three-phase relative permeabilities ... - universiteit utrecht · three-phase relative...

Three-phase relative permeabilities in porous media of heterogeneous

wettability

Rink van Dijke and Ken SorbieInstitute of Petroleum Engineering

Heriot-Watt University Edinburgh, Scotland

Utrecht, The NetherlandsAugust 2006

Outline1. Introduction– Upscaling using network models– 3-Phase relative permeabilities– Wettability– Capillary bundle model

2. Saturation-dependencies of 3-phase relative permeabilities (using a capillary bundle)

– Occupancies and filling sequences– Dependency regions– Probabilistic model for distributed contact angles

3. Applications– Saturation-dependencies from 3D networks– Matching 2-phase field data to predict 3-phase

properties using network model– Software demonstration

Introduction

• Upscaling using network models• 3-Phase relative permeabilities• Wettability• Capillary bundle model

Upscaling using network models• Network model:

Pore → Network (core, REV)

Upscaling using network models• Single-phase:

Pore → Network (core, REV)Conductance Absolute permeability

Poiseuille’s Law Darcy’s Law

(cylindrical tube)Resistor type calculation for network nodal

pressures

4

8p r pq gl l

πµ

∆ ∆= − = −

4mPa s

g⎛ ⎞⎜ ⎟⋅⎝ ⎠

2(m )K

A PQ KLµ∆

= −

Upscaling using network models• Multi-phase:

Pore → Network (core, REV)Conductance Absolute and

relative permeability

Poiseuille’s Law Darcy’s Law (extension)

For pore and subnetwork filled with phase i

ii i

pq gl∆

= −

4mPa sig

⎛ ⎞⎜ ⎟⋅⎝ ⎠ , ( )r ik −

,i

i r ii

PAQ KkLµ∆

= −


Pore → Network (core, REV)Conductance Absolute and

relative permeability

Poiseuille’s Law Darcy’s Law (extension)

For pore and subnetwork filled with phase i

ii i

pq gl∆

= −

4mPa sig

⎛ ⎞⎜ ⎟⋅⎝ ⎠

,i

i r ii

PAQ KkLµ∆

= −

, ( )r ik −


Pore → Network (core, REV)Capillary ‘entry’ Capillary pressure

pressurePressure differencebetween phases i and j

(Young-Laplace, cylinder)

Volume function of saturations Si, also

2,

cosij ijc ijP

rσ θ

= ( ), ,c ij i jP S S

( ), ,r i i jk S S

Upscaling using network models• Main aim of network modelling (multi-phase

flow):Determine core-scale relative permeabilities and capillary pressures as functions of phase saturations

• Depending on pore-scale parameters, such as:– Pore geometry– Pore connectivity– Interfacial tensions– Wettability (contact angles)

2,

cosij ijc ijP

rσ θ

=

100% water

3-Phase relative permeabilities• 3-Phase flow: injection of single phase into

system with 2 or 3 phases present, represented in

ternary diagram

Sg = 1.00

So= 1.00 Sw = 1.00

S w= 0

.50

S w=

0.00

So = 0.50

So = 0.00

Sg= 0.50

Sg= 0.00

100% gas

100% oil

1w o gS S S+ + =

3-Phase relative permeabilities• 3-Phase flow: injection of gas after two-phase

water floodinto oil

100% water

Sg = 1.00

So= 1.00 Sw = 1.00

100% gas

100% oil

water flood

gas flooddisplacing:

oil and water

oil only (Swconstant)

3-Phase relative permeabilities• 3-Phase flow: water-alternating-gas injection

(WAG) / fluctuating ground water table around NAPL spill– Infinite number

of possiblesaturationpaths

100% water

Sg = 1.00

So= 1.00 Sw = 1.00

100% gas

100% oil

water floodgas flood

3-Phase relative permeabilities• Relperms along saturation paths

100% WaterSw = 1

100% Oil, So = 1

100% GasSg = 1

krg

Swi

Cycle 1

Water floodGas flood

3-Phase relative permeabilities• Relperms along saturation paths:

– Different along each path?!

– Hysteresis

100% WaterSw = 1

100% Oil, So = 1

100% GasSg = 1

krg

Swi

Cycle 2

3-Phase relative permeabilities• Iso-relative permeabilities (isoperms)

100% water

Sg = 1.00

So= 1.00 Sw = 1.00

100% gas

100% oil

0 01, .r gk =

0 1.0 2.0 4.0 6.0 8.

3-Phase relative permeabilities• Measurements of 3-phase relperms extremely

difficult, examples:– Corey, Rathjens, Henderson and Wylie (AIME,

1956)– Lenhard and Parker (WRR, 1987)– Oak (SPE, 1990)– Skauge and Larsen (SCA, 1994)– Dicarlo, Sahni and Blunt (TIPM, 1999)

3-Phase relative permeabilities• Traditional example (Corey et al., 1956)

100% gas

100% oil100% water

3-Phase relative permeabilities• Traditional example (Corey et al., 1956)

•Curved oil isoperms

•Straight water and gas isoperms


– Saturation-dependency

100% water

Sg = 1.00

So= 1.00 Sw = 1.00

100% gas

100% oil

( ),r g gk S ( ), ,r g g ok S S


– Relation with 2-phaserelperms, e.g.

or

100% water

Sg = 1.00

So= 1.00 Sw = 1.00

100% gas

100% oil

( )2,go

r g ok S

( )2,gw

r g wk S

2, ,

gor g r gk k=

( )?

