Citation
Chaotic advection from stretching and folding in a Hele-Shaw cell

Material Information

Title:
Chaotic advection from stretching and folding in a Hele-Shaw cell
Creator:
Jones, Matt N. ( author )
Place of Publication:
Denver, CO
Publisher:
University of Colorado Denver
Publication Date:
Language:
English
Physical Description:
1 electronic file (pages). : ;

Thesis/Dissertation Information

Degree:
Master's ( Master of Science)
Degree Grantor:
University of Colorado Denver
Degree Divisions:
Department of Civil Engineering, CU Denver
Degree Disciplines:
Civil engineering

Subjects

Subjects / Keywords:
Groundwater ( lcsh )
Hydraulic engineering ( lcsh )
Stokes flow ( lcsh )
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Review:
Contamination of groundwater can lead to unsafe drinking water and ecological damage. Contamination can result from point sources such as leaking underground gasoline tanks, spills of industrial waste and many other sources. Because of this, remediation of contaminated groundwater is of great concern. In situ treatment has some attractive benefits, such as lower cost and a shorter treatment time, when compared with the most common current technique of extraction and treatment before reinjection. One of the main difficulties in attempting in situ groundwater treatment is that groundwater flows are very slow which maintains laminar flow. Laminar flow in an engineering reactor has the characteristic of poor or limited mixing when compared to turbulent flow. Because of this a treatment solution injected into the groundwater will experience little to no mixing with the contaminant. This in turn means that contaminant degradation is dependent on flow characteristics. Previous research has shown that chaotic advection can induce plume spreading. Plume spreading promotes mixing, which is needed to increase the quantity of reactions. More recent research has shown that injection and extraction can cause chaotic advection through the process of stretching and folding the contaminated groundwater plume. This research, however, has been in the analytical realm. What was needed at this point was to provide a physical demonstration of this process. The purpose of this study was first to build an apparatus that demonstrates the effects of injection and extraction on a plume in a two-dimensional plane. Once the apparatus was functional, the next step was to use the apparatus to demonstrate that injection and extraction can induce stretching and folding. This study has yielded a Hele-Shaw cell which displays the effect of injection and extraction on a source zone and on a contaminant plume still within a high permeability layer. With this Hele-Shaw cell it has been shown that the theorized stretching and folding from injection and extraction can be made to occur in a physical setting. Through visualization of the effects of injection and extraction it was observed that the results in the model were close to the theoretical results. Near Eulerian and near Lagrangian analysis showed that dispersion did not continuously increase as expected; rather, dispersion increased and then decreased.
Thesis:
Thesis (M.S.)--University of Colorado Denver. Civil engineering
Bibliography:
Includes bibliographic references.
System Details:
System requirements: Adobe reader.
General Note:
Department of Civil Engineering
Statement of Responsibility:
by Matt N. Jones.

Record Information

Source Institution:
University of Colorado Denver
Holding Location:
|Auraria Library
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
903218448 ( OCLC )
ocn903218448

Downloads

This item has the following downloads:


Full Text
CHAOTIC ADVECTION FROM STRETCHING AND FOLDING IN A HELE-SHAW
CELL
by
MATT N. JONES
BS, University of Colorado Denver, 2012
A thesis submitted to the Faculty of the Graduate School of University of Colorado in partial fulfillment of the requirements for the degree of Master of Science Civil Engineering
2014


This thesis for the Master of Science degree by Matt N. Jones
has been approved for the Civil Engineering program
by
David C. Mays, Chair Azadeh Bolhari James C.Y. Guo
July 23,2014
n


Jones, MattN. (M.S., Civil Engineering)
Chaotic Advection from Stretching and Folding in a Hele-Shaw Cell Thesis directed by Assistant Professor David C. Mays
ABSTRACT
Contamination of groundwater can lead to unsafe drinking water and ecological damage. Contamination can result from point sources such as leaking underground gasoline tanks, spills of industrial waste and many other sources. Because of this, remediation of contaminated groundwater is of great concern.
In situ treatment has some attractive benefits, such as lower cost and a shorter treatment time, when compared with the most common current technique of extraction and treatment before reinjection. One of the main difficulties in attempting in situ groundwater treatment is that groundwater flows are very slow which maintains laminar flow. Laminar flow in an engineering reactor has the characteristic of poor or limited mixing when compared to turbulent flow. Because of this a treatment solution injected into the groundwater will experience little to no mixing with the contaminant. This in turn means that contaminant degradation is dependent on flow characteristics.
Previous research has shown that chaotic advection can induce plume spreading. Plume spreading promotes mixing, which is needed to increase the quantity of reactions. More recent research has shown that injection and extraction can cause chaotic advection through the process of stretching and folding the contaminated groundwater plume. This research, however, has been in the analytical realm. What was needed at this point was to provide a physical demonstration of this process.
m


The purpose of this study was first to build an apparatus that demonstrates the effects of injection and extraction on a plume in a two-dimensional plane. Once the apparatus was functional, the next step was to use the apparatus to demonstrate that injection and extraction can induce stretching and folding.
This study has yielded a Hele-Shaw cell which displays the effect of injection and extraction on a source zone and on a contaminant plume still within a high permeability layer. With this Hele-Shaw cell it has been shown that the theorized stretching and folding from injection and extraction can be made to occur in a physical setting.
Through visualization of the effects of injection and extraction it was observed that the results in the model were close to the theoretical results. Near Eulerian and near Lagrangian analysis showed that dispersion did not continuously increase as expected; rather, dispersion increased and then decreased.
The form and content of this abstract are approved. I recommend its publication.
Approved: David Mays
IV


TABLE OF CONTENTS
CHAPTER
I. INTRODUCTION....................................................1
Literature Review.............................................6
Research Objective............................................6
Research Overview.............................................7
II. METHODS.........................................................8
Apparatus.....................................................8
Procedure....................................................11
Apparatus Loading..........................................11
Initial plume..............................................11
Injection and Extraction Method............................12
Image Analysis...............................................26
III. DISCUSSION.....................................................42
IV. CONCLUSION.....................................................46
REFERENCES...........................................................47
v


LIST OF TABLES
TABLE
1
13
vi


LIST OF FIGURES
FIGURE
1.......
2.......
3 .....
4 .....
5 .....
6 .....
7 .....
8 .....
9 .....
10 ....
11......
12......
13 ....
14 ....
15 ....
16 ....
17 ....
18 ....
19 ....
20 ....
21......
..4
.. 5
10
14
15
16
17
18
19
20
21
22
23
24
25
28
29
30
31
32
33


