Download - Compositional Simulation
L o g o
Compositional
Simulation
Applications
• Black oil models
Immiscible flow
Hydrocarbon phase has fixed composition of two component oil and gas
Fluid properties are a function of pressure
Fluid properties are a function of pressure and solution gas
Applications
• Compositional gas models
Significant mass transfer between oil and gas phases
Fluid properties are dependent on pressure and composition
Examples processes • Volatile oil reservoir depletion
• Gas condensate reservoir depletion
• Gas cycling
• Miscible flooding by CO2 or enriched gas
injection
Compositional Models
• Fussell and Fussell (1978)
• Coats (1980) : Fully implicit
• Nghiem et al. (1981) : IMPES-type
• Young and Stephens (1983)
• Acs et al. (1985) : IMPES–type
• Watts (1986) : Sequential implicit
• Collins et al. (1986) : Adaptive implicit
Formulations
• IMPES Type
Solve pressure and saturation separately
• Adaptive implicit
In each time step solve few gridblocks implicitly and the rest explicitly
• Fully implicit
Solve for pressure, saturation, and composition simultaneously
Applications
• Compositional chemical models
Injection of of liquid chemicals to displace oil
Phase behavior
Reduction in interfacial tension between oil
and liquid
Example processes are
• Surfactant/polymer
• Surfactant/alkaline/polymer
• Surfactant/foam
Displacements
• Immiscible flow under reservoir conditions
• fluids with different properties mix
Displacement of oil by miscible or partially
miscible fluids
Displacement using chemicals that can
affect fluid properties
Non isothermal flow or combustion reaction
Displacements
• Miscible
Two or more components flow within a
single phase
• Flow of oil by solvent
• Transport of salts
• Polymer
• Tracers
• Contaminant in water
Commercial Compositional Gas Reservoir Simulator
• VIP (Landmark Graphics)
• Eclipse 300 (Geoquest)
• CMG-GEM (Computer Modeling Group)
• Simbest (Formerly SSI and now Bakers)
Oil Reservoir Simulator
• Black oil model for immiscible
processes
• Miscible black oil model for first
contact miscible process where
injected fluid is directly miscible with
oil at reservoir pressure and
temperature
Reservoir Simulators
• Compositional gas models for multiple contact miscible displacements
Inject CO2 to mobilize and displace residual oil saturation through multiple contacts
between oil and CO2 phases
Intermediate and higher molecular weight hydrocarbons are contacted into the CO2
rich phase
Under proper conditions CO2 becomes miscible with the original reservoir oil
Reservoir Simulators
• Compositional chemical models for
chemical flooding processes
• Thermal models for steam flooding and
in situ combustion
Mathematical Models• Isothermal conditions
• No flow boundary
• Permeability tensor is orthogonal and aligned with the coordinate system
• No precipitation or chemical reaction
• Negligible adsorption