2 2, , ,,go gw

r g r g r gk f k k=

3-Phase relative permeabilities• Empirical relations (e.g. Stone, Lenhard & Parker)

– Stone (J. Pet. Tech., 1970), with normalised saturations by Aziz and Settari (Petroleum Reservoir Simulation, 1979)

( ) ( )2, ,

gor g g r g gk S k S=

( )( ) ( )( )

,2 2, ,

,,

r o w ggo ow

r o g r o w

k S Sf k S k S=

( ) ( )2, ,

owr w w r w wk S k S=

22

1 1

**,,* * *

* *

( )( )( , ) . .

goowr o gr o w

ro w g ow g

k Sk Sk S S S

S S⎛ ⎞⎛ ⎞

= ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟− −⎝ ⎠ ⎝ ⎠

3-Phase relative permeabilities• Empirical relations (Stone I)

– Reasonable agreement with some data sets

– Limits correctly to 2-phase relations:

( ) ( )2 20 0 1 1* * * *, , ,,ow go

w r o g o r o r o oS k S S k k S= → = − = → =

( )20* *, ,

owg r o r o wS k k S= → =

22

1 1

1

**,,* * *

* *

* * *

( )( )( , ) . . ,

goowr o gr o w

ro w g ow g

w o g

k Sk Sk S S S

S S

S S S

⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟− −⎝ ⎠ ⎝ ⎠

+ + =

3-Phase relative permeabilities• Traditional assumptions for saturation-dependencies

based on simple pore-scale view ofpore occupancy

( ) ( ) ( ), , ,, , ,r w w r g g r o w gk S k S k S S

frequ

ency

pore size r

(Pore size distribution, PSD)

water oil gas

3-Phase relative permeabilities• Traditional assumptions for saturation-dependencies

• Water-wet system: water wetting to oil wetting to gas → water in small pores, gas in big pores

(explained later)

( ) ( ) ( ), , ,, , ,r w w r g g r o w gk S k S k S S

pore

occ

upan

cy

(num

ber f

ract

ion)

pore size r

water oil gas

3-Phase relative permeabilities• Water occupancy same as in two-phase water-oil

system, depends only on water saturation, therefore

( )2, ,

owr w r w wk k S=

pore

occ

upan

cy

(num

ber f

ract

ion)

r

water oil

3-Phase relative permeabilities• Gas occupancy same as in two-phase gas-oil

system, depends only on gas saturation, therefore

pore

occ

upan

cy

(num

ber f

ract

ion)

oil gas

( )2, ,

gor g r g gk k S=

r

3-Phase relative permeabilities• Oil occupancy not identical to any two-phase

occupancy, therefore

( ), ,r o w gk S S

pore

occ

upan

cy

(num

ber f

ract

ion)

water oil gas

r

Wettability• Wettability of surface defined in terms of oil-water

contact angle (measured through water)

• Sign of determines wettability of the surface and wetting order of the fluid pair– water-wet if (strongly ww if )

– oil-wet if (strongly ow if )

SOLID SURFACE

wateroilowθ

cos owθ

0cos owθ >0cos owθ <

1cos owθ =1cos owθ = −

Wettability• Wettability of surface defined in terms of oil-water

contact angle (measured through water)

• Sign of determines wettability of the surface and wetting order of the fluid pair (oil and water):– water wetting to oil if

– oil wetting to water if

SOLID SURFACE

wateroilowθ

cos owθ

0cos owθ >0cos owθ <

Wettability• Wettability distributions in a porous medium

– Uniformly water-wet or oil-wet– Distribution of contact angles

0

-1

1water-wet

oil-wet

cos owθ

r 0

-1

1cos owθ

r

Wettability• Wettability distributions in a porous medium (often

correlated to pore size– Mixed-wet with larger pores oil-wet (MWL): may occur

after primary drainage, ageing (similarly MWS)

r0

-1

1water-wet

oil-wet

cos owθ

rwet

Kovscek et al., 1993; Blunt, 1997; McDougall and Sorbie, 1995

Wettability• Wettability distributions in a porous medium

– Fractionally-wet (FW): caused by mineral distributions, independent of pore size

– Varying fractions of wwet and owet pores

r0

-1

1water-wet

oil-wet

cos owθ

Wettability• Effect on 2-phase cap pressures and

relperms:

– Drainage and imbibition Pc: hysteresis !– Water-wet: positive Pc; oil-wet: negative Pc– Intermediate-wet (Mixed-wet): contact angle has changed in (large)

pores after drainage

Wettability• Effect on 2-phase cap pressures and relperms:

– Water-wet -> oil-wet: • reversal of wetting and non-wetting phases• water relperm larger, oil relperm smaller

Fertl (1978)

Wettability• Contact angles in 3-phase systems

– Young’s equation (1805) states (horizontal) force balance for liquid-vapour system, in terms of surface and interfacial tensions:

– Liquid saturated vapour (equilibrium)

