into the coarse sand below. b) Textural interfaces
and sub-surface layers may cause moisture and solutes to preferentially flow in a prescribed direction. c) Blue dye flowing vertically
(unsaturated flow) moves laterally when it encounters a sloping
coarse-textured lens (photographs from Cornell-Website).
cause moisture and solutes to preferentially flow in a prescribed direction. The
funnel flow phenomenon was coined as an example of preferential flow by Kung
(1990a;b).
2.4.4. Fingering flow
Instability in the wetting front leads water to find its way down
through a number of channels, so-called fingers (Cornell-Website).
Fingering (flow instability) is one form of preferential flow resulting from instability of infiltration through the unsaturated coarse soil. In general, fingering
29
2. Theoretical Background
Fingering Flow
j w Ks1
fine
layer
Ks1
layer
coarse
K
s2
Ks1
K
s2
ponded water
fingers
Figure 2.9.: Schematic diagram of fingering flow. Wetting front instability, or
gravity driven fingering, can occur during vertical infiltration in
homogeneous coarse grained materials when the infiltration rate is
below the saturated hydraulic conductivity.
occurs during water infiltration into a dry soil profile with a finer textured soil
layer rests on top of a coarse textured layer. The water infiltrates into regions where the saturated hydraulic conductivity increases in the direction of
positive flux (Fig. 2.9). When the wetting front in the finer textured medium
contacts the boundary with the coarser textured medium there is often insufficient absorbency or capillarity in the coarser medium to pull the water across
the boundary and into the coarser soil. Then, fingered flow can emerge as a
consequence of the development of an unstable wetting front in the coarsegrained sands, when the infiltration flux is less than the saturated hydraulic
conductivity of sand and gravitational influences on the imbibing solution must
dominate the forces of capillary.
Unstable flow of water during infiltration was first reported by Hill (1952).
In 1972 David Hill and Parlange (Hill and Parlange 1972) documented, for
the first time, preferential flow in homogeneous soil at low infiltration rates.
Consequently, flow instabilities have been studied extensively, both experimentally (Hill and Parlange 1972, Diment and Watson 1985, Glass et al. 1989a;b;c,
Baker and Hillel 1990, Liu et al. 1993; 1994b, DiCarlo 2004) and theoretically
30
2.4. Preferential Flow Phenomena
Fingering Flow
Figure 2.10.: Formation of water fingers in homogeneous sandy soils. These
images are produced by passing light through a ”sand sandwich”
and converting the different intensities to different colors by a
computer program. Black color represents regions of low moisture content and red soil-water saturation. The range of colors
between black and red represent degree of moisture saturation
(photographs from Cornell-Website).
(Eliassi and Glass 2001; 2002; 2003, Egorov et al. 2003, Jury et al. 2003).
The fingering phenomenon occurs in homogeneous sand. It has been demonstrated by Steenhuis and Parlange (1991) that layered soils are not necessarily
required to trigger fingering events, and that fingering can emerge in a relatively homogeneous sand at rainfall rates substantially below the saturated
hydraulic conductivity. The main experimental apparatus to monitor fingering
flow is the Hele-Shaw cells. It consists of a one-centimeter-thick layer of sand
between two glass plates and a light-intensity method. When water is applied
in excess at the top of the cell, it will enter and create flow fingers at a rate
depending on the water content in the cell and on the physical properties of
the sand used, like for example the permeability. Most of the water flows only
through a small portion of the cell. In a homogeneous sand profile, the fingers
are quite vertical, in contrast they deviate from their vertical paths as the sand
becomes more heterogeneous (Fig. 2.10).
Unstable finger flow forms during redistribution following the cessation of
ponded infiltration in porous sand and concrete surfaces under both dry and
wet initial conditions. Fingers form and propagate rapidly when the porous
media are initially dry, but form more slowly and are wider when the media
31
2. Theoretical Background
(a)
(b)
Figure 2.11.: Three-dimensional form of fingering flow: a) Frozen fingers of
blue-dyed water, exposed by removing dry sand (photograph from
Cornell-Website). b) 3-D illustration of a finger formed during
redistribution in a 10 cm diameter column with transparent walls.
The column was frozen at the end of the experiment to preserve
the shape of the finger. The picture was taken after the column
was removed from the freezer and the loose soil that had no water
in it fell out (photograph from Wang et al. (2003)).
are wet (Wang et al. 2003, Jury et al. 2003).
Some other experiments make use of a chamber that allows observation in
three dimensions. Since the light-intensity method cannot be employed in this
situation, flow paths are marked by water containing a blue dye. After the
water has had a chance to penetrate the soil, the sample is frozen, and when
the loose dry sand has fallen out, the congealed flow paths can be examined
(Fig. 2.11).
Fingering is an important mechanism causing agricultural herbicides and
fertilizers to move rapidly from the soil surface, through the crop root zone and
into the groundwater. Flow in these paths is still under extensive study because
of its random and its instability in wetting fronts. Classical convective-diffusion
models, however, have proven inadequate for the analysis of preferential and
multiphase flows in porous media (Glass et al. 1991). The effects of these flow
types are often cited as reasons for predictive error in multi-component flow
32
2.4. Preferential Flow Phenomena
models, such as those used by the oil industry and in predictions for the recovery
of pollutants in soils and aquifers. A better understanding of preferential flow
will in turn improve the predictions of oil field behavior, the development of
better remediation models and the design of pollutant recovery equipment.
While considerable effort has been made on the analysis of fingered flow in twodimensional air-water systems, and on the infiltration of water fingers into oil
saturated media, little advances have been achieved in the field of displacement
of water by oil or the analysis of fingered flows in three-dimensional systems in
real-time (Deinert et al. 2002).
The fingering phenomenon will be described extensively in chapter 5 and
will focus on the mechanism of finger developing in uniform and non-uniform
initial moisture content in a Hele-Shaw cell filled with sand.
33
3. Experimental Methods and
Materials
Fundamental studies on multiphase flow and transport processes of fingering
phenomena within porous media require experimental techniques as light transmission system to measure state-variable at high spatial and temporal resolutions. This chapter resumes the development and application of the Light
Transmission Method (LTM) for three phase flow systems as method for investigating unstable fingered flow in a sand-air-water system. This technique
involves placing a vertical two-dimensional experimental Hele-Shaw cell in front
of a uniform and stable light source. The transmitted light is recorded similar to the experiments done by Hoa (1981) and Glass et al. (1989c). These
experiments were performed in the laboratory. In the present study, methodology is extremely important, because unstable phenomena are very sensitive to
initial and boundary conditions. Hence, an experimental approach was developed and improved using a transparent porous Hele-Shaw cell and the visible
light transmission method to explore gravity-driven instability with high resolution.
In this chapter, the experimental setup together with the methodology are
presented in detail. Additionally, an improved method to measure the phase
saturation with high spatial and temporal resolution is described. In the following section 3.1, the experimental setup, material and measurement procedure are presented. Section 3.2 is about the methods with a brief overview
of LTM and X-ray absorption techniques used in this project for visualization
and imaging of the flow transport phenomena in Hele-Shaw cells. Within this
section, we describe an introduction of the physicals process arisen by light
propagation through porous media in an optical system used in this experiment. Section 3.3 focuses on the calibration of the measured transmitted light
and the water saturation measured in X-ray absorption.
35
3. Experimental Methods and Materials
3.1. Laboratory Materials and Setup
3.1.1. Materials and preparation of samples
The basic materials used were fine and coarse sand. The sands were sediments
and originally come from the river Rhein. The coarse sand consists of a small
amount of the fine sand. It was sieved to remove fine particles in order to
obtain homogeneous coarse-grained granular sand with a rather uniform grainsize distribution.
The grain sizes and saturated hydraulic conductivity of three types of sand
used are listed in Table 3.1.
Material
Fine sand
Coarse sand
Sieved coarse sand
a
Ks ± STDa [cm h−1 ]
12.0±0.7
265±8
873±24
Grain size [µm]
63-250
250-1250
630-1250
standard deviation
Table 3.1.: Grain sizes and saturated hydraulic conductivity of sands used in
experiments.
3.1.2. Saturated hydraulic conductivity
In a separate experiment the saturated hydraulic conductivity (K s ) of these
three types of sand was measured independently using the falling head method
(as introduced by Klute and Dirksen (1986)) applying a light transmission system (section 3.2.1). The porous medium were filled in a small Hele-Shaw cell
(30 × 30 × 0.3 cm) to a depth of L. Then the sample was ponded with water
which was allowed to drain freely due to gravity. During free drainage the
water level h(t) in the cell was monitored by taking sequential images with a
good time resolution. Brilliant blue solution was used as a tracer with very low
concentration for better visualize the water level. It was observed that when
h(t) > L, the height of the water level h(t) decreases exponentially with time
until it reaches the surface of the sample. With this information the saturated
hydraulic conductivity can be calculated using:
h(t) = h0 e
36
−Ks t
L
,
(3.1)
3.1. Laboratory Materials and Setup
h0
h(t)
sample
L
PSfrag replacements
free drainage outflow
Figure 3.1.: Sketch of falling head method used for determination of saturated
hydraulic conductivity.
where t is the duration of drainage, h 0 is the initial height of the water level
inside the cell at the time t = 0 and L is height of the sample as shown in
Figure 3.1.
Then the saturated hydraulic conductivity can be determined from plotting
log(h(t)/h0 ) versus time, i.e.,
Ks = −
h(t) L
log
.
t
h0
(3.2)
By this method, we obtain the saturated hydraulic conductivity of the fine,
coarse (heterogeneous structure) and sieved coarse (homogeneous structure)
sand as 12.0±0.7, 265±8 and 873±24 cm h −1 , respectively. The measured
values are listed in table 3.1.
3.1.3. Hele-Shaw cell
An attractive approach to get a high spatial and temporal resolution of phase
saturation is a Hele-Shaw cell. In this research, infiltration experiment was
conducted in a 2D Hele-Shaw cell (160 × 60 × 0.3 cm) which was made of two
parallel glass plates (8 mm thick, attached to an aluminum frame) with a small
distance apart (0.3 cm) and the porous material was placed inbetween. The
37
3. Experimental Methods and Materials
thickness of the cell, i.e. the separation of the glass plates, was small enough to
transmit visible light while the lengths of the other two dimensions were only
limited by mechanical stability.
In different types of experiments the cell was filled with different layers of
dry sand: (1) 5-10 cm fine-textured sand as a distribution layer on top characterized by a relatively low saturated hydraulic conductivity and with grain-size
diameter 63-250 µm, (2) the structured layer, with grain-size distribution in the
range from 250 to 1250 µm and (3) the coarse-grained granular homogeneous
layer, the same material for structured layer but previously sieved for grains
larger than 630 µm such as coarse-textured sand characterized with a high saturated hydraulic conductivity. The coarse homogeneous layers were filled into
the cell through the top using a funnel-randomizer in order to minimize segregation and make the most possible homogeneous and uniform distribution.
The textural interface between different layers was made as flat as possible specially between the fine and the coarse homogeneous layers. In contrast to sands
used, the structured layer was more heterogeneous with thin-grained filaments
separating coarse-grained regions. The structured layer was filled in a way that
grains of different size were allowed to separate during the filling procedure to
produce a heterogeneous structure of different textures. This is a consequence
of inevitable sorting that results from pouring of granular media with a wide
grain-size distribution.
At the top of the cell, water infiltration was applied through 6 hypodermic
dripper tubes (3 mm diameter) located near the top sand surface at fixed
positions. The flux was adjusted by a pump which sucks water from a reservoir
placed on a balance. The weight of water mass was frequently recoded with
computer which calculates the flux. The lower end of the cell was closed. A
single outlet, the black dot at the lower left corner (Fig. 3.2), allows water and
air to escape freely. The vertical boundaries were closed for water flow. Under
certain conditions the less conductive fine top layer may constrain the supply
of water to the more conductive coarse sublayer.
3.1.4. Light source
The cell was placed in front of a homogeneous illumination, from a bank
of high-output fluorescent light with four fluorescent lamps (OSRAM-55WWhite). These lamps were installed into a light box, a wooden box of dimensions 165 × 32 × 30 cm, where the cell containing the sand was mounted on
it. The light system provides a low-temperature and stable source for lighting
the sand pack from the back side. A homogeneous illumination of the entire
38
3.1. Laboratory Materials and Setup
0
homogeneous heterogeneous homogeneous
drippers
fine layer
0.2
digital camera
height [m]
Hele-Shaw cell
uniform light source
0.4
0.6
0.8
1.0
1.2
1.4
0
side view
0.2
0.4
width [m]
front view
Figure 3.2.: Sketch of the experimental setup (side and front view). A transparent Hele-Shaw cell with four layers of sand was placed in front
of a uniform light source. Transmitted light was recorded by a
digital camera.
surface of the Hele-Shaw cell was achieved by using a diffusion foil between
the lamps and the cell. Variations in the light source intensity occur each time
the system is switched on or off. A reasonably constant light source intensity
level is also required. In our system, the possible temporal variations in the
light intensity was calibrated out during the experiment through a region that
remains dry during the infiltrating. This adjustment was required for the light
technique to correct for variations in the light source stability (more details in
section 4.1).
A schematic representation of the front and side view of the light imaging
system used in experimental setup is shown in Figure 3.2.
3.1.5. Camera setting
To monitor the fingered flow and infiltration, images were captured and recorded
at predetermined intervals immediately after water application using a Canon
39
3. Experimental Methods and Materials
digital camera (Canon EOS 300D, 6 MPixel) with a 18-55 mm equivalent lens.
The camera was located approximately 1.5 m in front of the cell and focused
on the front of the experimental cell with array size of 691×1929 pixels image
with a pixel size of 55 cm×157 cm covering the entire cell. The spatial resolution of the acquired image is defined by the array size of the camera and
the size of the test medium which gives 0.6 mm 2 per pixel. The profiles were
photographed without any optical filters. Both the camera and the cell were
carefully aligned horizontally and fixed. The images were stored in a computer
simultaneously. A setup of USB board (hardware) was used to capture the
frame by using “ZoomBrowser”(software by Canon). The temporal resolution
was 15 seconds when the fingers were quickly traveling downward including an
exposure time 2 seconds (aperture F8) and storage of images. All images were
taken under identical conditions with manual setting.
3.1.6. Tensiometer construction
The experimental aspect of this study requires rapid point measurements of
phase pressure during the passage of fingers. The monitoring of internal dynamics of water pressure change was not possible with optical methods, hence
we employed the traditional instruments e.g. pressure sensors installed into
access ports. Many mini-tensiometers were installed at different locations in
the designated holes over the back wall of the cell according to Selker et al.
(1992a).
With this method the speed of the spreading pressure front and the actual
matric potential, as the difference between water pressure and atmospheric air
pressure outside the cell at a specific position, could be measured. The special
mini-tensiometer designed for these measurements is sketched in Figure 3.3.
This instrument consists of a cylindrical porous ceramic plate (P80), height
4 mm and diameter 8 mm, with a high air entry value less than 10 m. The
tip of ceramic plate had to be as flat as the surface of the sample on which it
was placed, for maximum contact. The tensiometers were flush with the inside
surface of the cell and fixed with a mounting boss. The transducer used to measure the pressure was from the Honeywell 26PC series (www.honeywell.com),
which features a sensing technology that utilizes a specialized piezoresistive
micro-machined sensing element. The transducer port was threaded and fastened directly to the tensiometer body, which yields the minimal hydraulic
path connection.
After installing the tensiometers, water was poured into the plastic tube
without entrapping air bubbles in the tube because this air could lead to a loss
40
3.1. Laboratory Materials and Setup
glass sand glass
8mm
40mm
ceramic
plate
mounting
boss
plastic tube
pressure
transducer
8mm
4mm
40mm
analog
output
Figure 3.3.: Sketch and Photo of the tensiometer installed over the cell and a
Sketched cross section of a mounted tensiometer. The porous ceramic plate (light gray) connects directly to the pressure transducer
via a firm plastic tube so that the ceramic plate was in contact with
the sand. It was necessary to avoid gaps between the glass and the
tube for install the tensiometer in the cell.
of hydraulic contact between the transducer and the sand. The response time
of tensiometers was less than 1 ms.
The calibration of tensiometers was carried out by using sequence defined
heights of water in a calibration tube. The different pressure values corresponding to different heights were evaluated by a computer. This yields the
calibration parameters for each sensor.
3.1.7. Experimental setting
The experimental procedure consists of several steps such as sand preparation,
cell cleaning, filling and packing of the cell with sand, injection of water into the
41
3. Experimental Methods and Materials
cell and recording light intensity and water pressure. Infiltration experiments
were carried out with an initially dry sand in the Hele-Shaw cell under varied
conditions for different flow rates. A set of many experiments was attempted
with constant and different flow rates.
After reaching stationary water flow to visualize the velocity field inside
the cell, additional infiltration experiments were performed by using a dye
tracer (0.5 g l−1 Brilliant Blue) which was applied to the almost stationary
flow field behind the water infiltration front. All experiments started with water
application on dry sand and after the fingered flow field was fully developed and
steady state was reached, the irrigation depending on the type of experiment
was stopped and dye was infiltrated.
3.2. Methods
In this study, we used the Light Transmission Method (LTM) to measure the
vertical redistribution and the dynamics of water content in the Hele-Shaw cell
according to Glass et al. (1989c), Tidwell and Glass (1994). The light intensity
was directly related to water content. Application of the LTM as a visualization
technique involves placing a two-dimensional experimental chamber in front of
a uniform light source and recording the transmitted light. This technique
requires that the image system is transparent to translucent and is based on
the fact that the intensity of transmitted light can be used as a proxy for water
content.
This thesis describes a moisture content visualization technique which is
based on the physical observation of the light transmission through sand. The
technique is developed to use in thin but extensive experimental systems. In
order to calibrate the light imaging system, we used X-ray absorption technique. In both the X-ray and light technique, electromagnetic energy is passed
through the test media and the water saturation distribution integrated over
the media’s thickness is measured as variations in the transmitted X-ray or light
intensity field. The difference between the techniques lies in the frequency of
the radiation used and in the physics governing the interaction that gives rise
to variations in the transmitted intensity field. When using low energy X-rays,
variations in the transmitted intensity field arise from the sensitivity of X-ray
absorption (photoelectric absorption) to the density of the media, which is directly related to liquid saturation (i.e., increase in saturation yields a decrease
in X-ray transmission). For the light technique an increase in saturation results
an increase in light transmission because of the closer matching of the index of
42
3.2. Methods
refraction of the matrix and water relative to the matrix and air.
A brief overview of the light transmission and X-ray absorption technique is
presented further below.
