a study of the effect of mobility ratios on ...1. a study of the effect of mobility ratios on...
TRANSCRIPT
![Page 1: A STUDY OF THE EFFECT OF MOBILITY RATIOS ON ...1. A study of the effect of mobility ratios on pattern displacement behavior It is well known, for unit mobility ratio, that the areal](https://reader031.vdocuments.net/reader031/viewer/2022040817/5e61c32167659d0b291248ba/html5/thumbnails/1.jpg)
A STUDY OF THE EFFECT OF MOBILITY RATIOS
ON PATTERN DISPLACEMENT BEHAVIOR AND
STREAMLINES TO INFER PERMEABILITY
FIELDS PERMEABILITY MEDIA
SUPRI TR 115
By
Yuandong Wang
Anthony R. Kovscek
William E. Brigham
December 1998
Work Performed Under Contract No. DE-FG22-96BC14994
Prepared forU.S. Department of Energy
Assistant Secretary for Fossil Energy
Thomas Reid, Project ManagerNational Petroleum Technology Office
P.O. Box 3628
![Page 2: A STUDY OF THE EFFECT OF MOBILITY RATIOS ON ...1. A study of the effect of mobility ratios on pattern displacement behavior It is well known, for unit mobility ratio, that the areal](https://reader031.vdocuments.net/reader031/viewer/2022040817/5e61c32167659d0b291248ba/html5/thumbnails/2.jpg)
ii
TABLE OF CONTENTS
List of Figures iii
Acknowledgments iv
Abstract v
Chapter 1 Effect of Mobility Ratio on Pattern Behavior of
A Homogeneous Porous Medium ..................................................... 1
1.1 Introduction .......................................................................................... 1
1.2 Sweep Efficiency ................................................................................. 3
1.3 Simulation Results ............................................................................... 4
1.4 Breakthrough Determination ............................................................... 7
1.5 Discussion of Pattern Behavior ........................................................... 9
1.5.1 Areal Sweep Efficiency ......................................................... 10
1.5.2 Unit Mobility Ratio ............................................................... 13
1.5.3 Five-Spot Pattern, Very Favorable Mobility Ratio ............... 15
1.5.4 Staggered Line Drive, Very Favorable Mobility Ratio ......... 17
1.6 Conclusions ......................................................................................… 19
Appendix A—Effect of Grid-block and Time-step Size .........................……….. 21
Appendix B—Effect of Grid-block Orientation .....................................………... 24
Appendix C—Numerical Dispersion ......................................................……….. 27
Chapter 2 Streamlines to Solve Inverse Problems .......................................... 29
2.1 Definition of the Research Topic ....................................................... 29
2.2 Importance of the Research ............................................................... 35
2.3 Future Work Plan and Expected Results ........................................... 35
Nomenclature 37
References 38
![Page 3: A STUDY OF THE EFFECT OF MOBILITY RATIOS ON ...1. A study of the effect of mobility ratios on pattern displacement behavior It is well known, for unit mobility ratio, that the areal](https://reader031.vdocuments.net/reader031/viewer/2022040817/5e61c32167659d0b291248ba/html5/thumbnails/3.jpg)
iii
LIST OF FIGURES
Chapter 1Fig. 1 Dyes et al (1954) Experimental Results ........................................ 2
Fig. 2 Comparison of Simulation Results and
Experimental Results by Dyes et al .....................................…… 6
Fig. 3 Fractional Flow(fw) vs tD at Producer—Determination of
Breakthrough Time ...................................……………………… 8
Fig. 4 Effect of Mobility Ratios and Pattern Geometry
on Areal Sweep Efficiency .....................................….................. 11
Fig. 5 Streamline and Saturation Distribution Unit Mobility Ratio
...............................................................……………………. 14
Fig. 6 Five-Spot Pattern, Very Favorable
Mobility Ratio, 1/M = 20 ...................................................……... 16
Fig. 7 Staggered-Line-Drive Pattern, Very Favorable Mobility Ratio,
1/M =20 ...........................………………………………………. 18
Fig. A Effect of Refining Grid and Increasing Time steps on the
simulation Accuracy ..............................………………..........…. 22
Fig. B.1 Grid Orientation………………………………………………. 24
Fig. B2 Effect of Grid Orientation on Simulation Results .............……. 26
Fig. C Displacing Front ..................................................................……. 28
Chapter 2Fig. 1 Comparison of Permeability Field ......................................……... 31
Fig. 2 Comparison on Breakthrough Curve ....................................…….. 32
Fig. 3 Error vs Iterations ..................................................................……. 34
![Page 4: A STUDY OF THE EFFECT OF MOBILITY RATIOS ON ...1. A study of the effect of mobility ratios on pattern displacement behavior It is well known, for unit mobility ratio, that the areal](https://reader031.vdocuments.net/reader031/viewer/2022040817/5e61c32167659d0b291248ba/html5/thumbnails/4.jpg)
iv
ACKNOWLEDGMENTS
This work was supported by the Assistant Secretary for Fossil Energy, Office of
Oil, Gas, and Shale Technologies of the U.S. Department of Energy under contract No.
DE-FG22-96BC14994 to Stanford University. Likewise, the support of the SUPRI-A
Industrial Affiliates is gratefully acknowledged.
![Page 5: A STUDY OF THE EFFECT OF MOBILITY RATIOS ON ...1. A study of the effect of mobility ratios on pattern displacement behavior It is well known, for unit mobility ratio, that the areal](https://reader031.vdocuments.net/reader031/viewer/2022040817/5e61c32167659d0b291248ba/html5/thumbnails/5.jpg)
v
ABSTRACT
In my MS research program I worked on two research topics. I finished the first
research topic and conducted some preliminary research on the second topic. The second
topic is actually a first step toward my Ph.D. program. Below is a brief description of
my work on the two topics.
1. A study of the effect of mobility ratios on pattern displacement behavior
It is well known, for unit mobility ratio, that the areal sweep efficiency of a
staggered line drive pattern is always better than a five-spot pattern. However, this
observation does not hold for very favorable mobility ratios.
I studied the effect of mobility ratios on pattern behavior by the means of
simulation using a streamline simulator. In this report, I present simulation results and,
with the help of streamline and saturation distributions, explain the differences between
displacements with unit and favorable mobility ratios. Simulations compare well with
experiments conducted elsewhere.
Accurate definition of breakthrough time is also discussed for multiphase,
streamline, simulation results. The exact definition of breakthrough is difficult due to
physical dispersion in experiments and numerical dispersion in simulations.
2. Streamline approach to the inverse problem of inferring permeability distribution
It is often useful to infer the permeability field of a porous medium, such as an oil
reservoir or a ground water aquifer from a tracer breakthrough curve at the production
well. There are many approaches to this inverse problem, such as simulated annealing,
sensitivity studies either analytical or numerical, genetic algorithms, and geo-statistical
approaches. Most of these approaches are very time consuming.
![Page 6: A STUDY OF THE EFFECT OF MOBILITY RATIOS ON ...1. A study of the effect of mobility ratios on pattern displacement behavior It is well known, for unit mobility ratio, that the areal](https://reader031.vdocuments.net/reader031/viewer/2022040817/5e61c32167659d0b291248ba/html5/thumbnails/6.jpg)
vi
An efficient approach is desirable. Here I propose an alternative approach—an
inverse streamline approach. I have conducted some preliminary work on this topic, to
be described later, and obtained some satisfactory results. These results indicate that this
project is robust and promising. However, there are simplifications in my current study,
such as piston-like displacement, unit mobility ratio and incompressible flow. If these
simplifications can be relaxed, then this can be a very efficient approach to the inverse
problem with broader applications.
![Page 7: A STUDY OF THE EFFECT OF MOBILITY RATIOS ON ...1. A study of the effect of mobility ratios on pattern displacement behavior It is well known, for unit mobility ratio, that the areal](https://reader031.vdocuments.net/reader031/viewer/2022040817/5e61c32167659d0b291248ba/html5/thumbnails/7.jpg)
1. EFFECT OF MOBILITY RATIO ON PATTERN BEHAVIOR
OF A HOMOGENEOUS POROUS MEDIUM
1.1 INTRODUCTION
Pattern geometry plays a major role in determining oil recovery during secondary
and enhanced oil recovery operations. Although simulation is an important tool for
design and evaluation, the first step often involves rough calculations based upon areal
sweep efficiencies of displacements in homogeneous, two-dimensional, scaled, physical
models (Dyes et al., 1954; Craig, 1971; Lake, 1989). These results are available as a
function of the displacement pattern and the mobility ratio, M. The mobility ratio is
simply the mobility of the displacing phase over that of the displaced, or resident, phase.
Because it is possible to compute sweep efficiency analytically when the displacing and
displaced phase have the same mobility (Morel-Seytoux, 1966), scaled physical model
results have been verified for unit mobility ratios.
Convincing verification of the non-unit mobility ratio cases does not appear in the
literature. Typical finite difference solutions of the reservoir flow equations suffer from
numerical dispersion, the effects of which are hard to evaluate. Furthermore, the scaled
physical model results at low mobility ratios (M << 1) are provocative. For instance,
Fig. 1a shows that recovery from a five-spot pattern at breakthrough for 1/M greater than
about 6 is virtually 100%, whereas recovery at breakthrough in Fig. 1b for a staggered-
line-drive pattern at an 1/M of 6 is only about 88.5%. This contradicts the common
notion that areal sweep efficiency from a staggered-line-drive pattern is always better
than that from a five-spot pattern.
We use a 3D streamline simulator (Batycky, et al., 1996) to analyze
displacements in five-spot and staggered-line-drive patterns for stable displacements, that
is M less than 1. In the following sections, we present streamline distributions, saturation
distributions, and fractional flow at the producer versus dimensionless time, tD. The
dimensionless time is the pore volumes of displacing fluid injected. With the streamline
![Page 8: A STUDY OF THE EFFECT OF MOBILITY RATIOS ON ...1. A study of the effect of mobility ratios on pattern displacement behavior It is well known, for unit mobility ratio, that the areal](https://reader031.vdocuments.net/reader031/viewer/2022040817/5e61c32167659d0b291248ba/html5/thumbnails/8.jpg)
2
Fig 1 Dyes et al(1954) Experimental Results
16
(a) Five Spot Pattern
(b) Staggered Line Drive, d/a = 1
0.5
0.6
0.7
0.8
0.9
1.0
Are
al S
wee
p E
ffici
ency
10-1 1 10Reciprocal of Mobility Ratio, 1/M
a
a
a
a
a
a
a
a
a
aa
a
a
a
a
a
a
a
a
a
a
a
a
aa
a
a
a
a
a
a
a
a
aa
a
a
a
a
a
a
a
aa
a
a
a
aa
a
a
a
a
a
aa
a
a
a
aa
Breakthrough
0.8
1.0
1.21.41.61.82.0
tD/∆S
0.5
0.6
0.7
0.8
0.9
1.0
Are
al S
wee
p E
ffici
ency
10-1 1 10Reciprocal of Mobility Ratio, 1/M
a
a
a
a
a
a
a
aa
a
aa
a
a
aa
a
a
a
a
a
aa
a a a a a
a
a
a
a
a
aa
aa
a
aa
a
aa
a
a
aa
a
aa
a
aa
a
a
a
aa
aa
Breakthrough
0.8
1.01.2
1.41.61.82.0
tD/∆S
![Page 9: A STUDY OF THE EFFECT OF MOBILITY RATIOS ON ...1. A study of the effect of mobility ratios on pattern displacement behavior It is well known, for unit mobility ratio, that the areal](https://reader031.vdocuments.net/reader031/viewer/2022040817/5e61c32167659d0b291248ba/html5/thumbnails/9.jpg)
3
and saturation distributions at different times, we explain why and at what mobility ratio
the five spot pattern can recover more oil than a staggered line drive pattern.
The streamline calculation method is advantageous in that the results suffer from
much less numerical dispersion than typical finite-difference approximations, but some
dispersion in simulation results is evident. Therefore, we discuss how to treat the
numerical dispersion to obtain accurate estimates of breakthrough times. We discuss the
proper way to calculate fractional flow based on the flow rates at the producer. In
comparing the simulation results with the experimental results of Dyes et al. (1954),
physical dispersion in the experiments is found even though a piston-like displacement
was assumed.
1.2 SWEEP EFFICIENCY
Before proceeding, it is useful to recall the representation of experimental data in
Fig. 1 and the meaning of sweep efficiency. Dyes et al. (1954) used various oils as both
the injected and displaced phases. These hydrocarbons were miscible and they assumed
piston-like displacement. An X-ray shadowgraph technique was used to observe the
position of the displacing front. Areal sweep efficiencies are plotted versus displacable
pore volumes injected for different mobility ratios. In the figure, the x axis is the
reciprocal of mobility ratio. Each curve in the graphs corresponds to a specific tD / ∆ S ,
or displacable pore volume injected. The bottom curves show sweep efficiencies at
breakthrough. It is assumed that the displacement has a piston-like front and there is no
physical dispersion. Likewise, the porous medium is assumed to be perfectly
homogeneous.
For piston-like displacement, the areal sweep efficiency is
EA = AS / AT (1)
![Page 10: A STUDY OF THE EFFECT OF MOBILITY RATIOS ON ...1. A study of the effect of mobility ratios on pattern displacement behavior It is well known, for unit mobility ratio, that the areal](https://reader031.vdocuments.net/reader031/viewer/2022040817/5e61c32167659d0b291248ba/html5/thumbnails/10.jpg)
4
where As is the swept area and AT is the total area. Before and at breakthrough, the
amount of displacing fluid injected is equal to the displaced fluid produced, disregarding
compressibility. Assuming piston-like displacement, injected volume is related to area
swept
VI = AS hφ∆ S (2)
where VI is the volume of displacing fluid injected, h is the thickness of the formation,
and φ is porosity. Hence,
EA = AS / AT = VI / (AT hφ ∆ S) = tD / ∆ S (3)
where tD = V I / (AT hφ) is the pore volume of fluid injected, also commonly called
dimensionless time. For ∆ S =1, EA = tD before and at breakthrough.
After breakthrough,
EA = (VI − VP ) / ( AT hφ ∆ S) (4)
where VP is volume of displacing fluid produced.
1.3 SIMULATION RESULTS
We use a three-dimensional streamline simulator, called 3DSL, written by
Batycky et al (1997) to simulate the displacement for the five spot and staggered line
drive patterns. In the simulations, we set the conditions identical to those in the
experiments and choose relative permeability curves that ensure a piston-like
displacement. The conditions are
1. Homogeneous permeability field, i.e., k is constant.
![Page 11: A STUDY OF THE EFFECT OF MOBILITY RATIOS ON ...1. A study of the effect of mobility ratios on pattern displacement behavior It is well known, for unit mobility ratio, that the areal](https://reader031.vdocuments.net/reader031/viewer/2022040817/5e61c32167659d0b291248ba/html5/thumbnails/11.jpg)
5
2. Straight line relative permeability curves with end point relative permeability of
1, i.e.,
krw = Sw, kro = Sw (5)
Therefore, krw + kro = 1 for any Sw .
4. Mobility is altered by changing viscosity, and the mobility ratio is the
reciprocal of viscosity ratio.
5. We set ∆ S =1 which means that ahead of the displacing front, the displacing
phase saturation is zero, and behind the front, it is unity.
3DSL is very fast compared to conventional finite difference simulators and
exhibits much less numerical dispersion (Batycky et al., 1997; Thiele, et al., 1996). For
our problem, it offers us the streamline distribution which facilitates explanation of
displacement behavior.
I use many pressure solves (time steps) and very fine grids (100 by 100 cells for
the five spot and 140 by 70 cells for the staggered line drive) to ensure converged
simulation results. A grid refinement study (refer to Appendix A for details) showed
these grids to be optimal in that further refinement of the grid did not yield noticeable
changes in breakthrough time, the oil production curve, or displacement patterns. For unit
mobility ratio (M = 1), we actually only need one pressure solve. But for mobility ratios
far from 1, we need many pressure solves. For 1/M = 20, we used up to 1000 pressure
solves to ensure that the results were converged. In the streamline approach, a pressure
solve is accompanied by a re-determination of streamline paths; hence, the flow field.
I also studied the effect of grid orientation on the simulation results (see Appendix
B) and found that the pattern behavior is independent of the grid orientation.
Figure 2 displays areal sweep efficiencies as a function of pore volume injected
for differing mobility ratios, and compares simulation and experimental results (Dyes et
al., 1954). The solid lines are simulation results, solid circles experimental results, and
dotted lines connect circles for ease of viewing. In this figure, we concentrate on only the
favorable mobility ratios, M < 1. We noticed more numerical dispersion for unfavorable
mobility ratio cases, not reported here.
![Page 12: A STUDY OF THE EFFECT OF MOBILITY RATIOS ON ...1. A study of the effect of mobility ratios on pattern displacement behavior It is well known, for unit mobility ratio, that the areal](https://reader031.vdocuments.net/reader031/viewer/2022040817/5e61c32167659d0b291248ba/html5/thumbnails/12.jpg)
6
Fig 2 Comparison of Simulation and Experimental
17
(a) Five Spot Pattern
(b) Staggered Line Drive, d/a = 1
0.70
0.75
0.80
0.85
0.90
0.95
1.00
Are
al S
wee
p E
ffici
ency
1 10Reciprocal of Mobility Ratio
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aa
a
SimulationDyes et alBreakthrough
0.8
1.0
1.21.41.61.82.0
tD/∆S
0.70
0.75
0.80
0.85
0.90
0.95
1.00
Are
al S
wee
p E
ffici
ency
1 10Reciprocal of Mobility Ratio
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aa
a
a
a
a
a
a
a
a
a
a
a
a
aa
aa
a
a
a
a
a
a
a
a
a
a
aa
a
a
SimulationDyes et al
Breakthrough0.8
1.0
1.21.41.6
tD/∆S
![Page 13: A STUDY OF THE EFFECT OF MOBILITY RATIOS ON ...1. A study of the effect of mobility ratios on pattern displacement behavior It is well known, for unit mobility ratio, that the areal](https://reader031.vdocuments.net/reader031/viewer/2022040817/5e61c32167659d0b291248ba/html5/thumbnails/13.jpg)
7
Figure 3 shows the displacing fluid fractional flow at the producer as a function of
dimensionless time for several mobility ratios. To compute fractional flow from the
numerical data we use a central finite-difference formula rather than backward
differences. The shapes of the fractional flow curves at breakthrough (tD from roughly
0.7 to 1) indicate some numerical dispersion. We expect the fractional flow to increase
sharply rather than gradually at water breakthrough. As expected, the numerical
dispersion decreases as M becomes more favorable (Peaceman, 1977). The most
numerical dispersion occurs for unit mobility ratio, as shown in Fig. 3. We modify the
breakthrough time by trimming the numerical dispersion as will be described next.
1.4 BREAKTHROUGH DETERMINATION
Due to numerical dispersion, injected fluid breaks through earlier at the producer
than it should. However, the numerical dispersion does not have much effect on the late-
time displacing fluid production. The fractional flow versus tD plots shown in Fig. 3
illustrate the early breakthrough caused by numerical dispersion.
To correct for numerical dispersion in breakthrough times and approximate the
breakthrough time more accurately, we use fractional flow data after breakthrough and
extrapolate back to breakthrough time. A least-squares method is used with second order
polynomials:
tD = a + bfw + cfw2 (6)
The data points employed lie between 0.1 < f w < 0.5. The dashed lines in Fig. 3 illustrate
this procedure. All the breakthrough times in Fig 2 are modified using this method.
Numerical dispersion is also related to the number of time steps (i.e., pressure
solves). In the streamline approach applied here, dispersion can be introduced through the
process of mapping the streamline saturation distribution onto the underlying Cartesian
grid used to compute the pressure field5. Hence, for a unit mobility ratio where the
![Page 14: A STUDY OF THE EFFECT OF MOBILITY RATIOS ON ...1. A study of the effect of mobility ratios on pattern displacement behavior It is well known, for unit mobility ratio, that the areal](https://reader031.vdocuments.net/reader031/viewer/2022040817/5e61c32167659d0b291248ba/html5/thumbnails/14.jpg)
8
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
f-w
0.5 0.6 0.7 0.8 0.9 1.0tD
5 spot,1/M=1
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
b
bbbbbbbbbbbbbbbbbbbbb
a
b
Original dataExtrapolation
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
f-w
0.5 0.6 0.7 0.8 0.9 1.0tD
5 spot, 1/M=2
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
a
a
a
a
a
a
a
a
a
a
a
aaaaaaaaaaaaaaaa
b
b
b
b
b
b
b
b
b
b
b
bb
a
b
Original dataExtrapolation
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8f-
w
0.5 0.6 0.7 0.8 0.9 1.0
5spot, 1/M=3
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
a
a
a
a
a
a
a
a
a
a
a
a
a
aaaaa
b
b
b
b
b
b
b
b
a
b
Original dataExtrapolation
0.0
0.2
0.4
0.6
0.8
1.0
f-w
0.5 0.6 0.7 0.8 0.9 1.0
5spot, 1/M=20
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
a
a
aaaaa
a
b
Original dataExtrapolation
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
f-w
0.5 0.6 0.7 0.8 0.9 1.0
Staggered, 1/M=1
a a a a a a a a a a a a aa
a
a
a
a
a
a
a
a
a
a
a
a
b
b
b
b
b
b
b
a
b
Original dataExtrapolation
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
f-w
0.5 0.6 0.7 0.8 0.9 1.0
Staggered, 1/M=2
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
a
a
a
a
a
a
a
a
a
a
a
a
aaaaaaaaaaaaaaaaaa
b
b
b
b
b
b
b
b
b
b
b
b
b
a
b
Original dataExtrapolation
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
f-w
0.5 0.6 0.7 0.8 0.9 1.0
Staggered, 1/M=3
a a a a a aa a a a a a a a a a a a a aa
a
a
a
a
a
b
b
b
b
a
b
Original dataExtrapolation
0.0
0.1
0.2
0.3
0.4
0.5
0.6
f-w
0.5 0.6 0.7 0.8 0.9 1.0
Staggered, M=1/20
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aa
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
b
bb
b
b
b
b
b
b
b
b
b
b
b
b
b
ba
b
Original dataExtrapolation
Fig. 3 Fractional Flow(fw) vs tD at Producer—Determination of Breakthrough Time
![Page 15: A STUDY OF THE EFFECT OF MOBILITY RATIOS ON ...1. A study of the effect of mobility ratios on pattern displacement behavior It is well known, for unit mobility ratio, that the areal](https://reader031.vdocuments.net/reader031/viewer/2022040817/5e61c32167659d0b291248ba/html5/thumbnails/15.jpg)
9
pressure field does not change, the most accurate results are obtained when a single time
step is used. By performing various single time step simulations, we determine a
breakthrough time of 0.7178 for the five spot with M equal to 1. This is in good
agreement with the analytical solution (Morel-Seytoux, 1966) of 0.7177. Likewise,
Morel-Seytoux analytically and numerically determined breakthrough times for a
staggered-line-drive pattern are both equal to 0.785. With multiple time steps (50)
breakthrough times were sooner, but using the method discussed above, we obtain the
same values for the breakthrough times, at M = 1.