22
23
24
25
26
27
28
29
30
34
35
36
37
38
39
40
41
44


CHAPTER I
INTRODUCTION
The field of Civil Engineering holds the responsibility of planning, designing, constructing, operating and protecting our infrastructure. This duty is applied to, among other things, the buildings in which we live our lives, the roads we travel, the air we breathe and the impact we have on the natural world around us.
The natural world includes the water falling from the skies, coursing across the land and slowly trickling beneath our feet. This water in the ground beneath our feet helps to sustain us and is simultaneously vulnerable to us. When we pollute the groundwater, we contaminate not only the soil and minerals beneath us, we also risk our own health as this water can eventually trickle into our streams, be pumped up in our wells or be drawn up by the plants we eat.
It is because of this interdependence with groundwater that we have the responsibility to remedy the damage that we cause. Contamination of the groundwater must be remedied. Remediation of the groundwater is not easy. The most common method of remediation used today is to extract the contaminated groundwater, treat the extracted water with an appropriate form of remediation, and then re-inject it back into the ground once its treatment is complete. This method is usually referred to as pump and treat remediation.
While pump and treat is effective, it is also very time consuming, labor intensive and costly. It would not only be beneficial to the populace to have groundwater remediation be quicker, it would also incentivize polluters to clean up their messes if it is less burdensome to them to do so. Additionally, having groundwater remediation be
1


cheaper could keep production and sales costs down. Combined, the listed benefits could lead to a safer and more productive society.
One idea for remedying groundwater contamination is to inject the appropriate treatment solution directly into the source zone or plume in a transmissive zone of contaminated groundwater, or in situ treatment. The transmissive zone of a contaminant plume is outlined in ITRC (2011). In situ treatment can be simple, quick and cheap.
Direct injection of the treatment solution has some limitations. As groundwater moves at a geologically slow pace, turbulence can never be achieved except in the unusual cases in the immediate vicinity of injection and extraction wells. So, when a treatment solution is introduced into a contamination source zone or plume in a transmissive zone the surface area where the two plumes meet will be the only place the chemical neutralization occurs, resulting in a mass of contaminated water, a smaller mass of the remedying chemical within it and an interfacial surface between them of treated water. This mass within a mass configuration would then carry on down the gradient with the interfacial layer of treated water growing very slowly; usually too slowly to prevent the contaminant from affecting its surroundings. This configuration is dependent on the treatment solution and source zone or plume having similar densities so as to prevent one flowing over the other. This is shown in Figure 1.
To cause more of the remediation interaction to occur, the contaminant and the treatment solution need to be brought together more often. A possible solution to that is postulated in Mays & Neupauer (2012).
Stretching and folding is one way to cause plume spreading. For this application,
plume spreading is the process of lengthening the interfacial surface between the
2


contaminant and the treatment solution. By increasing the length of interface we increase the possibility of chemical interaction between the contaminant and the treatment solution by increasing the actual physical space in which these reactions can occur. This study is a physical demonstration using a Hele-Shaw cell to show that stretching and folding can be induced by a sequence of injections and extractions arrayed around the contaminant as postulated in Mays & Neupauer (2012) and shown in Figure 2.
3


Figure 1: This is an example of contaminant and treatment solution in natural groundwater flow.
4


1 a ,NsteP 1 b Step 2 c Step 3 d Step 4
i 0 ID § -J f> * '$> <3 1
-1 s

1 e Step 5 f Step 6 g Step 7 h ^ Step 8
y/L o to 1 **s£^*
-1 1

1 i t Step 9 j t Step 10 k Step 11 1 t Step 12
y/L o -^Qfgr- . V ' V
-1 : :
-1 0 1 -1 0 1 -1 0 1 -1 0 1
x/L x/L x/L x/L
Figure 2: This is the computer example of stretching and folding induced by a sequence of injections and extractions from Mays and Neupauer (2012). Used with permission of the American Geophysical Union.
5


Literature Review
Early in the exploration of chaos theory Stephen Smale showed that stretching and folding could be accomplished using chaotic particle trajectories with his graphical simulation known as Smales horseshoe (Gleick, 1987), much like the action of a taffy puller. In 1984 it was shown that flows that contain chaos can be engineered. These engineered flows were plume spreading, which we now call chaotic advection (Aref, 1984). Later, Aref joined with Jones to prove that chaotic advection could be produced in irrotational flows which is the case in all laminar flows in homogeneous porous media (Jones & Aref, 1988). Mays and Neupauer then showed that this chaotic advection could occur without reinjection (Mays & Neupauer, 2012).
Research Objective
Previous research has shown that injection and extraction can cause stretching and folding during laminar flow in a two dimensional plane in computer simulations. While the theory is sound and the mathematics verified, the next logical step is to show that the theory works in a real world setting. The goal of this study is to physically demonstrate this theory. A physical demonstration would entail an actual apparatus to show what happens to a fluid when it is pushed and pulled by injection and extraction wells. The use of the apparatus in this demonstration has certain confining parameters.
The only effects allowed to occur for the results of this demonstration to be of value are those generated from the injections and extractions. The apparatus must be horizontal to eliminate gravitational effects. The plates in the apparatus must be truly smooth and flat to eliminate unpredictable distorting effects from the plates themselves. The Reynolds number for the fluid being injected and the fluid moving within the
6


apparatus must be below one to keep the flow laminar and thereby eliminate any dispersion or spreading effects from turbulence.
Research Overview
A physical apparatus has been constructed to demonstrate the effects of injection and extraction on a plume. A fluid was found to fill the apparatus and be used for injection and extraction that simulated the behavior water would have at this scale. Injection and extraction sequences were tested to find one that induced the desired stretching and folding. At each step of injection and extraction photographs were taken for demonstration of the desired stretching and folding and to analyze the dispersion that occurred during the sequence. These photographs were analyzed using image analysis software.
7


CHAPTER II
METHODS
Apparatus
The apparatus is a Hele-Shaw cell as shown in Figure 3. This means that it consists of two flat plates that define a quasi-two-dimensional area for the purpose of studying the flow paths of a liquid. This particular apparatus consisted of two flat plates of acrylic (Plasticare, Englewood, CO). The upper plate was 19 mm thick and clear while the lower plate was 19 mm thick and opaque white. Along all the edges of the defined area (400 mm x 400 mm) there is a foam rubber D-shaped gasket that prevents leakage. The two plates are pressed together using nut and bolt configurations at one at each of the four comers. Combined, the nuts and bolts squeeze the D-shaped gaskets making a constant aperture of 0.1 mm and forming a tight seal.
Wells were simulated by the use of intravenous (IV) therapy hubs (SAFSITE Valve and Cap RV-1000, B. Braun Medical Inc., Bethlehem, PA). The wells were fail-close, only opening to liquid flow when the internal plunger is depressed by the tip of other IV administrative devices, in this case the tip from a syringe (60 mL and 5 mL syringes, Becton, Dickinson and Company, Franklin, NJ), connected through IV valves were used to induce extraction and injection flows.
A single well is at the center. From the center well out in the direction perpendicular to the plate edges two wells were placed to divide the distance from the center well to the edge of the study area into two equal parts, each separated by a distance of 52 mm.
8


In order to simulate open boundary conditions, four syringes without plungers were inserted into holes on the top plate between the outermost wells and the edges of the study area, thus creating standpipes which act as reservoirs for overflow and resupply. The apparatus can be seen in Figure 3.
In order to adjust for scaling factors a fluid with a high viscosity was chosen. The fluid chosen was silicone oil. This silicone oil has a viscosity of 10.016 mPas. Injection and extraction times and quantities were measured to obtain the volumetric flow rate. As the flow generated from manipulation of the syringe and flowed outward from there, the syringe nozzles were the smallest radius of flow and therefore the point of highest fluid velocity for the injections and extractions. This means that the highest Reynolds number for each injection or extraction would also be at the opening of the syringe nozzle. Therefore the surface area for flow was calculated from the aperture of the syringe opening, radius of 1 mm, and the depth of the model space within the apparatus, 0.1 mm. The fluid velocity was calculated by dividing the volumetric flow rate for each injection and extraction by the aperture at the syringe nozzle opening. As long as there was a Reynolds number of less than one, the flow throughout the apparatus would be laminar. There have been four successful sequences showing that the injections and extractions can induce stretching and folding. The calculations of Reynolds numbers from a representative sequence are covered in Table 1.
The camera used was a Cyber-shot DSC-HX300 20.4 Megapixel Digital Camera (Sony Corporation of America, New York, NY).
9