SOLID SURFACE

liquidvapourvlθ

cosvs ls vl vlσ σ σ θ= +

, ,ow go owθ θ θ

vlσ

lsσvsσ

,vs lsσ σ vlσ


– Young’s equations for three 2-phase systems, ,ow go owθ θ θ

solid surface

σowoilwater

θow

σws

σos

cosos ws ow owσ σ σ θ= +

solid surface

σgwgaswater

θgw

σws

σgs

cosgs ws gw gwσ σ σ θ= +

solid surface

σgogasoil

θgo

σos

σgs

cosgs os go goσ σ σ θ= +

Eliminateto obtain:

,os ws gsσ σ σand

cos cos cosgw gw ow ow go goσ θ σ θ σ θ= +

(Bartell and Osterhof, 1927; Zhou and Blunt, 1997)


– 1 constraint: 2 independent contact angle values– To be measured:

• 2 contact angles (difficult)• 3 interfacial tensions

at 3-phase equilibrium: oil spreading coefficient non-positive (Rowlinson & Widom, 1982):

, ,ow go owθ θ θ

0,S o gw go owC σ σ σ= − − ≤

0,S oC <non-spreading: 0,S oC =spreading:

gw ow goσ σ σ> >


– Usual wetting orders:• water wetting to oil or vice versa both possible

or• oil wetting to gas• water wetting to gas ,

at least for water-wet surface …

, ,ow go owθ θ θ

0cos owθ >0cos goθ >

0cos owθ <

0cos gwθ >


– Wetting order water to gas:• strongly oil-wet surface:

with Bartell-Osterhof

and :

i.e. gas wetting to water !

, ,ow go owθ θ θ

1cos owθ = − 1cos goθ =

cos coscos go go ow ow go ow

gwgw gw

σ θ σ θ σ σθ

σ σ+ −

= =

ow goσ σ>

0cos gwθ <


– May use linear relations between (experimentally verified) endpoints for strongly water-wet and oil-wet

– Relations satisfy Bartell-Osterhof

, ,ow go owθ θ θ

+1

cos owθ+1-1

cos goθ

cos gwθ

gw ow

go

σ σσ−

go ow

gw

σ σσ−

wettability ‘below’ which gas wetting to waternon-spreading oil

Capillary bundle model• Invasion of a single tube (cylinder):

– Displacement of water by oil if

with capillary ‘entry’ pressure according to Young-Laplace

– NB: sign of Pc determined by(wettability)

2,

cosow owc owP

rσ θ

=

,ow o w c owP P P P= − >

cos owθ

oP wP

2,

cos( ) ow owc ow j

j

P rr

σ θ=

Capillary bundle model• Invasion of a bundle of tubes (oil into water):

– for increasing oil pressure, tube requiring smallest oil ‘entry’ pressure is invaded first, i.e. for which

is minimum• e.g. uniformly water-wet

oPwP

, ( )o w c owP P P r= +0cos owθ >

1r2r

3r

4r

Capillary bundle model• Invasion of a bundle of tubes (oil into water):

– define macroscopic (‘bundle-scale’) Pc as actual pressure difference Po -Pw (outlet pressure difference)

– used as starting point for gas flood

oP

wP

oP3, ( )ow c owP P r=

1r2r

3r

4r

5r

Capillary bundle model• Invasion of a bundle of tubes (gas into oil and

water):

– check for which pore (water-filled or oil-filled,size)is minimum

• e.g. uniformly water-wet

gP

wP

oP

, ,( ),i ig c gP P P r i o w= + =

1r2r

3r

4r

5r

3, ( )ow c owP P r=

Capillary bundle model• Invasion of a bundle of tubes:

– gas into water requires:

– gas into oil requires:

gP

wP

oP

22

1, ( ) ,cos

g w c gw j wgw gw

j

P P P r P jr

σ θ= + = + =

1r2r

3r

4r

5r

23 4 5,

cos( ) , ,g o c go j

go

go o

j

P P P r P jr

σ θ= + = + =

3, ( )ow c owP P r=



gP

wP

oP

1r2r

3r

4r

5r

3, ( )ow c owP P r=

3

, ,

, ,

( ) ( )

( )( )g c gw o c gw

o c gw

w ow

c ow

P P

P

P P r

r

P P r

P P r

= + = +

+−

=

=

−



– gas into oil requires:3

22 1 2coscos ,gw gwow ow

g oj

P P jr r

σ θσ θ= − + =

23 4 5

cos, ,go go

g oj

P P jr

σ θ= + =

– use Bartell-Osterhof :

and wettability

cos cos cosgo go gw gw ow owσ θ σ θ σ θ= −

0 0cos , cosow goθ θ> >



– gas into oil requires:3

2 2 1 2cos cos ,gw gw w

oj

gow oP j

rP

rσ θ σ θ

+ − ==

3

3 3

3

2 23 4

2 2 1 2

5

2 2. .