• Physical dispersion follows Fick’s law• Water is treated as a separate component present
only is water
• No mass transfer between water and oil or gas phases
• Hydrocarbon phase compose of nc hydrocarbon components which may include nonhydrocarbons such as CO2, N2, or H2S
• Assume instantaneous thermodynamic equilibrium between hydrocarbon phases
Mass Conservation Equation
termSourceR
FluxF
onAccumulatiW
RFt
W
i
i
i
iii
0
Mass Conservation Equation
jphaseinicomponentoffractionmolex
jphaseforsaturationS
jphasefordensitymolar
porosity
xSW
ij
j
j
ijj
n
jji
p
1
• Accumulation term
Mass Conservation Equation
111
2
,........1
SW
and
nhydrocarbonixSW
c
p
n
c
n
jijjji
Since there is no hydrocarbon component in the water phase
• Accumulation term
Mass Conservation Equation
• Phase Index
J = 1 water
J = 2 Oil
J = 3 Gas
Mass Conservation Equation
• Flux Term
ijijjjjij
n
jji xKSuxF
p
1
Convective Flux Dispersive Flux
Mass Conservation Equation
DepthD
jphaseofweightspecific
jphaseofitycosvis
typermeabilirelativek
mobilityrelative
tyPermeabiliK
DPku
j
j
rj
k
rj
jjrjj
j
rj
• Darcy's law
Mass Conservation Equation
1ccb
ii n,n,.....1i
V
qR
Source term:
Vb= Bulk volume
qi = Molar flow rate of component i
producer : negative value
Injector : Positive value
Physical Dispersion
• Diagonal terms
zzij
yyij
j
2zj
j
Tj
j
2yj
j
Tj
j
2xj
j
Ljijxxij
KandKfortermsSimilar
u
u
Su
u
Su
u
S
DK
zzzyzx
yzyyyx
xzxyxx
ij
KKK
KKK
KKK
K
tydispersiviTransverse
tydispersiviallongitudin
tortuosity
dispersionmolecularD
T
L
ij
Physical Dispersion
• Off Diagonal terms
2zj
2yj
2xjj
zyij
yzij
zxij
xzij
yxij
xyij
j
yjxj
j
TjLjxyij
uuuu
where
KK
KK
KK
u
uu
SK
L o g o
Compositional
Simulation
Mass Conservation Equation
Set of coupled, nonlinear, partial differential
equations with ncnp+6np+2 variables
1,,.........1
011
cc
b
in
jijijjjjijj
n
jijjj
nni
V
qxKSuxxS
t
pp
Pressure Equation
• Acs et al. formulation where total fluid volume is
equal to the pore volume
)(),( PVNPV pt
t
P
P
V
t
N
N
V
t
P
P
V pn
i
i
ikNPi
t
N
tc
ki
1
1)(,
reffref PPc 1 f
ref
p
pcV
P
V
Pressure EquationFrom mass conservation equation, we have
ij
n
1jjjbi
cc
i
n
1jijijjjjijjb
i
xSVNwhere
1n,n,.........1i
0qxKSuxVt
N
p
p
Define partial molar volume as
)(, ikNPi
tti
k
N
VV
Pressure Equation
Neglect physical dispersion and cjj PPP
1
1
1
1 1
c
c p
n
iiti
n
i
n
jjcjijjrjtib
tf
ref
p
qV
DPxkVVt
P
P
VcV
Initial conditions: known pressure and known no. of moles
for each component
Boundary conditions : No flow boundary
Variables No. of variables
1
Sj np
ij np
xij (np-1)nc
Krj np
j np
Pj np
j np
qi nc+1
Total nc np+6 np+2
List of Variables
Equations Name No. Of Eq.
Mass conservation nc+1
fij = fil Phase equilibrium nc(np-2)
Saturation constraint 1
Phase composition
constraint
np-1
Formation porosity 1
Equation of State np
Equation of State np
Viscosity np
11
pn
jjS
11
cn
iijx
P
),(
)(11
jjj xP
P
),(
)(11
jjj xP
P
),(
)(11
jjj xP
P
Number of Equations
Equations Name No. Of Eq.