1.5 DISCUSSION OF PATTERN BEHAVIOR
Comparing simulation results and experimental results in Fig. 2, we notice that
breakthrough occurs earlier in the experiments than predicted by the simulations. For the
staggered-line-drive pattern in Fig 2(b), experimental data indicates that EA at
breakthrough is roughly 0.75 for M equal to 1. In the experiments, there is physical
dispersion even though a piston-like front is assumed. In the simulation results presented
in Fig. 2a, the sweep efficiency at breakthrough is 99.7% when 1/M = 20. An almost
negligible area immediately around the producer is not completely swept at breakthrough
due to a very small amount of dispersion in the simulations (refer to Fig. 3, five spot, 1/M
= 20).
After breakthrough, the differences in areal sweep efficiencies between the
experiments and simulations become much smaller (Fig. 2). After breakthrough, the
numerical dispersion consists of only a portion of the displacing fluid produced. As time
increases, this portion decreases and the dispersion has less effect on areal sweep
efficiency. However, the differences between the experimental results and the simulation
results are consistent, i.e. the areal sweep efficiencies of the simulations are generally
higher than those of the experiments. As shown in Fig. 2(a), the simulated breakthrough
curve levels off at large 1 / M with zero slope and does not reach 1 at 1/M = 20. A plot
of displacing front position at late displacement time shows that a very small amount of
numerical dispersion causes slightly earlier breakthrough. Further details are given in
![Page 16: A STUDY OF THE EFFECT OF MOBILITY RATIOS ON ...1. A study of the effect of mobility ratios on pattern displacement behavior It is well known, for unit mobility ratio, that the areal](https://reader031.vdocuments.net/reader031/viewer/2022040817/5e61c32167659d0b291248ba/html5/thumbnails/16.jpg)
10
Appendix C. However, the experimental curve shows EA equal to 1 at 1 / M equal to
about 7.5. We note that in the plot drawn by Dyes et al, the point where the breakthrough
curve hits the EA equal to 1 line is only an extrapolation from other data points.
1.5.1 Areal Sweep Efficiency
Figure 4 plots computed breakthrough, tD , versus the conventional shape factor
d/a for various mobility ratios. The analytical solution for the unit mobility ratio (Morel-
Seytoux, 1966) is also plotted on the same figure for comparison. We find a good match
of the sweep efficiency at breakthrough between the analytical solution and simulation
results.
For unit mobility ratio, Fig. 4 teaches that a staggered-line-drive pattern always
has better areal sweep efficiency than a five-spot pattern. As the staggered-line-drive
pattern becomes longer relative to its width, the displacement pattern approaches linear
flow. High sweep efficiency results.
As the mobility ratio becomes more favorable, the advantage of staggered line
drive on sweep efficiency diminishes. When the mobility ratio decreases to 0.2, the five-
spot pattern becomes better than the staggered-line-drive pattern with d / a = 1. However,
if d / a is increased, the staggered line drive recovery is better than the five spot pattern
for this mobility ratio.
When the mobility ratio decreases to 0.05 or lower (very favorable), the areal
sweep efficiency for the five-spot pattern is very close to 1 at breakthrough. That is,
sweep out is complete at breakthrough. At this mobility ratio, the five spot pattern is as
good as a very long staggered line drive (d / a ≈ 15, almost linear flow), and much
better than the common staggered line drive (d / a = 1). The transition point for five
spot sweep efficiency exceeding that from a staggered line drive is around a mobility
ratio of 0.3. That is to say, if the mobility ratio is higher than 0.3, a staggered line drive
is always better than a five spot. If mobility ratio is lower than 0.3, then the five spot can
be higher in sweep efficiency than a staggered line drive.
![Page 17: A STUDY OF THE EFFECT OF MOBILITY RATIOS ON ...1. A study of the effect of mobility ratios on pattern displacement behavior It is well known, for unit mobility ratio, that the areal](https://reader031.vdocuments.net/reader031/viewer/2022040817/5e61c32167659d0b291248ba/html5/thumbnails/17.jpg)
11
Fig 4 Effect of Mobility Ratios and Patterns On Areal Sweep Efficiency
19
0.70
0.75
0.80
0.85
0.90
0.95
1.00
Are
al S
wee
p E
ffici
ency
1 10Aspect Ratio, d/a
a
a
a
a
a
a
a
aa
b
b
b
b
b
b
b
b
a
aa
a
a
aa
a
a
aa
a
aa
a
a a
a
a
a
b
a
a
a
a
M = 1.00M = 1.0, AnalyticalM = 0.33M = 0.30M = 0.20M = 0.05
![Page 18: A STUDY OF THE EFFECT OF MOBILITY RATIOS ON ...1. A study of the effect of mobility ratios on pattern displacement behavior It is well known, for unit mobility ratio, that the areal](https://reader031.vdocuments.net/reader031/viewer/2022040817/5e61c32167659d0b291248ba/html5/thumbnails/18.jpg)
12
The excellent displacement from a five spot pattern for very favorable mobility
ratios is explained with the help of streamline distributions. Every pair of streamlines
forms a stream tube, and the volumetric flow rate is the same in all the stream tubes. All
of the stream tubes connect with the same injector and producer, and the pressure drop
for all the streamtubes is the same. With the same pressure drop and the same volumetric
flow rate, the flow resistance is the same for all the streamtubes.
For our straight-line relative permeability assumption, we have
Ri = −∆p / q =µkA
dl0
Li∫ (7)
where Ri and Li are the resistance and length of streamtube i , respectively, k is the
homogeneous permeability, and A is the cross sectional area of the streamtube. The
cross-sectional area is
A = hw
where h is the constant thickness of the layer and w is the width of the streamtube.
Resistance in streamtube i is the same as that in streamtube j and thus
µkA
dl0
Li∫ = C (8)
for all the streamtubes at a given time, where C is constant. Moving the constant
parameters k and h to the right hand side, we have
µw
dl0
Li∫ = C (9)
For a piston-like displacing front,
µ µ1
0
2
1 1
wdl
wdl C
l
l
Lfi
fi
i
∫ ∫+ = (10)
![Page 19: A STUDY OF THE EFFECT OF MOBILITY RATIOS ON ...1. A study of the effect of mobility ratios on pattern displacement behavior It is well known, for unit mobility ratio, that the areal](https://reader031.vdocuments.net/reader031/viewer/2022040817/5e61c32167659d0b291248ba/html5/thumbnails/19.jpg)
13
where µ1 and µ2 are the constant viscosity of the displacing and displaced fluids,
respectively, and l fi is the distance from the injector to the displacing front.
1.5.2 Unit Mobility Ratio
For a unit mobility ratio, the pressure field remains unchanged throughout the
displacement, and so do the streamlines. The streamline distributions at M = 1 for the
five-spot and staggered-line-drive (d / a = 1) patterns are shown in Fig. 5.
For unit mobility ratio, Eq. (10) is
µ 1
w0
Li∫ dl = C (11)
From Eq. 11, we know that if the ith streamtube is longer than the jth streamtube, then the
average width of the ith streamtube wi is greater to maintain the same resistance and
flow rate. Therefore, the volume of the ith streamtube is larger than the jth streamtube.
The greater the difference in streamtube length, the bigger the difference in streamtube
width, and, when breakthrough happens in the jth streamtube, the front has not
progressed as far in the ith streamtube.
For a five-spot pattern, the longest streamline is that along the boundary, which is
2a. The shortest streamline is the one along the diagonal, at a length of 2 a . The ratio of
the longest streamtube length over the shortest is 2 . Since the width of the longest
streamtube is also 2 times as great as the shortest streamtube, the volume of the
longest streamtube is twice that of the shortest.
However, for a staggered-line-drive pattern with d/a = 1, the ratio of the length
along the boundary (3a/2) over the diagonal( 5a / 2)is 3 / 5 ≈ 1.34 . From the
streamline distribution in Fig. 5, we know that the shortest streamline is longer than
diagonal, and therefore the ratio of the longest streamline over the shortest is less than
1.34. This ratio is about 1.3 and, therefore, less than that ratio for a five spot pattern
which is 1.41. That is to say, the streamlines are more evenly distributed in the staggered
![Page 20: A STUDY OF THE EFFECT OF MOBILITY RATIOS ON ...1. A study of the effect of mobility ratios on pattern displacement behavior It is well known, for unit mobility ratio, that the areal](https://reader031.vdocuments.net/reader031/viewer/2022040817/5e61c32167659d0b291248ba/html5/thumbnails/20.jpg)
14
Fig 5 Saturation and Streamline Distributions, Unit Mobility Ratio
(a) Five Spot Pattern
(b) Staggered Line Drive
tD = 0.204
tD = 0.408
tD = 0.612
tD = 0.68
tD = 0.714
tD = 0.748
tD = 1.02
aa a a a a aa a a a aa
a a a a aa a a a a
a aa a a aa a a a a
a a aaa aa a a a a
a a aaa aa a a a
a aaa aa a a
a aaaa a a
aaaa a
aaaa a
aaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aa a aa a aa
a a aa a aa
a a aa a aa
a a aa aaa
a a aa aaa
a a aa aaa
a a aaaa a
a a aaa a
a aaaa
a aaaa
a aaa a
aaaa a
aaa aa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aa a aaaa
a aaaa
a aaaa
a aaaa
a aaaa
a aaaa
a aaa a
a aaa a
a aaa a
a aaa a
aaaa a
aa aaaa
a aaaa aaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aa aaaa
a aaa a
aa aa aa
a aaa a
a aaa a
aa aaaa
a aaa a
aaaa a
aaa aaa
a aaa a
aaa aaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aa aaa
aaaa aaa
aaaaaa
a aaa
aaaaaa
a aaa
aaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa a
aaaa aa
a aaaa
a aa aaa
a a aaaa a
a a a aaaa a a
a a a aa aaa a a
a a a a aaaa a a a
a a a a aaaa a a a
a a a a aa aaa a a
a a a a aa aa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
a aaa a
aa aa aa
a aaaa
a aaaa
a aa aaa
a a aaa a a
a a aaa a a
a aaaa a a
a aaaa a a
aaa aa a a
aaa aa a a
aaa aa a aa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaa a
aaa aaa
a aaa
a aaa a
a aaa a
aaaa a
aaaa a
aa aa a
aa aa aa
a aa aa
a aa aa
a aa aa
a aa aa
a aaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaa
aaaa aaa
aaaa a
aaa aaa
a aaa a
aa aaaa
a aaaa aaa
a aaa
a aaa
a aaa a
aaaa aa
a aa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaa
aaaa
aaaaaa a
aaaaaa
aaa aa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
a
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
tD = 0.208
tD = 0.416
tD = 0.624
tD = 0.763
tD = 0.783
tD = 0.797
tD = 0.832
tD = 1.04
aa a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a aa a a a a a a a a aa a
aaa
aa
aa
a
aaa
a
a
a
a
a
aaa
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aa
aa a a a a a a a a a a a aaa a a a a a a a a a a a a a a a a a a a aa a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a aaa a a a a a a a a a a a aa a a a a aa a
aaaaaaa
aaaaaa
aa
aaa
aaa
a
aaa
a
a
a
aa
a
a
a
a
a
aaa
a
a
a
a
a
a
a
aaa
a
a
a
a
a
a
a
a
a
a
aa
aa a a a a a aaa a a a a a a aa a a a a a a a a a aaa a a a a a a a a a a a a a a aa a a a a a a a a a a a a a a a a a a aaa a a a a a a a a a aaa a a a a aa a a a aaaaaaaaaaaaaaaaa
aa
aaa
aaa
aaa
a
aaa
a
aaa
a
a
a
aa
a
a
a
a
aaa
a
a
a
a
a
aa
a
a
a
a
a
aa
aa a a a aa a a a a aa a a a a a aaa a a a a aaa a a a a a a a aaa a a a a a a a a a a a aa a a a a a a a a a a a a a aa a a a a a a a a aaa a a a a aa a
a a aa aaaaaaaaaaaaaaaaaaaaaa
aaa
aa
a
aa
a
aaa
a
aaa
a
aaa
a
a
aaa
a
a
aaa
a
a
a
aa
a
a
a
aa
aa a a aaa a aaa
a a aaaa a a aa a a a a aa a a a a a aa a a a a a a a aa a a a a a a a a a a aa a a a a a a a a a a aa a a a a a a a aa a a a a aaa
a a aaaaaaaaaaaaaaaaaaaaaaaaaaa
aa
aaa
aaa
aaa
aaa
a
aaa
a
aa
a
a
aaa
a
aaa
a
a
aaa
a
aa
aa a aaaa aa
a a aa a a aaaa a aa a a a aa a a a a aa
a a a a aaa a a a a a aaa a a a a a a a aaa a a a a a a a a aa a a a a a a aa aa a a aaa
a a aaaaaaaaaa
aaa
aaaaaaaaaaaaa
aa
aa
aaa
aa
a
aa
a
aa
a
aa
a
aaa
aaa
a
aaa
a
aa
a
a
aa
a
aa
aa aaaaaaaaaaaaaaaaaaa aa a a aaa
a a aa a a a aaaa a a a aa
a a a a a aaa a a a a a a aaa a a a a a a aaa a a a a a aa a a a a aa a a a aaaa aaaaaaaaaaaaaa
aa
aaaaaaaa
aa
aa
aaa
aa
aaa
aaa
aaa
aaa
aaa
aaa
a
aa
a
aaa
aa
aaaaaaaaaaaaaaaaa
aaaaa aaaaaaa a aa
a a aaaa a aaa
a a a aaaa a a a aaa a a a a a a aa a a a a a a aa a a a a a aa a a a a aa a a aaa
aaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaa
aa
aaa
aa
aaa
aaa
aa
a
aa
a
aa
a
aa
a
aa
aa
aaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
a a aa a a a aa a a a a aaa a a a aaa a a a a a aa a a a a a aaa a a a a aaa
a a a aa a a a aaaa aaaaaaaaaaaaaaaaaaa
aaaaaaaa
aa
aa
aa
aaa
aa
aaaaa
aaa
aa
aaa
aaa
aa
aa
aaaa
aaaaa
aaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaa a aaa
a a aa aa a a aa
a a a a aa a a a a a aa a a a a a aaaa a a a aa a a a aaa
a a aa a aaaaa aaaaaaaaaaa
aaaaa
aa
aa
aaaaaa
aa
aa
aa
aaaaa
aa
aaa
aa
aaa
aa
aaaa
aaaaaa
aaaaaaaaaaaaaaaaaaaaaaa
aaa
aaaaaaaaaaa a aaa
a a aa a a a aaaa a a aaa
a a a a aaaa a a a aaa
a a a aaaa a a aa a a a aa
a a aaaaaaaaa
aaa
aaaaaaaa
aaaaaaaaaaaaaaaaaa
aa
aa
aaaaa
aa
aaaaaa
aaaaaaaaaa
aaaaaaa
aaaaaaaaaaaaaaaaaa
aaaaaaaaaaaa aa a a aaa
a a a aaa a a a aa
a a a a aaaa a a a aa a
a a a aaaa a a aa a a aaa
a aaaaaaaaaaaaa
aaa
aaaaa
aaaaaaa
aa
aa
aa
aa
aa
aa
aa
aa
aaaaaaaaaa
aaa
aa
aaaaaaaaaaaaa
aaaaaaa
aaaaaaaaaa
aaaaaaaaaaaaaaaaa
a a aa a a a aa a a a a aa a a a a aaaa a a a aa a
a a a aaaa a a aa a a aaa
a a aaaaaaaaaaaaaaaaaa
aaaaa
aaaaaaa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aaa
aa
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aaa
a
a
a
a
a
aaa
a
aa
aaaaaa aa a a a a a a a a aa a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a
a
aa
a
a
a
a
a
a
a
a
a
a
aa
a
a
a
a
a
a
a
a
aaa
a
a
a
a
a
aa
a
a
a
aaa
a
aaa
aaa
aa
aaaa
aaaaaaaaa a aa a a a a aaa a a a a a a a a a a a aa a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a aaa a a a a a a a a a a a a a a a a a a aaa a a a a a a a a a a a a
a
aa
a
a
a
a
a
aa
a
a
a
a
a
aaa
a
a
a
aaa
a
a
a
aa
a
a
aaa
a
aaa
aaa
aaa
aa
aaaaaa
aaaaa
aaaaaaa
a a aa a a a a a aa a a a a a a a a a a aa a a a a a a a a a a a a a a a a a a aaa a a a a a a a a a a a a a a aa a a a a a a a a a aa a a a a a a a aa a a a a a aa
aa
a
a
a
aa
a
a
a
aaa
a
a
aaa
a
a
aaa
a
aaa
a
aaa
a
aa
a
aa
aaa
aa
aaaaaaaaaa
aaaaaaaaaa aa a
a a aa aa a a a aa a a a a a a a a aaa a a a a a a a a a a a a aaa a a a a a a a a a a a aa a a a a a a a aaa a a a a a aaa
a a a a aa a a a a aa a a a aa
aa
a
aaa
a
a
aaa
a
aaa
a
aaa
a
aaa
a
aa
a
aaa
aaa
aaa
aa
aaaaaaaaaaaa
aaaaa
aaaaaaaa aa a a a aa a a a a aaa a a a a a a a aa a a a a a a a a a a aa a a a a a a a a a aaa a a a a a a aa a a a a a aa a a a a aa a a a aaa
a a aaaa a aa a
a aa
aa
a
aa
a
a
aa
a
aaa
a
aaa
aaa
a
aa
a
aa
a
aa
aaa
aaaaa
aa
aa
aa
aa
aaaaaaa
aaaaaaaaaaa aa a a a aa a a a a aaa a a a a a a aa a a a a a a a a aa a a a a a a a a aa a a a a a a aa a a a a aaa
a a a aa a a a aa a a aaaa a aa a
aaaaaaa a aa
aa
aaa
a
aa
a
aa
a
aaa
aaa
aaa
aaa
aa
aaa
aaaaa
aa
aa
aa
aa
aa
aaaaa
aaaaaaaaaaaaa aaa
a a aaaa a a a aa
a a a a a aaa a a a a a a aaa a a a a a a aaa a a a a a aaa a a a aa a a a aa a
a a aa a a aa aaaaaaaaaaaaaaaaaaaaa
aa
aa
a
aa
a
aa
a
aa
a
aa
aaa
aaa
aa
aaaaa
aa
aaaaaaaaaaaa
aaaaa
aaaaaaaaaa aaa a aa a a a aa
a a a aaaa a a a a aa a a a a a a aa a a a a a a aa a a a a a aa a a a a aa a a a aa a a aaa
aaaaaaaaaaaaaaaaaaaaa
aaa
aaa
aa
aa
aa
aaa
aaa
aa
aaa
aa
aaa
aa
aaaaa
aa
aa
aa
aa
aa
aaaaaaa
aaaaaaaa
aaaaaaaa a aa
a a aaaa a a aaa
a a a a aaa a a a a a aa a a a a a aaaa a a a aa a a a aaa
a a aaaa aaa
aaaaaaaaaaaaaa
aaaaaaaaa
aaa
aaaaaaa
aaaa
aa
aaa
aa
aaa
aa
aa
aaaaa
aa
aa
aa
aa
aa
aaaaaaa
aaaaa
aaaaaaaaaaaaa a aaa
a a aaaa a a aa a a a a aaa
a a a a a aa a a a a a aaa a a a aa a a a aaa
a a aa a aaaaaaaaaaaaaa
aaaaaaaaa
aaaaaaaa
aaaaaaaa
a
aaaaaa
aa
aaaaaaa
aa
aa
aaaaaaaaaaaa
aaaaaaa
aaaaa
aaa
aaaaaaaaaaa a aa a a aaa
a a a aaaa a a a aa
a a a a a aaa a a a aaa
a a a aaaa a aaa
a a aa aaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaa
aaaaa
aaaaaaaa
aaaaaaaaaaaa
aa
aa
aa
aa
aa
aaaaaaaaa
aaaaaaaaaa
aaa
aaa
aaaaa aa
a a aa a a a aa a a a aaaa a a a aa a a a a a aa
a a a a aa aa a a aa a a a aa a a aaa
aaaaaaaaaaaaaaaaaaaaaa
aaaaa
aaaaaaa
aaaaaaaaa
aaa
aaa
aa
aa
aa
aa
aa
aa
aa
aaaaaaaaa
aaaaa
aaaaa
aaaaaaaaaaaaaa aa
a a aa a a a aa a a a aaaa a a a aa a a a a a aa
a a a a aa aa a a aa a a a aa a a aa a
aaaaaaaaaa
aaa
aaa
aaaaa
aaaaa
aaaaaaaaaaaaaaa
aa
aa
aaa
![Page 21: A STUDY OF THE EFFECT OF MOBILITY RATIOS ON ...1. A study of the effect of mobility ratios on pattern displacement behavior It is well known, for unit mobility ratio, that the areal](https://reader031.vdocuments.net/reader031/viewer/2022040817/5e61c32167659d0b291248ba/html5/thumbnails/21.jpg)
15
line-drive than in a five-spot pattern. Therefore, when the shortest streamtube breaks
through, a larger portion of the other streamtubes have been swept in a staggered-line-
drive than in a five-spot pattern. When d / a increases, the streamtube length ratio
(longest to shortest) decreases. When breakthrough happens in the shortest streamtube, a
greater portion is swept in the longest streamtube resulting in higher sweep efficiency at
breakthrough.
1.5.3 Five-Spot Pattern, Very Favorable Mobility Ratio
For a favorable mobility ratio ( M < 1), the displacement is stable. For equal
volumetric flow rate streamtubes, Eq. 10 holds. Here, we consider the case of very
favorable mobility ratio, i.e., the displacing fluid viscosity is much higher than that of the
displaced fluid. When the front moves a portion of the way down the streamtube, the
pressure drop is mainly in the displacing phase. That is to say, after a short injection time
(compared to breakthrough), the pressure drop in the displaced phase is negligible.
