Pore-scale modelling of WAG:
impact of wettability
Rink van Dijke and Ken Sorbie
Institute of Petroleum Engineering
Heriot-Watt University
WAG Workshop FORCE, Stavanger, 18 March 2009
1
Introduction
• 3-phase (immiscible) flow processes, e.g.
– water-alternating-gas injection (WAG): improved oil recovery
– NAPL in unsaturated zone: ground water remediation
• modelled with Darcy’s law:
qi  
Kkr ,i
i
Pi ,
i  w , o, g
• capillary pressure and relative permeability functions



Pc ,ij Si , S j  Pi  Pj , kr ,i Si , S j

– difficult to measure
– pore-scale modelling
2
Introduction
• Pore-scale modelling:
– pore space structure:
• connectivity (topology)
• geometry (pore sizes and shapes)
– flow mechanisms:
• capillary forces
• conductance (viscous forces)
– wettability (contact angles)
– incorporated in
• idealized network models
(quasi-static “invasion percolation” or
dynamic)
• capillary bundle models
3
water, oil, gas
Introduction
• Capillary forces:
– invasion of a single tube (cylinder):
Po
Pw
– ‘rule’ for displacement of water by oil:
Pow  Po  Pw  Pc ,ow
with capillary ‘entry’ pressure according to
Young-Laplace:
2 cos 
Pc ,ow 
ow
ow
r
4
Introduction
• Wettability:
– wettability of pore surface defined in terms of oilwater contact angle (measured through water)
oil
 ow
cos ow  0
cos ow  0
water
SOLID
SURFACE
• water-wet if
• oil-wet if
5
Introduction
• Wettability:
– in 3-phase flow contact angles:  ow , go , gw
• related by Bartell-Osterhof equation:
 gw cos gw   ow cos ow   go cos go
• constitute capillary entry pressures for gas-water
and gas-oil displacements, e.g.
Pc , gw 
2 ow cos  gw
r
• determine presence of wetting films and spreading
layers
6
Introduction
• Micromodel experiments:
– understand flow mechanisms
– validate pore-scale network models
250 m
50 m
pore crosssection: wide and
shallow
– Sohrabi et al. (HWU)
7
Outline: effects of wettability
• Saturation-dependencies of three-phase capillary
pressures and relative permeabilities
• Intra-pore physics:
– fluid configurations
– capillary entry pressures and layer criteria
– non-uniform wettability
• Network displacement mechanisms:
– phase continuity and displacement chains
– WAG simulations
– comparison simulations and WAG micromodel experiments
• Concluding remarks
8
Saturation-dependencies
• Traditional example (Corey et al., 1956)
•Curved oil isoperms
•Straight water and
gas isoperms
9
Saturation-dependencies
• Traditional assumptions for saturation-dependencies
kr , w  S w  , kr , g  S g  , kr ,o  S w , S g 
pore occupancy
(number fraction)
• Water-wet system: water wetting to oil wetting to gas
 water in small pores, gas in big pores
water
oil
gas
pore size r
10
Saturation-dependencies
• Wettability distributions in porous medium often
correlated to pore size:
– mixed-wet with larger pores oil-wet (MWL): may occur
after primary drainage and aging (similarly MWS)
1
cos ow
water-wet
rwet
0
r
oil-wet
-1
11
Saturation-dependencies
• Paths in saturation space: gas flood into oil, followed
by water flood into gas and oil
water-wet
oil-wet
gas
water
flood
gas flood
oil
• capillary bundle model
I
II
III
water
12
Saturation-dependencies
• Regions in saturation space: iso-capillary pressure
curves
Pow ( Sw , So )
Pgo ( So )
II
II
II
gas is “intermediate-wetting”
13
Saturation-dependencies
• Regions in saturation space: iso-relative
permeability curves
k r , g ( S w , So )
kr ,o ( So )
II
II
II
gas is “intermediate-wetting”
14
Saturation-dependencies
• numerical example FW capillary bundle
wwet
owet
 0.6, cos  ow
 0.1
 gw  36,  go  14,  ow  29 mN/m cos ow
rmin  10 m, rmin  160 m
oil isoperms
gas isoperms
0.99
0.01
II
I
0.91
0.09
III
15
Intra-pore physics
• Films and layers:
– water-wet micromodel: WAG flood
• water wetting films around both oil and gas
• possible oil layers separating water and gas
16
Intra-pore physics
• Fluid configurations in angular pores:
– water-wet pores, e.g. strongly water-wet:
 ow , go , gw all close to 0
• water wetting films around both oil and gas
• possible oil layers separating water and gas:
affected by oil spreading coefficient
C S ,o   gw   ow   go
– oil-wet pores, e.g. weakly oil-wet:
 ow , gw close to 90 degrees,  go close to 0
