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 ...