Therefore, Eq. 10 can be simplified to the following form
1
w0
lti∫ dl ≅ C (12)
In this extreme case, the displacing phase does not feel the producer until it is very close
to it because the pressure drop between the front and the well plays a negligible role in
the displacement. Initially, flow around the injector is radial, because the pattern appears
to be infinite at short times. For example, examine Fig. 6b for tD = 0.583. However, after
the front reaches a corner of the pattern, the no-flow boundary condition along pattern
borders alters the radial flow pattern. Pressure isobars must intersect the no-flow
boundaries at 90o. This constrains the streamlines in the region adjacent to a boundary to
be parallel to it. Because the fluids are incompressible, streamlines cannot terminate. The
flow field in the region near the front transitions from radial to quasi-radial. From the
figure, we also see that streamtubes ahead of the front are narrower along the boundary
than those in the center, which makes the front in the boundary streamtubes move faster
![Page 22: A STUDY OF THE EFFECT OF MOBILITY RATIOS ON ...1. A study of the effect of mobility ratios on pattern displacement behavior It is well known, for unit mobility ratio, that the areal](https://reader031.vdocuments.net/reader031/viewer/2022040817/5e61c32167659d0b291248ba/html5/thumbnails/22.jpg)
16
Fig 6 Five Spot Pattern, Very Favorable Mobility Ratio(1/M=20)
(a) Saturation Distribution
(b) Streamline Distribution
tD = 0.583
tD = 0.796
tD = 0.837
tD = 0.878
tD = 0.918
tD = 0.938
tD = 0.969
tD = 0.998
tD = 0.583
aa a a a a a a a a a a a a a a aa a a a a a a a a a a a a a a a aa a a a a a a a a a a a a a a aaa a a a a a a a a a a aa
aaaa
aa
aa
aa
a
aa
a
a
aa
a
a
a
a
aaa
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aaa
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aa
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aa
aa a a a a a aa a a a a a a aa a a a a a a aa a a a a a a aaa a a a a a aaa a a a a a aaa a a a a a aa a a a a a aaa
a a aaaa
aa
aaa
aaa
a
aaa
a
a
a
aa
a
a
a
a
a
a
a
aa
a
a
a
a
a
a
a
a
a
aaa
a
a
a
a
a
a
a
aaa
a
a
a
a
a
aaa
a
a
a
a
a
a
aa
a
a
a
a
a
aa
aa a a aaaa a a aa a a a aaa
a a a aa a a a aaaa a a aa a a a aaa
a a a aa a a a aaaa a a aa a a a aa a a a aaa
a aaa a
aaa
aaa
a
aa
a
a
a
aa
a
a
a
a
aaa
a
a
a
a
a
a
aa
a
a
a
a
a
aaa
a
a
a
a
aa
a
a
a
aaa
a
a
a
aaa
a
a
a
aa
a
a
a
aaa
a
a
aa
aa a aaaa aaa
a a aaa a aa
a a aaaa a aa
a a aaa a aa
aa aaa
a a aaaaaaaaaaaaaaaaaaaaa aa
aa aa
aaaaaaa
aaa
a
a
aa
a
a
a
aaa
a
a
a
a
aaa
a
a
a
a
aa
a
a
a
aaa
a
a
aaa
a
a
aa
a
a
aaa
a
a
aa
a
a
aa
a
a
aaa
a
aaa
a
aa
aaa aaaaaaa aaaaaaa aa
aaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
a
a
aaa
a
a
a
a
aa
a
a
a
a
aa
a
a
a
aa
a
a
aaa
a
aaa
a
aa
a
a
aa
a
aaa
a
aa
a
aaa
a
aaa
a
aa
a
aaa
aa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaa
aaa
aaa
aaa
aaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
a
a
aaa
a
a
a
aaa
a
a
aaa
a
aaa
a
aa
a
aaa
a
aa
a
aa
a
aaa
aaa
aaa
aaa
aaa
aaa
a
aa
a
aa
aa
aaaa
aaaaaaaa
aaa
aaaaaaaa
aaaaaaaa
aaa
aaaaaaaa
aaaaaaaa
aaaaaaaa
aaa
aaaaa
aaa
aaaaa
aaa
aaaaa
aa
a
a
a
aa
a
a
a
aa
a
aaa
a
aa
a
aaa
aa
a
aa
a
aa
aaa
aaa
aaa
aa
aaa
aaa
aa
a
aa
aaa
aa
aa
aaaaaa
aaaaa
aaaaa
aaaaaaaaaaaa
aaaaa
aaaaa
aaaaaaa
aaaaa
aaaaa
aaaaaaaaaaaa
aaaaa
aaaaaaa
aaaaa
aaa
a
a
aaa
aaa
a
aa
aaa
aaa
aa
aaa
aa
aaa
aa
aaa
aa
aaa
aa
aaa
aa
aa
aaa
aa
aaaa
aaaaaaaa
aa
aaaaaaaaa
aa
aaaaaaa
aa
aaaaaaaaa
aa
aaaaaaa
aa
aaaaaaaaa
aa
aaaaaaa
aa
aaaaaaaaa
aa
aaaaaa
aaa
aaa
aaa
aa
aa
aaa
aa
aa
aaaaa
aa
aaaaaaa
aa
aa
aaaaa
aa
aa
aaaaaaaa
aaa
aa
aa
aa
aaaaaaaaaaaaaaaaaaaaaaaaa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aaaaaaaaaaaaaaaaaaaaaaaaaaa
aa
aa
aa
aa
aaaaaaaaa
aa
aa
aa
aa
aa
aa
aa
aaaaaaaaaaaaaaaaaaaaaaa
aa
aa
aa
aaa
aa
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aa
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aaa
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aa
a
a
a
a
a
a
a
a
a
a
a
aaa
aaa
aa
aa
aaaaaaa aa a a a a a aa a a a a a a a a a a a a a a a a aa a a a a a a a a a a a a a a a a a a aa a a a a a a a a a a a a a a a
a
aa
a
a
a
a
a
aa
a
a
a
a
a
a
aa
a
a
a
a
a
a
aa
a
a
a
a
a
a
aa
a
a
a
a
a
a
aa
a
a
a
a
a
a
aa
a
a
a
a
a
aaa
a
a
a
a
a
aa
a
a
a
aaaa
aaaaaaaaaaaa
a a a aa aa a a a a a aaa a a a a a a a a a aa a a a a a a a aaa a a a a a a aa a a a a a a aa a a a a a a
a
aa
a
a
aaa
a
a
a
aa
a
a
a
aaa
a
a
a
aa
a
a
a
aaa
a
a
a
aa
a
a
a
aaa
a
a
aaa
a
a
a
aa
a
a
a
aaa
a
a
aaa
a
a
a
aa
a
a
a
aa
aaaa
aaaaaaa a a aa a
a a a a aa a a a a a a aaa a a a a a aaa a a a aa a a a a aa a a a aaa
a a a aa a a a aaaa a aa
aa
a
aaa
a
aaa
a
a
aa
a
a
aa
a
a
aaa
a
aaa
a
aaa
a
a
aa
a
a
aa
a
a
aaa
a
aaa
a
a
aa
a
a
aa
a
a
aa
a
a
aaa
a
aaa
a
aaa
a
aaaaaa a aaa
a a a aa aa a a a aa a a a a aa a
a a a aaa a aaa
a a aa a a aaaa a aa
a aaaa a
aaa a
aaa a aa
aa
aaa
a
aa
a
aaa
a
aa
a
aaa
a
aa
a
aaa
a
aa
a
aaa
a
aaa
aaa
a
aaa
a
aa
a
aaa
aaa
a
aa
a
aaa
a
aa
a
aaa
a
aa
a
aaa
aaa
aaa a a aaa
a a a aaaa a a aaa
a a aaaaaaaaaaaaaaaaaaaa aaaaaaaaa
aaaaaaa
aaaaaa
aa
aa
a
aa
a
aa
a
aa
a
aaa
aaa
aaa
aaa
aaa
aaa
aaa
aaa
aaa
aaa
aaa
aaa
aaa
aaa
aaa
aaa
aaa
aaa
aaa
aaa
aaa
aaa
aaa a a a aa
a a a aaaa a aa
aaaaaaaaaaaaaaaaaaaaaaaa
aaa
aaa
aaa
aaa
aaa
aaa
aa
aa
aa
aaa
aa
a
aa
aaa
aa
a
aa
aaa
aa
a
aa
aaa
aaa
aa
aaa
aaa
aa
aaa
aaa
aa
aaa
aaa
aa
aaa
aaa
aa
aaa
aaa
aa
aaa
aa
aaaa
a a a aa a aaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaa
aaa
aaaaa
aaa
aaaa
aaaa
aa
aaaaa
aa
aaa
aa
aaa
aa
aaaaa
aa
aaa
aa
aaaaa
aa
aaa
aa
aaa
aa
aaaaa
aa
aaa
aa
aaaaa
aa
aaa
aa
aaaaa
aa
aaaaa
aaaaaaaaaaaaaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaaaa
aaaaa
aaaaa
a
aaaaaaaa
aa
aa
aaaaaaa
aa
aa
aaaaaaa
aa
aa
aaaaaaa
aa
aa
aaaaa
aa
aa
aa
aaaaa
aa
aa
aa
aaaaa
aa
aa
aa
aaaaa
aa
aaaaaaaaaaa
aaaaa
aa
aaaaaaa
aa
aaaaaaa
aa
aaaaaaaaa
aa
aaaaaaa
aaa
aaa
aa
aa
aa
aa
aa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aaaaaaaa
aa
aa
aa
aa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aa
aa
aa
aa
aa
aaa
tD = 0.796
aa a a a a a a a a a a a a aa a a a a a a a a a a a a a aaa a a a a a a a a a a a a aa a a a a a a a a a a a aa a a a a a a a a aa a a a a aa
aa
a
a
a
a
a
a
a
a
a
a
aaa
a
a
a
aaa
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aa
a
a
a
a
a
a
a
a
a
a
a
a
a
aa
aa a a a a a aa a a a a a aaa a a a a a aa a a a a a a aa a a a a a aaa
a a a a a aaa a a a a aa
a a a a aaaa a a aaa
a a a aa a a a aaa a aa
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aa
a
a
a
a
a
a
aa
a
a
a
a
a
aa
aa a a aaaa a a aa
a a a aa a a a aaaa a a aa a a a aa a a a aaa
a a aaaa a a aa
a a a aaa a aaa
a a aa a a aaaaa aa
aa aa
aaaaaaaaaa
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aa
a
a
a
a
a
a
a
a
a
a
a
a
a
aa
a
a
a
a
a
aaa
a
a
a
aa
a
a
a
aaa
a
a
aa
aa a aaaa aa
aaaaaaa aa
aa aa
aaaaaaaaaaaaaaaaaaa aaaa aa
aa aa
aa aa
aa aaaaaaaaaaaaa
aaaaaaa
aaaaaaaaaaaaaa
a
a
a
a
a
a
a
a
aa
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aaa
a
a
a
aa
a
a
aaa
a
aaa
a
aaa
a
aa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaa
aaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaa
aa
aaa
a
a
a
a
a
a
a
aa
a
a
a
a
a
a
a
a
a
a
a
aa
a
a
a
aa
a
a
aa
a
a
aa
a
aaa
a
aa
a
aaa
aa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaa
aaa
aaaaaaaaaaa
aaaaaaaa
aaaaa
aaaaa
aa
aaa
a
a
a
a
a
a
aaa
a
a
a
aaa
a
a
a
aaa
a
aa
a
aaa
aaa
a
aa
a
aa
a
aa
a
aa
aa
aaaa
aaaaaaaa
aaaaaaaa
aaaaaaaa
aaaaaaaa
aaaaaaaa
aaaaaaaa
aaaaaaaa
aaaaa
aaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaa
a
a
a
a
a
a
a
a
a
a
a
a
a
aaa
a
aa
a
aa
a
aa
a
aa
aaa
aa
a
aa
aaa
aa
aa
aaaaaa
aaaaa
aaaaaaa
aaaaa
aaaaa
aaaaaaa
aaaaa
aaaaaaa
aaaaa
aaaaaaa
aaaaaaa
aaaaa
aaaaaaa
aa
aaaaaaa
aaaaaaa
aa
aaaaaa
a
a
a
a
a
a
a
a
aaa
aaa
aaa
aa
aaa
aa
aaa
aa
aaaaa
aa
aaaa
aaaaaaaaaa
aa
aaaaaaa
aa
aa
aaaaaaa
aa
aaaaaaaaa
aa
aaaaaaaaa
aa
aa
aaaaaaa
aa
aa
aaaaaaaaa
aa
aa
aaaaaaaaaaa
aa
aa
aaaaaaaa
a
a
aaa
a
aa
aaa
aa
aaa
aa
aa
aaaaaaa
aa
aa
aaaaaaaa
aaa
aa
aa
aa
aa
aa
aa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aaaaaaaaaaaaaaaaaaaaaa
aa
aa
aaaaaaaaaaaaaaaaaaaaa
aa
aa
aa
aa
aa
aaa
aa
a
a
a
a
a
a
a
a
a
a
a
a
aa
a
a
a
a
a
a
a
a
a
a
a
a
a
aa
a
a
a
a
a
a
a
a
a
a
a
a
aaa
a
a
a
a
a
a
a
a
a
a
aaa
a
a
a
a
a
a
a
aaa
a
a
a
a
aa
aaa a a a a a a a a a a aaa a aa a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a aa a a a a a a a a a a a a a a
a
aa
a
a
a
a
aaa
a
a
a
a
a
aaa
a
a
a
a
a
aa
a
a
a
a
a
aaa
a
a
a
a
a
aa
a
a
a
a
a
aaa
a
a
a
a
aaa
a
a
a
a
aa
a
a
a
a
aa
a
a
a
aaa
a
a
aaa
a
a
aaa a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a aa a a a a a a aa a a a a a a
a
aa
a
a
aaa
a
a
aaa
a
a
a
aa
a
a
a
aaa
a
a
aaa
a
a
a
aa
a
a
a
aa
a
a
a
aa
a
a
a
aaa
a
a
aaa
a
a
aa
a
a
aaa
a
a
aa
a
a
aaa
a
aaa
a
aa
a
aaa
a
aaa a a a a a a a a a a a a a a a a aaa a a a a a a a a a a a a a aaa a a a a aaa
a a a aa aa a aaa
a a aa
aa
a
aaa
a
aaa
a
aaa
a
aaa
a
a
aa
a
a
aa
a
a
aa
a
a
aaa
a
aaa
a
aaa
a
aaa
a
aaa
a
aaa
a
aaa
a
aa
a
a
aa
a
aaa
a
aa
a
aaa
aaa
a
aa
a
aa
a
aa
aaa a a a a a a aaa a a a a a a a a a a a a a a a a a a a a aa a a a aa a a aaa
aaaaaa aa a aa
aa
aaa
a
aa
a
aaa
a
aa
a
aaa
a
aa
a
aaa
aaa
a
aa
a
aaa
a
aa
a
aaa
a
aa
a
aaa
aaa
a
aa
a
aaa
aaa
aaa
a
aa
a
aa
a
aa
a
aa
a
aa
a
aa
aaa
aaa
aa
aaaaa a a a a a a a aa a a a a a a a a a a a a aaa
a a aaaaaaaaaaaaaaaaaaaaaaa
aa
aa
a
aa
a
aa
a
aa
a
aa
a
aa
a
aa
a
aa
a
aa
a
aa
a
aa
a
aa
a
aa
a
aa
a
aa
a
aa
a
aa
aaa
aaa
aaa
aaa
aa
a
aa
a
aa
aaa
aaa
aa
aaa
aa
a
aa
aaa
aa
aaaaaaa a a a a a a aa a a a a a a aaa
a a a aaaaaaaaa
aaa
aaa
aaaaaaaaa
aa
aa
aa
aaa
aa
aaa
aaa
aa
aaa
aaa
aa
aaa
aaa
aa
aaa
aa
a
aa
aaa
aa
a
aa
aaa
aa
aaa
aa
a
aa
aaa
aa
aaa
aa
aaa
aa
aaa
aa
aaa
aa
aaaaa
aa
aaaaaa a a a a a aaa a aa a a a a aa
aaaaaaaaaaaa
aaa
aaaaaaaa
aaaa
aaaa
aa
aaaaa
aa
aaa
aa
aa
aaa
aa
aaa
aa
aa
aaa
aa
aaaaa
aa
aaa
aa
aaaaa
aa
aaa
aa
aa
aaa
aa
aa
aaa
aa
aa
aaaaa
aa
aaaaaaa
aa
aaaaaaa
aa a a aa a a aa a aa aaaaa
aaa
aaaaa
aaaaa
aaaaaaa
aaaaaa
aaaaaaaa
aa
aa
aa
aaaaaaa
aa
aa
aaaaaaa
aa
aa
aa
aaaaa
aa
aa
aa
aaaaaaa
aa
aa
aa
aaaaaaa
aa
aa
aa
aaaaaaaaa
aa
aa
aa
aaaaaaaaaaa
aaa a a aa
aaaaaaaaaaa
aa
aaaaaaa
aa
aa
aaaaaaa
aaa
aaa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aaa
tD = 0.837
aa a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a aa
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aa
aa a a a a a a a a a aaa a a a a a a a a a a aa a a a a a a a a a a aa a a a a a a a a a aaa a a a a a a a aa a a a a a aa a a a aaa
aaaaaaaaa
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aaa
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aa
a
a
a
a
a
a
a
a
a
a
aa
aa a a a a aa a a a a a aa a a a a a aa a a a a a aa a a a a a aa a a a a aaaa a a a aa a a a a aa a a a aaa
a a aa aa a a
aaaaaaaaaaaaaaaaa
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aa
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aaa
a
a
a
a
a
aa
a
a
a
a
aa
aa a a aa a a a aa a a a aa a a a aa a a a aa a a a aa a a a aa a a a aa a a aaaa a aa a
a a aaaa aa
aaaaaaaaaa aaaaaaaaaaaaaaaaaaaa
aa
a
a
a
a
a
a
a
a
a
aa
a
a
a
a
a
a
a
a
a
a
aa
a
a
a
a
a
a
a
a
aa
a
a
a
a
a
aaa
a
a
a
aa
a
a
a
aa
a
a
aa
aa a aaaaaaaaaaaaaaaaaaaaaaaaaaaa aa
aa aa
aa aaaaaaaaaaaaaaaaa
aaaaaaa
aaaaaaaaaa
aaa
aaa
aaaaaaaaaaa
aaaaa
a
a
a
a
aaa
a
a
a
a
a
a
aa
a
a
a
a
a
a
a
a
a
a
a
a
a
aaa
a
a
aaa
a
a
aa
a
a
aa
a
a
aa
a
aa
aaaaaa aa
aaaaaaaaaa
aaaaaaaaaa
aaaaaaaaaa
aaaaaaaaaa
aaa
aaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaa
aaaaa
aaaaa
aa
a
a
a
a
a
aaa
a
a
a
a
a
a
aa
a
a
a
aaa
a
a
a
a
aa
a
a
aa
a
aaa
a
aa
a
aa
a
aaa
aa
aaaaaaa
aaa
aaa
aaa
aaaaaaaaaaaaaaaaaaaaaaaaaa
aaa
aaaaaaaaaaaaaa
aaa
aaaaaaaa
aaaaa
aaaaaaaaaaaaaaa
aaaaa
aaaaaa
a
a
a
a
aaa
a
a
a
a
a
a
a
a
a
a
aa
a
a
aa
a
aa
a
aa
a
aa
a
aa
a
aa
a
aa
aa
aaaa
aaaaa
aaaaaaaa
aaaaa
aaaaaaaa
aaaaa
aaaaaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaaaa
aaaaaaa
aaaaaaa
aaaaaaa
aa
a
a
a
aaa
a
a
a
a
a
aaa
a
aaa
aaa
aaa
aaa
aa
aaa
aa
aaa
aaaa
aaaaaa
aaaaaaa
aaaaaaa
aaaaaaa
aaaaaaa
aaaaa
aaaaaaa
aaaaaaa
aaaaaaaaa
aaaaaaa
aa
aaaaaaa
aa
aaaaaaaaa
aa
aaaaaaaaa
aa
aa
aa
aaa
a
a
a
aa
a
a
aa
a
aa
aaa
aa
aaa
aa
aa
aaa
aa
aa
aaaa
aaa
aaaaaaaaa
aa
aa
aaaaaaaaa
aa
aa
aaaaaaaaaaa
aa
aa
aaaaaaaaaaa
aa
aa
aa
aaaaaaaaaaa
aa
aa
aa
aaaaaaaaaaaaaaa
aa
aa
aa
aa
aa
aa
aa
a
a
aaa
aaaaa
aa
aaaaaaa
aa
aa
aa
aaaaaaa
aaa
aa
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aaa
a
a
a
a
a
a
a
a
a
a
a
a
aaa a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a
a
aa
a
a
a
a
a
a
a
a
aaa
a
a
a
a
a
a
a
a
