Experiments that illustrate anomalous and non-local transport
A Thesis
SUBMITTED TO THE FACULTY OF THE
UNIVERSITY OF MINNESOTA
BY
Natasha Brynaert Filipovitch
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
MASTER OF SCIENCE
Vaughan Voller, Kimberly Hill
June 2017
Copyright c©Natasha Brynaert Filipovitch 2017
Abstract
In this work our focus is the related phenomena of anomalous diffusion and non-local
transport. The former refers to physical systems in which the spreading of a solute does
not exhibit the classic square root of time behavior; the later describes cases where the
flux of a solute at a point is controlled by features at locations removed from that point.
Two experimental systems that produce clear signals of anomalous diffusion and non-local
transport are presented. The first measures the infiltration of fluid into an obstacle filled
cavity. We show that when the obstacles are laid out in a fractal pattern the infiltration
measure exhibits an anomalous diffusion behavior. The second experiment studies the
steady state by-pass profile of a two-dimensional rice pile. We show that the pile’s profile
has a concave down curvature that, through modeling we argue, is a consequence of non-
local transport.
i
Contents
Contents ii
List of Tables iv
List of Figures v
1 Introduction 1
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Anomalous Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 Non-local Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Previous Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5 Intentions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Infiltration 11
2.1 Experimental Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.1 Visual images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.2 Comparison with theory . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.3 Anomalous results . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4 Infiltration Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
ii
3 Rice Pile Profile 25
3.1 Experimental Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3 Numerical Model Design and Results . . . . . . . . . . . . . . . . . . . . 36
3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4.1 Power-law avalanche analysis . . . . . . . . . . . . . . . . . . . . 41
3.4.2 Further profile analysis . . . . . . . . . . . . . . . . . . . . . . . . 43
3.5 Rice Pile Profile Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4 Evidence of Non-locality in the Active Layer 46
4.1 Search for the Active Layer and a Finite Thickness . . . . . . . . . . . . . 48
4.2 Evidence of Non-locality Involving the Bulk . . . . . . . . . . . . . . . . . 51
4.3 Evidence of Non-locality in the Active Layer Summary . . . . . . . . . . . 59
5 Conclusion 61
Bibliography 64
A Raw data for infiltration experiments 71
iii
List of Tables
2.1 Obstacle Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2 Experimental h0 values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1 Rice box (shown in Figure 3.5) and grain measurements . . . . . . . . . . . 32
4.1 Wedge experiment list . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
iv
List of Figures
1.1 Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Porous tube example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Normal vs Anomalous diffusion equations . . . . . . . . . . . . . . . . . . 6
1.4 Local vs Non-local slope equations . . . . . . . . . . . . . . . . . . . . . . 8
1.5 Küntz and Lavallée [2001] results of anomalous diffusion . . . . . . . . . . 9
1.6 Gerolymatou et al. [2006] results of anomalous diffusion . . . . . . . . . . 10
2.1 Sample Hele-Saw cell, showing all dimensions . . . . . . . . . . . . . . . 12
2.2 Sample Hele-Saw cell, view along z axis . . . . . . . . . . . . . . . . . . . 12
2.3 Schematic of fluid front with obstacles . . . . . . . . . . . . . . . . . . . . 14
2.4 Hele-Shaw tank design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5 Measurements of printed Hele-Shaw cell . . . . . . . . . . . . . . . . . . . 16
2.6 Video stills of an obstacle free cell . . . . . . . . . . . . . . . . . . . . . . 19
2.7 Video stills of a fractal obstacle cell . . . . . . . . . . . . . . . . . . . . . 19
2.8 Comparison of experimental data and theory for normal diffusion . . . . . . 21
2.9 Results showing anomalous diffusion . . . . . . . . . . . . . . . . . . . . 22
2.10 Graph of Hausdorff fractal dimension vs time exponent . . . . . . . . . . . 24
3.1 Schematic of a longitudinal profile . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Predicted profile curvatures for a by-pass environment . . . . . . . . . . . 29
v
3.3 A sketch of a three dimensional granular pile . . . . . . . . . . . . . . . . 30
3.4 A sketch of a two dimensional granular pile . . . . . . . . . . . . . . . . . 30
3.5 Rice box set up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.6 Build up process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.7 Rice pile in by-pass state . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.8 Experimental profile results . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.9 Schematic of a rice pile model . . . . . . . . . . . . . . . . . . . . . . . . 37
3.10 Schematic model stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.11 Schematic of a non-local rice pile model . . . . . . . . . . . . . . . . . . . 39
3.12 Rice pile profiles from the numerical models . . . . . . . . . . . . . . . . . 41
3.13 Experimental avalanche PDF plot . . . . . . . . . . . . . . . . . . . . . . 43
3.14 Long profile curvatures recovered from literature . . . . . . . . . . . . . . 44
4.1 Image Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2 Width of active layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.3 Rice box set up with wedge . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.4 Varying the angle of the wedge . . . . . . . . . . . . . . . . . . . . . . . . 54
4.5 Surface angles with varying wedge size . . . . . . . . . . . . . . . . . . . 55
4.6 Active Layer width with varying wedge size . . . . . . . . . . . . . . . . . 56
4.7 Active Layer width along x axis . . . . . . . . . . . . . . . . . . . . . . . 57
4.8 Active Layer transition shown on graph of surface angles . . . . . . . . . . 58
vi
Chapter 1
Introduction
1.1 Overview
The work we present here explores two sets of experiments completed with the goal of
improving our understanding of two concepts: anomalous diffusion and non-local trans-
port. With anomalous diffusion we want to specifically explore the conditions that produce
anomalous diffusion. With non-local transport we want to specifically recognize direct
signals of the presence and weighting direction of non-locality.
1.2 Anomalous Diffusion
To understand anomalous diffusion we must first discuss normal diffusion. The classic
model for normal diffusion is that of Brownian motion [Weeks et al., 1996]. Brownian
motion refers to the random motion of a particle suspended in a fluid that results due to
the collisions of that particle with the molecules or atoms of the fluid [Philibert, 2005].
It is with this motion that particles diffuse from areas of higher concentration to areas of
lower concentration. At first glance this motion may appear to be random, but with further
study scalings emerge. Einstein [1905] showed that the average displacement of a brownian
1
particle, from its starting position, is proportional to the square root of elapsed time,
x2 = 2Dt (1.1)
x∼ t 12 (1.2)
This time exponent of 12 is the defining characteristic of normal diffusion.
Figure 1.1: Brownian motion: (above) an example of a trajectory of a single particle and
(below) statistical distribution of displacements (the circles correspond to fractions and
multiples of the square root of the mean square displacement) [Philibert, 2005]
Anomalous diffusion, therefore, describes a system in which the time exponent does not
equal 12 [Korabel and Barkai, 2011]. The introduction of traps (hold ups), which impede
the motion of certain particles [Metzler and Klafter, 2004], or increased slopes (fast paths),
2
which accelerate the motion of some of the particles [Foufoula-Georgiou et al., 2010], may
result in such a time exponent. When the time exponent is less than 12 this is called sub-
diffusion; when the time exponent is greater than 12 this is called super diffusion. Therefore
anomalous diffusion may be induced by introducing hold ups, which leads to sub-diffusion,
or fast paths, which leads to super-diffusion.
One way to model anomalous transport is through non-local definitions of the fluxes in
the system [Schumer et al., 2009]. To describe this approach we will discuss an example of
moisture infiltration into a porous tube, (Figure 1.2). In this example, fluid flow is governed
by Darcy’s Law,
q =−K dh
dx
(1.3)
where q is the moisture flux [ m
3
m2s ], K is the hydraulic conductivity, and
dh
dx is the driving
head gradient. The fluid infiltrates a porous tube driven by a fixed head, h0, at x = 0. At
time t = 0 the moisture infiltration front is at x = 0, at time t > 0 the front is at x = s(t). At
x = 0, h = h0, and at x = s, h = 0. Between x = 0 and x = s, at any point in time, the head
gradient is fixed, meaning that dhdx =−h0s [Bear, 1988]. Since the moisture front is directly
proportional to this gradient (i.e. Darcy’s Law) we can present the following differential
equation for the position of the infiltration front, s(t).
ds
dt
=−K dh
dx
∣∣∣∣
s
= K
h0
s
, s(0) = 0 (1.4)
When we solve for s in terms of t the resulting time exponent is 12 , i.e.
s =
[√
2Kh0
]
t
1
2 (1.5)
indicating that this moisture infiltration has the signal of a normal diffusion process.
3
Figure 1.2: Porous tube example. Fluid from tank with a fixed head (h) infiltrates a tube
filled with porous media along the x direction. Speckled area represents porous medium,
grey area represents fluid filled regions.
Darcy’s Law (Eq 1.3) describes a local definition of flux since it depends on a constant
coefficient (hydraulic conductivity K) and the local gradient of the head (i.e. the value of
the head gradient is at the point where we are calculating the flux). We can, however, also
define a non-local flux, (with a new hydraulic conductivity, K∗) where the discharge at any
given point is determined not only by conditions at that point but also by conditions either
up or down stream from that point. This can be easily achieved through a weighted sum
(W ) where the effect on a point decreases with increasing distance from that point. In this
way, for example, a non-local flux can be defined as the weighted sum of the gradient at
points upstream of the point of interest,
qNL =−K∗∑W dhdx (1.6)
With a particular, (but still fairly general), choice of weights and letting the sum extend
over every upstream point back to the origin x = 0, we can write this flux as a fractional
derivative [Podlubny, 1999],
qNL,up =−K∗d
αh
dxα
, α ≤ 1 (1.7)
4
defined, in the Caputo sense [Caputo, 1967], as the convolution integral [Li et al., 2011]
qNL,up =−K∗
[
1
Γ(1−α)
x∫
0
(x−ξ )−α dh(ξ )
dξ
dξ
]
(1.8)
In Figure 1.8 we compare the solution of the tube infiltration problem using the normal
derivative with that using the fractional gradient in Eq 1.7. Here we see that the replace-
ment of the local flux, Eq 1.3, with the non-local flux, Eq 1.7 in our porous tube filling,
shows how the non-local flux modifies the nature of the movement of the moisture front.
In particular when α < 1 we see that the time exponent in the expression for the front
movement exhibits super-diffusion, i.e. the exponenton time n = 1α+1 >
1
2 . Thus, a frac-
tional definition of the head gradient in the porous tube leads to super-diffusive (fast path)
behavior. For completeness, we note that an alternative to introducing a fractional gradient
would be to define the time derivative of the problem as a fractional derivative. Analysis
with this setting [Voller, 2015] leads to sub-diffusive movement for the moisture front; in
this case the time exponent is less than 12 . The introduction of a fractional time derivative
represents "memory" and hold up, thus the observed sub-diffusion is to be expected.
5
Normal Diffusion Anomalous Diffusion
∂
∂x(q) = 0 in x< s
∂
∂x(q) = 0
q =−K ∂h∂x =constant q =−K∗ ∂
αh
∂xα =constant, 0< α ≤ 1
h =
(
s−x
s
)
h0 h∼ xαsα
ds
dt = q =
Kh0
s
ds
dt = q∼ 1sα
s∼ t 12 s∼ t 1α+1
Figure 1.3: A comparison of normal and anomalous porous tube equations. Note c and b
are constants and 0 < α ≤ 1 is a parameter that controls the weights and determines the
level of non-locality in the flux definition; a value of α = 1 recovers our local definition in
Eq 1.3.
1.3 Non-local Fluxes
Non-local definitions may also affect the spatial distribution of a system. Falcini et al.
[2013] discuss that the steady state longitudinal profile in sediment transport may show evi-
dence of non-locality. To demonstrate this let us consider a simple by-pass one dimensional
sedimentary fan in the absence of tectonics. In a unit length domain a standard model for
this, based on the Exner equation [Paola and Voller, 2005], is as follows
∂
∂x
(−q) = 0 0< x< 1 (1.9)
with the boundary conditions qx=0 = constant, η(1) = 0
In a simple approach, which neglects the critical shear stress, we can model the sediment
6
unit flux, q, as a diffusion like term, i.e. η , the elevation of the sediment deposit, takes on
a similar role to the pressure head h in our previous example.
q∼−K ∂η
∂x
(1.10)
where K is a "conductivity". When this flux definition is used in Eq 1.9 we get a linear
profile for the sediment deposit, i.e. η ∼ (1− x).
Moving away from the flux model in Eq 1.10 we can also use a particular non-local
model which treats the flux as a weighted sum of upstream slopes i.e. we define our flux
using Eqs 1.7 & 1.8 (we replace h with η , the height of the sediment deposit). In this case
as indicated in Figure 1.4 we get a concave up profile η ∼ 1− xα . Falcini et al. [2013]
also point out that one could use a non-local definition as a weighted sum of downstream
slopes, in this case the flux is given by the fractional derivative
qNL,down =−K d
αη
d(−x)α (1.11)
In a unit domain, using the Caputo definition, this can be written as
qNL,down =−K
[ −1
Γ(1−α)
1∫
x
(ξ − x)−α dη(ξ )
dξ
dξ
]
≡ ∂
αη
∂ (1− x)α (1.12)
Now as indicated in Figure 1.4 we will get a concave down profile η ∼ (1− x)α . Thus
in sediment and granular transport systems the long profile shape of a steady state deposit
may indicate the presence of non-local (anomalous) transport.
7
Local Slope Non-local Slope
Upstream weighted Downstream weighted
q∼−∂η
∂x
qNL,up ∼−∂
αη
∂xα
qNL,down ∼− ∂
αη
∂ (1− x)α
d
dx
(q) = 0, q(0) = qin, η(1) = 0
η ∼ 1− x η ∼ 1− xα η ∼ (1− x)α
0
1
0 1
η
x
η=1-x
Linear
0
1
0 1
η
x
η=1-x1/2
Linear
Upstream
0
1
0 1
η
x
η=(1-x)1/2
Linear
Downstream
Figure 1.4: An example of how the spatial relations of a system can be controlled by the
presence of non-locality in the fluxes. Here we show that when a non-local model is used
for flux the curvature of the slope provides information about the level and direction of
non-locality in the system. Note that 0 ≤ α ≤ 1 in all cases and α = 12 for purposes of
building these sample graphs.
1.4 Previous Observations
Very much related to our porous tube and non-local modeling examples, one physical
system where anomalous transport has been observed and measured is moisture infiltra-
tion into porous brick [Gerolymatou et al., 2006, Küntz and Lavallée, 2001]. Küntz and
Lavallée [2001] reanalyzed NMR (nuclear magnetic resonance) data from a number of ex-
periments involving infiltration into clay fired brick and limestone brick [Carpenter et al.,
1993, Pel et al., 1995, 1996, 1998]. They found that the infiltration results did not match
8
with expectations that include a time exponent of 12 . By using a theoretical model based
on a non-Fickian (anomalous) diffusion process, where the time exponent is not 12 , they
found they could fit the observed NMR data well. Figure 1.5 shows how the data from
the clay fired brick experiments aligns with the anomalous (non-Fickian) diffusion model,
whose time exponent is 0.58, rather than the Fick’s diffusion model, whose time expo-
nent is 0.50. Gerolymatou et al. [2006] noted a number of papers reporting anomalous
diffusion behavior during infiltration of building materials [El Abd and Milczarek, 2004,
Küntz and Lavallée, 2001, Taylor et al., 1999] and attempted to explain this anomalous
infiltration through the use of fractional calculus. They started with the classical Richards
equation (used to describe air-water flow through unsaturated zone in soils) and general-
ized this equation by introducing fractional time derivatives. In their investigation of the
resulting fractional Richards equation they note that data from the experiments performed
by Taylor et al. [1999] (on white silicious brick) fit much more closely with the fractional
Richards equation, whose time exponent is 0.43, than they do with the classical Richards
equation, whose time exponent is 0.5, as seen in Figure 1.6. They do note that while the
data seem more consistent with the fractional approach the uncertainty in the data is too
large to conclusively determine a time exponent.
9
Figure 1.5: Küntz and Lavallée [2001] show the experimental data (from Pel et al. [1995,
1996]) aligns much more closely with the non-Fickian anomalous diffusion model they pro-
posed (time exponent 0.58) than it does with Fick’s classic diffusion model (time exponent
0.50).
Figure 1.6: Gerolymatou et al. [2006] present data from infiltration experiments with white
silicious bricks [El Abd and Milczarek, 2004]. This data is presented as isochrones of θ ,
volumetric moisture content, and aligns more closely with the fractional Richards equation
they developed (time exponent 0.43) than it does with the classical Richards equation (time
exponent 0.5), this is particularly evident for the data collected in the later times of the
experiment.
10
1.5 Intentions
Following on, and extending, the infiltration observations in Küntz and Lavallée [2001]
and Gerolymatou et al. [2006] and non-local sediment transport theory presented by Fal-
cini et al. [2013], in this thesis we will present two systems where non-local and anomalous
transport can occur. Moving beyond observation and exponent fitting, however, our inten-
tion will be to understand what physical features in a given system induce and control the
observed anomalous behavior. Our first experiment is infiltration into a analogue porous
medium constructed by placing a fractal object pattern in a Hele-Shaw cell. This is an ex-
perimental arrangement that allows us to understand how heterogeneity controls observed
anomalous time exponents. The second experiment considers a rice pile as an analogue for
a sediment transport system. This experiment shows how the assumption of the appropri-
ate non-local flux (in terms of fractional derivative constructs) is needed to match observed
surface curvatures in the rice pile experiments.
11
Chapter 2
Infiltration
Our first set of experiments studies the relationship between the heterogeneity of a
porous medium and the resulting anomalous diffusion during infiltration. These experi-
ments were performed in the Multiphase Flow Laboratory in the CEGE Department at the
University of Minnesota. We use a Hele-Shaw cell as a analogue for this system. A Hele-
Shaw cell involves fluid flowing between closely spaced parallel plates (see Figure 2.1).
The plates are parallel to the xy plane and flow moves in the positive x direction. Our Hele-
Shaw cells are all square in plan view as seen in Figure 2.2. A Hele-Shaw cell may contain
obstacles between the plates and the size and arrangement of these obstacles may effect the
flow of the fluid. A Hele-Shaw system results in potential flow under the condition that
the viscous length scale (`v) is significantly smaller than the plan view length of the largest
obstacle (`min) [Batchelor, 1967].
`v =
gd4|∇h|
ν2
<< `min (2.1)
Where g is the gravitational acceleration, d is the distance between plates, ν is the kinematic
viscosity of the fluid (in our case glycerin), and ∇h is the gradient of the pressure head that
drives the flow; note when there are no obstacles `min = ` the cell plan view dimension.
12
Figure 2.1: A simple diagram of a Hele-Shaw cell. The two plates are parallel to the xy
plane and the distance between them is d, the fluid moves perpendicular to the z axis.
Figure 2.2: A simple diagram of a Hele-Shaw cell looking down the z axis. The fluid is
driven into the cavity by a fixed head h0, applied along the side x = 0, and moves in the
x direction. Note at the front of the infiltration fluid the pressure head is h = 0. In this
example the cell is a square with x and y dimensions of ` by `.
Here we will consider a Hele-Shaw system in which the flow is driven into the cell by
a fixed head along the side x = 0, see Figure 2.2. With the assumption that the flow is
potential flow in form, the governing equation for the average fluid front in Hele-Shaw
flow is identical to the governing equation for flow in a porous medium presented in the
introduction. In particular, for the uni-directional flow case shown in Figure 2.2 the fluid
flux(ms ) in the x direction is given by
qx =−K dhdx (2.2)
13
c.f. Eq 1.3. The parameter, K, in Eq 2.3 is a fluidic conductivity which, by the Hele-Shaw
theory [Batchelor, 1967], is given by
K =
1
12
d2g
ν
(2.3)
As with our porous tube the head gradient in the filled portion of the cavity (0 ≤ x ≤ s(t))
is a constant thus
ds
st
=−K dh
ds
= K
h0
s
,
(
0≤ x≤ s(t)) (2.4)
c.f. Eq 1.4 and
s =
[√
2Kh0
]
t
1
2 (2.5)
c.f. Eq 1.5. The time dependence here has the exponent = 12 which, as we discussed earlier,
is the hallmark of normal diffusion. This means that when the cell is obstacle free normal
diffusion can be expected and allows us to observe any changes in diffusive behavior as
heterogeneity (obstacles) is introduced. Note that in making those observations it is not
appropriate, due to interference of the obstacles, to track the flow as s(t), see Figure 2.3.
As an alternative we consider the infiltration F(t) defined as (for a rectangle or square cell),
F(t) =
Area Filled
`
(2.6)
i.e. the equivalent length of the fluid infiltration in the absence of obstacles. So with no
obstacles F(t) = s(t).
14
Figure 2.3: Schematic of infiltration with obstacles diagramming the non-planar location
for the fluid front.
