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Permeability Prediction of Shale Gas by the Monte Carlo Molecular
Simulations at Pore Scale
Jun LiResearch Engineer |||
Center for Petroleum & Mineral, Research InstituteKing Fahd University of Petroleum & Minerals
Contents
• Mathematical models for gas flows • Simulation methods • Numerical results of the DSBGK method
Division of gas flow regime
• Roughly, [50, 0.5] nm when p[0.1, 10] MPa. • Slip and transitional regimes dominate if L≈10 nm
BB
16, and
5 2
pKn n
L k Tn mk T
Shen C., Rarefied gas dynamics: fundamentals, simulations and micro flows, Springer, 2005.
Assumptions of Boltzmann Eq. for gas
• The number of molecules is so large that the statistical treatment is valid to handle the intermolecular collisions (elastic or reaction models) and the molecular collisions with the wall (diffuse reflection or, CLL model for polished surfaces)
• Except during collisions, the interactions among molecules are negligible molecular mean free path (p,m,T,)≈50 nm at STP
Boltzmann equation4
11 10
intermolecular collisionsmovements
2
( ' ' ) d d
( , , )d d : number in d d ,( )
hard sphere molecular model: .4
j rj
f fc f f ff c c
t x
f t x c x c x cD
������������������������������������������������������������������������������������
B
2
B
( , ) d
1( , ) d
2 ( )( , ) d
3 2
n t x f c p nk T
u t x cf cn
m c uT t x f c
k n
����������������������������
��������������������������������������� ���
����������������������������
eq
collision model2
3/2eq
B B
B B B
( )
( )where ( ) exp( )
2 25
and or = . 2
jj
f fc f f
t x
m m c uf n
k T k Tnk T k nk T
m
Simplified form: BGK equation
• Lose accuracy when perturbation is large• Improve accuracy for dense gas
Simulation methods
Direct Simulation Monte Carlo (DSMC)
• Converge to the Boltzmann equation
• Standard at high Kn• Very time-consuming
in the case of low speed
Lattice Boltzmann Method (LBM)
• Solve BGK equation with simplifications
• Converge to the N-S equation
• Recent developments are verified at high Kn in unidirectional flows
• Efficient
Direct Simulation of BGK Equation (DSBGK)
• Converge to the BGK equation
• Agree well with the DSMC method in several 2D benchmark problems over a wide range of Kn
• Efficient
DSBGK methodnew
new
new new
ne
0 eq eq
eqeq
w
unless reflect at the wall ( )exp( )
/( , , ) by conservation laws
m m mm
m mm
t m t t m m
m m m m m
x x tcxc cc
f F f F f F f tN N N F Ff f n u T
��������������������������������������������������������
MPI parallel computation
Numerical results: Couette flow
Li J., Comparison between the DSMC and DSBGK methods, arXiv: 1207.1040 [physics.comp-ph], 2012.
Numerical results: channel flow
Li J., Efficiency and stability of the DSBGK method, AIP Conference Proceedings 1501: 849-856, 2012.
Numerical results: lid-driven flow
Li J., Efficiency and stability of the DSBGK method, AIP Conference Proceedings 1501: 849-856, 2012.
Numerical results: thermal transpiration
Li J., Efficiency and stability of the DSBGK method, AIP Conference Proceedings 1501: 849-856, 2012.
Numerical results: artificial digital rock
Klinkenberg L.J., The permeability of porous media to liquids and gases, American Petroleum Institute,1941.
Experiments rely on / ( )?f Kn
Numerical results: real digital rock
Preparation for in-situ condition
Upscaling permeability for large RV
Divide a large representative
volume (RV) into many subdomains
Compute the permeability at the
pore scale for each subdomain using its digital rock sample
Compute the effective
permeability of RV using the obtained permeabilities of all
subdomains
Li J. and Brown D., Upscaled LBM for simulations of flows in heterogeneous porous media, under review.
Thank you!