two models for the simulation of multiphase flows in oil and gas pipelines
TRANSCRIPT
-
8/10/2019 Two models for the simulation of multiphase flows in oil and gas pipelines
1/35
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 1
Two models for the simulation of multiphaseflows in oil and gas pipelines
TACITE - TINA:E. Duret, I.Faille, M. Gainville, V. Henriot, H. Tran, F. Willien.
Consultants : T. Gallouet, H. Viviand
Division Technologie, Informatique et Mathematiques Appliquees
Institut Francais du Petrole, 1 et 4 avenue de Bois Preau, 92852 Rueil Malmaison
e-mail: [email protected]
-
8/10/2019 Two models for the simulation of multiphase flows in oil and gas pipelines
2/35
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 2
Multiphase production network
From the reservoir to the process installations
100 to 300 km
Transient Simulation
1000 to 5000 m
Gas
20 to 1000 m
Water
Liquid
-
8/10/2019 Two models for the simulation of multiphase flows in oil and gas pipelines
3/35
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 3
Offshore oilfields
Multiphase production
Marginal fieldsnear large decreasing reservoirs :
- small accumulations of hydrocarbons
- financially worthwhile if low cost developments- long subsea tiebacks to existing infrastructures
- ex : North sea
Deep water fields
- great water depths (1000m): no other choice- Gulf of Mexico, West Africa
Water, oil and gas
only two-phase flow in the following
-
8/10/2019 Two models for the simulation of multiphase flows in oil and gas pipelines
4/35
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 4
Two-phase flow regimes
ex : Horizontal pipe
Liquid Gas
with dropletsFlow
Annular
Stratifiedwavyflow
StratifiedFlow
DispersedBubbly
Flow
FlowSlug
LargeBubbleFlow
FlowSmall Slug
Depends on fluid velocities, gas fraction ...
-
8/10/2019 Two models for the simulation of multiphase flows in oil and gas pipelines
5/35
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 5
Gravity
Large differences in flow behavior between horizontal, inclined,
and vertical pipe flow
Gas density and gas/liquid disctribution change as a function
of pressure (depth)
Non uniform elevation of the pipeline
- pipe lying on the seabed
- riser reaching the platform
- can induce large scale instabilities :terrain slugging, severe slugging
-
8/10/2019 Two models for the simulation of multiphase flows in oil and gas pipelines
6/35
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 6
Severe-slugging phenomenon
It occurs for low velocity of gas and liquid phases.Cyclic phenomenonthat can be broken down into 4 parts :
1. liquid accumulates at the low point 2. blockage until the pressure becomes
sufficient to lift the liquid column
-
8/10/2019 Two models for the simulation of multiphase flows in oil and gas pipelines
7/35
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 7
3. the liquid slug starts to go upward
along the riser, the gas begins to flow
4. the gas arrives at the top of the riser
and the pressure rapidly decreases caus-
ing liquid flow down
Consequences:
- Large surges in the liquid and gas production rates
- Equipment trips and unplanned shutdowns if processing
facilities are not adequatly sized
Terrain slugging: low points in the topography of the pipe
-
8/10/2019 Two models for the simulation of multiphase flows in oil and gas pipelines
8/35
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 8
Industrial objectives
Simulation tool for the design of multiphase production networks
Maximise the production, minimize the risks and the costs
- it is difficult to avoid severe-slugging ( shut-downs ...)