2.1 Experimental Design
The apparatus used for these experiments includes a tank, gate and chamber (see Figure
2.4). The chamber is constructed of a base piece, or cell, (Figure 2.5) and a 6.35mm (14")
piece of transparent plastic bolted to the base piece that extends from the tank. The Hele-
Shaw cells were 3D printed using ABS filament. They were printed in white to provide for
maximum contrast. The fluid filled region of the cell is square, with a maximum area of
96mm x 96mm x 2mm. Any obstacles printed have a height (in the z direction) of 2mm.
Parallel to the x direction there is an additional 12mm on each side with holes to allow
for positioning of bolts to attach a 6.35mm (14") piece of transparent plastic to the top and
attach the whole cell to the apparatus. The only difference between printed cells was the
pattern of obstacles (or lack thereof) printed within.
15
Figure 2.4: Design for the apparatus for Hele-Shaw experiments. The half meter tall tank
was built from transparent plastic. The gate was manually operated using the lift rod with
a lever (not shown) to improve speed of gate opening. Insert shows view looking down the
z axis but does not include the 12mm sections on either side of the cell for bolt attachment.
16
Figure 2.5: Example of a printed Hele-Shaw cell. Fluid filled region in 96mm x 96mm
square. 12mm on each side with holes for bolts to affix transparent plastic above cell and
cell to apparatus. Fluid filled region and obstacles are 2mm in the z direction. The only
difference between printed cells is the obstacle arrangement within.
The half meter tall tank is made from clear transparent plastic, allowing for easy head
measurements before each experiment. Measured h0 values for each of the ten experiments
are shown in table 2.2. The fluid in the tank was glycerin with a small amount of food
coloring added to increase contrast between the fluid and the cell, we assume the value
of the kinematic viscosity of glycerin at 20◦ is ν = 0.00112 m
2
s [Anonymous, 1963]. We
positioned a video camera (COHU 4910 series CCD) directly above the cell in front of the
tank. During each experiment lamps were positioned at varying angles to decrease shad-
owing and increase contrast between fluid and cell in the videos. A gate lift rod manually
opens the gate to begin the experiment.
We started the filming before each experiment began and the video was taken at a rate
of 10 frames/second. Each experiment was considered complete when the fluid first exited
the cell at x = 96mm. Depending on the obstacle distribution and the applied tank head h0
the glycerin traversed the cell in texp = 38− 120s. It was determined that the volume of
fluid required to fill an obstacle free cell (∼ 2×104 mm3) was significantly smaller than the
17
initial volume (∼ 4× 105 mm3) in the tank thus a constant head was assumed throughout
each experiment.
By dividing the F(t) in Eq 2.6 by the x dimension of the cell we can discuss a non-
dimensionalized %F(t) representing the percentage of area infiltrated,
%F(t) =
AF
A
(2.7)
We used a threshold and binarized the images from the video of each experiment. We
quantitatively measured the filled area of a cell at each time step, raw data collected for
each time step included area filled (A)[square pixels] and area filled fraction (%F) [percent
of total area] .
In practice we found that a small portion (∼2%) of the plan-view area of the cell was
obscured by the gate and we adjusted all calculations accordingly. In addition, to avoid
possible bias from the initial entry into the cavity we neglected measurements taken at
times less than t = 1s.
Heterogeneity was introduced in the form of different obstacle patterns (table 2.1) with
the printed cells. We printed a blank cell, one with a repeating pattern, and five with differ-
ent Sierpinski carpet fractal patterns [Guo et al., 2006]. The fractal patterns are produced
by first looking at the whole plan-view area as one square. This square is then divided N
(pattern) number of times in the x and y directions, all but the outer row of (new smaller)
squares is filled as an obstacle, this is then a fractal of m (order) = 1. To build a fractal of
order 2 this process is repeated within each of the new smaller squares that were created in
the first process. As the fractal order increases the obstacle size decreases. The examples
in table 2.1 with a fractal order of 0 have such an indicator due to their patterns not being
fractal in nature.
18
N
(Pattern)
m
(Order)
Example N
(Pattern)
m
(Order)
Example
4 0 0 0
4 1 3 2
4 2 5 2
4 3
Table 2.1: Obstacle arrangements of Sierpinski carpet design used for experiments. Pattern
refers to the number of obstacles in a row or column, Order refers to the fractal order, and
the example pictures are not necessarily to scale.
Exp # N m h0[m]
1 0 0 0.040
2 3 2 0.040
3 5 2 0.039
4 4 0 0.041
5 4 0 0.040
6 4 1 0.039
7 4 2 0.041
8 4 2 0.039
9 4 3 0.037
10 4 3 0.039
Table 2.2: Measured h0 values for each experiment.
2.2 Results
2.2.1 Visual images
Figure 2.6 shows images from the experiment in the obstacle free cell at 1.8, 7, and
24 seconds. We can see the influence of an edge effect where the fluid along the side
edges is moving more slowly than the fluid slightly further inside the cell. We also see a
19
slight slowing in the middle of the cell which may be due to irregularities in the printing
process. Despite these effects we see a fluid front that can be fairly well approximated by
a horizontal line, s(t). Figure 2.7 shows images from an experiment in a fractal cell where
N=4 and m=2 at 1.8, 7, and 24 seconds. Here the edge effect is observed not only along the
outside edges but also along the edges of the obstacles but they appear to be the same order
as the non-obstacle case.
Figure 2.6: Video stills from an experiment using an obstacle free cell (1.8,7, and 24 sec-
onds into the experiment). ` = 96mm and s = the progress in the x direction of the average
front.
Figure 2.7: Video stills from an experiment using an fractal obstacle cell where N=4 and
m=2 (1.8,7, and 24 seconds into the experiment). ` = 96mm
20
2.2.2 Comparison with theory
As noted above, when we have no obstacles the infiltration will go as
%F(t) =
s× `
`2
=
[√
2Kh0
`
]
t
1
2 (2.8)
It has also been suggested [Filipovitch et al., 2016] that when we have a pattern (non-
fractal)
%F(t) =
[
µ
√
2h0µK
`
]
t
1
2 (2.9)
where µ is the pattern porosity, which is defined as the plan view area fraction of the cell
free of obstacles. We use the following values in these equations to determine what results
are expected by this theory; K = 0.0029ms by Eq 2.3 (using ν = 0.00112
m2
s , d = 0.002m,
and g = 9.8 ms2 ), ` = 0.096m, and h0 values found in table 2.2. We plotted these lines
alongside the infiltration data for the the empty cell (N = 0,m = 0) and repeating pattern
(N = 4,m = 0) experiments (Figure 2.8) in log log space where the time exponent (n) could
be easily determined by looking at the best fit line for each data set. Here we see the theory
and experimental data aligning very well which can be seen visually as well as through
the time exponent (n) values associated with a best fit for each experiment. The n values
for both the empty cell and repeating pattern experiments are very close to 12 which is a
clear sign of normal diffusion. What this comparison shows us is that these experiments
are behaving as they should, meaning that this experimental design is producing normal
diffusion when theory expects normal diffusion.
21
(i) (ii)
Figure 2.8: The dotted lines are measured data points and the solid line is the theory line
determined by Eqs 2.8 and 2.9 respectively. The n values shown on each graph are gathered
from the best fit for each set of experimental data. Infiltration results for both the empty
cell and the cell with a repeating pattern show a good match with theory. This is evidenced
visually by the close approximation the line gives for the data points as well as by the time
exponent (n) values all being approximately 12 .
2.2.3 Anomalous results
Now that we know this experimental design produces normal diffusion as expected we
look at the results from the experiments with cells containing fractal patterns. We plotted
the infiltration data for the fractal experiments in log log space where the time exponent (n)
could be easily determined by looking at the best fit line for each data set. In each of the
fractal experiments anomalous diffusion was observed (Figure 2.9). Time exponents range
from n = 0.3328 to n = 0.423 indicating sub-diffusive behavior for each of our Sierpinski
carpet patterns. These results indicate that the presence of system scale heterogeneity, (i.e.
the case where the obstacle size approaches the domain size and no repeating pattern can
be identified), will induce anomalous diffusion behavior; particularly that the presence of
heterogeneous obstacles will induce sub-diffusion.
22
Figure 2.9: The dotted lines are measured data points and the solid line is the best fit line
with the indicated time exponent (n). Infiltration results for all cells containing a fractal
pattern of obstacles show anomalous diffusive behavior. This is evidenced by the time
exponent (n) values all significantly less than 12 .
2.3 Discussion
While these results show that system scale heterogeneity leads to anomalous diffusion
behavior, we note that each obstacle arrangement, i.e. value of N (3,4,5), appears to have
a separate exponent and endeavor to quantify this relationship. We have already discussed
the mathematical significance of the time exponent (n). To quantify the heterogeneous
23
nature of each pattern we look to the Hausdorff fractal dimension [Guo et al., 2006].
H =
log(4N−4)
log(N)
(2.10)
This dimension is determined by the ratio of the log of two quantities. The quantity on
the bottom represents the number of sections any one side of the square is divided into.
The quantity on the top represents the number of new smaller squares created around the
previous large square in a single iteration. Note that this quantity is determined by the
pattern number of a fractal and is not dependent on the order number.
To investigate the connection between the nature of the heterogeneity of the system and
the resulting anomalous diffusion we plotted the H and n values of the experimental data
(Figure 2.10). The data was well fit by the quadratic
n = 1.29H2−4.17H +3.69 (2.11)
While this data set is not extensive enough to determine the ideal fit for this purpose it is
an improvement over the simple geometric model
n =
H−1
2
(2.12)
suggested by Voller [2015]. The fact that we see such a pattern indicates that the nature of
the heterogeneity of a porous medium is directly connected to the extent of the resulting
anomalous transport behavior.
24
Figure 2.10: Graph of Housdorff fractal dimension (H) vs time exponent (n) from experi-
mental data. The black dashed line is the best fit quadratic whose equation is displayed, the
solid grey line shows Eq 2.12
2.4 Infiltration Summary
The infiltration experiments demonstrated that anomalous transport behavior is calcu-
lably connected to the nature of heterogeneity of the system.
• Our experimental design produces normal diffusion when the Hele-Shaw cell is
empty or has a repeating pattern of obstacles which supports our conclusion that
the nature of the heterogeneity in the system is what leads to the anomalous transport
behavior.
• When system scale heterogeneity is introduced into the system anomalous transport
is seen. In this case we observe not only the the time exponents (n) are sub-diffusive
but that they vary based on the pattern (N) number of the fractal within the Hele-Shaw
cell.
• The connection between the Hausdorff fractal dimension (H) and the resulting diffu-
sive time exponent clearly indicates that anomalous diffusion can be induced by the
nature of the heterogeneity in the system.
25
Chapter 3
Rice Pile Profile
Our second experimental study is directed at identifying and modeling non-local trans-
port in granular piles. We can expect that the behavior of the rice pile may be related
to non-local behavior due to the heterogeneity of granular piles. Granular materials are
known to have high levels of heterogeneity relating to the development of discrete chains
or clusters of particles within the system [Drescher and de Josselin de Jong, 1972, Jaeger
et al., 1996]. This contributes to an understanding that forces in granular materials may
be effectively long-range as they are transmitted via these "force chains" to different parts
of the system. [e.g. Drescher and de Josselin de Jong, 1972, Jaeger et al., 1996]. In this
way, if one particle is disturbed in such a force chain, it could result in some response of all
particles in that chain [e.g. Nichol et al., 2010]. Previous studies have reported evidence of
non-local behaviors in sediment transport systems [Falcini et al., 2013, Foufoula-Georgiou
et al., 2010, Gabet and Mendoza, 2012, Martin et al., 2012, Nikora, 2002, Schumer et al.,
2009, Voller and Paola, 2010, Voller et al., 2012]. In particular, evidence for non-local
behaviors have been reported in details related to what is often called the long profile in a
fluvial landscape. This is the height profile of the region that extends from the upland head-
waters through the river system to the deltaic outlet in the ocean basin (see Figure 3.1). To
quantify this it is helpful to consider a very simple steady state treatment where, to first
26
order, we assume that unit sediment flux (Area/Time) is proportional to (and decreasing
with) the land surface slope, i.e.,
qs =−∂η∂x (3.1)
here η(x) is the land surface elevation above a datum at downstream position x. In this way
the Exner sediment transport equation for the long profile, see Eq 1.9 in the introduction,
can be written as
∂
∂x
(
∂η
∂x
)
= σ (3.2)
where we have used σ to represent an uplift rate (a negative value would be used for sub-
sidence). Broadly speaking there are three regions to consider, demarcated by the value of
σ . In the upland collection headwaters (region A in Figure 3.1) we would expect an uplift
(σ > 0). In order to remove the sediment provided by the uplift, satisfying the condition of
σ > 0 with the model in Eq 3.2 will require that the slope increases with distance down-
stream, i.e., the second derivative of η is greater than zero. Thus, in the upland we would
expect the fluvial sureface, η(x), to have a concave down profile. By contrast as the system
approaches the ocean region (region C in Figure 3.1) we would expect subsidence (σ < 0).
Here, in order to retain the sediment removed by the subsidence, satisfying the condition
of σ < 0 with the model in Eq 3.2 the slope decreases with distance downstream, i.e., the
second derivative of η is less than zero. This means that in the deltaic region we would
expect a concave up land surface profile. In the region connecting these two profile end
regions (region B in Figure 3.1) we would not expect tectonic activity (ie., σ ∼ 0), here
there is no need to remove or deposit sediment so we would expect the surface, η(x), to
have a linear slope.
27
Figure 3.1: Schematic of a sample longitudinal profile. Section A shows the approximate
location of the uplands where uplift would be occurring producing a concave down profile
(σ > 0). Section B shows the approximate location of the region with no tectonic activity
and a linear profile shape (σ ∼ 0). Section C shows the approximate location of the deltaic
region where subsidence would be occurring producing a concave up profile (σ < 0).
The exact modeling of the profiles in the uplands and deltaic regions have been areas
of intense study [Foufoula-Georgiou et al., 2010, Schumer et al., 2009, Voller and Paola,
2010, Voller et al., 2012]. This has included models that have used the concept of non-local
transport fluxes as defined in Eqs 1.6-1.12. In particular Foufoula-Georgiou et al. [2010]
have shown that an upstream weighted non-local treatment, where the sediment flux is
modeled by an upstream fractional derivative, (i.e. q∼ ∂αη∂xα , α < 1), provides a better fit to
observation of upland (hill slope) land surface profiles. In contrast, Voller and Paola [2010]
show that a downstream weight, (i.e. q ∼ ∂αη∂ (−x)α , α < 1), provides a better fit to experi-
mental observation of delta deposition systems. Further Voller et al. [2012] show that the
assumed direction of the non-locality has a binary behavior. With the non-local flux models
introduced in chapter 1 physically realistic predictions in a depositional environment can
only be reproduced if the non-locality is directed downstream, i.e. downstream features
in the system control the transport. In contrast realistic profiles in erosional (hill slope)
regions can only be predicted if the non-locality is directed upstream.
To date, not too much attention has been applied to the by-pass case [Postma et al.,
28
2008] but in a general study of non-local transport Falcini et al. [2013] used this system to
further investigate the consequences of the direction used in the non-local transport defini-
tion. Presented first in Figure 1.4 and summarized in Figure 3.2, this study suggested that,
in the absence of tectonics (σ = 0) an upstream non-local model would produce a profile
with the height (η), x relationship
η ∼ 1− xα (3.3)
where 0≤ α ≤ 1. Profile shapes produced here are concave up where α < 1. When α = 1
the profile becomes linear. A downstream non-local model in this same tectonically neutral
environment would produce a profile with the η , x relationship
η ∼ (1− x)α (3.4)
where 0≤ α ≤ 1. Profile shapes produced here are concave down where α < 1, with α = 1
once again producing a linear profile. This simple by-pass system with no uplift or subsi-
dence offers an opportunity to investigate the possible appearance of non-local behavior in
a particulate system. Falcini et al. [2013] investigate a number of physical situations which
might lead to profile curvature, e.g. non-linear transport coefficients; the conclusion was
that in a by-pass regime only this non-locality correctly curves the profile. Therefore, any
curvature seen in the profile of a by-pass system is an indication of non-locality. Further, if
non-locality is established the curvature of said profile will also determine the direction in
which that non-locality is weighted.
29
01
0 1
η
x
Downstream
Linear
Upstream
Figure 3.2: This figure shows predicted profile curvatures for a by-pass system as intro-
duced in Figure 1.4. The profile produced by downstream weighted non-locality is shown
as a solid line, the profile produced by upstream weighted non-locality is shown as a dotted
line, and a linear profile is shown as a dashed line for reference. It is clear that an observed
curvature in a by-pass system indicates a non-local influence and further that the nature of
that curvature indicates the direction of the non-local influence.
In a preliminary treatment, Falcini et al. [2013] used the steady state by-pass experi-
ments of Postma et al. [2008] to suggest that in a by-pass system (built through a deposi-
tion) the concave down profile suggests the presence of downstream directed non-locality,
i.e. experimental profiles show a concave down shape. Here we investigate this point fur-
ther but rather than using a sediment transport system we use an analogue system based
around the dynamics of a granular pile. Admittedly, one system is gravity driven and one
is fluid driven but both have grains that affect the surface profile so we feel comfortable
considering that insight gained in these experiments will translate to the field of fluvial sed-
iment transport. The most commonly studied granular pile is a sand pile. A sand pile is
often initiated by releasing grains from a single point of grain entry into an empty system.
These grains begin to build a pile whose grains periodically avalanche down the side of the
pile, extending the base and growing the pile. The behavior of sand piles has been studied
in both infinite and finite domains and in both three (e.g. [Juanico et al., 2008]) and two di-
mensions (e.g. [Alonso and Herrmann, 1996]). In an infinite domain there is no outlet and
the grains do not leave the system, the sand pile continues to build endlessly. Since there is
30
no outlet for the grains they must remain a part of the system. A finite domain is one that in-
cludes some sort of edge or drop off where the grains can (and do) leave the system. There
is a depositional stage before the grains reach the edge while the pile is building. Once the
grains begin to leave the system a by-pass stage begins when, on average, the same amount
of grains are entering and exiting the system. A three dimensional sand pile is built up in
the vertical direction (z direction in Figure 3.3 and out in the horizontal direction (as in the
xy plane in Figure 3.3. This produces a cone shape that is commonly associated with the
word "pile".
Figure 3.3: A sketch of a three dimensional granular pile with x, y, and z directions as well
as the angle of repose (θ ) indicated.
Two dimensional sand piles are built up in the vertical direction (the y direction in Figure
3.4) and out in the horizontal direction (the x direction in Figure 3.4) while the transverse
direction (the z direction in Figure 3.4) is kept sufficiently small so that the variance across
this dimension is negligible. In this study we will consider two dimensional finite piles.
31
Figure 3.4: A sketch of a two dimensional granular pile with x, y, and z directions as well
as the angle of repose (θ ) indicated
A critical measure of a granular pile is the so called angle of repose. The angle of repose
(e.g. θ in Figure 3.4) is the average angle at which the grains in the pile are at the threshold
of movement. As the grains attempt to build the pile beyond this angle avalanching begins,
bringing the pile back down to or below the angle of repose. Due to this process, there
is a range of angles that a granular pile displays in the steady state. Of particular interest
are the maximum angle of repose, which is the largest θ measured in a particular exper-
iment (and sometimes is referred to as the critical angle) [Denisov et al., 2012], and the
average angle of repose, which is an average of all of the measured θ for a particular exper-
iment [Frette et al., 1996]. The anisotropy of the grains affects the angle of repose and its
range; Denisov et al. [2012] report that "spherical grains have a lower angle of repose than
nonspherical grains and that for nonspherical grains the angle of repose fluctuates much
stronger". Angles of repose have been reported in the range of 30 degrees (spherical 65µm
diameter magnetite beads [Quintanilla et al., 2004]) to 51.1 degrees (polished rice with a
3.6 aspect ratio [Frette et al., 1996]) with a variety of values in between [e.g. Alonso and
Herrmann, 1996, Denisov et al., 2012, Juanico et al., 2008]. These reported angles have
been measured using a variety of granular media including rice, sand, lentils, mung beans,
polenta, and more.
All of these approaches have in common the presentation of one single angle stated as
the angle of repose for that system, this implies that the angle of repose is constant for all
32
x. If a system has such a constant angle of repose the surface of the pile must be linear.
Our hypothesis is that if there is non-locality involved in the building and maintaining of a
granular pile we may expect to see curvature in the pile and an angle of repose that is not
constant, θ(x). If the non-locality is upstream weighted, producing a concave up profile as
previously described, we expect to see θ decrease with increasing x and if the non-locality
is downstream weighted, producing a concave down profile as previously described, we
expect to see θ increase with increasing x. Our core idea is to see whether this rice pile
will reveal a non-linear profile shape, which would suggest non-local transport behavior, if
this is the case we further would like to use the observed shape of the profile to indicate the
direction of the non-locality.
