Numerical Modelling of Capillary Transition zones

Geir Terje Eigestad, University of Bergen, Norway

Johne Alex Larsen, Norsk Hydro Research Centre, Norway

Acknowledgments

Svein Skjaeveland and coworkers:

Stavanger College, Norway

I. Aavatsmark, G. Fladmark, M. Espedal:

Norsk Hydro Research Centre/

University of Bergen, Norway

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

Skjaeveland’s Hysteresis Model

• Mixed-wet reservoir

• General capillary pressure correlation

• Analytical expressions/power laws

• Accounts for history of reservoir

• Arbitrary change of direction

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

• Swr[k], Sor[k] adjustable parameters

( )1

w

w

aw wr

wr

cPc

S SS

Hysteresis curve generation

• Initial fluid distribution; primary drainage for water-wet system

• Imbibition starts from primary drainage curve

• Scanning curves• Closed scanning loops

Pc

Sw

Relative permeability

• Hysteresis curves from primary drainage

• Weighted sums of Corey-Burdine expressions

• Capillary pressure branches used as weights

kro

krw

Sw

Numerical modelling

• Domain for simulation discretized

• Block center represents some average

• Hysteresis logic apply to all grid cells

• Fully implicit control-volume formulation:

1n n n n nj

j

m m t f Q

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

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

Initial pressure gradients

• • OWC: Oil water contact• FWL: Free water level• Threshold capillary

pressure, wdc

Pc gh

Low rate: saturation distribution

• Production close to equilibrium

• Steep water-front; water sweeps much oil

• Small saturation change to reach equilibrium after shut off

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

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

Medium rate: capillary pressure

• Deviation from equilibrium

• Larger pressure drop in middle of the trans. zone

• Front behaviour explained by irreversibility

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

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

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

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

What about the real world?

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