- over dimension the processing facilities
Simulate transient phenomenainduced either by
- operating conditions : flowrates variations at the inlet,
pressure variations at the outlet
- the non uniform topography of the pipeline
Characteristics of the flow
- Long distance (10 to 100 kms), low velocities m/s
- Importance of gravity and compressibility: large
variations in the gas density, gas fraction
Accurate estimation of outlet flowrates
-
8/10/2019 Two models for the simulation of multiphase flows in oil and gas pipelines
9/35
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 9
Two models for oil and gas pipelines
I - Pipe and fluid representations
II - Drift Flux model
III - No pressure wave model
-
8/10/2019 Two models for the simulation of multiphase flows in oil and gas pipelines
10/35
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 10
Pipe and Fluid
Pipe
- large length, small diameters (10 to 30 cm)- 1D model with variable inclination (here : constant diameter)
inclination
Inlet
Outlet
Fluid description:
- Two-phase Immiscible Flow : compressible gas and liquid, no
mass transfer between phases
- Multiphase compositionnal Flow : mass tranfer between phases assuming thermodynamical
equilibrium
lumping preprocessing procedure to reduce the number of
components : 2 to 10 components
-
8/10/2019 Two models for the simulation of multiphase flows in oil and gas pipelines
11/35
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 11
Drift flux model
Immiscible two-phase flow :-Mass conservation of each phase
t(R) +
x(RV) = 0 = G, L
-Thermo: Phase properties .. as a function of (P,T)
Compositional two-phase flow :
-Mass conservation of each component k
t(=G,L
Ck R) +
x(=G,L
Ck RV) = 0 k= 1 . . . N
R volumetric fraction, V velocity, C
k mass fraction of component
k in phase
-Thermo:
Phase properties , R, Ck as a function of(P , T , C k, k= 1..N)
-
8/10/2019 Two models for the simulation of multiphase flows in oil and gas pipelines
12/35
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 12
Drift flux model
Momentum conservation equation for the mixture
t(
RV) +
x(RV
2 + P) =Tw gsin
Algebraic slip equation: dV =VG VL
(VM, xG, (P), d V , x) = 0
Temperature
T = cste or one mixture energy balance
-
8/10/2019 Two models for the simulation of multiphase flows in oil and gas pipelines
13/35
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 13
Characteristics of the DFM Model
Non linear hyperbolic system of conservation laws
tW(x, t) +
xF(x, W(x, t)) =Q(x, W(x, t))
- no algebraic expression ofF(x, W) and Q(x, W)
- Jacobian DF(x, W) computed numerically
Isothermal gas-liquid flow : 1< 2 < 3
- under simplifying assumptions (S. Benzoni) :
1 =vL w, 3 =vL+ w, 2 =vGw sound velocity from about 50 m/s to several 100m/s
- 1 0 : pressure pulses (sonic waves)
- 2 : gas volume fraction waves (fluid transport)
2 positive or negative, 1 m/s
- |1|>> |2|, |3|>> |2|
-
8/10/2019 Two models for the simulation of multiphase flows in oil and gas pipelines
14/35
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 14
Characteristics of the DFM Model
Isothermal compositional flow:Hyperbolic system (N+ 1)(N+ 1):
Eigenvalues1< ...k... < N+1
- 1 0 : pressure waves
- k : composition waves : m/s- |1|>> |k|, |N+1|>> |k|
Low Mach Number Flows
Main interest : fluid transport waves
responsible for the main dynamics in the pipeline
-
8/10/2019 Two models for the simulation of multiphase flows in oil and gas pipelines
15/35
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 15
Boundary Conditions (Two-phase immiscible flow)
- Inlet Boundary x= 0 : flow rates for each phase or for
each component
GRGVG(0, t) = QG(t)/Sect
LRLVL(0, t) = QL(t)/Sect
- Outlet boundary x= L : pressure, liquid can not go backinto the pipe
P(L, t) = Poutlet(t)
RL(L, t) = 0 si VL
-
8/10/2019 Two models for the simulation of multiphase flows in oil and gas pipelines
16/35
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 16
Numerical scheme
Cell centered Finite Volume schemecell ]xi 1
2
, xi+ 12
[ , unknown Wi W(xi)
xid
dtWi+ Hi+ 1
2
Hi 12
= xiQi
Numerical flux
- difficult to use algebraically constructed approximate
Riemann solver like Roe scheme
- Rusanov : uniform dissipation on all the waves, too
dissipative on void fraction waves
- Idea : introduce enough numerical dissipation but preservesthe different orders of magnitude
H(U, V) =1
2(F(U) + F(V)) +
1
2D(U, V)(U V)
D(U,V) =|DF(U)|+|DF(V)|
-
8/10/2019 Two models for the simulation of multiphase flows in oil and gas pipelines
17/35
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 17
Time discretisation
Explicit scheme- CFL based on sonic waves
- time steps too small / time of fluid transport
