Numerical Modelling of Capillary Transition zones

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

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Skjaevelands Hysteresis Model

• Mixed-wet reservoir
• General capillary pressure correlation
• Analytical expressions/power laws
• Accounts for history of reservoir
• Arbitrary change of direction

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Capillary pressure functions

• Capillary pressure for water-wet reservoir
• Brooks/Corey
• General expression water branch oil branch
• cs and as constants one set for drainage,
  another for imbibition
• Swrk, Sork adjustable parameters

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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
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Relative permeability
kro
krw

• Hysteresis curves from primary drainage
• Weighted sums of Corey-Burdine expressions
• Capillary pressure branches used as weights

Sw
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Numerical modelling

• Domain for simulation discretized
• Block center represents some average
• Hysteresis logic apply to all grid cells
• Fully implicit control-volume formulation

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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

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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

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Initial pressure gradients

• OWC Oil water contact
• FWL Free water level
• Threshold capillary pressure,

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Low rate saturation distribution

• Production close to equilibrium
• Steep water-front water sweeps much oil
• Small saturation change to reach equilibrium
  after shut off

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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

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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

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Medium rate capillary pressure

• Deviation from equilibrium
• Larger pressure drop in middle of the trans. zone
• Front behaviour explained by irreversibility

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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

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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

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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
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Comparison to Killoughs model

• Killoughs 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

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What about the real world?
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Conclusions

• Skjaevelands 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