Phase
Relative permeability np
Phase pressure np-1
Well model nc+1
Totalnc np+6 np+2
),,( xSPkk rjrj
),( xSPP jj
),,( xSPqq ii
Number of Equations
IMPES Computational Procedure
• Establish initial conditions
Phase saturation
Component moles
Viscosity
Density
Relative permeability
Capillary pressure
Original in place
Fluid volume
IMPES Computational Procedure
1. Solve pressure equation implicitly using
explicit saturation and phase composition
dependent properties
2. Compute flow rate and bottomhole pressure
for wells
3. Solve mass balance equation explicitly for
component mole number and overall
hydrocarbon composition
4. Solve for phase equilibrium and hydrocarbon
phase saturation
IMPES Computational Procedure
5. Compute phase densities
6. Calculate phase viscosities
7. Calculate phase relative
permeabilities
8. Compute capillary pressures
Special Case
• 2 hydrocarbon oil and gas phases
• No physical dispersion
• Use Darcy’s law
jrjj ku
Special Case
• Water equation is the same as
black oil model
w
www
w
w
B
S
tq
B
Mass balance for component i
ggiooiigggioooi SySxt
qyx
cowcoggw
wcoggwocow
coggoogcog
g
jjj
PPPP
PPPPPP
PPPPPP
PP
DPSubstitute
Reference phase
Special Case
Interfacial Tension
• Water and hydrocarbon phases = constant
• Hydrocarbon phases (Macleod –Sugden)
tensionerfacialintgas/oil
icomponentofParachor
yx016018.0
og
i
n
1i
igioiog
c
Physical Properties
• Relative permeability
Function of phase saturation
and interfacial tension
• Capillary pressure
Function of phase saturation
and interfacial tension
jljcjl SfP ,
jljrjl Sfk ,
Physical Properties
• Molar density
RTZ
P
v
1
jjj
• Mass density
gasoroiljWxcn
1itiijjj
Where Wti is the molecular weight of component i
waterref
PPc
111
111
Physical Properties
• Phase viscosity using linear mixing rule
gasoroiljx i
n
iijj
c
~1
Where i~ is the pure component viscosity
Phase Behavior
• Peng-Robinson Equation of State
bvbbvv
)T(a
bv
RTP
a and b are constants computed as a function of
critical properties.
Compressibility factor: RT
PVz
RT
bPB
RT
aPA
0BBABZB2B3AZB1Z
2
32223
Critical Properties
Comp. Composition P
(psi)
T
(R)
V
ft3/lbmole
MW Accentric
factor
Parachor
CO20.05 1073 547 1.5 44 0.23 49
C3-617.8 476 848 5.0 73 0.25 60
C715.5 453 985 6.0 96 0.28 100
C832.35 430 1040 8.8 102 0.30 300
Binary Interaction Coefficients
CO2 C3-6 C7 C8
CO2 0.0
C3-6 0.12 0.0
C7 0.12 0.0 0.0
C8 0.12 0.0 0.0 0.0
Phase Behavior
100% bubble
point curve
Bubble
point
Dew point
Liquid + vapor
Vapor
T
P
PC
TC
Phase Behavior
2 phase
1 phase
Methane
C7+ C2-C6
Tie line
Plait point
Binodal curve
T, P = constant
Ternary Diagram
Plait point
Binodal curve
Tie line
CO2
C13+ C5-12
L o g o
Three and Four Phase Flow
Compositional Gas Simulations
Phase Diagram
Liquid/
Liquid/
Vapor0
500
1000
1500
2000
2500
3000
0.0 0.2 0.4 0.6 0.8 1.0
Mole Fraction of CO2/NGL
Pre
ssu
re (
psi) Liquid
Liquid/Liquid
Liquid/Vapor
Permeability Field
100 200 300 400 500 600 700 800 900 1000
Length (ft)
-55
-50
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
Ele
va
tio
n (
ft)
0
50
100
150
200
250
300
350
400
450
500
550
600
650
700
Permeability(md)
Injector Producer
Phase Viscosities
0.01
0.1
1
10
100
1000
0 200 400 600 800 1000Length, ft
Vis
co
sit
y,
cp
Oil Viscosity (4-Phase Flow)Gas Viscosity (4-Phase Flow)Second HC Liquid Viscosity (4-Phase Flow)Oil Viscosity (3-Phase Flow)Gas Viscosity (3-Phase Flow)
Phase Densities
0
10
20
30
40
50
60
0 200 400 600 800 1000Length, ft
De
ns
ity
, lb
m/c
uft
Oil Density (4-Phase Flow)
Gas Density (4-Phase Flow)
Second HC Liquid Density (4-Phase Flow)
Oil Density (3-Phase Flow)
Gas Density (3-Phase Flow)
Number of Phases
0 100 200 300 400 500 600 700 800 900 1000
Length,ft
-55
-50
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
Ele
va
tio
n,f
t
2
3
4
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0 182.5 365 547.5 730
Total Injection Time (Days)
Cu
mu
lati
ve O
il R
eco
very
(F
racti
on
of
OO
IP)
4-phase flow (Run 4p2dh3b4)
3-phase flow (Run 3p2dh3b4)
Comparison of three and four phase flow results
SPE 89343
Reservoir Simulation of CO2 Storage in
Deep Saline Aquifers
A. Kumar, M. Noh, G.A. Pope, K. Sepehrnoori,
S.L. Bryant and L.W. Lake.