Figure 3: This is the apparatus fabricated for this study. The camera is positioned as it was during sequences to record images of the result of each injection and extraction and A) is centered B) is from the left and C) is from the right.
10


Procedure
Apparatus Loading
Initial loading of the apparatus began with injecting silicone oil (Viscosity Reference Standard RT10000, Fungilab, Inc., New York, NY) into the center well. To avoid air bubbles in the apparatus, the well was filled from the edge with clear oil to displace the air. Then immediately once the air was displaced, the syringe was brought to a vertical position and attached by screwing it onto the well. This method was used for every injection step.
First the center well was filled until the oil passed the first ring of outside wells. Next, silicone oil was injected from each of the inner ring of wells until the oil reached the next ring. Then silicone oil was injected from each of the outer ring of wells until the oil reached the stand pipes.
This sequence of loading was necessary to fill the apparatus to the stand pipes. Loading only from the center well could not reach the stand pipes as the high viscosity and friction due to the small aperture could not be overcome.
Initial plume
The contaminant plume or source zone was simulated by mixing silicone oil with paint pigment (Pearl Ex Powdered Pigments Carbon Black, Rupert, Gibbon & Spider, Inc., Healdsburg, CA). For this experiment black pigment was used. Pigment was added until the colored oil was completely opaque.
We began by filling the center well with colored oil to displace the air as in the initial loading. After displacing the air and attaching the syringe, some oil was withdrawn to capture any remaining air inside the well. The air will no longer interfere once it has
11


reached the top of the syringe; at this point is when the air bubble will be visible as the colored oil is opaque. 0.5 mL of the colored oil was injected into the apparatus to create the simulated contaminated plume.
Clear oil was injected into the well until the well was full to prevent attraction between the colored oil in the well and the colored oil in the apparatus (otherwise there will be an unexpected result of a trail of colored oil, resembling that left by a garden snail, that is thought to be an experimental artifact but not be experienced in an actual setting).
Injection and Extraction Method
The air bubble removal method mentioned earlier was necessary every time oil was injected into the apparatus. It was not necessary for extractions. The amount injected or extracted varied between sequence steps in order to achieve the desired shapes. As the shapes in individual steps had to be formed in order to cause the intended stretching and folding, the shape to be achieved was more important than the quantity to be injected or extracted. In Figures 4 through 10 are A) the intended shape from the computer simulation, B) the photograph of the shape created by injection or extraction and C) the outline from the computer simulation on top of the photograph to demonstrate how close the physical demonstration came to the computer simulation.
To determine the Reynolds number the amount injected/extracted and the time of injection/extraction was recorded.
12


Table 1: These are the calculations of the Reynolds number for each step in the sequence
at the opening of the syringes.
CD P£h =p*u*L/ju
Step Volume injected (mL) Volume Injected (mmA3) Injection Time (sec) Volumetric Flow Rate (mmA3/s) Velocity1 u (mm/s) Reynolds Number
1 14 14000 140 100.0 159.2 0.00154
2 19 19000 131 145.0 231.0 0.00223
3 16 16000 63 254.0 404.4 0.00390
4 11 11000 242 45.5 72.4 0.00070
5 5 5000 57 87.7 139.7 0.00135
6 4 4000 146 27.4 43.6 0.00042
7 16 16000 135 118.5 188.7 0.00182
8 9 9000 155 58.1 92.5 0.00089
9 6 6000 118 50.8 81.0 0.00078
10 14 14000 251 55.8 88.8 0.00086
11 4 4000 52 76.9 122.5 0.00118
12 5 5000 76 65.8 104.8 0.00101
'The velocity shown is the maximum velocity calculated at the point of injection and extraction.
13


Figure 4: In these images are A) the theoretical shape, B) the actual photograph and C)
the photograph with theoritical shape overlay.
14


Figure 5: In these images are A) the theoretical shape, B) the actual photograph and C)
the photograph with theoritical shape overlay.
15


Figure 6: In these images are A) the theoretical shape, B) the actual photograph and C)
the photograph with theoritical shape overlay.
16


Figure 7: In these images are A) the theoretical shape, B) the actual photograph and C)
the photograph with theoritical shape overlay.
17


Figure 8: In these images are A) the theoretical shape, B) the actual photograph and C)
the photograph with theoritical shape overlay.
18


Figure 9: In these images are A) the theoretical shape, B) the actual photograph and C)
the photograph with theoritical shape overlay.
19


g Step 7
Figure 10: In these images are A) the theoretical shape, B) the actual photograph and C)
the photograph with theoritical shape overlay.
20


Figure 11: In these images are A) the theoretical shape, B) the actual photograph and C)
the photograph with theoritical shape overlay.
21


i # Step 9
Figure 12: In these images are A) the theoretical shape, B) the actual photograph and C)
the photograph with theoritical shape overlay.
22


Figure 13: In these images are A) the theoretical shape, B) the actual photograph and C)
the photograph with theoritical shape overlay.
23


k # Step 11
V
t
A

0
\ U
i O' i i
Figure 14: In these images are A) the theoretical shape, B) the actual photograph and C)
the photograph with theoritical shape overlay.
24


Figure 15: In these images are A) the theoretical shape, B) the actual photograph and C)
the photograph with theoritical shape overlay.
25


Image Analysis
To obtain an accurate area of dispersion imaging software was used. The software used was ERDAS Imagine 2013 (Intergraph Corporation, Englewood, CO), which was developed to analyze satellite images. To analyze the photos the software must be trained on what are non-dispersed plume, dispersed plume and non-plume areas. This was done using a signature editor and a nearest neighbor grow technique with limits of 100,000 pixels and a Euclidean distance of 10 (units not specified in software). The image alarm was used with a setting of two standard deviations of discrimination. The image alarm created a more clearly defined border for the dispersed regions.
Once the dispersed regions were clearly defined, the images were loaded into AutoCAD 2012 and a grid was used to determine where to start measuring dispersion lengths following the methods of Neupauer et al (2014). The results of the analyses are shown in Figures 16-27.
Another method of selecting locations form dispersion length was performed.
This method was closer to a Lagrangian analysis in that it was an attempt to follow the lifespan of a particle or group of particles rather than the entire plume. The symbols on the theoretical shapes in Figures 4 through 15 were used to designate where to measure the dispersion length. Then the progression of lengths for each symbol was graphed as in
26


6.0000
Dispersion Length [mm]
0.0000
4.0000
2.0000
3.0000
5.0000
1.0000
Diamond
0 2 4 6 8 10 12 14
Step in Sequence
Figure 29: This graph shows the change in dispersion length throughout the sequence of injections and extractions using a near Langrangian method. This additional method of analysis confirmed that dispersion did not continue to grow throughout the sequence of injections and extractions.
Wherever the outer edge of the dispersion met a grid line the length was measured from that point perpendicularly to the inner edge of the dispersion. These lengths were averaged and the change in dispersion is show in the graph in Figure 28.
27


Step 1
Dispersion Lengths Average Length
0.03 0.03
0.03
0.03
0.03
0.02
0.02
0.04
Figure 16: The photographs displayed are A) the actual photograph, B) the analyzed
image, C) the image measured for dispersion, D) the dispersion measurement and E) the
table of the measured lengths and the average of the measured length.
28


