Two models for the simulation of multiphase flows in oil and gas ...
Transcript of Two models for the simulation of multiphase flows in oil and gas ...
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]
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 2
Multiphase production network
From the reservoir to the process installations
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100 to 300 km
Transient Simulation
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1000 to 5000 m
Gas
20 to 1000 m
WaterLiquid
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 3
Offshore oilfields
Multiphase production
• Marginal fields near 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 gasonly two-phase flow in the following
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
Stratified Flow
DispersedBubbly
Flow
FlowSlug
LargeBubble Flow
FlowSmall Slug
Depends on fluid velocities, gas fraction ...
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 functionof 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
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 phenomenon that 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
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 processingfacilities are not adequatly sized
• Terrain slugging : low points in the topography of the pipe
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 phenomena induced 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: largevariations in the gas density, gas fraction
• Accurate estimation of outlet flowrates
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
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)
θinclination
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
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(ραRα) +
∂
∂x(ραRαVα) = 0 α = G, L
- Thermo : Phase properties ρα.. as a function of (P,T)
• Compositional two-phase flow :
- Mass conservation of each component k
∂
∂t(
∑α=G,L
Cαk ραRα) +
∂
∂x(
∑α=G,L
Cαk ραRαVα) = 0 k = 1 . . . N
Rα volumetric fraction, Vα velocity, Cαk mass fraction of component
k in phase α
- Thermo :Phase properties ρα, Rα, Ck
α as a function of (P, T, Ck, k = 1..N)
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 12
Drift flux model
• Momentum conservation equation for the mixture
∂
∂t(∑α
ραRαVα) +∂
∂x(ραRαV 2
α + P ) = Tw − ρgsinθ
• Algebraic slip equation : dV = VG − VL
Φ(VM , xG,Γ(P ), dV , x) = 0
• TemperatureT = cste or one mixture energy balance
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 of F (x, W ) and Q(x, W )
- Jacobian DF (x, W ) computed numerically
• Isothermal gas-liquid flow : λ1 < λ2 < λ3
- under simplifying assumptions (S. Benzoni) :λ1 = vL − w, λ3 = vL + w, λ2 = vG
w ' sound velocity from about 50 m/s to several 100m/s
- λ1 < 0, λ3 > 0 : pressure pulses (sonic waves)
- λ2 : gas volume fraction waves (fluid transport)
λ2 positive or negative, ' 1 m/s
- |λ1| >> |λ2|, |λ3| >> |λ2|
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):Eigenvalues λ1 < ...λk... < λN+1
- λ1 < 0, λN+1 > 0 : “pressure waves”
- λk : composition waves : m/s
- |λ1| >> |λk|, |λN+1| >> |λk|
• Low Mach Number FlowsMain interest : fluid transport “waves”responsible for the main dynamics in the pipeline
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 foreach component
ρGRGVG(0, t) = QG(t)/Sect
ρLRLVL(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 < 0
• Initial conditionSteady flow with BC at t = 0 QG(0), QL(0), Poutlet(0)
• Transient Flow induced by BC variations, instabilities(severe-slugging)
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 16
Numerical scheme
• Cell centered Finite Volume schemecell ]xi− 1
2, xi+ 1
2[ , unknown Wi ' W (xi)
∆xid
dtWi + Hi+ 1
2−Hi− 1
2= ∆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 preserves
the different orders of magnitude
H(U, V ) =12(F (U) + F (V )) +
12D(U, V )(U − V )
D(U, V ) = |DF (U)| + |DF (V )|
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 schemeLinear implicitation of the source term and of the flux
H(Un+1, V n+1) ' H(Un, V n) + approx(
∂
∂UH(Un, V n)
)(Un+1 − Un)
+approx(
∂
∂VH(Un, V n)
)(V n+1 − V n)
∂
∂UH(U, V ) ' 1
2(DF (U) + D(U, V ))
Does not account for the different wave velocities
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 fastwaves
• CFL based on fluid transport waves, time steps in agreementwith the main phenomena
• Splitting of the flux into a “slow waves” part and a “fastwaves” one
- cf. G. Fernandez PHD thesis for the Euler equations
- The small eigenvalues are cancelled in the fluxderivatives, similar to the “diagonal” approach of Fernandezex : DF (U) replaced by DF (U) = T ΛT−1,Λk = λk(U) for k = 1, N + 1, Λk = 0 for k = 2, ..., N
• CFL :∆t < min(CFLv
∆x2maxλk,k=2,...,N+1 , CFLp
∆x2maxλk,k=1,N+1 ),
CFLv = 0.8, CFLp = 20
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 19
Second order scheme
Wall friction and gravitational terms induce large ∂∂x (P ) : second
order 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
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 : W0HG(W0,W1) = QG(t)HL(W0,W1) = QL(t)Lt
1W0 = Lt1W1
- Non linear system, sometimes very difficult to solve
• Outlet BC
- Pressure BC : fictitious I + 1 cell, WI+1P (WI+1) = Poutlet(t)Lt
2WI+1 = Lt2WI
Lt3WI+1 = Lt
3WI
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 21
• Change in the flow direction at the oulet : vL < 0
- failure of any numerical treatment based on the sign ofthe eigenvalues or on a adequat fictitious state (RI+1 = 0)
- Empirical and simple approach :if HL(WI ,WI+1) < 0 then (HL)I+ 1
2= 0
- approach justified theoritically on a scalar model (T.Gallouet) : leads to a monotone continuous numerical flux andsatisfies the BC.