3.1 Experimental Design
We used a rice pile experimental set up based on Frette et al. [1996] (Figure 3.5) built
by Douglas Jerolmack [Jerolmack and Paola, 2010]. The rice was brown, long grain rice
that has an average length of 7.20 ± 0.48 mm, an average width of 2.15 ± 0.34 mm and
an average weight of .02 g per grain. The rice pile box is made up of two vertical parallel
transparent plastic panels of height (η) = 36.50 cm spaced 2.54 cm apart making a long
thin box with two closed and two open sides secured at all four corners. A styrofoam block
is used at L = 0 giving the set up the ability to change L if desired; we ran all experiments
with length (L) = 20.50 cm (exceptions described in Chapter 4).
Rice Pile Box Dimensions Rice Grain Characteristics
Height 36.50 cm Length 7.20 ± 0.48 mm
Length 20.50 cm Width 2.15 ± 0.34 mm
Width 2.54 cm Average weight .02 g
Table 3.1: Rice box (shown in Figure 3.5) and grain measurements
33
Each experiment begins with an empty rice box (exceptions described in Chapter 4)
and it takes about 14 hours to reach the by-pass state. A toothed wheel picks up rice at
an average of one grain per tooth and drops the grain into a paper funnel secured within
the box. The paper funnel decreases the velocity of the grains as they are added to the
pile as well as increases the precision of the grain entrance location onto the pile. For
these experiments the speed at which the wheel turns was set such that the grains entered
the system at a rate of approximately one grain per second. A camera with 6 megapixel
resolution took time lapse photographs of the η−L plane of the box every 5 minutes. We
used these photographs to analyze the profile shape. There is a scale below the box at x = 1
that collects data at 1 Hz. Initially the set up included a blower connected to a timer to
periodically blow rice off of the scale and into a large bin as had been used in previous
studies to manage the scale’s 400 g capacity [Monbouquette, 2008]. Due to the need to
discount the data during the time the blower was running it was eventually determined
unnecessary for these experiments and was not active for all experiments. The data from
the scale was used for avalanche analysis discussed in Section 3.4.1.
34
Figure 3.5: Sketch of rice pile box. The box consists of two large transparent plastic plates
a distance w apart, with transparent plastic between the plates on the bottom and styrofoam
between the plates on the left side. The edges of the box drawn in black are closed and the
edges drawn in grey are open. There is also a transparent plastic peg of length w at the open
corner. There is a wheel above the box with teeth that pick up an average of one grain of
rice as each tooth passes the supply and rotates at a programmable speed to introduce rice
into the system at whatever rate an experiment requires. From the wheel the rice enters into
a paper funnel to ensure a more precise entry point. The height of the rice pile is shown as
h and the length is shown as L.
When the rice first begins to enter the empty system a large majority of the rice stays in
the system and builds a pile. Because, on average, more rice is entering the system than ex-
iting this phase is referred to as a depositional environment. In the photographs of the build
up stage we see a concave up profile as is expected in a depositional environment (Figure
3.6). This build up occurs through alternating periods of deposition and avalanching. We
define deposition as the entrance of a grain that does not dislodge other grains as it comes
to a rest inside the system. We define avalanching as the entrance of a grain whose addi-
tion causes some portion of the pile to exceed its critical state resulting in the movement
of grains previously at rest. Avalanching is what causes the pile’s width to increase and
35
grains to exit as the pile nears the edge of the system. As the width of the pile increases
more and more grains leave the system. When, on average, the number of grains entering
the system equals the number of grains leaving, the system is in a by-pass (or steady state)
environment.
(i) (ii) (iii) (iv)
Figure 3.6: Build up process
Using the photographs from the by-pass stage, we analyzed the shape of the experi-
mental profile. In our analysis, we consider each surface profile over the downstream 90%
of the rice pile. We discount the first 10%, influenced by inertial effects from the impacts
of the grains as they drop into the box. We use ξ to denote the horizontal dimension of
the experiments; 0 ≤ ξ ≤ L and the top surface of the pile is h(ξ ) (Figure 3.5). We nor-
malize these variables according to the remaining length of the box and h0 =h(ξ = 0.1L):
x = (ξ −0.1L)/0.9L and η(x) =h(x)/h0 (Figure 3.7).
Figure 3.7: Snapshot of a ricepile in by-pass state. The thick white line indicates the
approximate surface profile.
This system is consistent with similar systems in literature shown by the power law
nature of the avalanches (Section 3.4.1) and the thickness of the active layer (Chapter 4).
36
3.2 Experimental Results
If the system dynamics are local we would expect a constant angle of repose, i.e. η ∼
(1− x) and that the profiles, when plotted on a log-log plot of η vs (1− x), would appear
as approximately linear and scattered about the line of slope =1. To analyze the shape of
the observed profiles we plotted the normalized profiles on such a log-log plot (Figure 3.8).
We then fit each profile with the power-law η = (1− x)α using a linearized least squares
fit. On this plot we see the profiles do appear as approximately linear but are not around
the line α = 1. The range of experimental α values is 0.8 < α < 1. The mean α value is
0.90±0.02 using the standard deviation about the mean for uncertainty. The fact that none
of the data shows an α = 1 indicates a persistent curved profile surface. This curvature
signals the presence of non-locality. Further we see that the curvature of this profile is a
concave down curvature which indicates that if non-locality is present it is weighted in the
downstream direction. Therefore our average profile can be described by
η = (1− x)0.9 (3.5)
Figure 3.8: Normalized experimental profiles. Nonlinear least squares fitting indicates the
results are well-fit by η = (1− x)α , α = 0.90± 0.02. For reference, lines representing
α = 1 (linear) and α = 0.8 are shown.
37
3.3 Numerical Model Design and Results
To further investigate these results and ensure they were not merely a result of the the
way we truncated the profiles, my research group, led by Anthony Longjas, developed a
model of a dynamic rice pile that explores the effect of using local or non-local treatment.
The starting point of the model follows classic rice pile model of Frette [1993]. This
models the rice pile as a set of columns (usually ∼100) each with a number of "grains"
modeled as blocks. Each column is labeled by j, i.e. j = 1,2,3, ...,L and an integer height
value η( j) is assigned to each column. There is a solid wall at x = 0 and the drop off is
at L+ 1. Starting from a stable pile, in a time step, we add a grain at j = 1, i.e. η(1)→
η(1)+1, and then iteratively track how this grain, and those it interacts with, move down
the pile until a new stable state is reached. Essentially at each iteration in a time step, we
progressively (from top to bottom) testing the stability of each column in turn. Stability
in the classic rice pile model is defined by the difference in height between neighboring
columns, SL. The Frette [1993] model begins by using a constant critical slope, Sc = 1
(block) but also allows for the addition of randomization, or noise, by allowing the critical
slope Sc to take on a value of either 1 or 2, they do so with equally probability. For the
constant critical slope model if SL≤ Sc = 1 (block) (a difference in height between columns
of one block or less) this portion of the pile would be stable (Figure 3.10 i) and if SL > Sc =
1 (a difference in height between columns greater than one) this portion of the pile would
not be stable and failure would occur. In the model failure occurs by moving the top block
from column j to the top of column j + 1, as shown in Figure 3.10 ii. The net result of
this model is, after appropriate normalization, a pile with an average profile of η = (1−x).
Interestingly the avalanches (i.e. the total grains leaving the system in a time step) in the
system are in a power-law distribution as will be discussed further in Chapter 3.4.1.
38
Figure 3.9: Schematic of a possible stable rice pile configuration of the model where noise
is included.
(i) (ii)
Figure 3.10: Schematic of rice pile model stability criteria. i) a stable configuration where
SL ≤ 1. ii) an unstable configuration where SL > 1 and thus failure occurs in the form of
the top block in the jth column moving to the ( j+1)th column.
The extension by Anthony Longjas to this model was to use a non-local behavior to
determine the stability of a column. As before if the measured slope (SNL) exceeds the
critical slope, i.e. SNL > Sc = 1, failure occurs. Where the local slope was determined
39
only by the difference in η between the jth and the ( j+ 1)th column, the non-local slope
is determined by a weighted sum of the local downstream slopes, i.e. SNL = ω1SL j +
ω2SL j+1 +ω3SL j+2 + ... .
Figure 3.11: For the non-local rice pile model the stability of a column is determined not
only by the difference in η between columns j and j+1 (SL) but rather by a weighted sum
of all downstream slopes where the weights decrease in increasing distance from column
j. i represents the points between columns where local slopes are measured.
The local slope of each point j downstream of x contributes to the expression for SNL(x)
with a weight ωi that decreases with increasing distance from x
SNL(x) =
M−1
∑
i=1
ωiSi (3.6)
Where the length of the domain downstream of x is divided into M−1 sections of horizontal
length ∆x, and M = 1−x∆x + 1, and i = 1 corresponds tothe jth column. A possible set of
weights to be used in Equation 3.6 are in terms of the Grünwald weights
ωi = ∆x1−β
i
∑
k=1
gk (3.7)
40
where 0 < β ≤ 1 the Grünwald weights are (see page 23 in Meerschaert and Sikorskii
[2012]):
g1 = 1; gi =
i−2−β
i−1 gi−1, i≥ 2 (3.8)
This is choice of weights is motivated by noting that in the limit that ∆x→ 0 it can be
shown that
SNL(x) =
M−1
∑
i=1
ωiSi =
1
∆xβ
M
∑
i=1
giηi ≈ 1Γ(1−β )
1∫
x
(ξ − x)−βSLdξ ≡ ∂
βη
∂ (1− x)β (3.9)
where ∆x= 1−xM−1 , ηi =η(x+(i−1)∆x), i.e. this choice make the rice pile model compatible
with our previous non-local fractional derivative treatment. The net result of this non-
local model is a pile with an average profile of η = (1− x)α and β ≈ α . Interestingly
the avalanches in this system also appear in a power-law distribution as will be discussed
further in Chapter 3.4.1.
When the results of these two models are compared to our experimental results (Fig-
ure 3.12) we can see that our choice of non-local model reproduces our experimentally
observed profile while the local model does not. The local model produces an approxi-
mately linear profile with an average α = 0.98 while the non-local model produces the
same concave down profile observed in our experiments with an average α = 0.90. This
result verifies not only our conclusion that non-locality was observed in our experiments
but also that the observed non-locality was downstream weighted.
41
η1-x
Figure 3.12: Rice pile profile data from numerical models. Sample profiles of the curvature
are obtained using the previously presented rice pile model of Christensen et al. [1996] as
the local model and the modified form of this model incorporating a non-local failure cri-
terion. The numerical results reported here use an array of M = 250 cells and time steps
t = 2M2 = 1.25x105. The results obtained using the local model exhibit linear profiles,
while the results obtained using the non-local form of the model exhibit the experimen-
tally observed surface profiles with a power-law exponent of α = 0.90. Non-linear least
squares fitting indicates the results for both sets of data are well-fit by η = (1− x)α . Lines
representing α = 1 (linear) and α = 0.8 are shown for reference.
3.4 Discussion
3.4.1 Power-law avalanche analysis
When the sizes of avalanches, (measures of the pile output from one input event), ob-
served in granular piles are plotted against the frequency of occurrence a power-law pattern
is evident for many granular piles [Amaral and Lauritsen, 1997, Bak et al., 1987, Frette
et al., 1996, Quintanilla et al., 2004]. This pattern demonstrates that in these systems there
is an expected relative frequency of an avalanche of a given size inversely related to the
size of an avalanche. A representative equation for such a power-law system in which the
probability distribution of x is shaped by the experimentally determined exponent τ is
p(x)∼ x−τ (3.10)
The power-law distribution, however, is not present for all granular piles [Feder, 1995].
42
For instance, Bak et al. [1987] predicted power-law distribution of avalanches in a sand
pile and, while the description of a sand pile can be useful in picturing a process that would
produce this pattern, Nagel [1992] shows clearly that the expected pattern is not seen with
sand. One important characteristic is the anisotropy of the grains in the system [Denisov
et al., 2012, Frette et al., 1996]. Although the aspect ratio of the rice grains was chosen
specifically to ensure power-law behavior it is important to gather avalanche data to verify
the expected patterns.
Our avalanche data is that of drop statistics. After removing known erroneous data
from the set (e.g. large negative numbers during times when the blower was active) we
plotted data collected during equilibrium. We plotted the avalanche size versus frequency.
We did only one experiment for a length of time sufficient for power-law analysis, since
large data sets are preferable to minimize fluctuations in this analysis [Clauset et al., 2009].
The avalanche data from the scale was plotted on PDF plots (Figure 3.13 a) where patterns
consistent with power-law distributions were observed. A linearized least squares fit to
the experimental data yields a power-law exponent τ =−1.94 which is consistent with re-
sults seen in literature [Amaral and Lauritsen, 1997, Frette, 1993, Frette et al., 1996]. The
power-law exponent cited in Frette et al. [1996] of τ = −2.02 is reasonably close to our
experimental value. This is an indication that the rice pile system studied here is consis-
tent with other systems discussed in literature. It is interesting to note that the avalanche
statistics for local and non-local models were also recorded (Figure 3.13 b) and power-law
avalanche sizes were observed in both cases. This not only goes towards validating the
model as we see this pattern with the model under both local and non-local conditions but
it also goes towards showing that the presence of these power-law sized avalanches may
not be a signal of non-locality.
43
Figure 3.13: Avalanche size power-law distribution for experimental and modeled data. (a)
The linearized least squares fit to the experimental data yields a power-law exponent τ =
−1.94, which is reasonably close with the τ =−2.02 reported in the experiment of Frette
et al. [1996]. (b) The local model yields a power-law exponent of τ =−1.57/pm0.02, and
the model with the non-local correction yields τ =−1.56±0.01; both are consistent with
values reported in literature [Amaral and Lauritsen, 1997, Christensen et al., 1996]. The
broken line serves as a guide to the eye and has a value of τ =−2.0 for (a) and τ =−1.55
for (b).
3.4.2 Further profile analysis
In conjunction with Anthony Longjas we analyzed surface profiles presented for the
rice pile in Frette et al. [1996] and the sediment by-pass in Postma et al. [2008] for profile
curvature, and a concave down profile was seen in both instances (Figure 3.14). The rice
pile profiles in Frette et al. [1996] produced an α value of 0.86±0.08 whereas the Postma
et al. [2008] by-pass sediment profile produced an α value of 0.74. For each of these eval-
uations limited numbers of profiles were available but the results show clear consistency
with our experiments. These results further strengthen the experimental results of profile
curvature and the inclusion of non-local dynamics in the modeling of such systems.
44
(i) (ii)
Figure 3.14: Long profile curvature results recovered from i) the rice pile presented in Frette
et al. [1996] and ii) the by-pass sediment profile presented in Postma et al. [2008]. The
recovered profiles were fitted with η = (1−x)α resulting in α = 0.86±0.08 and α = 0.74,
respectively. Lines representing α = 1 (linear) and α = 0.8 are shown for reference.
3.5 Rice Pile Profile Summary
The rice pile demonstration recognized a clear signal of non-local dynamics as was
evidenced by a surface curvature, the shape of which further signals the direction of the
non-locality to be downstream weighted.
• Falcini et al. [2013] predict a curvature in such a by-pass system if and only if non-
locality is present.
• All experimental profiles present a concave down curvature, which demonstrates
anomalous transport behavior caused by the presence of downstream non-local dy-
namics.
• This anomalous behavior is reproduced in a model through the use of downstream
weighted non-local dynamics.
• Our model produces power-law sized avalanches using both local and non-local fail-
ure criteria, this result goes towards validating the model.
45
• The fact that both the local and non-local model reproduce power-law sized avalanches
suggests, however, that there is no strong connection between power-law sized avalanches
and non-locality.
46
Chapter 4
Evidence of Non-locality in the Active
Layer
One of the goals of our work is not only to build experiments that exhibit strong sig-
nals of anomalous behavior and non-local transport but also to develop an explanation for
the observed behaviors. In the infiltration experiments we were successful at linking our
observation of anomalous diffusion to the level of heterogeneity in the system. A working
hypothesis for the expected presence of non-local transport in the rice pile was the oc-
currence of power-law sized avalanches. The observation that both a local and non-local
model were able to reproduce the same power-law statistics for observed avalanches, how-
ever, made us question if this feature was a direct signal of non-local transport dynamics.
This lead us to look for other features in the dynamics of the rice pile that might have a
more direct connection to non-local behavior.
Nichol et al. [2010] noted non-local behavior in a granular system where the dynamics
in the creeping section were to some level controlled by the dynamics of the shear zone.
The experimental design of Nichol et al. [2010] was a drum whose shear zone is at the
bottom unlike our two dimensional rice pile. However, a rapidly flowing layer above a
creeping layer was seen in a two dimensional granular pile by Komatsu et al. [2001]. Casual
47
observation of our rice pile appears to show a rapidly flowing layer on the surface with little
or no movement beneath. Jaeger et al. [1996] noted that granular piles previously at rest
whose angles are changed adjust with movement of only the surface grains where the grains
deeper within the pile appear at rest rather than creeping. This leads us to explore whether
our rice pile does contain a rapidly flowing layer (active layer) that reaches a distinct depth
within the pile and whether the bulk of the rice pile is a zone of creeping or immobile
grains.
The avalanching grains of the rice pile move with a non-uniform velocity profile wherein
the grain velocity is greatest near the surface of the pile. The depth to which this movement
extends defines an active layer. In grain flows with shear zones a zone of shear flow may be
accompanied by a zone of creep [Nichol et al., 2010] and Komatsu et al. [2001] claim that
some degree of movement extends throughout the entire granular pile but it is not debated
that the velocity profile is non-uniform where the top grains move more quickly than the
grains below. The terminology surrounding the active layer is not used consistently in all
of the literature; for instance Jaeger and Nagel [1992] referred to it as a zone of rapid shear
flow. In this paper we will be using the term active layer to refer to the layer in which sig-
nificant movement of grains, as determined by the process of image subtraction, is detected
during the time frame of consideration.
While an active layer and a shear band are not identical the width of shear bands can be
used as an analogy for the depth of the active layer. The width of shear bands in granular
materials is not affected by any geometrical concerns of the system other than the grain
dimensions themselves [Muhlhaus and Vardoulakis, 1987]. There are reports that the cor-
relation between grain diameter and shear zone width are anywhere between a factor of 3
and a factor of 18. Roscoe [1970] observes in experiments with sand that shear zones are
on the order of 10 grains thick. This value is also used in calculations by Boutreux et al.
[1998]. Aradian et al. [1999] argue that a value of 3 is more appropriate for their calcula-
tions while Muhlhaus and Vardoulakis [1987] see experimental evidence of shear zones up
48
to 18 grains thick.
4.1 Search for the Active Layer and a Finite Thickness
To analyze the thickness of the active layer we performed image subtraction of two
photographs representing various time intervals (Figure 4.1). Any area of the two pho-
tographs that are identical appears black on the final image. Areas showing rice deposited
in previously empty space will show up bright (light grey to white) and clear as in Figure
4.1 i. Areas that contained rice in the first image but contained nothing in the second image
will show as empty space (black). In Figure 4.1 ii, iii, & iv the mid to dark grey represents
areas where rice was present in both of the images but has changed position. We defined
the active layer in each subtracted image pair as the region that appeared white to dark grey,
representing rice grains that had moved.
49
(i) Build up stage (ii) 10 Minutes at steady state
(iii) 2 hours at steady state (iv) 12 hours at steady state
Figure 4.1: Examples of image subtraction for a depositional environment, 10 minutes in
a steady state environment, 2 hours in a steady state environment, and 12 hours in a steady
state environment
We considered the width of the active layer to be represented by thickness of this grey
region in the area. We measured this in five different locations on each image subtraction
pair using standard image processing tools (image j). We averaged these results and then
plotted the averages against the length of time between photos subtracted to investigate how
the apparent width of the active layer changed with length of time between images (similar
to the shutter speed in Komatsu et al. [2001]) (Figure 4.2). The apparent width of the active
layer initially grows with this time interval, similar to Komatsu et al. [2001]. However,
50
unlike this previous work, in the rice piles, the width appears to approach a constant, (i.e.
saturated), of approximately 2.44 cm as the length of time considered is increased. The
range (showing lowest and highest recorded values) of average measurements (represented
with black error bars) is very close to the width of a single grain of rice (represented with
grey error bars).
Figure 4.2: Width of active layer as determined through image subtraction plotted opposite
the time difference between the subtracted images. Black error bars represent range in
measurements, grey error bars represent the width of a grain of rice.