cos cos, ,

cos co

cos cos ,

s

go go go gog o o

j

gw gw ow owo

gw gw ow owo

B O

j

P P P jr r

Pr

Pr

r

jr

σ θ σ θ

σ θ σ θ

σ θ σ θ

= + ≤ + = =

= + −

− =

<

< +

Capillary bundle model• Invasion of a bundle of tubes (gas into oil and

water):– oil-filled pores preferred over water-filled pores– starting with the largest pore

gP

wP

oP

1r2r

3r

4r

5r

3, ( )ow c owP P r=

Capillary bundle model• Also presented in pore occupancy plot: pore

filling sequence• Water-wet system: water wetting to oil wetting

to gas → water in small, gas in big pores

pore

occ

upan

cy

oilwater gas

r

2. Saturation-dependencies of 3-phase relative permeabilities (using a

capillary bundle model)

• Occupancies and filling sequences• Dependency regions• Probabilistic model for distributed contact

angles

Occupancies and filling sequences• Example mixed-wet system:

– MWL– constant contact angles– (strongly) water-wet / (weakly) oil-wet pores

r0

-1

1water-wet

oil-wet

cos owθ

rwetrmin rmax

1cos wwetowθ =

0cos owetowθ <

rmax

Occupancies and filling sequences• 2-phase oil invasion into water

– Minimum entry pressure?• water-wet:

or

rrwetrmin

pore

oc

cupa

ncy water-wet oil-wet

2

min

cos wwetowow

o wPr

P θσ= +

2 cos wwetoo w

wet

wo w r

P P σ θ= +

0 mincos ,wwetow wetr rθ > <

Po minimum for rwet

rmax


– Minimum entry pressure?• oil-wet:

or

rrwetrmin

pore

oc

cupa


2 cos owetoo w

wet

wo w r

P P σ θ= +

2

max

cos owetowow

o wPr

P θσ= +

0 maxcos ,owetow wetr rθ < <

Po minimum for rwet

rmax


– Minimum entry pressure?• water-wet or oil-wet:

or

rrwetrmin

pore

oc

cupa


2 cos wwetoo ww

o wwet

P Pr

σ θ= +

2 cos owetoo ww

o wwet

P Pr

σ θ= +

0 0cos , coswwet owetow owθ θ> <

Po minimum for oil-wet pores

rmax


– Filling occurs in oil-wet pores in increasing size, followed by water-wet pores in decreasing size

– Finishes at row,1 , e.g. related to connate water, with

rrwetrmin

pore

oc

cupa


row,1

1

2

,

cos wwetow ow

owow

Pr

σ θ=

rmax

Occupancies and filling sequences• 3-phase gas invasion into oil and water

– Starting with oil-water filling up to row,1– Candidate pores water-wet:

• water-filled rmin, row,1• oil-filled row,1, rwet

– Candidate pores oil-wet: • water-filled rwet, rmax

rrwetrmin

pore

oc

cupa


row,1

rmax


– Minimum entry pressure water-wet pores?rrwetrmin

pore

oc

cupa


row,1

min , min: ( )g w c gwr P P P r= +

1 1water, , ,( ) : ( )ow g w c gw owr P P P r= +

,: ( )wet g o c go wetr P P P r= +1 1oil, , ,( ) : ( )ow g o c go owr P P P r= +

rmax



pore

oc

cupa


row,1

1,min , in, m: (( ) )o c og w w go c wP P rr P P r+−=

1 1 1water, , ,, ,( ) (): )(o c oow g c gwow owwP PP rr P r−= +


rmax



pore

oc

cupa


row,1

1min , , , min: ( ) ( )g o c ow ow c gwr P P P r P r= − +

1 11 1water , , , , , ,,

. .( ) ( ) ( )( ) : c ow ow c gw ow c goow wo og o

B OP r P r P rr P P P+− == +


consistency

rmax



pore

oc

cupa


row,1

1min , , , min: ( ) ( )g o c ow ow c gwr P P P r P r= − +

1 1water, , ,( ) : ( )ow g o c go owr P P P r= +


increasing pressure: filling order

minimum

rmax


– Minimum entry pressure oil-wet pores?rrwetrmin

pore

oc

cupa


row,1

,: ( )wet g o c go wetr P P P r= +

max , max: ( )g o c gor P P P r= +increasing pressure: filling order

minimum

rmax


– Minimum entry pressure water-wet vs. oil-wet pores?rrwetrmin

pore

oc

cupa


row,1

water w2

- et ,( ) : (c s

)o

wet g o c go wet o

wwetgo go

wet

r P P rr

P Pσ θ

= + = +

oi - e2

l w tmax , maxmax

( ) : ( )cos owet

go gg o c go o

or P P P r Pr

σ θ= + = +


and

• No definiteordering ofentry pressures!– gas may invade

water-wet andoil-wet poressimultaneously

maxwetr r<

cos coswwet owetgo goθ θ<

cos owθ

cos goθ

cos owetowθ cos wwet

owθ

cos wwetgoθ

cos owetgoθ

water-wet2

( )cos

:wet g o

wwetgo go

wetrr P P

σ θ= +

oil-wet2

mmax

ax

( ) :cos

g o

owetgo gor P P

rσ θ

= +

rmax


– Filling occurs in decreasing size in both oil-wet and water-wet pores and may occur simultaneously