3.2.1. Light Transmission Method (LTM)
The color can be expressed in different formats and quantification of color
difference can be complicated. The color for video cameras, color monitors and
computer graphics is defined in terms of a vector with the intensities of the three
components of Red, Green and Blue (RGB). Another system to specify the
color vector is Hue, Saturation and Intensity (HSI). Hue is the attribute that
describes the pure color and is what we are typically referring to when we use
the term color. Saturation is the attribute that describes the degree to which
the color is diluted with white. Intensity is the attribute which corresponds to
the gray level (black and white) of the color image (Darnault et al. 1998). The
HSI color model owes its usefulness to two principal facts. First, the intensity
component, I, is decoupled from the color information in the image and second,
the hue and saturation components are intimately related to the way in which
human beings perceive color. The advantage of the HSI space is that it treats
color roughly the same way that human perceive and interpret color.
Using RGB format was not sufficient to give a unique relationship with water
content. The difficulty in using RGB is the interdependence of color saturation
with values of the components of the color vector. Thus, even if the eye can
see a difference in color, RGB is unable to pick out the colors in a simple
predictable manner (Wilson 1988).
These features make the HSI model, an ideal tool for developing image
processing algorithms, based on some of the color sensing properties of the
human visual system (Gonzales and Woods 1993). Therefore, if the human eye
is able to see, these differences can be quantified using the HSI space.
The HSI model can be defined by a transformation of RGB color component
as (Gonzales and Woods 1993):


h
i


1


2 (R − G) + (R − B)
−1
H = cos
1
i  ,
h
2

2
(R − G) + (R − B)(G − B) 
h
i
3
min(R, G, B) ,
S = 1−
(R + G + B)
1
(R + G + B) ,
(3.3)
I =
3
43
3. Experimental Methods and Materials
PSfrag replacements
RGB
Hue
Saturation
Intensity
Figure 3.4.: Visualization of flow fingering experiment through initially dry
porous Hele-Shaw cell using RGB, Hue, Saturation and Intensity
formats calculated according to Eq. 3.3. The intensity image shows
the direct relation with highly localized flow paths that originate
from the flow instability in uniform media.
where H, S, and I are the hue, saturation and intensity of the output HSI
image; and R, G, and B are the red, green, and blue components of the RGB
image.
Image processing was done using the C/C++ library Quantim written by
Vogel (2006). A complete description of processing on images is presented in
chapter 4. In image processing recorded images were converted from RGB to
HSI format. Figure 3.4 shows RGB and HSI spaces of a sample experiment
image. In a dry porous medium the moisture content is immediately visible
to the eye from the other side. The recorded color differences are visible between the different formats. In each format, (RGB and HSI), the color is not
completely uniform and slight spatial differences exist due to sand and water
ganglia formation (Fig. 3.4). The color attributes of pixels in HSI space are
compared with water content in RGB space. This comparison exhibits a better
direct relation between intensity and water flow fingers.
The moisture visualization technique is based on the principle that the transmitted light intensity measured in a RGB image (Eq. 3.3) increases with an
increase in moisture content (i.e. saturation). At any given location in the
medium through which light is transmitted, we found that the brighter loca-
44
3.2. Methods
tion represents the higher porosity and/or moisture content (Fig. 3.4). This
technique also allows a rapid assessment of the homogeneity in the porous cell
before the onset of infiltration.
Visible light transmission was first used quantitatively by Hoa et al. (1977),
Hoa (1981) to measure the saturation within a thin, sand-filled slab chamber
using a small movable light transmission sensor. Hoa’s method was expanded
by Glass et al. (1989c) and Bell et al. (1991) to visualize moisture content in
an entire two-dimensional flow field by supplying a diffuse light source. This
method has been successfully used for the study of rapid variations of the
water content distribution in thin samples of porous media. We applied directly this development using the relationship between intensity of transmitted
light through a Hele-Shaw cell and water content within the context of this
study.
Here, before describing the next method (X-ray absorption) that we used
for calibration of the measured light intensity in order to get the water saturation, we present a brief overview of the the physical process arisen by light
propagation through porous media in an optical system used in this experiment.
Light Propagation through Porous Media: During the propagation of
light through a transparent porous medium with densely packed particles, light
is deviated from its course. According to the physical nature of light and its
behavior at interfaces of different media, the deviation is dependent on the
angle of incidence, the wavelength of the light beam and the refractive indices
of that medium. In three phase system used in this study (air-water, sand-air,
sand-water), for example, light that passes through these phases and media
encounters sand, air and water phases. In each of these phases, light is absorbed
exponentially. In addition, it can be scattered, reflected, and refracted at the
interfaces between the different phases. Through these deviations of light,
the scattering usually referred to as a random disturbance of light induced by
passage through regions of different refractive indices. As the light is scattered,
it becomes anisotropic, and hence, the direction in which it leaves the material
is essentially random.
The effect of water saturation on transmittance light, which is depended
not only water saturation but also on the spatial arrangement of water and
air in the pore space of the scattering medium is also important property of
the influence of water content on transmission. Hoa (1981) reported that the
transmission is a function of the size distribution of the water-filled pores. A
decrease of reflectance is caused either by a decrease in scattering leading to
an increase of light propagation into forward direction or by light absorption in
45
3. Experimental Methods and Materials
water. The last cause is pronounced in the absorption bands of water.
Tidwell and Glass (1994), Niemet and Selker (2001) and Niemet et al. (2002)
discussed the influence of water content on transmission. They describe light
absorption with Lambert-Beer’s law. In these studies the scattering medium
was assumed to be homogeneous, i.e. the arrangement of water in the pore
space was not considered. To calculate transmittance, the authors used the
following refractive indices: sand ns = 1.6 , water nw = 1.33 , and air na = 1.0 .
The model presumes normal incidence and homogeneous distribution of water
and solid phase. The experiments were designed in such a way that the samples were sufficiently translucent, yielding a quantifiable amount of transmitted
light through a medium of significant thickness.
Bänninger (2004), Bänninger et al. (2005) considered that the properties
which influence the reflectance and transmittance, and degree of the scattering
is highly dependent upon the media thickness, size of the particles, the optical
properties of the phases, porosity, wavelength, and the surrounding materials.
They studied the principles of optics for inferring textural attributes of media surfaces from light scattering and developed a radiative transfer model to
describe the physical processes of light propagation e.g., the geometry of multilayered film particles, orientation of light transfer, changing transmitted light
configuration with water content, and the actual light paths through porous
media.
A crucial issue in the porous Hele-Shaw cell is multi-light-scattering which
reduces spatial resolution of transmitted light and also affects water saturation
measurements. The qualitative behavior of a light beam passing through a
sand-filled Hele-Shaw cell is illustrated in Fig 3.5 showing that light can be
transmitted, reflected, scattered, or attenuated due to multiple reflection from
the particles.
In all previous applications of the LTM only the transmitted light intensity is
considered without any correction for light scattering (Tidwell and Glass 1994).
However, multiple light scattering reduces the contrast between zones of different water contents and hence it affects one of the most important aspects which
we intend to correct it in our experiments. Therefore a correction of the images
is advisable. In this study, an image processing algorithm has been developed
that exploits the multiple light scattering in observed image. This technique is
the deconvolution using an experimentally measured response function (point
spread function). We will describe this process in image processing chapter (4).
46
3.2. Methods
Figure 3.5.: Sketch of the optical paths of a light beam through a porous HeleShaw cell filled with sand. The possible events for light propagation
are shown for a small section of the cell: light can be absorbed
within each phase and scattered, reflected, and refracted at the
interfaces between the different phases and the sand particles.
3.2.2. X-ray absorption
During the experiments simultaneous measurements of water saturation inside
the cell were done by monitoring X-ray absorption. For the X-ray measurements, unlike LTM, there is a fundamental relationship between the attenuation
of X-rays and moisture content. The X-ray consists of a polychromatic medical X-ray tube operated at 141 KV and 5 mA together with a 12 bit CCD
line detector with 1280 pixel of 0.4 mm side length. Tube and detector were
mounted at a distance of ∼2 m and can be moved synchronously in vertical
direction from the bottom to the top of the cell.
The cell was placed between the X-ray tube and the detector. The images
were acquired by directing a beam of X-rays to the face of the test media while
47
3. Experimental Methods and Materials
recording the transmitted intensity field with the detector behind the cell. To
monitor the vertical distribution of water within a sample, the line sensor and
the focus of the X-ray tube were placed in the same vertical position, at the
bottom of the cell. Then both were moved synchronously from the bottom to
the top of the sample. During the movement the intensity on the detector was
measured at several positions. The positions and the velocity of the movement
were adjusted to useful values. Measured X-ray intensities of each pixel I(d)
after passage of the thickness of the cell were related to the emitted intensities
I0 which were measured without the cell. The intensities for each pixel was
corrected for the dark current of the respective detector pixel. This dark current
was determined in an independent measurement. The absorption coefficient, µ,
can be calculated as the product of the effective absorption coefficient, µ ∗ , and
the pass length d of path of the photons through the material by the following
relation:
I0
) .
(3.4)
µ = µ∗ (x) d = ln(
I(d)
In Eq. 3.4, the effective absorption coefficient, µ ∗ (x), at sample height x is
the sum of the length-fraction weighted absorption coefficients of the different
components, hence:
µ(x) = µ∗ (x)d = (1 − φ)dµsand + dglass µglass + θ(x)dµH2 O + (φ − θ(x))dµair ,
(3.5)
where µsand , µglass , µH2 O and µair are the absorption coefficient of sand, the
glass plates of the cell, water and air, respectively. Where d and d glass are
thicknesses of the cell and its glass, respectively, φ is the porosity and θ(x) is
the distribution of the volumetric water content. Neglecting absorption of air,
whose absorption coefficient is two orders of magnitude lower than that of the
other present materials, the last term in Eq. 3.5 disappears. After measurements of the dry and the fully water saturated sample the water saturation,
S(x), can be calculated from:
S(x) =
µ(x) − µdry
,
µsat − µdry
(3.6)
where µdry is the calculated value using Eq 3.4 of the dry and µsat of the
completely saturated sample (Bayer et al. 2004, Bayer 2005).
3.2.3. Limitations of techniques
Measurements of saturation with either technique require achievement of suitable image contrast. Contrast is defined here as the difference between the
48
3.3. Water Content Calibration
intensity field transmitted through the test media and the transmitted through
the dry test media.
For the X-ray technique, difficulties arise because of the low density of water relative to that of the minerals composing the porous media. Hence a
large change in saturation produces only a small decrease in transmitted X-ray
intensity. For the light transmitting technique, image contrast is governed by
differences in the refractive indices of the air-sand and water-sand interfaces. In
the air-water-sand systems contrast is exceptional without modification. The
main disadvantage of the LTM in comparison to X-ray, is that the measurements depend on the translucence of the porous medium. Therefore the use
of silica sands or glass beads and the thickness of the test media is limited on
the order of a centimeter for most cases. Since this drawback can not easily
be solved, three dimensional transient fluid observations will still have to wait
a while. The principle advantage of this method is that it does not involve
radiation and it enables us to visualize fluid content changes over the whole
flow field with high spatial and temporal resolution. The X-ray measurements
can also be performed with high spatial resolution but only at one location
at one time. The high temporal measurement of the light transmission makes
it especially suitable for transient flow experiments and compensates for that
limitation.
3.3. Water Content Calibration
The motivation for calibrating the measured light intensity and water content
is the physical processes of light propagation through porous media e.g., the
geometry and orientations of multilayered film particles, changing transmitted
light configuration with water content within the medium, and the actual light
paths. The calibration of the LTM in order to get the water saturation was done
by simultaneous measurements of X-ray absorption and light transmission. For
this purpose we filled the homogeneous coarse sand into a smaller calibration
cell with dimensions 30× 30× 0.3 cm. The water table within this small cell was
adjusted with an inlet at the bottom of the cell. The cell was placed between
the X-ray source and the detector with the glass plates perpendicular to the
center ray. Simultaneous X-ray (vertical scan) and light transmission images
were taken for three different situations: completely dry, water saturated and
at hydrostatic equilibrium stage with several heights of the water table at the
lower boundary. X-ray and LTM profiles were recorded for each water table
through parallel measurements. The completely dry image as first reference was
49
PSfrag replacements
3. Experimental Methods and Materials
1.0
0.8
Intensity (LTM)
0.4
0.6
0.5
0.4
0.3
0.2
0.1
0
0.6
0.8
1.0
0.8
Saturation (X-ray)
RGB
1.0
0 0.2 0.4 0.6 0.8 1.0
450
450
450
450
400
400
400
400
350
350
350
350
300
300
300
300
250
250
250
250
200
200
200
200
150
150
150
150
100
100
100
100
50
50
50
50
0
0
0
0.4
0.4
0.6
0.6
0.8
1.0
normalized intensity
height [pixel]
0.7
height [pixel]
saturation [–] (X-ray)
0.9
0
0 0.2 0.4 0.6 0.8 1.0
saturation [–]
normalized intensity (LTM)
Figure 3.6.: Correlation between measured water saturation by relative X-ray
absorption and the normalized intensities by LTM. Although the
relation between transmission and saturation is nonlinear, the
transmitted light is still a good proxy for water saturation.
taken before the sample came in touch with water and the image at saturation
as second reference was measured at the end of the experiment, after the cell
had been saturated by stepwise imbibition (to minimize air entrapments) of
water through the bottom to the top of the cell. After fully saturating of
the cell, the water table was decreased again in several steps and profile data
recorded.
The vertical profile measurements of relative X-ray absorption and normalized light intensity as a function of cell height were compared with each other
and agreement between two profiles of these two independence techniques was
very close (Fig. 3.6). To decrease the signal to noise ratio the vertical homogeneity was improved and saturation and intensity were calculated by averaging
of 400 pixels along a centered horizontal line. Since neither measurement technique gives the true value, we can only compare statistically how well both
50
3.3. Water Content Calibration
methods agree. A regression analysis between the water saturation measurements by the relative X-ray absorption and by the normalized light intensities
of the LTM indicates a relation by fitting a third order polynomial to the data
with a correlation coefficient of 0.999 and a standard deviation of 0.010. We
obtained:
2
3
S = 1.7 − 9ILTM + 15.3ILTM
− 7.1ILTM
,
(3.7)
where S is the distribution of water saturation calculated from the relative Xray absorption coefficient using Eq. 3.6 and ILTM is the normalized intensity
with two reference measurements which can be expressed as:
ILTM =
I − Idry
,
Isat − Idry
(3.8)
where I, Idry , Isat are the light intensities of the transmitted light for the partially saturated, dry and saturated sample, respectively. Equation 3.7 is valid
only when the normalized intensity is between 0.47 and 1.00. The resulting
S, ILTM and calibration curves are shown in Figure 3.6. More details of image analysis to produce the intensity and normalized intensity images data are
presented in chapter 4.
Obviously, the general shape of this fitted curve is non-linear. It starts with
an initial non-linear shape in the range of low transmitted intensity and saturation, then in the range between 0.6 and 0.85 of intensity values looks linear
and in the range of high values of intensity and saturation, it has the nonlinear shape. The calibration curve has the only significant deviations, from
the ideal data, in the range of non-linear shapes. This is because of comparison of these two independent techniques in the part of heterogeneity structure
of sand. Fine-textured sands show higher saturation by X-ray measurements
and lower transmitted intensity by LTM, and also in the range of high values, it could be resulted by pore spaces between the grains which shows much
higher transmitted intensity in comparison with the real saturation measured
by X-ray. By this correlation, the light intensity can be used as a proxy variable of water saturation. This calibration was used for all experiments in this
study.
For sand the intensity of transmitted light increases with grain size. Notice
that the constant values in Eq. 3.7 are valid only for the specific type of sand
used in this project. The system needs to be calibrated each time a different
camera or sand are used. This is the disadvantage of this calibration that holds
only for exactly the material used. Other materials with different transmission
properties have to be measured again with the procedure described above. Each
51
3. Experimental Methods and Materials
accurate determination of saturation from light intensities requires a careful
calibration depending on the materials.
3.3.1. Calibration error
Errors that influence the results of the saturation calibration can originate from
the comparison experiments of the X-ray and light transmission. The error due
to the comparison experiment stem from the need to move the experiment between two facilities (X-ray and LTM). Slight cell movement can be translocated
during the transport of the experiment carried about ∼2 m distance between
the facilities. For each method the experiment also had to be aligned properly
with respect to the source and detector. We adjusted all positions for X-ray and
LTM measurements to examine exactly the same location of the cell and this
was done using marked points on the aluminum frame of the cell as reference.
A perfect alignment was impossible. A misalignment of above 1 − 2 mm can
arise each time the cell was moved. Such misalignment can cause errors of few
percent of saturation where moisture conditions are rapidly varying. Although,
each of the X-ray and light images was adjusted and scaled for their geometric
distortion by image processing applications on images.
Another source of error is given by technical differences between measurements techniques. There was a time lag of 20 − 30 min between X-ray and
light transmission imaging, so the two distributions do not completely coincide.
52
4. Image Processing
In the present study, determination of water saturation from a sand profile was
done by digital image analysis. The image adjustment, intensity conversion,
inhomogeneous illumination correction, subtraction from background, and finally deconvolution were the processes of images analysis. An image processing
code was written to analyze the images in order to transfer the optical proxy
information (intensity) in each pixel/cell to the quantity of interest (phase saturation) using independently determined calibration functions. The purpose of
this chapter is to provide an accurate and complete presentation of the image
processing that leads to the results in next chapter. In the first section, we
describe the procedures of pre-processing of an image. These procedures are:
(i) image adjustment using space references, (ii) filtering, (iii) converting each
pixel of image based on red, green, and blue value to intensity format, (iv)
correcting inhomogeneous illumination and (v) subtracting from background
(dry) image. In section 4.2, we will demonstrate how we correct the images
from light scattering effect occurring during the light transmission through
porous material by deconvolution.
4.1. Pre-processing of Images
This section describes the pre-processing of the acquired data, i.e. the digital
images. The aim of this process is to transform the captured light intensity
information into a corresponding set of intensity values which is then the basis
for further evaluations. The value assigned to each pixel in the digital image
which has been captured as a RGB format is generally coded on 24 bits per
pixel. The pre-processing steps are presented below.
1- Noise Reduction: Digital sequence images exhibits temporal variability
(i.e., noise) at each pixel as well as spatial variability. Because the spatial variability is constant in time, it influences measurements only if there is a camera
shift relative to the fractured cell, whereas noise leads to an uncertainty in
measurements at each pixel in every image. In the first step of pre-processing,
53
4. Image Processing
an adjustment of images is required to correct variations in the space. They
are referenced to assure that the images from the experimental period could
be compared. First, the serial images were adjusted for geometrical distortion using the spatial coordinates of the edges which are identified by marked
points on the fixed aluminum frame of the cell as reference. By this adjustment, all images were corrected to represent exactly the same locations within
the cell as well as for orthogonal directions and non-rotation of images. It
was required because of slight movements of the imaging devices during the
experiment, which produce noise on images. These corrections were done by
cross-correlation analysis on images. After this procedure, the intensity values
across the image are smoothed in order to eliminate noise. A common method
for image smoothing and to remove noise effects is the median low pass filtering, which replaces each pixel value with the median gray scale value of its
immediate neighbors. For our data evaluation, we applied a 5×5 median filter
on each image.
