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Black oil vs. compositional simulation in
gas injection processes
PETROBRAS-PEMEX IOR-EOR WORKSHOP
March 8-12 th
Rio de Janeiro
• Conservation of mass in one dimension
• Generalized isothermal black-oil model
• Isothermal compositional model
• Black-oil vs. compositional
• Simulation exercise
• Final remarks and discussion
Outline
Conservation of mass in one dimension (1/3)
c = componente
i = célula
n = início do incremento de tempo ∆t
n+1 = final do incremento de tempo ∆t
M = mass
w = fonte ou sumidouro
qc = vazão mássica do componente c
A = área de fluxo
ρp = massa específica da fase p
yc,p = fração mássica do componente c na fase p
up = velocidade da fase p
Mas o componente c pode estar presente em cada fase pi:
Assumindo que o fluxo entre células é somento por convecção:
Sendo que pela lei de Darcy:k = permeabilidade absoluta do meio
kr = permeabilidade relativa à fase
µp = viscosidade da fase p
pp = pressão da fase p
D = profundidade vertical
= peso específico da fase ppγ
Conservation of mass in one dimension (2/3)
V = volume total da célula (rocha + poros)
Ф = porosidade
Sp = saturação da fase p
Massa do componente c na célula i:
Sendo que a massa do componente c em cada fase, e na célula i, é:
Substituindo tudo na equação inicial temos, para o componente c:
This is the general compositional model for one dimensional flow.
Conservation of mass in one dimension (3/3)
Onde a transmissibilidade do componente c na fase p entre as células i+1 e i:
A equação anterior está na forma contínua (diferencial), portanto pode-se neste ponto apenas a título de
ilustração continuar a derivação aproximando as derivadas na lei de Darcy utilizando:
Na forma compacta, utilizando os deltas:
Conservation of mass in one dimension (4/4)
Generalized isothermal black-oil model (1/2)
Vw
Vo
Vg
oil p
hase
Sto
ck ta
nk o
il
gas
phas
ew
ater
phas
e
P, T StandardConditions
ggV
,
ogV
,
ooV
,
wwV
,
wgV
,
• Three-component system (c = , , )
= water component
= liquid hydrocarbon component (oil)
= gaseous hydrocarbon component (gas)
• Three-phase system (p = w, o, g)
w = water phase
o = oil phase
g = gas phase
• The gas component can dissolve in the oil and water
phases
• Oil and water components are not allowed to vaporize
into the gas phase at reservoir conditions
o
w
g
w o g
• Fração mássica do componente água na fase água:
• Fração mássica do componente óleo na fase óleo:
• Frações mássicas do componente gás:
• na fase água:
• na fase óleo:
• na fase gás: Vw
Vo
Vg
oil p
hase
Sto
ck ta
nk o
il
gas
phas
ew
ater
phas
e
P, T StandardConditions
ggV
,
ogV
,
ooV
,
wwV
,
wgV
,
wwww
ww
ww BV
Vy
ww
ρ
ρ
ρρ
== ,
,
oo
o
oo
ooo
oo BV
Vy
ρρ
ρρ
== ,
,
ww
wgg
ww
wgg
wg B
R
V
Vy
ρρ
ρρ
,,
,==
oo
ogg
oo
ogg
og B
R
V
Vy
ρρ
ρρ
,,
,==
gg
g
gg
ggg
gg BV
Vy
ρρ
ρρ
== ,
,
Generalized isothermal black-oil model (2/2)
• In writing the above equation, we have assumed that mass transfer by diffusion
and dispersion can be neglected and there are no chemical reactions
• It is also assumed, in the standard black-oil simulation model, that the gas come
instantaneously into solution in the oil phase
• Besides mass conservation equation for each component there are some
constraint equation:
Final black-oil mass conservation equation
∑
∑
−
=
=
−
∂∂−
∂∂
−
∂∂−
∂∂
∆
+
−+
p
n
pp
pc
n
pp
pc
p
wpc
i
pp
p
rp
p
pc
i
pp
p
rp
p
pc
SB
RVS
B
RV
qx
D
x
pkk
B
RA
x
D
x
pkk
B
RAt
i
,
1
,
,,,
φφ
γµ
γµ
wocow ppP −= ogcog ppP −=
1=++ wgo SSS
• The black-oil model, as presented previously, is just a particular case of the
compositional simulation model
• In the compositional model we can predict phase compositions, amount in each
phase and all other thermodynamic properties. To do this we need a
thermodynamic model of the system. In the reservoir simulation such a model is
either a K-value (equilibrium ratio) correlation or table, or an equation of state
(EoS)
• So the differences between black-oil and compositional model are basically the
number of components and how the thermodynamic equilibrium of reservoir
fluids is characterized
Isothermal compositional model (1/4)
• In the standard compositional model, besides the mass conservation equation for each
component (hydrocarbons + water) there are nc+3 system constraints:
• Once the solution of the phase equilibrium equations is itself a complex duty (flash
calculation) the simulator (GEM) solves separately the mass conservation and the phase
equilibrium equations.