– Arrives at rgo,1 , and rgo,2, with

rrwetrmin

pore

oc

cupa


1 2

2 2

, ,

cos coswwet owetgo go go go

gogo go

Pr r

σ θ σ θ= =

rgo,1 rgo,2row,1

rmax


– May continue to invade water-filled pores– Finishes at rgw,1 , and rgo,2, say at residual oil, with

rrwetrmin

pore

oc

cupa


1 2

2 2

, ,

cos coswwet owetgw gw go go

go wgw go

P Pr r

σ θ σ θ+ ==

rgw,1 rgo,2

rmax

Occupancies and filling sequences• 3-phase water invasion into gas, oil and water

– Starts invading gas-filled water-wet pores at rgw,1– Minimum entry pressure oil-wet pores?

rrwetrmin

pore

oc

cupa


rgw,1 rgo,2

rmax


– Minimum entry pressure oil-wet pores? ( )

rrwetrmin

pore

oc

cupa


rgo,2

,: ( )wet w o c ow wetr P P P r= −

2 2oil, , ,( ) : ( )go w o c ow gor P P P r= −

max , max: ( )w g c gwr P P P r= −2 2gas, , ,( ) : ( )go w g c gw gor P P P r= −

rgo,2 preferred over rwet

identical (Bartell-Osterhof)

, ,c ji c ijP P= −

rmax


– Minimum entry pressure oil-wet pores?

rrwetrmin

pore

oc

cupa


rgo,2

2

maxmax , max: ( )

cosgww

ow

g c gww

g

etgr P P P r

rP

θσ= − = −

22

2

2gas, , ,

,

c) )

s(

o( : gw

go w g c gw go g

owetgw

go

r P P P r Pr

σ θ= − = −


and

– Water wetting to gas in weaklyoil-wet pores

– Minimum entry pressure oil-wet pores?

cos owθ

cos gwθ2, maxgor r<

0cos owetgwθ >

cos owetowθ

cos wwetgwθ

2

mama

xx :

cosgwowetg

w gw

rr P P

σ θ= −

22

2gas

,, ( ) :

cosgwgo w

owetgw

gogr P P

rσ θ

= − minimum

rmax


– Water invades oil-filled and gas-filled pores simultaneously, starting from rgo,2

rrwetrmin

pore

oc

cupa


rgo,2

2 2gas, , ,( ) : ( )go w g c gw gor P P P r= −

2 2 2 2

2

oil. .

,

, , , , , ,

,

,( ) : ( ) ( ) ( )

( )B O

g c gw go

go w o c ow go g c go go c ow gor P P P r P P r P

P P r

r= − = − − =

= −

rmax

Occupancies and filling sequences• Types of occupancies

– Type I: oil has ‘boundaries’ with both water and gas i.e. oil invasion leads to displacement of both gas and water

⇒ oil is the intermediate-wetting phase

rrwetrmin

pore

oc

cupa



– Type II: gas has ‘boundaries’ with both oil and water

⇒ gas is the intermediate-wetting phase

rmax rrwetrmin

pore

oc

cupa



– Type III: water has ‘boundaries’ with both oil and gas

⇒ water is the intermediate-wetting phase

rrmaxrwetrmin

pore

oc

cupa


Saturation-dependencies• For the different types of occupancies

– how do the relperms depend on phase saturations?– relation between 2-phase and 3-phase relperms?

• Definitions of saturation and relperm for capillary bundle:

– Pore size distribution– Phase j occupancy function– Volume function– Conduction function (e.g. Poiseuille)

( ) ( ) ( )j jo

S r r V r drπ ϕ∞

= ∫

( )j rπ

, ( ) ( ) ( )r j jo

k r r g r drπ ϕ∞

= ∫( )rϕ

2( )V r r∝4( )g r r∝

Saturation-dependencies• Definitions of saturation and relperm for capillary

bundle, for example:

rmax rrwetrmin

pore

oc

cupa


rgo,2

21 for

0 otherwise,( ) wet go

o

r r rrπ

< <⎧= ⎨⎩

2,

( ) ( )go

wet

r

or

S r V r drϕ= ∫2,

, ( ) ( )go

wet

r

r or

k r g r drϕ= ∫

Saturation-dependencies• Paths in saturation space: gas flood into oil, followed

by water flood into gas and oil

oil water

gas

gas flood

water flood

I

III

II

Saturation-dependencies• Paths in saturation space: water floods starting at

different gas saturations

oil water

gas

lines of constant Pgo

3 regions in saturation space

III

III

Saturation-dependencies• Paths in saturation space: gas floods starting at

different water saturations

oil water

gas

lines of constant Pow

3 regions in saturation space

III

III

Saturation-dependencies• Regions in saturation space: iso-capillary pressure

curves

II II( )go oP S ( , )ow w oP S S

II

gas is “intermediate-wetting”

Saturation-dependencies• Regions in saturation space: iso-relative

permeability curves

II II, ( )r o ok S , ( , )r g w ok S S

gas is “intermediate-wetting”

II

rmax

Saturation-dependencies• Type I: oil is intermediate-wetting

rrwetrmin

pore

oc

cupa


,

, , ,

( ), ( ), ( )

( ), ( ), ( , )ow w go g gw w g

r w w r gg og r w

P S P S P S

k

S

Sk S SS k

Saturation-dependencies• Type II: gas is intermediate-wetting

rmax rrwetrmin

pore

oc

cupa


,

, , ,

( ), ( ), ( )