2.44 cm is only slightly greater than ten times the width (2.15 mm) of our rice grains. If
we consider our shear zones analogous to shear bands as discussed at the beginning of this
chapter, this is consistent with results from Muhlhaus and Vardoulakis [1987] who found
that shear bands are typically approximately 8-10 grains thick. This is an indication that
this system is consistent with comparable systems in literature. Since a grain of rice is not
spherical it is interesting to note that this rule indicates that in a rice pile it is the width of
the grain, rather than the length, that relates to the thickness of the active layer.
51
4.2 Evidence of Non-locality Involving the Bulk
When describing the active layer it is unclear how discontinuous the movement is from
the rest of the pile. That is, to the naked eye there appears to be a static section of rice
below the active layer, this section also appears immobile when using the image subtraction
technique. However Komatsu et al. [2001] noted that in a system with spherical particles
and steady surface flow the layer beneath the surface flow has a creep velocity that decays
exponentially with depth beneath the free surface. Comparing their "creep zone" with our
apparent fixed depth flowing layer motivated us to investigate this question in more detail
in our grain piles. We did so by introducing a completely static wedge to the base of the
pile to represent the apparently immobile rice grains in the system. We reason that if there
is truly no creep in our pile beneath the flow, a sufficiently small wedge would have no
effect. If there is a creep velocity of the grains within this wedge region we would expect
that replacing these mobile grains with a completely immobile section would change the
characteristics of the rice pile and/or statistics of the flow layer. To investigate this question
of creep, for this thesis, we considered two characteristics of our rice pile, the active layer
width and the average angle of repose.
52
Wedge Experiments
Wedge Angle (θ ) Wedge Length (Lw) Box Length (L) Wedge Surface Texture
39◦ 17.5 cm 20.5 Rough
41◦ 17.5 cm 20.5 Rough
43◦ 17.5 cm 20.5 Rough
41◦ 27.2 cm 30.5 Smooth
41◦ 25.3 cm 30.5 Smooth
41◦ 29.0 cm 30.5 Rough
41◦ 28.8 cm 30.5 Rough
41◦ 28.1 cm 30.5 Rough
41◦ 27.6 cm 30.5 Rough
41◦ 27.2 cm 30.5 Rough
41◦ 26.6 cm 30.5 Rough
41◦ 25.9 cm 30.5 Rough
41◦ 25.3 cm 30.5 Rough
41◦ 24.5 cm 30.5 Rough
41◦ 23.1 cm 30.5 Rough
0◦ 0 cm 30.5 N/A
Table 4.1: Experiments conducted with a styrofoam wedge in the box to investigate active
layer characteristics.
We performed the experiments in table 4.1 in the rice box monitoring outputs and in-
stantaneous bed heights as described in Chapter 3 with two important changes : (1) for
greater resolution we used a longer box than the one described in Chapter 3; (2) we in-
serted a styrofoam wedge that varied in size from one experiment to the next (Figure 4.3).
We ran experiments with two different surfaces at the boundary between the wedge and the
rice grains, smooth and rough. The smooth surface was the surface of the styrofoam sanded
flat. The rough surface was developed by gluing rice grains (dropped onto the surface at
random angles) to the contact surface. The experiments with the smooth wedge exhibited a
flow-through environment with no rice remaining in the box whatsoever. The experiments
with the roughened wedge surface exhibited a build up of rice in front of and on top of the
wedge. This indicated that these experiments require a wedge contact surface that mimics
the roughness of a static wedge of rice. The rough surface was used for all reported data.
53
Figure 4.3: A styrofoam wedge was placed within the box simulating a static section of the
rice pile.
We were interested in the steady state so we began each experiment with an empty
box except for the styrofoam wedge (or completely empty). The addition of the wedge
decreased the time it took to reach steady state (from the 14 hours for the experiments in
Chapter 3) and varied inversely with the size of the wedge.
We conducted two sets of experiments. The first was to investigate whether or not, for
a given wedge length Lw < L there is a minimum θ below which the active layer would not
be affected by the presence of the wedge. The second was to investigate whether or not, for
a given wedge angle θ there is a minimum wedge length Lw below which the active layer
would not be affected by the presence of the wedge.
For the first set of experiments we used a box length of L(x) = 20.5cm and a wedge
length Lw = 17.5cm and three different wedge angles: 39, 41, and 43 degrees. The mobile
grains on top of the wedge change their behavior with different values of θ (Figure 4.4).
As is visible in Figure 4.4 the presence and characteristics of the static wedge can affect
both the thickness of the active layer and the angle of the surface of the pile of rice. At 39◦
54
the layer of rice above the wedge extends along the entire surface and appears to be slightly
thicker at small x than at large x. At 41◦ the layer of rice above the wedge extends along
the entire surface and appears similar in thickness from small x to large x. At 43◦ the layer
of rice above the wedge does not extend all the way to small x, is thicker at small x than at
large x, and even the thickest portion is thinner than either of the previous angle tests. The
41◦ wedge surface is closest to parallel to the surface of the rice and is not so large that is
interferes with the development of an active layer. From this we concluded that 41◦ was
below the minimum θ that would affect the active layer, also note that this value is similar
to the average angle of the surface of the rice pile with no wedge (See Figure 4.5).
(i) (ii) (iii)
Figure 4.4: The angle of the wedge was varied until an angle was found where the active
layer was clearly affected. Above are wedges with angles of θ = 39◦ (i), θ = 41◦ (ii), and
θ = 43◦ (iii). When θ reaches 43◦ the mobile layer of grains no longer reaches the top of
the wedge and the layer that is present is much thinner, even at the bottom of the wedge.
Based on these results, we performed the second set of these experiments using 41◦
wedges in a box where L = 30.5 to allow for more data points near the active layer tran-
sition. As one might expect, as the size of the wedge was decreased, the surface of the
wedge moved further away from the visual approximation of the bottom of the active layer.
We used two physical parameters to quantitatively evaluate the effect of the wedge on the
flowing layer and how that effect varied with wedge size: (1) surface slope, and (2) active
layer thickness.
As we have discussed, the surface of the rice pile contains a curvature rather than being
a linear slope, so for this analysis we visually approximated an average slope to determine
55
how the average slope changed with wedge size (Figure 4.5). We recorded such slopes
for 12 consecutive images from each wedge experiment, thus representing data from a two
hour period. The average and spread of these slopes is presented in Figure 4.5. For the
experiment with no wedge in the box with L = 30.5cm we recorded visually determined
linear average slopes for two sets of 12 consecutive images. In other words two two hour
periods of data are represented by the no wedge average and range.
These data indicate that while the wedge is large the surface angle of the pile is less than
the surface angle of a pile with no wedge (and indeed often truncates on the wedge rather
than extending to the top) but as the wedge size decreases the measured surface angles are
very close to or within a normal range of values for a pile with no wedge. Surprisingly,
for the smallest wedges measured the surface angle is lower than that without the wedge
(Figure 4.5). We discuss this shortly.
Figure 4.5: Approximate angle of rice surface with different sizes of styrofoam wedges
below the flowing layer. Both the grey band, which represents the range of angles mea-
sured, and the dashed line, which represents the average surface angle measured, were
measured in experiments without a wedge and are not associated with a Length value. All
experiments presented here were preformed in a box with L = 30.5cm.
The width of the active layer was measured using the image subtraction technique de-
56
scribed earlier. The subtracted images were taken two hours apart. Each data point plotted
here is calculated from the average of three two hour image subtraction image pairs, with
black error bars representing the range of measured averages. In these data the disturbance
from the presence of the wedge is, once again, observed with the three largest wedge sizes,
after which the 2 hour active layer width falls within a normal range for a rice pile with no
wedge. However, unlike the results from the measured surface angle, the thickness of the
flowing layer is constant once the wedge is sufficiently small.
Figure 4.6: Approximate active layer width for rice piles with varying wedge sizes. Widths
were calculated on image subtraction products with two hour time differences. Black error
bars show the range of measured average values. Both the grey band, which represents the
range of widths measured, and the dashed line, which represents the average active layer
width measured, were measured in experiments without a wedge and are not associated
with a Length value. All experiments analyzed here were performed in a box where L =
30.5cm.
One might expect the wedge to have a clear effect if it is protruding into the region
where without a wedge the active layer is present. Indeed, as illustrated in Figure 4.7
when we consider the results in Figures 4.6 &4.5, the presence of the wedge within this
57
active layer region decreases the thickness of the mobile layer and the angle of the surface
compared to these details in experiments without a wedge. Once the wedge drops below
what one might call a "buffer layer" below the measured bottom of the flowing layer with
no wedge present, one might expect the presence of the wedge to have little or no effect.
E.g., If we consider the active layer to be 2.15 cm thick (the average measurement for a
2 hour portion of the experiment) then it would extend along the x axis x∗ = 2.15sin41◦ and
x∗ = 3.28 cm (Figure 4.7). Given that the full length of the box is 30.5 cm this puts the
approximate transition point at 27.22 cm. If we consider the active layer to be 2.44 cm
thick (the width the active layer appears to approach in Figure 4.2) then it would extend
along the x axis x∗ = 2.44sin41◦ and x
∗ = 3.72 cm (Figure 4.7). Given that the full length of the
box is 30.5 cm this puts the approximate transition point at 26.78 cm. Both transitions are
represented in Figure
Figure 4.7: Schematic of a rice pile with an active layer shaded in grey. We can approximate
how far along the x axis the active layer extends (x∗)with geometric arguments. In our
experiments we measured both the width of the active layer (WAL) and the average slope
(θ ). Dividing WAL by sinθ will give us x∗.
58
Figure 4.8: Lines representing possible active layer transitions are shown along with the
active layer width data from Figure 4.6. The vertical dashed line is at x = 27.22cm which
could represent the transition suggested by a 2.15 cm wide active layer. The vertical broken
dashed line is at x = 26.78cm which could represent the transition suggestion by a 2.44 cm
wide active layer.
In our experiments we found the flowing layer thickness and surface angle to be sta-
tistically similar to that without a wedge for the wedge length Lw in an intermediate range
between 27.6 cm and 25.3 cm. This is surprising because our geometric arguments indicate
that the first (and possibly second) wedge represented in this range (Lw = 27.6cm) extends
beyond the transition and into the active layer. We also note that while the measurements
for active layer width are within the range of what is measured with no wedge they all
fall below the average until we reach a wedge size of L = 24.5cm which could indicate a
buffer region between the bottom of the measured active layer and the bulk of 2.28-2.72
cm depending on where the transition is considered to occur.
Another surprising result is that for sufficiently small values of Lw, while the flowing
layer thickness was the same as that from experiments without a wedge, the steady surface
angle was smaller than the results without a wedge. A cursory inspection of this experiment
59
shows us that the large avalanches taking place result in a larger ∆η at x = 0 than is seen
in experiments with no wedge. While we cannot draw a direct connection, perhaps these
surprising results are somehow connected to the previously described non-locality in the
rice pile. Non-locality in grain flows has been seen to alter the flow of grains far from the
primary zone of movement, e.g. a non-local rheology where grains removed from the shear
zone were affected by the shear zone was seen by Nichol et al. [2010]. It is possible that
what we see here shows the converse of what was seen by Nichol et al. [2010] in that we
may be seeing a change in the creep layer have a non-local effect on the grains in the active
layer.
These data are preliminary, leaving much room for further expansion. Such possible
studies include analysis of how such a wedge affects drop statistics, longer experiments to
discern whether there is such a time where image subtraction will see movement throughout
the pile, profile shape analyses, an investigation of the velocity profile within the active
layer, experiments with smaller wedges, or smaller intervals between wedges. All of these
studies could expand our understanding of the process and the role of the active layer.
4.3 Evidence of Non-locality in the Active Layer Summary
These active layer analyses demonstrated that while this rice pile has a discrete and
measurable active layer where a majority the movement in the by-pass state is contained,
the mobile properties of the bulk are important to the overall dynamics of the pile.
• Image subtraction shows a discrete active layer width approaching ∼ 2.44 cm with
increasing time between images.
• The inclusion of a static wedge most dramatically affected the observed width and
angle when some portion of this wedge was within the previously measured active
layer.
60
• Once the wedge was minimally within the active layer all active layer width mea-
surements were consistent with no-wedge experiments but may show a buffer region
where the active layer width stays below average but still within range.
• The smallest wedges produced surface angles lower than seen in no-wedge experi-
ments.
• While not directly linked to the non-locality discussed in Chapter 3, the surprisingly
low angle of a portion of our results may indicate some non-locality which we think
should be investigated.
61
Chapter 5
Conclusion
In this work we discussed the related phenomena of anomalous diffusion and non-local
transport. Anomalous diffusion refers to physical systems in which the spreading of a solute
does not exhibit the classic square root of time behavior. Non-locality is the circumstance
in which conditions removed (in time or space) from a given location affect the transport
process at that location. It is very difficult to observe and experimentally isolate anomalous
transport and non-locality since the signals it produces are commonly driven by multiple
phenomena. The main objective here was to develop experiments to show a clear, clean
signal of anomalous transport and non-locality.
The work we presented here looked at two systems, fluid infiltration into a thin cavity
containing flow obstacles and the profile of a stable two dimensional rice pile. Both sets of
experimental results confirmed the presence of non-locality or anomalous transport behav-
ior. In each of these instances the presence of non-locality or anomalous transport behavior
was tied to specific features which induce or control this behavior. In the case of the in-
filtration experiments into fractal obstacle patterns anomalous diffusion was confirmed by
a transport time exponent n < 12 . This anomalous behavior was shown to be connected to
the level of heterogeneity in the obstacle pattern. This is shown, first, by the observation
that homogeneous patterns (empty cell or repeating obstacle pattern) confirmed normal
62
diffusion with time exponents n = 12 . Second, and more directly, when the obstacle is a
fractal pattern we see a connection between the Hausdorff fractal dimension (H) and the
observed time exponents (n) defined by a quadratic function. In the case of the rice pile
experiment non-locality was indicated by the observation of a curvature of the long profile,
η = (1−x)α , α 6= 1. The concave down shape of this curvature further indicates that the
non-locality is downstream weighted. These observations were further validated by a nu-
merical model that produces a linear surface when using local failure criteria and a surface
whose curvature is consistent with our experimental results when using non-local failure
criteria. This anomalous behavior might be controlled by the failure mechanisms which we
show produce a discrete active layer for the total length of our experiments.
There are many possible directions for future work on these projects. The infiltration
experiments could be improved by creating the cells in a manner that decreases or elimi-
nates the possibility of warping. One possible extension to these infiltration experiments
is to include additional patterns to increase the number of data points available to refine
the connection between the Hausdorff fractal dimension (H) and the observed time expo-
nents. Another possible extension is to alter the patterns to create fast paths rather than
hold ups and thus investigate super-diffusion. It is suggested in [Voller, 2015] that placing
cavities in the locations where obstacles were in our experiments is one way to produce
super-diffusion in this system. Another suggestion to produce a super-diffusive environ-
ment [Reis et al., In Preparation 2017] is to stagger the pattern of obstacles so that y = 0
and y = 1 lines run through the center of the large obstacle (what was previously y = 0.5)
with a narrow area in the middle where there are no obstacles. Since each of these sug-
gestions for producing super-diffusion contains a fractal pattern of obstacles (and thus a
Hausdorff fractal dimension, H) the results from these experiments can be used to better
understand the connection between the heterogeneity of the medium and observed time
exponents.
The rice pile profile shape experiments could be improved by increasing the variation
63
of the boundary conditions. For example, one possible extension to the rice pile profile
shape experiments is to run experiments for many different L values to see how this might
affect the curvature of the surface. Another possible extension is to use grains with different
shapes. Denisov et al. [2012] demonstrated that the shape of the grains affects the angle of
repose and here we argue that we must approach the concept of angle of repose in a manner
which includes the pile’s surface curvature. We also argue that the level of profile curvature
indicates the level of non-locality in the rice pile dynamics. Therefore by observing not
only the average surface angle with relation to the x axis but also the profile curvature it
may be possible to determine whether grain shape affects the level of non-locality in the
dynamics of the grain pile.
We believe the insights presented in this thesis help to improve our understanding of
both anomalous diffusion and non-local transport and improve our ability recognize and
predict direct signals of both.