2- Image Conversion: As we described in section 3.2, a pixel’s color can be
described by a RGB (Red, Green, Blue) vector or by an HSI (Hue, Saturation,
Intensity) vector. In the RGB color model, the value of each pixel corresponds
to the intensity of three primary color components. Each of these colors in
original captured images were saved in a separate plane with 256 gray values.
The direct application of the RGB color space is not suitable because this
operation produces color shifts, which alter the image. A possible approach
is a conversion of the RGB image into the HSI color space. Transforming a
RGB image to an HSI format image was performed as a second step of image
pre-processing using Eq. 3.3. Figure 4.1 shows an example of conversion for a
partially wet sample. The best correlation to water saturation was found for
the intensity component (I = 31 (R + G + B)) of the HSI system, hereafter
referred to color saturation. The variable I shows the brightness of a pixel and
higher brightness represents the higher saturation in the cell.
3- Dark Intensity Correction: Dark intensity contributions (image taken
with a closed camera shutter: the so-called “dark image”) were removed from all
images for subsequent processing. This was done by subtracting the intensities
of “dark image” from an original captured image with an equal exposure time.
Measurements of the dark intensity have shown that it is fairly uniform in
space and time with an intensity value between 0 and 4. Once the image of
the experiment is corrected for any dark intensity, a flat-field correction can be
54
4.1. Pre-processing of Images
0
calculation of
Hue, Saturation and
Ri
Intensity
Gi
Bi
determination of image
information Ri, Gi, and Bi
for each pixel
RGB image
255
Hue
Saturation
basic triangle
Intensity
color space
I
RGB color space
HSI color space
Figure 4.1.: Converting RGB image (top left) to HSI color space (top right)
using Eq. 3.3 for a partially wet sample. Bottom left: the RGB
color model has three basic primary colors (shown with P in cube):
red, green, and blue. All other colors are obtained by combining
them. Bottom right: in the HSI model, H (Hue) is the angle
of the vector over the basic triangle, S (Saturation) is the proportional size of the module of the projection of the vector over the
basic triangle and I (Intensity) is the distance from the end of the
vector to the basic triangle. The intensity image (top right) shows
better proxy for water saturation.
done.
4- Flat-field Correction: The visible light sources used for the transmission experiments should possess the qualities of diffusivity and spatial uniformity. But even under the best imaging conditions, the illumination across a
field of cells is not quite uniform. This is due to imperfections contributed
to each optical element within light source. Therefore a correction or rather
55
4. Image Processing
a compensation for inhomogeneous illumination was necessary. Flat-field correction was used to overcome the uneven illumination. The starting point
for determining the flat-field illumination is having a background illumination
emitted intensity which should be homogeneous throughout the view field at
all pixels. Therefore, the background emitted intensities were measured for
each experiment, separately. It was taken at a position where no sample was
between light source and camera using the identical exposure times and acquisition settings. Any image I(x, y) can be corrected, on a pixel-per-pixel basis,
for inhomogeneous lighting by dividing it by a flat-field image I 0 (x, y) relative
to the dark image Id (x, y) and rescaling as
IF (x, y) =
I(x, y) − Id (x, y)
× 255 ,
I0 (x, y) − Id (x, y)
(4.1)
where IF (x, y) is the corrected image for inhomogeneous illumination by flatfield correction. Figure 4.2 illustrates the intensity image of a porous material
sample before and after the flat-field correction.
5- Background Subtraction: A background image (dry) of the cell before the onset of infiltration was subtracted from the digitized image to yield
an image that shows only the moisture content in the cell. Hence, the images are displayed according to light intensity at each location which allows
the visualization of the relative moisture content. This was performed by an
image processing procedure where a large portion of the original image was
removed in the pictures by pixelwise subtraction the original image values by
the corresponding background values. This process often is called “background
subtraction”. The processed images provide a pixel-by-pixel spatial distribution of the changes in water saturation which we call normalized. Then, the
images were filtered again using a 5 by 5 median filter to produce the final
normalized image.
We applied these pre-processing steps to each of the captured images. Figure 4.3 presents these pre-processing steps for two images captured from a dry
and a partially wet sample.
4.2. Deconvolution Processes
As we described in chapter 3 (subsec. 3.2.1), during light propagation through
porous media it leads to a specific spreading because of multiple light scattering.
Deconvolution is being increasingly utilized for improving the contrast and
56
4.2. Deconvolution Processes
correction
correction
correction
correction
223
191
400
159
300
128
96
200
250
before flat-field correction
32
0
0
0
100
200
300
PSfrag replacements
position [pixel]
depth [pixel]
after flat-field correction
depth [pixel]
PSfrag replacements 600
255
500
150
100
223
400
159
300
128
intensity
before flat-field correction
191
96
200
before flat-field correction after
flat-field correction
64
after flat-field correction before
100
flat-field correction
32
before flat-field correction
0
0
0
before flat-field correction
after flat-field correction
200
64
100
intensity
flat-field
flat-field
flat-field
flat-field
500
intensity
before
after
before
after
depth [pixel]
before flat-field correction
PSfrag replacements 600
255
100
200
50
0
0
50
100
150
200
250
300
position [pixel]
300
position [pixel]
Figure 4.2.: Intensity image of a porous material sample before and after the
flat-field correction for inhomogeneous illumination. The graph
on the right shows the comparison of intensity values for a cross
section at fixed pixel depth 330 before and after the flat-field correction. It results that illuminated flat-field image improves image
uniformity.
correct the distortion of digital images to demand a true representative of
the original image in presence of a determined Point Spread Function (PSF).
Deconvolution is both a mathematical concept and an important operation
used in digital image processing to recover an object from an image that is
degraded by physical process of the optical system. This operation is an inverse
filtering and its task consists in despreading the observed image in order to
recover the original shape of the object. Deconvolution is a computationally
intensive image correction and it is inherently unstable and highly sensitive to
the effective measurement noise. It is capable of: removing noise, increasing
contrast, and increasing resolution.
57
4. Image Processing
dry sample
median
filtering
intensity
conversion
dark and
flat-field
correction
subtraction of
images
partially wet sample
median
filtering
intensity
conversion
dark and
flat-field
correction
normalized image
Figure 4.3.: Sequential steps of the image pre-processing for two-dimensional
distribution from two images captured from a dry and a partially
wet sample. The more water in the sand, the brighter location,
allows to quantify the water saturation. The intensity of subtracted
(normalized) image is rescaled to better visualization.
4.2.1. Point Spread Function (PSF)
The PSF plays an important role in the image formation and it is of fundamental importance to deconvolution and should be clearly understood in order to
avoid imaging artifacts. Consequently, the image of transmitted light is convoluted by a PSF which is expected to be constant for homogeneous porous
media and a fixed thickness of the transmitted layer.
We start with mathematical description for PSF. The one-dimensional PSF
process can be mathematically described by a convolution equation. Let us
assume that during an image acquisition stage, the original image is blurred by
a PSF. Then, an observed light distribution f may usually be mathematically
expressed as a convolution of the original (real) light distribution h and the
58
4.2. Deconvolution Processes
point spread function w as:
f (x) =
Z
+∞
0
0
0
h(x ) w(x − x ) dx .
(4.2)
−∞
In the first step to determine the PSF, we want to obtain w from observed
f . The easiest way would be to use a very small slit, where the real image
would be expected as the Dirac delta function δ(x). Experimentally, that is
not realizable. Although, a very small slit would lead to additional diffractions
of the slit itself. Therefore, a slit with a width much larger than the used
wavelength must be used, where the real image can be mathematically expected
as
with
h(x) = u(x − x1 ) − u(x − x2 )
0
for x < 0
and x1 < x2 .
u(x) =
1
for x > 0
(4.3)
Hence, the PSF can be determined using first derivative of f (x), which yields:
Z +∞
0
0
0
∂
(4.4)
∂x f (x) =
w(x − x ) dx
h(x )
∂x
−∞
Z +∞
0
∂
0
0
=−
h(x ) 0 w(x − x ) dx
∂x
−∞
Z +∞ h
i
0
0
0
∂
=
0 h(x ) w(x − x ) dx ,
∂x
−∞
assuming
lim
h(x0 ) w(x − x0 ) = 0 ,
∀x
(4.5)
∂x h(x) = ∂x (u(x − x1 ) − u(x − x2 ))
(4.6)
x0 →±∞
= δ(x − x1 ) − δ(x − x2 ) ,
herewith,
∂x f (x) = w(x − x1 ) − w(x − x2 ) ,
(4.7)
where w(x) is the PSF. Figure 4.4 shows the process graphically.
In our application, the specific PSF was measured experimentally using narrow slit with dimension 1×13 cm. To obtain more accurate PSF, we used 8 slits
59
4. Image Processing
f (x)
∂
∂x f (x)
slit cell
x
source
PSfrag replacements
w(x1 )
light spreading
w(x2 )
Figure 4.4.: Schematic representation of the method for measurement of PSF.
The observed light distribution due to passage of light through a
narrow slit leads that the light diffracts outwardly in all directions.
∂
f (x) is the
f (x) is the measured light intensity outside the slit, ∂x
differential form of f (x) and w(x1 ) and w(x2 ) are the PSFs.
installed over on the back glass of the Hele-Shaw cell not only to the vertical
but also to the horizontal directions at different locations. We installed 5 slits
in vertical and 3 slits in horizontal direction. A photograph of the installed
slits and porous medium into the cell corresponding to this situation, is shown
in Fig 4.5(a1 -a3 ).
We identified 3 lines along each slit through both horizontal and vertical
directions. All derivative curves of the measured decreasing light intensities
outside the slits, obtained by derivation of the cross-section transmittance data,
were fitted using a Gaussian fitting procedure (Fig. 4.5(b-e)). According to
mean Gaussian fit determined for both one-dimensional x and y direction, the
two-dimensional Gaussian PSF can be determined as below:
w(x, y) = w(x) × w(y) ,
60
(4.8)
derivative of intensity [1/pixel]
4.2. Deconvolution Processes
250
slit
slit
dry
wet
200
wet
intensity
dry
150
100
50
a1
0
vertical slit
0
b
100
200
300
position [pixel]
2
6
R=
Ax=
x1 0 =
W 1x=
B1 x=
x2 0 =
W 2x=
B2 x=
( )
4
2
0.98742
0.06986
127.0728
29.00898
238.42252
158.67285
46.74113
-257.88029
±0.01817
±0.46829
±0.86106
±25.00109
±2.53553
±2.69214
±26.2378
0
-2
( )
d
-4
0
w(x, y)=w(x) × w(y)
50
100
150
200
250
300
position [pixel]
derivative of intensity [1/pixel]
position [pixel]
position [pixel]
f
mean 2D
Gaussian PSF
( )
200
2
300
300
e
R=
Ay=
Y10=
W1y=
B1y=
Y20=
W2y=
B2y=
( )
250
c
6
100
4
slit
dry
wet
200
horizontal slit
250
f
150
wet
200
100
dry
2
0
slit
0
150
-2
100
50
a2
50
0
0
-4
-6
intensity
0.99254
0.01653
179.45174
28.47407
-177.39563
118.28989
28.30508
173.14779
±0.01513
±0.14991
±0.32211
±1.92646
±0.15225
±0.32691
±1.91872
a3
Figure 4.5.: The process procedure to measure the PSF. a 1 ) Eight slits installed
over the cell. a2 ) Dry porous material+slits. a3 ) Partially wet
porous material after water infiltration from top of the cell+slits.
b,c) Cross-section transmittance curves for an example of vertical
and horizontal slit. d,e) A Gaussian fitted (red line) to the derivative curve of cross-section transmitted data (black line) measured
for a vertical and horizontal slit. f ) The mean 2D Gaussian PSF
image obtained by combination of x and y direction fitted Gaussian
distributions.
with Gaussian functions of the form
Bx
w(x) := Ax + q
exp
2
2πWx
By
exp
w(y) := Ay + q
2
2πWy
)
,
)
,
(
−(x − x0 )2
(
−(y − y0 )2
2 Wx
2 Wy
2
2
(4.9)
where w(x, y) is the mean 2D Gaussian PSF, w(x) is the average of 15 Gaussian
61
4. Image Processing
functions obtained by 3 lines cross-section measurement through each of the
vertical slits, and w(y) is the average of 9 Gaussian functions obtained by 3
lines cross-section measurement through each of the horizontal slits. These two
functions, w(x) and w(y), are averaged over all fitted Gaussian parameters of
both x and y direction where the mean parameters p x and py can be defined
as:
5
w(x) = w(x; p x ) ,
3
1 XX
px =
px,ij ,
5×3
i=1 j=1
px,ij = (Ax,ij , Bx,ij , Wx,ij , x0 )
with
3
w(y) = w(y; p y ) ,
py =
i = 1...5 ,
j = 1...3 ,
(4.10)
3
1 XX
py,kj ,
3×3
k=1 j=1
py,kj = (Ay,kj , By,kj , Wy,kj , y0 )
with
k = 1...5 ,
j = 1...3 ,
(4.11)
where the Gaussian parameter A is the baseline offset, B is the total area under
the curve from the baseline, W is the width of the peak at half height, x 0 and
y0 are the center of the peak.
Finally, to remove the noise, the baseline offset A of the Gaussian was set to
zero and using obtained two-dimensional Gaussian PSF, we generated an image
that is called the PSF image for further deconvolution analysis (Fig. 4.5(f)).
Note that the quality of the PSF is critical to the performance of a deconvolution algorithm and should be come under a very close scrutiny. A noisy
PSF will have a disproportionate effect on the results of deconvolution and
substantial noise will appear in the deconvolved image.
4.2.2. 2D deconvolved image
The two-dimensional image f (x, y) can be represented by a convolution of the
0
0
0
0
real image h(x , y ) with the point spread function w(x , y ) as the following
double integral:
Z +∞ Z +∞
0
0
0
0
0
0
f (x, y) =
h(x , y ) w(x − x , y − y ) dx dy .
(4.12)
−∞
−∞
The aim of deconvolution may be stated in the following way: given the image
(f ) and the point spread function (w), recover the original light distribution
(h). The Fourier transform of this convolution from the spatial domain x, y to
the frequency domain u, v is given by F (u, v), where:
F (u, v) = H(u, v) × W (u, v) .
62
(4.13)
4.2. Deconvolution Processes
2D normalized
image f (x, y)
Fourier transform (F)
F (u, v)
H(u, v)=
regularization
1
2D Gaussian PSF F W (u, v)
W (u,v)
0
0
w(x , y )
h(x, y)
real image
1
W (u,v) >α
1
W (u,v) <α
F (u, v)
W (u, v)
1
W (u,v)
regularization
R
m0 = P f (x, y) dxdy
R 0
m1 = P h (x, y) dxdy
m0 0
h(x, y)=
h (x, y)
m1
P : picture domain
F −1 H(u, v)
h0 (x, y)
Figure 4.6.: Flowchart to illustrate the procedure of the image deconvolution.
A deconvolved image generates from a input image and a PSF.
Herewith, the inverse of the PSF (1/W ) was obtained so that it can be
multiplied by the image F image to get the real image H in frequency domain,
H(u, v) = F (u, v) ×
1
.
W (u, v)
(4.14)
After an inverse Fourier transform of the H(u, v) we get the deconvolved image h(x, y). During the deconvolution process, we used low pass filters for
regularization in the frequency domain to reduce noise in which the higher
frequencies are only partially lost. A more extensive description of the deconvolution theory and applications can be found in several textbooks and
literatures, e.g. (Banham and Katsaggelos 1997, Biemond and Mersereau 1990,
Kundur and Hatzinakos 1996, Bones et al. 1992, Press et al. 1992, Jähne 2002).
All images were deconvoluted using the same PSF. Figure 4.6 illustrates the
procedure flowchart of deconvolution on an image using PSF. This figure should
hopefully make this process a bit more clear. Figure 4.7 shows the correction
of multiple light scattering in an observed image for a porous material sample
by deconvolution process. In this figure, the result of deconvolution before and
after the process is compared with the slit. Figure 4.8 illustrates the sequen-
63
PSfrag replacements
observed slit without
porous material
observed image with
porous material
mean Gaussian PSF
deconvolved image
observed image+slit
intensity
4. Image Processing
deconvolved image+slit
Figure 4.7.: Example of deconvolution results for an observed image with
porous material using the Gaussian point spread function obtained
from averaging eight horizontal and vertical slits distributed over
the cell.
tial image processing for an example image of fingering flow experiment. The
image after deconvolution process (real image) shows the despreading of the
observed image in order to recover the original (real) shape of the flow pattern
image.
In a separate experiment, we compared the width of finger using light and Xray transmission measurements. In this experiment, for X-ray measurements,
the cell was placed between the X-ray source and detector which both moved
vertically. First image was taken by X-ray transmission, then the light source
was moved to behind the cell for light transmission measurement. The results
are presented in Fig. 4.9 that show the water contents along a cross-section
through two fingers as measured with X-ray, LTM with deconvolution, and
LTM without deconvolution. Clearly, the observed width of the fingers is reduced through deconvolution which better reflects reality. This is supported
by the X-ray results in which, due to the less scattering effect of X-ray transmission through porous material in comparison with visible light. A perfect
match between X-ray measurements and the deconvolved LTM image cannot
be expected because of the technical reasons. As measurements were done first
64
4.2. Deconvolution Processes
Width[m]
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
depth[m]
0.1
digital color captured image
(original)
0.2
0.3
0
0.4
0.5
0.1
0.2
0.3
0.4
0
digital color captured image
(background (dry))
0.1
0.6
0.2
0
0.4
0.1
0.5
0.2
0.1
0.2
0.3
digital color captured image
(example of flow pattern)
0
0.3
0.4
0.1
0.2
0.3
0.4
normalized image
(before deconvolution)
0
0.4
0.1
0.5
0.2
0
0.3
0.1
0.2
0.3
0.4
0
1.00
0.4
0.5
0.1
0.88
0.75
0.2
0.3
0.4
0.62
0.50
0.38
0.25
0.12
0.5
relative saturation [-]
0
0.3
0
real image (after deconvolution)
Figure 4.8.: Sequence of the image processing steps. The digital color background and example of flow pattern images are after geometrical
correction from original captured image. The normalized image
is the same flow pattern image but referenced to the dry image
to visualize the water saturation. The real image achieved after
deconvolution using the PSF.
by X-ray and then LTM, there was a time lag of 20 minutes between these two
techniques when we moved the light source to behind the cell for the LTM measurements and hence, the two distributions do not completely coincide.
Finally, the images are prepared to proceed to the next step, quantification
that will greatly depends upon the application. The digitized intensity level
values are converted to water saturation through application of non-linear calibration curve (Eq. 3.7).