CJ D
Step 2
Dispersion Lengths Average Length
0.04 0.02
0.02
0.01
0.01
0.01
0.02
0.01
0.02
0.02
0.02
0.04
0.04
0.04
Figure 17: The photographs displayed are A) the actual photograph, B) the analyzed image, C) the image measured for dispersion, D) the dispersion measurement and E) the table of the measured lengths and the average of the measured length.
29


C D
Step 3
Dispersion Lengths Average Length
0.08 0.05
0.03
0.02
0.03
0.04
0.04
0.04
0.03
0.06
0.07
0.09
Figure 18: The photographs displayed are A) the actual photograph, B) the analyzed image, C) the image measured for dispersion, D) the dispersion measurement and E) the table of the measured lengths and the average of the measured length.
30


c; d
Step 4
Dispersion Lengths Average Length
0.01 0.04
0.03
0.02
0.04
0.06
0.03
0.05
0.05
0.12
0.03
Figure 19: The photographs displayed are A) the actual photograph, B) the analyzed
image, C) the image measured for dispersion, D) the dispersion measurement and E) the
table of the measured lengths and the average of the measured length.
31


* i
c
B
dM
D
Step 5
Dispersion Lengths Average Length
0.01 0.04
0.04
0.01
0.04
0.03
0.06
0.12
Figure 20: The photographs displayed are A) the actual photograph, B) the analyzed image, C) the image measured for dispersion, D) the dispersion measurement and E) the table of the measured lengths and the average of the measured length.
32


C: D
Step 6
Dispersion Lengths Average Length
0.04 0.10
0.12
0.11
0.05
0.15
0.13
Figure 21: The photographs displayed are A) the actual photograph, B) the analyzed image, C) the image measured for dispersion, D) the dispersion measurement and E) the table of the measured lengths and the average of the measured length.
33


L D Step 7
Dispersion Lengths Average Length
0.16 0.14
0.15
0.38
0.09
0.14
0.05
0.03
Figure 22: The photographs displayed are A) the actual photograph, B) the analyzed image, C) the image measured for dispersion, D) the dispersion measurement and E) the table of the measured lengths and the average of the measured length.
34


C .D
Step 8
Dispersion Lengths Average Length
0.03 0.09
0.04
0.13
0.14
0.06
0.19
0.06
Figure 23: The photographs displayed are A) the actual photograph, B) the analyzed image, C) the image measured for dispersion, D) the dispersion measurement and E) the table of the measured lengths and the average of the measured length.
35


D
Step 9
Dispersion Lengths Average Length
0.05 0.06
0.05
0.03
0.02
0.02
0.14
0.03
0.13
0.05
0.05
0.05
Figure 24: The photographs displayed are A) the actual photograph, B) the analyzed
image, C) the image measured for dispersion, D) the dispersion measurement and E) the
table of the measured lengths and the average of the measured length.
36


C D
Step 10
Dispersion lengths Average Length
0.03 0.08
0.09
0.03
0.15
0.06
0.07
0.04
0.18
0.11
0.04
0.03
Figure 25: The photographs displayed are A) the actual photograph, B) the analyzed
image, C) the image measured for dispersion, D) the dispersion measurement and E) the
table of the measured lengths and the average of the measured length.
37


Step 11
Dispersion Lengths Average Length
0.15 0.04
0.04
0.09
0.03
0.02
0.03
0.03
0.03
0.03
0.01
0.01
0.01
0.02
Figure 26: The photographs displayed are A) the actual photograph, B) the analyzed
image, C) the image measured for dispersion, D) the dispersion measurement and E) the
table of the measured lengths and the average of the measured length.
38


B
Step 12
Dispersion Lengths Average Length
0.01 0.04
0.04
0.03
0.03
0.02
0.06
0.03
0.07
0.03
0.03
0.05
0.03
0.08
0.07
0.06
0.09
Figure 27: The photographs displayed are A) the actual photograph, B) the analyzed image, C) the image measured for dispersion, D) the dispersion measurement and E) the table of the measured lengths and the average of the measured length.
39


4.00000
0.00000 -----------------1--------------1----------------1---------------1--------------1-
1 3 5 7 9 11
Sequence Step
Figure 28: This graph shows the change in dispersion length throughout the sequence of
injections and extractions using a near Eularian method.
40


Dispersion Length [mm]
North
South
East
West
Square
Circle
Star
Diamond
Figure 29: This graph shows the change in dispersion length throughout the sequence of injections and extractions using a near Langrangian method.
41


CHAPTER III
DISCUSSION
The amount of colored oil injected was 0.5 mL. At a thickness of 0.1 mm the diameter of the plume would be:
As you can see from Figure 30 the actual diameter is 22.6 mm with a corresponding volume of 0.5 mL. The difference is that 0.38 mL of the colored oil is not accounted for. This could result from the aperture being larger than the intended 0.1 mL, or from colored oil still within the well, or from some other unknown effect.
The apparatus is an idealization. It is a Newtonian fluid in a laminar state.
Because of this idealized state the dispersion is not continually gaining. As the plume was moved by the injections and extractions a trail of dispersion followed the main body of the plume. Dispersion following a plume tends to increase with more movement. In this study, however the dispersion increased and decreased with a final result of less dispersion than the maximum that occurred during the stretching and folding. This can be seen in the graph in Figure 30. This appeared to be caused by the plume retracing its path. Because of the plume retracing its path it went back over the regions that previously had dispersion and recollected the dispersed material. So as dispersion was left behind by the plume, it was recaptured in front of the plume.
As mentioned earlier, the pigment appeared to decrease the effective viscosity of the colored oil and therefore a trail of semi-colored oil lagged back, connecting the plume to the well. Attempts were made to diminish this effect by injecting enough clear oil into
42


the well to prevent there being any left behind. It was difficult to inject an exact amount where the oil was no longer connected to the well and not produce an area within the center of the plume that appeared to be dispersion.
43


44


There was another unexpected result. During extractions a filament of the colored plume would shoot from the center of the plume toward the well much faster than the rest of the plume. While the center of the plume was intended to move toward the well faster (this is how folding is induced) the far greater speed of the tendril was unexpected. This was thought to have been caused by a difference in viscosity from addition of the pigment. As was shown in Jha et al. (2011), a fluid with a lower viscosity displacing a fluid with a higher viscosity will cause viscous fingering.
Future study will need to be done to overcome some of the hurdles not fully addressed in this physical demonstration.
Further on the research will have to go into the third dimension. Once working in the third dimension, porous media can be used. This will begin the process for determining the proper sequences and site specific variable adjustment needed for practical application.
45


CHAPTER IV
CONCLUSION
The intent of this experimental venture was to demonstrate a new technique for management of contamination of groundwater. This physical demonstration shows that injection and extraction wells can cause stretching and folding in laminar, two-dimensional flows. Stretching and folding has been shown in earlier work to cause chaotic advection.
As chaotic advection promotes mixing, this demonstration therefore shows that injection and extraction can effectively cause mixing in two-dimensional, laminar fluid flows. This mixing can increase the effectiveness of in situ groundwater remediation in transmissive zones such as the source zone or the plume in a highly permeable layer.
46