( ), ( ), ( , )gw w go o ow w o

r w w r oo go r w

P S P S P S

k

S

Sk S SS k

Saturation-dependencies• Type III: water is intermediate-wetting

rrmaxrwetrmin

pore

oc

cupa


,

, , ,

( ), ( ), ( )

( ), ( ), ( , )gw g ow o go o g

r g g r go wo r o

P S P S P S

k

S

Sk S SS k

rmax

Saturation-dependencies• Link 2-phase and 3-phase relperms

– Type I: oil is intermediate-wetting

rrwetrmin

pore

oc

cupa


rmax



rrwetrmin

pore

oc

cupa


water occupancy same as for oil-water

rmax



rrwetrmin

pore

oc

cupa


water occupancy same as for oil-watergas occupancy same as for gas-oil

oil occupancy different from any 2-phase occupancy

rmax

Saturation-dependencies• Link 2-phase and 3-phase relationships


rrwetrmin

pore

oc

cupa


2,, ,( ) ( )ow

r w w r w wk S k S=

rmax



rrwetrmin

pore

oc

cupa


2,, ,( ) ( )ow

r w w r w wk S k S=2,

, ,( ) ( )gor g g r g gk S k S=

2 21 , ,, , , ,( ) ( ) ( )go ow

r o w g r g g r w wk S S k S k S= − −


– Type II: gas is intermediate-wetting

rmax rrwetrmin

pore

oc

cupa




rmax rrwetrmin

pore

oc

cupa


2,, ,( ) ( )ow

r w w r w wk S k S=



rmax rrwetrmin

pore

oc

cupa


2,, ,( ) ( )ow

r w w r w wk S k S=2,

, ,( ) ( )gor o o r o ok S k S≠

Non-genuine dependency: result of simultaneous invasion water-wet and oil-wet pores


– Type III: water is intermediate-wetting

rrmaxrwetrmin

pore

oc

cupa


2,, ,( ) ( )gw

r g g r g gk S k S=2 21 , ,

, , , ,( ) ( ) ( )gw owr w o g r g g r o ok S S k S k S= − −

2,, ,( ) ( )ow

r o o r o ok S k S=

Saturation-dependencies• Three phase occupancy / dependency types:

– Type I: oil is intermediate-wetting phase– Type II: gas is intermediate-wetting phase– Type III: water is intermediate-wetting phase

• Relperm of intermediate-wetting phase depends on two saturations (similar conclusion for cap pressures)

• Relperms of remaining two phases depend on their own saturations (single-phase dependency)

• Single-phase dependencies can be non-genuine: no link between 2-phase and 3-phase relperms

• In saturation space regions of each dependency may occur

Saturation-dependencies• Features of dependency regions for MWL

II or IIII

II

non-linear boundary between regions I & II

cos gwθregion III exists only if water wetting to gas in oil-wet pores: sign of

gas

oil water

r

water-wet

oil-wet

cos owθ

r

2,, ,( ) ( )go

r o o r o ok S k S≠

2,, ,( ) ( )gw

r w w r w wk S k S≠

Saturation-dependencies• Features of dependency regions for FW

location of boundaries also determines non-genuine behaviour

gas

oil water

r

water-wet

oil-wet

cos owθ

r

IIII

II

2 non-linearboundaries

• numerical example FW

gas isoperms

II

IIII

0.09

0.99

Saturation-dependencies

36 14 29 mN/m, ,gw go owσ σ σ= = = 0 6 0 1cos . , cos .wwet owetow owθ θ= = −

10 m 160 mmin min,r rµ µ= =

oil isoperms

0.01

0.91

Distributed contact angles

• Multiple phase occupancy for given pore size

0

-1

1water-wet

oil-wet

cos owθ

minrwetr maxr

r

rmax rrwetrmin

pore

oc

cupa


,( ) ( )owo ow c owr P P rπ ⎡ ⎤= >⎣ ⎦P

Distributed contact angles• View occupancy that pore is occupied by a

given phase as the probability that the corresponding entry pressure(s) are overcome– for example in 2-phase oil-water system, pore

occupied by oil if– then oil occupancy given by

,ow c owP P>

( )owo rπ

Distributed contact angles• In 3-phase flow, two entry pressures need

to be overcome– for example gas entry pressure conditions are

– then occupancy by gas in three-phase system can be expressed as joint probability

,

,

gw c gw

go c go

P P

P P

>⎧⎪⎨ >⎪⎩

( ) ( ), ,( ) ( ) ( )g go c go gw c gwr P P r P P rπ ⎡ ⎤= > ∧ >⎣ ⎦P

Distributed contact angles• Using Young-Laplace equation

and linear relation between andeach condition can be expressed as a (linear) inequality on– for example

cos ( ; )ow go goh r Pθ >

2,

cos( ) ij ij

c ijP rr

σ θ=

+1

cos owθ+1-1

cos goθ

cos gwθ

gw ow

go

σ σσ−

go ow

gw

σ σσ−

cos ,cosgo gwθ θ cos owθ

cos owθ

,go c goP P>

Distributed contact angles• Three-phase occupancies are joint

probabilities in terms of the randomly distributed variable– e.g. the three-phase gas occupancy becomes