65
4. Image Processing
1.0
1.0
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
relative absorption [–](X-ray)
PSfrag replacements
normalized intensity (LTM)
.... deconvolution data
0
0
100
200
300
400
500
600
horizontal distance [pixel]
Figure 4.9.: Comparison of X-ray saturation (blue dots) and light intensity
without deconvolution (red dots) and with deconvolution (green
dots) for a horizontal cross-section through two fingers. There was
a time lag of 20 min between X-ray imaging and light intensity
measurements so the two distributions do not completely coincide.
66
5. Results and Discussion
Using the experimental setup described in the chapter 3, several experiments
were performed with the aim to illuminate and analyze, qualitatively and quantitatively, the fingering flow in sand-air-water systems. Qualitative data, such
as observation of the phenomena and quantitative data, such as the dimensions,
velocity and water content dynamics of the fingers are of primary importance
to understand the type of flow patterns which are presented in this chapter.
The experimental evidences demonstrating gravity-driven instability in porous
Hele-Shaw cell and development of the finger structure are considered in the
first section, following with qualitative descriptions of saturation overshoot and
physical explanation of the finger initiation and spatial path of the fingers. After the qualitative description, quantitative analysis of measurements of the
finger behavior, including dynamics of water saturation and pressure in flow
fingers, dynamics and stabilization of the fingers, finger width, and finger tip
velocity following with further experiments under different conditions are presented.
5.1. Experimental Evidences
Laboratory experiments have shown that the redistribution process following
the infiltration in homogeneous coarse sand is unstable, producing a series of
the fingers. The flow instability is easily observed in a porous Hele-Shaw cell
where a fine textured layer overlies a coarse layer. Fingering occurs when the
infiltrated water meets, during its vertical movement, an interface of great variation of hydraulic conductivity and the flux is less than the saturated hydraulic
conductivity of layers (Fig. 2.9). A capillary barrier effect exists between these
two-layer structure where an inclined coarse layer is overlaid by a fine layer
which the coarse layer tend to impede the flow from fine layer. In this study,
2 − 5 cm of water (depending to used flow rate) was ponded on the surface
of the fine layer and redistribution of the fingers began at about 10 − 15 min
(depending on the interface shape between the fine and the coarse layer) after
starting of infiltration. Qualitative observations imply that finger development
67
5. Results and Discussion
is strongly correlated to the structure of the imbibition front at the onset of
flow redistribution.
5.1.1. Observations
In all experiments, the onset of a stable wetting front in the upper fine-textured
layer characterized by a relatively low saturated hydraulic conductivity was observed which supplies uniformly water to the underlying coarse-textured layer
characterized by a high value of saturated hydraulic conductivity. At the textural interface between the fine and the coarse layer the water moves in preferred
paths or “fingers” at many discrete points induced by infiltrating flow. After visualization of the stable infiltration fronts and subsequent transition to
instability in homogeneous coarse layer, the flow fingers disturbed within the
underlying heterogeneous layer. Figure 5.1 shows a qualitative record of the
advance of the downward growing fingers through an initially dry porous HeleShaw cell by time-lapse images during an experiment with constant flux of
1.2 mm min−1 . The elapsed time is indexed from t = 3 min, defined as the moment when water is uniformly infiltrating downward with a nearly horizontal
wetting front in the depth of 5 cm of the fine layer indicating that the flow
process was stable. When the wetting front reached the critical depth (interface between the fine and the coarse layer shown in Fig. 5.2), the downward
movement of the wetting front was observed to slow down and a pause for few
minutes (t = 12 min). During the pause suction gradients at the interface will
continue to diminish until the water pressure of the bottom layer is archived.
Once the water pressure is archived, the wetting front crosses the interface into
the coarse-grained sublayer and the front becomes unstable and fingering forms
at several locations and penetrated deeper into the coarse layer (more details
of the finger initiation are described in section 5.2). These fingers grew and
propagated in depth interval between 0.07 m and 0.58 m with different velocity,
width and spacing and encountered to middle heterogeneous layer. Obviously,
the increased capillary forces within the fine textured areas of the heterogeneous middle-layer (depth interval between 0.58 m and 1.07 m) were sufficient
to disturb the flow fingers. In this layer finger movement in the vertical direction was significantly reduced and became horizon gradually. Generally, the
overall effect of heterogeneous media is to make the finger less regular. However, as soon as the fine textured heterogeneities disappeared, the conditions
again became favorable for flow fingers and they reappear in the homogeneous
layer below and moved downwards to the cell bottom.
68
5.1. Experimental Evidences
width [m]
width [m]
depth [m]
0
0.2
0
0.4
0.2
width [m]
0.4
0
0.2
0
0
0
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
1.0
1.0
1.0
depth [m]
time:3 min 1.2
1.2
1.4
depth [m]
time:3 min 1.2
time:12 min 1.4
1.4
time:15 min
time:3
min
time:20
min
0
depth [m]
0
0
PSfrag replacementsPSfrag
PSfrag
PSfrag
0.8 replacements
0.8 replacements
0.8 replacements
time:12 min
time:15 min
time:20 min
width [m]
0.4
0.2
time:12
time:20min
min
0
0.4
0.2
depth [m]
time:3 min
time:12 min
time:15 min
0
0.2
1.0
1.2
1.4
time:20 min
0.4
0
0
0
0
0
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
PSfrag replacementsPSfrag
PSfrag
PSfrag
0.8 replacements
0.8
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width [m]
width [m]
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time:30 min 1.2 time:30 min 1.2
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replacements 0.8
width [m]
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time:30 min 1.2
time:50 min 1.4
time:75 min
time:75 min
time:100 min
Figure 5.1.: Photographic sequence of images showing the downward growth of
water fingering observed by light transmission into dry Hele-Shaw
cell at eight different times. Water infiltration with constant water
flux of 1.2 mm min−1 into the uniform medium leads to infiltrating fingers due to a flow instability. Flow instability is induced at
the transition from the fine-textured sand (dark top layer) to the
coarse-textured sand. The fingers are disturbed within the heterogeneous middle-layer and reappear in the uniform layer below.
Fingers in homogeneous coarse layer sand, show several instances
of the fingers merging, as well as a case of splitting. (see more
pictures of this experiment in appendix A.1)
.
69
5. Results and Discussion
capillary barrier
effect region
fine layer
critical depth
homogeneous
coarse layer
heterogeneous
layer
Figure 5.2.: Redistribution of uniformly applied water through fine and coarse
sand. In the left picture, dashed line shows the critical depth of
infiltration during redistribution in a fine sand and extended to
a coarse sand. Unstable flow develops when a wetting front in
the coarse sand exceed the critical depth of wetting. Right picture
shows the destruction of the finger flow channel when it encounters
heterogeneous structured layer.
Observation of the fingering process have repeatedly shown that some fingers do not carry enough water to keep downward growing downwards, and
the number of the fingers decreases with increasing depth. In sand that is not
completely homogeneous, the fingers deviate from a strictly vertical path, and
merged to form larger and faster moving fingers which continued to move downward. When this happens, they come together like the arms of a Y (Fig. 5.1).
These merging and splitting created by small media heterogeneity show the
convergence and divergence structure of flow fingers. After a merger, the continuing finger carries considerably more water than either of the contributors
and the extra water increases its conductivity and, in accordance with the
conservation of mass, its speed grows.
5.1.2. Flow field structure using tracer experiment
After fingers had fully developed, a dye tracer (Brilliant Blue) was added to
visualize local flow velocities. It demonstrates the high separation of a mobile
70
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5.1. Experimental Evidences
width [m]
width [m]
0.2
0
0.4
0
0
0.1
0.1
0.2
0.3
0.4
0.5
depth [m]
depth [m]
0
0.2
0.4
0.2
0.3
0.4
0.5
mobile core
immobile fringe
Figure 5.3.: Infiltration of dye tracer into stabilized water fingers highlights the
separation of the water phase into a mobile component (core) and
an immobile one (fringe).
and an immobile component of water within flow pathways of the fingers which
has implications for solute transport. This shows the high velocity in the
center and stagnant water at the periphery of the fingers. Here it could be
hypothesized as separation of a portion inside the finger (from here on called
the “finger core”) with convective gravity driven flow and a portion at the
boundaries of the finger (from here on called the “finger fringe”) with slow
and diffusive flow (Fig. 5.3).
Figure 5.4 shows photographs of the dye wetting front patterns taken at different times during the second cycle of infiltration into previous water fingers
of the experiment shown in Fig. 5.1 with the same flow rate. The redistribution time is indexed from t = 110 min, defined as the moment when dye was
penetrating into top fine-textured layer. As soon as the dye reached to the
fine-coarse interface, at t = 115 min, dye started flow into the old fingers paths.
Two additional fingers were emerged in the coarse layer between the previously
generated fingers, one in the middle and the other at the third position from
the right side. Dye fingers gradually diffused into the heterogeneous layer and
found the previous finger pathways at the bottom of the cell.
71
5. Results and Discussion
width [m]
width [m]
depth [m]
0
0.2
0
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0
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0
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time:110 min
time:110 min
1.4 time:115 min
time:120 min
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time:125min
min
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min
0
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0
0
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0
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0.8 replacements
width [m]
width [m]
width [m]
width [m]
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1.0
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depth [m]
depth [m]
1.2
time:140 min 1.2 time:140 min 1.2 time:140 min 1.2
time:150 min 1.4
1.4 time:150 min 1.4 time:150 min 1.4
time:175 min
time:175 min
time:175 min
time:150
time:140
time:200 min
time:175 min
time:200 min
time:200min
min
time:200min
min
Figure 5.4.: Development of the dye fingers in the experiment with Brilliant
Blue tracer infiltration with constant flux of 1.2 mm min −1 . The
dye flows only through the core and follows the same paths as the
water. The locations of the core region as highlighted by the blue
dye tracers were almost the same as in the water infiltration experiment. It shows the persistence of dye fingers into previously
established path by water fingers. (see more pictures of this experiment in appendix A.2)
72
0
relative saturation [–]
depth [m]
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5.1. Experimental Evidences
depth
[m]
time:11.5 min
time:11.5 min
time:11.5 min
time:115 min
time:115 min
time:115 min
time:11.5 min
width [m]
width [m]
width [m]
width [m]
time:120 min
time:120
time:120
min
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min
1.00
0
0.1 0.2 0.3
0
0.1 0.2 0.3
0 0.1 0.2 0.3
0 0.1 0.2min
0.3
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time:125 min
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min
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min
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min
0
0
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a
a
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0.1 time:125 min 0.1
0.1
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b 0.2
b 0.2
a 0.2
0.2
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c 0.3
c 0.3
b
0.38
0.3
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d 0.4
d 0.4
d 0.4
c
0.4
0.12
relative saturationrelative
[–]
saturation
relative
[–]
saturation
[–]
a
c
b
d
Figure 5.5.: Sequence illustrating the development of instability in an initially
dry sand. a) advancement of stable front (t = 11.5 min), b) onset
of instability (t = 12.5 min), c) development of the fingers (t =
13.5 min), and d) developed fingers (t = 17 min). The times refer
to the time after starting of water infiltration.
5.1.3. Finger development in initially dry sand
To generalize the description of the finger flow field structure, an illustrative
example of the transition to instability and subsequent development of the fingers is shown in Figure 5.5. The initial wetting front begins from many separate
points that rapidly influences into a stable front (Fig.5.5a). Behavior of the
stable wetting front changes entirely with continuous slug of infiltration and
redistribution begins and the onset of instability is observed (Fig.5.5b). The
front shown in Figure 5.5b exhibits four growing fingers that have developed
from stable front. The second finger from the left-hand has the smallest length
of the three fingers and it shows less competition to develop. After the onset of
instability, individual fingers begin to rapidly develop, grow and forming distinct fingers (Fig.5.5c). The developed fingers are more likely separated fingers
with very fast vertical movement and a downward growing tip consist of fully
water spanning that will be referred to as the finger tip. Finger tips are essentially short, locally saturated water that partially drain along their trailing
edge as they propagate downward. This phenomenon is called “saturation
overshoot”.
5.1.4. Saturation overshoot
As a finger grows downwards from the textural interface, there is a narrow
zone of about 5 − 7 cm (depends on the applied flux) at the finger tip where
the water saturation increases rapidly. These maximum values are required to
73
5. Results and Discussion
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depth [m] 0.4
0.5relative saturation[–] 0.5
depth [m] 0.4
0.25
0.3
0.4
0.5
0.12
0
relative saturation[–]
width [m]
Figure 5.6.: Fingered flow in a homogeneous initially dry sand as observed with
transmitted light (left). The same image but referenced to the dry
image as normalized image to visualize the water saturation (middle) and image after deconvolution using the PSF (right) shows the
highly localized flow paths that originate from the flow instability
in the uniform part of the medium. Note the saturation overshoot
in the tips.
advance the water front into the dry porous medium. This pattern consists of
a region directly behind the wetting front with a high water saturation called
finger tip, followed by another region with a lower water saturation called finger
tail. The water saturation in the tip of the finger is close to saturation (∼ 0.8)
and drops at a short distance behind the advancing tip to ∼ 0.5 (these values
are for the experiment with flux jw = 1.2 mm min−1 and depend on the applied
flux). An example of the downward growth of the fingers in an initially dry
porous medium and typical finger moisture content structure during the water
infiltration after 20 min infiltration is shown in Figure 5.6. This figure shows
the typical finger moisture content structure during the rapid growth stage
with saturation overshoot in the tip. The bluest color at the tip indicates the
highest water saturation.
5.2. Physical Explanation of the Finger Initiation
The current understanding is that fingers develop in coarse-textured media if
the driving force, typically gravity, dominates the capillary force. The relative
weight of these two forces is given by the bond number, Bo = ρ g r 2 /σ, described
in section 2.2. The characteristic length scale r which is related to the size of the
pores, indicates that fingers occur only in coarse-textured porous media having
74
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5.2. Physical Explanation of the Finger Initiation
fine sand (0.025-0.25 mm), Ks=12 cm/h
15 cm
2 µm
100 µm
1 mm
pore interface
between two layers
3 mm
coarse sand (0.63-1.25 mm), Ks=873 cm/h
15 cm
1 mm
100 µm
2 µm
Figure 5.7.: Photographs of the grain size distribution and particle shapes for
the fine (top) and the coarse (bottom) sands used in the experiment. The pore interface acts as a water flow gate between two
less permeable fine toplayer and more permeable coarse sublayer.
The length scale for images are given below of images.
pores above a critical size. In experiments the fingering phenomenon is actually
found in granular media with a mean grain size above 0.5 − 1.0 mm in diameter
(Diment and Watson 1985, Glass et al. 1989b). A developed conceptual model
of Jury et al. (2003) for predicting the development of unstable flow during
redistribution predicts that all soils are unstable during redistribution, but
shows that only coarse-textured soils and sediments will form fingers capable
of moving appreciable distances. It was shown by Flekkøy et al. (2002) that
the width of the fingers increases with decreasing Bo. A microscopic view of
the grain size distribution as well as particle shapes of the fine and coarse sands
used in the experiment is presented in Fig. 5.7.
The dynamics of water pressure may be the most important in the finger
initiation and formation process because of its role in determining the boundary condition at the wetting front. Wang et al. (2000) measured the water-
75
5. Results and Discussion
entry pressure with a water ponding method for water-repellent soil and with a
tension-pressure infiltrometer method for both water-repellent and wettable
soils. Using the concept of dynamic capillary pressure, DiCarlo and Blunt
(2000) obtained a self-similar solution to a moving finger with a curved interface, where the capillary pressure depends on the velocity of the moving
interface.
A physical argument can be used to illustrate why the process of water
infiltration into a vertical profile of two-layer sand (fine-over-coarse) initiates
unstable fingered flow. We consider the wetting front propagation into a twolayer system consist of a fine overlies a coarse sand, separately for each layer.
In our applied set-up, the dynamics of water saturation inside the fine toplayer
that is important to understand the finger propagation was not possible to
measure using LTM, because light could not pass through the very fine sand
particles. For this purpose, we measured the water pressure (p w ) inside the
fine toplayer using three pressure sensors installed at different locations over
the 20 cm depth of the fine toplayer, indicated by red, green, and blue circles
in Fig. 5.8 (top right).
5.2.1. Distribution of flow in the fine-textured toplayer
Continuous water infiltration creates a ponded water over the fine layer surface (z = 0). A wetting front forms within the initially dry toplayer and
advances downward continously, in response to the combined matric and gravitional potential gradients. The results of water pressure measurements into the
fine toplayer show that when the wetting front reaches the interlayer interface
(z = z2 ), it pauses at that interface (1 in Fig. 5.8, top right and bottom left).
During that pause, potential gradients in the toplayer continue to induce flow
toward the interface. This process acts as a capillary barrier system. Under
this condition, when the pores in the fine layer above the interface are water
filled, while the coarse sand is nearly dry, the hydraulic conductivity of the fine
sand can be several orders of magnitude larger than in the coarse sand. The
wetting front that builds up above the interface flows up within the toplayer.
In this state a quasi-hydrostatic pressure distribution in the fine layer is established while in the coarse-textured layer matric potential is very negative.
Due to continuous infiltration in the vertical direction, at a certain distance the
toplayer is sufficiently saturated as well as its hydraulic conductivity increased,
that capillary forces in coarse layer pull forward the moisture from fine into the
coarse layer.
76
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jw
water pressure
wetting front distribution into fine layer
w
z2
A
jw=0.6 [mm/min]
jw=0.6 [mm/min]
jw=0.6 [mm/min]
jw=0.6 [mm/min]
10 cm
fine layer
jw=0.6 [mm/min]
A
B
C
C
1
pwt
4
3
1
jw=0.6 [mm/min] jw=1.2-4.8 [mm/min]
2
3
4
start of fingers
initiation
coarse layer (dry)
experiment data
jw=0.6 [mm/min]
0
numerical simulation
5
-5
4
-10
3
-15
2
-20
1
-25
0
4000
time [sec]
6
flow rate [mm/min]
2
B
flow rate [mm/min]
matric pressure [cm]
z
fingers form
depth
wetting front
flow up
matric pressure [cm]
0
z1
5.2. Physical Explanation of the Finger Initiation
8000
12000
16000
20000
24000
time [sec]
Figure 5.8.: Evaluation of wetting front development into a vertical profile consisting of two-layer sand. Top left: Formation and propagation of
the finger during redistribution, when the pressure decreases toward the surface. The dry region below the wetting front must
have a threshold water pressure (pwt ), which allows water to enter the dry region. Top right: Sketch of wetting front distribution in the overlying fine-textured layer during infiltration starting
with constant flux 0.6 mm min−1 and additional infiltration with
multi-step changing of flux in the range 1.2 − 4.8 mm min −1 . Bottom left: The real measured water pressure using three pressure
sensors installed at three locations inside the fine toplayer indicated by red, green, and blue circles in top right sketch. In the
overlying fine layer, pressure drops upon initialization of new fingers. In the case of both developing and initiating of the fingers,
the water pressure inside the fine layer decreases slightly. Bottom right: One-dimensional numerical simulation modeled data
of wetting front distribution into the fine toplayer obtained with
the HYDRUS model. The time refers to the time after starting of
water infiltration.