64
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Appendix A
Raw data for infiltration experiments
N 0 4 4 3
m 0 0 0 2
run 1 1 2 1
frame %F frame %F frame %F frame %F
34 2.382 47 0.789 35 2.208 35 1.717
35 4.227 48 3.545 36 4.139 36 3.979
36 5.77 49 5.297 37 5.251 37 5.693
37 7.085 50 6.381 38 6.193 38 7.163
38 8.249 51 7.278 39 6.938 39 8.25
39 9.314 52 8.008 40 7.58 40 9.145
40 10.327 53 8.625 41 8.151 41 9.995
41 11.244 54 9.216 42 8.704 42 10.712
42 12.092 55 9.738 43 9.196 43 11.413
43 12.859 56 10.23 44 9.675 44 12.041
44 13.632 57 10.697 45 10.129 45 12.62
72
45 14.407 58 11.131 46 10.536 46 13.177
46 15.041 59 11.552 47 10.945 47 13.709
47 15.799 60 11.977 48 11.354 48 14.246
48 16.39 61 12.375 49 11.721 49 14.703
49 17.053 62 12.772 50 12.11 50 15.191
50 17.678 63 13.15 51 12.475 51 15.648
51 18.26 64 13.525 52 12.846 52 16.085
52 18.824 65 13.898 53 13.218 53 16.513
53 19.406 66 14.251 54 13.587 54 16.956
54 19.948 67 14.619 55 13.937 55 17.357
55 20.494 68 14.978 56 14.285 56 17.764
56 21.007 69 15.328 57 14.635 57 18.172
57 21.515 70 15.671 58 14.975 58 18.563
58 22.016 71 16.024 59 15.316 59 18.945
59 22.508 72 16.388 60 15.653 60 19.328
60 22.989 73 16.735 61 15.994 61 19.702
61 23.463 74 17.063 62 16.33 62 20.073
62 23.922 75 17.394 63 16.671 63 20.446
63 24.373 76 17.71 64 17.002 64 20.812
64 24.819 77 18.042 65 17.341 65 21.181
65 25.253 78 18.381 66 17.674 66 21.527
66 25.677 79 18.709 67 18.014 67 21.888
67 26.091 80 19.041 68 18.339 68 22.246
68 26.516 81 19.385 69 18.681 69 22.605
69 26.932 82 19.716 70 19.037 70 22.965
70 27.34 83 20.041 71 19.358 71 23.311
73
71 27.736 84 20.369 72 19.682 72 23.664
72 28.132 85 20.658 73 19.993 73 24.009
73 28.519 86 20.954 74 20.33 74 24.342
74 28.91 87 21.26 75 20.661 75 24.652
75 29.288 88 21.55 76 20.997 76 24.978
76 29.666 89 21.85 77 21.305 77 25.315
77 30.028 90 22.144 78 21.611 78 25.631
78 30.4 91 22.453 79 21.931 79 25.935
79 30.777 92 22.745 80 22.229 80 26.247
80 31.125 93 23.029 81 22.527 81 26.561
81 31.471 94 23.305 82 22.805 82 26.878
82 31.834 95 23.582 83 23.103 83 27.161
83 32.178 96 23.842 84 23.385 84 27.436
84 32.514 97 24.117 85 23.661 85 27.702
85 32.859 98 24.364 86 23.955 86 27.966
86 33.19 99 24.613 87 24.205 87 28.24
87 33.527 100 24.857 88 24.451 88 28.497
88 33.848 101 25.11 89 24.717 89 28.756
89 34.181 102 25.35 90 24.955 90 29.001
90 34.499 103 25.591 91 25.183 91 29.26
91 34.813 104 25.82 92 25.424 92 29.51
92 35.135 105 26.057 93 25.648 93 29.752
93 35.445 106 26.28 94 25.876 94 30.007
94 35.762 107 26.531 95 26.107 95 30.247
95 36.069 108 26.754 96 26.326 96 30.49
96 36.365 109 26.952 97 26.539 97 30.726
74
97 36.686 110 27.162 98 26.755 98 30.96
98 36.981 111 27.361 99 26.968 99 31.198
99 37.272 112 27.559 100 27.181 100 31.427
100 37.559 113 27.756 101 27.389 101 31.658
101 37.863 114 27.956 102 27.599 102 31.884
102 38.16 115 28.159 103 27.802 103 32.121
103 38.438 116 28.349 104 28.004 104 32.347
104 38.73 117 28.528 105 28.197 105 32.566
105 39.014 118 28.717 106 28.389 106 32.8
106 39.289 119 28.899 107 28.58 107 33.014
107 39.564 120 29.092 108 28.763 108 33.229
108 39.844 121 29.278 109 28.95 109 33.454
109 40.127 122 29.455 110 29.133 110 33.67
110 40.4 123 29.639 111 29.314 111 33.876
111 40.674 124 29.822 112 29.485 112 34.085
112 40.943 125 29.996 113 29.668 113 34.276
113 41.207 126 30.178 114 29.85 114 34.474
114 41.473 127 30.354 115 30.023 115 34.674
115 41.723 128 30.526 116 30.201 116 34.863
116 41.998 129 30.698 117 30.372 117 35.039
117 42.249 130 30.871 118 30.545 118 35.22
118 42.52 131 31.05 119 30.72 119 35.402
119 42.775 132 31.222 120 30.894 120 35.58
120 43.038 133 31.397 121 31.066 121 35.753
121 43.296 134 31.581 122 31.182 122 35.937
122 43.54 135 31.75 123 31.353 123 36.12
75
123 43.798 136 31.914 124 31.518 124 36.285
124 44.039 137 32.088 125 31.689 125 36.456
125 44.289 138 32.255 126 31.858 126 36.619
126 44.538 139 32.432 127 32.028 127 36.786
127 44.779 140 32.598 128 32.199 128 36.949
128 45.028 141 32.779 129 32.375 129 37.111
129 45.267 142 32.941 130 32.547 130 37.27
130 45.513 143 33.119 131 32.713 131 37.429
131 45.744 144 33.29 132 32.864 132 37.591
132 45.987 145 33.456 133 33.04 133 37.745
133 46.223 146 33.625 134 33.208 134 37.906
134 46.467 147 33.795 135 33.38 135 38.063
135 46.699 148 33.91 136 33.553 136 38.213
136 46.937 149 34.08 137 33.704 137 38.353
137 47.167 150 34.252 138 33.88 138 38.512
138 47.408 151 34.416 139 34.05 139 38.66
139 47.634 152 34.586 140 34.221 140 38.806
140 47.868 153 34.762 141 34.377 141 38.908
141 48.093 154 34.967 142 34.543 142 39.061
142 48.323 155 35.083 143 34.714 143 39.199
143 48.547 156 35.247 144 34.871 144 39.346
144 48.763 157 35.419 145 35.04 145 39.492
145 48.991 158 35.578 146 35.198 146 39.634
146 49.212 159 35.746 147 35.374 147 39.778
147 49.441 160 35.921 148 35.544 148 39.914
148 49.664 161 36.137 149 35.694 149 40.05
76
149 49.883 162 36.302 150 35.864 150 40.191
150 50.107 163 36.464 151 36.024 151 40.376
151 50.326 164 36.628 152 36.196 152 40.466
152 50.539 165 36.795 153 36.361 153 40.605
153 50.755 166 36.975 154 36.531 154 40.741
154 50.959 167 37.13 155 36.7 155 40.875
155 51.184 168 37.305 156 36.863 156 41.018
156 51.393 169 37.477 157 37.033 157 41.153
157 51.611 170 37.668 158 37.196 158 41.288
158 51.82 171 37.842 159 37.373 159 41.426
159 52.032 172 38.024 160 37.542 160 41.56
160 52.245 173 38.199 161 37.72 161 41.695
161 52.461 174 38.35 162 37.893 162 41.837
162 52.671 175 38.501 163 38.064 163 41.967
163 52.87 176 38.655 164 38.229 164 42.093
164 53.075 177 38.804 165 38.402 165 42.224
165 53.269 178 38.952 166 38.546 166 42.353
166 53.476 179 39.129 167 38.708 167 42.487
167 53.682 180 39.287 168 38.877 168 42.612
168 53.888 181 39.449 169 39.049 169 42.741
169 54.099 182 39.613 170 39.224 170 42.878
170 54.288 183 39.78 171 39.401 171 43.006
171 54.49 184 39.946 172 39.576 172 43.136
172 54.694 185 40.114 173 39.753 173 43.262
173 54.897 186 40.271 174 39.919 174 43.398
174 55.104 187 40.435 175 40.089 175 43.526
77
175 55.297 188 40.597 176 40.25 176 43.664
176 55.492 189 40.752 177 40.418 177 43.789
177 55.683 190 40.916 178 40.567 178 43.921
178 55.881 191 41.092 179 40.727 179 44.059
179 56.08 192 41.253 180 40.899 180 44.187
180 56.271 193 41.426 181 41.069 181 44.318
181 56.465 194 41.564 182 41.211 182 44.444
182 56.657 195 41.691 183 41.355 183 44.578
183 56.858 196 41.853 184 41.52 184 44.706
184 57.049 197 42.02 185 41.672 185 44.833
185 57.241 198 42.177 186 41.838 186 44.961
186 57.431 199 42.316 187 42 187 45.092
187 57.612 200 42.467 188 42.157 188 45.217
188 57.807 201 42.605 189 42.313 189 45.345
189 57.998 202 42.756 190 42.463 190 45.482
190 58.183 203 42.899 191 42.623 191 45.615
191 58.367 204 43.043 192 42.772 192 45.754
192 58.551 205 43.209 193 42.925 193 45.884
193 58.746 206 43.344 194 43.088 194 46.011
194 58.934 207 43.486 195 43.216 195 46.138
195 59.112 208 43.628 196 43.381 196 46.262
196 59.288 209 43.767 197 43.532 197 46.394
197 59.471 210 43.909 198 43.68 198 46.53
198 59.656 211 44.045 199 43.823 199 46.657
199 59.838 212 44.179 200 43.97 200 46.78
200 60.014 213 44.311 201 44.103 201 46.901
78
201 60.196 214 44.445 202 44.248 202 47.028
202 60.389 215 44.58 203 44.38 203 47.147
203 60.561 216 44.712 204 44.514 204 47.287
204 60.732 217 44.799 205 44.653 205 47.408
205 60.913 218 44.937 206 44.784 206 47.535
206 61.098 219 45.061 207 44.927 207 47.667
207 61.286 220 45.193 208 45.053 208 47.789
208 61.45 221 45.32 209 45.18 209 47.914
209 61.624 222 45.445 210 45.308 210 48.044
210 61.751 223 45.577 211 45.437 211 48.168
211 61.917 224 45.696 212 45.571 212 48.29
212 62.102 225 45.827 213 45.697 213 48.418
213 62.278 226 45.937 214 45.824 214 48.544
214 62.451 227 46.063 215 45.95 215 48.662
215 62.628 228 46.197 216 46.079 216 48.788
216 62.802 229 46.323 217 46.208 217 48.91
217 62.973 230 46.444 218 46.325 218 49.034
218 63.144 231 46.573 219 46.451 219 49.153
219 63.322 232 46.692 220 46.58 220 49.275
220 63.497 233 46.818 221 46.698 221 49.41
221 63.67 234 46.933 222 46.825 222 49.523
222 63.835 235 47.056 223 46.94 223 49.65
223 64.01 236 47.176 224 47.061 224 49.764
224 64.178 237 47.297 225 47.181 225 49.887
225 64.353 238 47.421 226 47.298 226 50.005
226 64.512 239 47.534 227 47.421 227 50.131
79
227 64.69 240 47.654 228 47.531 228 50.251
228 64.86 241 47.772 229 47.659 229 50.362
229 65.026 242 47.889 230 47.773 230 50.487
230 65.197 243 48.007 231 47.886 231 50.604
231 65.355 244 48.122 232 47.999 232 50.724
232 65.525 245 48.23 233 48.118 233 50.843
233 65.695 246 48.344 234 48.233 234 50.964
234 65.854 247 48.455 235 48.348 235 51.079
235 66.026 248 48.57 236 48.46 236 51.195
236 66.192 249 48.683 237 48.571 237 51.316
237 66.357 250 48.796 238 48.688 238 51.441
238 66.52 251 48.906 239 48.804 239 51.543
239 66.678 252 49.021 240 48.923 240 51.671
240 66.845 253 49.125 241 49.033 241 51.788
241 67.014 254 49.238 242 49.141 242 51.893
242 67.169 255 49.35 243 49.261 243 52.02
243 67.335 256 49.459 244 49.37 244 52.138
244 67.495 257 49.574 245 49.481 245 52.25
245 67.658 258 49.682 246 49.593 246 52.365
246 67.818 259 49.781 247 49.711 247 52.49
247 67.977 260 49.895 248 49.812 248 52.604
248 68.137 261 49.999 249 49.932 249 52.721
249 68.299 262 50.114 250 50.042 250 52.83
250 68.46 263 50.214 251 50.155 251 52.947
251 68.614 264 50.328 252 50.256 252 53.072
252 68.776 265 50.442 253 50.367 253 53.188
80
253 68.935 266 50.547 254 50.477 254 53.303
254 69.091 267 50.661 255 50.592 255 53.417
255 69.255 268 50.772 256 50.692 256 53.536
256 69.412 269 50.874 257 50.809 257 53.657
257 69.566 270 50.981 258 50.922 258 53.775
258 69.721 271 51.091 259 51.03 259 53.893
259 69.885 272 51.201 260 51.141 260 54.021
260 70.039 273 51.306 261 51.246 261 54.145
261 70.192 274 51.414 262 51.359 262 54.254
262 70.351 275 51.528 263 51.473 263 54.379
263 70.505 276 51.63 264 51.587 264 54.479
264 70.659 277 51.734 265 51.692 265 54.6
265 70.811 278 51.848 266 51.797 266 54.72
266 70.968 279 51.937 267 51.901 267 54.821
267 71.126 280 52.05 268 52.012 268 54.919
268 71.277 281 52.158 269 52.121 269 55.012
269 71.428 282 52.265 270 52.234 270 55.114
270 71.579 283 52.373 271 52.335 271 55.224
271 71.732 284 52.467 272 52.451 272 55.329
272 71.887 285 52.576 273 52.549 273 55.437
273 72.035 286 52.683 274 52.659 274 55.537
274 72.182 287 52.772 275 52.771 275 55.642
275 72.336 288 52.876 276 52.878 276 55.747
276 72.485 289 52.983 277 52.987 277 55.859
277 72.636 290 53.086 278 53.104 278 55.971
278 72.793 291 53.193 279 53.212 279 56.079
81
279 72.938 292 53.288 280 53.316 280 56.183
280 73.087 293 53.389 281 53.426 281 56.289
281 73.237 294 53.484 282 53.525 282 56.397
282 73.383 295 53.603 283 53.639 283 56.502
283 73.533 296 53.699 284 53.744 284 56.61
284 73.677 297 53.809 285 53.854 285 56.713
285 73.827 298 53.918 286 53.969 286 56.816
286 73.972 299 54.007 287 54.069 287 56.919
287 74.118 300 54.118 288 54.186 288 57.022
288 74.266 301 54.22 289 54.297 289 57.126
289 74.411 302 54.329 290 54.401 290 57.229
290 74.558 303 54.422 291 54.511 291 57.34
291 74.71 304 54.521 292 54.613 292 57.448
292 74.85 305 54.644 293 54.726 293 57.548
293 74.994 306 54.744 294 54.834 294 57.653
294 75.135 307 54.853 295 54.941 295 57.761
295 75.284 308 54.955 296 55.05 296 57.859
296 75.433 309 55.062 297 55.165 297 57.97
297 75.566 310 55.166 298 55.265 298 58.063
298 75.718 311 55.263 299 55.377 299 58.171
299 75.854 312 55.369 300 55.49 300 58.27
300 75.993 313 55.472 301 55.591 301 58.376
301 76.14 314 55.569 302 55.7 302 58.483
302 76.276 315 55.687 303 55.816 303 58.586
303 76.419 316 55.784 304 55.911 304 58.687
304 76.562 317 55.894 305 56.032 305 58.79
82
305 76.701 318 55.985 306 56.144 306 58.885
306 76.839 319 56.098 307 56.251 307 58.986
307 76.983 320 56.198 308 56.364 308 59.09
308 77.121 321 56.31 309 56.483 309 59.189
309 77.264 322 56.418 310 56.6 310 59.294
310 77.402 323 56.512 311 56.722 311 59.396
311 77.537 324 56.63 312 56.832 312 59.502
312 77.68 325 56.738 313 56.951 313 59.598
313 77.822 326 56.84 314 57.044 314 59.703
314 77.955 327 56.955 315 57.148 315 59.798
315 78.097 328 57.063 316 57.262 316 59.896
316 78.233 329 57.185 317 57.338 317 60.001
317 78.371 330 57.281 318 57.444 318 60.096
318 78.512 331 57.384 319 57.556 319 60.195
319 78.649 332 57.494 320 57.677 320 60.298
320 78.786 333 57.599 321 57.779 321 60.395
321 78.922 334 57.69 322 57.898 322 60.492
322 79.05 335 57.774 323 58.007 323 60.591
323 79.191 336 57.866 324 58.125 324 60.696
324 79.322 337 57.964 325 58.237 325 60.79
325 79.462 338 58.071 326 58.339 326 60.898
326 79.602 339 58.188 327 58.451 327 60.988
327 79.726 340 58.293 328 58.574 328 61.1
328 79.873 341 58.395 329 58.696 329 61.19
329 80.005 342 58.499 330 58.805 330 61.294
330 80.137 343 58.599 331 58.924 331 61.396
83
331 80.274 344 58.715 332 59.049 332 61.497
332 80.397 345 58.813 333 59.156 333 61.591
333 80.539 346 58.931 334 59.265 334 61.703
334 80.662 347 59.029 335 59.388 335 61.789
335 80.799 348 59.148 336 59.5 336 61.9
336 80.929 349 59.261 337 59.602 337 61.987
337 81.061 350 59.368 338 59.718 338 62.082
338 81.198 351 59.474 339 59.82 339 62.178
339 81.328 352 59.588 340 59.928 340 62.282
340 81.458 353 59.699 341 60.006 341 62.375
341 81.586 354 59.814 342 60.108 342 62.477
342 81.726 355 59.924 343 60.222 343 62.58
343 81.843 356 60.032 344 60.325 344 62.678
344 81.981 357 60.136 345 60.443 345 62.778
345 82.106 358 60.245 346 60.546 346 62.882
346 82.242 359 60.358 347 60.652 347 62.974
347 82.361 360 60.471 348 60.757 348 63.072
348 82.502 361 60.583 349 60.864 349 63.167
349 82.626 362 60.673 350 60.976 350 63.272
350 82.758 363 60.756 351 61.072 351 63.366
351 82.884 364 60.883 352 61.178 352 63.461
352 83.011 365 60.976 353 61.288 353 63.557
353 83.145 366 61.086 354 61.396 354 63.657
354 83.273 367 61.192 355 61.494 355 63.76
355 83.395 368 61.294 356 61.603 356 63.86
356 83.529 369 61.398 357 61.712 357 63.956
84
357 83.659 370 61.499 358 61.805 358 64.054
358 83.781 371 61.603 359 61.912 359 64.153
359 83.912 372 61.698 360 62.016 360 64.259
360 84.038 373 61.795 361 62.119 361 64.339
361 84.165 374 61.899 362 62.229 362 64.445
362 84.295 375 62 363 62.326 363 64.532
363 84.419 376 62.09 364 62.42 364 64.624
364 84.548 377 62.19 365 62.522 365 64.723
365 84.667 378 62.287 366 62.609 366 64.815
366 84.792 379 62.357 367 62.718 367 64.917
367 84.922 380 62.458 368 62.807 368 65.012
368 85.04 381 62.548 369 62.899 369 65.112
369 85.167 382 62.647 370 62.991 370 65.211
370 85.285 383 62.733 371 63.087 371 65.309
371 85.408 384 62.831 372 63.176 372 65.411
372 85.537 385 62.912 373 63.268 373 65.501
373 85.662 386 63 374 63.362 374 65.586
374 85.78 387 63.097 375 63.454 375 65.687
375 85.904 388 63.186 376 63.542 376 65.782
376 86.027 389 63.277 377 63.639 377 65.867
377 86.16 390 63.361 378 63.739 378 65.966
378 86.278 391 63.447 379 63.828 379 66.054
379 86.399 392 63.526 380 63.92 380 66.146
380 86.519 393 63.624 381 64.005 381 66.24
381 86.646 394 63.711 382 64.091 382 66.329
382 86.763 395 63.803 383 64.193 383 66.425
85
383 86.888 396 63.892 384 64.272 384 66.522
384 87.005 397 63.971 385 64.366 385 66.612
385 87.136 398 64.056 386 64.456 386 66.717
386 87.248 399 64.158 387 64.545 387 66.806
387 87.373 400 64.242 388 64.635 388 66.901
388 87.485 401 64.338 389 64.722 389 66.992
389 87.615 402 64.425 390 64.818 390 67.087
390 87.725 403 64.513 391 64.904 391 67.197
391 87.849 404 64.602 392 64.996 392 67.288
392 87.966 405 64.683 393 65.085 393 67.39
393 88.09 406 64.784 394 65.172 394 67.488
394 88.201 407 64.861 395 65.264 395 67.582
395 88.322 408 64.941 396 65.36 396 67.676
396 88.442 409 65.038 397 65.439 397 67.77
397 88.558 410 65.108 398 65.533 398 67.87
398 88.67 411 65.209 399 65.623 399 67.962
399 88.795 412 65.299 400 65.711 400 68.06
400 88.905 413 65.372 401 65.794 401 68.169
401 89.033 414 65.443 402 65.878 402 68.268
