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Supplemental Materials for:
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Tracking colloid transport in real pore structures: comparisons with
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correlation equations and experimental observations
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Zhelong Li1,2, and Dongxiao Zhang2,, Xiqing Li1,2,
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Laboratory of Earth Surface Processes, College of Urban and
Environmental Sciences, Peking University, Beijing, 100871, P. R. China
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Department of Energy and Resources Engineering, College of
Engineering, Peking University, Beijing, 100871, P. R. China
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9 pages of text
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2 figures
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Co-corresponding author, e-mail: dxzhang@coe.pku.edu.cn, phone: 86-10-62757432, fax: 86-10-62757532
Corresponding author, e-mail: xli@urban.pku.edu.cn, phone/fax: 86-10-62753246
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Algorithm to extract pore structures from XMT images
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In this work, an algorithm was developed to automatically extract grain center
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coordinates and radii from XMT data, which are a set of brightness values at each
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pixel in three dimensions. For a chosen domain for particle tracking, a domain
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extending 44 pixels out in each dimension is selected for pore structure extraction.
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This is done because some grains may partially fall outside the chosen domain but
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these grains also need to be considered in particle tracking. Manual reading from the
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XMT images indicates that the maximum grain diameter is no more than 44 pixels. As
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such, extending 44 pixels out from the chosen domain guarantees that all the grains
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falling (even if partially) within the chosen domain are extracted. From the XMT data,
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the pixels that represent solid grains and pore spaces were first assigned with a value
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of 1 and 0, respectively. The pixels at grain surfaces (solid boundaries) were identified
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by finding the solid pixels whose 26 neighboring pixels include at least one non-solid
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pixel.
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Next, for every solid pixel, the number of solid boundary pixels that fall between
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the sphere with a radius of r-1 pixels and the sphere with a radius of r+1 pixels (both
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spheres are centered at the solid pixel) is counted and denoted as Nr. Here r is an
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integer that ranges from 13 to 22, which are the minimum and maximum number of
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pixels that a grain radius covers, respectively. A test function is then assigned to the
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solid pixel and is defined as Nr divided by the volume of a sphere with a radius of r.
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By incrementing r from 13 to 22, the maximum value (with regard to r) that the test
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function yields is recorded. If the solid pixel is at the center of a grain, the maximum
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value occurs at an r that is approximately equal to the radius of the grain. If r is very
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small, the two test spheres would both fall within the grain. The number of boundary
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pixels falling between the two spheres would be zero. If r is slightly larger than the
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grain radius, the number of boundary pixels may not be zero, as boundary pixels of
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the neighboring grains may fall between the two test spheres. However, the number
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would be much smaller (compared to the case where r is close to the true grain radius),
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as the test spheres only cut a small portion of the neighboring grains (Figure S1). If r
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is much larger, the number of boundary pixels falling between the two test spheres
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could be large. However, the value of the test function that involves a denominator
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related to r would still be small. For solid pixels that are not at the center of the grain,
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the maximum test function values would be also small, as the test spheres only
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encircle a portion of boundary pixels.
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To identify the solid pixel at the grain center, for each solid pixel with a
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coordinate of (x, y, z), the maximum test function values within the region of [x-2,
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x+2] [y-2, y+2]  [z-2, z+2] were compared to identify the pixel at which the
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maximum test function value is the greatest. The solid pixels whose maximum test
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function values are not locally the greatest are excluded (not recorded). After this step,
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there may be solid pixels left whose test spheres (at a proper r) contact a number of
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neighboring grains (Figure S2). In this case, the test spheres could also encircle a
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significant number of boundary pixels and yield a locally greatest test function value.
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Therefore, the algorithm counts the number of the non-solid pixels falling within the
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sphere that is centered at each remaining solid pixel with a radius of r and that yields
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the locally greatest test function value. As illustrated in Figure S1 and Figure S2, for a
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center pixel, the majority of the pixels falling within the sphere are solid pixels,
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whereas for a non-center pixel, a significant fraction of the pixels falling within the
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sphere are non-solid pixels. Thus, a solid pixel with the locally greatest test function
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value is considered as a non-center pixel and excluded if the ratio of the number of
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non-solid pixels to the total number of pixels in the sphere is greater than 20%. The
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remaining solid pixels are all center pixels and their coordinates were recorded.