77
5. Results and Discussion
This break down of the interface happens in the crossing of the hydraulic conductivity function between the fine and the coarse textured sands is shown in
Fig. 2.3a. With entering the moisture into the coarse layer, pressure in the
overlying fine-textured layer drops upon initialization of a new finger (2 in
Fig. 5.8, bottom left).
A simple one-dimensional numerical simulation using a hydraulic model of
van Genuchten-Mualem parameterization showed the same results as found
in the experiment measurements. The simulation results obtained with the
HYDRUS model of Ŝimûnek et al. (2005) using a profile consisting of two layers
and with the upper fine sand layer having a depth of 20 cm and the lower
coarse sand layer having a depth of 130 cm. The hydraulic parameters values
of the used sand were adapted from Cheng (2004). Two graphs in Figure 5.8
(bottom left and right) show the observed and modeled pressure head in the
fine layer. The numbers indicate the wetting front distribution in the fine layer
in comparison with the numbers indicated in top left sketched figures. This
simulation shows that Richards’ equation can explain the stable wetting front
distribution into fine toplayer. We found that in the case of both developing
and initiating of the fingers, the water pressure in the fine layer decreases (see
the slight pressure decreasing after drops in Fig. 5.8, bottom left).
5.2.2. Initiation of the finger in the coarse-textured sublayer
With percolating of the infiltrating water from fine into the coarse layer the
suction at the interface falls below a threshold of water pressure (p wt ) into the
pores in the sublayer, and the wetting front then begins to move into the sublayer. Because of inevitable spatial variability, what typically happens below
the interface is that water first penetrates the sublayer at distinct randomly
distributed locations, rather than uniformly over the entire area of the interface. Such locations may not immediately admit the full flux deliverable by
the toplayer; hence the suction at the interface may continue to decrease. The
water pressure at the interface between wet and dry zones of developed wetting front is at a threshold water pressure, which allows water to enter the
dry region. Because so much of the void space suddenly fills as this threshold
is reached, the conductivity of the medium changes from negligible to high.
Hence, the wetting front advances into coarse-textured sublayer and the water
pressure behind it is lower than the pressure near the wetting front.
The downward advance of the wetting front within the sublayer becomes
“unstable” with a spatially rapid movement and the front will grow into commonly called fingers. The system described is illustrated in Fig. 5.8 (top left).
78
5.3. Why does the Saturation Overshoot Occur?
This figure shows the formation and propagation of two fingers during redistribution. Three wetting front positions during passage of the finger have been
indicated by A, B, and C. The graph in the left side of this figure shows that
during finger advancement, the water pressure behind the wetting front is lower
than the pressure near the wetting front and the lateral flow behind the wetting
front decreases slightly the matric potential of the sand adjacent to the finger.
For example, in position C of the wetting front, the pressure near wetting front
is higher than behind the wetting front (point B), then the lateral flow at point
B occurs and finger develops more.
5.3. Why does the Saturation Overshoot Occur?
Several experimental observations by Glass et al. (1989c), Selker et al. (1992b),
Liu et al. (1994a), Geiger and Durnford (2000), DiCarlo (2004) have shown
that gravity-driven fingers exhibit saturation overshoot. From these observations, it has been proposed that saturation overshoot is a necessary and sufficient prerequisite for gravity-driven fingering, i.e., gravity-driven fingering will
occur if and only if saturation overshoot occurs. Thus a correct understanding
of the physics controlling saturation overshoot is necessary for understanding the larger question of gravity-driven fingers (Eliassi and Glass 2001; 2002,
DiCarlo 2004). Figure 5.9 (left) shows sketch of a fingering flow path and the
associated saturation along flow fingers with saturation overshoot in the tip.
Selker et al. (1992b) have shown that saturation overshoot is associated with
an overshoot in water potential which decreases from the tip to tail of the
fingers. Geiger and Durnford (2000) found pressure overshoot for initially dry
coarse sands at large range of flow rates, and for initially dry fine sands at
moderate flow rates, but with no pressure overshoot for initially wet sands or
for the fine sands and low flow rates. Note that saturation overshoot occurs
only during the unsaturated infiltration, causing unstable flow. It has been
found that saturation and pressure overshoot are eliminated with increasing
initial water content (Glass and Nicholl 1996, Bauters et al. 2000).
DiCarlo (2004) summarized the newer developments which focus on the
nature of the saturation overshoot in the finger tip. He worked with lighttransmission in a narrow column that corresponds to a one-dimensional system
and showed that saturation overshoot is strongest for intermediate fluxes. He
found that saturation overshoot ceases below a certain minimum and above a
certain maximum infiltrating flux. This limit flux depends greatly on the grain
smoothness and initial water content of the media and slightly on the mean
79
5. Results and Discussion
saturation
tail
tip
v
z
Figure 5.9.: Left: Sketch of a fingering flow path and the associated saturation
within the flow path. Saturation overshoot occurs when the tip
saturation is greater than the tail saturation. Right: Saturation
profile versus depth for six different applied fluxes in initially dry
porous column measured using light transmission. At the highest (11.8 cm min−1 ) and lowest (7.9 × 10−4 cm min−1 ) fluxes the
profiles are monotonic with distance and no saturation overshoot
is observed, while all of the intermediate fluxes exhibit saturation
overshoot. (Figure 1 and 5 from DiCarlo (2004))
grain size. He measured the saturation overshoot as a function of infiltrating
flux, mean grain size, grain sphericity, and initial water content. The observations of DiCarlo (2004) can be readily understood: (1) At very low flow rates
(in one-dimensional) the wetting of the solid surface is fast compared to the
speed of the advancing front. Therefore, the pressure overshoot is smaller and
hence, no overshoot is generated. Conversely, for very high fluxes the porous
medium is already completely saturated also behind the tip, so no overshoot
can occur even if the matric potential is positive (Fig. 5.9, right). (2) Saturation overshoot decreases quickly with increasing initial water content. (3)
Saturation overshoot is much less for angular sand grains than for spherical
sand grains.
A crucial question for understanding fingering flow is: On a physical level,
why does the saturation overshoot occur (i.e., why is the tip saturation greater
than the tail?) To answer this, we consider two quasi-equilibrium and rapid
infiltration regimes from the microscopic perspective of water-interface illus-
80
5.3. Why does the Saturation Overshoot Occur?
Figure 5.10.: Microscopic view of water-air interface (upper row) and pressure
(lower row) for infiltration into an initially dry, water-wet medium
with different fluxes: (left) static equilibrium (red lines) and low
flux (blue lines) and (right) very high flux. Notice that the slope
of p(x) is related to the flux and to the viscosity of the fluid.
Hence it is steeper for water than for air. (Figure 5.36 from Roth
(2006))
trated in Figure 5.10. The below description is adapted from Roth (2006)
(chapter 5).
“First consider the regime sketched in frame A of Figure 5.10. In static
equilibrium, with the fluids at rest, the pressure is constant within both phases
and discontinuous at their interfaces. The pressure jump, the matric potential
ψm = pw − pa is determined by the mean curvature as described by the YoungLaplace equation (2.15). Slowly increasing the pressure gradient in the situation
shown in Figure 5.10, at first only leads to a readjustment of the interface but
not to a water flux: the interface is pinned. Further increase leads to a sudden
jump of one of the interfaces across the large void and the subsequent rapid
invasion of smaller pores ahead until the accumulated pressure is released.
81
5. Results and Discussion
With the gradient sufficiently high, many temporally overlapping jumps occur
at different locations and give rise to a continuous flux. Increasing the gradient
further will lead to a proportional increase of the flux as described by the
Buckingham-Darcy law (Eq. 2.8).
Next, consider frame B of Figure 5.10 which represents a much higher water
flux. The velocity here is so high that the water-air interface moves faster than
the water-solid interface, despite the fact that water is the wetting fluid. As a
consequence, the mean curvature of the water-air interface becomes positive.
Hence, the very small value of the capillary number, pressure builds up in the
water phase such that the matric potential ψ m becomes positive. Notice that
this is another instance where ψm does not reflect pore size and wettability
alone, as is the case for static equilibrium, but that it contains a dynamic
component. Notice that these two cases are fundamentally independent of
each other. A common feature of all situations where a dynamic component
of ψm becomes significant is that the relation between θ and ψ m , as it is given
by the soil-water characteristic, breaks down because the latter is a relation
for static equilibrium. As a consequence, also Richards’ equation breaks down
since it is based on quasi-static equilibrium states”.
We use this description to explain the saturation overshoot in flow fingers
when water is rapidly infiltrated into a coarse dry sand. At the given velocity
of water, capillary suction is too weak and the hydraulic conductivity in dry
coarse textured sand is too low to remove the invading water. It piles up in
the tip, thereby increasing the pressure gradient, until the sum of capillary and
pressure gradient forces balance the invading force. At the invading front the
mean curvature of the air-water interface may even become positive and hence,
also ψm becomes positive. Despite the high velocity of water (≈ 1 mm s −1 )
the capillary number of the air-water system, Ca = ηv/σ, is very low (≈
10−5 ) meaning that capillary forces still dominate viscous forces. The increased
pressure leads to a higher water saturation, and consequently to the observed
saturation overshoot in the finger tip.
Weitz et al. (1987) performed an ingenious experiment to reveal the dependence of ψm on velocity. They found that when an immiscible wetting fluid
displaces an immiscible non-wetting fluid in a long tube of a porous medium
through forced flow, the wetting angle is dependent on the forced velocity of
the interface. They measured the dynamic pressure versus the local interface
velocity for large range of velocities. They found that, depending on the flow velocity, the pressure jump across the interface, i.e., the matric potential ψ m , was
positive, zero, or negative, in accordance with the sketch in Figure 5.11.
Many approaches have been undertaken to explain saturation overshoot.
82
5.3. Why does the Saturation Overshoot Occur?
Figure 5.11.: Left: Measured pressure drop across the porous medium as
a function of interface position for three different velocities.
The pressure jumps as the interface moves through the porous
medium, and reflects the magnitude of the pressure. Right: Pressure as a function of velocity. At low velocity, reflecting the fact
that wetting properties cause the water to imbibe into the porous
medium. However, at high velocity an additional driving pressure
forces the interface through the porous medium. The dyne is the
unit of force in the cgs system and equals 10 −5 N. (Figure 1 and
2 from Weitz et al. (1987))
It has been shown that the overshoot cannot be described by the conventional Richards’ equation. Recently, Eliassi and Glass (2001) have argued
that Richards’ equation with standard non-monotonic pressure saturation and
relative permeability curves cannot produce solutions with saturation overshoot. Nevertheless, Eliassi and Glass (2002) considered three possible additional continuum terms to be added to the Richards’ equation and from
numerical simulations using one of them (Eliassi and Glass 2003) produced
saturation patterns which qualitatively matched up with those seen in gravity. Glass and Yarrington (2003) considered a mechanistic approach based on
forms of modified invasion percolation to simulate gravity-driven fingers using
pore-scale modeling. DiCarlo (2004) also discussed how the measured saturation overshoot is inconsistent with a continuum description of porous media
but qualitatively matches well observations and predictions from discrete porefilling mechanisms.
The next question then arises: Why does the saturation, and presumably
also the matric potential, decrease behind the tip? The additional pressure is
83
5. Results and Discussion
required to overcome the “entrance resistance” posed by the slow wetting of the
wetting. Behind the tip, no such resistance is present anymore, the medium is
already wet. The tip saturation is basically the minimum saturation available
for a certain wetting front velocity and media. If this saturation is less than the
tail saturation, the saturation increase will be continuous, but if it is greater
than tail saturation, saturation overshoot will take place.
5.4. Dynamics of Water Saturation and Pressure
In this section we describe simultaneous measurements of the water saturation
and water potential within fingers during the redistribution.
The dynamics of water saturation within the finger was analyzed at different
locations during the passage through the cell. Figure 5.12 shows the temporal
dynamics of water saturation for a small area inside the finger core at four fixed
locations during the passage of the finger the tip at a temporal resolution of 2
minutes. Two distinct zones within the vertically elongated core area can be
distinguished. At the tip, there is a zone with a water content near saturation (∼ 0.8), followed by a drier zone, where the water saturation of the finger
core decreased to ∼ 0.5. We observed a minimum of water saturation within
the finger core immediately behind the tip which would not be predicted by
classical theory. With increasing from distance to the tip, the water content
increases slightly to arrive at a steady state (the difference in water saturation
is by the spatial heterogeneity of the small area selected inside the finger core
at different depths). This clear experimental finding was not reported in previous works, however, it can be explained using standard theory. Behind the
tip there is an lateral gradient which induces horizontal flow. This could be
measured by monitoring a horizontal transect of water saturation at a fixed
location during the passage of a finger (see Fig. 5.16). With increasing from
distance to the tip this lateral flux decreases as will be discussed further in section 5.5. Since the water flux into the finger is constant the additional lateral
flux component behind the tip reduces the vertical flow component and leads
to a reduced water content according to the law of Buckingham-Darcy. The
graph on top right in Fig. 5.12 shows the vertical water profiles produced from
water distribution inside a finger at four different depths. Again, finger has a
wet tip and drier tail during its traveling downward. Figure 5.13 shows the
water saturation versus time for three small areas within a finger during the
passage through the cell. The water saturation in area 1, which exists inside
the finger core, changes as described above with a saturated tip, a minimum
84
5.4. Dynamics of Water Saturation and Pressure
width [cm]
width [cm]
width [cm]
width [cm]
0
0
0
0
5
10
5
10
5
10
5
saturation [–]
10
0
0.5
1.0
depth [cm]
PSfrag replacements
PSfrag replacements
PSfrag replacements
PSfrag replacements
PSfrag replacements
0
0
0
0
0
width
[cm]
10
10
10
10
10
depth [cm] depth [cm] depth [cm] depth [cm]
20
saturation [–] 20
saturation [–] 20
saturation [–] 20
saturation [–] 20
30time:17 min 30 time:17 min 30 time:17 min 30 time:17 min 30
time:19 min 40
40 time:19 min 40 time:19 min 40 time:19 min 40
time:21 min
time:21 min
time:21 min time:21 min
time:19 min
time:23 min
time:17 min
time:23 min
time:23
min time:23
min time:21 min
time:23
min
0
10
saturation [–]
1.0
PSfrag replacements
20
30
40
50
60
70
80
90
100
110
1.0
I
III
II
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0
10
20
30
40
50
60
70
80
90
100
110
time [min]
Figure 5.12.: The temporal dynamics of water saturation inside the finger core
during the passage of a finger at four fixed depths at a temporal resolution of 2 minutes. In bottom graph, region I represents
the period of rapid passage of the finger tip with a water content
near saturation; region II is the finger tail portion with onset of
a minimum value behind the tip followed by a slightly increasing;
and region III represents the period in which water saturation
becomes to a quasi-stable state. The depth profiles of the water saturation distribution in progressing finger at four different
depths are shown on the graph top right. The difference in water
saturation is by the spatial heterogeneity within the finger.
behind the tip and a slightly increase again towards a constant value at large
distance from the tip. The area 2 exists inbetween the core and the fringe.
Its saturation starts to change before stabilization of the finger core. After an
increase in saturation, it decreases during lateral movement of moisture from
the core to the fringe until finger is developed. The saturation in area 3, which
85
5. Results and Discussion
area 1
area 2
area 3
PSfrag replacements
saturation [–]
0.8
0.6
PSfrag replacements
saturation [–]
time [min]
area 1
area 2
area 3
0.4
0.2
0
0
20
40
60
80
core
fringe
1.0
3
2
1
tip
100
time [min]
Figure 5.13.: Water saturation versus time within three small areas: 1) inside
the core, 2) inbetween the core and the fringe, and 3) inside the
fringe, at depth 10 cm for the finger shown in Fig. 5.12 during
the passage through the cell.
exists inside the fringe region, changes when the moisture hits to this area.
During lateral expansion of finger that is by suppling water from the core to
the fringe zones of developed finger, the saturation inside this area does not
change as this area plays as a gate between the core and fringes with uniform
moisture content.
Many experimental runs were attempted to measure the water pressure using
tensiometers. We measured the water potentials along many positions within
the cell, but only few resulted in accurate tensiometer readings (no bubbles or
leaks from any tensiometers). The largest problem for these measurements was
that the finger trajectory was impossible to control and hence, the hitting of
the fingers with tensiometers was uncertain. In our experiments, we measured
water pressure behind the finger tip. Because of two reasons, we could not
measure the water pressure in finger tip with saturation overshoot. First, with
water pouring into the plastic tube of tensiometer, the dry sands, in front of
sensor, would suck the moisture into the cell and the area around the sensor
is wetted and herewith the saturation overshoot does not occur in pre-wetted
sand. Second is that the suction head for water in initially dry sand is relatively
high and hence, the finger tends to travel, to dry regions of sand and thus, the
trajectory of the finger changes. Therefore, tensiometers were saturated when
86
5.4. Dynamics of Water Saturation and Pressure
-1
1.0
-2
-3
-4
0.6
-5
-6
0.4
-7
matric head [cm]
saturation [–]
0.8
-8
0.2
-9
PSfrag replacements
0
-10
0
20
40
60
80
100
120
time [min]
Figure 5.14.: Plot of a simultaneous measurement of water saturation inside a
finger core from initial finger tip and water potential inside the
core after the initial finger tip passed through the tensiometer
position at 50 cm depth below the sand surface. The jump in the
pressure results at the start of measurements for a short time,
which is the reaction time of the tensiometer, was for tensiometer
reading reached equilibrium.
the finger tip passed from the position that tensiometers were placed.
Figure 5.14 gives the results of water saturation versus time inside a finger
core from initial finger tip and a simultaneous measurement of matric head
(h)a inside the core after the initial finger tip passed through the tensiometer
position. Upon hitting the tensiometer with the water finger with inserting
the transducer into the tensiometer tube, after the passage of the finger tip,
the tensiometers momentarily came up from the extreme negative matric head
value of the air dry sand (we measured over −50 cm) to a value of approximately
∼ −6 cm. This jump was during a short time which is the reaction time of
a
In soil physics, often the matric potential is expressed in terms of “matric head” (h =
ψm /ρg). The idea is that h can be interpreted as equivalent height of a water column;
negative values correspond to a hanging column.
87
5. Results and Discussion
the tensiometer where the pressure quickly increases to about -1 cm and after
∼ 1 minute (reaction time), the tensiometer reading reached equilibrium. After
this increase, the matric head slightly increased due to the horizontal widening
of the finger and at the same time, water saturation was increasing slightly.