• At each iteration of each time-step in each block the simulator calculate the overall mass
fractions and the pressure, then a flash calculation is performed to obtain the phases
compositions and the amount in each phase
gcoc ff ,, =
Isothermal compositional model (2/4)
∑ =c
ocy 1,
∑ =c
gcy 1,
1=++ wgo SSS
phase equilibrium equations
Isothermal compositional model (3/4)• One possible flowchart for flash calculation is:
6. Have L/F , xc and yc
changed since the last interaction?
7. L/F , xc and yc
1. Given zc (overal mass fractions), P and
T, calculate Pb and Pd to ensure the
system has two phases in equilibrium. As
a first guess compute the equilibrium
ratios (Kc) or xc=yc=zc
2. Compute , and
L
c
^
φV
c
^
φV
c
L
ccK
^
^
φ
φ=
3. Compute
)1( cc
cc
KF
LK
zx
++=
∑ ∑−= cc yxD
ccc xzy =
4. D = 0 ?
5. New estimate for L/F
If D > 0, increase L/F
If D < 0, decrease L/F
no
yes
no
yes
c = component
zc = overall mole fraction of component c
xc = mole fraction of component c into de liquid phase
yc = mole fraction of component c into de gaseous phase
P = pressure
T = temperature
Pb = bubble point pressure
Pd = dew point pressure
Kc = equilibrium ratio of component c
= fugacity coefficient of component c in the liquid phase
= fugacity coefficient of component c in the gaseous phase
L = moles of liquid phase
F = total moles
L
c
^
φ
V
c
^
φ
Isothermal compositional model (4/4)• From flash calculation it is obtained (L/F). Then it is possible to
compute the number of mols in the liquid and in the gas, and the
saturations:
• The number of mols of the component are then calculated from
the saturations
No = numbers of mols in the liquid phase
Ng = numbers of mols in the gas phase
Nc,o = numbers of mols of the component c in the liquid phase
Nc,g = numbers of mols of the component c in the liquid phase
∑= io NF
LN
∑
−= ig NF
LN 1
( )F
LSS wo −= 1
( )
−−=F
LSS wg 11
coooc xSN φρ=,
cgggc ySN φρ=,
Black-oil vs. compositional
63n°of primary variables
166Total
166Total
5 x 20Compositions (Compositions (xxcc, , yycc))
33Saturations (So, Sw, Sg)
33Pressures (po, pw, pg)
n°of unkowns
20Phase constraintsPhase constraints
11Volume (saturation) constraint
22Capillary pressure relations
50Phase equilibriumPhase equilibrium
11Mass balance of water component
52Mass balance of each “hydrocarbon” component
n°of equations(per block per iteration)
gcoc ff ,, =
wocow ppP −= ogcog ppP −=
1=++ wgo SSS
∑ =c
ocy 1, ∑ =c
gcy 1,
B.O. Comp
• Black-oil– nº of components
– nº of equations
– choice of primary variables and equations
Black-oil
– Basic assumption: The composition (and hence the properties) of the two pseudo-components (stock tank oil and surface gas) remain constant during reservoir production - this is rarely true!
– It is usually selected, instead of compositional, based on the volality of the oil:
Rs < 750 scf/stb
Bo < 1.4 bbl/stb
API gravity < 30
– But it is also selected based on the drainage strategy for the reservoir:
DepletionWater injection Aquifer water influx
Compositional
– Basic assumption: The composition (and hence the properties) of the n pseudo-components (ex.: CO2; C1; C2-C6; C7
+) do not remain constant during reservoir production – more realistic!
– Drainage strategy for the reservoir:
EOR processes:Hydrocarbon Gas injection
CO2 injection
WAG-HC
WAG-CO2
Integration with down stream facilities
Black-oil vs. compositional
Simulation exercise
Simulation exercise - depletion
Producer constraints:
• BHPMIN = 450 kgf/cm2
• Maximum liquid rate = 8000 m3 std /d
Injector constraint:
• gas injection rate = 5500 m3 res/d
Simulation exercise – gas injection
Simulation exercise – gas injection
Sg
Composicional gás injetado
17%CO2 83%C1
So
Composicional gás injetado
17%CO2 63%C1 20% C2-C6Black-oil
Sor = 34% Swr = 16%Miscibilidade: Sg > 50% So < 50%
P
Simulation exercise – gas injection (2 years)
• Petrobras has been needed to run EOR predictions on million cells models (mainly
carbonate fields with a huge area and with a high rock heterogeneity)
• It has been assumed that the compositional simulation is the best model to run EOR
predictions: Gas injection, CO2 injection, WAG-HC and WAG-CO2
• However these full field simulations are very slow in the compositional approach, so
Petrobras has considered the following options:
• Sub-model compositional simulations
• Simulate SWAG processes in a full field compositional model to represent WAG
• Pseudo compositional models (Todd-Longstaff empirical model) may also be an option,
but it hasn’t been teste yet
Final remarks and discussion
Gracias!