which varies with the actual pressure combinations

• Occupancy calculated according to

cos owθ

( ) ( ; ) cos ( ; )g go go ow gw gor h r P h r Pπ θ⎡ ⎤= < <⎣ ⎦P

,go gwP P

[ ] cosP cos ( , )h

ow h r x dxθθ ψ−∞

< = ∫

• Conditions for these probabilities can be expressed in the ‘wettability’ plane

rrm a x

w etrrm in

o wco sθ

1

1−

watergas

gasoil

oil

o wh

P( ) ( ) cos ( )g go ow gwr h r h rπ θ⎡ ⎤= < <⎣ ⎦

( ,cos )owr θ

g oh

g oc

g wh

g wc


gas

III

I

iso-relperms• Regions I and III

arise• However, mainly

multipledependencies– all 3 relperms

depend on more than one saturation

• Cusp-shape part of region I non-genuine


oil

III

water

I

gas

III

I


Distributed contact angles• Probabilistic model basis for software tool

3. Applications

• Saturation-dependencies from a 3D network• Matching 2-phase field data to predict 3-

phase properties using network model• Software demonstration

Saturation-dependencies in network• Dependencies will change in 3D network, mainly

because of:– interconnectivity (causing trapping)– flow in films and layers (directly related to wettability)both reducing phase continuity

WAG flood in water-wet glass micromodel(Sohrabi et al. HWU)

– wetting films around both oil and gas– oil layers separating water and gas

Saturation-dependencies in network• Features of HW network model:

– rectangular 3-D network, random pore size distributions

– variable coordination number– variable pore wettability

(constant or distributed)– wetting films and spreading

layers– capillary dominated flow– consecutive flood cycles

(WAG): drainage (piston-like) and imbibition (snap-off) events

– multiple displacements

Saturation-dependencies in network• High phase continuity simulations: saturation paths

– MWL, z=6, water non-wetting in owet pores, wetting films for both oil and water 21x14x14 nodes

network capillarybundle

gas injection

III

gas injection

III

good agreement

water injection

Saturation-dependencies in network• High phase continuity simulations: saturation paths

– MWL, z=6, water non-wetting in owet pores, wetting films for both oil and water 21x14x14 nodes


good agreement, but gas trapping causes deviationswater injection

o-w-go-w-g

Saturation-dependencies in network• Low phase continuity simulations: saturation paths

– MWL, z=4, water non-wetting in owet pores, no wetting films, non-spreading oil 21x14x14 nodes


gas injection: remnant of analytical pattern

o-g-w

III

o-g-w

Saturation-dependencies in network• Low phase continuity simulations: saturation paths

– MWL, z=4, water non-wetting in owet pores, no wetting films, non-spreading oil 21x14x14 nodes


water injection: very large residuals

Match and predict with network model• Main parameters (cubic grid)

– connectivity defined by coordination number – pore radius distribution controlled by exponent

– geometry of “pore elements” defined by:• volume exponent

• conductance exponent

– wettability parameters: oil-water contact angle vs. radius

– films and layers

( ) nf r r∝

zn

min maxr r r< <

( )V r rν∝

( )g r rλ∝

ν

λ

with

cos ( )ow rθ wetrinvolving

Match and predict with network model• Field case experimental data (single and two phase)

– absolute permeability– mercury intrusion data (PSD)– initial (connate) water saturations– constant (high) rate water flood data + in situ saturations → imbibition oil-water relperms and cap pressure (indicates non water-wet)

– constant differential pressure gas flood data (ambient) + in situ saturation → gas-oil relperms

– trapped gas saturations to oil and water from oil and water imbibition

– centrifuge gas-oil cap pressure and oil relperm– IFTs not measured !

Match and predict with network model• Anchoring network parameters

– use gas flood data: independent of wettabilitywhen assuming spreading oil(IFT estimate: )

– network parameters

– estimate from mercury injection data:

40 15 25 mN/m, ,gw go owσ σ σ= = =

( ) nf r r∝( )V r rν∝( )g r rλ∝

, , ,n zν λ(PSD)

(volume)

(conductance)

0 05 m 15 mmin max.r rµ µ= =

Match and predict with network model• connate water and oil spreading layers present

– all oil drained at connate water ( )

4 0 040 4 3 4

.. .

z nν λ= == =

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1Sg

Krg

/Kro

Kro

Krg

Sim - Kro

Sim - Krg

0 88.gS =

0.00001

0.0001

0.001

0.01

0.1

10 0.2 0.4 0.6 0.8 1

Sg

Krg

/Kro

Kro

Krg

Sim - Kro

Sim - Krg

Match and predict with network model• connate water and oil spreading layers present

– match of only -> prediction (!) of

4 0 040 4 3 4

.. .

z nν λ= == =

,r gk ,r ok

Match and predict with network model• centrifuge gas-oil capillary pressure

– correct trend, precise fit depends on “weight” of oil layers

4 0 040 4 3 4

.. .

z nν λ= == =

0

20000

40000

60000

80000

100000

0 0.2 0.4 0.6 0.8 1Sg

Pcg

o (P

a)