At a large distance behind the tip, the matric head came to an equilibrium
state of −3 cm (at time about 90 minutes). These differences in moisture
content inside the tip, core, and fringe zones can be explained simply in terms of
hysteresis in the pressure-saturation curve, as the finger cores are on the drying
branch of the pressure-saturation curve and thus hold much more water at equal
pressure than outside the fingers, where the sand is on the wetting branch. Our
experimental data offer a good opportunity to examine this further. This is
detailed in below section.
5.4.1. Dynamics and stabilization of the fingers
One of the crucial ingredients for understanding the mechanism of fingered
flow is the hydraulic state (θ, ψm ) during the passage of the finger tip governed
by the hysteretic soil-water characteristic as sketched in Fig. 5.15. This figure
shows the hysteretic behaviors of moisture conditions in the finger core and
the surrounding finger-fringe zone. We consider three locations: x 0 on the
centerline of the core, x1 at the outer limit of the core, and x2 at the outer
limit of the fringe. Initially, the sand is dry hence ψ m is strongly negative and
θ very small. As the finger moves downward and with the tip approaching
x0 , both θ and ψm increase rapidly and move below the static wetting branch
state of the soil-water characteristic until they reach the maximal values θ tip
tip
and ψm
, respectively, with the center of the tip passing over x 0 . This is a
non-equilibrium process. Note, that during this highly dynamic process ψ m
in the finger tip does not reflect the pore-size or the contact angle and the
tip
values of θ tip and ψm
are not located on the static soil-water characteristic
(as implied by Fig. 5.15 for simplification). This is also the reason why the
observed overshoot cannot be reproduced by Richards’ equation which relies
on a static θ(ψ) relation.
After the passage of the finger tip, the water content drops immediately behind the tip to a minimum value for a very short time and brings the hydraulic
0
core ). As it cannot stay
state on a point above the static curve (θ<θ core , ψm
outside the static state, it has to back to an equilibrium state. Hence, the water content increases at x0 which brings the hydraulic state on the desorption
branch of the soil-water characteristic. After the tip passes, the material is wet
and no additional force for pushing the phase boundary through the porous
88
fringe
5.4. Dynamics of Water Saturation and Pressure
PSfrag replacements
core
x2 x1 x0
core
ψm
0
tip
tip
core
ψm
tip
ψm
θ fringe
θ core θ tip
θ
ψm
Figure 5.15.: Sketch of the evolution of hydraulic states during the passage of a
finger: at location x0 , where the center of the tip passes through,
(θ, ψm ) moves from the initial very dry state to the maximum
tip
(θ tip , ψm
) along the wetting branch of the water characteristic
core ) along the dryand then, behind the tip, towards (θ core , ψm
ing branch with an immediately drop to a minimum value for a
very short time. During the same time, the state at x 1 evolves
core ). As the states at x and x
monotonically towards (θ fringe , ψm
0
1
core on different hysteresis loops,
approach the same potential ψm
the radial water flux ceases even though the water contents are
quite different. This prevents further rapid radial growth of the
finger and stabilizes it. The gradient between x 1 and x2 causes
only minor water flux because of the low hydraulic conductivity
and by the limited supply of water. The gray area represents the
main wetting and drying boundaries at equilibrium state of the
soil-water characteristic curve.
medium is required anymore. At this point, water infiltrates radially, driven
by the large hydraulic gradient. Thereby, (θ, ψ m ) approaches the metastable
core ), whose time-scale for change is very much longer than the
state (θ core , ψm
one for reaching it. During the same time of approaching to the metastable
89
5. Results and Discussion
state, the state at location x1 evolves monotonically along the wetting branch
core ). With the matric potential at x and x approaching the
towards (θ fringe , ψm
0
1
core , the radial gradient decreases and with it the corresponding
same value, ψm
flux, and hence, the finger becomes stabilized.
On a time-scale that is much longer than that of the finger’s creation, it
continues to expand. This is caused by the large radial gradient in the fringe of
the finger, between x1 and x2 . However, with the water content already low at
x1 and further decreasing in the fringe, hydraulic conductivity decreases very
rapidly and with it the radial flux.
This hysteretic behavior on the soil-water characteristic curve describes the
mechanism of the development of a saturated finger tip, and the subsequent
drainage behind the tip. Since the finger core is initially wetter than the fringe
region, its hydraulic conductivity is higher than the fringe zone. Thus, tip, core
and fringe regions having essentially different water characteristic curves. The
tip and the fringes of the finger are on a wetting curve, while the remaining
core of the finger is on a drying branch.
5.5. Finger Width
Based on Hele-Shaw cell experiments, many scientists have studied the theory
of wetting front instability to describe and predict the finger width in homogeneous soils (e.g., Raats (1973), Philip (1975a;b), Parlange and Hill (1976a),
Diment et al. (1982), Hillel and Baker (1988), Glass et al. (1989a;c)) and verified their predictions experimentally (e.g., Baker and Hillel (1990), Liu et al.
(1994b), Bauters et al. (1998)). A long-term lateral spreading of the fingers
emplaced in dry media has also been widely observed. Glass et al. (1989c),
Glass and Nicholl (1996) and DiCarlo et al. (1999), for example, showed that
water flow continuing for tens or hundreds of hours results in large lateral expansion of the fingers. DiCarlo et al. (1999) noted that the lateral expansion
of the fingers was linearly related to the square root of time. They suggested
that the lateral spreading was mainly due to vapor diffusion, and presented an
analysis which appeared to show a correspondence between the lateral growth
of water wetted area, and the expected rate of vapor transport between a saturated to dry division.
In our experiments, as expected in the initially dry sand, the finger pattern began to widen immediately after infiltration. We observed that during
downward movement of a finger under continuous infiltration, the width increases with time. We found a gradual change in the water saturation across
90
5.5. Finger Width
boundaries between the finger core and the finger-fringe zone. According to
the core and fringe regions development having essentially different moisture
characteristic curves, the finger width growths can be classified into two lateral
expansion stages:
Stage I:
Core Stabilization: A fast expansion of the finger width to stabilization of the finger core that conducts most of the flow by effects
of infiltrating forces on a short time scale. In this stage, water
infiltrates radially driven by the large hydraulic gradient between
the center of the finger core and the outer limit of the core. The
radial gradient decreases over time and with it the corresponding
flux, approaching the center and the outer limit of the finger core to
the same pressure. This stage of the development can be described
by hysteretic behavior on soil-water characteristic curve where the
tip is on a wetting branch and the core behind the tip on a drying branch. The radial water flux ceases when core and the outer
limit of the core approach the same potential on different hysteresis
loops, and hence, the finger core becomes stabilized.
Stage II:
Fringe Expansion: A slow expansion associated with growth
by effects of capillary forces and hydraulic gradient between the
outer limit of the core and fringe zone on a time scale much longer
than core stabilization. This low hydraulic gradient decreases the
hydraulic conductivity very rapidly and with it the radial flux. In
this stage, water flows vertically in the stabilized core and laterally
in the fringe zone which means that wetting fronts leave the fingers
and move laterally into the dry sand on either side of the finger core
areas. Over time, slow lateral movement of moisture from finger
core regions creates a less saturated surrounding fringe region and
the finger continues to expand.
A time series of intensity measurements of a lateral cross-section at a fixed
location during the passage of a finger is shown in Fig. 5.16 (a), which shows a
maximum in the center of the finger (core areas) and slow water lateral movement from core regions to the less saturated surrounding resulting in significant
lateral expansion. This plot illustrates the dynamics of water saturation within
a finger from the initially saturated finger tip to a quasi-stable state. The width
of the finger is stabilized after some 75 minutes (stage I). This corresponds to
the time to arrive at approximately constant water contents within the finger
core as shown in (Fig. 5.16, blue curve in the plot c).
91
5. Results and Discussion
time [min]
finger width [mm]
saturation [–]
core zone
fringe zone
0.8
PSfrag
0.6
t:17 min
t:19 min
t:21 min
t:23 min
t:50 min
t:75 min
t:100 min
width [mm]
0.4
normalized intensity
time [min]
finger width [mm]
0.2
0
0.2
0
core35zone
40
fringe
zone
horizontal distance [mm]
0
(c)
0.4
5
10
15
20
25
30
0
5
saturation [–]
15
20
25
30
35
40
horizontal distance [mm]
25
1.0
core zone
PSfrag replacements
time [min]
width [mm]
horizontal distance [mm]
normalized intensity
10
fringe zone
0.8
20
0.6
15
0.4
10
0.2
5
0
finger width [mm]
PSfrag replacements
time [min]
width [mm]
(b) 1.0
t:17 min
t:19 min
t:21 min
t:23 min
0.8
t:50 min
replacements
t:75 min
time
[min] 0.6
t:100 min
saturation [–]
normalized intensity
(a) 1.0
0
0
10
20
30
40
50
60
70
80
90
100
110
time [min]
Figure 5.16.: a) Horizontal transverse intensity profiles at a fixed location for
different times during the passage of a finger in an experiment
with constant flux infiltration of 1.2 mm min −1 showing the dynamics of water saturation within a finger from the initial finger
tip to a quasi-stable state. b) The same horizontal transverse
profiles for water saturation calculated by the calibration equation (Eq. 3.7) between normalized intensity and water saturation.
As described in section 3.3, due to the fundamental problems, the
calibration is valid only when the normalized intensity is between
0.47 and 1.0 with water saturation between 0.1 and 1.0. c) Water
saturation changes inside the finger core (blue dots) and measured
optical width for a finger (red dots) using horizontal transverse
intensity profiles as a function of time during the passage of a
finger at the same fixed location of top plots. The time numbers
refer to the time after starting of water infiltration.
92
5.5. Finger Width
water flow
b) t+ t
a) t
generating of
finger
stabilizing of
finger core
tip
...
fringe zone
...
fringe zone
...
...
...
...
...
...
lateral water movement
d) t+n t
lateral water movement
lateral water movement
lateral water movement
stabilized finger core
c) t+2 t
Figure 5.17.: Schematic diagrams of core and fringe development of a single
finger during downward growth with lateral water movement from
the core to the fringe zone. After the passage of the finger tip and
the stabilization of the finger core (stage I), the finger starts to
grow radially (stage II).
Figure 5.16 (b) shows the same horizontal transverse profiles for water saturation calculated by the calibration equation (Eq. 3.7) between normalized
intensity and water saturation. As described in section 3.3, due to the fundamental problems, the calibration is valid only when the normalized intensity is
between 0.47 and 1.0.
We measured the optical width of a finger by calculating the difference between points corresponding to half the maximum intensity of the horizontal
transverse intensity profiles (Fig. 5.16, red curve in the plot c). These measurements show that the finger width increases from 11 mm after the passage of
the finger tip to 21 mm after 100 min of continuous constant infiltration. The
measurements of the finger width using the horizontal transverse saturation
profiles lead to narrower widths in comparison with intensity profiles.
Figure 5.17 is a schematic drawing of the finger development showing the
93
5. Results and Discussion
t=30 min
t=3 days
t=30.5 min
t=4 days
t=31 min
t=5 days
lateral water
movement
lateral water
movement
1.0
t=36 min
finger
fringe
~10 days
0.9
t=6 days
t=41 min
t=46 min
t=1 day
t=7 days
t=8 days
t=9 days
normalized intensity
0.8
finger
core
16 min
finger
fringe
~10 days
0.7
0.6
0.5
0.4
0.3
I
II
Stage I:
core stabilization
0.2
Stage II:
fringe expansion
0.1
t=2 days
t:time
--- t=30 min
--- t=30.5 min
--- t=31 min
--- t=36 min
--- t=41 min
--- t=46 min
--- t=1 d
--- t=2 d
--- t=3 d
--- t=4 d
--- t=5 d
--- t=6 d
--- t=7 d
--- t=8 d
--- t=9 d
--- t=10 d
t=10 days
0
0
10
20
30
40
50
60
70
80
90
horizontal distance [mm]
Figure 5.18.: Sixteen different times series visualizing the finger core and fringe
areas development and the lateral cross-section intensity profiles
of a finger at a fixed location (indicated by black dash line in
images top left) during the passage for the duration of 10 days
continuous infiltration. Over time, slow lateral movement of moisture from finger core regions creates a less saturated surrounding
fringe region and finger continues to expand. The width of the
fingers increases from 11 mm after the passage of tip to 60 mm
after 10 days continuous infiltration.
downward growth followed by two stages I and II of lateral growth into the core
and fringe areas, respectively. Progressive growth of a finger can be divided
into following steps: In step 1, a finger is generated (Fig. 5.17a). In step 2,
it moves downward and the finger core is stabilizing (Fig. 5.17b). In step 3,
water flows vertically in the stabilized core and laterally in the fringe zone
(Fig. 5.17c). After several steps, the finger grows with lateral expansion for a
long time (Fig. 5.17d).
If the water flow continues for a long period of time, lateral water movement
leads to significant lateral expansion of the fingers. Fig. 5.18 illustrates the
94
5.6. Finger Tip Velocity
development of the finger core and fringe areas in time, for an experiment with
the duration of the 10 days of continuous infiltration with a constant flux of
1.2 mm min−1 . These intensity measurements show that the finger core stabilized 16 minutes after the tip wetting front passed (stage I) and the lateral
movement in the fringe zones continues to expand for a long time after stabilization of the core (stage II). As long as there is a radial hydraulic gradient
between the outer limit of core and fringe zones, it continues to expand. This
is an example showing that the growth of the fingers is a very slow process.
Hence, the wetting and drying to equilibrium inside the finger core took place
in less than an hour, while the wetting to equilibrium outside the fingers took
place over several days.
5.6. Finger Tip Velocity
In a quantitative analysis, the finger tip velocity was defined as the rate of
change in position of the leading edge of the tip. Finger tip velocity is expected
to be a function of the gravity gradient, capillary gradient, media heterogeneity,
applied flux, initial moisture content and number of the fingers. As the finger
tip advances, the small-scale heterogeneity in the sand layers acts to continuously alter the shape of the wetting front. These fluctuations in wetting front
structure are affected by dynamic variation in the capillary gradient, and result
in velocity variations. On the other hand, when a new finger is initiated the
flux in the existing fingers decrease and the loss of flow is expected to slow the
fingers. The fingers, during flowing in the Hele-Shaw cell, have very sensitive
distribution and their movement is not continuous. Because the change of the
wetting front shape and loss of fluid are coupled dynamic processes the actual
behavior is quite complex.
The images shown in Figure 5.19a are the finger advancements of an experiment with constant flux infiltration of j w = 1.2 mm min−1 . In this experiment,
the cell was filled without heterogeneous layer to calculate the tip velocity during the passage of the fingers for a long depth (150 cm) through the cell. The
dynamic change of the local finger tip velocity of these four fingers is shown in
Figure 5.19b. The velocity of the fingers shows a large local variation at early
times and a more smoothly gradual trend toward deceleration over the time.
These large variations at early times arise from initiation of new fingers in the
interface between the fine and the coarse layer, during that time with constant
infiltration and uniform suppling of water from top fine layer. Both deceleration
and local tip velocity fluctuations are apparent in Figure 5.19b.
95
5. Results and Discussion
width [m]
width [m]
width [m]
width [m]
width [m]
0
0
0
0
0
0.2
0.4
0.4
0
0
depth [m]
0.2
0.2
0.2
0.4
0.4
0.6
0.6
0.2
0.4
0
3
4
0.8
0.2
0.4
0
0.2
0.4
0
0.2
0.2
0.2
0.4
0.4
0.4
0.6
0.6
0.6
0.8
0.8
(a)
0.8
1 2
PSfrag replacements
PSfrag 1.0
replacements
PSfrag 1.0
replacements
PSfrag 1.0
replacements
PSfrag 1.0
replacements 1.0
0.8
1.2
1.2
1.4
depth [m]
finger tip velocity [cm min−1 ]
t:15 min
PSfrag replacements
1.4
1.2
depth [m]
1.4
t:25 min
1.2
depth [m]
t:35 min
1.4
1.2
depth [m]
t:45 min
7
1.4
t:55 min
finger 1
finger 2
finger 3
finger 4
6
5
(b)
4
3
2
0
10
20
30
40
50
time [min]
Figure 5.19.: a) Images illustrating the advancement of the fingers into
the cell for an experiment with constant infiltration of j w =
1.2 mm min−1 . The indexed time below the images is the real
time after starting of infiltration. b) Finger tip velocity variation
as a function of time for four fingers.
The velocity measurements collected from the two significant fingers of an
experiment with constant flux of jw = 1.2 mm min−1 are presented in Fig 5.20.
For two major fingers, the velocity of their tips are in general not constant and
they are anti-correlated. The dashed line labeled finger 1 corresponds to the
right-hand finger and the solid line labeled finger 2 to the left-hand finger. The
temporal distribution of the images corresponds to the time scale along the
bottom of the images. Measurements made at early times (t < 12 min) show
96
PSfrag replacements
5.6. Finger Tip Velocity
finger tip velocity [cm min−1 ]
6
Finger 1
Finger 2
5
4
3
1
2
2
t: 5.5 min
0
5
t: 12 min
t: 7.5 min
10
t: 17 min
15
t: 21 min
20
t: 25 min
25
t: 29 min
30
t: 33 min
35
time [min]
Figure 5.20.: Finger tip velocity for two major fingers as a function of time
for an experiment with constant flow rate infiltration j w =
1.2 mm min−1 . The velocities of their tips are anti-correlated and
the two fingers remained in hydraulic communication during the
passage of the fingers.
a high degree of variability and on average when finger 1 accelerates, finger 2
decelerates. These two fingers are in hydraulic communication and gradient
changes in one finger will affect the other. At approximately t = 12 min,
hydraulic communication between the fingers is broken, and a second flow
regime begins when a third finger initiates into the cell. In this case, the velocity
profile appears to be much smoother, showing some variation of a distinct trend
toward deceleration. At t = 23 min, the velocity of finger 1 increases, because
at that time this finger is merged with a new finger generated in the left side
of the cell.
97
5. Results and Discussion
5.7. Fingering Flow under Different Flux Rate
Infiltration
A series of water infiltration experiments was conducted in six separate HeleShaw cells filled with initially dry sand using six different fluxes through upper
boundary thus varying from 0.6 to 19.2 mm min −1 . All experiments were run
for 5 hours and images were recorded with an interval time of 1 minute. The
purpose of these experiments were to quantify the effects of infiltration rate on
flow instability and to compare the redistribution of flow fingers for different
infiltration rates. Based on the experimental results, we present the effect of
the flow rate on the behavior of the finger width development, velocity, water
content, and number of the fingers that form. Figure A.3 illustrates sequence
of digital images for these experiments. All infiltration flow rates between 0.6
and 19.2 mm min−1 produced wetting front instabilities that generated many
fingers during redistribution through the cell.
Table 5.1 presents the results of final number of the fingers, average width,
average tip velocity, average saturation inside the core, and the percent total
area of the cell occupied by fingers for these series of experiments.