a
aaa
a
a
a
a
a
a
a
a
aaa
a
a
a
a
a
a
a
a
aaa
a
a
a
a
a
a
a
aa
a
a
a
a
a
aaa
a
a
a
aaa
a
a
aa
a
aaa
aaaa a a a a a a a a a a a a a a a a aa a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a aa a a a a a a a a a a
a
aa
a
a
a
a
aa
a
a
a
a
aaa
a
a
a
a
aaa
a
a
a
a
aa
a
a
a
a
aaa
a
a
a
a
aaa
a
a
a
aaa
a
a
a
aaa
a
a
a
aa
a
a
a
aa
a
a
aaa
a
aaa
a
aa
a
aaa
aaa
aaa a a a a a a a a a a a a a aaa a a a a a a a a a a a a a a a a a a a a a a a a a a a aa a a a a a aa a a a a aa
aa
a
a
aa
a
a
a
aa
a
a
a
aa
a
a
aaa
a
a
aaa
a
a
aaa
a
a
aaa
a
a
aa
a
a
a
aa
a
a
aaa
a
a
aa
a
a
aa
a
a
aaa
a
aa
a
a
aa
a
aaa
aaa
aaa
aaa
aaa
aa
aaa a
a a a a a a a aaa a a a a a a a a aaa a a a a a a a a a a aa a a a a a aa a a a aa a a a aa a a aa
aa
a
aa
a
a
aa
a
a
aa
a
a
aa
a
a
aa
a
aaa
a
aaa
a
aaa
a
aaa
a
aaa
a
aa
a
a
aa
a
aaa
a
aaa
a
aa
a
aaa
aaa
a
aa
a
aa
a
aa
a
aa
a
aa
aaa
aaa
aa
aaa
aa
aa a a a aa a a a a a a aaa a a a a a a a a a a a a a a aa a a a aaaaaaaaaaaaaaa aa
aa
aaa
aaa
a
aa
a
aaa
aaa
a
aa
a
aaa
aaa
a
aa
a
aaa
aaa
a
aa
a
aa
a
aaa
aaa
aaa
a
aa
a
aa
a
aa
a
aa
a
aa
aaa
aaa
aaa
aa
aaa
aaa
aa
aaa
aa
aaaaaa a a a a aa a a a a a a aa a a a a a aaa a a a aa
aaaaaaaaaaa
aaaaaaaaa
aa
aa
a
aa
a
aa
aaa
aaa
aaa
aaa
aaa
aa
a
aa
a
aa
a
aa
a
aa
aaa
aaa
aaa
aa
a
aa
a
aa
aaa
aaa
aa
aaa
aaa
aa
aaa
aa
aaa
aa
aaa
aa
aaa
aa
aaaaa
aa
aaa a a a aa a a a a a a a a a a a aa a
aaa
aaaaaaaaaaaaaaaaaaaa
aaaa
aaa
aa
aaa
aa
aaa
aa
a
aa
aaa
aa
aaa
aa
aaa
aaa
aa
aaa
aa
aaa
aa
aaa
aa
aaa
aa
aaa
aa
aaa
aa
aaa
aa
aaaaa
aa
aaaaa
aa
aaaaa
aa
aa
aaaaaa a a aa a a a a a a aaa
aaaaaaa
aaa
aaaaaaaa
aaaaa
aaaa
aaaaaa
aa
aaa
aa
aa
aaa
aa
aa
aaa
aa
aa
aaa
aa
aa
aaa
aa
aaaaa
aa
aa
aaa
aa
aa
aaa
aa
aa
aaaaa
aa
aa
aaaaa
aa
aa
aaaaaaa
aa
aa
aaaaaaa
aa
aa
a a a a a a aaaaaaaaa
aaa
aaaaaaa
aaaaa
aaaaaaaa
aaa
aaaaaaa
aa
aa
aa
aa
aaaaaaa
aa
aa
aa
aaaaaaaaa
aa
aa
aa
aaaaaaaaa
aa
aa
aa
aa
aaaaaaaaa
aa
aa
aa
aa
aa
aaaaaaaaaaa
aa
aa
aa
aa
aa
aa
aa a aaaaaaaaa
aaaaaaaaa
aa
aa
aaaaaaaaa
aaa
tD = 0.878
aa a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a aaa a a a a a a a a a a a a a a a a aa a a a a aa
aaaa
aa
aa
aaa
aa
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aa
aa a a a a a a a a aa a a a a a a a a aaa a a a a a a a a aa a a a a a a a a aa a a a a a a aa
a a a a aa a a a aa aaaaaaaaaaaaaaa
aa
aaaaaaa
aa
aaa
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aaa
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aa
a
a
a
a
a
a
a
a
aa
aa a a a aaaa a a a aa a a a a aaa
a a a a aa a a a a aaaa a a aaa
a a a aa a a a aaaa a aa a a
aaaaaaaaaaaaaaaaaa
aaaaaaa
aa
aa
aa
aa
aa
aa
aa
a
a
aaa
a
a
a
a
a
a
a
aaa
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aaa
a
a
a
a
aaa
a
a
a
aa
aa a a aaa a aaa
a a aaaa a aa a
a a aa a a a aaa a aaa
a a aa a a aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaa
aaaaaaaaa
aa
aa
aaaaaaaa
aaa
a
a
a
a
a
aaa
a
a
a
a
a
a
a
a
a
aaa
aa
a
a
a
a
a
a
a
aa
a
a
a
aaa
a
a
aa
a
a
aa
aa a aaaa aa
aaaaaaaaaaaaaaaaaaaaaaa
aa aaaaaaaaaaaaaaaaaaaaaaa
aaa
aaa
aaa
aaaaaaaa
aaaaa
aaaaaaaaa
aaaaaaaaa
aa
aa
a
a
a
a
a
aa
a
a
a
a
a
a
aa
a
a
a
a
a
a
a
aaa
a
a
a
aa
a
a
aa
a
a
aa
a
a
aa
a
aa
aaaaaaaaaaa
aaaaaaaaaa
aaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaa
aaaaaaa
aaaaaaa
aa
aaaaaaa
aa
aaaaaaaa
aa
a
a
a
a
aaa
a
a
a
a
a
a
a
a
a
a
aa
a
a
aa
a
aaa
a
aa
a
aaa
aaa
aa
aaaaaaa
aaa
aaaaaaaaaaaaaaaaa
aaa
aaaaaaaaaaaaaa
aaa
aaaaaaaa
aaaaa
aaaaaaaaaa
aaaaa
aa
aaaaaaa
aaaaaaaaa
aa
aaaaaaaaa
aa
a
aaa
a
a
a
a
a
aa
a
aa
a
a
a
a
aa
a
aaa
aaa
aaa
aaa
aa
a
aa
aa
aaaa
aaaaa
aaaaa
aaaaaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaaaa
aaaaaaa
aaaaaaa
aa
aaaaaaaaa
aa
aa
aaaaaaaaa
aa
aa
aa
a
aa
a
a
a
a
a
a
a
aa
a
aaa
aaa
aaa
aa
aaa
aaa
aa
aaaa
aaaaaa
aa
aaaaa
aaaaaaa
aaaaaaaaa
aaaaaaa
aaaaaaa
aaaaaaa
aa
aaaaaaa
aa
aaaaaaaaa
aa
aa
aaaaaaaaa
aa
aa
aa
aaaaaaaaaaaaa
aa
aa
aa
a
a
a
a
a
aaa
a
aa
aaa
aa
aaa
aa
aaaaa
aa
aaaaaa
aaa
aaaaaaaaaaa
aa
aa
aaaaaaaaaaa
aa
aa
aa
aaaaaaaaaaa
aa
aa
aa
aaaaaaaaaaaaa
aa
aa
aa
aa
aa
aa
aaaaaaaaaaaaaaaaaaaaaaaaa
aa
aa
aa
aa
a
a
aa
a
aa
aa
aaaaa
aa
aa
aa
aaaaaaaaa
aaa
aa
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aa
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aa
a
a
a
a
aaa
aaa
aa
aaaaa
aa aa a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a
a
aa
a
a
a
a
a
a
a
a
aa
a
a
a
a
a
a
a
a
aaa
a
a
a
a
a
a
a
a
aa
a
a
a
a
a
a
a
aaa
a
a
a
a
a
aaa
a
a
a
a
aa
a
a
a
aa
a
a
aa
a
aa
a
aa
aaaaaaaaaaaaaa
aaaaa a a a a a a a a a a a a a a aa a a a a a a a a a a a a a a a a a a a a a a a a a aa a a a a a a a a a
a
aa
a
a
a
aaa
a
a
a
a
aa
a
a
a
a
aaa
a
a
a
a
aa
a
a
a
a
aa
a
a
a
a
aa
a
a
a
a
aa
a
a
a
aa
a
a
a
aa
a
a
aa
a
a
aa
a
aa
a
aa
a
aa
aaa
aa
aaaaaaaaaaaaaaaaaaaa a aa a a a a a a a a aaa a a a a a a a a a a a a a a a a a a a a aaa
a a a a aaaa a a aa
aa
a
a
aa
a
a
aaa
a
a
aaa
a
a
aa
a
a
a
aa
a
a
aaa
a
a
aa
a
a
aaa
a
a
aa
a
a
aaa
a
aaa
a
aa
a
aaa
a
aa
a
aa
a
aa
a
aa
aaa
aa
aaa
aa
aa
aaaaaaa
aa
aa
aa
aaaa aaa a a a a a aa a a a a a a a a a aa a a a aa a a a a a a aa a a a aaa
a a aa a a aa
aa
a
aa
a
aaa
a
aaa
a
aaa
a
aaa
a
aa
a
a
aa
a
aaa
a
aaa
a
aa
a
aaa
a
aaa
aaa
a
aa
a
aaa
aaa
aaa
aaa
aa
a
aa
aaa
aa
aaaaa
aa
aaaaa
aa
aa
aa
aaaaa
aaa a a a a aa a a a a a aa a a a a a a a a a aa
a a a aa aaaaaaaaaaaaa aa
aa
aaa
aaa
a
aa
a
aa
a
aaa
aaa
a
aa
a
aa
a
aaa
aaa
aaa
a
aa
a
aa
a
aa
a
aa
a
aa
a
aa
a
aa
aaa
aaa
aa
aaa
aa
aaa
aa
aaa
aa
aa
aaaaa
aa
aa
aaaaa
aa
aa
aaaaa a a a a a aa a a a a a a a a a a a aa a
aaaaaaa
aaaaaaaaaa
aa
aa
aa
a
aa
aaa
aaa
aaa
aa
a
aa
a
aa
aaa
aaa
aaa
aa
a
aa
aaa
aaa
aaa
aa
aaa
aaa
aa
aaa
aa
aaa
aa
aaa
aa
aaaaa
aa
aa
aaaaa
aa
aa
aaaaaaa
aa
aa
aaaaaa aa a a a a a aa a a a aa a a a
aaaaaaaaaaaaaaaaaaaaaa
aaaa
aa
aaa
aa
a
aa
aaa
aa
aaa
aa
aaa
aa
aaa
aa
aaa
aa
aaa
aa
aaa
aa
aaa
aa
aaa
aa
aa
aaa
aa
aa
aaaaa
aa
aa
aaaaa
aa
aa
aa
aaaaaaa
aa
aa
aa
aaaaaaaaa a a a a a a a a aa
aaaaaaa
aaa
aaaaa
aaaaa
aaa
a
aaaaaa
aa
aaaaa
aa
aa
aaa
aa
aa
aaaaa
aa
aaaaa
aa
aa
aaa
aa
aa
aaaaa
aa
aa
aaaaaaa
aa
aa
aaaaaaa
aa
aa
aa
aa
aaaaaaaaa
aa
aa
aa
aa
aa
aaaaaaaa a a a a a aa
aaaaaa
aaaaa
aaaaaaa
aaaaaaaa
aaa
aaaaaaaaaaa
aa
aa
aa
aa
aaaaaaaaaaa
aa
aa
aa
aa
aaaaaaaaaaa
aa
aa
aa
aa
aa
aa
aaaaaaaaaaaaaaaaa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aaaaaaaaaa
aaaaaaaaaaa
aa
aa
aaaaaaaaaaa
aa
aaa
tD = 0.918
aa a a a a a a a a a a a a a a a a a a a a a a a aa a a a a a a a a a a a a a a a a a aaa a a a a a a a aaa
a a a aa aaaaaaa
aa
aa
aa
a
aa
a
aaa
a
a
aaa
a
a
a
a
a
a
aa
a
a
a
a
a
a
a
a
aaa
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aa
aa a a a a a a aaa a a a a a a a aa a a a a a a a aa a a a a a a aa a a a a a aaa
a a a aaaa a aa
aaaaaaaaaaaaa
aaaaaa
aa
aaa
aa
a
aa
a
aaa
a
aa
a
a
aaa
a
a
aa
a
a
aa
a
aaa
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aaa
a
a
a
a
a
a
aa
aa a a a aa a a a a aa a a a a aa a a a a aa a a a a aaa a a aa a
a a aaaa a aa a a
aaaaaaaaaaaaa
aaaaaaaaaa
aa
aaaaaaa
aa
a
aa
aaa
aaa
a
aa
a
aa
a
aa
a
aa
a
aa
aa
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aaa
a
a
a
a
aa
a
a
a
aa
aa a aaaa a aa a a aaa
a a aa a a aaaa a aa a a aaa
aa aaaa aaaaaaaaaaa aaaaa
aaa
aaa
aaaaa
aaaaaaaaaaaa
aa
aa
aaa
aa
aaa
aa
a
aa
aaa
aa
aaa
aa
aa
aa
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aa
a
a
a
aa
a
a
aaa
a
aa
aa a aaaaaaaaaaaaa
aaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaa
aa
aa
aa
aa
aa
aaaaa
aa
aa
aaa
aa
aa
aaa
aa
aa
aaaaaa
aa
a
a
a
a
aaa
a
a
a
a
aaa
a
a
a
aaa
a
aaa
a
aaa
a
aa
a
aa
aaaaaaaaaaaaaa
aaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaa
aaaaaaaaa
aa
aa
aa
aaaaaaaaa
aa
aa
aaaaaaa
aa
aa
aa
aa
aa
aa
aa
aa
a
a
a
a
a
a
a
a
aa
a
a
aaa
a
aa
a
aaa
aaa
aaa
aa
aaaa
aaa
aaaaaaaa
aaa
aaaaaaaaaaa
aaa
aaaaaaaa
aaaaa
aaaaaaaa
aaaaaaa
aaaaaaaaa
aa
aa
aa
aa
aa
aa
aa
aa
aaaaaaaaa
aa
aa
aa
aa
aa
aa
aa
aaaa
a
a
a
a
a
a
a
a
aa
a
a
aa
a
aa
a
aa
aaa
aaa
aa
aa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaa
aaaaa
aaaaaaa
aaaaaaa
aaaaaaaaa
aa
aa
aa
aaaaaaaaaaaaaa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
a
a
a
a
aa
a
a
aa
a
aa
a
aa
aaa
aa
aaa
aa
aaaa
aaaaaaaa
aaaaaaaaa
aaaaaaa
aa
aaaaaaa
aa
aaaaaaa
aa
aaaaaaaaa
aa
aa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aa
aaa
a
a
aa
a
aa
aaa
aa
aaaaa
aa
aa
aaaaaa
aaa
aa
aaaaaaaaaaaaaaa
aa
aa
aa
aa
aa
aaaaaaaaaaaaaaaaa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aaaaaaaaaaaaaa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aaaaaaaa
aaa
aa
aa
aa
aaaaaaaaaaaaa
aa
aaa
aa
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aaa
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aa
a
a
a
a
a
a
a
a
a
aa
a
a
a
a
aaa
a
a
aa
a
aa
aaa
aa
aaaaa
aaa
aa aa a a a aaa
a a a a a a aa a a a a a a a aaa a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a
a
aa
a
a
a
a
a
a
aa
a
a
a
a
a
a
a
aa
a
a
a
a
a
a
aaa
a
a
a
a
a
a
aa
a
a
a
a
a
aa
a
a
a
a
aa
a
a
a
aa
a
a
aa
a
aaa
aaa
aa
aaaaaaaaaa
aaaaa
aaa
aaaaaaaa a
aaaa a a
aaa aaaaa
a a a a a a a a a a a a a a a a a a a a a a a a aa a a a a a a aa
aa
a
a
a
aa
a
a
a
a
aa
a
a
a
aaa
a
a
a
aaa
a
a
a
aa
a
a
a
aaa
a
a
aaa
a
a
aaa
a
aaa
a
aaa
a
aa
a
aa
a
aa
aaa
aa
aa
aaaa
aa
aaaaaaa
aaaaaaaaaaaaaaaaaa
aaa
aaaaa a a a a a a a a a a a a a a a a a a a a aa a a a a aa a a a aa
aa
a
aaa
a
a
aa
a
a
aaa
a
a
aa
a
a
aaa
a
aaa
a
a
aa
a
a
aa
a
a
aaa
a
aa
a
aaa
a
aa
a
aaa
aaa
aa
a
aa
aaaaa
aa
aa
aa
aa
aa
aaaaaaa
aaaaa
aaa
aaaaa
aaaaa
aa
aa
aaa a a a a a a a aa a a aa a a a a a a aa a a a aa a a aaa
a aa
aa
a
aa
a
aaa
a
aa
a
aaa
a
aaa
a
aa
a
aaa
a
aaa
a
aa
a
aaa
aaa
a
aa
a
aa
a
aa
a
aa
a
aa
aaa
aa
aaa
aa
aaaaaaaaaaaaaaaaaa
aaaaaaa
aaaaaaa
aaaaaaaaaaaaa a a a aa a a a a a a aa a
a a a aaaa aa
aaaaaaaa aa
aa
aaa
aaa
aaa
aaa
a
aa
a
aa
a
aa
a
aa
a
aaa
aaa
aaa
aaa
aaa
aaa
aa
a
aa
aaa
aa
aaa
aa
aaa
aa
aa
aaaaaaaaaaaaaaaaaa
aa
aa
aaaaaaa
aa
aa
aa
aa
aa
aa
aa
a a a a a a a a a a aaaaa aaaaaaaaaaaa
aaa
aa
aa
aa
aaa
aaa
aa
a
aa
aaa
aaa
aa
a
aa
aaa
aaa
aa
a
aa
aaa
aa
a
aa
aaa
aa
aaa
aa
aaaaa
aa
aa
aaaaaaaaa
aa
aa
aa
aaaaaaaaaaaaa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aaa a a a a a a a aa
aaaa
aaaaaa
aaa
aaaaaaa
aaaa
aa
aaa
aa
aaa
aa
aaa
aa
aaa
aa
aaa
aa
aaaaa
aa
aaa
aa
aaa
aa
aa
aaaaa
aa
aaaaaaa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aaaaaaaa
aa
aa
aa
aaa a a a aa a
aaaaaaaaa
aaaaaaaaaaaaa
a
aaaaaa
aa
aa
aaaaa
aa
aa
aaaaa
aa
aa
aaaaa
aa
aa
aaaaa
aa
aa
aaaaaaa
aa
aa
aa
aa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa a a
aaaaaaaa
aaaaaaa
aa
aaaaaaaa
aaa
aa
aaaaaaaaaaaaaaa
aa
aa
aa
aa
aa
aa
aaaaaaaaaaaaaaa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aaaaaaaa
aaaaa
aa
aaaaaaaaaaaaa
aa
aa
aaa
tD = 0.938
aa a a a a a a a a a a a a a a a a a a a aa a a a a a a a a a a a a a a a a aaa a a a a a a a a aaa a a a a aa a a
aaaaaa
aaaaaaaa
aa
a
aaa
a
aaa
a
a
a
a
aa
a
a
a
a
a
a
a
aaa
a
a
a
a
a
a
a
a
a
aa
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aa
aa a a a a a a aa a a a a a a a aa a a a a a a aaa a a a a a aaa
a a a a aaaa a a aaa
a a aaaaaaaaaa
aaa
aaaaa
aa
aa
aa
aaa
aa
aaa
a
aa
a
aaa
a
a
aa
a
a
a
aa
a
a
a
aaa
a
a
aaa
a
aaa
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aa
a
a
a
a
a
a
aa
aa a a a aa a a a a aaa a a aaa
a a a aa aa a a aa a a a aa a a a aa a
a a aaaaaaaaaa
aaaaaaaaaaaa
aaaaaaaaaaaa
aa
aaa
aa
a
aa
a
aa
a
aaa
a
aa
a
aaa
a
aa
a
aaa
a
aa
aaa
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aa
a
a
a
a
aa
a
a
a
aa
aa a aaaa a aa
a a aaaa a aa
a a aa a a aaaa a a
aaa aa
aaaa
aa aaaaaaaaa
aaa
aaa
aaa
aaaaa
aaaaaaaaaaaa
aa
aa
aaa
aa
a
aa
aaa
aaa
aaa
aaa
aaa
aaa
aa
aaaaa
a
a
a
a
a
a
a
a
a
a
a
a
a
aa
a
a
a
aa
a
a
aaa
a
aa
aa a aaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaa
aaa
aaa
aaaaaaaa
aaaaaaa
aa
aa
aa
aa
aaaaaaa
aa
aaa
aa
aaa
aa
a
aa
aaa
aa
aaa
aa
aaaaaaa
a
a
a
a
a
a
a
a
a
a
aaa
a
a
aa
a
aaa
a
aa
a
aa
aaaaaaaaaaaaaaaaaaaa
aaa
aaaaaaaaaaaaaaaaaaaaa
aaa
aaaaaaaa
aaaaa
aaaaaaaaa
aa
aa
aa
aa
aaaaaaa
aa
aaaaa
aa
aaa
aa
aa
aaa
aa
aa
aaaaaaaaa
a
aa
a
aaa
a
a
a
aaa
a
aaa
aaa
aaa
aaa
aa
aaaa
aaaaaaaaaaa
aaaaaaaaaaa
aaaaaaaa
aaa
aaaaa
aaaaaaaaaaaaaaa
aaaaaaaaa
aa
aa
aa
aa
aa
aa
aa
aa
aaaaaaa
aa
aa
aaaaa
aa
aa
aaaaaaaaaaaaa
a
a
a
a
aaa
a
a
aa
a
aa
a
aa
aaa
aaa
aa
aa
aaaaaaaaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaaaa
aaaaa
aaaaaaa
aaaaaaaaa
aa
aaaaaaaaaaaaaaaaaaaaaaaa
aa
aa
aa
aaaaaaa
aa
aa
aa
aa
aa
aa
aa
aa
aa
a
a
a
a
aa
a
aa
a
aa
aaa
aa
aaa
aa
aaaa
aaaaaaaa
aa
aaaaaaa
aa
aaaaaaa
aa
aaaaaaa
aa
aaaaaaaaa
aa
aa
aaaaaaaaa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aaaaaaaaaaaaaaa
aa
aa
aa
aaaaaa
aa
a
a
aa
aaa
aa
aaaaa
aa
aa