402 89.138 415 65.554 403 65.964 403 68.363
403 89.26 416 65.626 404 66.053 404 68.453
404 89.372 417 65.714 405 66.146 405 68.539
405 89.489 418 65.798 406 66.232 406 68.634
406 89.603 419 65.867 407 66.312 407 68.732
407 89.713 420 65.961 408 66.393 408 68.821
408 89.837 421 66.044 409 66.48 409 68.917
86
409 89.948 422 66.125 410 66.567 410 68.999
410 90.063 423 66.204 411 66.645 411 69.091
411 90.171 424 66.281 412 66.73 412 69.183
412 90.294 425 66.351 413 66.812 413 69.278
413 90.402 426 66.428 414 66.896 414 69.379
414 90.518 427 66.514 415 66.977 415 69.475
415 90.627 428 66.592 416 67.07 416 69.569
416 90.742 429 66.669 417 67.142 417 69.657
417 90.856 430 66.75 418 67.223 418 69.756
418 90.965 431 66.835 419 67.315 419 69.848
419 91.082 432 66.905 420 67.395 420 69.938
420 91.192 433 66.986 421 67.478 421 70.03
421 91.306 434 67.069 422 67.563 422 70.129
422 91.419 435 67.157 423 67.644 423 70.227
423 91.534 436 67.235 424 67.734 424 70.363
424 91.64 437 67.328 425 67.821 425 70.459
425 91.755 438 67.401 426 67.908 426 70.575
426 91.874 439 67.485 427 67.992 427 70.676
427 91.972 440 67.561 428 68.091 428 70.774
428 92.093 441 67.63 429 68.193 429 70.874
429 92.195 442 67.719 430 68.288 430 70.966
430 92.31 443 67.793 431 71.162
431 92.413 444 67.873 432 71.265
432 92.53 445 67.969 433 71.369
433 92.632 446 68.032 434 71.474
434 92.747 447 68.117
87
435 92.855 448 68.192
436 92.97 449 68.279
437 93.074 450 68.354
438 93.183 451 68.432
439 93.295 452 68.516
440 93.402 453 68.6
441 93.511 454 68.68
442 93.624 455 68.765
443 93.731 456 68.85
444 93.845 457 68.942
445 93.944 458 69.024
446 94.069 459 69.114
447 94.164 460 69.2
448 94.282 461 69.275
449 94.387 462 69.398
450 94.493
451 94.605
452 94.709
453 94.821
454 94.92
455 95.04
456 95.148
457 95.257
458 95.371
459 95.489
460 95.606
88
461 95.738
462 95.868
N 5 4 4 4
m 2 1 2 2
run 1 1 1 2
frame %F frame %F frame %F frame %F
39 0.592 34 1.485 36 2.234 33 2.12
40 2.047 35 3.526 37 4.035 34 3.751
41 2.86 36 5.178 38 5.176 35 4.79
42 3.498 37 6.601 39 6.086 36 5.676
43 4.055 38 7.854 40 6.814 37 6.388
44 4.541 39 8.992 41 7.468 38 7.019
45 4.987 40 9.998 42 8.052 39 7.558
46 5.399 41 10.94 43 8.593 40 8.067
47 5.778 42 11.813 44 9.098 41 8.542
48 6.132 43 12.647 45 9.565 42 8.973
49 6.494 44 13.42 46 10.008 43 9.393
50 6.851 45 14.156 47 10.446 44 9.773
51 7.182 46 14.873 48 10.863 45 10.136
52 7.466 47 15.575 49 11.266 46 10.5
53 7.751 48 16.221 50 11.632 47 10.84
54 8.031 49 16.889 51 11.989 48 11.199
55 8.314 50 17.483 52 12.36 49 11.529
56 8.589 51 18.068 53 12.718 50 11.861
57 8.863 52 18.704 54 13.036 51 12.205
58 9.136 53 19.257 55 13.342 52 12.53
89
59 9.418 54 19.786 56 13.67 53 12.815
60 9.663 55 20.333 57 13.979 54 13.089
61 9.861 56 20.883 58 14.296 55 13.368
62 10.047 57 21.426 59 14.607 56 13.645
63 10.21 58 21.965 60 14.924 57 13.934
64 10.369 59 22.31 61 15.194 58 14.207
65 10.528 60 22.565 62 15.427 59 14.475
66 10.688 61 22.879 63 15.655 60 14.756
67 10.838 62 23.22 64 15.882 61 15.016
68 11 63 23.526 65 16.1 62 15.286
69 11.162 64 23.82 66 16.32 63 15.577
70 11.334 65 24.107 67 16.539 64 15.855
71 11.47 66 24.391 68 16.757 65 16.072
72 11.601 67 24.671 69 16.983 66 16.253
73 11.752 68 24.941 70 17.199 67 16.474
74 11.875 69 25.206 71 17.41 68 16.669
75 11.987 70 25.467 72 17.619 69 16.858
76 12.095 71 25.716 73 17.833 70 17.05
77 12.179 72 25.972 74 18.041 71 17.25
78 12.281 73 26.214 75 18.246 72 17.461
79 12.373 74 26.449 76 18.429 73 17.659
80 12.459 75 26.681 77 18.616 74 17.849
81 12.551 76 26.913 78 18.798 75 18.034
82 12.633 77 27.141 79 18.984 76 18.196
83 12.718 78 27.366 80 19.162 77 18.359
84 12.801 79 27.587 81 19.339 78 18.526
90
85 12.876 80 27.806 82 19.514 79 18.688
86 12.951 81 28.014 83 19.679 80 18.849
87 13.033 82 28.232 84 19.839 81 19.015
88 13.105 83 28.446 85 19.985 82 19.171
89 13.181 84 28.645 86 20.131 83 19.335
90 13.253 85 28.854 87 20.272 84 19.483
91 13.321 86 29.053 88 20.411 85 19.614
92 13.397 87 29.257 89 20.544 86 19.755
93 13.464 88 29.456 90 20.675 87 19.893
94 13.543 89 29.641 91 20.807 88 20.021
95 13.607 90 29.83 92 20.926 89 20.145
96 13.672 91 30.021 93 21.044 90 20.277
97 13.751 92 30.206 94 21.164 91 20.398
98 13.812 93 30.387 95 21.277 92 20.523
99 13.88 94 30.568 96 21.391 93 20.639
100 13.95 95 30.75 97 21.5 94 20.753
101 14.012 96 30.927 98 21.612 95 20.863
102 14.078 97 31.105 99 21.713 96 20.975
103 14.158 98 31.277 100 21.819 97 21.075
104 14.221 99 31.452 101 21.924 98 21.187
105 14.284 100 31.629 102 22.023 99 21.289
106 14.361 101 31.798 103 22.13 100 21.393
107 14.422 102 31.965 104 22.231 101 21.491
108 14.493 103 32.135 105 22.336 102 21.592
109 14.554 104 32.298 106 22.431 103 21.687
110 14.63 105 32.459 107 22.525 104 21.779
91
111 14.702 106 32.628 108 22.621 105 21.874
112 14.766 107 32.789 109 22.719 106 21.964
113 14.839 108 32.952 110 22.816 107 22.057
114 14.911 109 33.113 111 22.907 108 22.149
115 14.975 110 33.267 112 23.007 109 22.239
116 15.046 111 33.433 113 23.098 110 22.323
117 15.117 112 33.583 114 23.191 111 22.417
118 15.182 113 33.738 115 23.283 112 22.504
119 15.252 114 33.906 116 23.374 113 22.593
120 15.318 115 34.053 117 23.46 114 22.68
121 15.397 116 34.195 118 23.558 115 22.765
122 15.462 117 34.353 119 23.647 116 22.856
123 15.534 118 34.501 120 23.736 117 22.939
124 15.609 119 34.651 121 23.82 118 23.022
125 15.674 120 34.793 122 23.915 119 23.112
126 15.742 121 34.944 123 24.006 120 23.189
127 15.806 122 35.087 124 24.098 121 23.28
128 15.876 123 35.227 125 24.194 122 23.368
129 15.937 124 35.384 126 24.285 123 23.449
130 16.009 125 35.518 127 24.374 124 23.535
131 16.082 126 35.666 128 24.463 125 23.62
132 16.158 127 35.811 129 24.556 126 23.704
133 16.219 128 35.949 130 24.647 127 23.789
134 16.298 129 36.087 131 24.739 128 23.871
135 16.373 130 36.221 132 24.827 129 23.951
136 16.421 131 36.357 133 24.92 130 24.042
92
137 16.484 132 36.504 134 25.009 131 24.121
138 16.542 133 36.638 135 25.073 132 24.208
139 16.59 134 36.77 136 25.16 133 24.29
140 16.652 135 36.863 137 25.256 134 24.373
141 16.722 136 36.996 138 25.343 135 24.464
142 16.783 137 37.125 139 25.433 136 24.533
143 16.842 138 37.258 140 25.525 137 24.623
144 16.895 139 37.387 141 25.613 138 24.704
145 16.94 140 37.522 142 25.71 139 24.783
146 16.991 141 37.651 143 25.801 140 24.873
147 17.046 142 37.78 144 25.886 141 24.956
148 17.072 143 37.911 145 25.974 142 25.034
149 17.13 144 38.037 146 26.069 143 25.123
150 17.177 145 38.169 147 26.156 144 25.201
151 17.221 146 38.296 148 26.249 145 25.291
152 17.272 147 38.424 149 26.331 146 25.364
153 17.321 148 38.548 150 26.43 147 25.454
154 17.369 149 38.679 151 26.514 148 25.536
155 17.412 150 38.802 152 26.606 149 25.617
156 17.46 151 38.928 153 26.702 150 25.69
157 17.506 152 39.048 154 26.786 151 25.771
158 17.572 153 39.176 155 26.884 152 25.855
159 17.603 154 39.294 156 26.984 153 25.935
160 17.644 155 39.416 157 27.087 154 26.016
161 17.7 156 39.539 158 27.177 155 26.107
162 17.743 157 39.661 159 27.279 156 26.186
93
163 17.785 158 39.78 160 27.355 157 26.266
164 17.832 159 39.907 161 27.431 158 26.342
165 17.877 160 40.023 162 27.507 159 26.425
166 17.915 161 40.147 163 27.6 160 26.503
167 17.963 162 40.264 164 27.681 161 26.585
168 18.002 163 40.383 165 27.78 162 26.67
169 18.059 164 40.503 166 27.86 163 26.758
170 18.089 165 40.617 167 27.942 164 26.846
171 18.13 166 40.738 168 28.03 165 26.93
172 18.171 167 40.85 169 28.117 166 27.015
173 18.21 168 40.969 170 28.205 167 27.096
174 18.258 169 41.081 171 28.289 168 27.173
175 18.295 170 41.194 172 28.371 169 27.256
176 18.334 171 41.313 173 28.462 170 27.338
177 18.377 172 41.432 174 28.544 171 27.407
178 18.412 173 41.546 175 28.626 172 27.491
179 18.457 174 41.657 176 28.709 173 27.562
180 18.509 175 41.769 177 28.792 174 27.639
181 18.546 176 41.886 178 28.875 175 27.709
182 18.584 177 41.997 179 28.957 176 27.778
183 18.623 178 42.114 180 29.032 177 27.845
184 18.669 179 42.228 181 29.103 178 27.917
185 18.704 180 42.332 182 29.178 179 27.998
186 18.758 181 42.446 183 29.247 180 28.067
187 18.797 182 42.544 184 29.321 181 28.14
188 18.837 183 42.657 185 29.392 182 28.22
94
189 18.879 184 42.762 186 29.462 183 28.291
190 18.926 185 42.877 187 29.537 184 28.37
191 18.967 186 42.976 188 29.607 185 28.445
192 19.009 187 43.092 189 29.675 186 28.52
193 19.059 188 43.19 190 29.736 187 28.594
194 19.105 189 43.299 191 29.805 188 28.663
195 19.15 190 43.403 192 29.881 189 28.744
196 19.197 191 43.518 193 29.944 190 28.805
197 19.235 192 43.621 194 30.011 191 28.872
198 19.282 193 43.729 195 30.073 192 28.934
199 19.329 194 43.844 196 30.138 193 29.003
200 19.365 195 43.948 197 30.206 194 29.069
201 19.415 196 44.056 198 30.269 195 29.144
202 19.457 197 44.151 199 30.336 196 29.211
203 19.508 198 44.261 200 30.399 197 29.281
204 19.551 199 44.358 201 30.464 198 29.339
205 19.598 200 44.459 202 30.523 199 29.398
206 19.64 201 44.566 203 30.584 200 29.462
207 19.682 202 44.667 204 30.646 201 29.521
208 19.732 203 44.775 205 30.704 202 29.587
209 19.777 204 44.876 206 30.766 203 29.647
210 19.824 205 44.984 207 30.828 204 29.712
211 19.875 206 45.074 208 30.888 205 29.776
212 19.912 207 45.183 209 30.948 206 29.814
213 19.963 208 45.281 210 31.007 207 29.876
214 20.003 209 45.385 211 31.065 208 29.939
95
215 20.05 210 45.488 212 31.129 209 29.996
216 20.089 211 45.583 213 31.185 210 30.061
217 20.14 212 45.69 214 31.242 211 30.116
218 20.18 213 45.782 215 31.303 212 30.176
219 20.224 214 45.881 216 31.36 213 30.235
220 20.272 215 45.981 217 31.417 214 30.291
221 20.315 216 46.08 218 31.482 215 30.348
222 20.364 217 46.183 219 31.531 216 30.407
223 20.41 218 46.277 220 31.585 217 30.461
224 20.452 219 46.381 221 31.642 218 30.521
225 20.501 220 46.478 222 31.695 219 30.579
226 20.551 221 46.609 223 31.755 220 30.636
227 20.595 222 46.668 224 31.812 221 30.689
228 20.642 223 46.765 225 31.868 222 30.741
229 20.691 224 46.868 226 31.92 223 30.804
230 20.736 225 46.963 227 31.984 224 30.858
231 20.787 226 47.06 228 32.038 225 30.912
232 20.84 227 47.157 229 32.107 226 30.967
233 20.887 228 47.255 230 32.148 227 31.023
234 20.933 229 47.352 231 32.208 228 31.077
235 20.973 230 47.443 232 32.261 229 31.122
236 21.026 231 47.548 233 32.321 230 31.183
237 21.067 232 47.665 234 32.376 231 31.227
238 21.115 233 47.741 235 32.428 232 31.29
239 21.166 234 47.826 236 32.484 233 31.34
240 21.205 235 47.919 237 32.536 234 31.395
96
241 21.246 236 48.016 238 32.595 235 31.44
242 21.292 237 48.109 239 32.648 236 31.493
243 21.331 238 48.204 240 32.705 237 31.551
244 21.378 239 48.291 241 32.761 238 31.6
245 21.422 240 48.381 242 32.816 239 31.652
246 21.462 241 48.476 243 32.865 240 31.704
247 21.508 242 48.567 244 32.926 241 31.753
248 21.559 243 48.664 245 32.981 242 31.801
249 21.602 244 48.754 246 33.033 243 31.857
250 21.658 245 48.845 247 33.09 244 31.907
251 21.704 246 48.93 248 33.149 245 31.961
252 21.743 247 49.026 249 33.199 246 32.009
253 21.783 248 49.126 250 33.256 247 32.059
254 21.821 249 49.209 251 33.31 248 32.109
255 21.851 250 49.295 252 33.368 249 32.162
256 21.892 251 49.381 253 33.425 250 32.214
257 21.936 252 49.474 254 33.476 251 32.261
258 21.982 253 49.566 255 33.537 252 32.313
259 22.023 254 49.653 256 33.589 253 32.359
260 22.06 255 49.747 257 33.652 254 32.412
261 22.092 256 49.827 258 33.698 255 32.464
262 22.113 257 49.916 259 33.759 256 32.518
263 22.149 258 50.006 260 33.809 257 32.567
264 22.183 259 50.099 261 33.866 258 32.621
265 22.214 260 50.193 262 33.915 259 32.667
266 22.248 261 50.27 263 33.978 260 32.718
97
267 22.27 262 50.36 264 34.027 261 32.767
268 22.31 263 50.452 265 34.085 262 32.823
269 22.343 264 50.538 266 34.139 263 32.871
270 22.38 265 50.62 267 34.197 264 32.922
271 22.41 266 50.706 268 34.255 265 32.969
272 22.446 267 50.802 269 34.315 266 33.018
273 22.476 268 50.885 270 34.365 267 33.068
274 22.515 269 50.967 271 34.419 268 33.118
275 22.545 270 51.06 272 34.478 269 33.163
276 22.574 271 51.148 273 34.535 270 33.222
277 22.61 272 51.224 274 34.59 271 33.27
278 22.645 273 51.317 275 34.653 272 33.32
279 22.68 274 51.406 276 34.702 273 33.365
280 22.709 275 51.494 277 34.758 274 33.419
281 22.745 276 51.586 278 34.819 275 33.468
282 22.772 277 51.665 279 34.878 276 33.52
283 22.804 278 51.755 280 34.928 277 33.566
284 22.836 279 51.842 281 34.982 278 33.619
285 22.871 280 51.927 282 35.041 279 33.669
286 22.903 281 52.009 283 35.09 280 33.722
287 22.936 282 52.104 284 35.148 281 33.77
288 22.967 283 52.189 285 35.202 282 33.825
289 22.994 284 52.274 286 35.255 283 33.866
290 23.027 285 52.36 287 35.318 284 33.925
291 23.062 286 52.448 288 35.369 285 33.969
292 23.087 287 52.527 289 35.437 286 34.028
98
293 23.12 288 52.619 290 35.485 287 34.078
294 23.15 289 52.701 291 35.545 288 34.125
295 23.186 290 52.787 292 35.602 289 34.18
296 23.213 291 52.874 293 35.656 290 34.217
297 23.239 292 52.961 294 35.715 291 34.279
298 23.27 293 53.043 295 35.772 292 34.328
299 23.305 294 53.126 296 35.83 293 34.377
300 23.34 295 53.212 297 35.88 294 34.423
301 23.365 296 53.295 298 35.941 295 34.481
302 23.396 297 53.377 299 35.995 296 34.529
303 23.429 298 53.462 300 36.045 297 34.583
304 23.455 299 53.549 301 36.114 298 34.634
305 23.488 300 53.629 302 36.176 299 34.68
306 23.514 301 53.712 303 36.229 300 34.729
307 23.549 302 53.795 304 36.293 301 34.778
308 23.574 303 53.865 305 36.353 302 34.835
309 23.607 304 53.962 306 36.411 303 34.882
310 23.634 305 54.03 307 36.486 304 34.931
311 23.669 306 54.12 308 36.55 305 34.979
312 23.696 307 54.21 309 36.607 306 35.026
313 23.716 308 54.286 310 36.659 307 35.082
314 23.749 309 54.373 311 36.718 308 35.127
315 23.785 310 54.45 312 36.766 309 35.18
316 23.809 311 54.53 313 36.8 310 35.236
317 23.842 312 54.612 314 36.857 311 35.279
318 23.871 313 54.691 315 36.918 312 35.33
99
319 23.907 314 54.783 316 36.97 313 35.387
320 23.933 315 54.857 317 37.026 314 35.438
321 23.964 316 54.951 318 37.083 315 35.486
322 23.993 317 55.016 319 37.145 316 35.544
323 24.025 318 55.112 320 37.197 317 35.591
324 24.056 319 55.186 321 37.256 318 35.646
325 24.086 320 55.271 322 37.309 319 35.697
326 24.116 321 55.355 323 37.367 320 35.74
327 24.152 322 55.433 324 37.42 321 35.785
328 24.177 323 55.512 325 37.483 322 35.847
329 24.21 324 55.595 326 37.535 323 35.896
330 24.248 325 55.677 327 37.585 324 35.939
331 24.276 326 55.758 328 37.645 325 35.996
332 24.307 327 55.837 329 37.703 326 36.056
333 24.34 328 55.927 330 37.756 327 36.111
334 24.372 329 56 331 37.81 328 36.161
335 24.405 330 56.085 332 37.87 329 36.22
336 24.433 331 56.163 333 37.924 330 36.275
337 24.462 332 56.248 334 37.978 331 36.313
338 24.498 333 56.326 335 38.039 332 36.369
339 24.531 334 56.41 336 38.085 333 36.413
340 24.557 335 56.489 337 38.138 334 36.454
341 24.595 336 56.572 338 38.206 335 36.495
342 24.628 337 56.654 339 38.249 336 36.555
343 24.657 338 56.73 340 38.312 337 36.588
344 24.704 339 56.814 341 38.364 338 36.634
100
345 24.72 340 56.891 342 38.42 339 36.686
346 24.764 341 56.971 343 38.478 340 36.737
347 24.789 342 57.057 344 38.541 341 36.79
348 24.82 343 57.132 345 38.59 342 36.84
349 24.854 344 57.212 346 38.639 343 36.895
350 24.887 345 57.293 347 38.687 344 36.953
351 24.917 346 57.378 348 38.733 345 36.998
352 24.951 347 57.453 349 38.78 346 37.054
353 24.986 348 57.534 350 38.83 347 37.1
354 25.016 349 57.609 351 38.882 348 37.147
355 25.054 350 57.687 352 38.93 349 37.187
356 25.089 351 57.767 353 38.984 350 37.24
357 25.117 352 57.844 354 39.032 351 37.278
358 25.157 353 57.929 355 39.081 352 37.319
359 25.184 354 58.006 356 39.133 353 37.37
360 25.22 355 58.089 357 39.178 354 37.422
361 25.263 356 58.169 358 39.222 355 37.466
362 25.293 357 58.241 359 39.279 356 37.521
363 25.327 358 58.318 360 39.323 357 37.564
364 25.357 359 58.394 361 39.375 358 37.614
365 25.389 360 58.474 362 39.417 359 37.669
366 25.425 361 58.55 363 39.465 360 37.717
367 25.456 362 58.628 364 39.513 361 37.764