Applied
flow rate
[mm min−1 ]
0.6
1.2
2.4
4.8
9.6
19.2
Final number
of fingers
Tb
Mc
Bd
4
4
4
4
4
3
4
5
4
8
8
8
12
11
11
16
15
15
Average
width±STDa
[cm]
2.15±0.15
2.75±0.21
3.56±0.41
3.81±0.49
4.05±0.50
4.27±0.68
% of cell
occupied
by fingers
35±2.05
44±2.14
49±2.16
70±2.62
88±3.13
91±3.21
Average
velocity ±STD
[cm min−1 ]
3.09±0.68
5.41±0.95
8.12±1.16
10.83±1.23
13.12±1.86
16.25±2.04
Average
saturation
±STD [–]
0.60 ±0.05
0.65±0.05
0.73±0.06
0.78±0.06
0.82±0.07
0.87±0.07
a
Standard deviation
Top cross-section
c
Middle cross-section
d
Bottom cross-section
b
Table 5.1.: The final number of the fingers, average fingers width, average fingers tip velocity, average saturation inside the core and percent total
area of the cell occupied by fingers for a series of experiments with
different flow rates.
Figure 5.21 illustrates the number of the fingers as a function of flow rate
through upper boundary for three different times during passage from three
cross-sections (top, middle, bottom) of the cell. From a qualitative point of
view, from these plots and from sequence images presented in Fig. A.3, the total
98
5.7. Fingering Flow under Different Flux Rate Infiltration
number of the finger
PSfrag replacements
middle cross-section
bottom cross-section
t: 10min
t: 60min
20
15
10
20
PSfragt: replacements
5min
number of the finger
15
10
top cross-section
middle cross-section
5
bottom cross-section
0
5
10 t:15 5min
20
flow rate [mm min−1 ]
t: 60min
top cross-section
t: 60min
15
10
top cross-section
5
0
20
PSfrag
replacements
t: 10min
number of the finger
0
5
0
0
20
t: 5min
−1 ]
flow rate [mm min
t: 10min
5
10
15
middle cross-section
0
5
10
15
20
flow rate [mm min−1 ]
bottom cross-section
Figure 5.21.: The number of the fingers as a function of flow rate for three different times (5, 10, and 60 minutes) after initiation of the fingers
during water infiltration through the cell. The number of the fingers was counted in three different cross-sections at depths of 0.3,
0.8, and 1.3 m of the cell.
number of the fingers increases as flux increases. For all applied fluxes, counted
number of the fingers decreased following passage of the fingers over depth
during downward movement from three cross-sections as shown in Fig. 5.21.
This is because some fingers do not carry enough water to keep downward
growing and they merge together for further movement and thus the number
of the fingers reduces with depth. For an individual flux, the total number of
the fingers increases over the time. The amount of water flux applied through
the system is constant through the top layer which controls the infiltrating
flow. The total flux at z component (qw,z ) is divided into N number of the
fingers, hence
n
X
i
Ni jw,z
(5.1)
qw,z =
i=1
i
where jw,z
is the water flux within the generated finger i. We observed that
when the initial fingers reached to bottom of the cell, some new fingers start to
initiate. When the wetting fronts find ways to penetrate interface between the
fine and the coarse layer, the flux into other fingers decreases and according to
Eq. 5.1, the number of the fingers has to increase to sustain the constant flux
qw,z . Figure 5.22, presents the plots of measured values for average finger width,
tip velocity, saturation inside the core and total area occupied by fingers as a
function of flow rate. Top left plot, shows that percentage of cell occupied by
fingers increases linearly with flux. This linear relation between the occupied
99
100
1.0
80
60
40
PSfrag replacements
average finger width [cm]
20
average finger tip velocity [cm min−1 ]
average saturation [–]
PSfrag replacements
erage finger width [cm]
tip velocity [cm min−1 ]
average saturation [–]
% of cell occupied by fingers
5. Results and Discussion
0
0.8
0.6
0.4
0.2
0
0
% 5of
15
cell10occupied
by20
fingers
0
PSfrag replacements
rage finger width [cm]
average saturation [–]
ell occupied by fingers
20
16
12
8
PSfrag replacements
4
average finger tip velocity [cm min−1 ]
0
average saturation [–]
0
15 by20
%5of cell10occupied
fingers
flow rate [mm min−1 ]
10
15
20
flow rate [mm min−1 ]
average finger width [cm]
average finger tip velocity [cm min−1 ]
flow rate [mm min−1 ]
5
7
6
5
4
3
2
1
0
0
5
10
15
20
flow rate [mm min−1 ]
Figure 5.22.: Plots of average finger width, average tip velocity, average saturation inside the core, and percent of cells occupied by fingers as
a function of flow rate through upper boundary.
area and the flow through the system, suggests that an increase in flow rate
through the system increases the fraction of the cell occupied by fingers. The
non-linearity shape at high flux (19.2 mm min −1 ) is because there was no more
space to development of the fingers in the cell. The average saturation inside
the core shown in Fig. 5.22, top right, increases rapidly at low flux, the slope off
and approach to near saturation, which shows to be near the porosity, at high
100
width [mm]
30
40
width [mm]
width [mm]
50
60
0
0
10
20
30
40
50
60
0
0
10
10
10
20
20
1.0
1.0
finger
core
finger fringe zone
30
40
finger fringe zone
30
40
finger fringe zone
30
40
0.8
1.0
0.8
0.6
0.6
0.6
0.4
0.4
0.2
0.2
0.2
10
20
30
40
50
jw =0.6 mm min−1
60
30
40
50
40
50
finger
core
60 1.00
0.88
0.75
0.62
0.50
0.38
0.25
0.12
0
0.8
0.4
0
20
0
20
finger
core
10
finger fringe zone
20
finger fringe zone
10
finger fringe zone
saturation [-]
depth [mm]
0
relative saturation [–]
5.7. Fingering Flow under Different Flux Rate Infiltration
0
10
20
30
40
50
60
jw =4.8 mm min−1
0
10
20
30
60
jw =19.2 mm min−1
Figure 5.23.: Typical example of three horizontal water profiles for three fingers
generated at three different flow rates infiltration. Higher flux
shows the wider core with more saturation within the fingers.
flow rate. The experimental results indicate that finger velocity at low flow rate
was slow and very fast at high flow rates. The slope of the finger velocity versus
flux steadily decreases with increasing flux as shown in Fig. 5.22, bottom left.
The average width versus flux is shown in Fig. 5.22, bottom right. Generally,
the slope of the measured width as a function of flow rate becomes less as the
applied flux increases.
Water saturation measured within the finger core was found to be an increasing function of applied flux. Figure 5.23 is a plot showing the structure of
the finger width and horizontal transverse saturation profile for three fingers at
three fluxes (low, intermediate, and high) at the end of the experiment. These
cross-section profiles of water saturation measurements clearly show the nonuniform saturation profiles within the fingers and illustrates that the saturation
in the finger tail is controlled by the applied flux, where the finger width and
saturation inside the finger core are higher when the flux is higher.
The width growing of six arbitrary fingers obtained from horizontal transverse saturation profiles for six different forcing flow rates are presented in
101
5. Results and Discussion
PSfrag replacements
finger width [mm]
60
jw = 19.2 mm min−1
jw = 9.6 mm min−1
50
jw = 4.8 mm min−1
40
jw = 2.4 mm min−1
jw = 1.2 mm min−1
30
jw = 0.6 mm min−1
20
10
0
0
50
100
150
200
250
300
time [min]
Figure 5.24.: The width of six growing fingers as a function of time, for six different forcing flow rates measured using the horizontal transverse
saturation profile.
Fig. 5.24. It shows that the finger width tends to increase with time and increasing the flux influences on width expansion and under all applied flux it
continues to increase slowly for a long time. Because of large number of the
fingers at high flow rate infiltration, the width of the fingers was possible to
measure only before filling the cell by water moistures.
Thus, unstable phenomena are very sensitive to initial and boundary conditions and the flow rate appears to play an important role in width, velocity,
saturation and number of the fingers and referring to these results, the increasing of flow rates leads to increasing finger width, number, velocity and the
saturation inside the core.
5.8. Finger Persistence
Glass et al. (1989c) studied the finger persistence over a long periods of time.
They showed the persistence of fingered flow structure upon a subsequent infiltration event after the full development of the two regions core-fringe flow field.
Wang et al. (2003) have shown in their experiments that the porous medium
retained a memory of the fingers formed in the first experiment, so fingers
formed in subsequent redistribution cycles followed the old finger paths, even
after 28 days had elapsed.
102
5.8. Finger Persistence
In this study, another experiment was performed to determine the persistence
of fingered flow paths and structure of core and fringe for infiltration after an
extensive period of time. Experiment was conducted with onset of a constant
water infiltration (1.2 mm min−1 ) and when the fingers reached the bottom of
the cell, the infiltration was stopped. Five days after, it was restarted using
dye tracer infiltration at the same flow rate.
Figure 5.25 shows the illustration of the growth and persistence of fingered
flow paths. At the start of the experiment, fingers formed and reached the bottom of the cell during the first water application period. Following the cessation
of water application, the water profile inside the cell changed much more slowly
and fingers were drained. Five days after the water flow in the experiment was
stopped, the fingers had steadily grown to fill the entire cell and the initial moisture content field appeared almost uniform to the eye. During the five days
cessation of infiltration, water in the finger domains gradually expanded into
the surrounding profile by the mechanism described in section 5.4.1. However,
upon reinfiltration using dye, the flow still tended to flow preferentially down
along the previously formed finger pathways and core structure in the fingers
again reappeared. The locations of the core regions as highlighted by the blue
pulses were almost the same as in the first water infiltration cycle.
The question then arises: Why does the persistence in flow fingers occur?
By analyzing the horizontal distribution of water saturation within and around
the fingers, we found the water content in center of the finger is higher than
fringe area even after 5 days of no inflow (Fig. 5.25, bottom graph). After
stopping the infiltration, the water content within the core decreased slowly.
The horizontal saturation profiles did not reach to a steady state after 5 days.
In the first day after cessation of water infiltration, the saturation distribution
area in the fringes (lateral movement) was more than expected according to
decreased amount of water in core. This can be explained by the fact that the
distribution of water in the fine toplayer and cores were very slow to reach a
quasi-equilibrium state. After first day, the decreased amount of water within
the core was consistent with distribution area in fringes. Thus, core and fringe
regions have essentially different moisture characteristic curves. The core area
is preserved as a relatively wetter zone than fringe areas and until the water
potential at the fringe is approximately smaller than the core, lateral expansion
will continue.
103
5. Results and Discussion
width [m]
width [m]
0
0.2
0
0.4
0.2
width [m]
0.4
0
0.2
width [m]
0.4
0
0
0
0
0
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.2
0.4
depth [m]
0.6 replacements
0.6 replacements 0.6
0.6 replacements
PSfrag replacementsPSfrag
PSfrag
PSfrag
0.8
0.8
0.8
0.8
depth [m]
depth [m]
depth [m]
1.0
1.0
1.0
1.0
(stop of water
infiltration)
(stop of water infiltration)
(stop of water infiltration)
1.2
t:67 h 1.2
t:67 h 1.2
t:67 h 1.2
t:5 days
t:5 days 1.4
t:5
days 1.4
1.4
1.4
t:60 min
t:60 min
t:60 min
t:5 days+32 min t:5 days+32
t:5 days+32 min
t:60 min min t:5 days+32
t:67 h min
t:5 days
(dye infiltration) (stop
(dyeof infiltration)
(dye infiltration)
water infiltration)
(dye infiltration)
t:25 min
t:26 min
t:27 min
t:29 min
t:45 min
t:60 min
t:1 day
t:2 days
t:3 days
t:4 days
1.0
saturation [–]
0.9
0.8
0.7
0.6
0.5
0.4
PSfrag replacements
0.3
0
10
20
30
40
50
60
70
80
horizontal distance [mm]
Figure 5.25.: A sequence of photographs of an experiment demonstrating finger persistence. It represents the distribution of water content
in the Hele-Shaw cell during an event in which water was applied for 60 min, then stopped for 5 days, and then dye tracer
applied for 32 min. The image of the moisture content field before second cycle (t = 5 days) showed that the variation in initial
moisture content was very small and uniform. The subsequent
dye infiltration events demonstrates the persistence of the corefringe region structure during the rapid growth of dye fingers. The
bottom graph shows the horizontal transverse saturation distribution at a fixed location for different times during passage of a
finger and after cessation of infiltrating water. When infiltration
is stopped, the amount of water within the cores is distributed
into the fringes. The time numbers refer to the time after starting of water infiltration.
104
PSfrag replacements
5.8. Finger Persistence
PSfrag replacements
K
core
ψm
fringe
θ
core
x2 x1
x0
00
core
ψm
θ
core
ψm
core
ψm
θ core
0
tip
tip
ψ0core
m
θ core
θ fringe
00
0
core
ψm
θ fringe
0
tip
ψm
K
θ fringe
ψm
θ core θ tip
tip
xψ
0m
0
0
θ fringe θ core
x1
x2 ψm
K
θ tip
Figure 5.26.: Finger persistence demonstration using the evolution of hydraulic
states of a finger described in Fig. 5.15. After cessation of the infiltrating water to the finger, the water content and matric poten0
00
core ) to (θ core , ψ core ) along
tial at location x0 move from (θ core , ψm
m
the main or adjacent drying branch because of the slow draining
process following the interruption of flow. During the same time,
0
00
core ). The state at
the state at x1 , evolves towards (θ fringe , ψm
x2 also decreases with limited supply of water from the core. At
these new states after stopping inflow, the core area is wetter than
in the surrounding area and hence, the hydraulic conductivity is
higher than fringe zones.
Following the cessation of water supply and subsequent drainage to the finger, the water content in the finger core, the outer limit of the core, and fringe
areas will drain to new states on the soil-water characteristic curve with different drainage curves as shown in Fig. 5.26. The dye infiltration clearly show that
the non-uniform saturation profile observed after fingering exists for long periods of time and that the locations of the original core areas were slightly wetter
than in the surrounding fringe areas. Since the finger core is initially wetter
than the fringe region, its hydraulic conductivity is higher than the fringe zone
at the same suction. The persistence of fingering is a direct consequence of
the hydraulic conductivity of the core area and hence, the subsequent infil-
105
5. Results and Discussion
tration enters through the finger core areas established previously. Therefore,
fingers have a water phase memory formed in the first experiment, so fingers
formed in subsequent redistribution cycles followed the same flow paths that
had been produced by the fingers in the previous cycle, even after 5 days had
elapsed.
5.9. Effects of High Initial Water Content
Only a limited number of experiments exists on the effects of initial moisture
content on gravity-driven instability, most likely due to the difficulty of measuring and controlling initial moisture fields. However, the limited available
evidence does imply that initial moisture content has a fundamental impact on
finger behavior, and as such, warrants additional investigation. Experiments
in porous media suggest that the occurrence of wetting front instability in uniform moisture field is a function of the initial moisture content. The effects
of uniform initial moisture content on finger development were explored in a
series of experiments by Diment and Watson (1985) where they found that as
the uniformly distributed initial moisture content increased, gravity fingering
became less distinct. They noted that wetting front is stabilized and eliminated the instability during redistribution when the initial moisture content
was raised above a few percent. Recent experiments by Wang et al. (2003),
however, show the fingers generated under initial moisture condition and the
observed fingers were significantly slower and wider than when the soil was
dry.
We designed an experiment to see the effects of initial water moisture on wetting front instability and if fingering in initially and uniformly quite wet sand
would occur. In this experiment, we started with water infiltration through
initially dry sand and observed the flow fingers. Then the cell was saturated
with water from below of the cell to depth 0.3 m step by step (to minimize
the air entrapments). After saturation, the cell was allowed to slowly drain
for 24 hours to obtain of the hydrostatic pre-wetted system. The moisture
content field at the end of the drainage cycle formed the initial moisture field
for the second infiltration experiment in pre-wetted sand. A uniform moisture
saturation of about 0.5 in the pre-wetted layer resulted and was more or less
constant with depth. For the second infiltration cycle we added water in the
cell but the wetting front was not visible in pre-wetted sand. To this end in
the subsequent infiltration, we used dye tracer to better visualization of flow
through the cell (Fig. 5.27). Both applied water and dye were at flow rate of
106
5.9. Effects of High Initial Water Content
width [m]
0
0.2
0
width [m]
0.4
0
0.2
width [m]
0.4
0
0
0.2
width [m]
0.4
0
0
0.2
0
width [m]
0.4
0
0.2
0.4
0
depth [m]
PSfrag replacements
PSfrag
replacements
PSfrag
replacements
PSfrag
replacements
PSfrag
replacements 0.2
0.2
0.2
0.2
0.2
b
c
d
e
0.4
0.6
0.4
depth [m]
a 0.6
0.8
0.8
1.0
a
c
1.0
d
e
depth [m]
a 0.6
b 0.8
0.4
depth [m]
a 0.6
b 0.8
c
1.0
1.0
b
d
e
0.4
c
e
0.4
depth [m]
a 0.6
b 0.8
c
1.0
d
d
e
Figure 5.27.: Photographs of the experiment for effects of uniform initial water content on flow fingers. Images a,b and c are dry, saturated
and desaturated (uniform pre-wetted) cell, respectively. Images
d and e are the second cycle of water and dye infiltration where
dye tracer redistribution shows the transverse dispersion and diffused flow in pre-wetted region. Black points distributed over the
cell are the clips that we used to avoid any bending of the glass
plates due to the hydrostatic pressure during the imbibition to
saturation of the cell.
1.2 mm min−1 .
The dye flow redistribution was stable in fine layer and then at the top dye
followed the water fingers path established during the first cycle in the initially
dry sand. When the fingers penetrate the pre-wetted sand, a much stronger
lateral diffusion than in the dry sand emerges (Fig. 5.27). Thus, we observed
that the dye flow was non-uniform and transported with a small transverse
dispersion and more diffuse in the pre-wetted sand.
The behavior of the flow in uniform pre-wetted condition in this experiment
directly corroborated the results observed by Diment and Watson (1985), however, realizing at a higher initial moisture saturation (0.5) than their experiments.
The common description of flow saturation in initially dry sand cannot be
used in the sand with non-uniform (as formed by full development of the tworegion core and fringe flow by fingering in initially dry sand) initially moisture
content. The finger flow features that form initially non-uniform moisture
content do not exhibit a saturated tip with drainage behind, as is seen under
initially dry conditions (Glass and Nicholl 1996).
