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Simulation of oil recovery with miscible gas injection

using a black oil model

R. Masson

04/06/2014

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1 Black oil model

2 Discretization of Black Oil models

3 Solution of the discrete system of equations

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Motivations of compositional two phase oil gas models

Saturated or under saturated oil reservoirs: volatile HydrocarbonComponents (HC) evaporate with pressure drop

Oil recovery by miscible gas injection

Gas condensate reservoirs: liquid phase appears with pressure drop

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Black Oil Model: definition

Two phases: oil and gasTwo components: heavy HC (h), and light HC: (l)

The light component can dissolve into the oil phase

The heavy component cannot evaporate into the gas phase

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Black Oil Model: conservation equations

t

o (1 c) So

+div

(1 c) o

kr,o(So)

o Vo

= 0,

to c So +g Sg+ divc o kr,o(So)

o Vo

+ div

g kr,g(Sg)

g Vg

= 0,

Vo = KP og

, Vg = K

P gg

,

So + Sg = 1.

Oil phase properties: o(P, c), o(P, c)

Gas phase properties: g(P), g(P)

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Black Oil Model: thermodynamical equilibrium

If the oil and gas phases are both present, the thermodynamical equilibriumimposes that c= c(P).

Sg (c(P) c) = 0,Sg 0,

(c(P) c) 0.

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Black Oil Model

t

o(P, c) (1 c) So

+div

(1 c) o(P, c)

kr,o(So)

o(P,c)Vo

= 0,

t

o(P, c) c So +g(P) Sg

+div

c o(P, c)

kr,o(So)

o(P,c) Vo

+ div

g(P)

kr,g(Sg)

g(P) Vg

= 0,

Vo = KP o(P, c)g

, Vg = K

P g(P)g

,

So + Sg = 1,Sg (c(P) c) = 0,Sg 0,(c(P) c) 0.

Remarks:

Only one set of unknowns P, So, Sg, cand equations because the oil phaseis always present.

No flash needed.

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Black Oil Model: Bo, Bg and Rs

Bo

= Vo

Vo,h,s=

h,s

o(P, c) (1 c), B

g=

Vg

Vg,l,s=

l,s

g(P)

Rs= Vo,l,s

Vo,h,s=

1 c

c

l,s

h,s

h,s and l,sare the oil and gas densities at surface conditions. T is fixed.

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Black Oil Model: phase properties in terms ofBo, Bg andRs

Bo(P), Rs(P), o

(P) are given at equilibrium (saturated oil).Bg(P) and

g(P) are given.

For saturated oil (c= c(P) orP=Pb(c))and undersaturated oil (c< c(P) orP>Pb(c))

Bo(P, Pb) = Bo(Pb) + cBo(P Pb), o(P, Pb) = o(Pb) + co(P Pb)

o(P, c) = h,s

(1 c) Bo(Pb(c)) + cBo(P Pb(c)),

o

(P, c) = o

(Pb(c)) + co(P Pb(c)),c(P) =

l,s

h,s Rs(P) +l,s,

Pb(c) = c1(c),

g(P) = l,s

Bg(P).

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Discretization of Black Oil model: 2D fivespots test case

2D horizontal reservoir with 4 gas injectors at the corners and 1 producerat the center.Prescribed pressures Pinj for the injection wells and Pprod for theproduction wellInitial state: under saturated oil at Pprod

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Discretization of Black Oil model: diffusion fluxes andupwinding (without gravity)

Diffusion fluxes (without gravity for simplicity)

Fint1(), = F

int2(),

=Tint p1() p2(), int,Fprodw = WIprodw

pprod(w) pprodw

, w prod,

Finjw = WIinjw

pinj(w) p

injw

, w

inj,

Upwinding (without gravity): for all int

up() =

1() = si Fint1(), >0,

2() = si Fint1(),

0.

Five Spots example: prod = one producer, inj = the four injectors.