aaaaaa
aaa
aa
aa
aaaaaaaaaaaaaaaaa
aa
aa
aa
aa
aa
aa
aa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aa
aa
aaaaaaaaaaaaa
aa
aa
aaa
aa
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aaa
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aaa
a
a
a
a
a
a
a
a
a
aa
a
a
a
a
a
aaa
a
a
aaa
a
aa
a
aa
aaaaaa
aaaaa
aaaa a a aa a a a a a aa a a a a a a a aaa a a a a a a a aa a a a a a a a a a a a a a a a a a a a a a a a a a a a a
a
aa
a
a
a
a
a
aaa
a
a
a
a
a
a
aa
a
a
a
a
a
a
aaa
a
a
a
a
a
aa
a
a
a
a
a
aa
a
a
a
a
aa
a
a
a
aa
a
a
aaa
a
aa
a
aa
aaa
aa
aaaaaaaa
aaaaa
aaa
aaaaa aa
a a aa aa a aa a a a aa a a a aa
aa aa
a a a a a a a a a a a a a a a a a a aa a a a a a a aa
aa
a
a
a
aa
a
a
a
aaa
a
a
a
aa
a
a
a
aaa
a
a
a
aa
a
a
a
aaa
a
a
aaa
a
a
aa
a
a
aaa
a
aa
a
aaa
aaa
aaa
aaaaa
aa
aa
aa
aa
aaaaaaa
aaa
aaaaaaaaa
aaaaaaaaa aaaaaaaaaaa
a a a a a a a a a a a a a a a aa aa a a aa a a a aa
aa
a
aaa
a
a
aa
a
a
aa
a
a
aaa
a
a
aa
a
a
aa
a
a
aaa
a
aaa
a
aaa
a
aa
a
a
aa
a
aa
a
aaa
aaa
aa
a
aa
aaaaa
aa
aa
aa
aa
aaaaaaa
aaaaa
aaaaaaaaaaaaaaaaaaaa
aaaaa
aaa a a a a a a a a a a a a aa a
a a aa a a aaa a aa
aa
aaa
a
aaa
a
aa
a
aaa
a
aa
a
aaa
a
aa
a
aaa
a
aaa
aaa
a
aa
a
aaa
aaa
aaa
aaa
aa
a
aa
aaa
aa
aa
aaaaaaaaaaaaaaaa
aaaaaaa
aaaaa
aaaaaaaa
aaaaa
aaaaaaaaa a a a a a a a a a a a aa
aaaaaaaaaaaa aa
aa
aa
a
aaa
aaa
aaa
aaa
aaa
aaa
a
aa
a
aa
a
aa
a
aa
a
aa
a
aa
aaa
aaa
aa
a
aa
aaa
aa
aaa
aa
aa
aaaaaaaaaaaaaaaaaa
aa
aaaaaaa
aaaaa
aaaaaaa
aaaaaaaaaaaaa aa a a a aaa a a a aa
aaaaaaaaaa
aaa
aa
aa
aa
aaa
aaa
aa
aaa
aaa
aa
a
aa
aaa
aaa
aa
a
aa
aaa
aa
aaa
aaa
aa
aaa
aa
aaaaa
aa
aa
aaaaa
aa
aa
aa
aa
aa
aa
aaaaaaaaa
aa
aaaaaaaaa
aaaaaaaaa
aa
aa
aaaaa a a a a aa a
aaa
aaaaaa
aaa
aaaaaaa
aaaa
aa
aaa
aa
aaa
aa
aaaaa
aa
aaa
aa
aaa
aa
aaa
aa
aa
aaa
aa
aaaaa
aa
aaaaaaa
aa
aa
aa
aaaaaaaaaaaaaaaaaa
aa
aa
aa
aaaaaaaaa
aa
aa
aa
aa
aa
aa
aa
aa
aaa a a a
aaaaaaaaaaa
aaaaa
aaaaa
a
aaaaaa
aa
aa
aaaaaaa
aa
aa
aaaaa
aa
aa
aaaaa
aa
aa
aaaaaaa
aa
aa
aa
aaaaaaaaa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aaaaaaaaaaaaaaa
aa
aa
aa
aa
aa
aaaaaa a
aaaaaa
aaaaaaa
aa
aaaaaaaa
aaa
aa
aa
aaaaaaaaaaaaaaaaa
aa
aa
aa
aa
aa
aa
aa
aa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaa
aa
aa
aa
aaa
tD = 0.969
aa a a a a a a a a a a a a a a aa a a a a a a a a a a a a a aaa a a a a a a a a a aaa a a a a a aa a
a a aaaaa aaaaa
aaaaa
aa
aaa
aa
a
aaa
a
aaa
a
a
aaa
a
a
a
a
a
aaa
a
a
a
a
a
a
a
a
a
aa
a
a
a
a
a
a
a
a
a
a
aaa
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aa
aa a a a a a aa a a a a a a aa a a a a a a aa a a a a a aaa
a a a a aa a a a a aa a a a aa a aaaaaaaaaaaaaaaa
aa
aa
aa
aa
aaa
aa
aaa
a
aa
a
aaa
a
aaa
a
a
aaa
a
a
a
aaa
a
a
a
aaa
a
a
a
a
aa
a
a
a
a
aa
a
a
a
a
a
a
a
a
a
aaa
a
a
a
a
a
aa
aa a a aaaa a a aa a a a aaa
a a a aa a a a aa a a a aa aa a aa a a a aa
aa aa
aa aaaaaaaaaaaaaaaaaaa
aa
aa
aa
aa
aaa
aa
aaa
aaa
aaa
a
aa
a
a
aa
a
a
aa
a
a
aaa
a
a
aa
a
a
aaa
a
aaa
a
aaa
a
a
a
a
a
aaa
a
a
a
a
aaa
a
a
aa
aa a aaaa aa
aa a aa
aa aa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaa
aa
aa
aa
aa
aaa
aa
aaa
aa
aaa
aaa
aaa
a
aa
a
aaa
a
aa
a
aaa
a
aaa
a
aa
a
aa
a
aaa
a
a
aa
a
a
a
a
a
aaa
a
aaa
a
aa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaa
aaaaa
aa
aaaaaaaaaaaaaa
aa
aaaaa
aa
a
aa
aaa
aaa
aaa
aaa
aaa
aaa
aaa
aaa
aaa
aaa
aa
aaa
a
a
a
a
a
aa
a
a
aa
a
aaa
aa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaa
aaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaa
aa
aa
aaa
aa
aaa
aa
aaa
aa
a
aa
aaa
aaa
aa
a
aa
aaa
aa
aaa
aa
aa
a
a
a
a
aa
a
aaa
a
aa
a
aa
aa
aaaa
aaaaaaaa
aaaaaaaa
aaaaaaaa
aaa
aaaaa
aaaaaaaaaaaaaaaaaaaa
aa
aaaaaaa
aa
aa
aa
aa
aa
aa
aaaaaaa
aa
aa
aaa
aa
aaaaa
aa
aaa
aa
aaa
aa
aaa
aa
aaaaa
aa
aa
a
a
aaa
a
aa
a
aa
aaa
aa
aa
aaaaaa
aaaaa
aaaaaaaaaaaa
aaaaa
aaaaaaa
aaaaaaa
aaaaaaa
aaaaaaa
aa
aa
aaaaaaaaaaaaaaaaaaaa
aa
aa
aa
aaaaaaa
aa
aaaaa
aa
aa
aaa
aa
aa
aaaaa
aa
aa
aaaa
a
aaa
aaa
aa
aaa
aa
aaaa
aaaaaaaaaa
aa
aaaaaaa
aa
aa
aaaaaaa
aa
aa
aaaaaaaaa
aa
aaaaaaaaaaa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aaaaaaaaa
aa
aa
aa
aa
aaaaaaa
aa
aa
aa
aa
aa
aa
aa
a
aa
aaa
aa
aa
aaaaaaaa
aaa
aa
aa
aa
aa
aa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aa
aa
aa
aa
aa
aaa
aa
a
a
a
a
a
a
a
a
a
a
a
a
aaa
a
a
a
a
a
a
a
a
a
a
a
a
aa
a
a
a
a
a
a
a
a
a
aaa
a
a
a
a
a
a
aa
a
a
a
aaa
a
a
aa
a
aaa
aa
aaaaaaaaaa
aaaaaaaa aa a
a a a aa a a a a a a aa a a a a a a a a a aa a a a a a a a a a aa a a a a a a a a a a a a a a a a a aa
aa
a
a
a
a
a
aa
a
a
a
a
a
aaa
a
a
a
a
a
aa
a
a
a
a
a
aaa
a
a
a
a
aa
a
a
a
a
aa
a
a
a
aa
a
a
aaa
a
aa
a
aaa
aaa
aa
aaaaaaaaaa
aaaaa
aaaaaa
aaaaa a aa
a a a aaa a a aaa
a a a aaaa a a a aa a a a aa a a a a a a a a a a aa a a a a a a
a
aa
a
a
aaa
a
a
a
aa
a
a
a
aa
a
a
a
aaa
a
a
aaa
a
a
aaa
a
a
aaa
a
a
aa
a
a
aa
a
a
aa
a
aaa
aaa
aaa
aaa
aa
aaaaaaaaaaaa
aaaaa
aaaaaaaaaa aaaaaaaaaaa a aa
a a aaaa a aa
aaaaaaaa
a a a a a a aa a a a a aaaa a aa
aa
a
aaa
a
aaa
a
aaa
a
a
aa
a
a
aa
a
a
aa
a
a
aa
a
a
aa
a
a
aa
a
aaa
a
aa
a
aaa
aaa
aaa
aaa
aa
aaa
aa
aaaaaaaaaaaaaa
aaaaa
aaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaa a a aa a a a a aaa
aaaa a aa
aa
aaa
a
aa
a
aaa
a
aa
a
aaa
a
aa
a
aaa
a
aa
a
aaa
aaa
a
aa
a
aa
a
aa
a
aa
a
aa
aaa
aaa
aa
aaaaa
aa
aa
aaaaaa
aa
aa
aaaaaaaaaaaaaaa
aaa
aaaaaaaaaaaa
aaa
aaa
aaa
aaa
aaaaa a a a a a aa
aaaa
aaaaaa
aa
aa
a
aa
a
aa
a
aa
a
aa
a
aa
a
aa
a
aa
a
aa
a
aa
a
aa
a
aa
a
aa
aaa
aaa
aa
aaa
aaa
aa
aa
aaa
aa
aa
aaaaaaaaaaaa
aa
aaaaaaa
aaaaaaaaaa
aaaaaaaa
aaaaaaaa
aaa
aaaaa
aa
aaa a a a aa
aaaa
aaa
aaa
aa
aa
aa
aaa
aa
aaa
aaa
aa
a
aa
aaa
aa
a
aa
aaa
aa
aaa
aaa
aa
aaa
aa
aaa
aa
aa
aaa
aa
aa
aaaaaaaaa
aa
aa
aaaaaaaaaaa
aaaaaaa
aaaaaaa
aaaaa
aaaaa
aaaaa
aaaaaaa
aa
aaa a aa
aaaaaaa
aaa
aaaa
aaaa
aa
aaaaa
aa
aaa
aa
aaaaa
aa
aaa
aa
aaaaa
aa
aaa
aa
aa
aaa
aa
aa
aaaaaaa
aa
aa
aa
aaaaaaaaaaaaaa
aa
aa
aa
aaaaaaaaa
aaaaaaa
aaaaaaaaa
aaaaaaaaa
aa
aa
aaa aa
aaaa
aaaaa
aaaaaa
aaaaaaaa
aa
aa
aaaaaaa
aa
aa
aaaaaaa
aa
aa
aaaaaaa
aa
aa
aa
aaaaaaaaaaa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aaaaaaaaaaa
aa
aa
aa
aaaaaaaaa
aa
aa
aa
aa
aa
aa
aaaaa
aaa
aa
aaaaaaa
aaa
aaa
aa
aa
aa
aa
aa
aa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aa
aa
aa
aa
aa
aaa
tD = 0.998
aa a a a a a a a a a a a a aa a a a a a a a a a a a a aaa a a a a a a a a a aa a a a a a a aaa
a a a aaaaa aaaaaaaaaaaaaaaa
aa
a
aa
a
aa
a
a
aaa
a
a
a
aaa
a
a
a
a
a
aaa
a
a
a
a
a
a
a
a
a
aaa
a
a
a
a
a
a
a
a
a
a
a
aa
a
a
a
a
a
a
a
a
a
a
a
a
a
aa
aa a a a a a aa a a a a a aaa a a a a a aa a a a a a aa a a a a a aa
a a a aaaa a a aa
a aaaaaaa
aaaaaaaaa
aaaaaaaa
aa
aaa
aa
a
aa
a
aa
a
a
aa
a
a
aaa
a
a
aaa
a
a
a
aa
a
a
a
a
a
aa
a
a
a
a
a
aa
a
a
a
a
aaa
a
a
a
aaa
a
a
a
a
a
a
a
aa
aa a a aaaa a aaa
a a a aa a a a aa a a a aaaa a aaa
a a aa a a a aaaa aa
aa aaaaaaaaaaaaaaaaa
aa
aaaaaa
aa
aa
aaa
aa
aaa
aaa
aaa
a
aaa
a
aa
a
a
aa
a
a
aaa
a
a
aaa
a
a
aaa
a
a
aaa
a
a
aa
a
a
aa
a
a
a
a
a
aaa
a
a
aa
aa a aaaaaaaaaaaaaaaaa aa
aa aa
aa aa
aa aa
aa aaaaaaaaa
aaaaaaaaaaaaaaa
aaaaa
aaaaaaaaaa
aa
aa
aaa
aa
aaa
aaa
aaa
aaa
a
aa
a
aaa
a
aa
a
a
aa
a
a
aa
a
a
aa
a
aaa
a
aaa
a
aa
a
a
aa
a
a
a
aaa
a
aa
aaaaaaaaaaaaaaa
aaaaaaa
aaaaaaaaaaaaaaaaaaaa
aaa
aaa
aaa
aaaaaaaa
aaaaaaa
aa
aa
aa
aa
aa
aaaaa
aa
aaa
aa
aaa
aaa
aa
a
aa
a
aa
a
aaa
aaa
a
aa
a
aa
a
aaa
aaa
a
aa
a
aa
aaa
a
a
aaa
a
aaa
aa
aaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaa
aaa
aaa
aaaaaaaaaaa
aaaaaaaaaa
aaaaaaa
aa
aa
aa
aa
aa
aa
aa
aaaaa
aa
aaa
aa
aaa
aa
a
aa
aaa
aaa
aa
a
aa
a
aa
a
aa
aaa
aaa
aaa
aa
aaa
aaa
a
a
aa
a
aa
aa
aaaa
aaaaaaaa
aaaaaaaa
aaaaa
aaa
aaaaa
aaaaaaaa
aaaaa
aaaaaaa
aaaaaaa
aa
aaaaaaaaaaaaaaaa
aa
aa
aa
aaaaa
aa
aaa
aa
aaa
aa
aaa
aa
aaa
aa
aaa
aaa
aa
aaa
aa
aaaaaaa
a
aaa
aaa
aa
aa
aaaaaa
aaaaa
aaaaaaa
aaaaa
aaaaaaa
aaaaaaa
aaaaaaa
aaaaaaa
aa
aaaaaaaaa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aaaaaaa
aa
aa
aaaaa
aa
aa
aaaaa
aa
aaa
aa
aa
aaa
aa
aa
aaa
aa
aa
aa
aaa
aaa
aa
aaaa
aaaaaaaaaa
aa
aaaaaaaaa
aa
aa
aaaaaaaaa
aa
aaaaaaaaaaa
aa
aa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aa
aa
aa
aa
aaaaaaa
aa
aa
aa
aaaaaaa
aa
aa
aa
aaaaaaaaa
aa
aaaaaaaa
aaa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aaa
aa
a
a
a
a
a
a
a
a
a
a
a
aa
a
a
a
a
a
a
a
a
a
a
a
aa
a
a
a
a
a
a
a
a
a
aaa
a
a
a
a
a
a
aa
a
a
a
a
aaa
a
a
aa
a
aaa
aaa
aaaaaaaaaa
aaaaaaaaa
a a aaaa a a a aa
a a a a a a aa a a a a a a a a a aaa a a a a a a a a a a aa a a a a a a a a a a a a aa
aa
a
a
a
a
aaa
a
a
a
a
a
aa
a
a
a
a
a
aa
a
a
a
a
a
aa
a
a
a
a
aaa
a
a
a
aaa
a
a
aaa
a
a
aa
a
a
aa
a
aaa
aaa
aa
aaaaaaaaaaaaaaaaa
aaa
aaaaa aa
a a aaaa a aaa
a a a aa aa a a a aa
a a a a aaaa a a a aa a a a a a aa a a a a a a aa
aa
a
a
aaa
a
a
aaa
a
a
aaa
a
a
a
aa
a
a
a
aa
a
a
a
aa
a
a
aaa
a
a
aa
a
a
aa
a
a
aa
a
aaa
a
aa
a
aa
aaa
aa
aaaaa
aa
aa
aaaaaaa
aaaaaaaaaaaaaaa
aa aaaa aa
a a aaaa a aa a a a aa a a a aa
a a aa aaaaa a a a a aaa
a a aa
aa
a
aa
a
a
aaa
a
aaa
a
aaa
a
aaa
a
aaa
a
aaa
a
aaa
a
aaa
a
aa
a
aaa
a
aa
a
aa
a
aa
a
aa
aaa
aa
aaaaa
aa
aa
aa
aa
aaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaa aa a a a a a
a a aa
aa
aaa
a
aa
a
aaa
a
aa
a
aa
a
aaa
a
aa
a
aaa
aaa
a
aa
a
aaa
aaa
aaa
aaa
aaa
aa
a
aa
aaa
aa
aa
aaaaaaaaaaaaaaaaaa
aaaaa
aaaaa
aaa
aaaaaaaaaaaaaaa
aaaaaaaaaaaaa
aaaaaa
aaa
a a a aaaaaaaa
aa
aa
a
aa
a
aa
a
aa
a
aa
a
aa
a
aa
a
aa
a
aa
aaa
aaa
aaa
aaa
aa
a
aa
aaa
aa
a
aa
aaaaa
aa
aaaaaaa
aa
aa
aa
aa
aaaaaaa
aaaaaaa
aaaaa
aaaaaaaa
aaa
aaaaaaaaaaaaaa
aaa
aaaaaaaaaaa
a aaa
aaa
aa
aa
aa
aaa
aa
aaa
aaa
aa
aaa
aa
a
aa
aaa
aa
aaa
aaa
aa
aaa
aa
aaa
aa
aaaaa
aa
aaaaa
aa
aa
aa
aaaaaaaaaa
aa
aa
aaaaaaaaa
aaaaa
aaaaaaaaaaaaaaa
aaaaa
aaaaa
aaaaaaaaaaaa
a aaaaaa
aaaa
aaaa
aa
aa
aaa
aa
aaa
aa
aa
aaa
aa
aaaaa
aa
aaa
aa
aa
aaa
aa
aa
aaaaa
aa
aa
aaaaaaaaa
aa
aa
aa
aa
aa
aa
aaaaaaaaaaa
aa
aaaaaaaaa
aaaaaaa
aaaaaaa
aaaaaaa
aaaaaaa
aa
aa
aaaaaa
aaaaaa
aaaaaaaa
aa
aa
aa
aaaaaaa
aa
aa
aaaaaaa
aa
aa
aa
aaaaaaaaa
aa
aa
aa
aa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aa
aa
aa
aa
aaaaaaaaa
aa
aa
aaaaaaaaa
aa
aa
aaaaaaaaaaaaaaaaaa
aaa
aaa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aaa
![Page 23: A STUDY OF THE EFFECT OF MOBILITY RATIOS ON ...1. A study of the effect of mobility ratios on pattern displacement behavior It is well known, for unit mobility ratio, that the areal](https://reader031.vdocuments.net/reader031/viewer/2022040817/5e61c32167659d0b291248ba/html5/thumbnails/23.jpg)
17
than in the central streamtubes. Little area is unswept and the sweep efficiency at
breakthrough approaches unity.
Similar to before, when the path of displacing fluid in the ith streamtube is longer
than that in the jth streamtube, then the ith streamtube is wider in the swept region to
keep the same resistance (same flow rate and pressure). Therefore, the streamtubes along
the boundary become wider near the corner where the streamtube changes direction.
Streamlines remain smooth. This streamline distribution trend is apparent in Fig. 6b.
In summary, the very favorable mobility ratio conspires with boundary conditions
to determine the way that streamlines evolve, and makes the sweep efficiency at
breakthrough near unity. If the mobility ratio is very favorable, the pressure drop is
mainly in the displacing phase, and it does not feel the well, but is affected by the
boundary.
1.5.4 Staggered Line Drive, Very Favorable Mobility Ratio
For the staggered line drive pattern, the displacement at the beginning is similar to
that in the five-spot pattern. That is, the displacement pattern is radial around the injector
before the front reaches the nearest corner. The differences in displacement behavior
between the two patterns occurs after the front reaches the near corner. For a five spot,
because of the symmetry, the front reaches the two corners at the same time. However,
for a staggered line drive, the front reaches the closest corner first.
After the front passes the near corner, the streamlines evolve in a way similar to
the five spot. The streamtubes along the boundary are wide near the corner but narrow
near the front. This makes the front near the boundary move faster because the
streamtubes are narrower than those in the center of the pattern. Therefore, the front near
the boundary on the near no-flow corner side catches up, and the displacement
approaches linear flow (see the relevant streamline distribution in Fig. 7 at tD = 0.60 and
0.80). If the aspect ratio is large, flow in the center of the pattern must become nearly
linear because the pressure isobars are nearly straight and intersect the pattern boundary
at 90o.
