**Paulo C. Philippi**

#### Porous Media and Thermophysical Properties Laboratory (LMPT). Mechanical Engineering Department. Federal University of Santa Catarina. 88040‐900. Florianópolis. Santa Catarina. Brazil

The correct understanding of fluid flow and phase transitions in porous media is important in both environmental and industrial research related to the development of technologies in agriculture, hydrology, soil contamination, petroleum production, bioengineering, buildings, heat‐pipes, capillary pumps and many other technologies. In particular, the numerical simulation of immiscible displacement in porous rocks is a challenging problem whose solution is receiving an increasing attention in the last two decades due to the importance it represents for the recovering of oil from petroleum rocks, using water as the displacing fluid. Simulations based on the spatial and temporal discretization of the macroscopic fluid dynamics equations showed to be difficult to be achievable, specially when based on singular interfaces. Indeed, in addition to the geometric complexity of the porous structure, the triple contact line between the phases gives rise to a singularity that requires, in this approach, the use of empirical models based on lubricant films of questionable validity for understanding the complex physical phenomena related to the interaction between the fluids and the solid surface. The same difficulties are found when trying to describe the

dynamic processes related to the interface formation and breakdown, including the segregation processes, the coalescence and the bubbling of the invader into the resident phase.

In this aspect, the use of the LB method has three main advantages with respect to classical CFD methods in simulating immiscible flows: a) owning a mesoscopic framework it is able to solve macroscopic problems that require information from the underlying molecular scale; b) it is intrinsically Lagrangian, avoiding to capture the interface at each time step; c) as it is the case for all LB algorithms, it is based on an hyperbolic equation, the field information travelling along characteristic lines and enabling the simulator to avoid the solution of very large systems of algebraic equations, as it happens with numerical schemes based on elliptic equations.

LB models for immiscible flows were proposed in the yearly 90’s by Shan & Chen¹ and by Gunstensen et al.2,3 . As pointed out by He and Doolen⁴ and Philippi et al.⁵, segregation and interface tension are predicted in Shan‐Chen model as related to effects with the same order of magnitude of the errors of the discrete approximation, when the isotropy of the lattice tensors used in the discrete calculation of the spatial derivatives is not assured up to a given rank. In addition, it is thermodynamic inconsistent because the kinetic and thermodynamic pressure are not the same and because it fails in predicting an interfacial tension in accordance with the one derived from the thermodynamic Helmholtz free‐energy. The addition of a volume exclusion term⁵ or a gradient force⁶, can make the model compliant with the free‐energy functional. On the other hand, Gunstensen model suffers from interface pinning. Pinning was investigated by Latva‐Kokko and Rothman⁷ who proposed a new segregation rule based on the ideas of d’Ortona et al.⁸ and Tölke et al.⁹ Later Halliday and co‐workers¹⁰⁻¹² proposed improvements of the model.

The knowledge of the interface dynamics inside a porous structure requires to understand the influence of the underlying molecular physics on the macroscopic phenomena that are apparent at the porous scale and this is difficult to be achieved without detailed kinetic models. In this direction, the recognition of LB equations as finite Hermite polynomial expansions of the kinetic equations¹³⁻¹⁵, enabled to relate the order of approximation of the LBE to the set velocities in the discrete space and to independently treat the velocity discretization problem and the problem of constructing kinetic models suitable for a given purpose.

In this tutorial, a review is presented focusing on recent aspects and contributions in the lattice‐

Boltzmann description of isothermal immiscible flows in porous media. Emphasis is given to the main

properties the LB equations are required to satisfy ‐ thermodynamic consistency, Galilean invariance and

avoidance of compressibility effects ‐ and to the chief challenges in this field.

References

[1] X. Shan and H. Chen, Phys. Rev. E 47, 1815 (1993).

[2] A.K. Gunstensen, D.H. Rothman, S. Zaleski, and G. Zanetti, Europhys. Lett. 43, 4320 (1991).

[3] A.K. Gunstensen, D.H. Rothman, S. Zaleski, and G. Zanetti, Europhys. Lett.18, 157 (1992).

[4] X. He and G.D. Doolen, J. Stat. Phys. 107, 309 (2002).

[5] P. C. Philippi, K. K. Mattila, D. N. Siebert, L. O. E. dos Santos, L. A. Hegele Jr. and R. Surmas, J. Fluid Mech., 713, 564 (2012).

[6] M. Sbragaglia, H. Chen, X. Shan and S. Succi, Europhys. Lett. 86, 24005 (2009).

[7] M. Latva‐Kokko and D. H. Rothman, Phys. Rev. E 71, 056702 (2005).

[8] U. D’Ortona, D. Salin, M. Cieplak, R. Rybka, and J. Banavar,Phys. Rev. E 51, 3718 (1995).

[9] J. Tölke, M. Krafczyk, M. Schultz, and E. Rank, Philos. Trans. R. Soc. London, Ser. A 360, 535 (2002).

[10] I. Halliday, A. P. Hollis and C. M. Care, Phys. Rev. E 76, 026708 (2007).

[11] T. J. Spencer, I. Halliday, C. M. Care, Phys. Rev. E 82, 066701 (2010).

[12] T. J. Spencer, I. Halliday and C. M. Care, Phil. Trans. R. Soc. A 369, 2255 (2011).

[13] X. He and L‐S. Luo, L.‐S., Phys. Rev. E 56, 6811 (1997).

[14] P. C. Philippi, L. A. Hegele Jr., L. O. E. dos Santos, and R. Surmas, Phys. Rev. E 73, 56702 (2006).

[15] X. Shan, X‐F. Yuan and H. Chen, J. Fluid Mech. 550, 413 (2006