Multicomponent two-phase flow in porous media:

Macro-kinetics of oscillatory regims

Laboratoire d'Énergétique et de Mécanique Théorique et Appliquée (LEMTA – CNRS UMR 7563)

International Conference: Scaling Up and Modeling for Transport and Flow in Porous Media

Dubrovnik, Croatia, 13-16 October 2008

Mojdeh Rassoulzadeh – LEMTAIrina Panfilova – LEMTA/SchlumbergerMichel Panfilov – LEMTA

Gaz

Water

2

2

2

light

H

N

O

I :

2: heavyH OII

FLUIDE

Subsurface waste storage

Components :

Gaz

Components :

FLUIDE

Oil and natural gas

2

4

2 6

light

N

CH

C H

I :

3 8

4 10

2

neutral

C H

C H

CO

III :

5 12

20 42

: heavy

C H

C H

II

Oil

Gaz

Components :

FLUIDE

Oil and CO2

4

2 6

lightCH

C H

I :

2 neutralCOIII :

5 12

20 42

: heavy

C H

C H

II

Oil

Liquid

Gas + Liquid

Gas

PT-Diagram

Liquid

Gas + Liquid

Gas

Initial state

Classic systems

Liquid

Gas + Liquid

Gas

Initial state

Retrograde systems

MAIN PROBLEM OF MULTICOMPONENT FLOW

Non-equilibrium behaviour

1. Oscillatory regimes

Non-equilibrium behaviour

MAIN PROBLEM OF MULTICOMPONENT FLOW

1. Oscillatory regimes

Non-equilibrium behaviour

2. Over-saturated zones

MAIN PROBLEM OF MULTICOMPONENT FLOW

PROBLEM 1 :

Oscillatory regimes

RETROGRADE GAS-OIL RESERVOIRS

Liquid

Gas + Liquid

Gas

RETROGRADE GAS-OIL RESERVOIRS

Theory

composition

flow rate

RETROGRADE GAS-OIL RESERVOIRS

Field data

composition

flow rate

TWO TIME SCALES IN OSCILLATIONS

Ganglion character of flow (V. E. Gorbunov, 1990) Each fluid becomes mobile only when it reachesits representative elementary volume (REV)

Thermodynamic instability (V. Mitlin, 1990)Stability analysis of the compositional flow modelshows that the system becomes instable when

is the total mixture density,

P is the pressure

HYPOTHESES ON THE MECHANISMOF OSCILLATIONS

0dP

d

double phase transition:

condensation

coagulation of liquid

internal evaporation

internal gas evacuation

OUR THEORY

condensation coagulation of liquidP leads to evaporation

OUR THEORY

P condensation liquid coagulation

internal evaporation

Phase diagram for the initial fluid

Phase diagram for the secondary liquid aggregates

OUR THEORY

Double phase transition

Liquid

Gas + Liquid

Gas

Initial state

Liquid

Gas + Liquid

Gas

Initial state

Double phase transition

Liquid

Gas + Liquid

Gas

Initial state

Double phase transition

Initial stateCapillary condensation

Double phase transition

Initial stateCapillary condensation

Double phase transition

Liquid coagulation

Double phase transition

Liquid coagulation

Liquid aggregate

Double phase transition

Double phase transition

Transition to the second phase diagram

Double phase transition

Internal evaporation (boiling)

Transition to the second phase diagram

Double phase transition

Gas Evacuation

TOTAL COMPOSITION OF THE SYSTEM: 4 PHASES

2

3

4

1

Classic phases

MODEL of DOUBLE PHASE TRANSITION

Capillary condensation Minimisation of free Gibbs energy

Coagulation Smoluchowski + effective media

Evaporation Kinetics of Frenkel-Zeldovich

Evacuation Gravity segregation + volume exceed mechanism

CAPILLARY CONDENSAIONPore-scale modeling

Correlated capillary network Liquid aggregates 1 anddispersed condensate 2, 3

Results of modeling the liquid COAGULATION

Dynamics of the averaged size of liquid aggregates

COAGULATON: Effective medium approach

Mean vale of particle for power law probability of coagulation

Comparison of the effective medium theory and the network simulations

2 1dnn

dt

2 1 2,d

dt

kinetic of coagulation

SECONDARY EVAPORATION (BOILING)

