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

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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

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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

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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!