Linearly implicit scheme
Linear implicitation of the source term and of the flux
H(Un+1, Vn+1) H(Un, Vn) + approx
UH(Un, Vn)
(Un+1 Un)
+approx
VH(Un, Vn)
(Vn+1 Vn)
UH(U, V)
12
(DF(U) + D(U, V))
Does not account for the different wave velocities
-
8/10/2019 Two models for the simulation of multiphase flows in oil and gas pipelines
18/35
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 18
Semi-implicit scheme
explicit on the slow waves and linearly implicit on the fast
waves
CFL based on fluid transport waves, time steps in agreement
with the main phenomena
Splitting of the flux into a slow waves part and a fast
waves one- cf. G. Fernandez PHD thesis for the Euler equations
- The small eigenvalues are cancelled in the flux
derivatives, similar to the diagonal approach of Fernandez
ex : DF(U) replaced by DF(U) =TT1,
k =k(U) fork= 1, N+ 1, k = 0 for k= 2,...,N
CFL :
t < min(CF Lvx
2maxk,k=2,...,N+1, C F Lp
x2maxk,k=1,N+1
),
CF Lv = 0.8, CF Lp = 20
-
8/10/2019 Two models for the simulation of multiphase flows in oil and gas pipelines
19/35
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 19
Second order scheme
Wall friction and gravitational terms induce large x (P) : secondorder scheme necessary
MUSCL approach :
- linear reconstruction on physical variables : P, ci, VM
- minmod limiter
RK2 type scheme to enable CFL 0.8 on slow waves
- second order time scheme on the slow waves
- first order on the fast waves
-
8/10/2019 Two models for the simulation of multiphase flows in oil and gas pipelines
20/35
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 20
Boundary conditions (immiscible flow)
Inlet BC imposed on the numerical fluxes
- Fictitious cell : W0
HG(W0, W1) = QG(t)
HL(W0, W1) = QL(t)Lt1W0 = Lt1W1
- Non linear system, sometimes very difficult to solve
Outlet BC
- Pressure BC : fictitious I+ 1 cell, WI+1
P(WI+1) = Poutlet(t)
Lt2WI+1 = Lt2WI
Lt3WI+1 = Lt3WI
-
8/10/2019 Two models for the simulation of multiphase flows in oil and gas pipelines
21/35
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 21
Change in the flow direction at the oulet: vL
-
8/10/2019 Two models for the simulation of multiphase flows in oil and gas pipelines
22/35
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 22
Application : Inlet Gas Flowrate Increase
Horizontal pipeline of 5000m length, immiscible two-phase flow
16
18
20
22
24
26
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
LiquidM
assFlow
Rate
length
QL t=0.QL t=100.QL t=200.QL t=500.
QL t=1000.QL t=1500.
Oil Mass Flowrate at different times
16
18
20
22
24
26
0 500 1000 1500 2000 2500 3000 3500 4000
OilMassF
low
Ratex=3000m
Time
SEMI-IMPLI
IMPLI
EXPLI
REF
Oil Flowrate /time at x = 3000m
Explicit, Implicit and Semi-implicit
Nb Step Expli = 50 * nbStep Semi-Impli
CPU Expli = 45 * CPU Semi-Impli
-
8/10/2019 Two models for the simulation of multiphase flows in oil and gas pipelines
23/35
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 23
Application : 7 components
Ascending pipeline of 5000m length and 0.146m diameter.Boundary conditions : Poutlet(t) = 10 bar
At the inlet: Q1..Q5 constant, Q6,Q7 increased from 0 to 2 in 50 s
0
0.5
1
1.5
2
2.5
3
3.5
4
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
MassFlowRate
length
Q1 t=0.Q1 t=50.
Q1 t=200.Q1 t=400.
Q1 t=1000.Q7 t=0.
Q7 t=50.Q7 t=200.Q7 t=400.
Q7 t=1000.
Q1 and Q7 at different times
-400
-300
-200
-100
0
100
200
300
400
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Eigenvalues
t=
400s
length
EV1EV2EV3EV4EV5EV6
EV7EV8
Eigenvalues at time t = 400s
-
8/10/2019 Two models for the simulation of multiphase flows in oil and gas pipelines
24/35
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 24
Vertical pipe : liquid fraction and flowrate
0 10 20 30 40 50 60 70 80
Length (m)
0.0
0.2
0.4
0.6
0.8
1.0
( )
Oilvolumetricfract
0 10 20 30 40 50 60 70 80
Length (m)
0.0
0.2
0.4
0.6
0.8
1.0
( )
Oilvolumetricfract
0 10 20 30 40 50 60 70 80
Length (m)
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
(kg/s)
O
ilmassflow
rate
t = 0.000000E+00
t = 50.0000 s
t = 70.0000 s
t = 100.000 s
t = 200.000 s
t = 225.000 s
t = 250.000 s
0 10 20 30 40 50 60 70 80
Length (m)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
(kg/s)
Oilmassflow
rate
t = 250.000 s
t = 300.000 s
t = 400.000 s
t = 500.000 s
t = 600.000 s
-
8/10/2019 Two models for the simulation of multiphase flows in oil and gas pipelines
25/35
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 25
No Pressure Wave Model
Two phase immiscible flow
Modify the DFM model to account for the low Mach Number
flow
Analytical study for a simplified slip law (H. Viviand )
- = UaG , UaL = K
a sound velocity in phase , U characteristic velocity