![Page 24: A STUDY OF THE EFFECT OF MOBILITY RATIOS ON ...1. A study of the effect of mobility ratios on pattern displacement behavior It is well known, for unit mobility ratio, that the areal](https://reader031.vdocuments.net/reader031/viewer/2022040817/5e61c32167659d0b291248ba/html5/thumbnails/24.jpg)
18
Fig 7 Staggered Line Drive, Very Favorable Mobility Ratio(1/M=20)
(a) Saturation Distribution
(b) Streamline Distribution
tD = 0.20
tD = 0.40
tD = 0.60
tD = 0.80
tD = 0.88
tD = 0.90
tD = 0.92
tD = 0.94
tD = 1.0
tD = 0.20
aaaaaaaaaaaaaaa
aaaaaaaaaaaaa
aaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaa
aaaaaaa
aaaaa
aaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaa
aaaaaaa
aaaaaaa
aaaaaaa
aaaaaa
aaaaaaa
aaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaa
aaaaaaaaaa
aaaaaaaa
aaaaa
aaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaa
aaaaa
aaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaa
aaaaaaaa
aaaaa
aaaaa
aaaaaa
aaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaa
aaaaa
aaaaaaaaa
aaaa
aaaaaa
aaaaa
aaaaa
aaaaaaaaa
aaaaaa
aaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaa
aaaaaaaa
aaaaaa
aaaaa
aaaaaa
aaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaa
aaaa
aaaaaaa
aaaaaaa
aaaaaa
aaaaaaa
aaaaaaa
aaaaaaa
aaaaaaa
aaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaa
aaaaaaaaaa
aaaaaaa
aaaaaa
aaaaa
aaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaa
aaaaaaaaa
aaaaaaa
aaaaa
aaaaa
aaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaa
aaaaaaaaaa
aaaaaaa
aaaaaa
aaaaa
aaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaa
aaaaaaaaaa
aaaaaaa
aaaaaaa
aaaaa
aaaaaa
aaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaa
aaaaaaaaa
aaaaaaa
aaaaaaa
aaaaa
aaaaa
aaaaaa
aaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaa
aaaaaaaaaaaaaa
aaaaaaaaaa
aaaaaaaa
aaaaaaa
aaaaa
aaaaa
aaaaa
aaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaa
aaaaaaaa
aaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaa
aaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaa
aaaaaa
aaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaa
aaaaaaaaaaaa
aaaaaaaaa
aaaaaaaaa
aaaaaaaa
aaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaa
aaaaaa
aaaaaaaa
aaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaa
aaaaaaaaaaaa
aaaaaaaaa
aaaaaaaa
aaaaaaa
aaaaaa
aaaaaa
aaaaa
aaaaaa
aa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaa
aaaaaa
aaaaaaaaa
aaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaa
aaaaaaaaaa
aaaaaaaa
aaaaaaa
aaaaaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaa
aaaaa
aaaaaaa
aaaaaaaaaa
aaaaaaaaaaaaaaa
aaaaaaaaaaaaaa
aaaaaaaaaaaa
aaaaaaaaa
aaaaaaa
aaaaaaa
aaaaa
aaaaa
aaaaa
aaaaaa
aaaaa
aaaaaaa
aaaaaaa
aaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaa
aaaaaa
aaaaaaaaa
aaaaaaaaaaaa
aaaaaaaaaaaaaaa
aaaaaaaaaaaa
aaaaaaaaaa
aaaaaaa
aaaaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaaa
aaaaaaa
aaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaa
aaaaaaa
aaaaaaaaaa
aaaaaaaaaaaaa
aaaaaaaaaaaaa
aaaaaaaaaa
aaaaaaaa
aaaaaaa
aaaaaa
aaaaa
aaaaa
aaaaa
aaaaaa
aaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaa
aaaaaaaaaa
aaaaaaaaaaaaaa
aaaaaaaaaaaaa
aaaaaaaaaa
aaaaaaaa
aaaaaaa
aaaaaaa
aaaaa
aaaaa
aaaa
aaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaa
aaaaaaaaaaaaaa
aaaaaaaaaaaaaa
aaaaaaaaaa
aaaaaaaa
aaaaaaa
aaaaa
aaaaa
aaaaaaaa
aaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
tD = 0.40
aaaaaaaaaaaaaaa
aaaaaaaaaaaa
aaaaaaaaaaaaaaa
aaaaaaaaaaaaaa
aaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaa
aaaaaa
aaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaa
aaaaaaa
aaaaaaa
aaaaaaa
aaaaaaa
aaaaaaaa
aaaaaaaaa
aaaaaaaaa
aaaaaaaaa
aaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaa
aaaaa
aaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaaa
aaaaaa
aaaaaa
aaaaaa
aaaaaaa
aaaaaaa
aaaaaaa
aaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaa
aaaaa
aaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaa
aaaaa
aaaaaaaaa
aaaaaa
aaaaa
aaaaa
aaaaa
aaaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaaa
aaaaa
aaaaa
aaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaa
aaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaa
aaaa
aaaaaaa
aaaaaaa
aaaaaaa
aaaaaaa
aaaaaaa
aaaaa
aaaaaa
aaaaa
aaaaaa
aaaaa
aaaaaa
aaaa
aaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaa
aaaaa
aaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaa
aaaa
aaaaaa
aaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaa
aaaaa
aaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaa
aaaaa
aaaaaa
aaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaa
aaaaa
aaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaa
aaaaa
aaaaaa
aaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaa
aaaaa
aaaaa
aaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaa
aa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaa
aaaaaaaaa
aaaaaaaa
aaaaaaa
aaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaa
aaaaaaaa
aaaaaaa
aaaaaaa
aaaaaaa
aaaaa
aaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaa
aaaaaaaa
aaaaaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaa
aaaaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaaa
aaaaa
aaaaaa
aaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaa
aaaaaaa
aaaaa
aaaaa
aaaaaa
aaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaa
aaaaaa
aaaaa
aaaaa
aaaaaa
aaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaa
aaaaaa
aaaaaa
aaaaa
aaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaa
aaaaaa
aaaaa
aaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
tD = 0.60
aaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaa
aaaaaaa
aaaaa
aaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaa
aaaaaaaaa
aaaaaaaaa
aaaaaaaaaa
aaaaaaaaaaaa
aaaaaaaaaaaaaa
aaaaaaaaaaaaaaa
aaaaaaaaaaaaaaa
aaaaaaaaaaaa
aaaaaaaa
aaaaaa
aaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaa
aaaaa
aaaaaaa
aaaaaaa
aaaaaaa
aaaaaaa
aaaaaaaaa
aaaaaaaaaa
aaaaaaaaaa
aaaaaaaaaaa
aaaaaaaaaaa
aaaaaaaaaa
aaaaaaaa
aaaaaaaa
aaaaa
aaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaa
aaaaa
aaaaa
aaaaaa
aaaaa
aaaaa
aaaaa
aaaaaaa
aaaaaaa
aaaaaaa
aaaaaaaaa
aaaaaaaaa
aaaaaaaaa
aaaaaaaaa
aaaaaaaaa
aaaaaaaaa
aaaaaa
aaaaa
aaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaa
aaaaaaa
aaaaaaa
aaaaa
aaaaa
aaaaaa
aaaaa
aaaaa
aaaaaa
aaaaaaa
aaaaaaa
aaaaaaa
aaaaaaa
aaaaaaa
aaaaaaa
aaaaaaa
aaaaaaa
aaaaaa
aaaaa
aaaaa
aaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaa
aaaaa
aaaaa
aaaaaa
aaaaa
aaaaa
aaaaaa
aaaaaaa
aaaaaaa
aaaaaa
aaaaaaa
aaaaaaa
aaaaaaa
aaaaaa
aaaaaaaa
aaaaa
aaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaa
aaaaaaa
aaaaa
aaaaaa
aaaaa
aaaaa
aaaaa
aaaaaaa
aaaaaa
aaaaaa
aaaaaa
aaaaa
aaaaa
aaaaa
aaaaaaa
aaaaaaa
aaaaaa
aaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaa
aaaaaaa
aaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaaaa
aaaaaaa
aaaaa
aaaaaa
aaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaa
aaaaa
aaaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaaaaaaa
aaaaaaaa
aaaaa
aaaaa
aaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaa
aaaaa
aaaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaaa
aaaaaaaaaaaaa
aaaaaaa
aaaaaa
aaaaa
aaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaa
aaaaaaa
aaaaaaaaaa
aaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaa
aaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaa
aaaaa
aaaaaaa
aaaaaaaa
aaaaaaaaaa
aaaaaaaaaaa
aaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaa
aaaaaaaaa
aaaaaaaaa
aaaaaaa
aaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaa
aaaaa
aaaaa
aaaaaaa
aaaaaaa
aaaaaaaa
aaaaaaa
aaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaa
aaaaaaaa
aaaaaaa
aaaaaaaa
aaaaaa
aaaaa
aaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaa
aaaaa
aaaaa
aaaaaaa
aaaaaaa
aaaaaa
aaaaaaaa
aaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaa
aaaaaaa
aaaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaaa
aaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaa
aaaaaa
aaaaa
aaaaaa
aaaaaaa
aaaaa
aaaaa
aaaaa
aaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaa
aaaaaaa
aaaaa
aaaaa
aaaaa
aaaaaa
aaaaaaa
aaaaaaaa
aaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaaa
aaaaa
aaaaaaaaaaaaaaaaaaaaaaa
aaaaaaa
aaaaa
aaaaa
aaaaa
aaaaaa
aaaa
aaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaa
aaaaaa
aaaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaaa
aaaaaaaaaaaaaaaaaaaaaa
aaaaaa
aaaaa
aaaaa
aaaaaa
aaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaa
aaaaa
aaaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaaa
aaaaaaaaaaaaaaaaaa
aaaaaaaaa
aaaaa
aaaaa
aaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaa
aaaaa
aaaaa
aaaaa
aaaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaaaaaaaaaaaa
aaaaaaaa
aaaaa
aaaaa
aaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
tD = 0.80
aaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaa
aaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaa
aaaaaaaaa
aaaaaaaaa
aaaaaaaaaaa
aaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaa
aaaaa
aaaaaaa
aaaaaaa
aaaaaaa
aaaaaaaa
aaaaaaaaaa
aaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaa
aaaaa
aaaaa
aaaaaa
aaaaa
aaaaa
aaaaaa
aaaaaaa
aaaaaaaa
aaaaaaaaa
aaaaaaaaaaaa
aaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaa
aaaaaaa
aaaaaaaa
aaaaa
aaaaaa
aaaaa
aaaaa
aaaaaa
aaaaaaa
aaaaaaaa
aaaaaaaaaa
aaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaa
aaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaa
aaaa
aaaaaaaaaaa
aaaaa
aaaaa
aaaaaa
aaaaaaa
aaaaaaaa
aaaaaaaaaa
aaaaaaaaaaaa
aaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaa
aaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaa
aaaaaaa
aaaaa
aaaaa
aaaaa
aaaaaaaa
aaaaaaa
aaaaaaaaa
aaaaaaaaaaaa
aaaaaaaaaaaaaa
aaaaaaaaaaaaaa
aaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaa
aaaaa
aaaaa
aaaaa
aaaaaaa
aaaaaaa
aaaaaaaaa
aaaaaaaaaaa
aaaaaaaaaaaaa
aaaaaaaaaaaaa
aaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaa
aaaaa
aaaaa
aaaaaa
aaaaaaa
aaaaaaaa
aaaaaaaaaa
aaaaaaaaaaaa
aaaaaaaaaaaaa
aaaaaaaaaaaa
aaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaa
aaaaa
aaaaa
aaaaaaa
aaaaaaaa
aaaaaaaaa
aaaaaaaaaaa
aaaaaaaaaaaa
aaaaaaaaaaaa
aaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaa
aaaaaaaa
aaaaaaaaaa
aaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaa
aaaaaaaaaa
aaaaaaaaaaaaa
aa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaa
aaaaaa
aaaaaaa
aaaaaaaaaa
aaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaa
aaaaa
aaaaa
aaaaaaa
aaaaaaa
aaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaa
aaaaaa
aaaaaaa
aaaaaaaaaa
aaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaa
aaaaaaaaaa
aaaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaa
aaaaa
aaaaaa
aaaaaaa
aaaaaaaaaa
aaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaa
aaaaaa
aaaaa
aaaaaa
aaaaa
aa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaa
aaaaa
aaaaaa
aaaaaaa
aaaaaaaaa
aaaaaaaaaaaa
aaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaa
aaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaa
aaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaa
aaaaa
aaaaa
aaaaaaa
aaaaaaaa
aaaaaaaaaa
aaaaaaaaaaaaaa
aaaaaaaaaaaaaaa
aaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaa
aaaaa
aaaaaa
aaaaaaa
aaaaaaaaa
aaaaaaaaaaaa
aaaaaaaaaaaaaa
aaaaaaaaaaaaaa
aaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaa
aaaaaa
aaaaa
aaaaaaa
aaaaaaa
aaaaaaaaa
aaaaaaaaaaaa
aaaaaaaaaaaaa
aaaaaaaaaaaa
aaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaa
aaaaaa
aaaaaa
aaaaaaa
aaaaaaaa
aaaaaaaaaa
aaaaaaaaaaaa
aaaaaaaaaaaaa
aaaaaaaaaaa
aaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
tD = 0.88
aa a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a aaa a a a a a a a a a a a a a a a a a a a a a a aaa
aaaaa
aa
a
aaa
a
a
a
a
a
a
aa
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aa
aa a a a a a a a aaa a a a a a a a a a a aa a a a a a a a a a a a a a a a a a a a aa a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a aa a
aaa
aaaaaaaa
aaa
a
aa
a
a
a
aaa
a
a
a
a
a
a
aaa
a
a
a
a
a
a
a
a
a
a
a
a
aaa
a
a
a
a
a
a
a
a
aa
aa a a a aaaa a a a aa a a a a a aaa a a a a a a aaa a a a a a a a a a a a aaa a a a a a a a a a a a a a a a a a a a a a a a a a a a a aaa a a a a a a a a a a a a a a a a a aa
aaa
aa
aa
aa
a
aa
a
aa
a
a
aa
a
a
a
aaa
a
a
a
a
a
aa
a
a
a
a
a
a
a
aaa
a
a
a
a
a
aaa
a
a
a
aa
aa a a aaa a aaa
a a aaaa a a aa a a a a aaa
a a a a a aa a a a a a a a a aa a a a a a a a a a a a a a a a a aa a a a a a a a a a a a a a a a a a a a a a a a a a a a aa a a a aaaaaaaaa
aa
a
aa
a
aaa
a
aa
a
a
a
aa
a
a
a
a
aaa
a
a
a
a
aaa
a
a
a
aaa
a
a
a
aa
a
a
aa
aa a aaaa aaaaaaa a aa
a a aa aa a aa a a a aaa
a a a a aa a a a a a a aa a a a a a a a a a a aaa a a a a a a a a a a a a a a a a a a a aaa a a a a a a a a a a a a a a a a aaaaaa
aa
aaa
aaa
aaa
a
aaa
a
a
aa
a
a
a
aaa
a
a
a
aa
a
a
a
aaa
a
a
aa
a
a
aa
a
aa
aaaaaaaaaaa
aaaaa aaaaaaaaaa
a a aaa a aaa
a a a aaaa a a a aa a a a a a a a a aa a a a a a a a a a a a a a aaa a a a a a a a a a a a a a a a a a a aa a a a a a a a a
aaa
aa
aaa
aaa
aaa
a
aa
a
a
aaa
a
a
aa
a
a
aaa
a
a
aaa
a
aaa
a
aaa
aaa
aa
aaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaa aa
a a aaaa a aaa
a a a aaaa a a a a aa a a a a a a a a a a aa a a a a a a a a a a a a a a a a aa a a a a a a a a a a a a aaa a a a
aaa
aaa
aa
a
aa
a
aaa
a
aaa
a
aaa
a
aaa
a
a
aa
a
aaa
a
aa
a
aa
a
aa
aa
aaaa
aaaaa
aaaaaaaa
aaaaaaaaaaaaaaa
aaaaa aaa
a a aa a a a aaaa a a a aa a a a a a a a aaa a a a a a a a a a a a aaa a a a a a a a a a a a a a a aaa a a a a a a a aa
aaaa
aa
a
aa
a
aaa
a
aa
a
a
aa
a
aaa
a
aaa
a
aa
a
aa
a
aa
aaa
aaaa
aaaaaa
aaaaaaa
aaaaa
aaaaa
aaaaaaaaaaaaa aaaaaa a aaa
a a aaaa a a aaa
a a a a a aa a a a a a a a a a aa a a a a a a a a a a a a a a aa a a a a a a a a a a a a aa a a aaa
aaa
aa
a
aaa
aaa
a
aaa
a
aa
a
aaa
aaa
aaa
aaa
aa
aaaaaa
aaa
aaaaaaaaa
aa
aaaaaaa
aaaaa
aaaaa
aaaaaaaaaaaaaa aa
a a aaaa a aaa
a a a a aa a a a a a a a aa a a a a a a a a a a a aaa a a a a a a a a a a a a a aaa a a a a a a a aaa
a
aa
a
aa
a
aaa
aaa
a
aa
a
aaa
aaa
aaaaa
aaaaaaa
aaa
aa
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aa
a
a
a
a
a
a
a
a
a
aaa
a
a
aaa
a
aa
aaaaaaaaaa aa a a a a a aaa a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a aa a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a
a
aa
a
a
a
a
a
a
a
aa
a
a
a
a
a
a
aaa
a
a
a
a
aaa
a
a
a
aa
a
a
aaa
a
aa
a
aa
aaa
aa
aa
aaaaa
aaaaaa a a a aa a a a a a aa a a a a a a a a a a a a a a aa a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a
aaaaa a a aa a a a a a a a
a
aa
a
a
a
aaa
a
a
a
aa
a
a
a
aaa
a
a
aaa
a
a
aa
a
aaa
a
aa
a
aa
aaa
aaaaaaaaaa
aaaaa
aaaaaaa
a a aa a a a a aa a a a a a a a a aaa a a a a a a a a a a a a a a a a a a a a a a a a a aa a a a a a a a a a a a a a a a a a a a aaaaaaaaaa a
a a aa a a a aa
aa
a
aaa
a
a
aaa
a
aaa
a
aaa
a
aaa
a
aa
a
aaa
aaa
aa
aaa
aa
aaaaaaaa
aaaaaaaaaaaaaa a a aa a a a aaa
a a a a a aaa a a a a a a a a a a a a aaa a a a a a a a a a a a a a a a a a a a a a a a a a a a a aa a a aaaaa
aaaaaaaaaaa a aaa
a aa
aa
a
aa
a
aaa
a
aa
a
aaa
a
aa
a
aaa
aaa
aa
a
aa
aaaaa
aa
aa
aaaa
aaaaaaa
aaaaaaa aaaa aa a a a aa
a a a a aaa a a a a a a a a aa a a a a a a a a a a a a a a a a a a aa a a a a a a a a a a a a a a a a a a aaaa
aa
aaaaaa
aaaaa
aaa
aaaa aa
aa
aaa
aaa
aaa
aaa
aaa
aaa
aaa
aa
aaa
aa
aaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaa a a aaa
a a a aaaa a a a a aaa a a a a a a a a a a a aa a a a a a a a a a a a a a a a a a a a aaa a a a a a a a a
aaa
aa
aaa
aa
aa
aaaaaaa
aaa
aaa
aa
aa
aa
aaa
aaa
aa
a
aa
aaa
aa
aaa
aa
aa
aaaaaaaaaaaaaa
aaaaa
aaaaaaaaaaaaa a aa a
a a aaaa a a a aa a a a a a a a a aa a a a a a a a a a a a a a a a aa a a a a a a a a a a a a a a a aaa a a
aaa
aa
aaa
aaaaaaaaaaaaaa
aaaaaaa
aaaa
aa
aaa
aa
aaa
aa
aa
aaaaa
aa
aa
aa
aa
aaaaaaaaa
aaaaaaaa
aaaaaaaa aaa
a a aa a a a a aa a a a a a a aa a a a a a a a a a a aaa a a a a a a a a a a a a a a a aaa a a a a a a a a a aa
aa
a
aa
aaa
aaa
aa
aaaaaaaaaaaa
aaaaa
a
aaaaaa
aa
aa
aaaaaaaaa
aa
aa
aa
aaaaaaaaa
aaaaa
aaaaaaaaaaaaaaaa
a aaaa a a aa a a a a a aa a a a a a a a aaa a a a a a a a a a a a a a aa a a a a a a a a a a a a a aaa a a a a
aaa
aaa
aaa
aaa
aaa
aa
aaaaaaa
aaaaaaaaaa
aaa
aaaaaaaaaaaaaaaaaa
aa
aa
aaaaaaaaaaaaaaaaaaaaaaa aaa a aa a a a aa
a a a a aa a a a a a a aa a a a a a a a a a a aa a a a a a a a a a a a a a a aaa a a a a a a a a a a aaaa
aaa
aaa
a
aa
a
aa
a
aa
a
aa
aaa
aa
aa
aa
aa
aa
aaa
tD = 0.90
aa a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a aaa a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a aa
aaaaaa
aaa
a
aa
aa a a a a a a a aa a a a a a a a a a aa a a a a a a a a a a a a a a a aaa a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a aaa a a a a a
aaaaaa
aa
aaa
aa
a
aaa
a
a
aaa
a
a
a
a
a
a
aa
a
aa a a a aa aa a a aaa
a a a a aaa a a a a a a aa a a a a a a a a a a aa a a a a a a a a a a a a a a a a a a a a a a a aa a a a a a a a a a a a a a a a a a a a a a a a a a a a a aaaaa
aa
aaa
aa
aaa
a
aaa
a
a
aaa
a
a
a
a
aaa
a
a
a
a
a
a
a
a
aa
a
aa a a aaa a aaa
a a aa aa a aaa
a a a aaaa a a a aaa a a a a a a a aa a a a a a a a a a a a a a aa a a a a a a a a a a a a a a a a a a a a a a a a a a aa a a a a a a a a a a a aa
aaaa
aa
aaa
aaa
a
aa
a
a
aaa
a
a
a
a
aa
a
a
a
a
a
aaa
a
a
a
a
a
aa
aa a aaaaaaaaaaaaaa a a aa
a a aaaa a aaa
a a a a aaa a a a a aaa a a a a a a a a aaa a a a a a a a a a a a a a a a a a aaa a a a a a a a a a a a a a a a a a a aa a a a a a
aaa
aa
aaa
aaa
a
aa
a
a
aaa
a
a
aaa
a
a
a
a
aa
a
a
a
a
aaa
a
a
a
aa
aaaaaaaaa aaaaaaaaaaaaaaaaa a aa a
a a aa a a a aaaa a a a aa a a a a a a aaa a a a a a a a a a a a aaa a a a a a a a a a a a a a a a a a a aaa a a a a a a a a a a a aa
aaaa
aa
a
aa
a
aaa
a
aaa
a
a
aa
a
a
a
aaa
a
a
a
aa
a
a
a
aa
a
a
aa
aaaa
aaa
aaaaaaaaaaaaaaaaaaaaaaaaaa a aaa
a a aaaa a a aa a
a a a a aaa a a a a a a a a aa a a a a a a a a a a a a a a a aa a a a a a a a a a a a a a a a aaa a a a a aaa
aaa
aa
a
aaa
a
aa
a
a
aaa
a
a
aaa
a
a
aaa
a
a
aa
a
aaa
a
aa
aaaaaaaaa
aaaaa
aaaaaaaaaaaaaaaaaaaaaaa aa
a a aaaa a aaa
a a a a aaa a a a a a aa a a a a a a a a a a a aa a a a a a a a a a a a a a a a a aaa a a a a a a a a a a a aa
aa
aa
a
aaa
a
aa
a
a
aa
a
a
aaa
a
aaa
a
a
aa
a
aaa
a
aa
aa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaa
aa aa
a a aaaa a a aa a a a a a aa a a a a a a a a aaa a a a a a a a a a a a a a aa a a a a a a a a a a a a a a aaa a a a a a
aa
a
aa
a
aaa
a
aa
a
a
aa
a
a
aaa
a
aaa
a
aa
a
aa
aaa
aa
aaa
aaaaaaaaaaa
aa
aaaaaaa
aaaaa
aaaaaaaaaaaaaaaaaaa a aaa
a a aaaa a a aaa
a a a a a a aa a a a a a a a a a a aaa a a a a a a a a a a a a a a a aa a a a a a a a a a a a aaaa
a
aa
a
aaa
a
aa
a
a
aa
a
aaa
a
aa
a
aaa
aa
aaaa
aa
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aaa
