1. Numerical Modelling of Capillary Transition zones Geir Terje Eigestad, University of Bergen, Norway Johne Alex Larsen, Norsk Hydro Research Centre, Norway

2. Acknowledgments Svein Skjaeveland and coworkers: Stavanger College, Norway I. Aavatsmark, G. Fladmark, M. Espedal: Norsk Hydro Research Centre/ University of Bergen, Norway

3. Overview Capillary transition zone: Both water and oil occupy pore-space due to capillary pressure when fluids are immiscible Numerical modeling of fluid distribution Consistent hysteresis logic in flow simulator Better prediction/understanding of fluid behavior

4. Skjaeveland’s Hysteresis Model Mixed-wet reservoir General capillary pressure correlation Analytical expressions/power laws Accounts for history of reservoir Arbitrary change of direction

5. Capillary pressure functions Capillary pressure for water-wet reservoir: Brooks/Corey: General expression: water branch + oil branch c’s and a’s constants; one set for drainage, another for imbibition S wr [k], S or [k] adjustable parameters

6. Hysteresis curve generation Initial fluid distribution; primary drainage for water-wet system Imbibition starts from primary drainage curve Scanning curves Closed scanning loops PcPc SwSw

7. Relative permeability Hysteresis curves from primary drainage Weighted sums of Corey- Burdine expressions Capillary pressure branches used as weights k ro k rw SwSw

8. Numerical modelling Domain for simulation discretized Block center represents some average Hysteresis logic apply to all grid cells Fully implicit control-volume formulation:

9. Numerical issues Discrete set of non-linear algebraic equations Use Newtons method Convergence: Lipschitz cont. derivatives Assume monotone directions on time intervals ‘One-sided smoothing’ algorithm

10. Numerical experiment Horizontal water bottom drive Incompressible fluids Initial fluid distribution; water-wet medium Initial equilibrium gravity/capillary forces Given set of hysteresis-curve parameters Understanding of fluid (re)distribution for different rate regimes

11. Initial pressure gradients OWC: Oil water contact FWL: Free water level Threshold capillary pressure,

12. Low rate: saturation distribution Production close to equilibrium Steep water-front; water sweeps much oil Small saturation change to reach equilibrium after shut off

13. Low rate: capillary pressure Almost linear relationship cap. pressure-height Low oil relative permeability in lower part of trans. zone Curve parameters important for fronts

14. Medium rate: saturation distribution Same trends as for lowrate case Water sweeps less oil in lower part of reservoir Redistribution after shut- off more apparent

15. Medium rate: capillary pressure Deviation from equilibrium Larger pressure drop in middle of the trans. zone Front behaviour explained by irreversibility

16. High rate: saturation distribution Front moves higher up in reservoir Less oil swept in flooded part of transition zone Front behaviour similar to model without capillary pressure

17. High rate: capillary pressure Large deviation from equilibrium Bigger pressure drop near the top of the transition zone Insignificant effect for saturation in top layer

18. Comparison to reference solution Compare to ultra-low rate Largest deviation near new FWL Same trends for compressed transition zone Relative deviations from ultra-low rate

19. Comparison to Killough’s model Killough’s model in commercial simulator More capillary smoothing with same input data Difference in redistribution in upper part Scanning curves different for the models Convergence problems in commercial simulator

20. What about the real world?

21. Conclusions Skjaeveland’s hysteresis model incorporated in a numerical scheme ‘Forced’ convergence Agreement with known solutions Layered medium to be investigated in future Extension to 3-phase flow