-asymptotic expansion of the solution with respect to
Simplified momentum conservation without
momentum time derivative and flux terms
-
8/10/2019 Two models for the simulation of multiphase flows in oil and gas pipelines
26/35
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 26
NPW
Mass and simplified momentum conservation
t(R) +
x(RV) = 0 = G, L
x(P) =Tw gsin
- thermo, slip laws
Properties
B
tW+ A
xW =Q
B is singular. Find (a, b) s.t. det(aB bA) = 0 :
- = ab u : one fluid wave velocity
- b= 0 : one double infinite wave velocity
Sonic waves approximated by infinite velocity waves
Mixed parabolic/hyperbolic system of PDE
-
8/10/2019 Two models for the simulation of multiphase flows in oil and gas pipelines
27/35
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 27
Compositional NPW Model
Mass conservation of each component i
t(=G,L
Ci R) +
x(=G,L
Ci RV) = 0 i= 1 . . . N
Thermo:
Phase properties , R, Ci as a function of (P , T , C i)
Simplified momentum conservation
x(P) =Tw gsin
Algebraic slip equation: dV =VG VL
(VM, xG, (P), d V , x) = 0
Same initial and Boundary conditions as DFM
-
8/10/2019 Two models for the simulation of multiphase flows in oil and gas pipelines
28/35
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 28
Numerical scheme
VF scheme on staggered mesh- implicit centered scheme for the parabolic part (sonic waves)
- explicit upwind scheme for the hyperbolic part (slow waves)
- CFL based on phase velocities (0.4)
Mass conservation+ thermo : cell [xi 12 , xi+ 12 ]xit
(Ck R)
n+1i (C
k R)
ni
+ Hi+ 1
2
Hi 12
= 0
Hi+ 12
=
(Ck R)n
iVn+1
i+ 12
if (V)i+ 12
>0
(C
k
R
)
n
i+1Vi+
1
2 otherwise
Momentum conservation + slip law: cell [xi, xi+1]
Pn+1i+1 Pn+1i = xiQ
n+1i+ 1
2
i=1..I-1
-
8/10/2019 Two models for the simulation of multiphase flows in oil and gas pipelines
29/35
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 29
Boundary conditions
Two-phase immiscible flow
Inlet Boundary: given mass flowrates
H 12
=Q
Outlet Boundary
- Mass Fluxes :
HLI+ 12
=
LI(RL)IVLI+ 12
if (VL)I+ 12
>0
0 otherwise
- Pressure
Poutlet PI=xI
2 QI+ 1
2
-
8/10/2019 Two models for the simulation of multiphase flows in oil and gas pipelines
30/35
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 30
Application : Inlet Gas Flowrate Increase
Horizontal pipeline of 5000m length and 0.146m diameter.
16
18
20
22
24
26
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
LiquidMassFlow
Rate
length
DFM t=0.NPW t=0.
DFM t=500.NPW t=500.
DFM t=1000.NPW t=1000.DFM t=1500.NPW t=1500.
Oil Mass Flowrate in the pipe : NPW/DFM comparison
-
8/10/2019 Two models for the simulation of multiphase flows in oil and gas pipelines
31/35
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 31
Severe Slugging
60 m
14 m
60m long horizontal pipe, 14m long riser
D=5cm
Constant inlet mass flowrates and outlet pressure
- QG= 1, 96104kg/s, QL = 2, 8510
4kg/s
- Poutlet = 1bar
-
8/10/2019 Two models for the simulation of multiphase flows in oil and gas pipelines
32/35
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 32
Severe Slugging : RL
NPW
RL(t)
x= 0, 60, 74m
0 200 400 600 800 1000 1200 1400
Time (s)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
( )
Oilvolumetricfractio
n
DFM
RL(t) pour
x= 0, 60, 74m
0 200 400 600 800 1000 1200 1400
Time (s)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
( )
Oilvolumetricfraction
x = 0.000000E+00
x = 60.0000 m
x = 74.0000 m
DFM CPU 9 NPW CPU (single-phase flow)
-
8/10/2019 Two models for the simulation of multiphase flows in oil and gas pipelines
33/35
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 33
Severe Slugging : RL in the riser
NPW
56 58 60 62 64 66 68 70 72 74
Length (m)
0.0
0.2
0.4
0.6
0.8
( )
Oilvolumetricfractio
n
DFM
56 58 60 62 64 66 68 70 72 74
Length (m)
0.0
0.2
0.4
0.6
0.8
( )
Oilvolumetricfraction
t = 370.000 s
t = 385.000 s
t = 390.000 s
t = 410.000 s
t = 420.000 s
t = 440.000 s
t = 460.000 s
-
8/10/2019 Two models for the simulation of multiphase flows in oil and gas pipelines
34/35
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 34
Conclusion
Two models and schemesadapted to low Mach Number
compositional two-phase flow
- DFM : scheme explicit for the slow wave and
implicit for the fast waves
- NPW : accoustic waves approximated by infinite valocitywaves, semi-implicit scheme
Larger time steps for NPW, less CPU time
Extension to
-more complex fluid flow:
Three-phase flow : water phase
Four-phase flow : solid phase (hydrate, wax) for cold
deep sea production
-Complex networks
-
8/10/2019 Two models for the simulation of multiphase flows in oil and gas pipelines
35/35
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 35
The Girassol ( West Africa) production network