1. Permeability Prediction of Shale Gas by the Monte Carlo Molecular Simulations at Pore Scale Jun Li Research Engineer ||| Center for Petroleum & Mineral, Research Institute King Fahd University of Petroleum & Minerals

2. Contents Mathematical models for gas flows Simulation methods Numerical results of the DSBGK method

3. 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 Shen C., Rarefied gas dynamics: fundamentals, simulations and micro flows, Springer, 2005.

4. 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

5. Boltzmann equation

6. Simplified form: BGK equation Lose accuracy when perturbation is large Improve accuracy for dense gas

7. 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

8. DSBGK method

9. MPI parallel computation

10. Numerical results: Couette flow Li J., Comparison between the DSMC and DSBGK methods, arXiv: 1207.1040 [physics.comp-ph], 2012.

11. Numerical results: channel flow Li J., Efficiency and stability of the DSBGK method, AIP Conference Proceedings 1501: 849-856, 2012.

12. Numerical results: lid-driven flow Li J., Efficiency and stability of the DSBGK method, AIP Conference Proceedings 1501: 849-856, 2012.

13. Numerical results: thermal transpiration Li J., Efficiency and stability of the DSBGK method, AIP Conference Proceedings 1501: 849-856, 2012.

14. Numerical results: artificial digital rock Klinkenberg L.J., The permeability of porous media to liquids and gases, American Petroleum Institute,1941.

15. Numerical results: real digital rock

16. Preparation for in-situ condition

17. 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.

18. Thank you!