• no oil wetting films around water
• only oil wetting films around gas
– ensures phase continuity along pores
17
Intra-pore physics
• true 3-phase capillary entry pressures (improved
Y-L)
bulk displacement


– gas-oil entry pressure Pc , go Pow depends on water
wetting film pressure
– determined by free energy calculation (MS-P)
– also criterion for (oil) layers
layer displacement
18
Intra-pore physics
• consistent relation 3-phase pressure differences
and occupancies
oil-water bulk
1.5
displacement
gas-oil bulk
displacement
(true varying)
rgo
Pgo
1

go
0.5
0
0
layer
displacement
0.5
1
Pow
row
1.5
gas-oil bulk
displacement,
with layer
(constant)
19
Intra-pore physics
• mixed-wet bundle of triangular pores:
– small pores strongly water-wet
– large pores weakly oil-wet: cos ow  0.1
rw
rd
5  m  rd  20  m
20  m  rd  55  m 20
Intra-pore physics
• water injection
– no difference true (3-phase)
and constant (2-phase)
during invasion of water-wet
pores
– huge differences
during invasion of
oil-wet pores
– true: simultaneous
w->o and
w->g
– volume effect
oil films
true
constant
21
o-g-w
Intra-pore physics
• nonuniform wettability:
– after primary
drainage
surface rendered
oil-wet: aging
(Kovscek)
- after imbibition
oil layers
(2-phase)
oil
water
– strongly affects water flood Sor (Ryazanov et al.,
2009)
22
Intra-pore physics
• non-uniform wettability
• layers in 3-phase configuration
• consistent entry pressures and layer
criteria
23
Intra-pore physics
1
high Pow
drainage
gas injection
0
-6
-5
-4
Pgw
-7
-3
-2
-1
-1
-2
-3
P ow
24
D->
D->
D->
D->
G->
G->
A->
B->
B->
B->
B->
D->
A->
Network displacement mechanisms
• phase continuity:
– connectivity
– films and layers (wettability)
– water-wet micromodel: WAG flood
25
Network displacement mechanisms
• connected, trapped and disconnected phases
disconnected
oil cluster
invading
gas cluster
water cluster
connected to
outlet
outlet
inlet
trapped
oil
cluster
disconnected disconnected
water cluster gas cluster
– phase cluster map
oil cluster
connected
to outlet
26
Network displacement mechanisms
• multiple displacement chains displace disconnected clusters
invading
gas cluster
rgo
disconnected
oil cluster
water cluster
connected to
outlet
outlet
inlet
trapped
oil cluster
rog
disconnected disconnected
water cluster gas cluster
• based on “target” pressure difference
rgw
oil cluster
connected to
outlet
e.g. gas->oil->gas->water
target
Pginvading  Pwout  Pgw
 Pc , go (rgo )  Pc ,og (rog )  Pc ,ow (row )
• determining lowest target requires shortest path algorithm
27
Network simulations
• 3-phase flow simulator 3PhWetNet: regular lattice,
arbitrary wettability, capillary-dominated flow
• few free parameters describing essence of porescale displacements (needs “anchoring”)
– coordination number z
– pore size distribution
– volume and conductance
exponents
– wettability (contact angle
distribution)
– film and layers (notional)
28
Network simulations
• Network model:
– parameters “anchored” to easy-to-obtain data:
network structure and wettability
– example mixed-wet North Sea reservoir data
1
1
Krw
gas flood
water flood
Kro
0.8
Sim - Krw - rw et=1
Sim - Kro - rw et=1
0.8
0.6
Kro/Krw
Krg/Kro
Sim - Krw - rw et=2
0.4
Kro
Sim - Kro - rw et=2
0.6
mixed-wet
(MWL)
0.4
water-wet
Krg
Sim - Kro
0.2
0.2
Sim - Krg
0
0
0
0.2
0.4
0.6
Sg
0.8
1
0
0.2
0.4
0.6
Sw
0.8
29
1
Network simulations
• Network model:
– predict difficult-to-obtain data, e.g. 3-phase kr and Pc
gas
1
Sw i = 0.1
80
70
60
50
40
Sw i = 0.3
10
0.8
Sw i = 0.65
30
Sw i = 0.7
0.6
40
50
0.4
60
30
20
70
0.2
80
10
oil
Sw i = 0.5
20
Krg
90
90
90 80 70 60 50 40 30 20 10
three-phase gas injection
displacement paths
water
0
0
0.2
0.6
0.4
1
0.8
Sg
three-phase gas relperms
30
WAG network simulations
• mixed-wet
• no films or layers
• varying coordination
number z
z=5
z=3
• high residual, but additional recovery during WAG
for z=3
31
WAG network simulations
• displacement statistics (chain lengths), z=5
chain length fraction
1.00
0.75
5
4
3
2
1
0.50
0.25
0.00
water 1
gas 1
water 2
gas 2
water 3
gas 3
• few multiple, many double displacements
• continuing phase “movement” but no additional
recovery
32
WAG network simulations
• displacement statistics (types), z=5
displacement type fraction