107
5. Results and Discussion
PSfrag replacements
width [mm]
20
depth [mm]
time [min]
saturation [–]
0
depth [mm]
time [min]
saturation [–]
20
relative saturation [–]
horizontal distance [mm] horizontal
distance [mm]
40
normalized intensity
normalized intensity
width [mm]
width [mm]
60
initially dry
initially dry
pre-wetted
pre-wetted
80
tip in initially dry
tip in initially dry
tail in pre-wetted 100 tail in pre-wetted
0
0.2
tip in initially dry
tail in pre-wetted
30
25
pre-wetted
time [min]
relative saturation [–]
horizontal distance [mm]
normalized intensity
width [mm]
0.6
0.8
20
15
10
20
1.00
0.75
40
0
0.6
0.62
0.50
60
0.38
0.25
80
0.12
0
100
0
0.8
20
40
1.0
tip in initially dry
0.8
0.6
0.6
0.4
0
0.4
0.88
20
0.4
tail in pre-wetted
15
width [mm]
initially 10
dry 0.2
pre-wetted 0
5
0.2
40
1.0
5
0
20
0
40
PSfrag replacements
width [mm]
35
depth [mm]
1.0
30
time [min]
saturation25[–] 0.8
relative saturation [–]
35
PSfrag replacements
width [mm]
depth [mm]
0.4
initially dry
40
width [mm]
0
40
0.2
normalized intensity
0
relative saturation [–]
PSfrag replacements
0
0
20
40
horizontal distance [mm]
1.0
saturation [–]
Figure 5.28.: The longitudinal and transverse dynamics of water saturation
through one finger in initially dry and non-uniform pre-wetted
sand. The longitudinal saturation measurements were done at
the small area inside the finger tip and the transverse saturation
was taken at 74 cm depth. Note that the maximum displayed
saturation is ∼0.8 which is measured in initially dry condition.
The small difference in saturation for finger tail between initially
dry and pre-wetted condition (bottom right) cab be explained by
entrapped air from the initial moisture content within the initial
water finger.
108
5.9. Effects of High Initial Water Content
Figure 5.28 compares the longitudinal and transverse saturation through one
finger at a fixed location (indicated by black line) in initially dry and nonuniform initial pre-wetted condition. In the longitudinal temporal dynamics of
water saturation inside the finger core at a fixed location, the large difference
in saturation at the finger tip between these two conditions can be clearly seen.
Thus saturated zone at the finger tip, which is characteristic for dry sand, did
not occur in the pre-wetted sand. Comparison of the water saturation inside
the finger core between these two conditions shows that the water saturation
along the finger in the pre-wetted sand remained approximately the same as
dry sand. The horizontal transect of the water saturation through flow finger
shows the water saturation inside the finger core reduces in pre-wetted sand in
comparison with dry sand, i.e, there is no more finger tip in pre-wetted sand.
The cross-sectional area of the finger in the pre-wetted sand is larger than in
the dry sand.
109
6. Summary and Conclusions
This thesis is a new look on unstable fingering flow generated from ponded
infiltration into an initially dry two-layered sand, where a fine-textured layer
overlies a coarse-textured layer. For multi-layered systems, the fingers disappear when they encounter heterogeneous layer and reappear as the flow enters
a homogeneous coarse region.
We developed a unique, high resolution, 2D transmitted light imaging system
to use for quantitative imaging of transient and steady state flows in porous
media. The Light Transmission Method (LTM) is a nondestructive and simple
tool that permits visualization and measurement of water saturation in HeleShaw cells with a high spatial (millimeters) and temporal (seconds) resolution.
This technique also opens promising perspectives for investigation of multiphase phenomenon. We further improved the LTM to measure the dynamics
of water in Hele-Shaw cells. This was achieved by adding a deconvolution
procedure to correct the measurements for light scattering. This technique
was used to visualize and analyze, qualitatively and quantitatively, fingering
phenomena. The transmitted light intensity was calibrated to absolute water
content values calculated from X-ray attenuation data. Thereby, we used intensity of transmitted light as a proxy for water content and hence, the changing
water content within flow fingers could be measured in great detail.
After the water fingers fully developed, we used dye tracer to visualize the
velocity field within the flow fingers. Infiltration of dye tracer into stabilized
water fingers highlights the separation of the water phase into a mobile component (core) and an immobile one (fringe).
The experiments to investigate the structure of the fingering flow were examined in different conditions. Individual fingers initiated from interface between
the fine and the coarse layer are typically observed to separate into two regions, a saturated tip that advances downward and a partially drained region
behind the finger tip. The longitudinal saturation dynamics profile shows that
it decreases immediately behind the tip and slightly increases again towards
111
6. Summary and Conclusions
longer distance from the tip and then becomes steady over time in finger core.
The transverse saturation profile shows a maximum in the center of the fingers
and slow lateral movement of the moisture from finger core regions creates less
saturated surrounding finger. Hence, the fully developed fingers consist of a
high saturated tip, a core with mobile water and a hull with immobile water
fringe and once a finger is developed, the saturation profile along its core is
invariant as the tip progresses.
The simultaneous measurements of pressure and saturation definitively show
the nonuniform moisture profiles in flow paths created by fingered flow. In the
evaluation of hydraulic states to describe the dynamics and stabilization of
fingers, we demonstrated that the water content and matric potential at the
wetting front of unstable fingered flow are on the wetting curve and behind the
front are on the drying curve of the soil-water characteristic curve.
Based on the experimental results, we propose two stages of lateral expansion for fingers: a fast expansion in a short time to core stabilization and a
slow and steadily expansion for long time to fringe lateral growth. The finger
core stabilization well expressed by the interface between the finger core and
the fringe zone approaching to the same potential on different hysteresis loop
of soil-water characteristic curve, even through the water contents are quite
different. A large fringe lateral expansion, however, is expressed by a hydraulic
conductivity and water pressure gradient between the outer limit of the core
and the fringe region of finger. This is severely hindered by the low conductivity in this dryer range and by the limited supply of water and as long as
the matric potential in finger core is smaller than the surrounding zone, finger
grows. Hence, within the core of the fingers, the fast convective flow is driven
by gravity while at the boundaries (fringes), flow is slow and diffusive.
The fingered flow are highly dependent on the porous medium properties,
infiltration fluxes, and the initial and boundary conditions. The dependence
of the finger moisture content, width, velocity, and occupied area by fingers as
a function of the flux was examined through experiments under different flow
rates infiltration. High flow rate produces wider, faster, and more fingers. The
finger merging is stated to be an important process, increasing the velocity and
width of continuing finger and reducing the number of fingers with depth.
Fingers can persist in the same location for a long time. It is shown that
the water content at the finger core and the position of the hydraulic states of
112
the wetting curve relative to the drying curve are key factors in the persistence
of the fingers. Therefore, fingers have a water phase memory formed in the
first cycle of infiltration and the subsequent infiltration events persists into the
core-fringe region structure followed on the previous finger paths, even after 5
days had elapsed.
The instability of the wetting fronts is eliminated and does not exhibit a saturated tip during redistribution when the initial moisture content of medium
raises above a few percent.
In outlook of this work, we note a few main findings:
Flow fingers in coarse textured sand are destroyed by finer textured inhomogeneities, but they reappear as the flow enters a uniform region.
We improved the light transmission method to get the real visualization
of fingers through deconvolution using point spread function.
Through parallel measurements of X-ray absorption and light transmission method, we found a good agreement between these two techniques.
Our experiments confirm the frequently observed overshoot in water saturation within the tip of flow fingers.
As a new observation, the water saturation in the core of the finger has
a minimum immediately behind the tip and increases again towards a
constant value at larger distance from the tip.
The width of flow fingers increases initially and reaches a quasi-stable
state at the same time when the water saturation within the core becomes
quasi-stable.
The water within the finger is sharply separated into a core and a fringe
with fast convective flow in the core and slow diffusive flow in the fringe.
When two fingers are present, they are in hydraulic communication and in
general the velocities of their tips are not constant and are anti-correlated.
In the overlying fine-textured layer, pressure drops upon initialization of
a new finger.
113
6. Summary and Conclusions
Once the finger is formed, its core is preserved as a wet zone for persistence and the fingers formed in subsequent redistribution cycles followed
the same flow paths that had been produced by the fingers in the previous
cycle.
As the uniformly distributed initial moisture content increased, gravity
fingering becomes border and less distinct.
All observed phenomena, with the exception of saturation overshoot, could
be consistently explained based on the hysteretic behavior of the soil-water
characteristic. This research was an effort of laboratory experiment to investigate the unstable flow in unsaturated porous media and to identify the
processes and mechanism of fingered flow.
114
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125
A. Appendix
The following figures show the sequence digital images of the fingering patterns observed by light transmission method during water infiltration (A.1),
dry tracer infiltration after stabilization of the water fingers (A.2), and experiments of water infiltration under different flow rates (A.3) into an initially dry
layered porous medium. For each experiment, the time is indicated below the
images.
Readers can find video clip of the fingering flow experiment during water
and dye tracer infiltration into a multi-layered medium (A.1 and A.2) in below
link:
http://www.hydrol-earth-syst-sci-discuss.net/3/2595/2006/hessd-3-2595-2006supplement.zip
127
A. Appendix
width [m]
0
0.2
width [m]
0.4
0
0.2
width [m]
0.4
0
0.2
width [m]
0.4
0
0.2
width [m]
0.4
0
0
0
0
0
0
0.2
0.2
0.2
0.2
0.2
0.2
0.4
depth [m]
PSfrag replacements
PSfrag 0.4
replacements
PSfrag 0.4
replacements
PSfrag 0.4
replacements
PSfrag 0.4
replacements 0.4
t:5
t:7.5
t:10
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Figure A.1.: Digital images of water fingering experiment into an initially dry
porous Hele-Shaw cell observed by light transmission system described in section 5.1 (Fig. 5.1). The time numbers refer to the
time after starting of water infiltration.
A. Appendix
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Figure A.2.: Digital images of dye tracer infiltration experiment into stabilized
water fingers described in section 5.1 (Fig. 5.4). The time numbers
refer to the time after starting of water infiltration.
A. Appendix
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0.4
0.4
jw = 1.2 mm min−1
PSfrag replacements
PSfrag 0.2
replacements
PSfrag 0.2
replacements
PSfrag 0.2
replacements
PSfrag 0.2
replacements 0.2
0.4
depth [m]
depth [m] 0.6 depth [m] 0.6 depth [m] 0.6 depth [m] 0.6
0.6
t:5 min 0.8
t:5 min 0.8
t:5 min 0.8
t:5 min 0.8
0.8
t:10
min
t:10
min
t:10
min 1.0
1.0
1.0
1.0
1.0
t:60 min
t:60 min
t:60 min
1.2
1.2
1.2
1.2
1.2
t:120 min
t:120 min
t:120 min
1.4
1.4
1.4
1.4
1.4
t:270 min
t:270 min
t:270 min
−1
−1
−1
1.2 mm
jw = 1.2 mm
jw = 1.2 mm
t:10 min
min
t:60 min
min
t:120 min
t:5 min
min
t:10 min
t:60 min
t:120 min
t:270 min
jw = 1.2 mm min
jw−1
=
width [m]
0
0
0.2
width [m]
0.4
0
0
0.2
width [m]
0.4
0
0
0.2
t:270 min
width [m]
0.4
0
0
0.2
width [m]
0.4
0
0.2
0.4
0
0.4
0.4
0.4
0.4
depth [m] 0.6 depth [m] 0.6 depth [m] 0.6 depth [m] 0.6
t:5 min 0.8
t:5 min 0.8
t:5 min 0.8
t:5 min 0.8
0.8
t:10
min
t:10
min
t:10
min 1.0
1.0
1.0
1.0
1.0
t:60 min
t:60 min
t:60 min
1.2
1.2
1.2
1.2
1.2
t:120 min
t:120 min
t:120 min
1.4
1.4
1.4
1.4
1.4
t:270 min
t:270 min
t:270 min
−1
−1
−1
2.4 mm
jw = 2.4 mm
jw = 2.4 mm
t:10 min
min
t:60 min
min
t:120 min
t:5 min
min
depth [m]
t:10 min
t:60 min
t:120 min
t:270 min
jw = 2.4 mm min
jw−1
=
0.4
jw = 2.4 mm min−1
PSfrag replacements
PSfrag 0.2
replacements
PSfrag 0.2
replacements
PSfrag 0.2
replacements
PSfrag 0.2
replacements 0.2
0.6
t:270 min
width [m]
0
0.2
width [m]
0.4
0
0
0.2
width [m]
0.4
0
0
0.2
width [m]
0.4
0
0
0.2
width [m]
0.4
0
0
0.2
0.4
0
0.4
0.4
0.4
0.4
jw = 4.8 mm min−1
PSfrag replacements
PSfrag 0.2
replacements
PSfrag 0.2
replacements
PSfrag 0.2
replacements
PSfrag 0.2
replacements 0.2
0.4
depth [m]
depth [m] 0.6 depth [m] 0.6 depth [m] 0.6 depth [m] 0.6
0.6
t:5 min 0.8
t:5 min 0.8
t:5 min 0.8
t:5 min 0.8
0.8
t:10 min 1.0 t:10 min 1.0 t:10 min 1.0
1.0
1.0
t:60 min
t:60 min
t:60 min
1.2
1.2
1.2
1.2
1.2
t:120 min
t:120 min
t:120 min
1.4
1.4
1.4
1.4
1.4
t:270 min
t:270 min
t:270 min
−1
4.8 mm
jw =−14.8 mm
jw =−14.8 mm
t:5 min
min
t:10 min
min
t:60 min
min
t:120 min
t:10 min
t:60 min
t:120 min
t:270 min
jw = 4.8 mm min
jw−1
=
width [m]
0
0.2
width [m]
0.4
0
0
0.2
width [m]
0.4
0
0
0.2
width [m]
0.4
0
t:270 min
0
0.2
width [m]
0.4
0
0
0.2
0.4
0
0.4
0.4
0.4
0.4
jw = 9.6 mm min−1
PSfrag replacements
PSfrag 0.2
replacements
PSfrag 0.2
replacements
PSfrag 0.2
replacements
PSfrag 0.2
replacements 0.2
0.4
depth [m]
depth [m] 0.6 depth [m] 0.6 depth [m] 0.6 depth [m] 0.6
0.6
t:5 min 0.8
t:5 min 0.8
t:5 min 0.8
t:5 min 0.8
0.8
t:10
min
t:10
min
t:10
min 1.0
1.0
1.0
1.0
1.0
t:60 min
t:60 min
t:60 min
1.2
1.2
1.2
1.2
1.2
t:120 min
t:120 min
t:120 min
1.4
1.4
1.4
1.4
1.4
t:270 min
t:270 min
t:270 min
−1
−1
−1
9.6 mm
jw = 9.6 mm
jw = 9.6 mm
t:5 min
min
t:10 min
min
t:60 min
min
t:120 min
t:10 min
t:60 min
t:120 min
t:270 min
jw = 9.6 mm min
jw−1
=
width [m]
0
0
0.2
width [m]
0.4
0
0
0.2
width [m]
0.4
0
0
0.2
t:270 min
width [m]
0.4
0
0
0.2
width [m]
0.4
0
0.2
0.4
0
0.4
0.4
0.4
0.4
0.4
depth [m] 0.6 depth [m] 0.6 depth [m] 0.6 depth [m] 0.6
t:5 min 0.8
t:5 min 0.8
t:5 min 0.8
t:5 min 0.8
0.8
t:10 min 1.0
t:10
min
t:10
min
t:10
min 1.0
1.0
1.0
1.0
t:60 min
t:60 min
t:60 min
t:60 min
1.2
1.2
1.2
1.2
1.2
t:120 min
t:120 min
t:120 min
t:120 min
1.4
1.4
1.4
1.4
1.4
t:270 min
t:270 min
t:270 min
t:270 min
−1
−1
−1
−1
jw = 19.2 mm min
jw = 19.2 mm
= 19.2 mm
t:60 min
min
t:120 min
t:10jmin
min
t:5jmin
min
w = 19.2 mm
w
depth [m]
jw = 19.2 mm min−1
PSfrag replacements
PSfrag 0.2
replacements
PSfrag 0.2
replacements
PSfrag 0.2
replacements
PSfrag 0.2
replacements 0.2
0.6
t:270 min
Figure A.3.: Digital images of six separate experiments under different flux infiltration into initially dry porous Hele-Shaw cells. The flow rates
are indicated in the right side of the images for each experiment.
The time numbers refer to the time after starting initiation of
fingers in interface between the fine and the coarse layer.
Acknowledgments
In this project I got help of an enormous number of people which supported
and encouraged me in making this work a success. I hope that I didn’t forget
someone in the following list. So for safety’s sake:
I would like to express my heartfelt thanks to everyone
who helped me with this project as a
specialist or as a friend!!!
Prof. Kurt Roth who gave me the opportunity to work on a very interesting project in his group and I would like to extend my heartfelt thanks for
his continual support. I appreciate his scientific ideas and contributions
for experimental investigations which set this work on a firm basis. He
also created a unique and enjoyable working environment as I could not
have asked for better supervisor. Thank you so much.
Hans-Jörg Vogel for the scientific impact and his efforts in understanding
of the flow in porous media to a PhD student with no experience in this
field. He took the most stupid questions seriously, so I dared to contact
him for any problems and questions. He provided a friendly environment
for discussion and interpretation of data through this project. His advanced library (Quantim) was very useful for image processing analysis.
Prof. Bernd Jähne that agreed to act as second referee on this thesis. I
always used his helpful comments from his book and lectures for image
processing part of this project.
Ute, Olaf, Volker, Andreas, Zhuhua, Holger, Klaus, Carolin, Moritz,
Philip, Alexandra, Benedikt, Patrick, Jörg, Anatja, Nadija, Tobias, David,
Mathias and Angelika, my former and current colleagues in soil physics
group at the Institute of Environmental Physics, Heidelberg for their support and a friendly atmosphere and fruitful discussions during the last
years. And special thanks to Andreas, Holger and Klaus for helping me
for my computer and programming problems. Thanks friends.
Guilia, Björn, Florian, Steffan, Jens, Dominik and Xia who helped me
for many successful experiments during their Mini-Research course on
this project.
A. Appendix
Andreas Bayer for experiments and processing of the X-ray data and
Felix Heimann for writing of the deconvolution algorithm.
Carolin, Holger, Ute, Klaus, Sreejith, Masoumeh, Sarah, Fahim for making valuable hints and corrections of my dissertation.
Staff of the workshop at the Institute of Environmental Physics for the
enormous efforts for the construction of the experimental Set-Up.
Homa, my lovely wife, who with great patience and tolerance carried
with me the burden of my PhD and supported me during the stressful
times. She has also endured many long hours waiting for me to come
from institute and has provided stability to our family. Without you,
none of these could happen. I love you forever!
My wonderful son Hooman, his birth brought exceptional joy and happiness to my life.
And last but not least, my family for their encouragements from so far away
and all my friends who accompanied me during the last years.
Financial support for this research was provided by Deutsche Forschungsgemeinschaft (DFG) through project RO: 1080-9/1&2.
Fereidoun Rezanezhad, March 2007
136
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