1. Modelling immiscible displacement in porous rocks with LBM models

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

3. 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)

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

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

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

7. Simplified Version for Isothermal Problems
Molecular interaction terms

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

9. In addition:

10. Simplified Version for Isothermal Problems

11. Simplified Version for Isothermal Problems
difusivity p-q
viscosity

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

13. 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

14. 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

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

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

17. 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

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

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

20. 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

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

22.  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

23. 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

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

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

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

27. Carbonates
Multi-scale systems
pore
solide ?

28. 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)

29. 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)

30. 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 , 1995