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Finally, all the radii that yield the maximum test function values at the center pixels
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were slightly adjusted by multiplying an identical factor (ranging between 1 and 1.03
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for different domains) to yield a porosity of the extracted pore structure that is equal
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to the porosity of the original structure.
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Justification of deriving flow fields at different superficial velocities and for
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porous media of different grain diameters
The flow of an uncompressible fluid is described by the Navier-Stokes equation:
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v  0
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
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where v is fluid velocity,  is density, t is time, p is the pressure, and  is viscosity.
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For steady state flow, the first term of the left side of equation 2 equals zero and can
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be omitted. According to Happel and Brenner [1974], fluid with a Reynolds number
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less than 1 can be treated with low Reynolds number hydrodynamics and the second
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term (which represent the inertial force) of the left side of equation 2 can be neglected.
(1)
v
  (v )v  p   2 v
t
(2)
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The highest superficial velocity used in our simulations was 1.36 10-3 m s-1. This
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velocity corresponds to a maximum Reynolds number of 0.68, assuming a
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characteristic length of 510 m, which is the maximum average grain diameter
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encountered in simulations. As such, our simulations all involved low Reynolds
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number flows that can be described by equation 1 and the following equation:
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p   2 v
(3)
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For a particular domain, say Domain 1, flow field at a superficial velocity V was
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obtained by LB simulation by a pressure difference of P1 applied at the inlet and
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outlet boundary. The flow field derived from LB simulation satisfies the following
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boundary conditions:
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v1  x1 , y1 , z1   0
x1 , y1 , z1  S1
(4)
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p1  x1 , y1 , z1   P1
x1 , y1 , z1  Sin1
(5)
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p1  x1 , y1 , z1   0
x1 , y1 , z1  Sout1
(6)
p1 are the simulated velocity and pressure at ( x1 , y1 , z1 ),
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where v1 and
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respectively. S1 is the solid boundary (the bounding x and y planes). Sin1 , Sout1 are the
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inlet and outlet plane, respectively.
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For the same domain, the flow field at a superficial velocity of V
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that differs
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from V1 by a factor of k corresponds to a pressure difference of P1 . The flow field
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,
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(represented by v1 and p1 ) at this superficial velocity should satisfy the following
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boundary conditions:
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v1,  x1 , y1 , z1   0
x1 , y1 , z1  S1
(7)
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p1,  x1 , y1 , z1   P1,
x1 , y1 , z1  Sin1
(8)
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p1,  x1 , y1 , z1   0
x1 , y1 , z1  Sout1
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Assume:
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P1,
v ( x1 , y1 , z1 )  v1 ( x1 , y1 , z1 )
P1
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,
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(10)
P1,
p1 ( x1 , y1 , z1 )
P1
p1, ( x1 , y1 , z1 ) 
(9)
(11)
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,
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Given that v1 and p1 satisfy equation 1, 3, 4, 5, 6, v1 and p1 with expressions
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in equation 10 and 11 would satisfy equation 1, 3, 7, 8, 9 (this can be easily verified
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by plugging the two expressions into these equations). Because equation 1 and 3
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would have a single solution (given the boundary conditions in equation 7, 8, 9) at
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V1, , the fact that the v1, and p1, in equation 10 and 11 satisfy equation 1 and 3
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indicates that the expressions in equation 10 and 11 represent the flow field at V1 .
By definition, the two superficial velocities can be expressed as:
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V 
,
 v  x , y ,z  dx dy
z1
1
1
1
1
1
(12)
A1
 v  x , y ,z  dx dy

,
z1
1
1
1
1
1
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V
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where vz1 and vz1 are the z-direction velocities at ( x1 , y1 , z1 ) at the two superficial
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velocities; A1 is the area of a cross section of the domain perpendicular to
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z-direction. Note that the cross section of integration can be randomly picked due to
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fluid continuity. Plugging the expression in equation 10 into equation 13 yields:
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V, 
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Hence:
(13)
A1
,
P1,
V1
P1
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1
P1,
k
P1
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v1, ( x1 , y1 , z1 )  kv1 ( x1 , y1 , z1 )
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Equation 14 verifies that a flow field at a superficial velocity can be derived from the
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flow field at another superficial velocity by multiplying the ratio of the two velocities.