Evaporation has 2 stages:

A : formation and growth of germs of bubbles (Frenkel, Zeldovich)

B : coagulation of bubbles

, ,boil boil A boil Bd d d

, * 2 1 * 2, ,boil B

boil boil B boil boil

d

dt

*,

, , ,

( ),

2boil A

boil A boil A boil Aaggr

D g

t M N kT

is the mass concentration of the aggregate Is the mass concentration of the boiling gas

EVACUATION: gravity segregation + volume exceed mechanism

Internal exchange:

formation of gas bubbles leads to the reduction of the liquid mass

External exchange:

geometrical “volume exceed”

gravity-induced uplift of bubbles

1/3 1/3 2/3

0

/ 316 4

8g l g l gext boil l boil l boil

aggrg g g aggr

gKd dV

dt N N dt M

General kinetic for the external exchange

Volterra generalized model

2 11

*1 3

da

dt

da a

dt

= mass of liquid aggregates

= mass of interior gas

aggregation evaporation

evaporation

evacuation by volume exceed

evacuation by segregation

Rapid gas evacuation:( 1) Phase portrait

Rapid gas evacuation:( 1) Phase portrait

CENTER

(case of rapid gas evacuation)

( 1) Stable Oscillations

System oscillation: a =0.2 b =5. c =1.

0

0,2

0,4

0,6

0,8

1

0 20 40 60 80 100

t

sigma

theta

sigma+theta

Slow gas evacuation: ( 1)Phase portrait

Slow gas evacuation: ( 1)Phase portrait

FOCUS

(case of slow gas evacuation)

( 1)Attenuating Oscillations

System oscillation:

0

0,2

0,4

0,6

0,8

1

0 50 100 150 200

t

sigma

theta

sigma+theta

FLOW with DOUBLE PHASE TRANSITION

FOUR-PHASE MODEL: Numerical tests

, 1,...4i ii i ij

j i

Sdiv V H i

t

Volterra kinetics

Total liquid Saturation

Radial coordinate

FLOW

production well

PSEUDO THREE-PHASE MODEL

- Mobile liquid is neglecting- Two-component system (light & heavy components)

1

1

( )S F SS a S

t

a St

0 0

0 0, ,

t tS S

LIQUID SATURATION

Flow direction

CLASSIC MODEL

CLASSIC MODEL

LIQUID SATURATION

MODEL with DOUBLE PHASE TRANSITION

LIQUID SATURATION

The macroscale oscillations – whether this is possible ?

TWO TIME SCALES IN OSCILLATIONS

Two scales of time

Sat

SaSx

SFV

t

S

2

12

1

1)(

),,(~

),(

),,(~

),(

xtxt

xtSxtS

/t

t= fast time,

= slow time

Two-Scale Formulation

~~~1~~

1

~~1)~(

~~1

2

12

Sat

S

SaSx

SFV

t

SS

...),,(),(),,(~

...),,(),(),,(~

210

210

xtxtxt

xtSxtSxtS

Additional condition : peridocity w.r.t.

Zero-order Model

Volterra modelin the fast time

00020

00100

Sa

SaSS

Nonlinear oscillations

First-order Model

12212121

11211111

SAt

SAt

S

Linear oscillator

2122121

1112111

1

1

AS

AS

d

dS

121*1

12221221

11121111

,

,

,

ASA

aSa

Saa

Explanation to the macroscale oscillations

Increae of Liquid (S) leads to the increase of Gas but The increase of Gas leads to rthe decrease of liquid

Typical linear oscillator

Global Possible Behaviour

Macroscale (slow) linear oscillations

superposed with

nonlinear (Volterra) microscale (fast) oscillations

CONCLUSIONS

Alain,

you are the best ...