Supplementary Information for
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Nanoscale simulation of shale transport properties using the
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lattice Boltzmann method: permeability and diffusivity
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Li Chen a,b, Lei Zhang c, Qinjun Kang b, Hari S Viswanathan b, Jun Yao c, Wenquan
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Tao a
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a: Key Laboratory of Thermo-Fluid Science and Engineering of MOE, School of
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Energy and Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi, 710049,
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China
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b: Earth and Environmental Sciences Division, Los Alamos National Laboratory,
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Los Alamos, New Mexico, 87545, USA
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c: School of Petroleum Engineering, China University of Petroleum, Qingdao,
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Shandong, 266580, China
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Corresponding author: Qinjun Kang: [email protected]
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Context:
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PS1: Multi-relaxation-time (MRT) lattice Boltzmann model for fluid flow
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PS2: Single-relaxation-time (SRT) lattice Boltzmann model for mass
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transport
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PS3: Boundary conditions
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PS4: Validations
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PS1. Multi-relaxation-time lattice Boltzmann model for fluid flow.
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In the Single-relaxation-time (SRT)-Lattice Boltzmann method (LBM) model, the
evolution equation for the distribution functions can be written as
fi (x  ei t , t  t )  fi (x, t )  D[ fi eq (x, t )  f i (x, t )] i  0 ~ N ,
(S1)
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where fi(x,t) is the ith density distribution function at the lattice site x and time t.
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D is the relaxation matrix which is a diagonal matrix with all the components as
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1/τ. For the D3Q19 (three-dimensional nineteen-velocity) lattice model with
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N=18, the discrete lattice velocity ei is given by
0

ei  (1,0,0), (0, 1,0), (0, 1,0)
 (1, 1,0), (0, 1, 1), (1,0, 1)

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i0
i 1~ 6 .
(S2)
i  7  18
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feq is the ith equilibrium distribution function and is a function of local density and
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velocity
 e  u (e  u)2 u  u 
fi eq  wi  1  i 2  i 4 
,
2(cs )
2(cs )2 
 (cs )
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(S3)
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with the weight coefficient wi as wi=1/3, i=0; wi=1/18, i=1,2,…, 6; wi=1/36, i
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=7,8,…,18. cs  1/ 3 is the speed of sound. By multiplying a transformation
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matrix Q (a (N +1)×(N +1) matrix) in Eq. (S1), the evolution equation in the
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moment space can be expressed as
m(x  ct , t  t )  m(x, t )  S[meq (x, t )  m(x, t )] ,
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where
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(S4)
m  Q  f , meq  Q  f eq , D  Q  D  Q1 ,
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(S5)
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with m and meq as the velocity moments and equilibrium velocity moments,
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respectively. The equilibrium velocity moments meq can be found inS1. Q-1 is the
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inverse matrix of Q. The transformation matrix Q is constructed based on the
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principle that the relaxation matrix D (a (N +1)× (N +1) matrix) in moment space
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can be reduced to the diagonal matrix2, namely
D  diag(s0 ,s1 ,...,s17 ,s18 ) ,
(S6)
1
2  1
s0  s3  s5  s 7  0, s1  s 2  s915  , s 4  s6  s8 =s1618  8
,

8  1
(S7)
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with
τ is related to the fluid viscosity by

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(S8)
The equilibrium velocity moments meq are as follows S1
m eq
0  
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
 0.5
c t
2
s
m1eq  11  19
j j
0
(S9a)
, m eq
2  3 
11 j  j
2 0
(S9b)
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m3eq  jx , m eq
4 
2
jx
3
(S10c)
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m 5eq  j y , m 6eq  
2
jy
3
(S10d)
2
jz
3
(S10e)
eq
m eq
7  jz , m8  
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m 
eq
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3 jx2  j  j
0
3 jx2  j  j
, m 
2 0
eq
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3
(S10f)
eq
m11

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eq
m13

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jx j y
0
j y2  jz2
0
eq
, m14

j y jz
0
(S10g)
2 0
eq
, m15

jx jz
0
eq
m16
18  0
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j y2  jz2
eq
, m12