REFERENCES
Aref, H. (1984). Stirring by chaotic advection. J. Fluid Mech., 143, 1-21.
Gleick, J. (1987). Chaos: Making a New Science. Penguin Group, New York.
Jha, B., Cueto-Felgueroso, L., Juanes, R. (2010) Fluid mixing from viscous finger Amer. Phy. Society, PRL 106, 194502, doi: 10.1103/PhysRevLett. 106.194502
Jones, S. W. and Aref, H. (1987). Chaotic advection in pulsed source-sink systems. Phys. Fluids, 31(3), 469-485.
Mays, D. C., and R. M. Neupauer (2012), Plume spreading in groundwater by stretching and folding, Water Resource. Res., 48, W07501, doi: 10.1029/2011WR011567.
Neupauer, R.M., J.D. Meiss, and D.C. Mays (2014), Chaotic advection during
engineered injection and extraction in heterogeneous porous media. Water Resources Research, 50(2), 1433-1447
ITRC (Interstate Technology & Regulatory Council). (2011). Integrated DNAPL Site Strategy. IDSS-1.
47


Full Text

PAGE 1

CHAOTIC A DVECTION FROM STRETC H ING AND FOLDING IN A HELE SHAW C ELL by MATT N. JONES BS, University of Colorado Denver, 2012 A thesis submitted to the Faculty of the Graduate School of University of Colorado in partial fulfillment o f the requirements for the degree of Master of Science Civil Engineering 2014

PAGE 2

ii This thesis for the Master of Science degree by Matt N. Jones has been approved for the Civil Engineering program by David C. Mays Chair Azadeh Bol hari James C.Y. Guo July 23, 2014

PAGE 3

i ii Jones, Matt N. (M.S., Civil Engineering) Chaotic Advection from Stretching and F olding in a Hele Shaw C ell Thesis directed by Assistant Professor David C. Mays ABSTRACT Contamination of groundwater can lead to unsafe drinking water and ecological damage. Contamination can result from point sources such as leaking underground gasoline tanks, spills of industrial waste and many ot her sources. Because of this remediation of contaminated groundwater is of great concern. In situ treatment has some attractive benefits such as lower cost and a shorter treatment time when compared with the most common current technique of extraction a nd treatment before reinjection. One of the main difficulties in attempting in situ groundwater treatment is that groundwater flows are very slow which maintains laminar flow Laminar flow in an engineering reactor has the characteristic of poor or limited mixing when compared to turbulent flow Because of this a treatment solution injected into the groundwater will experience little to no mixing with the contaminant This in turn means that contaminant degradation is dependent on flow characteristics. Previous research has shown that chaotic advection can induce plume spreading. Plume spreading promotes mixing which is needed to increase the quantity of reactions More recent research has shown that injection and extraction can cause chaotic advection through the process of stretching and folding the contaminated groundwater plume. This research, however, has been in the analytical realm What wa s needed at this point wa s to provide a physical demonstration of this process.

PAGE 4

iv The purpose of this study wa s first to build an apparatus that demonstrates the effects of injection and extraction on a plume in a two dimensional plane Once the apparatus was functional, the next step was to use the apparatus to demonstrate that injection and extraction can induce stretching a nd folding This study has yielded a Hele Shaw c ell which displays the effect of injection and extraction o n a source zone and on a contaminant plume still within a high permeability layer With this Hele Shaw c ell it has been shown that the t heorized stretching and folding from injection and extraction can be made to occur in a physical setting. Through visualization of the effects of injection and extraction it was observed that the results in the model were close to the theoretical results. Near Eulerian a nd near Lagrangian analysis showed that dispersion did not continuously increase as expected; rather, dispersion increased and then decreased. The form and content of this abstract are approved. I recommend its publication. Approved: David Mays

PAGE 5

v TABLE OF CONT ENTS CHAPTER I. INTRODUCTION ................................ ................................ ................................ .. 1 Literature Review ................................ ................................ ................................ 6 Research Obj ective ................................ ................................ .............................. 6 Research Overview ................................ ................................ ............................. 7 II. METHODS ................................ ................................ ................................ ............. 8 Apparatus ................................ ................................ ................................ ............ 8 Procedure ................................ ................................ ................................ ........... 11 Apparatus Loading ................................ ................................ ........................ 11 Initial plume ................................ ................................ ................................ ... 11 Injection and Extraction Method ................................ ................................ ... 12 Image Analysis ................................ ................................ ................................ .. 26 III. DISCUSSION ................................ ................................ ................................ ....... 42 IV. CONCLUSION ................................ ................................ ................................ ..... 46 REFERENCES ................................ ................................ ................................ ................. 47

PAGE 6

vi LIST OF TABLES TABLE 1. ................................ ................................ ................................ ................................ ........ 13

PAGE 7

vii LIST OF FIGURES FIGURE 1 ................................ ................................ ................................ ................................ ........... 4 2 ................................ ................................ ................................ ................................ ........... 5 3 ................................ ................................ ................................ ................................ ......... 10 4 ................................ ................................ ................................ ................................ ......... 14 5 ................................ ................................ ................................ ................................ ......... 15 6 ................................ ................................ ................................ ................................ ......... 16 7 ................................ ................................ ................................ ................................ ......... 17 8 ................................ ................................ ................................ ................................ ......... 18 9 ................................ ................................ ................................ ................................ ......... 19 10 ................................ ................................ ................................ ................................ ....... 20 11 ................................ ................................ ................................ ................................ ....... 21 12 ................................ ................................ ................................ ................................ ....... 22 13 ................................ ................................ ................................ ................................ ....... 23 14 ................................ ................................ ................................ ................................ ....... 24 15 ................................ ................................ ................................ ................................ ....... 25 16 ................................ ................................ ................................ ................................ ....... 28 17. ................................ ................................ ................................ ................................ ...... 29 18 ................................ ................................ ................................ ................................ ....... 30 19 ................................ ................................ ................................ ................................ ....... 31 20. ................................ ................................ ................................ ................................ ...... 32 21. ................................ ................................ ................................ ................................ ...... 33

PAGE 8

viii 22 ................................ ................................ ................................ ................................ ....... 34 23. ................................ ................................ ................................ ................................ ...... 35 24 ................................ ................................ ................................ ................................ ....... 36 25 ................................ ................................ ................................ ................................ ....... 37 26 ................................ ................................ ................................ ................................ ....... 38 27 ................................ ................................ ................................ ................................ ....... 39 28 ................................ ................................ ................................ ................................ ....... 40 29 ................................ ................................ ................................ ................................ ....... 41 30 ................................ ................................ ................................ ................................ ....... 44

PAGE 9

1 CHAPTER I INTRODUCTION The field of Civil Engineering holds the responsibility of planning, designing, constructing, operating and protecting our infrastructure. This duty is applied to, among other things, the buildings in which we live our lives, the roads we travel, the air w e breathe and the impact we have on the natural world around us. The natural world includes the water falling from the skies, coursing across the land and slowly trickling beneath our feet. This water in the ground beneath our feet helps to sustain us and is simultaneously vulnerable to us. When we pollute the groundwater we contaminate not only the soil and minerals beneath us, we also risk our own health as this water can eventually trickle into our streams, be pumped up in our wells or be drawn up by t he plants we eat. It is because of this interdependence with groundwater that we have the responsibility to remedy the damage that we cause Contamination of the groundwater must be remedied. Remediation of the groundwater is not easy. The most common meth od of remediation used today is to extract the contaminated groundwater treat the extracted water with an appropriate form of remediation, and the n re inject it back into the ground once its treatment is complete. This method is usually referred to as pum p and treat remediation While pump and treat is effective, it is also very time consuming, labor intensive and costly. It would not only be beneficial to the populace to have groundwater remediation be quicker, it would also incentivize polluters to clean up their messes if it is less burdensome to them to do so. Additionally, having groundwater remediation be

