Modelling immiscible displacement in porous rocks with LBM models

Attractive van der Waals forces phase p phase q transition Random thermal motion X Attractive van der Waals forces

Full Version from Liouville Equation using the Enskog volume exclusion for dense gases Short distance collision term without volume effects Long range mean field gravity volume exclusion (Enskog)

Short distance collision term Diffusivity Viscosity of component p Facin, Philippi and Santos, FGCS (2004)

Full Version from Liouville Equation using the Enskog volume exclusion for dense gases Molecular interaction terms

Advection-diffusion equation Momentum balance equation Pressure tensor Surface tension tensor

Simplified Version for Isothermal Problems Molecular interaction terms

Simplified Version for Isothermal Problems Advection diffusion equation Momentum balance equation Equation of state

In addition:

Simplified Version for Isothermal Problems

Simplified Version for Isothermal Problems difusivity p-q viscosity

Simplified Version for Isothermal Problems p-q atractive forces External forces p-p attractive forces Volume exclusion 12

Model parameters from PVT data a(p) a(q) Force parameters b(p) b(q) Volume parameters (p) (q) Viscosities water oil a(pq) Cross force parameter D(pq) diffusivity

Model parameters from PVT data We consider that in the transition region the mixture follows, for instance, a Peng-Robinson equation of state [n]= number density of molecules k= Boltzmann constant=R/A 14

Model parameters from PVT data Equation of state for the model phase p phase q transition 15

Model parameters in terms of the surface and interfacial tensions Goniometer Tensiometer

Model parameters in terms of the surface and interfacial tensions The intermolecular potential energy for a single p-particle due to the interaction with an s-particle close to it was considered as The surface tension is related to the work for moving a layer of liquid from a distance d→∞ from the surface (where the interaction is null) to the liquid surface Parameter “a” was defined as

Model parameters in terms of the surface and interfacial tensions s-s potential energy Fvert d= distância de interação s=p,q 18

Model parameters in terms of the surface and interfacial tensions Fvert 19

Interaction with the solid wall atenuation mediators Strength lattice symmetry D2Q9 F. G. Wolf, L. O. E. dos Santos. and P. C. Philippi Capillary rise between parallel plates under dynamic conditions, Journal of Colloid and Interface Science 344 (2010) ,171–179 20

Interaction with the solid wall p-q atraction External forces Volume exclusion p-p atraction Interaction with the solid wall 21

 Interaction with the solid wall  Oil Water The strengths (ps) and (qs) are to be related to the surface energies g(ps) and g(qs) in the Young-Dupré equation

Interaction with the solid wall F. G. Wolf, L. O. E. dos Santos. and P. C. Philippi Capillary rise between parallel plates under dynamic conditions, Journal of Colloid and Interface Science 344 (2010) ,171–179 23

Comparison with Shan-Chen model Present model diffusion control volume without diffusion control viscosity diffusivity

Comparison with Shan-Chen model Present model =massa virtual

Comparison with Shan-Chen model: Equation of state Present model Virtual mass of the original model

Carbonates Multi-scale systems pore solide ?

Capillary flow, water is the wetting fluid and the larger pores do not form a continuous phase Oil (PO) The displacement will happen after PO-PW<Ppercolation pore Larger pores with oil will be partially blocked solide Water (Pw)

Capillary flow, water is the non-wetting fluid and the smaller pores do not form a continuous phase Oil (PO) smaller pores with oil will be blocked pore solide Water (Pw)

The worst case (but with better recovery): water is the non-wetting fluid and the larger pores do not form a continuous phase Mixed approach: LBM for finding k related to the smaller pores and to simulate the flow inside the larger ones. Finite volume for the second scale. PO Pw 2R Oil (PO) k(P) pore solide Displacement of the interface will be conditioned by the transfer into the smaller pores Water (Pw) The displacement will happen after PW-PO>Ppercolation for the smaller pores PHILIPPI, P. C. ; SOUZA, H. A. . Modeling Moisture Distribution and Isothermal Transfer In a Heterogeneous Porous Material. INTERNATIONAL JOURNAL OF MULTIPHASE FLOW, v. 21, n. 4, p. 667-691, 1995