compositional x black oil simulation

22
Black oil vs. compositional simulation in gas injection processes PETROBRAS-PEMEX IOR-EOR WORKSHOP March 8-12 th Rio de Janeiro

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Page 1: Compositional x Black Oil Simulation

Black oil vs. compositional simulation in

gas injection processes

PETROBRAS-PEMEX IOR-EOR WORKSHOP

March 8-12 th

Rio de Janeiro

Page 2: Compositional x Black Oil Simulation

• 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

Page 3: Compositional x Black Oil Simulation

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

Page 4: Compositional x Black Oil Simulation

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)

Page 5: Compositional x Black Oil Simulation

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)

Page 6: Compositional x Black Oil Simulation

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)

Page 7: Compositional x Black Oil Simulation

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

Page 8: Compositional x Black Oil Simulation

• 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)

Page 9: Compositional x Black Oil Simulation

• 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

Page 10: Compositional x Black Oil Simulation

• 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)

Page 11: Compositional x Black Oil Simulation

• 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

Page 12: Compositional x Black Oil Simulation

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

Page 13: Compositional x Black Oil Simulation

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=,

Page 14: Compositional x Black Oil Simulation

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

Page 15: Compositional x Black Oil Simulation

• 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

Page 16: Compositional x Black Oil Simulation

Simulation exercise

Page 17: Compositional x Black Oil Simulation

Simulation exercise - depletion

Page 18: Compositional x Black Oil Simulation

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

Page 19: Compositional x Black Oil Simulation

Simulation exercise – gas injection

Page 20: Compositional x Black Oil Simulation

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)

Page 21: Compositional x Black Oil Simulation

• 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

Page 22: Compositional x Black Oil Simulation

Gracias!