PAGE 10

2 cheaper could keep production and sales costs down. Combined, the listed benefits could lead to a safer and more productive society. One idea for remedying groundwater contamination is to inject the appropriate treatment solution directly into the source zone or plume in a transmissive zone of contaminated groundwater or in situ treatment. The transmissive zone of a contaminant plume i s outlined in ITRC (2011). In situ treatment can be simple, quick and cheap. Direct injection of the treatment solution has some limitations. As groundwater moves at a geologically slow pace, t urbulence can never be achieved except in the unusual cases in the immediate vicinity of injection and extraction wells. So, when a treatment solution is introduced into a contamination source zone or plume in a transmissive zone the surface area where the two plumes meet will be the only place the chemical neutraliza tion occurs, resulting in a mass of contaminated water, a smaller mass of the remedying chemical within it and an interfacial surface between them of treated water This mass within a mass configuration would then carry on down the gradient with the interf acial layer of treated water growi ng very slowly; usually to o slowly to prevent the contaminant from affecting its surroundings This configuration is dependent on the treatment solution and source zone or plume having similar densities so as to prevent on e flowing over the other. This is shown in Figure 1 To cause more of the remediation interaction to occur, the contaminant and the treatment solution need to be broug ht together more often A possible solution to that is postulated in Mays & Neupauer (2012). Stretching and folding is one way to cause plume spreading For this application, plume spreading is the process of lengthening the interfacial surface between th e

PAGE 11

3 contaminant and the treatment solution. By increasing the length of interface we increase the possibility of chemical interaction between the contaminant and the treatment solution by increasing the actual physical space in which these reactions can occu r. This study is a physical demonstration using a Hele Shaw cell to show that stretching and folding can be induced by a sequence of injections and extraction s arrayed around the contaminant as postul ated in Mays & Neupauer (2012) and shown in Figure 2

PAGE 12

4 Figure 1 : This is an example of contaminant and treatment solution in natural groundwater flow.

PAGE 13

5 Figure 2 : This is the computer example of stretching and folding induced by a sequence of injections and extractions from Mays and Neupauer (2012). Used with permission of the American Geophysical Union.

PAGE 14

6 Literature Review Early in the exploration of c haos t heory Stephen Smale showed that stretching and folding could be accomplish ed using chaotic particle trajectories with his graphical orseshoe (Gleick, 1987), much like the action of a taffy puller. In 1984 it was shown that flo ws that contain chaos can be engineered. These engineered flows were plume spreading, which we now call chaotic advection (Aref, 1984). Later, Aref joined with Jones to prove that chaotic advection could be produced in irrotational flows which is the case in all laminar flows in homogeneous porous media (Jones & Aref, 1988). Mays and Neupauer then showed that this chaotic advection could occur without reinjection (Mays & Neupauer, 2012). Research Objective Previous research has shown that injection and extr action can cause stretching and folding during laminar flow i n a two dimensional plane in computer simulations. While the theory is sound and the mathematics verified, the next logical step is to show that the theory works in a real world setting. The goal of this study is to physically demonstrate this theory A physical demonstration would entail an actual apparatus to show what happens to a fluid when it is pushed and pulled by injection and extraction wells. The use of the apparatus in this demonstratio n has certain confining parameters. The only effects allowed to occur for the results of this demonstration to be of value are those generated from the injections and extractions. The apparatus must be horizontal to eliminate gravitational effects. The pla tes in the apparatus must be truly smooth and flat to eliminate unpredictable distorting effects from the plates themselves. The Reynolds number for the fluid being injected and the fluid moving within the

PAGE 15

7 apparatus must be below one to keep the flow lamin ar and thereby eliminate any dispersion or spreading effects from turbulence. Research Overview A physical apparatus has been constructed to demonstrate the effects of injection and extraction on a plume. A fluid was found to fill the apparatus and be use d for injection and extraction that simulated the behavior water would have at this scale. I njection and extraction sequences were tested to find one that induced the desired stretching and folding. At each step of injection and extraction p hotographs were taken for demonstration of the desired stretching and folding and to analyze the dispersion that occurred during the sequence. These photographs were analyzed using image analysis software

PAGE 16

8 CHAPTER I I METHODS Apparatus The apparatus is a Hele Shaw cell as shown in Figure 3 This means that it consists of two flat plates that define a quasi two dimensional area for the purpose of stu dying the flow paths of a liquid. This particular apparatus consisted of two flat plates of acrylic (Plasticare, Englewood, CO). The upper plate was 19 mm thick and clear while the lower plate was 19 mm thick and opaque white. Along all the edges of the de fined area ( 400 mm 400 mm) there is a foam rubber D shaped gasket that prevents leakage. The two plates are pressed together using nut and bolt configurations at one at each of the four corners. Combined, the nuts and bolts squeeze the D shaped gaskets m aking a constant aperture of 0.1 mm and forming a tight seal. Wells were simulated by the use of intravenous (IV) therapy hubs (SAFSITE Valve and Cap RV 1000, B. Braun Medical Inc., Bethlehem, PA). The wells were fail close, only opening to liquid flow wh en the internal plunger is depressed by the tip of other IV administrative devices, in this case the tip from a syringe (60 mL and 5 mL syringes, Becton, Dickinson and Company, Franklin, NJ), connected through IV valves were used to induce extraction and i njection flows. A single well is at the center. From the center well out in the direction perpendicular to the plate edges two wells were placed to divide the distance from the center well to the edge of the study area into two equal parts, each separated by a distance of 52 mm.

PAGE 17

9 In order to simulate open boundary conditions, four syringes without plungers were inserted into holes on the top plate between the outermost wells and the edges of the study area, thus creating standpipes which act a s reservoir s for overflow and resupply. The apparatus can be seen in Figure 3 In order t o adjust for scaling factors a fluid with a high viscosity was chosen. The fluid chosen wa s silicone oil. This silicone oil has a viscosity of 10.016 mPas. Injection and extraction times and quantities were measured to obtain the volumetric flow rate. As the flow generated from manipulation of the syringe and flowed outward from there, the syr inge nozzles were the smallest radius of flow and therefore the point of highest fluid velocity for the injections and extractions. This means that the highest Reynolds number for each injection or extraction would also be at the opening of the syringe noz zle. Therefore the surface area for flow was calculated from the aperture of the syringe opening, radius of 1 mm, and the depth of the model space within the apparatus, 0.1 mm. The fluid velocity was calculated by dividing the volumetric flow rate for each injection and extraction by the aperture at the syringe nozzle opening. As long as there was a Reynolds number of less than one, the flow throughout the apparatus would be laminar. There have been four successful sequences showing that the injections and extractions can induce stretching and folding. The calculations of Reynolds numbers from a representative sequence are covered in Table 1. The camera used was a Cyber shot DSC HX300 20.4 Megapixel Digital Camera (Sony Corporation of America, New York, NY).

PAGE 18

10 Figure 3 : This is the apparatus fabricated for this study. The camera is positioned as it was during sequences to record images of the result of each injection and extraction and A) is centered B) is from the left and C) is from the right.