(S10h)
(S10i)
where density and momentum are determined by
   f i , j   f i ei
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i
(S11)
i
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ρ0 in Eq. (S10) is the mean density of the fluid, which is employed to reduce the
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compressibility effects of the model
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Navier-Stokes equations using Chapman–Enskog multiscale expansion under the
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low Mach number limitation.
S1,2.
Eqs. (S5) and (S8) can be recovered to
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PS2: Single-relaxation-time (SRT) lattice Boltzmann model for mass
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transport
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For pure methane diffusion in shales, the evolution equation for the
concentration distribution function is as follows
gi (x  ei t , t  t )  gi (x, t )  
1
g
( gi (x, t )  gieq (x, t ))
(S12)
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where gi is the concentration distribution function. It is worth mentioning that for
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simple geometries, D3Q7 lattice model (or D2Q5 model in 2D) is sufficient to
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accurately predict the diffusion process and transport properties, which can
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greatly reduce the computational resources, compared with D3Q19 (or D2Q9 in
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2D), as proven by our previous work S3,4. However, for complex porous structures,
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especially for those with low porosity such as shales, using reduced lattice model
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will damage the connectivity of the void space, thus leading to underestimated
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effective diffusivity. Therefore, in coincidence with the 19 direction labeling
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algorithm (see Results), D3Q19 lattice model is adopted. The equilibrium
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distribution function geq is thus given by
g  CCH4 / ai ,
eq
i
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1/ 3

ai  1/18
1/ 36

i0
(S13)
i 1~ 6
i  7  18
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The concentration and the diffusivity are obtained by CCH4   gi and
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D  ( g  0.5)x2 / 3t , respectively.
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PS3: Boundary conditions
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In the LB framework, for no slip boundary condition, the half way bounce-back
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scheme is employed; for periodic boundary conditions, the unknown distributions
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at one boundary (y=0 for example) is set to that at the other boundary (y= Ly, for
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example);
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extrapolation method proposed by Guo et al. S5 is adopted for its good accuracy.
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For methane diffusion, for the concentration boundary condition, the unknown
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distribution function is determined using the equilibrium distribution functions;
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for the no flux boundary condition, bounce-back scheme is adopted.
and
for
pressure
boundary
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condition,
the
non-equilibrium
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PS4: Validations
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For validation of the MRT-LBM fluid flow model, simulation is performed for flow
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through an 3D open cube in which equal-sized spheres of diameter d are
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arranged in periodic BCC (body-centered cubic) arrays and Re number is much
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less than unity, as shown in Fig. S1. 100×100×100 lattice is employed, with
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pressure difference applied on the left and right boundaries and periodic
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boundary conditions on the other four directions. The insert of Fig. S1 shows the
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simulated streamlines inside the periodic BCC arrays with a porosity of 0.915. A
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comparison between the simulated permeability and Kozeny-Carman (KC)
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equation S6 for the BCC arrays with different porosities is shown in Fig. S1. The
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KC equation, a widely used semi-empirical equation for predicting permeability, is
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given by kd  A 3 / (1   ) 2 , with A is the KC constant and is set as A  (d 2 /180) for
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packed-spheres porous media. Analytic solutions S7 are also displayed in Fig. S1.
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As shown in the figure, the MRT-LBM simulation results agree well with the
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analytical solution for the whole range of ε. However, the KC equation presents
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upward discrepancy and downward discrepancy for high and low porosity,
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respectively.
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Further, our SRT-LBM diffusion model is validated by predicting the effective
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diffusivity in a cubic domain containing a sphere whose diameter is the same as
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the side length of the cube S8. For such configuration, the porosity is always 1-π/6.
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100×100×100 lattice is employed. The predicted effective diffusivity (0.3317 in
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lattice units) is in good agreement with the results in the literatureS8.
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Fig. S1 Fluid flow and permeability for periodic BBC arrays
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References
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