a
a
a
a
a
a
a
a
a
aaa
a
a
a
aaa
a
aaa
aa
aaaa
aaaaaa a a aa a a a a a a a a aa a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a
aa
aa
a
a
a
a
a
a
aaa
a
a
a
a
a
aaa
a
a
a
a
aaa
a
a
a
aa
a
a
aaa
a
aa
a
aa
aaa
aa
aaaa
aaaaa
aaa
aaaa
a a a aa a a a a a a aaa a a a a a a a a a a a a a a a a a a a a aa a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a aaaaa
a aa
aa
a
a
a
aa
a
a
a
aaa
a
a
a
aa
a
a
a
aa
a
a
aa
a
a
aa
a
aaa
aaa
aa
aaa
aa
aa
aaaaaaa
aaaa aaaaaa
a a aaaa a a a aaa a a a a a a a a a a aa a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a aa
aa
aa
aaaaaa aa
aa
a
aaa
a
a
aa
a
a
aaa
a
aaa
a
aa
a
aaa
a
aa
a
aa
aaa
aaaaa
aa
aa
aa
aaaaaaaaaa
aaaaaaa a a aa a
a a a aa a a a a a a a aaa a a a a a a a a a a a a a a a a aaa a a a a a a a a a a a a a a a a a a a a a a a a a a a aa
aaaaa
aa
aaaaaaaa
aa
a
aa
a
aaa
a
aa
a
aaa
aaa
a
aa
a
aa
aaa
aaa
aa
aaaaa
aa
aa
aaaaaaa
aaaaaaaaaa aaa a aa a a a aaa
a a a a aaa a a a a a a a a a a aa a a a a a a a a a a a a a a a a a a a a a a aaa a a a a a a a a a a a aaa
aaa
aaaaa
aaaaaaaaaa
aa
aaa
aaa
aaa
aaa
aaa
aaa
aa
aaa
aa
aaa
aa
aa
aaaaaa
aa
aaaaaaaaaa
aaa
aaaaaaa a a a aa a a a a aa a a a a a a a aaa a a a a a a a a a a a a a aaa a a a a a a a a a a a a a a a a a a a a aaa a a a
aa
a
aa
aaa
aaaaa
aaaaaaa
aa
aa
aa
aaa
aaa
aa
aaa
aaaaa
aaaaa
aa
aa
aaaaaaaa
aa
aaaaa
aaaaa
aaa
aaaaa aa
a a aaaa a a aa
a a a a a aa a a a a a a a a a a aa a a a a a a a a a a a a a a a a a aaa a a a a a a a a a a a a a aa
aa
a
aa
a
aa
a
aa
aaa
aa
aa
aaaaa
a
aaaa
aa
aaa
aa
aaaaa
aa
aaaaaaaaaaaaaaaaaaaa
aaaaa
aaaaaaaaaaaaaa aaa
a a aaaa a a aaa a a a a a a a aa a a a a a a a a a a a a aaa a a a a a a a a a a a a a a a a aa a a a a a a a
aaa
a
aa
a
aa
a
aaa
aa
aaa
aa
aa
aa
aaa
aaaaaa
aa
aa
aaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaa
aaaa aaaaaaaaa a a aaa
a a a aaaa a a a a aa a a a a a a a a a aa a a a a a a a a a a a a a a a aaa a a a a a a a a a a a a aa a a
aa
a
aaa
a
aa
a
aaa
aaa
aaa
aa
aaaaaaaa
aaa
aa
aaaaaaaaaaaaaa
aa
aa
aaaaaaa
aaaaa
aaaaaaaaaaaa
aaaa a a aa a a a aa a a a a aaa a a a a a a aaa a a a a a a a a a a a a aa a a a a a a a a a a a a a a a aa a a a a a a a aa
aa
a
aa
a
aaa
a
aa
a
aaa
a
aa
a
aa
aaa
aa
aa
aa
tD = 0.92
aa a a a a a a a a a a a a a a a a a a a aa a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a aaa
aaaaaa
aaa
a
aa
a
a
a
a
a
a
a
aaa
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aa
aa a a a a a a aa a a a a a a a aaa a a a a a a a a a a aa a a a a a a a a a a a a a a a a a a a a a a a a a a aa a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a aa
aaaa
aa
a
aa
a
a
aaa
a
a
a
a
aaa
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aaa
a
a
a
a
a
a
a
a
a
a
a
a
aa
aa a a a aaa a a a aa
a a a a aa aa a a a aaa a a a a a a a aaa a a a a a a a a a a a a a aa a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a aaa a a a a a a a a a
aa
aa
aaa
aaa
a
aaa
a
a
a
a
aa
a
a
a
a
a
a
a
a
aa
a
a
a
a
a
a
a
a
a
a
a
aaa
a
a
a
a
a
aa
aa a aaaa a aa a a aaa
a a aaaa a a aa a a a a aaa
a a a a a aa a a a a a a a a a aaa a a a a a a a a a a a a a a a a a a a a aa a a a a a a a a a a a a a a a a a a a a a a a a a a aa aaa
aaa
aaa
a
aaa
a
a
aaa
a
a
a
a
a
aa
a
a
a
a
a
a
a
a
aa
a
a
a
a
a
a
a
aa
a
a
a
aa
aa a aaaaaaaaaaa aa
a a aa a a aaaa a aa a
a a a aaa a a a aaa a a a a a a aaa a a a a a a a a a a a a aa a a a a a a a a a a a a a a a a a a a a a a a a aa a a a a a a a a a a a a aa
aa
aaa
a
aaa
a
aaa
a
a
a
a
aa
a
a
a
a
a
aaa
a
a
a
a
a
a
aa
a
a
a
a
aaa
a
aa
aaaaaaaaaaaaaaaaaaaaaaaaaa aa a
a aaaa a aaa
a a a aaaa a a a a aa a a a a a a a a aaa a a a a a a a a a a a a a a a a a aa a a a a a a a a a a a a a a a a a a aaa a a a a a
aaa
a
aa
a
a
aa
a
a
a
aa
a
a
a
a
aaa
a
a
a
a
aaa
a
a
a
a
aaa
a
a
aa
a
aa
aaaa
aaaaaaaaaaaaaaaaaaaaaa aaaaaaa aaa
a a aa a a a aa a a a a aaa a a a a a a aa a a a a a a a a a a a a aa a a a a a a a a a a a a a a a a a a aaa a a a a a a a a a a a a aa a
aaa
a
aaa
a
aaa
a
a
a
aa
a
a
a
a
aaa
a
a
a
aaa
a
a
aaa
a
aaa
aa
aaaaaaaaaaaaaa
aaaaa
aaaaaaaaaaaaaaaaa aaa a aa a a aaa
a a a aa aa a a a aaa a a a a a a a a aa a a a a a a a a a a a a a a a aa a a a a a a a a a a a a a a a a aa a a a a a a aa
aaa
a
aaa
a
a
aa
a
a
a
aaa
a
a
a
aa
a
a
a
aaa
a
aaa
a
aa
aa
aaaaaaaa
aaaaaaa
aaaaa
aaaaa
aaaaaaaaaaaaa aaaaaa a a aa a a a aa a a a a aa a a a a a a aaa a a a a a a a a a a a aa a a a a a a a a a a a a a a a a aaa a a a a a a a a a a aaa a a
a
aa
a
a
aaa
a
a
aa
a
a
a
aaa
a
a
aaa
a
aaa
a
aaa
aaaa
aaa
aa
aaaaaaaaaaa
aaaaaaaaa
aaaaa
aaaaaaaaaaaaaaaaa aa a a aaa
a a a aa a a a a a aa a a a a a a a a aaa a a a a a a a a a a a a a a aa a a a a a a a a a a a a a a aaa a a a a a a aa
a
aa
a
a
aa
a
a
aaa
a
a
aaa
a
a
aa
a
aaa
a
aa
aaaa
aa
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aaa
a
a
a
a
a
a
a
a
a
aa
a
a
a
a
a
aa
a
a
a
aa
a
aaa
aa
aa
aaaaaaaaa a aa
a a a a a aa a a a a a a a a a a a a a a a a a a a a aaa a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a aa a aaaaaa a
a
aa
a
a
a
a
a
aaa
a
a
a
a
a
aa
a
a
a
a
aaa
a
a
a
aa
a
a
aaa
a
aa
a
aaa
aa
aaaaaaaaaaaaaaa
aaaa a a aa a a a a aaa a a a a a a a a a a aa a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a aa
a
aaaa
aaa aa
aa
a
a
aaa
a
a
a
aa
a
a
a
aa
a
a
aaa
a
a
aa
a
aaa
a
aa
a
aa
a
aa
aaaaa
aa
aaaaaaa
aaaaaaa aa
a a aaaa a a aa a a a a a a a aa a a a a a a a a a a a a a a a a a a a aa a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a aa
aaaaaaaaaa
aa
aa
a
aaa
a
aaa
a
aaa
a
aaa
a
aaa
aaa
a
aa
a
aa
aaa
aaaaa
aa
aa
aa
aa
aaaaa
aaaaaaa aaa a aa a a a aa a a a a a aaa a a a a a a a a a a aa a a a a a a a a a a a a a a a a a a a a a a a a a a a aa a a a a a a a a a a a a a a a
aaa
aaa
aa
aaaa
aaaa
aa
aaa
a
aa
a
aaa
a
aa
a
aaa
aaa
aaa
aa
aaa
aaaaa
aa
aa
aa
aa
aaaaaaa
aaaaaaa aaaaaa
a a aaaa a a aaa a a a a a a a aa a a a a a a a a a a a a a a a aaa a a a a a a a a a a a a a a a a a a a a a a a a aa a a a a a
a
aaa
aaa
aaaaa
aaaaaa
aa
aa
a
aaa
aaa
aa
a
aa
a
aa
aaa
aa
aaa
aa
aaaaaaaaaaaaaa
aaaaaaaaaaaaaaa aa
a a aa aa a a aa
a a a a a aa a a a a a a a a a a aa a a a a a a a a a a a a a a a a a a a a aa a a a a a a a a a a a a a a a a aaaa
a
aa
a
aaa
aaa
aaaaaaaaa
aa
aa
aaa
aa
aaa
aaa
aa
aaaaa
aa
aaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaa aaa
a a aaaa a a a aa a a a a a a a aa a a a a a a a a a a a a a a aa a a a a a a a a a a a a a a a a a a aa a a a a a a a a aa
aa
a
aaa
a
aaa
aaa
aaa
aa
aaaaa
aaaa
aa
aaaaa
aa
aaaaa
aa
aa
aaaaaaaaaa
aa
aaaaaaaaaa
aaaaaaaaaaa aa a a aaa
a a a aa a a a a a aaa a a a a a a a a a aa a a a a a a a a a a a a a a a a a aa a a a a a a a a a a a a a aa a a a aa
aa
a
a
aa
a
a
aa
a
aaa
a
aa
aaa
aa
aaa
aaaaaaaa
aa
aa
aa
aaaaaaaaaaaa
aa
aaaaaaa
aaaaa
aaaaaaaaaaaaa a a aa a a a aa a a a a aaa a a a a a a aaa a a a a a a a a a a a a aaa a a a a a a a a a a a a a a a aaa a a a a a a a a aaa
a
aa
a
a
aa
a
a
aaa
a
aaa
a
aa
a
aa
aaa
aaa
aaa
aa
aa
aa
aa
aa
aa
aa
aa
aaaaaaa
aaaaaaa
aaaaaaaaaaaaaaaaa aa a a a aa
a a a aaaa a a a a aa a a a a a a a a a a aa a a a a a a a a a a a a a a a aaa a a a a a a a a a a a a aa a a a a a
a
aaa
a
a
aa
a
a
aaa
a
aaa
a
a
aa
a
aaa
aaaaaa
tD = 0.94
aa a a a a a a a a a a a a a a a aaa a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a aa a a a a a a a a a a a a a a a a a a a a a a a
aaa
aa
aa
a
aaa
a
a
a
a
a
a
aaa
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aa
aa a a a a a aaa a a a a a a aaa a a a a a a a a a aa a a a a a a a a a a a a a a a a a a aaa a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a aa a aaa
aa
aaa
a
aaa
a
a
a
a
aaa
a
a
a
a
a
a
a
a
a
a
a
a
a
aa
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
aa
aa a a aaaa a a aaa
a a a aaaa a a a aaa a a a a a a aaa a a a a a a a a a a a aa a a a a a a a a a a a a a a a a a a a a a a a a a a a a aa a a a a a a a a a a a a a a a a a a a a a a aa
aa
a
aa
a
aaa
a
a
a
aa
a
a
a
a
a
a
a
aa
a
a
a
a
a
a
a
a
a
a
a
a
a
aa
a
a
a
a
a
a
a
a
aa
aa a aaaa a aa
a a aa aa a aa a a a aaa
a a a aaaa a a a a aa a a a a a a a a aa a a a a a a a a a a a a a a a aaa a a a a a a a a a a a a a a a a a a a a a a a a a aaa a a a a a a a a a aa
aaa
aaa
a
a
aaa
a
a
a
a
aa
a
a
a
a
a
a
a
a
aa
a
a
a
a
a
a
a
a
a
aaa
a
a
a
a
aa
aa aaaaaaaa aa
aaaaaaaa
a a aa a a aaaa a a aa a a a a aa a a a a a a aaa a a a a a a a a a a aa a a a a a a a a a a a a a a a a a a a a aa a a a a a a a a a a a a a a a a a aa a a a a a
a
aaa
a
aaa
a
a
aaa
a
a
a
a
a
aaa
a
a
a
a
a
a
aaa
a
a
a
a
a
a
aa
a
a
a
aa
aaaaaaaaaaaaaaaaa
aaaaa aa
aa aa a a aa a a a aa
a a a aaaa a a a aa a a a a a a a a aa a a a a a a a a a a a a a aaa a a a a a a a a a a a a a a a a a a aaa a a a a a a a a a a aaa a
a
aaa
a
a
aa
a
a
a
a
aa
a
a
a
a
a
aaa
a
a
a
a
a
aaa
a
a
a
a
aa
a
a
aa
aaaa
aaaaaaaaaaa
aaa
aaa
aaaaaaaaaaaa aa
a a aa aa a aa a a a a aa a
a a a a a aa a a a a a a a a a aaa a a a a a a a a a a a a a a a a aa a a a a a a a a a a a a a a aa a a a a a a a aa
a
aaa
a
a
aaa
a
a
a
aaa
a
a
a
a
aaa
a
a
a
a
aaa
a
a
aaa
a
aa
aaaaaaaaaaa
aaaaaaaa
aaaaaaaaaaaaaaaaaaaa aa a
a a aaa a a aa a a a a a aa a a a a a a a aa a a a a a a a a a a a a aaa a a a a a a a a a a a a a a a aaa a a a a a a a a a aa a a a a
a
a
aaa
a
a
a
aa
a
a
a
aaa
a
a
a
a
aa
a
a
a
aaa
a
a
aa
a
aa
aaaaaaaa
aaaaaaa
aaaaaaa
aaaaaaaa
aaa
aaaaaaaaaaa a aa a a a aa a a a aaa
a a a a a aaa a a a a a a a a aaa a a a a a a a a a a a a a a aaa a a a a a a a a a a a a aa a a a a a a a aaa
a
a
aaa
a
a
aaa
a
a
a
aa
a
a
a
aaa
a
a
aaa
a
aaa
aa
aaa
aa
aaaaaaaaaaaaa
aaaaaaa
aaaaa
aaaaaaaa
aaaaaaaaaa a
a aaaa a a aa
a a a a aaa a a a a a a aaa a a a a a a a a a a a aaa a a a a a a a a a a a a a a aaa a a a a a a a a aaa a a a aa
a
a
aa
a
a
a
aa
a
a
a
aa
a
a
a
aa
a
a
aaa
a
aa
aa
aa
a
a
a
a
a
a
a
a
a
a
a
a
aaa
a
a
a
a
a
a
a
a
a
aa
a
a
a
a
a
aaa
a
a
aa
a
a
aa
aaa
aa
aa
aaaaa
aaaa
a a aaaa a a a a a aaa a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a aaa a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a
aaaa
aa
a
a
a
a
a
aa
a
a
a
a
a
aaa
a
a
a
aaa
a
a
a
aa
a
a
aaa
a
aaa
aaa
aaa
aa
aa
aa
aaaaaaaaaaaa aa
a a a aa a a a a a aaa a a a a a a a a a a a a a aaa a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a aa a aaaa aaaaaaaa
aa
a
a
aaa
a
a
aaa
a
a
aaa
a
a
aaa
a
aaa
a
aa
a
aaa
aaa
aaa
aa
aa
aaaaaa
aaaaaaaaaaaaaa a a aaa
a a a a aa a a a a a a a a aaa a a a a a a a a a a a a a a a a a a a a a a a aaa a a a a a a a a a a a a a a a a a a a a a a a a a a a aa
aaa
aa
aa
aaa
aa
a
aaa
a
aaa
a
aaa
a
aaa
a
aa
a
aaa
aaa
aaa
aaa
aa
aaaaaaaaaaaaaa
aaaaaaaaaa aa a a aa a a a a aa
a a a a a a aa a a a a a a a a a a a a aaa a a a a a a a a a a a a a a a a a a a a a a a a a aaa a a a a a a a a a a a a aa
aa
a
aa
aaa
aa
aaa
aa
aaa
a
aa
a
aaa
aaa
a
aa
a
aa
a
aa
aaa
aaa
aa
aaaaaaaaaaaaaaaa
aaaaaaaaaaaaa a aa
a a a aaa a a a aaa a a a a a a a a aa a a a a a a a a a a a a a a a a a aaa a a a a a a a a a a a a a a a a a a aaa a a a a a a a
a
a
aa
a
aaa
aa
aaa
aaa
aa
aa
a
aa
a
aa
a
aa
a
aa
aaa
aaa
aa
aaaaa
aa
aaaaaaaaaa
aaaaaaaaaa
aaaaaaaa aa
a a aaaa a a aa a
a a a a a aaa a a a a a a a a a a a aa a a a a a a a a a a a a a a a a a a a aa a a a a a a a a a a a a aa a a aaa
a
a
aa
a
aaa
aaa
aaaaaa
aaaa
aaa
aa
aaa
aaaaa
aa
aaa
aa
aa
aaaaaaaaaaaa
aaaaaaa
aaaaaaa aaaaaa a aaa
a a a aa a a a a aaa a a a a a a a a aa a a a a a a a a a a a a a a a aa a a a a a a a a a a a a a a a aa a a a a a a a a aa a
aaa
a
aaa
a
aaa
a
aa
a
aa
aaaa
aaaa
aa
aaaaa
aa
aaaaaaa
aa
aa
aa
aa
aa
aa
aaaaaaaaaaaaaaaaaaaaaaa aaa
a a aaaa a a aaa
a a a a a aaa a a a a a a a a a a aaa a a a a a a a a a a a a a a a aaa a a a a a a a a a a aaa a a a a aa
a
aa
a
a
aaa
a
aaa
a
aaa
aaa
aaaa
aaaaaaaaaa
aa
aa
aa
aaaaaaaa
aa
aaaaaaa
aaaaa
aaaaaaaaaaaaaaaa a a aa
a a a aa a a a a a aa a a a a a a a a aa a a a a a a a a a a a a a a aa a a a a a a a a a a a a a aaa a a a a a a a a aa a aa
a
aaa
a
a
aa
a
a
aaa
a
aaa
a
aaa
aa
aa
aaa
aa
aa
aa
aa
aa
aa
aa
aaaaaaaaa
aaaaa
aaaaa
aaa
aaa
aaaaa aaa a aa a a a aa
a a a a aa a a a a a a aa a a a a a a a a a a a aa a a a a a a a a a a a a a a a aa a a a a a a a a a a aaa a a a a a aa
a
a
aaa
a
a
aa
a
a
a
aa
a
a
aaa
a
aaa
a
aa
aa
tD = 1.0
aaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaa
aaaaaaaaa
aaaaaaaaa
aaaaaaaaaa
aaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaa
aaaaaaaa
aaaaaaa
aaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaa
aaaaa
aaaaaaa
aaaaaaa
aaaaaaa
aaaaaaaa
aaaaaaaaa
aaaaaaaaaaaa
aaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaa
aaaaaaaaaaa
aaaaaaaa
aaaaaaa
aaaaa
aaaaa
aaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaa
aaaaa
aaaaa
aaaaaa
aaaaa
aaaaa
aaaaaa
aaaaaaa
aaaaaaa
aaaaaaaaa
aaaaaaaaaaa
aaaaaaaaaaaaaa
aaaaaaaaaaaaaaa
aaaaaaaaaaaa
aaaaaaaaaa
aaaaaaaa
aaaaaa
aaaaa
aaaaa
aaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaa
aaaaaaa
aaaaaaaa
aaaaa
aaaaaa
aaaaa
aaaaa
aaaaaa
aaaaaaa
aaaaaaaa
aaaaaaaaa
aaaaaaaaaaa
aaaaaaaaaaaaa
aaaaaaaaaaaa
aaaaaaaaaa
aaaaaaaa
aaaaaaa
aaaaa
aaaaa
aaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaa
aaaa
aaaaaaa
aaaaa
aaaaa
aaaaaa
aaaaa
aaaaaaa
aaaaaaa
aaaaaaaaa
aaaaaaaaaaa
aaaaaaaaaaa
aaaaaaaaaaa
aaaaaaaaa
aaaaaaa
aaaaaaa
aaaaa
aaaaa
aaaaa
aaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaa
aaaaaaa
aaaaa
aaaaa
aaaaaa
aaaaa
aaaaaa
aaaaaaaaa
aaaaaaaaa
aaaaaaaaaaa
aaaaaaaaaaa
aaaaaaaaa
aaaaaaa
aaaaaaa
aaaaa
aaaaa
aaaa
aaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaa
aaaaa
aaaaaa
aaaaa
aaaaaa
aaaaaa
aaaaaaa
aaaaaaaaa
aaaaaaaaa
aaaaaaaaa
aaaaaaaaa
aaaaaaaa
aaaaaaa
aaaaaa
aaaaa
aaaaaaaaaa
aaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaa
aaaaa
aaaaaa
aaaaa
aaaaaa
aaaaaaa
aaaaaaaa
aaaaaaaaa
aaaaaaaaa
aaaaaaaaa
aaaaaaaa
aaaaaaa
aaaaaaa
aaaaa
aaaaaa
aaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaa
aaaaa
aaaaa
aaaaaa
aaaaaaa
aaaaaaaa
aaaaaaaa
aaaaaaaaa
aaaaaaaaa
aaaaaaaaa
aaaaaaa
aaaaaa
aaaaa
aaaaa
aaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaa
aaaaaaaa
aaaaaaaaaa
aaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaa
aaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaa
aaaaa
aaaaaaa
aaaaaaaaaa
aaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaa
aaaaaaaaa
aaaaaaa
aaaaaa
aaaaa
aa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaa
aaaaaa
aaaaaaa
aaaaaaaaaa
aaaaaaaaaaaaa
aaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaa
aaaaaaaaaaaa
aaaaaaaa
aaaaaaa
aaaaaa
aaaaa
aaaaa
aaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaa
aaaaa
aaaaaa
aaaaaaa
aaaaaaaaa
aaaaaaaaaaaa
aaaaaaaaaaaaaa
aaaaaaaaaaaaa
aaaaaaaaaa
aaaaaaaa
aaaaaa
aaaaa
aaaaa
aaaaa
aaaaaa
aaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaa
aaaaa
aaaaa
aaaaaaa
aaaaaaaa
aaaaaaaaaa
aaaaaaaaaaaa
aaaaaaaaaaaa
aaaaaaaaaa
aaaaaaaa
aaaaaaa
aaaaa
aaaaa
aaaaaa
aaaa
aaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaa
aaaaa
aaaaaaa
aaaaaaa
aaaaaaaaa
aaaaaaaaaaa
aaaaaaaaaaa
aaaaaaaaaa
aaaaaaaaa
aaaaaaa
aaaaaa
aaaaa
aaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaa
aaaaaa
aaaaa
aaaaaaa
aaaaaaa
aaaaaaaaa
aaaaaaaaaa
aaaaaaaaaa
aaaaaaaaa
aaaaaaaa
aaaaaaa
aaaaa
aaaaa
aaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaa
aaaaa
aaaaa
aaaaaaa
aaaaaaa
aaaaaaaaaa
aaaaaaaaa
aaaaaaaa
aaaaaaaaa
aaaaaaa
aaaaaaa
aaaaaa
aaaaa
aaaaa
aaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaa
aaaaa
aaaaa
aaaaaa
aaaaaaa
aaaaaaaa
aaaaaaaaa
aaaaaaaaa
aaaaaaaaa
aaaaaaaaa
aaaaaaa
aaaaaa
aaaaa
aaaaa
aaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
![Page 25: A STUDY OF THE EFFECT OF MOBILITY RATIOS ON ...1. A study of the effect of mobility ratios on pattern displacement behavior It is well known, for unit mobility ratio, that the areal](https://reader031.vdocuments.net/reader031/viewer/2022040817/5e61c32167659d0b291248ba/html5/thumbnails/25.jpg)
19
Similar to the discussion above for a five-spot pattern, for a very favorable
mobility ratio, the displacing front is perpendicular to the borders of the pattern both
before and after the front passes the nearest corner. As a result, we see linear
displacement for some time until the front on the near-corner side approaches the
producer. We see these front shapes in Fig. 7.
When the front approaches the producer, the streamtubes narrow due to the
confinement of the pattern boundary and the well. And therefore, with the same flow rate,
the displacing fluid will break through relatively quickly in the streamtubes closest to the
producer. The front on the far no-flow side progresses more slowly. This streamline
distribution does not change greatly as the mobility ratio becomes more favorable.
Sweepout of the pattern is not complete at breakthrough. For instance, a small amount of
the resident fluid remains along the right hand boundary as shown in Fig 7a at tD = 0.94 .
If the length of staggered line drive is increased (increasingd / a ), then the
displacement will approach linear flow and the sweep efficiency will approach unity. The
proportion of unswept area decreases as d / a increases.
1.6 CONCLUSIONS
Pattern performance changes with mobility ratio. For unit mobility ratio,
unfavorable mobility ratios and some favorable mobility ratios (M > 0.3), a staggered-
line-drive pattern has higher areal sweep efficiency than a five-spot pattern. However, for
very favorable mobility ratios (M < 0.3), a five-spot pattern has better sweep efficiency
than a common staggered-line-drive.
The reason for this behavior is the change of streamline and pressure distributions
with mobility ratios. For very favorable mobility ratios, the displacing front is near an
isobar and intersects the pattern boundary at 90o. This causes the fronts at times near
breakthrough to become radial around the producer for a five-spot pattern. This
displacing front shape is due to the symmetry of the five spot pattern.
![Page 26: A STUDY OF THE EFFECT OF MOBILITY RATIOS ON ...1. A study of the effect of mobility ratios on pattern displacement behavior It is well known, for unit mobility ratio, that the areal](https://reader031.vdocuments.net/reader031/viewer/2022040817/5e61c32167659d0b291248ba/html5/thumbnails/26.jpg)
20
For a staggered line drive, the displacing front is also perpendicular to the border
of the pattern. However, because the pattern is not symmetric, sweepout at breakthrough
is not complete. Only in the limit of very large d/a will the areal sweep efficiency
approach 1.
The simulation results are quite close to the analytical solutions for unit mobility
ratio. The results are also very close to the experimental data, Dyes et al. (1954), after
breakthrough at various mobility ratios. We find physical dispersion in the Dyes et al.
experimental results that cause earlier breakthrough time.
We observed some numerical dispersion in our simulation results. For very
favorable mobility ratios, the dispersion is small. We corrected the simulation results by
fitting the fractional flow curve with a second order polynomial to estimate breakthrough
time.