PAGE 19

11 Procedure Apparatus Loading Initia l loading of the apparatus began with injecting silicone oil (Viscosity Reference Standard RT10000, Fungilab, Inc., New York, NY) into the center well. To avoid air bubbles in the apparatus, the well was filled from the edge with clear oil to displace the air. Then immediately once the air wa s displaced, the syringe was brought to a vertical position and attached by screwing it onto the well. This method wa s used for every injection step. First the center well was fill ed until the oil passe d the first ring of outside wells. Next, silicone oil was injected from each of the inner ring of wells until the oil reache d the next ring. Then silicone oil was injected from e ach of the outer ring of wells until the oil reache d the stand pipes. This sequence of loading was necessary to fill the apparatus to the stand pipes. Loading only from the center well could not reach the stand pipes as the high viscosity and friction due to the small aperture could not be overcome. Initial plume The contaminant plume or source zone was simulated by mixing silicone oil with paint pigment ( Pearl Ex Powdered Pigments Carbon Black Rupert, Gibbon & Spider, Inc. Healdsburg, CA ) For this experiment black pigment was used. Pigment was added until the colored oil wa s completely opaque. We bega n by filling the center well with colored oil to displace the air as in the initial loading. After displacing the air and attaching the syri nge, some oil was withdraw n to capture any remaining air inside the well. The air will no longer interfere once it has

PAGE 20

12 reac hed the top of the syringe; at this point is when the air bubble will be visible as the colored oil is opaque. 0. 5 mL of the colored oil was injected into the apparatus to c reate the simulated contaminated plume Clear oil was injected into the well until the well wa s full to prevent attraction between the colored oil in the well and the colored oil in the apparatus (otherwise there wi ll be a n unexpected result of a trail of colored oil resembling that left by a garden snail, that is thought to be an experimental artifact but not be experienced in an actual setting ). Injection and Extraction Method The air bubble removal method mentioned earlier was necessary every time oil wa s injected into the apparatus. It wa s not necessary for extractions. The amount injected or extracted varied between sequence steps in order to achieve the desired shapes. As th e shapes in individual steps had to be formed in order to cause the intended stretching and folding, t h e shape to be achieved was more important than the quantity to be injected or extracted In Figures 4 through 10 are A) the intended shape from the compu ter simulation, B) the photograph of the shape created by injection or extraction and C) the outline from the computer simulation on top of the photograph to demonstrate how close the physical demonstration came to the computer simulation. To determine the Reynolds number the amount injected/extracted and the time of injection/extraction was recorded

PAGE 21

13 Table 1 : These are the calculations of the Reynolds number for each step in the sequence at the opening of the syringes. Step Volume injected (mL) Volume Injected (mm^3) Injection Time (sec) Volumetric Flow R ate (mm^3/s) Velocity u (mm/s) Reynolds Number 1 14 14000 140 100.0 159.2 0.00154 2 19 19000 131 145.0 231.0 0.00223 3 16 16000 63 254.0 404.4 0.00390 4 11 11000 242 45.5 72.4 0.00070 5 5 5000 57 87.7 139.7 0.00135 6 4 4000 146 27.4 43.6 0.00042 7 16 16000 135 118.5 188.7 0.00182 8 9 9000 155 58.1 92.5 0.00089 9 6 6000 118 50.8 81.0 0.00078 10 14 14000 251 55.8 88.8 0.00086 11 4 4000 52 76.9 122.5 0.00118 12 5 5000 76 65.8 104.8 0.00101 The velocity shown is the maximum velocity calculated at the point of injection and extraction.

PAGE 22

14 Figure 4 : In these images are A) the theoretical shape, B) the actual photograph and C) the photograph with theoritical shape overlay.

PAGE 23

15 Figure 5 : In these images are A) the theoretical shape, B) the actual photograph and C) the photograph with theoritical shape overlay.

PAGE 24

16 Figure 6 : In these images are A) the theoretical shape, B) the actual photograph and C) the photograph with theoritical shape overlay.

PAGE 25

17 Figure 7 : In these images are A) the theoretical shape, B) the actual photog raph and C) the photograph with theoritical shape overlay.

PAGE 26

18 Figure 8 : In these images are A) the theoretical shape, B) the actual photograph and C) the photograph with theoritical shape overlay.

PAGE 27

19 Figure 9 : In these images are A) the theoretical shape, B) the actual photograph and C) the photograph with theoritical shape overlay.

PAGE 28

20 Figure 10 : In these images are A) the theoretical shape, B) the actual photograph and C) the photog raph with theoritical shape overlay.

PAGE 29

21 Figure 11 : In these images are A) the theoretical shape, B) the actual photograph and C) the photograph with theoritical shape overlay.

PAGE 30

22 Figure 12 : In these images are A) the theoretical shape, B) the actual photograph and C) the photograph with theoritical shape overlay.

PAGE 31

23 Figure 13 : In these images are A) the theoretical shape, B) the actual photograph and C) the photograph with theoritical shape overlay.

PAGE 32

24 Figure 14 : In these images are A) the theoretical shape, B) the actual photograph and C) the photograph with theoritical shape overlay.

PAGE 33

25 Figure 15 : In these images are A) th e theoretical shape, B) the actual photograph and C) the photograph with theoritical shape overlay.

PAGE 34

26 Image Analysis To obtain an accurate area of dispersion imaging software was used. The software used was ERDAS Imagine 2013 ( Intergraph Corporation, Englew ood, CO) which was developed to analyze satellite images To analyze the photos the software must be trained on what are non dispersed plume, dispersed plume and non plume areas. This was done using a signature editor and a nearest neighbor grow techniq ue with limits of 100,000 pixels and a Euclidean distance of 10 (units not specified in software) The image alarm was used with a setting of two standard de viations of discrimination. The image alarm create d a more clear ly define d border for the dispersed region s Once the dispersed regions were clearly defined, the images were loaded into AutoCAD 2012 and a grid was used to determine where to start measuring d ispersion lengths following the methods of Neupauer et al (2014 ) The results of the analyses are shown in Figures 16 27. Another method of selecting locations form dispersion length was performed. This method was closer to a Lagrangian analysis in that it was an attempt to follow the lifespan of a particle or group of particles rather than the entir e plume. The symbols on the theoretical shapes in Figures 4 through 15 were used to designate where to measure the dispersion length. Then the progression of lengths for each symbol was graphed as in

PAGE 35

27 Figure 29 : This graph shows the change in dispersion length throughout the sequence of injections and extractions using a near Langrangian method. This additional method of analysis confirmed that dispersion did not continue to grow throughout the sequence of injections and extractions. Wher ever the outer edge of the dispersion met a grid line the length was measured from that point perpendicularly to the inner edge of the dispersion. These lengths were averaged and t he change in dispersion is show in the graph in Figure 28 0.0000 1.0000 2.0000 3.0000 4.0000 5.0000 6.0000 0 2 4 6 8 10 12 14 Dispersion Length [mm] Step in Sequence North South East West Square Circle Star Diamond

PAGE 36

28 Step 1 Dispersion L engths Average Length 0.03 0.03 0.03 0.03 0.03 0.02 0.02 0.04 Figure 16 : The photographs displayed are A) the actual photograph, B) the analyzed image, C) the image measured for dispersion, D) the dispersion measurement and E) the table of the measured lengths and the average of the measured length.

PAGE 37

29 Step 2 Dispersion Lengths Average Length 0.04 0.02 0.02 0.01 0.01 0.01 0.02 0.01 0.02 0.02 0.02 0.04 0.04 0.04 Figure 17 : The photographs displayed are A) the actual photograph, B) the analyzed image, C) the image measured for dispersion, D) the dispersion measurement and E) the table of the measured lengths and the average of the measured length.

PAGE 38

30 Step 3 Dispersion Lengths Average Length 0.08 0.05 0.03 0.02 0.03 0.04 0.04 0.04 0.03 0.06 0.07 0.09 Figure 18 : The photographs displayed are A) the actual photograph, B) the analyzed image, C) the image measured for dispersion, D) the dispersion measurement and E) the table of the measured lengths and the average of the measured length.