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This means that for a particular domain, LB simulation can be performed only once
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for various superficial velocities.
(14)
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Now, consider a domain (denoted as Domain 2) that is obtained by scaling from
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Domain 1 by a factor k1. Assuming after scaling a node at ( x1 , y1 , z1 ) in Domain 1
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yields a node with a coordinate of ( x2 , y2 , z 2 ) in Domain 2, x2  k1 x1 , y2  k1 y1 ,
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and z2  k1 z1 . At an identical superficial velocity V, the flow field in Domain 2
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(represented by v2 and p2 ) should satisfy following boundary conditions (in
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addition to equation 1 and 3):
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v2  x2 , y2 , z2   0,
x2 , y2 , z2  S2
(15)
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p2  x2 , y2 , z2   P2
x2 , y2 , z2  Sin2
(16)
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p2  x2 , y2 , z2   0
x2 , y2 , z2  Sout 2
(17)
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where S2 is the solid boundary of Domain 2. Sin 2 , Sout 2 are the inlet and outlet plane,
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respectively.
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Assume:
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v2 ( x2 , y2 , z2 )  k1
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p2 ( x2 , y2 , z2 ) 
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P2
v1 ( x1 , y1 , z1 )
P1
(18)
P2
p1 ( x1 , y1 , z1 )
P1
(19)
Given that v1 and p1 satisfy equation 1, 3, 4, 5, 6, v2 and p2 with expressions
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in equation 18 and 19 would satisfy equation 1, 3, 15, 16, 17. This can be verified first
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by plugging the expressions in equation 18 and 19 into equation 15, 16, 17. Then
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apply x1 
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19 and plug the resulting expressions into equation 1 and 3. Applying the chain rule of
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differentiation will verify that equation 1 and 3 are also satisfied. Thus the expressions
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in equation 18 and 19 represent the flow field in Domain 2. By definition, the
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superficial velocity in Domain 2 can be expressed as:
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V
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where vz 2 is the z-direction velocity at ( x2 , y2 , z 2 ); A2 is the area of the cross
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x2
v x
z2
2
y 
k1 , 1
z 
k1 and 1
z2
k1 to the right sides of equation 18 and
, y2 ,z2  dx2 dy2
(20)
A2
section of integration in Domain 2. Comparing equation 12 and 20 gives:
 v  x , y ,z  dx dy
z1
1
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y2
1
A1
1
1
1

v x
z2
2
, y2 ,z2  dx2 dy2
(21)
A2
Since Domain 2 is scaled from Domain 1 by a factor of k1 ,
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A2  k12 A1
(22)
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dx2  k1 dx1
(23)
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dy2  k1dy1
(24)
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Plugging equation 18, 22, 23, 24 into equation 21 yields:
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k1
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Thus, equation 18 reduces to:
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v2 ( x2 , y2 , z2 )  v1 ( x1 , y1 , z1 )
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Equation 25 indicates that the corresponding nodes in the two scaled domains have an
P2
1
P1
(25)
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identical velocity and LB simulation only needs to be performed in one domain.
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References
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Happel, J., H. Brenner (1974), Low Reynolds number hydrodynamics: with special
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applications to particulate media. Noordhoff International Publishing,.
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9
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3
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Figure S1. Schematic demonstration (in 2-D) of test spheres (pink circles) that
encircle the grain surface boundary pixels. The red circles represent test spheres that
have too large radii and encircle a smaller number of surface boundary pixels. The
green circles are grains. Red dots represent surface boundary pixels.
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45
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40
35
3
30
r’+1
25
r-1
r’-1
20
1
r+1
15
10
2
5
0
0
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10
15
20
25
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Figure S2. Schematic demonstration of test spheres (blue circles) that encircle a
significant number of surface boundary pixels (and yield a locally greatest maximum
test function value) but whose center is not located at a grain center.
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45
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40
35
30
3
r-1
25
r+1
20
1
15
10
2
5
0
0
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10
15
20
25
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11
30
35
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45
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