368 25.493 363 58.712 365 39.556 362 37.807
369 25.523 364 58.788 366 39.608 363 37.867
370 25.552 365 58.872 367 39.655 364 37.909
101
371 25.587 366 58.945 368 39.703 365 37.953
372 25.628 367 59.023 369 39.748 366 38.011
373 25.656 368 59.104 370 39.8 367 38.054
374 25.695 369 59.178 371 39.85 368 38.098
375 25.727 370 59.255 372 39.893 369 38.145
376 25.766 371 59.331 373 39.938 370 38.192
377 25.796 372 59.419 374 39.988 371 38.249
378 25.826 373 59.49 375 40.036 372 38.286
379 25.861 374 59.564 376 40.08 373 38.338
380 25.906 375 59.644 377 40.13 374 38.382
381 25.937 376 59.725 378 40.174 375 38.423
382 25.973 377 59.801 379 40.221 376 38.466
383 26.008 378 59.884 380 40.271 377 38.507
384 26.048 379 59.956 381 40.313 378 38.557
385 26.076 380 60.032 382 40.367 379 38.6
386 26.107 381 60.112 383 40.41 380 38.644
387 26.141 382 60.19 384 40.454 381 38.693
388 26.173 383 60.266 385 40.509 382 38.741
389 26.205 384 60.343 386 40.557 383 38.792
390 26.234 385 60.417 387 40.606 384 38.837
391 26.267 386 60.498 388 40.653 385 38.882
392 26.3 387 60.567 389 40.701 386 38.945
393 26.336 388 60.653 390 40.753 387 38.988
394 26.355 389 60.719 391 40.799 388 39.023
395 26.391 390 60.808 392 40.844 389 39.06
396 26.424 391 60.883 393 40.884 390 39.099
102
397 26.457 392 60.964 394 40.94 391 39.139
398 26.484 393 61.037 395 40.984 392 39.184
399 26.514 394 61.117 396 41.037 393 39.226
400 26.551 395 61.197 397 41.084 394 39.265
401 26.578 396 61.263 398 41.13 395 39.313
402 26.61 397 61.347 399 41.17 396 39.357
403 26.641 398 61.422 400 41.218 397 39.4
404 26.673 399 61.494 401 41.254 398 39.443
405 26.713 400 61.581 402 41.307 399 39.49
406 26.737 401 61.651 403 41.359 400 39.534
407 26.769 402 61.736 404 41.4 401 39.579
408 26.805 403 61.812 405 41.445 402 39.619
409 26.845 404 61.889 406 41.499 403 39.665
410 26.886 405 61.971 407 41.537 404 39.724
411 26.914 406 62.041 408 41.59 405 39.755
412 26.94 407 62.127 409 41.632 406 39.799
413 26.974 408 62.271 410 41.681 407 39.84
414 27.005 409 62.354 411 41.728 408 39.886
415 27.033 410 62.427 412 41.776 409 39.92
416 27.06 411 62.501 413 41.812 410 39.974
417 27.086 412 62.589 414 41.863 411 40.008
418 27.108 413 62.663 415 41.906 412 40.048
419 27.124 414 62.76 416 41.951 413 40.104
420 27.154 415 62.846 417 42.002 414 40.137
421 27.182 418 42.043 415 40.191
422 27.205 419 42.093 416 40.225
103
423 27.233 420 42.136 417 40.264
424 27.262 421 42.182 418 40.312
425 27.285 422 42.228 419 40.351
426 27.31 423 42.274 420 40.391
427 27.339 424 42.317 421 40.438
428 27.373 425 42.364 422 40.48
429 27.39 426 42.415 423 40.525
430 27.413 427 42.456 424 40.571
431 27.436 428 42.507 425 40.609
432 27.474 429 42.549 426 40.65
433 27.502 430 42.594 427 40.7
434 27.536 431 42.637 428 40.742
435 27.565 432 42.684 429 40.786
436 27.588 433 42.72 430 40.826
437 27.625 434 42.769 431 40.869
438 27.647 435 42.82 432 40.909
439 27.677 436 42.86 433 40.962
440 27.707 437 42.91 434 41.003
441 27.735 438 42.944 435 41.043
442 27.76 439 42.997 436 41.09
443 27.794 440 43.041 437 41.134
444 27.824 441 43.082 438 41.171
445 27.853 442 43.127 439 41.214
446 27.879 443 43.178 440 41.259
447 27.911 444 43.216 441 41.299
448 27.942 445 43.271 442 41.342
104
449 27.972 446 43.311 443 41.381
450 28.003 447 43.349 444 41.428
451 28.03 448 43.404 445 41.461
452 28.065 449 43.445 446 41.51
453 28.092 450 43.486 447 41.553
454 28.113 451 43.534 448 41.587
455 28.144 452 43.573 449 41.629
456 28.173 453 43.619 450 41.674
457 28.191 454 43.667 451 41.711
458 28.232 455 43.712 452 41.753
459 28.255 456 43.762 453 41.799
460 28.284 457 43.807 454 41.836
461 28.315 458 43.85 455 41.886
462 28.339 459 43.894 456 41.921
463 28.369 460 43.937 457 41.967
464 28.401 461 43.98 458 42.004
465 28.432 462 44.027 459 42.043
466 28.457 463 44.071 460 42.081
467 28.48 464 44.118 461 42.124
468 28.51 465 44.163 462 42.166
469 28.538 466 44.205 463 42.207
470 28.569 467 44.245 464 42.245
471 28.595 468 44.295 465 42.289
472 28.621 469 44.333 466 42.327
473 28.643 470 44.381 467 42.374
474 28.675 471 44.428 468 42.405
105
475 28.705 472 44.473 469 42.449
476 28.73 473 44.511 470 42.496
477 28.759 474 44.553 471 42.535
478 28.793 475 44.601 472 42.573
479 28.821 476 44.64 473 42.619
480 28.84 477 44.69 474 42.659
481 28.866 478 44.731 475 42.692
482 28.899 479 44.773 476 42.738
483 28.929 480 44.825 477 42.782
484 28.939 481 44.872 478 42.823
485 28.971 482 44.914 479 42.861
486 28.994 483 44.942 480 42.903
487 29.023 484 44.986 481 42.947
488 29.047 485 45.026 482 42.985
489 29.069 486 45.074 483 43.023
490 29.098 487 45.113 484 43.05
491 29.123 488 45.16 485 43.095
492 29.15 489 45.214 486 43.128
493 29.187 490 45.25 487 43.171
494 29.205 491 45.293 488 43.214
495 29.235 492 45.339 489 43.25
496 29.265 493 45.379 490 43.291
497 29.291 494 45.423 491 43.345
498 29.318 495 45.466 492 43.386
499 29.345 496 45.508 493 43.431
500 29.376 497 45.552 494 43.464
106
501 29.4 498 45.595 495 43.51
502 29.426 499 45.636 496 43.548
503 29.454 500 45.687 497 43.587
504 29.478 501 45.723 498 43.624
505 29.507 502 45.774 499 43.668
506 29.528 503 45.822 500 43.707
507 29.558 504 45.865 501 43.745
508 29.594 505 45.907 502 43.785
509 29.615 506 45.954 503 43.827
510 29.642 507 46.003 504 43.865
511 29.671 508 46.044 505 43.905
512 29.694 509 46.093 506 43.948
513 29.723 510 46.131 507 43.979
514 29.753 511 46.186 508 44.018
515 29.776 512 46.225 509 44.062
516 29.811 513 46.278 510 44.096
517 29.834 514 46.33 511 44.149
518 29.858 515 46.381 512 44.187
519 29.886 516 46.435 513 44.223
520 29.915 517 46.483 514 44.27
521 29.946 518 46.537 515 44.304
522 29.975 519 46.598 516 44.346
523 30 520 46.649 517 44.387
524 30.021 521 46.704 518 44.427
525 30.05 522 46.744 519 44.467
526 30.078 523 46.787 520 44.509
107
527 30.104 524 46.842 521 44.545
528 30.133 525 46.894 522 44.588
529 30.16 526 46.936 523 44.625
530 30.182 527 46.981 524 44.662
531 30.205 528 47.026 525 44.706
532 30.235 529 47.067 526 44.744
533 30.262 530 47.111 527 44.783
534 30.295 528 44.825
535 30.322 529 44.865
536 30.349 530 44.904
537 30.366 531 44.949
538 30.403 532 44.986
539 30.422 533 45.027
540 30.458 534 45.065
541 30.487 535 45.101
542 30.51 536 45.146
543 30.535 537 45.182
544 30.564 538 45.222
545 30.595 539 45.262
546 30.613 540 45.3
547 30.642 541 45.341
548 30.669 542 45.384
549 30.698 543 45.418
550 30.72 544 45.476
551 30.758 545 45.515
552 30.785 546 45.555
108
553 30.809 547 45.597
554 30.84 548 45.632
555 30.862 549 45.677
556 30.903 550 45.719
557 30.929 551 45.753
558 30.987 552 45.792
559 31.016 553 45.83
560 31.05 554 45.86
555 45.899
556 45.934
557 45.978
558 46.013
559 46.061
560 46.101
561 46.139
562 46.172
563 46.217
564 46.26
565 46.296
566 46.341
567 46.387
568 46.424
569 46.474
570 46.502
571 46.55
572 46.595
109
573 46.63
574 46.675
575 46.713
N 4 4
m 3 3
run 1 2
frame %F frame %F
37 1.257 35 0.644
38 3.018 36 1.804
39 3.316 37 2.627
40 3.659 38 3.046
41 3.916 39 3.413
42 4.121 40 3.762
43 4.323 41 4.003
44 4.517 42 4.247
45 4.734 43 4.469
46 4.941 44 4.72
47 5.151 45 4.939
48 5.326 46 5.139
49 5.469 47 5.311
50 5.629 48 5.493
51 5.801 49 5.662
52 5.947 50 5.846
53 6.071 51 5.996
54 6.192 52 6.146
55 6.282 53 6.27
110
56 6.385 54 6.408
57 6.511 55 6.54
58 6.632 56 6.688
59 6.767 57 6.826
60 6.884 58 6.965
61 7.009 59 7.087
62 7.12 60 7.228
63 7.246 61 7.344
64 7.37 62 7.449
65 7.469 63 7.545
66 7.58 64 7.651
67 7.668 65 7.76
68 7.762 66 7.884
69 7.853 67 8.014
70 7.944 68 8.132
71 8.042 69 8.257
72 8.148 70 8.392
73 8.25 71 8.515
74 8.362 72 8.634
75 8.492 73 8.764
76 8.6 74 8.874
77 8.705 75 8.967
78 8.794 76 9.076
79 8.894 77 9.184
80 8.986 78 9.291
81 9.064 79 9.403
111
82 9.151 80 9.477
83 9.236 81 9.571
84 9.328 82 9.671
85 9.417 83 9.763
86 9.508 84 9.871
87 9.601 85 9.964
88 9.691 86 10.058
89 9.789 87 10.131
90 9.869 88 10.208
91 9.958 89 10.291
92 10.036 90 10.388
93 10.133 91 10.477
94 10.221 92 10.571
95 10.295 93 10.671
96 10.369 94 10.762
97 10.426 95 10.836
98 10.491 96 10.93
99 10.561 97 11.018
100 10.634 98 11.112
101 10.715 99 11.212
102 10.791 100 11.3
103 10.868 101 11.387
104 10.951 102 11.441
105 11.026 103 11.503
106 11.105 104 11.571
107 11.184 105 11.639
112
108 11.265 106 11.698
109 11.35 107 11.756
110 11.448 108 11.831
111 11.527 109 11.9
112 11.601 110 11.979
113 11.677 111 12.052
114 11.748 112 12.132
115 11.803 113 12.214
116 11.87 114 12.283
117 11.928 115 12.356
118 11.989 116 12.431
119 12.053 117 12.516
120 12.106 118 12.594
121 12.163 119 12.678
122 12.231 120 12.778
123 12.295 121 12.84
124 12.369 122 12.918
125 12.439 123 12.985
126 12.521 124 13.039
127 12.58 125 13.124
128 12.645 126 13.183
129 12.703 127 13.242
130 12.761 128 13.295
131 12.808 129 13.346
132 12.877 130 13.403
133 12.958 131 13.432
113
134 13.007 132 13.496
135 13.077 133 13.547
136 13.175 134 13.622
137 13.244 135 13.661
138 13.291 136 13.717
139 13.324 137 13.767
140 13.358 138 13.82
141 13.409 139 13.879
142 13.462 140 13.905
143 13.519 141 13.955
144 13.591 142 14.007
145 13.659 143 14.065
146 13.709 144 14.113
147 13.752 145 14.16
148 13.797 146 14.199
149 13.832 147 14.24
150 13.873 148 14.28
151 13.933 149 14.333
152 13.983 150 14.38
153 14.044 151 14.43
154 14.089 152 14.482
155 14.124 153 14.537
156 14.166 154 14.581
157 14.217 155 14.618
158 14.268 156 14.665
159 14.329 157 14.708
114
160 14.386 158 14.748
161 14.447 159 14.8
162 14.498 160 14.85
163 14.542 161 14.902
164 14.579 162 14.941
165 14.612 163 14.984
166 14.659 164 15.016
167 14.695 165 15.047
168 14.741 166 15.086
169 14.777 167 15.126
170 14.825 168 15.145
171 14.879 169 15.174
172 14.921 170 15.206
173 14.958 171 15.232
174 15 172 15.269
175 15.041 173 15.295
176 15.082 174 15.326
177 15.128 175 15.357
178 15.166 176 15.39
179 15.219 177 15.428
180 15.258 178 15.474
181 15.298 179 15.495
182 15.335 180 15.549
183 15.379 181 15.589
184 15.415 182 15.631
185 15.449 183 15.668
115
186 15.49 184 15.694
187 15.515 185 15.718
188 15.545 186 15.737
189 15.578 187 15.755
190 15.609 188 15.78
191 15.626 189 15.816
192 15.654 190 15.836
193 15.683 191 15.87
194 15.714 192 15.895
195 15.742 193 15.93
196 15.773 194 15.954
197 15.788 195 15.987
198 15.822 196 16.019
199 15.85 197 16.045
200 15.883 198 16.067
201 15.918 199 16.103
202 15.95 200 16.133
203 15.975 201 16.162
204 16.004 202 16.195
205 16.038 203 16.221
206 16.07 204 16.237
207 16.109 205 16.256
208 16.129 206 16.284
209 16.158 207 16.307
210 16.2 208 16.333
211 16.239 209 16.357
116
212 16.267 210 16.382
213 16.294 211 16.405
214 16.33 212 16.426
215 16.362 213 16.457
216 16.392 214 16.486
217 16.419 215 16.509
218 16.457 216 16.541
219 16.476 217 16.577
220 16.495 218 16.613
221 16.525 219 16.645
222 16.55 220 16.68
223 16.573 221 16.717
224 16.592 222 16.748
225 16.622 223 16.778
226 16.65 224 16.804
227 16.679 225 16.835
228 16.699 226 16.857
229 16.725 227 16.882
230 16.746 228 16.908
231 16.773 229 16.915
232 16.799 230 16.95
233 16.823 231 16.977
234 16.851 232 17.003
235 16.88 233 17.026
236 16.907 234 17.055
237 16.938 235 17.074
117
238 16.965 236 17.098
239 16.99 237 17.136
240 17.026 238 17.168
241 17.06 239 17.189
242 17.087 240 17.227
243 17.115 241 17.244
244 17.136 242 17.267
245 17.155 243 17.291
246 17.172 244 17.315
247 17.198 245 17.344
248 17.217 246 17.374
249 17.248 247 17.391
250 17.273 248 17.421
251 17.301 249 17.443
252 17.325 250 17.465
253 17.343 251 17.488
254 17.364 252 17.513
255 17.385 253 17.539
256 17.41 254 17.564
257 17.437 255 17.598
258 17.463 256 17.611
259 17.485 257 17.641
260 17.515 258 17.668
261 17.528 259 17.694
262 17.558 260 17.718
263 17.576 261 17.742
118
264 17.598 262 17.769
265 17.622 263 17.792
266 17.646 264 17.817
267 17.674 265 17.845
268 17.696 266 17.877
269 17.721 267 17.896
270 17.752 268 17.929
271 17.781 269 17.953
272 17.805 270 17.981
273 17.835 271 18.01
274 17.863 272 18.048
275 17.89 273 18.07
276 17.914 274 18.102
277 17.943 275 18.132
278 17.977 276 18.16
279 17.999 277 18.196
280 18.034 278 18.228
281 18.062 279 18.255
282 18.099 280 18.293
283 18.12 281 18.315
284 18.151 282 18.346
285 18.18 283 18.376
286 18.202 284 18.403
287 18.235 285 18.427
288 18.266 286 18.442
289 18.293 287 18.46
119
290 18.327 288 18.486
291 18.354 289 18.505
292 18.379 290 18.539
293 18.408 291 18.571
294 18.43 292 18.604
295 18.444 293 18.629
296 18.468 294 18.66
297 18.502 295 18.681
298 18.528 296 18.723
299 18.561 297 18.75
300 18.583 298 18.788
301 18.607 299 18.815
302 18.633 300 18.852
303 18.669 301 18.878
304 18.703 302 18.904
305 18.731 303 18.942
306 18.759 304 18.974
307 18.783 305 18.997
308 18.812 306 19.032
309 18.843 307 19.053
310 18.866 308 19.077
311 18.893 309 19.122
312 18.923 310 19.142
313 18.943 311 19.172
314 18.963 312 19.207
315 18.988 313 19.236
120
316 19.011 314 19.258
317 19.027 315 19.281
318 19.044 316 19.308
319 19.056 317 19.333
320 19.085 318 19.352
321 19.106 319 19.365
322 19.136 320 19.384
323 19.161 321 19.406
324 19.188 322 19.429
325 19.223 323 19.451
326 19.239 324 19.473
327 19.27 325 19.504
328 19.293 326 19.528
329 19.32 327 19.548
330 19.349 328 19.577
331 19.365 329 19.606
332 19.4 330 19.633
333 19.42 331 19.656
334 19.447 332 19.68
335 19.472 333 19.712
336 19.501 334 19.74
337 19.53 335 19.768
338 19.551 336 19.796
339 19.574 337 19.823
340 19.594 338 19.855
341 19.613 339 19.876
121
342 19.647 340 19.9
343 19.669 341 19.919
344 19.696 342 19.947
345 19.723 343 19.97
346 19.746 344 20.003
347 19.776 345 20.031
348 19.802 346 20.053
349 19.823 347 20.073
350 19.851 348 20.095
351 19.867 349 20.13
352 19.896 350 20.14
353 19.919 351 20.163
354 19.939 352 20.192
355 19.968 353 20.21
356 19.992 354 20.239
357 20.016 355 20.262
358 20.035 356 20.286
359 20.06 357 20.318
360 20.076 358 20.343
361 20.095 359 20.381
362 20.127 360 20.404
363 20.156 361 20.425
364 20.174 362 20.459
365 20.199 363 20.489
366 20.227 364 20.507
367 20.261 365 20.541
122
368 20.291 366 20.568
369 20.325 367 20.597
370 20.356 368 20.62
371 20.377 369 20.644
372 20.412 370 20.668
373 20.433 371 20.698
374 20.452 372 20.723
375 20.482 373 20.744
376 20.499 374 20.769
377 20.524 375 20.792
378 20.552 376 20.821
379 20.584 377 20.846
380 20.608 378 20.881
381 20.633 379 20.905
382 20.657 380 20.925
383 20.681 381 20.952
384 20.695 382 20.979
385 20.724 383 21.002
386 20.75 384 21.02
387 20.776 385 21.046
388 20.798 386 21.064
389 20.829 387 21.085
390 20.852 388 21.093
391 20.873 389 21.115
392 20.885 390 21.14
393 20.905 391 21.155
123
394 20.92 392 21.183
395 20.948 393 21.205
396 20.971 394 21.222
397 20.989 395 21.243
398 21.02 396 21.257
399 21.044 397 21.299
400 21.069 398 21.317
401 21.094 399 21.335
402 21.111 400 21.345
403 21.131 401 21.366
404 21.151 402 21.388
405 21.171 403 21.41
406 21.195 404 21.421
407 21.216 405 21.461
408 21.24 406 21.493
409 21.268 407 21.501
410 21.29 408 21.525
411 21.313 409 21.541
412 21.334 410 21.562
413 21.359 411 21.582
414 21.372 412 21.605
415 21.398 413 21.624
416 21.42 414 21.646
417 21.452 415 21.668
418 21.482 416 21.684
419 21.503 417 21.712
124
420 21.526 418 21.72
421 21.544 419 21.742
422 21.566 420 21.761
423 21.584 421 21.779
424 21.602 422 21.808
425 21.625 423 21.829
426 21.649 424 21.857
427 21.666 425 21.876
428 21.696 426 21.903
429 21.714 427 21.924
430 21.734 428 21.946
431 21.758 429 21.969
432 21.782 430 21.992
433 21.8 431 22.019
434 21.823 432 22.044
435 21.843 433 22.075
436 21.857 434 22.099
437 21.885 435 22.116
438 21.904 436 22.147
439 21.925 437 22.178
440 21.937 438 22.193
441 21.957 439 22.218
442 21.98 440 22.236
443 22 441 22.262
444 22.014 442 22.282
445 22.028 443 22.316
125
446 22.047 444 22.339
447 22.065 445 22.356
448 22.078 446 22.37
449 22.09 447 22.39
450 22.126 448 22.406
451 22.142 449 22.425
452 22.16 450 22.454
453 22.176 451 22.469
454 22.2 452 22.483
455 22.217 453 22.503
456 22.238 454 22.527
457 22.259 455 22.538
458 22.277 456 22.556
459 22.287 457 22.578
460 22.304 458 22.597
461 22.32 459 22.605
462 22.34 460 22.619