PAGE 39

31 Step 4 Dispersion Lengths Average Length 0.01 0.04 0.03 0.02 0.04 0.06 0.03 0.05 0.05 0.12 0.03 Figure 19 : The photographs displayed are A) the actual photograph, B) the analyzed image, C) the image measured for dispersion, D) the dispersion measurement and E) the table of the measured lengths and the average of the measured length.

PAGE 40

32 Step 5 Dispersion Lengths Average Length 0.01 0.04 0.04 0.01 0.04 0.03 0.06 0.12 Figure 20 : The photographs displayed are A) the actual photograph, B) the analyzed image, C) the image measured for dispersion, D) the dispersion measurement and E) the table of the measured lengths and the average of the m easured length.

PAGE 41

33 Step 6 Dispersion Lengths Average Length 0.04 0.10 0.12 0.11 0.05 0.15 0.13 Figure 21 : The photographs displayed are A) the actual photograph, B) the analyzed image, C) the image measured for dispersion, D) the dispersion measurement and E) the table of the measured lengths and the average of the measured length.

PAGE 42

34 Step 7 Dispersion Lengths Average Length 0.16 0.14 0.15 0.38 0.09 0.14 0.05 0.03 Figure 22 : The photographs displayed are A) the actual photograph, B) the analyzed image, C) the image measured for dispersion, D) the dispersion measurement and E) the table of the measured lengths and the average of the measured length.

PAGE 43

35 Step 8 Dispersion Le ngths Average Length 0.03 0.09 0.04 0.13 0.14 0.06 0.19 0.06 Figure 23 : The photographs displayed are A) the actual photograph, B) the analyzed image, C) the image measured for dispersion, D) the dispersion measurement and E) the table of the measured lengths and the average of the measured length.

PAGE 44

36 Step 9 Dispersion Lengths Average Length 0.05 0.06 0.05 0.03 0.02 0.02 0.14 0.03 0.13 0.05 0.05 0.05 Figure 24 : The photographs displayed are A) the actual photograph, B) the analyzed image, C) the image measured for dispersion, D) the dispersion measurement and E) the table of the measured lengths and the average of the measured length.

PAGE 45

37 Step 10 Dispersion lengths Average Length 0.03 0.08 0.09 0.03 0.15 0.06 0.07 0.04 0.18 0.11 0.04 0.03 Figure 25 : The photographs displayed are A) the actual photograph, B) the analyzed image, C) the image measured for dispersion, D) the dispersion measurement and E) the table of the measured lengths and the average of the measured length.

PAGE 46

38 Step 11 Dispersion Lengths Average Length 0.15 0.04 0.04 0.09 0.03 0.02 0.03 0.03 0.03 0.03 0.01 0.01 0.01 0.02 Figure 26 : The photographs displayed are A) the actual photograph, B) the analyzed image, C) the image measured for dispersion, D) the dispersion measurement and E) the table of the measured lengths and the average of the measured length.

PAGE 47

39 Step 12 Dispersion Lengths Average Length 0.01 0.04 0.04 0.03 0.03 0.02 0.06 0.03 0.07 0.03 0.03 0.05 0.03 0.08 0.07 0.06 0.09 Figure 27 : The photographs displayed are A) the actual photograph, B) the analyzed image, C) the image measured for dispersion, D) the dispersion measurement and E) the table of the measured lengths and the average of the measured length.

PAGE 48

40 Figure 28 : This graph shows the change in dispersion length throughout the sequence of injections and extractions using a near Eularian method 0.00000 0.50000 1.00000 1.50000 2.00000 2.50000 3.00000 3.50000 4.00000 1 3 5 7 9 11 Average Dispersion Length [mm] Sequence Step

PAGE 49

41 Figure 29 : This graph shows the change in dispersion length throughout the sequence of injections and extractions using a near Langrangian method. 0.0000 1.0000 2.0000 3.0000 4.0000 5.0000 6.0000 0 2 4 6 8 10 12 14 Dispersion Length [mm] Step in Sequence North South East West Square Circle Star Diamond

PAGE 50

42 CHAPTER I II DISCUSSION The amount of colored oil injected was 0.5 mL. At a thickness of 0.1 mm the diameter of the plume would be: d=2 =79.79 mm As you can see from Figure 30 the actual diameter is 22.6 mm with a corresponding volume of 0.5 mL The difference is that 0.38 mL of the colored oil is not accounted for. This could result from the aperture being larger than the intended 0.1 mL, or from colored oil still within the well, or from some other unknown effect. The apparatus is an idealization. It is a Newtonian fl uid in a laminar state. Because of this idealized state the dispersion is not continually gaining. As the plume was moved by the injections and extractions a trail of dispersion followed the main body of the plume. Dispersion following a plume tends to inc rease with more movement. In this study, however the dispersion increased and decreased with a final result of less dispersion than the maximum that occurred during the stretching and folding. This can be seen in the graph in Figure 30 This appeared to be caused by the plume retracing its path. Because of the plume retracing its path it went back over the regions that previously had dispersion and recollected the dispe rsed material. So as dispersion was left behind by the plume, it was recaptured in front of the plume. As ment ioned earlier the pigment appeared to de crease the effective viscosity of th e colored oil and therefore a trail of semi colored oil lagged back connecting the plume to the well. Attempts were made to diminish this effect by injecting enough clear oil into

PAGE 51

43 the well to prevent there being any left behind. It was difficult to inject an exact amount where the oil was no longer connected to the well and not produce an area within the center of the plume that appeared to be dispersion.

PAGE 52

44 Figure 30 : This is the initial plume before any manipulation has occurred.

PAGE 53

45 There was another unexpected result. During extractions a filam ent of the colored plume would shoot from the center of the plume toward the well much faster than the rest of the plume. While the center of the plume was intended to move toward the well faster (this is how folding is induced) the far greater speed of th e tendril was unexpected. This was thought to have been caused by a difference in viscosity from addition of the pigment. As was shown in Jha et al. (2011), a fluid with a lower viscosity displacing a fluid with a higher viscosity will cause viscous finger ing. Future study will need to be done to overcome some of the hurdles not fully addressed in this physical demonstration Further on the research will have to go into the third dimension. Once working in the thir d dimension, porous media can be used. T his will begin the process for determining the proper sequences and site specific variable adjustment needed for practical application.

PAGE 54

46 CHAPTER I V CONCLUSION The intent of this experimental venture was to demonstrate a new technique for management of contamination of groundwater. This physical demonstration shows that injection and extraction wells can cause stretching and folding in laminar, two dimensional flows Stretching and folding has been shown in earlier work t o cause chaotic advection. As chaotic advection promotes mixing, this demonstration therefore shows that injection and extraction can effectively cause mixing in two dimensional laminar fluid flows This mixing can increase the effectiveness of in situ g roundwater remediation in transmissive zones such as the source zone or the plume in a highly permeable layer.

PAGE 55

47 REFERENCES Aref, H. (1984). 1 21. Gleick, J. (1987). Chaos: Making a New Science. Penguin Group, New York. Jha, B. Cueto Amer. Phy. Society, PRL 106, 194502, doi: 10.1103/PhysRevLett.106.194502 sin Phys. Fluids, 31(3), 469 485. ce Res., 48, W07501, doi:10.1029/2011WR011567. Neupauer, R.M., J.D. Meiss, and D.C. Mays (2014), Chaotic advection during engineered injection and extractio Water Resources Research, 50(2), 1433 1447 1.