Pcgo

Sim - Pcgo - layers

Sim - Pcgo - no layers

Match and predict with network model• Anchoring wettability parameters

– use water flood data: imbibition after ageing– network parameters determined with gas flood

data– assume MWL system + constant contact angles:

– vary size of largest water-wet pore

– assume water wetting films absent

4 0 04 0 4 3 4. . .z n ν λ= = = =

0 5

0 1

cos .

cos .

wwetow

owetow

θ

θ

=

= −water-wet oil-wet

minr

0 5 0 1cos . cos .w oow owθ θ= = −

wetr maxr

PSD

wetr

Match and predict with network model• water flood relperms in MWL system

– values of bounded by

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1Sw

Kro

/Krw

KrwKroSim - Krw - rw et=1Sim - Kro - rw et=1Sim - Krw - rw et=2Sim - Kro - rw et=2

,r wk wetr

water-wet

0 05 m15 m

min

max

.rr

µµ

==


– good match of small

0.0001

0.001

0.01

0.1

10 0.2 0.4 0.6 0.8 1

Sw

Kro

/Krw

KrwKroSim - Krw - rw et=1Sim - Kro - rw et=1Sim - Krw - rw et=2Sim - Kro - rw et=2

,r ok


– location of steep decline of well matched

-5000

-4000

-3000

-2000

-1000

00 0.2 0.4 0.6 0.8 1

Sw

Pco

w (P

a)

Pcow

Sim - Pcow - rw et=1

Sim - Pcow - rw et=2

,c owP

0 05 m15 m

min

max

.rr

µµ

==

Match and predict with network model• Prediction of 3-phase properties:

– network parameters from anchoring– IFTs (estimated):– wettability:

• MWL system, choose = 1•

– films and layers:• suppress wetting films and spreading layers in water-

wet pores (not relevant)• probably oil wetting films in gas filled oil-wet pores

( ), no films around water ( )

40 15 25 mN/m, ,gw go owσ σ σ= = =

0 5 0 1cos . , cos .w oow owθ θ= = −

wetr mµ

1cos goθ = 0 1cos .owθ = −

Match and predict with network model• 3-phase gas floods (following water flood):

– significant water displaced (except for )– saturation-dependency as expected (except for )

10 90

8020

30 70

6040

50 50

4060

70 30

2080

90 10

102030405060708090wateroil

gas

0 1.wiS =0 7.wiS =

saturation paths starting at range of Swi

indicates region III: water intermediate-wetting

Indicates region I: oil iw

indicates region II: gas iw

• gas-water cap pressures:– same along different paths for small Sg– different for large Sg

10 90

8020

30 70

6040

50 50

4060

70 30

2080

90 10

102030405060708090wateroil

gas

0

5000

10000

15000

20000

0 0.2 0.4 0.6 0.8 1

Sg

Pcg

w (P

a)

Sw i = 0.1

Sw i = 0.3

Sw i = 0.5

Sw i = 0.7

( ),c gw gP S

Match and predict with network model

( ), ,c gw w gP S S

• water relperms different along different paths(intermediate-wetting phase)

0

0.2

0.4

0.6

0 0.2 0.4 0.6 0.8

Sw

Krw

Sw i = 0.1

Sw i = 0.3

Sw i = 0.5

Sw i = 0.7

10 90

8020

30 70

6040

50 50

4060

70 30

2080

90 10

102030405060708090wateroil

gas

( ), ,r w o gk S S


• oil relperms same along different paths(mostly wetting phase)

10 90

8020

30 70

6040

50 50

4060

70 30

2080

90 10

102030405060708090wateroil

gas

0

0.2

0.4

0.6

0 0.2 0.4 0.6 0.8 1

So

Kro

Sw i = 0.1

Sw i = 0.3

Sw i = 0.5

Sw i = 0.7

( ),r o ok S


• gas relperms:– same along different paths for large Sg (non-wetting)– different for small Sg (intermediate-wetting)?!

10 90

8020

30 70

6040

50 50

4060

70 30

2080

90 10

102030405060708090wateroil

gas

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Sg

Krg

Sw i = 0.1

Sw i = 0.3

Sw i = 0.5

Sw i = 0.7( ),r g gk S


( ), ,r g o wk S S

• Conclusions– good anchoring of network parameters using gas-oil data – indication of wettability state from match to water flood

data: mixed-wet with larger pores weakly oil-wet– uncertainties: interfacial tensions, presence of oil wetting

films around gas in ow pores; films have significant effect– qualitative behaviour of predicted 3-phase relperms

mostly as expected– assumed mixed-wet system rather unfavourable for gas

flooding (gas displacing water), although lack of water continuity may force oil displacement


Discussion• Simple capillary bundle model to predict saturation-

dependencies of 3-phase relperms and cap pressures– complex dependency regions found, which no empirical

models a priori include in their formulations (except strongly water-wet case)

– limitations: no interconnectivity (trapping, multiple displacements), limited intra-pore effects (e.g. film flow); however, saturation-dependency regions broadly correct

– applicability:• indication of “need-to-know information” (IFT’s, degree of oil-water

wettability, ranges of pore sizes): measuring wettability !• criteria for combining 2-phase & 3-phase data

– process-based approach necessary: network model

three-phase relative permeabilities ... - universiteit utrecht · three-phase relative...

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