463 22.356 461 22.632
464 22.375 462 22.643
465 22.392 463 22.671
466 22.405 464 22.68
467 22.418 465 22.705
468 22.433 466 22.728
469 22.452 467 22.736
470 22.462 468 22.751
471 22.487 469 22.764
126
472 22.498 470 22.779
473 22.52 471 22.799
474 22.536 472 22.814
475 22.551 473 22.831
476 22.582 474 22.847
477 22.594 475 22.864
478 22.617 476 22.872
479 22.637 477 22.888
480 22.659 478 22.91
481 22.668 479 22.918
482 22.687 480 22.929
483 22.697 481 22.95
484 22.717 482 22.961
485 22.733 483 22.984
486 22.754 484 22.999
487 22.773 485 23.022
488 22.788 486 23.034
489 22.805 487 23.053
490 22.828 488 23.067
491 22.845 489 23.088
492 22.864 490 23.108
493 22.883 491 23.128
494 22.904 492 23.153
495 22.914 493 23.171
496 22.933 494 23.19
497 22.951 495 23.211
127
498 22.971 496 23.22
499 22.982 497 23.246
500 23.01 498 23.267
501 23.027 499 23.292
502 23.043 500 23.307
503 23.06 501 23.327
504 23.074 502 23.366
505 23.091 503 23.388
506 23.108 504 23.406
507 23.123 505 23.417
508 23.142 506 23.432
509 23.159 507 23.45
510 23.173 508 23.473
511 23.19 509 23.482
512 23.2 510 23.497
513 23.214 511 23.515
514 23.228 512 23.545
515 23.243 513 23.562
516 23.256 514 23.576
517 23.268 515 23.588
518 23.284 516 23.605
519 23.296 517 23.625
520 23.315 518 23.626
521 23.333 519 23.651
522 23.347 520 23.664
523 23.365 521 23.676
128
524 23.387 522 23.7
525 23.405 523 23.727
526 23.425 524 23.73
527 23.437 525 23.746
528 23.454 526 23.766
529 23.475 527 23.781
530 23.489 528 23.795
531 23.504 529 23.807
532 23.518 530 23.824
533 23.545 531 23.842
534 23.562 532 23.872
535 23.578 533 23.887
536 23.593 534 23.907
537 23.608 535 23.926
538 23.618 536 23.937
539 23.64 537 23.962
540 23.657 538 23.972
541 23.671 539 23.988
542 23.684 540 24.002
543 23.698 541 24.028
544 23.714 542 24.04
545 23.723 543 24.056
546 23.741 544 24.07
547 23.753 545 24.089
548 23.767 546 24.1
549 23.776 547 24.115
129
550 23.791 548 24.127
551 23.814 549 24.137
552 23.825 550 24.15
553 23.841 551 24.179
554 23.861 552 24.194
555 23.876 553 24.209
556 23.893 554 24.218
557 23.907 555 24.23
558 23.92 556 24.241
559 23.935 557 24.256
560 23.959 558 24.276
561 23.974 559 24.287
562 23.985 560 24.295
563 24.002 561 24.33
564 24.02 562 24.334
565 24.036 563 24.343
566 24.059 564 24.362
567 24.082 565 24.378
568 24.096 566 24.389
569 24.11 567 24.4
570 24.135 568 24.419
571 24.149 569 24.432
572 24.164 570 24.444
573 24.183 571 24.477
574 24.19 572 24.498
575 24.213 573 24.505
130
576 24.231 574 24.525
577 24.244 575 24.539
578 24.26 576 24.551
579 24.275 577 24.564
580 24.288 578 24.586
581 24.305 579 24.603
582 24.321 580 24.607
583 24.341 581 24.63
584 24.357 582 24.651
585 24.372 583 24.66
586 24.382 584 24.675
587 24.4 585 24.697
588 24.41 586 24.707
589 24.424 587 24.732
590 24.444 588 24.746
591 24.458 589 24.766
592 24.473 590 24.791
593 24.492 591 24.808
594 24.51 592 24.827
595 24.521 593 24.847
596 24.538 594 24.863
597 24.552 595 24.88
598 24.573 596 24.883
599 24.586 597 24.906
600 24.599 598 24.913
601 24.617 599 24.929
131
602 24.628 600 24.961
603 24.647 601 24.984
604 24.659 602 25.006
605 24.678 603 25.02
606 24.688 604 25.039
607 24.707 605 25.061
608 24.715 606 25.079
609 24.734 607 25.108
610 24.744 608 25.124
611 24.757 609 25.142
612 24.772 610 25.146
613 24.784 611 25.177
614 24.796 612 25.187
615 24.818 613 25.211
616 24.833 614 25.22
617 24.842 615 25.24
618 24.857 616 25.254
619 24.864 617 25.271
620 24.872 618 25.282
621 24.891 619 25.299
622 24.909 620 25.307
623 24.928 621 25.337
624 24.943 622 25.35
625 24.957 623 25.364
626 24.969 624 25.38
627 24.988 625 25.395
132
628 25.002 626 25.41
629 25.019 627 25.435
630 25.035 628 25.446
631 25.049 629 25.466
632 25.063 630 25.467
633 25.081 631 25.482
634 25.097 632 25.49
635 25.116 633 25.51
636 25.127 634 25.53
637 25.138 635 25.541
638 25.154 636 25.559
639 25.172 637 25.573
640 25.196 638 25.587
641 25.205 639 25.594
642 25.224 640 25.613
643 25.235 641 25.622
644 25.254 642 25.64
645 25.27 643 25.636
646 25.29 644 25.645
647 25.298 645 25.679
648 25.315 646 25.691
649 25.328 647 25.694
650 25.348 648 25.704
651 25.36 649 25.721
652 25.379 650 25.735
653 25.391 651 25.748
133
654 25.416 652 25.769
655 25.437 653 25.767
656 25.458 654 25.786
657 25.476 655 25.796
658 25.483 656 25.81
659 25.495 657 25.815
660 25.513 658 25.831
661 25.531 659 25.85
662 25.543 660 25.858
663 25.556 661 25.869
664 25.572 662 25.89
665 25.596 663 25.894
666 25.613 664 25.9
667 25.629 665 25.919
668 25.648 666 25.925
669 25.661 667 25.945
670 25.684 668 25.965
671 25.695 669 25.973
672 25.71 670 25.983
673 25.721 671 26.002
674 25.726 672 26.019
675 25.741 673 26.033
676 25.757 674 26.05
677 25.778 675 26.052
678 25.795 676 26.067
679 25.81 677 26.069
134
680 25.821 678 26.077
681 25.842 679 26.099
682 25.857 680 26.112
683 25.872 681 26.12
684 25.886 682 26.136
685 25.9 683 26.147
686 25.924 684 26.148
687 25.942 685 26.168
688 25.954 686 26.183
689 25.976 687 26.198
690 25.989 688 26.207
691 26.005 689 26.215
692 26.017 690 26.226
693 26.033 691 26.232
694 26.05 692 26.243
695 26.059 693 26.265
696 26.081 694 26.273
697 26.092 695 26.283
698 26.113 696 26.28
699 26.129 697 26.289
700 26.142 698 26.294
701 26.16 699 26.309
702 26.169 700 26.324
703 26.184 701 26.344
704 26.202 702 26.359
705 26.219 703 26.377
135
706 26.235 704 26.392
707 26.258 705 26.405
708 26.269 706 26.422
709 26.287 707 26.439
710 26.303 708 26.457
711 26.32 709 26.48
712 26.343 710 26.496
713 26.358 711 26.511
714 26.374 712 26.523
715 26.399 713 26.545
716 26.411 714 26.558
717 26.431 715 26.578
718 26.45 716 26.593
719 26.463 717 26.608
720 26.48 718 26.626
721 26.494 719 26.644
722 26.507 720 26.665
723 26.525 721 26.687
724 26.546 722 26.697
725 26.56 723 26.711
726 26.564 724 26.732
727 26.582 725 26.742
728 26.595 726 26.757
729 26.618 727 26.77
730 26.627 728 26.783
731 26.648 729 26.803
136
732 26.662 730 26.814
733 26.681 731 26.833
734 26.695 732 26.854
735 26.712 733 26.861
736 26.724 734 26.879
737 26.744 735 26.891
738 26.756 736 26.908
739 26.767 737 26.926
740 26.787 738 26.949
741 26.802 739 26.962
742 26.816 740 26.974
743 26.824 741 26.988
744 26.84 742 27.005
745 26.856 743 27.018
746 26.868 744 27.039
747 26.887 745 27.049
748 26.9 746 27.064
749 26.92 747 27.082
750 26.935 748 27.09
751 26.954 749 27.109
752 26.969 750 27.119
753 26.984 751 27.132
754 26.999 752 27.15
755 27.014 753 27.166
756 27.027 754 27.183
757 27.044 755 27.192
137
758 27.081 756 27.201
759 27.097 757 27.22
760 27.106 758 27.232
761 27.128 759 27.244
762 27.141 760 27.268
763 27.157 761 27.274
764 27.167 762 27.282
765 27.188 763 27.303
766 27.202 764 27.312
767 27.22 765 27.329
768 27.219 766 27.344
769 27.257 767 27.361
770 27.268 768 27.378
771 27.277 769 27.391
772 27.297 770 27.402
773 27.317 771 27.416
774 27.331 772 27.427
775 27.338 773 27.447
776 27.351 774 27.465
777 27.353 775 27.479
778 27.372 776 27.491
779 27.397 777 27.516
780 27.406 778 27.529
781 27.425 779 27.541
782 27.446 780 27.551
783 27.462 781 27.574
138
784 27.466 782 27.593
785 27.492 783 27.607
786 27.499 784 27.619
787 27.514 785 27.639
788 27.521 786 27.659
789 27.538 787 27.677
790 27.562 788 27.696
791 27.567 789 27.71
792 27.576 790 27.73
793 27.586 791 27.738
794 27.606 792 27.754
795 27.616 793 27.783
796 27.631 794 27.795
797 27.656 795 27.808
798 27.677 796 27.829
799 27.682 797 27.833
800 27.701 798 27.856
801 27.726 799 27.867
802 27.747 800 27.889
803 27.754 801 27.907
804 27.764 802 27.921
805 27.77 803 27.935
806 27.79 804 27.961
807 27.801 805 27.974
808 27.802 806 27.984
809 27.815 807 28.006
139
810 27.832 808 28.019
811 27.854 809 28.033
812 27.864 810 28.041
813 27.877 811 28.064
814 27.894 812 28.068
815 27.902 813 28.082
816 27.909 814 28.098
817 27.916 815 28.112
818 27.928 816 28.133
819 27.94 817 28.154
820 27.955 818 28.181
821 27.965 819 28.19
822 27.977 820 28.201
823 27.99 821 28.212
824 27.998 822 28.233
825 28.019 823 28.247
826 28.027 824 28.266
827 28.047 825 28.279
828 28.052 826 28.28
829 28.061 827 28.297
830 28.077 828 28.305
831 28.098 829 28.308
832 28.109 830 28.322
833 28.119 831 28.343
834 28.129 832 28.352
835 28.142 833 28.361
140
836 28.154 834 28.38
837 28.169 835 28.391
838 28.187 836 28.393
839 28.189 837 28.411
840 28.213 838 28.433
841 28.228 839 28.443
842 28.242 840 28.458
843 28.252 841 28.46
844 28.269 842 28.47
845 28.283 843 28.483
846 28.298 844 28.491
847 28.306 845 28.513
848 28.325 846 28.524
849 28.335 847 28.538
850 28.347 848 28.557
851 28.356 849 28.556
852 28.38 850 28.58
853 28.389 851 28.594
854 28.4 852 28.619
855 28.416 853 28.622
856 28.431 854 28.632
857 28.45 855 28.644
858 28.463 856 28.661
859 28.472 857 28.677
860 28.486 858 28.697
861 28.501 859 28.702
141
862 28.516 860 28.721
863 28.53 861 28.73
864 28.546 862 28.754
865 28.555 863 28.76
866 28.569 864 28.777
867 28.585 865 28.795
868 28.605 866 28.808
869 28.62 867 28.819
870 28.637 868 28.826
871 28.652 869 28.846
872 28.664 870 28.855
873 28.702 871 28.863
874 28.713 872 28.879
875 28.735 873 28.899
876 28.744 874 28.903
877 28.757 875 28.918
878 28.778 876 28.934
879 28.798 877 28.94
880 28.809 878 28.951
881 28.827 879 28.968
882 28.845 880 28.986
883 28.856 881 29
884 28.87 882 29.004
885 28.885 883 29.018
886 28.898 884 29.041
887 28.915 885 29.054
142
888 28.929 886 29.067
889 28.939 887 29.082
890 28.953 888 29.099
891 28.972 889 29.112
892 28.983 890 29.124
893 29.004 891 29.142
894 29.015 892 29.152
895 29.035 893 29.177
896 29.049 894 29.186
897 29.062 895 29.19
898 29.08 896 29.208
899 29.095 897 29.219
900 29.106 898 29.234
901 29.122 899 29.242
902 29.138 900 29.258
903 29.147 901 29.276
904 29.167 902 29.296
905 29.187 903 29.298
906 29.195 904 29.318
907 29.209 905 29.343
908 29.224 906 29.354
909 29.232 907 29.356
910 29.245 908 29.376
911 29.253 909 29.402
912 29.268 910 29.403
913 29.275 911 29.43
143
914 29.298 912 29.445
915 29.304 913 29.456
916 29.31 914 29.474
917 29.329 915 29.493
918 29.336 916 29.511
919 29.349 917 29.526
920 29.37 918 29.542
921 29.371 919 29.554
922 29.383 920 29.565
923 29.393 921 29.579
924 29.395 922 29.601
925 29.421 923 29.619
926 29.431 924 29.636
927 29.444 925 29.641
928 29.455 926 29.659
929 29.47 927 29.682
930 29.479 928 29.696
931 29.487 929 29.707
932 29.495 930 29.721
933 29.516 931 29.732
934 29.533 932 29.753
935 29.553 933 29.772
936 29.561 934 29.783
937 29.567 935 29.807
938 29.574 936 29.825
939 29.589 937 29.837
144
940 29.601 938 29.854
941 29.618 939 29.868
942 29.626 940 29.885
943 29.638 941 29.901
944 29.656 942 29.901
945 29.656 943 29.907
946 29.68 944 29.918
947 29.695 945 29.939
948 29.706 946 29.959
949 29.72 947 29.965
950 29.734 948 29.986
951 29.742 949 29.999
952 29.757 950 30.022
953 29.765 951 30.029
954 29.781 952 30.043
955 29.782 953 30.07
956 29.804 954 30.074
957 29.817 955 30.106
958 29.826 956 30.126
959 29.839 957 30.124
960 29.85 958 30.136
961 29.867 959 30.149
962 29.875 960 30.153
963 29.888 961 30.195
964 29.899 962 30.202
965 29.92 963 30.217
145
966 29.941 964 30.238
967 29.952 965 30.245
968 29.968 966 30.25
969 29.973 967 30.266
970 29.986 968 30.284
971 30.006 969 30.282
972 30.013 970 30.3
973 30.027 971 30.308
974 30.046 972 30.322
975 30.057 973 30.344
976 30.06 974 30.357
977 30.083 975 30.375
978 30.096 976 30.389
979 30.105 977 30.395
980 30.113 978 30.402
981 30.129 979 30.425
982 30.141 980 30.438
983 30.155 981 30.453
984 30.157 982 30.456
985 30.164 983 30.492
986 30.171 984 30.502
987 30.188 985 30.515
988 30.198 986 30.527
989 30.215 987 30.542
990 30.224 988 30.552
991 30.241 989 30.563
146
992 30.25 990 30.572
993 30.259 991 30.596
994 30.276 992 30.614
995 30.291 993 30.624
996 30.306 994 30.641
997 30.321 995 30.651
998 30.327 996 30.666
999 30.343 997 30.681
1000 30.36 998 30.71
1001 30.38 999 30.722
1002 30.391 1000 30.776
1003 30.403 1001 30.793
1004 30.42 1002 30.802
1005 30.429 1003 30.823
1006 30.447 1004 30.831
1007 30.461 1005 30.85
1008 30.481 1006 30.866
1009 30.497 1007 30.869
1010 30.512 1008 30.879
1011 30.52 1009 30.907
1012 30.54 1010 30.918
1013 30.555 1011 30.932
1014 30.56 1012 30.943
1015 30.582 1013 30.965
1016 30.603 1014 30.976
1017 30.613 1015 31.001
147
1018 30.629 1016 31.007
1019 30.642 1017 31.035
1020 30.653 1018 31.049
1021 30.668 1019 31.051
1022 30.68 1020 31.068
1023 30.685 1021 31.059
1024 30.697 1022 31.094
1025 30.714 1023 31.088
1026 30.723 1024 31.113
1027 30.734 1025 31.141
1028 30.756 1026 31.127
1029 30.769 1027 31.154
1030 30.78 1028 31.137
1031 30.794 1029 31.163
1032 30.81 1030 31.176
1033 30.819 1031 31.184
1034 30.828 1032 31.183
1035 30.837 1033 31.226
1036 30.85 1034 31.236
1037 30.863 1035 31.239
1038 30.882 1036 31.27
1039 30.899 1037 31.25
1040 30.905 1038 31.287
1041 30.917 1039 31.302
1042 30.92 1040 31.303
1043 30.942 1041 31.32
148
1044 30.955 1042 31.342
1045 30.969 1043 31.351
1046 30.976 1044 31.361
1047 30.99 1045 31.375
1048 31.003 1046 31.377
1049 31.018 1047 31.403
1050 31.023 1048 31.412
1051 31.036 1049 31.423
1052 31.048 1050 31.446
1053 31.054 1051 31.468
1054 31.07 1052 31.48
1055 31.09 1053 31.485
1056 31.102 1054 31.499
1057 31.118 1055 31.501
1058 31.113 1056 31.526
1059 31.132 1057 31.551
1060 31.152 1058 31.596
1061 31.163 1059 31.632
1062 31.177 1060 31.636
1063 31.192 1061 31.637
1064 31.204 1062 31.669
1065 31.214 1063 31.683
1066 31.228 1064 31.7
1067 31.239 1065 31.708
1068 31.249 1066 31.72
1069 31.264 1067 31.725
149
1070 31.269 1068 31.744
1071 31.275 1069 31.767
1072 31.285 1070 31.762
1073 31.296 1071 31.773
1074 31.311 1072 31.81
1075 31.317 1073 31.816
1076 31.337 1074 31.834
1077 31.354 1075 31.854
1078 31.367 1076 31.861
1079 31.37 1077 31.884
1080 31.395 1078 31.897
1081 31.402 1079 31.905
1082 31.418 1080 31.915
1083 31.418 1081 31.931
1084 31.434 1082 31.948
1085 31.457 1083 31.96
1086 31.461 1084 31.967
1087 31.472 1085 31.983
1088 31.487 1086 31.999
1089 31.499 1087 32.011
1090 31.508 1088 32.026
1091 31.524 1089 32.031
1092 31.534 1090 32.04
1093 31.55 1091 32.057
1094 31.56 1092 32.07
1095 31.576 1093 32.091
150
1096 31.585 1094 32.09
1097 31.6 1095 32.107
1098 31.607 1096 32.116
1099 31.624 1097 32.145
1100 31.635 1098 32.158
1101 31.65 1099 32.159
1102 31.687 1100 32.17
1103 31.695 1101 32.201
1104 31.713 1102 32.206
1105 31.727 1103 32.225
1106 31.737 1104 32.227
1107 31.75 1105 32.242
1108 31.76 1106 32.253
1109 31.78 1107 32.262
1110 31.791 1108 32.271
1111 31.798 1109 32.279
1112 31.81 1110 32.298
1113 31.827 1111 32.308
1114 31.84 1112 32.303
1115 31.848 1113 32.329
1116 31.858 1114 32.354
1117 31.879 1115 32.345
1118 31.891 1116 32.365
1117 32.371
1118 32.386
1119 32.399
151
1120 32.41
1121 32.424
1122 32.43
1123 32.448
1124 32.454
1125 32.473
1126 32.476
1127 32.5
1128 32.506
1129 32.522
1130 32.525
1131 32.546
1132 32.56
1133 32.569
1134 32.59
1135 32.614
1136 32.619
1137 32.64
1138 32.661
1139 32.664
1140 32.681
1141 32.691
1142 32.702
1143 32.732
1144 32.758
1145 32.758
152
1146 32.774
1147 32.784
1148 32.796
1149 32.815
1150 32.812
1151 32.838
1152 32.843
1153 32.863
1154 32.877
1155 32.885
1156 32.89
1157 32.908
1158 32.923
1159 32.945
1160 32.95
1161 32.959
1162 32.986
1163 33.007
1164 33.012
1165 33.031
1166 33.035
1167 33.05
1168 33.064
1169 33.087
1170 33.103
1171 33.125
153
1172 33.13
1173 33.137
1174 33.154
1175 33.161
1176 33.186
1177 33.196
1178 33.225
1179 33.211
1180 33.235
1181 33.266
1182 33.283
1183 33.287
1184 33.307
1185 33.309
1186 33.324
1187 33.342
1188 33.346
1189 33.381
1190 33.378
1191 33.402
1192 33.41
1193 33.42
1194 33.444
1195 33.446
1196 33.464
1197 33.47
154
1198 33.509
1199 33.52
1200 33.541
1201 33.539
1202 33.552
1203 33.579
1204 33.584
1205 33.599
1206 33.595
1207 33.631
1208 33.652
1209 33.655
1210 33.668
1211 33.678
1212 33.694
1213 33.711
1214 33.712
1215 33.74
1216 33.741
1217 33.746
1218 33.77
1219 33.752
1220 33.79
1221 33.796
1222 33.817
1223 33.817
155
1224 33.833
1225 33.841
1226 33.851
1227 33.852
1228 33.89
1229 33.906
1230 33.912
156