1.00
0.75
g->w
g->o
w->g
w->o
o->g
o->w
0.50
0.25
0.00
water 1 gas 1 water 2 gas 2 water 3 gas 3
• mainly 3 displacement types, corresponding to doubles,
33
e.g. g->o and o->w during gas flood
WAG network simulations
pore occupancy fraction
• WAG occupancy statistics (z=5): end gas flood 2
1.00
0.75
gas
water
oil
0.50
0.25
0.00
5
9
13
17
21
25
r (m)
• oil and gas in both water-wet and oil-wet pores
38
WAG network simulations
• Chain lengths
(z=3)
chain length fraction
1.00
0.50
0.25
5
ga
s
5
4
at
er
w
ga
s
4
at
er
3
w
ga
s
3
at
er
2
w
ga
s
2
at
er
1
1
w
ga
s
at
er
w
1.00
chain length fraction
5
4
3
2
1
0.00
z=5
•
0.75
0.75
5
4
3
2
1
0.50
0.25
significant number of multiple chains
39
0.00
water 1
gas 1
water 2
gas 2
water 3
gas 3
WAG network simulations
0.50
0.25
g->w
g->o
w->g
w->o
o->g
o->w
0.50
0.25
5
s
ga
r5
4
at
e
s
w
ga
r4
3
at
e
s
w
ga
r3
2
at
e
s
w
ga
r2
at
e
s
w
at
e
w
0.75
1
0.00
z=5
1.00
displacement type fraction
g->w
g->o
w->g
w->o
o->g
o->w
r1
•
0.75
ga
• Displacement
types (z=3)
displacement type fraction
1.00
additional types of displacements
g->o for water and o->g for gas floods
0.00
water 1 gas 1 water 2 gas 2 water 3 gas 3
40
WAG simulation micromodel experiment
water-wet
oil-wet
• weakly wetted: little evidence of (continuous)
water and oil wetting films (around water)
• spreading oil: assume oil layers and oil wetting
41
films around gas  gw  56  ow  41  go  15 mN/m
WAG simulation micromodel experiment
• Fractionally-wet
– 50% water-wet & oil-wet pores
– angles distributed between 60-120 degrees
– oil layers and oil wetting films around gas
– recovery
ceases after
WAG 2
0,4
60
l [%]
OilResidual
recoveryoi%Sorw
• Comparison
simulated and
experimental
recoveries
0,35
50
0,3
Neutral-wet
Neutral-wet
Moreoil-wet
oil-wet
More
40
0,25
0,2
30
Morewater-wet
water-wet
More
0,15
20
0,1
10
0,05
Case77
Case
00
wfwf 1g1g1w 1w2g 2g2w 2w
3g 3g
3w 3w
4g
4w
4g
4w
42
WAG simulation micromodel experiment
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
5
4
3
2
1
wa
te
rfl
oo
d
ga
sf
lo
od
wa
1
te
rfl
oo
d
1
ga
sf
lo
od
wa
2
te
rfl
oo
d
2
ga
sf
lo
od
wa
3
te
rfl
oo
d
3
ga
sf
lo
od
wa
4
te
rfl
oo
d
4
Fraction
• Displacement chain lengths
– many multiples (few films: low phase continuity)
– multiples dying out after WAG 3
43
WAG simulation micromodel experiment
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
g->w
g->o
w->g
w->o
o->g
1
ga
sfl
oo
d
wa
2
te
rfl
oo
d
ga
2
sf
loo
d
wa
3
te
rfl
oo
d
ga
3
sf
lo
od
wa
4
te
rfl
oo
d
4
od
er
fl o
wa
t
fl o
od
ga
s
er
fl o
wa
t
1
o->w
od
Fraction
• Type of displacements
– all types of displacements occur
– many displacements involving oil movement
– after WAG 3 mainly w->g, g->w
44
WAG simulation micromodel experiment
• fluid distributions after
gas flood 1
– narrow gas finger in both
simulation and experiment
– significant amount of oil
displaced
– multiple displacements:
e.g. gas->oil->gas->water
45
WAG simulation micromodel experiment
• fluid distributions after
water flood 1
– water disperses gas
– slightly more extensive in
experiment
46
WAG simulation micromodel experiment
• fluid distributions after
gas flood 2
– different gas finger
appears
– additional oil production
47
WAG simulation micromodel experiment
• fluid distributions after
gas flood 3
– new gas finger in
simulation
– some additional oil
displaced (“jump” in
recovery)
– after this flood mainly
water displacing gas and
vice versa
48
Conclusions
• Mixed wettability leads to three types of pore occupancy and
corresponding saturation-dependencies of three-phase capillary
pressures and relative permeabilities:
– difficult to capture in empirical model
• True three-phase capillary entry pressures and layer criteria essential
for consistent and accurate modelling
• Phase continuity driver for WAG at pore-scale
– strongly affected by network connectivity and presence films and layers:
precise wettability
– multiple displacement chains
– new fluid patterns during each cycle (micromodels)
– recovery ceases after few WAG floods, oil movement may continue
49
Near-miscible WAG: micromodel
After 1 hour
After 2 hours
Continued gas injection in strongly water-wet experiment:
• Much oil displaced through film flow + mass transfer (?)
50