Center for Petroleum & Geosystems Engineering,
The University of Texas at Austin.
Carbon Sequestration
CO2
Deep Saline Aquifer
Aquifer/Depleted
Oil or Gas Reservoir
Oil/Gas Producing Reservoir
Unmineable Coal
CO2
Deep Saline Aquifer
Aquifer/Depleted
Oil or Gas Reservoir
Oil/Gas Producing Reservoir
Unmineable Coal
Evaluate Storage Potential of Aquifers
CO2
Brine Dissolution Traps Significant CO2 Volume
1000 Years50 Years
CO2 mole fraction in
aqueous phase
0
20
40
60
80
100
0 200 400 600 800 1000
Time, years
CO
2 S
tore
d i
n V
ari
ou
s P
ha
se
s,
%CO2 Capture
Percentage CO2
as Residual Gas
Percentage CO2 in
Aqueous Phase
Percentage CO2
as Free Gas
Injection
ends
L o g o
SPE 80253
Effect of Dispersion on Transport and
Precipitation of Barium and Sulfate In Oil
Reservoirs
2003 SPE International
Symposium on Oilfield
Chemistry
February 5-7, 2003
Houston, Texas
Objectives
• Determine sensitivity to sulfate concentration
in injected seawater for 150-300 ppm barium
concentration in formation brine
• Estimate how much sulfate will need to be
removed from injected water
• Determine sensitivity to dispersion and mixing
in reservoir
• Determine how much solid precipitates in
reservoir and in wellbore
Reservoir Model Description
• Three - dimensional
22x40x22 (19,360 gridblocks)
• Pore volume
123 million m3 (774 million bbls)
• OOIP
6 million m3 (38 million bbls)
• 4 vertical injectors, 3 slanted producers
• 10-year simulation
Water Analysis
Ion Formation Water, Mg/L Injected Seawater, Mg/L
Na+ 37,400 11,400
K+ 329 400
Ca++ 3,067 435
Mg++ 1,114 1,370
HCO3- 1,029 0
Cl- 62,380 20,500
Li+ 3.4 0
Sr++ 153 0
Ba++ 151 0
Fe 1.7 0
B 41 0
Br- 325 0
SO4-- 0 2,800
Simulation Variables
• Initial Barium ion concentration (ppm)
151 and 300
• Physical longitudinal dispersivity (m)
100, 12, 0 (only numerical dispersion)
• Injected sulfate ion in water (ppm)
2800 ( seawater), 200, 120, 80, 40, 20
Simulation Grid
Vertical exaggeration = 3
Permeability Distribution150 m
Permeability, md
1 50 100+25 75
Permeabilities from 100 to 3720: red
Permeabilities below 20: transparent
Barium, mole/L
0.0000 0.0010 0.00200.0005 0.0015
Barium Ion Concentration
Barium ion concentrations less than 0.0007 are transparent
After One Year of
Seawater Flooding
After Ten Years of
Seawater Flooding
Initial Ba++ = 300 ppm
Dispersivity = 12 m
2800200
120
80
40
20100
12
00
50
100
150
200
250
300
350
400
Initial Barium
Concentration = 300 ppm
Total Barium Sulfate Precipitated in Wells
Mass of Barium Sulfate Precipitated in Formation
0
500
1000
1500
2000
2500
3000
3500
4000
0 1000 2000 3000 4000
Time, days
Ma
ss
of
Pre
cip
ita
te, to
ns
0 m
12 m
100 m
Dispersivity
120 ppm SO4
--and 151 ppm Ba
++