![Page 27: A STUDY OF THE EFFECT OF MOBILITY RATIOS ON ...1. A study of the effect of mobility ratios on pattern displacement behavior It is well known, for unit mobility ratio, that the areal](https://reader031.vdocuments.net/reader031/viewer/2022040817/5e61c32167659d0b291248ba/html5/thumbnails/27.jpg)
21
Appendix A—Effect of Grid-Block and Time-Step Size
The number and size of grid blocks and time steps has a strong effect on
simulation results. The extent of the effect is related to the mobility ratio. This study is
performed on five-spot pattern cases and the results are shown in Fig. A. However, the
conclusions drawn from this study do not lose any generality.
A.1 EFFECT OF TIME-STEP SIZE
Figure A plots the fractional flow at the producer versus time; changes in this
curve versus the number of grid blocks or time step are a convenient measure of the
accuracy of a solution. Comparing the effect of time-step size between M=1 and M=1/20
in Fig A, it is obvious that the effect of time-step size is much stronger in the very
favorable mobility ratio case (M=1/20). For unit mobility ratio (M=1), simulation results
stop changing at 50 time steps. However, for M=1/20, the difference between simulation
results and the converged solution for the same number of time steps is much greater. The
reason is described below.
For unit mobility ratio, the pressure field as well as the streamline distribution
does not change with time. If we map the analytical solution along the streamline, then,
time-step size does not have any effect on the simulation result. However, I mapped the
numerical solution to be consistent with the very favorable mobility rstio cases in which
mapping the analytical solution is not applicable.
As mobility ratio departs from unity, the pressure field changes more intensively.
For a very favorable mobility ratio, we must solve the pressure field enough times to get
a converged solution. We can see that, for M=1/20, the simulation result converges at
100 pressure solves (time steps).
For the calculations presented in the main portion of the report, I used 100 time
steps for unit mobility ratio and 300 time steps for M=1/20. The simulation result is
converged in terms of number of time steps.
![Page 28: A STUDY OF THE EFFECT OF MOBILITY RATIOS ON ...1. A study of the effect of mobility ratios on pattern displacement behavior It is well known, for unit mobility ratio, that the areal](https://reader031.vdocuments.net/reader031/viewer/2022040817/5e61c32167659d0b291248ba/html5/thumbnails/28.jpg)
22
![Page 29: A STUDY OF THE EFFECT OF MOBILITY RATIOS ON ...1. A study of the effect of mobility ratios on pattern displacement behavior It is well known, for unit mobility ratio, that the areal](https://reader031.vdocuments.net/reader031/viewer/2022040817/5e61c32167659d0b291248ba/html5/thumbnails/29.jpg)
23
A.2 EFFECT OF GRID-BLOCK AND TIME-STEP SIZE
Figure A also shows that the effect of grid-block size is much stronger in the unit
mobility ratio cases than it is in the very favorable mobility ratio cases. For M=1/20, the
simulation results converge when the number of grid blocks is greater than 10 by 10.
However, for unit mobility ratio, results converge for much finer grid blocks (100 by
100).
In very favorable mobility ratio cases, the simulation result is not sensitive to the
grid-block size. The reason can be the very sharp displacing front for very favorable
displacement.
I used 100 by 100 grid blocks for all the cases in this study, and therefore, the
simulation results are converged in terms of grid-block size.
A.3 CONCLUSION
In the unit mobility ratio cases, simulation results are sensitive to the grid-block
size but not very sensitive to time-step size. In the very favorable mobility ratio cases,
the simulation results are sensitive to grid-block size. For the calculations presented in
the main portion of the report, I used enough number of time steps and grid blocks for a
converged simulation result. Therefore, the results above demonstrate that simulations
are free from effects of the size of grid blocks and time steps.
![Page 30: A STUDY OF THE EFFECT OF MOBILITY RATIOS ON ...1. A study of the effect of mobility ratios on pattern displacement behavior It is well known, for unit mobility ratio, that the areal](https://reader031.vdocuments.net/reader031/viewer/2022040817/5e61c32167659d0b291248ba/html5/thumbnails/30.jpg)
24
Appendix B—Effect of Orientation of Grid-Blocks
We thought that the orientation of the grid-block with respect to the flow direction
may affect the simulation results, especially the streamline distribution. Therefore, I
studied this effect. In the work presented before, the injector and producer are aligned
with the diagonal as shown in Fig. B1a. It was not clear whether the streamline
distribution changes if streamlines align with the grid block (see Fig. B1b). Again, I only
studied the five-spot pattern. However, the conclusion is general for other patterns.
(a) (b)
Fig. B1 Grid Orientation.
I ran the simulation for both cases illustrated in Fig. B1 with time steps and grid-
block size remaining the same.
In 3DSL only rectangular grid blocks can be used, in the rotated grids. Therefore,
I have a bigger domain with the quarter of five-spot pattern sitting in the middle. A very
low permeability was assigned to regions outside the five-spot domain. In doing so, there
is essentially no flow outside the five-spot domain. The no-flow boundary is
approximated by zigzag lines along the boundary. However, as I used many grid
blocks(100 by 100), the boundary is quite smooth.
Because the grid block in the no-flow corner is closed on three sides, it is not
open to flow and it is not filled with the displacing phase when the surrounding blocks in
the five-spot pattern domain are swept by the displacing phase (see Fig. B2). It causes
the breakthrough time to be slightly earlier (about 0.1%). However, it does not make
much difference to the overall pattern behavior.
![Page 31: A STUDY OF THE EFFECT OF MOBILITY RATIOS ON ...1. A study of the effect of mobility ratios on pattern displacement behavior It is well known, for unit mobility ratio, that the areal](https://reader031.vdocuments.net/reader031/viewer/2022040817/5e61c32167659d0b291248ba/html5/thumbnails/31.jpg)
25
There is also some dispersion into the region outside the five-spot domain, as
illustrated in Fig. B2, due to the nonzero permeability in that region. 3DSL does not
work for zero permeability. This dispersion is small and does not affect the pattern
behavior.
Comparing Fig. B2 with Fig. 6, we can easily see that the direction of the grid
does not effect the pattern behavior. Therefore, any difference in the pattern behavior at
different mobility ratios is not due to improper grid orientation.
![Page 32: A STUDY OF THE EFFECT OF MOBILITY RATIOS ON ...1. A study of the effect of mobility ratios on pattern displacement behavior It is well known, for unit mobility ratio, that the areal](https://reader031.vdocuments.net/reader031/viewer/2022040817/5e61c32167659d0b291248ba/html5/thumbnails/32.jpg)
26
![Page 33: A STUDY OF THE EFFECT OF MOBILITY RATIOS ON ...1. A study of the effect of mobility ratios on pattern displacement behavior It is well known, for unit mobility ratio, that the areal](https://reader031.vdocuments.net/reader031/viewer/2022040817/5e61c32167659d0b291248ba/html5/thumbnails/33.jpg)
27
Appendix C—Numerical Dispersion
Fig. C shows the position of the displacing front at times near breakthrough (tD =
0.95) for the case of a five-spot pattern and a very favorable mobility ratio, M=1/20. The
solid line plots front location in the streamline closest to the boundary, whereas the
dashed line corresponds the diagonal line between injector and producer. The purpose of
this plot is to study the numerical dispersion for this case, because we think that the
breakthrough time should be unity in the absence of numerical dispersion.
I assumed piston-like displacement in choosing relative permeability curves.
However, because I mapped the numerical solution along the streamlines, this sharp front
may be smeared by numerical dispersion. However, due to the numerical-sharpening
effect for very favorable mobility ratios, the front is still quite sharp.
From Fig. C, it is obvious that the displacing front is very sharp. However, we
can still clearly see that it is not exactly a piston-like displacement due to the numerical
dispersion. The dispersion is very small. If there is no dispersion, the piston-like
displacing front bisects the numerically predicted front. The dimensionless width of the
front is 0.02, and the perfect piston-like front is in the middle. A dimensionless distance
of 0.01 from the front in the simulation result. Therefore, the numerical dispersion
causes 0.01 (tD) earlier breakthrough in the earliest breakthrough streamline. Referring to
Fig. 2(a), the calculated breakthrough tD for this case is over 0.99. Therefore, without
numerical dispersion, there will be perfect displacement for the five-spot pattern at very
favorable mobility ratio.
Fig. C also tells us the front shape along the boundary is almost the same as that
along the diagonal. Therefore, the slanted front is only due to the numerical dispersion,
not the boundary effect.
![Page 34: A STUDY OF THE EFFECT OF MOBILITY RATIOS ON ...1. A study of the effect of mobility ratios on pattern displacement behavior It is well known, for unit mobility ratio, that the areal](https://reader031.vdocuments.net/reader031/viewer/2022040817/5e61c32167659d0b291248ba/html5/thumbnails/34.jpg)
28
Fig. C Displacing FrontFive Spot Pattern, 1/M = 20
30
0.0
0.2
0.4
0.6
0.8
1.0
Dis
plac
ing
phas
e sa
tura
tion
0.0 0.2 0.4 0.6 0.8 1.0(a) Dimensionless distance from injector, xD
aa aa
a
a
a
a
aaaaaaaaaaaaaaaaaa
bb b
b
bbbbbbbbbbbbb
a
b
Along boundaryAlong diagonal
0.0
0.2
0.4
0.6
0.8
1.0
Dis
plac
ing
phas
e sa
tura
tion
0 20 40 60 80 100 120 140(b) Distance from Producer
aaa
a
a
a
a
aaaaaaaaaaaaaaaaaa
bb
b
bbbbbbbbbbbbb
a
b
Along boundaryAlong diagonal
![Page 35: A STUDY OF THE EFFECT OF MOBILITY RATIOS ON ...1. A study of the effect of mobility ratios on pattern displacement behavior It is well known, for unit mobility ratio, that the areal](https://reader031.vdocuments.net/reader031/viewer/2022040817/5e61c32167659d0b291248ba/html5/thumbnails/35.jpg)
29
2 STREAMLINES TO SOLVE INVERSE PROBLEMS
2.1 DEFINITION OF THE RESEARCH TOPIC
The proposed research is to apply the concept of streamlines to infer permeability
fields based on the tracer breakthrough curve and pressure differences between the
injector and producers. The basic idea of my approach is to adjust the streamlines to
match the reference fractional flow and pressure data, and through streamlines to modify
the underlying permeability field.
Because there is no flow across a streamline, we can represent the flow field by
1D flow streams along the streamlines. Each streamline is associated with a time of flight
which indicates the breakthrough time. In the 3DSL streamline simulator (Batycky et al),
we solve the pressure field at given saturation distribution, obtain the streamline
distribution, then map the Buckley-Leverett solution along the 1D streamlines for a short
time period, and then solve the pressure field again and repeat the process. Because the
pressure equation and saturation equation are decoupled in 3DSL, the simulation is
speeded up significantly.
Conventional reservoir simulation history matching is time consuming because of
the large number of grid blocks. However, if we know the time of flight of the
streamline, then we will know the breakthrough time of an individual streamline.
Therefore, if the breakthrough curve for our computed permeability field does not match
the reference breakthrough curve, then we know along which streamline and in which
way to modify the permeability for a better match.
This approach is described below. To start with this project, I made the
following simplifications:
• Incompressible flow, because the streamline simulator (3DSL by Batycky et al) that I
am using works for incompressible fluid flow only;
• Piston-like displacement;
• Unit mobility ratio, valid for tracer flow study;
![Page 36: A STUDY OF THE EFFECT OF MOBILITY RATIOS ON ...1. A study of the effect of mobility ratios on pattern displacement behavior It is well known, for unit mobility ratio, that the areal](https://reader031.vdocuments.net/reader031/viewer/2022040817/5e61c32167659d0b291248ba/html5/thumbnails/36.jpg)
30
• Two dimensional;
• No constraint of the permeability value or its distribution.
Steps of the Streamline Approach to the Inverse Problem:
1. Given a reference permeability field, use the 3DSL streamline simulator for forward
simulation to obtain the reference water breakthrough curves at producers and
pressure drops between injector and producers.
2. Start from an initial uniform permeability field. Do the same simulation. Check the
match—both breakthrough and pressure drop. If it does not match the reference data,
modify the permeability as in the following steps;
3. Work on the streamlines: Calculate the time of flight for all streamlines using 3DSL
to output the coordinates of the streamlines. Sort the streamlines in the order of time
flight.
4. Compare the computed and the reference breakthrough and pressure drops. For the
breakthrough curve, check where in the curve the difference lies. Relate it to the
corresponding streamline, and then change the permeability value depending on the
difference of the breakthrough curves;
5. Repeat the simulation using the new permeability distribution. This completes one
iteration;
6. Repeat steps 2 to 5 until the match is good enough.
I obtained very promising results with this approach. In the reference field, there
are one injector and two producers. Between the injector and producer there exists a high
permeability channel, and between the injector and producer 2 there is a low permeability
barrier (See Fig. 1). Starting from a homogeneous permeability field, both the
breakthrough and pressure drops are far away from the reference (see Fig. 2). After three
iterations, I obtained a very good match both in pressure drops and breakthrough curves
(Fig. 2), which indicates that this approach converges very fast, much faster than most of
other approaches to this inverse problem.
![Page 37: A STUDY OF THE EFFECT OF MOBILITY RATIOS ON ...1. A study of the effect of mobility ratios on pattern displacement behavior It is well known, for unit mobility ratio, that the areal](https://reader031.vdocuments.net/reader031/viewer/2022040817/5e61c32167659d0b291248ba/html5/thumbnails/37.jpg)
31
Fig. 1 Comparison on Permeability Field
Producer 1
Producer 2
36
Reference Permeability Field
Computed Permeability Field after 3rd Modification
![Page 38: A STUDY OF THE EFFECT OF MOBILITY RATIOS ON ...1. A study of the effect of mobility ratios on pattern displacement behavior It is well known, for unit mobility ratio, that the areal](https://reader031.vdocuments.net/reader031/viewer/2022040817/5e61c32167659d0b291248ba/html5/thumbnails/38.jpg)
32
Fig. 2 Comparison on Breakthrough Curve
37
0.00
0.12
0.24
0.36
0.48
0.60
0.72
0.84
wat
er c
ut
0 80 160 240 320 400 480time, days
Producer 1
a a a a a a a a a a a a a a a a a a a a a a a a a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a a
a
a a
a
a a
a
a a a
a a a
a a a
a a
a a a
a a a a
a
a a a a a
a
aa a a
a
a a a a
a
a a a a a a
a a aa
a a a a
aa
a a a a aa a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a a
a
a
a
a
a a
a
a a
a
a a
aa a
a
a a a a a
a a a a a
a a a aa
a a a
a a
a a a a a a
a a a a a a a a a a a a a a a a a a a a a a a a a a
a
a
a
a
a
a
a
a
a
a
a
a a
a a
a a
a
a a a
a a a
a
a a a
a
a a a a
a a a a a
a
a a a a a
aa a aa
a a a a
a a a a a
a a a a a
a aa a a
a
a aa a
a
a
a
a
Referenceuniform3rd modi
0.00
0.12
0.24
0.36
0.48
0.60
0.72
wat
er c
ut
0 80 160 240 320 400 480time, days
Producer 2
a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a a
a
a a
a a
a
a a
a a
a
a
a
a
a a
a a
a a
a
aa
a
a a a a
a a a a
a a
a
a
a a
a aa
a a a a
aa
a a a a aa a a a a a a a a a a a a a a a a a a a a a a a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a a
a
a a
a a
a a
a a
a
a
a a
a a a
a a
a
a
a a
a a a
a
a a
a a a
a a a
a a
a a a a
a aa a a
a
a a a a
a
a a a
a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a a a
a a
a
a a
a
a
a
a a
a a
aa
a
aa a
a a a a
a a a a
a
a a a
aa aa
a a
a a a
a a a
a
a
a
Referenceuniform3rd modi
![Page 39: A STUDY OF THE EFFECT OF MOBILITY RATIOS ON ...1. A study of the effect of mobility ratios on pattern displacement behavior It is well known, for unit mobility ratio, that the areal](https://reader031.vdocuments.net/reader031/viewer/2022040817/5e61c32167659d0b291248ba/html5/thumbnails/39.jpg)
33
Fig. 3 plots the error versus the number of iterations. The error is defined as
[ ]E E Efw i p ii
n prod
= +=∑ , ,
_
∆1
(1)
where Efw is the average absolute error in the fractional flow at different times, and E p i∆ ,
is the relative error of the pressure drops between the injector and producers at producer i.
They are defined below
En
f ffw i w i jC
w i jR
j
n
, , , , ,= −=
∑1
1
(2)
where fw i jC, , and fw i j
R, , are the computed and reference water fractional flow during time
step j at producer i, respectively.
Ep p
pp iiC
iR
iR∆
∆ ∆∆, = −
(3)
where ∆piC and ∆pi
R are calculated and reference pressure drop between the injector and
producer i, respectively.
I normalize the error in pressure by the reference pressure so that it will be within
the range of 0 to 1 and it is of the same scale as the error in fractional flow. Then I can
sum up them up to have a unified error estimation.
The solid line in Fig. 3 (a) is the sum of error at the two producers. The dashed
and dotted line are for each producer, respectively.
Comparing the computed permeability field with the reference field in Fig. 1, we
find that, with this approach, the high permeability channel is retrieved quite accurately.
For the barrier, the resolution is not very good. However, we have good matches for the
breakthrough curve and pressure drop.
![Page 40: A STUDY OF THE EFFECT OF MOBILITY RATIOS ON ...1. A study of the effect of mobility ratios on pattern displacement behavior It is well known, for unit mobility ratio, that the areal](https://reader031.vdocuments.net/reader031/viewer/2022040817/5e61c32167659d0b291248ba/html5/thumbnails/40.jpg)
34
(b)
Fig. 3 Error vs Iterations
38
0.0
0.2
0.4
0.6
0.8
1.0
Err
or
0 1 2 3(a)
a
a
a
a
b
b
b
b
a
a
a
a
a
b
a
TotalProducer 1Producer 2
0.0
0.2
0.4
0.6
Err
or
0 1 2 3Iteration
a
a
a
a
b
b
b
b
a
a a
a
b
b
b
b
a
b
a
b
E∆p at Producer 1Efw at Producer 1E∆p at Producer 2Efw at Producer 2
![Page 41: A STUDY OF THE EFFECT OF MOBILITY RATIOS ON ...1. A study of the effect of mobility ratios on pattern displacement behavior It is well known, for unit mobility ratio, that the areal](https://reader031.vdocuments.net/reader031/viewer/2022040817/5e61c32167659d0b291248ba/html5/thumbnails/41.jpg)
35
2.2 IMPORTANCE OF THE RESEARCH
History matching plays an important role in reservoir engineering. It is important
for prediction and data interpretation. In many cases, we have water or tracer
breakthrough and pressure information at the producers. This information contains much
information about the permeability distribution of a reservoir, but it is difficult to infer
this distribution. There are many approaches to this inverse problem. Most approaches
manipulate parameters at the grid-block level that corresponds to conventional simulation
grids. Because there are many cells, the optimization problem is large. By conducting the
optimization at the streamline level, the opportunity for speed improvement is great.
Since we know how each streamline affects water cut, we can speed up the inverse
process tremendously. A more typical approach would be to perturb parameters in a
more random fashion to gauge the effect of the parameter.
2.3 FUTURE WORK PLAN AND EXPECTED RESULTS
There are many simplifications in my current study. Simplification means
limitations in the application. Therefore, my future work will be focused on relaxing the
simplifications to broaden its application. My plan is to
• Release the piston-like displacement and unit mobility ratio assumptions. It is not
difficult to do since I can simply map the Buckley-Leverett solution instead of the
sharp front, and I can map the numerical solution for one-dimensional displacement
instead of the analytical one along the streamline.
• Seek a streamline simulator for compressible flow. Study my current method for
modifying a permeability field to see whether it is still valid for compressible flow.
Also study what modification of the current approach should be made for
compressible flow. Then I can make this approach work for primary recovery and
well-test problems. This will be the most difficult portion and therefore the key part
of this project.
![Page 42: A STUDY OF THE EFFECT OF MOBILITY RATIOS ON ...1. A study of the effect of mobility ratios on pattern displacement behavior It is well known, for unit mobility ratio, that the areal](https://reader031.vdocuments.net/reader031/viewer/2022040817/5e61c32167659d0b291248ba/html5/thumbnails/42.jpg)
36
• Put constraints on the permeability distribution from other information such as
seismic data, and broaden its use to three-dimensional problems. Improve the
method.
• For comparison, develop examples that use a conventional history-matching approach
to infer heterogeneity.
I will continue to work on this topic in my Ph.D. When this project is finished, I
expect that this approach will work for very general history matching purposes.
![Page 43: A STUDY OF THE EFFECT OF MOBILITY RATIOS ON ...1. A study of the effect of mobility ratios on pattern displacement behavior It is well known, for unit mobility ratio, that the areal](https://reader031.vdocuments.net/reader031/viewer/2022040817/5e61c32167659d0b291248ba/html5/thumbnails/43.jpg)
37
NOMENCLATURE
A area, L2
AS area swept, L2
AT total area of the pattern, L2
a distance between like wells (injection or production) in a row, L
d distance between adjacent rows of injection and production wells, L
EA areal sweep efficiency
fw fractional flow of water
h bed thickness, L
k permeability, L2
kro relative permeability of oil
krw relative permeability of water
l length, L
Li length of stream tube i, L
lfi distance of the displacing front from the injector in stream tube i, L
M mobility ratio
p pressure, M/L • T2
q flow rate, L3/T
Ri flow resistance of stream tube i, M/L4 • T
S saturation
Sw water saturation
tD dimensionless time
VI volume of displacing phase injected, L3
w width of a stream tube, L
φ porosity
µ viscosity, M/L• T
![Page 44: A STUDY OF THE EFFECT OF MOBILITY RATIOS ON ...1. A study of the effect of mobility ratios on pattern displacement behavior It is well known, for unit mobility ratio, that the areal](https://reader031.vdocuments.net/reader031/viewer/2022040817/5e61c32167659d0b291248ba/html5/thumbnails/44.jpg)
38
REFERENCES
1. Batycky, R. P., Blunt, M. J., and Thiele, M. R.: “A 3D Streamline-Based ReservoirSimulator” SPE Reservoir Engineering, 12, 246, (1997).
2. Craig, F. F. Jr.: “The Reservoir Engineering Aspect of Water Flooding,” Society ofPetroleum Engineers Monograph, Dallas, TX, 1971.Dyes, A. B., Caudle, B. H., andErickson, R. A., “Oil Production after Breakthrough—as Influenced by MobilityRatio”, Petroleum Transactions, AIME, 201, 27 (1954).
3. Dyes, A.B., Caudle, B.H., and Erickson, R.A.: “Oil Production AfterBreakthrough—as Influenced by Mobility Ratio,” Petroleum Transactions, AIME,201, 27 (1954).
4. Lake, L. W.: Enhanced Oil Recovery. Prentice Hall Inc., New Jersey, 1989.
5. Morel-Seytoux, Hubert J.: “Unit Mobility Ratio Displacement Calculations forPattern Floods in Homogeneous Medium,” SPE J, 6, 217, (1966).
6. Peaceman, D. W., Fundamentals of Numerical Reservoir Simulation. ElsevierScientific Publishing Co., New York, 1977.
7. Thiele, M. R., Batycky, R. P., Blunt, M. J. and Orr Jr, F. M. Jr.: “Simulating Flow inHeterogeneous Systems Using Streamtubes and Streamlines,” SPE. Reser. Eng, 11, 5,(1996).