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
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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:
cw
• Brooks/Corey: Pc 
S w  S wr a
(
)
1  S wr
• 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
w
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 CoreyBurdine expressions
• Capillary pressure
branches used as weights
kro
krw
Sw
Numerical modelling
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Domain for simulation discretized
Block center represents some average
Hysteresis logic apply to all grid cells
Fully implicit control-volume formulation:
n 1
m  m  t
n
n
f
n
j
j
 Q
n
Numerical issues
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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
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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
• Pc   gh
• OWC: Oil water contact
• FWL: Free water level
• Threshold capillary
pressure, c
wd
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 shutoff 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
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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