Di i i f Bl k Oil M d l di

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Discretization of Black Oil Model: discrete system(without gravity)

Heavy HC residual

Rh,

= o(P

, c

) So

(1 c

) mn1h, ||

t

+

int

(1 cup()) o(Pup(), cup())

kro(Soup()

)

o(Pup(), cup()) Fint,

+

wprod|prod(w)=

o(P, c) (1 c) kro(So)

o(P, c)

Fprodw

+ = 0,

Di i i f Bl k Oil M d l di

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Discretization of Black Oil Model: discrete system(without gravity)

Light HC residual

Rl, =

o(P, c) So c+

g(P) Sg m

n1l,

||t

+ int

cup() o(Pup(), cup())

kro(Soup()

)

o

(Pup(), cup())

Fint,

+

int

g(Pup())krg(S

g

up())

g(Pup()) Fint,

+

wprod|prod(w)=

c

o(P, c) kro(So)

o(P, c)+g(P)

krg(Sg )

g(P)

Fprodw

+= 0,

+ winj|inj(w)=

g(Pinjw

) krg(1)

g(Pinj

w)Finjw

= 0,

Di i i f Bl k Oil M d l di

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Discretization of Black Oil Model: discrete system(without gravity)

Local closure laws (volume conservation an thermodynamical equilibrium)

So

+ Sg

= 1,

Sg (c(P) c) = 0,

Sg 0,

(c(P) c) 0.

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Discretization of Black Oil Model: Newton type algorithm

The oil phase is always present in all cells (Sor >0)

Gas phase indicator for all cell K

I = 1 if gas phase is present0 if gas phase is absent

Given I for all cells, the closure laws are eliminated at the non linear level: ifI= 1 : S

o = 1 S

g , c = c(P),

ifI= 0 : Sg = 0, S

o = 1,

the unknowns X for the Newton linearization depend on I for all cells:

ifI= 1 : X =

P, S

g

,

ifI= 0 : X =

P, c

.

S l ti f th di t t f ti N t t

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Solution of the discrete system of equations: Newton typealgorithm

The conservation equations are linearized with respect to the set of

unknownsX =

X

K

taking into account the previous elimination of

the closure laws for a given I:

R(X) =

Rh,(X)

Rl,(X)

R= RK, dX = RX

1

R

On each cell update of the unknowns P, S, c and of the Gas phaseindicatorI to satisfy the inequality constraints S

g 0 and c c(P),

K.

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Solution of the discrete system of equations: algorithm

P= (P)K

, I= (I)K

, S= (S =Sg

)K

,c= (c)K

.

R=

Rh,, Rl,

K

.

Initialization ofP=P0,I= I0, S=S0, c=c0 and t at t= 0.

Time loop: while t then Restart the time step: t= tt,

t= t/2, P= P0,I=I0,S= S0, c= c0

Else new time step: update t

and setP

0

=P

,I

0

=I,S

0

=S

,c

0

=c

Solution of the discrete system of equations: update of

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Solution of the discrete system of equations: update ofP, S, c andI

dX = J1 R

Newton update: for = 1, , N:

P P+ dX(21),ifI = 1 then S S+ dX(2), c = c(P)else c c+ dX(2), S = 0.

Gas phase indicator update: for = 1, , N:IfI = 1 then

ifS < 0 thenI 0, S 0, c c(P)

else ifS > 1Sor thenS 1 Sor,

Else ifI = 0 then

ifc > c(P) then

I 1, S 0, c c(P)

Solution of the discrete system of equations: computation

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Solution of the discrete system of equations: computationof the residual R(P, S, c,P0, S0, c0, t)

numbering of the cells: = 1, , N

numbering of the equations:

Rh R(2k 1), = 1, , N

Rl

R(2

k)

, = 1

,

, N

Loop on cells = 1, , N

o = o(P, c),

g = g(P),

R(2 1) R(2 1) +

o (1S) (1c)o(P0,c0) (1S0) (1c0)

||

t

R(2) R(2) +

o

(1S) c+g S

o(P0,c0

) (1S0

) c0g(P0

) S0

||

t

Solution of the discrete system of equations: computation

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Solution of the discrete system of equations: computationof the residual R(P, S, c,P0, S0, c0, t)

Loop on interior faces:

Fh = (1 cup()) oup()

kro(1Sup())

o Fint1(),

Fl =

cup() oup()

kro(1Sup())

o +g

up()

krg(Sup())

g

Fint1(),

R(21() 1) R(21() 1) + Fh

R(22() 1) R(22() 1) Fh

R(21()) R(21()) + FlR(22()) R(22()) F

l

Solution of the discrete system of equations: computation

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Solution of the discrete system of equations: computationof the residual R(P, S, c,P0, S0, c0, t)

Loop on injector wells w:

R(2inj(w)) R(2inj(w)) +g(Pinj

w)

krg(1)

gFinj

w

Producer well w

R(2prod(w) 1) R(2prod(w) 1) +

o (1 c)

kro(1S)o F

prodw

+

R(2prod(w)) R(2prod(w)) +

c o kro(1S)

o +g krg(S)g

Fprodw

+

Solution of the discrete system of equations: Newton

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Solution of the discrete system of equations: Newton

linearization R(X) +

RX

dX = 0

Unknowns numbering for the Newton linearization: = 1, , N

dX(2 1) = dP,

dX(2) =

dS if I = 1,

dc if I = 0.

Notation X1, X2

=

P, S

if I= 1 c= c(P),

P, c

if I= 0 S= 0.

Derivatives: example ofo

(P, c):

o

X1 =o

P(P, c) +

o

c (P, c)

c

P(P)I +

oP

(P, c)

(1 I)

o

X2 = o

c (P, c)(1 I)

Solution of the discrete system of equations: Newton

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Solution of the discrete system of equations: Newton

linearization R(X) +

RX

dX = 0

Jacobian: example of the accumulation term for Rh:o (1 S) (1 c)

o(P0, c0) (1 S

0) (1 c

0)||

t

Loop on cells

J(2 1, 2 1) J(2 1, 2 1) +I

o (1 S)

cP

(P)

+(1 S) (1 c)

o

P(P, c) +

o

c(P, c)

cP

(P)

||t

+(1 I)oP(P, c) (1 S) (1 c)

||t

J(2 1, 2) J(2 1, 2) +I

o (1 c)

||t

+(1 I)

o

c(P, c) (1 S) (1 c)

o (1 S)

||t

Solution of the discrete system of equations: Newton

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Solution of the discrete system of equations: Newton

linearization R(X) +

RX

dX = 0

Jacobian: example of the interior face flux term:

Fh = (1 cup()) oup()

kro(1Sup())

o Fint1(),

Loop on interior faces : 1 =1(), 2 =2(), up=up()

Gh = (1 cup) oup

kro(1Sup )

o

J(21 1, 21 1) J(21 1, 21 1) + Gh T

int

J(22 1, 22 1) J(22 1, 22 1) + Gh T

int

J(21 1, 22 1) J(21 1, 22 1) Gh T

int

J(22 1, 21 1) J(22 1, 21 1) Gh Tint

to be continued ...

Solution of the discrete system of equations: Newton

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y q

linearization R(X) +

RX

dX = 0

suite of the interior flux term Fh

=I

(1 cup)

o

P(Pup , cup) + (1 cup)

o

c(Pup , cup)

cP

(Pup)

cP

(Pup)oup

+ (1 I) (1 cup)

o

P(Pup , cup)

kro(1Sup )

o Fint1,

= I

(1cup )oup

okroS

((1 Sup)

Fint1,

+(1 I)oup + (1 cup)

o

c(Pup , cup)

kro(1Sup )

o Fint1,

J(21 1, 2up 1) J(21 1, 2up 1) +J(22 1, 2up 1) J(22 1, 2up 1) J(21 1, 2up) J(21 1, 2up) +J(22 1, 2up) J(22 1, 2up)

to be continued for all remaining terms of the Jacobian ...

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