• Time discretization
- BC solved during the “explicit” step of the scheme
- W0,WI+1 kept constant during the semi-implicit stepLoss of accuracy as Hn+1 does not satisfy the BCReasonnable results
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
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18
20
22
24
26
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Liqu
id M
ass
Flo
w R
ate
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 4000O
il M
ass
Flo
w R
ate
x=30
00m
Time
SEMI-IMPLIIMPLIEXPLI
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
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
Mas
s F
low
Rat
e
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
Eig
enva
lues
t=
400
s
length
EV1 EV2 EV3 EV4 EV5 EV6 EV7 EV8
Eigenvalues at time t = 400s
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
( )
Oil
vo
lum
etr
ic f
ract
0 10 20 30 40 50 60 70 80
Length (m)
0.0
0.2
0.4
0.6
0.8
1.0
( )
Oil
vo
lum
etr
ic f
ract
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)
Oil
ma
ss f
low
ra
te
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)
Oil
ma
ss f
low
ra
te
t = 250.000 s
t = 300.000 s
t = 400.000 s
t = 500.000 s
t = 600.000 s
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 Numberflow
• Analytical study for a simplified slip law (H. Viviand )
- ε = UaG
, UaL
= εK
aα sound velocity in phase α, U characteristic velocity
- asymptotic expansion of the solution with respect to ε
• Simplified momentum conservation withoutmomentum time derivative and flux terms
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 26
NPW
• Mass and simplified momentum conservation
∂
∂t(ραRα) +
∂
∂x(ραRαVα) = 0 α = G, L
∂
∂x(P ) = Tw − ρgsinθ
- thermo, slip laws
• Properties
B∂
∂tW + A
∂
∂xW = Q
B is singular. Find (a, b) s.t. det(aB − bA) = 0 :
- λ = ab ' u : one “fluid wave” velocity
- b = 0 : one “double” infinite wave velocitySonic waves approximated by infinite velocity wavesMixed parabolic/hyperbolic system of PDE
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 27
Compositional NPW Model
• Mass conservation of each component i
∂
∂t(
∑α=G,L
Cαi ραRα) +
∂
∂x(
∑α=G,L
Cαi ραRαVα) = 0 i = 1 . . . N
• Thermo :Phase properties ρα, Rα, Ci
α as a function of (P, T,Ci)
• Simplified momentum conservation
∂
∂x(P ) = Tw − ρgsinθ
• Algebraic slip equation : dV = VG − VL
Φ(VM , xG,Γ(P ), dV , x) = 0
• Same initial and Boundary conditions as DFM
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+ 1
2]
∆xi
∆t
((Cα
k ραRα)n+1i − (Cα
k ραRα)ni
)+ Hαi+ 1
2−Hαi− 1
2= 0
Hαi+ 12
=
(Cαk ραRα)n
i Vαn+1
i+ 12
if (Vα)i+ 12
> 0
(Cαk ραRα)n
i+1Vαi+ 12
otherwise
• Momentum conservation + slip law : cell [xi, xi+1]
Pn+1i+1 − Pn+1
i = ∆xiQn+1i+ 1
2i=1..I-1
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 29
Boundary conditions
Two-phase immiscible flow
• Inlet Boundary : given mass flowrates
Hα 12
= Qα
• Outlet Boundary
- Mass Fluxes :
HLI+ 12
=
ρLI(RL)IVLI+ 12
if (VL)I+ 12
> 0
0 otherwise
- Pressure
Poutlet − PI =∆xI
2QI+ 1
2
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
Liqu
id M
ass
Flo
w R
ate
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
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, 9610−4kg/s, QL = 2, 8510−4kg/s
- Poutlet = 1bar
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 32
Severe Slugging : RL
NPWRL(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
( )
Oil
volu
metr
ic fra
ction
DFMRL(t) pourx = 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
( )
Oil
volu
metr
ic fra
ction
x = 0.000000E+00
x = 60.0000 m
x = 74.0000 m
DFM CPU ' 9 NPW CPU (single-phase flow)
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
( )
Oil
volu
metr
ic fra
ction
DFM
56 58 60 62 64 66 68 70 72 74
Length (m)
0.0
0.2
0.4
0.6
0.8
( )
Oil
volu
metr
ic fra
ction
t = 370.000 s
t = 385.000 st = 390.000 s
t = 410.000 st = 420.000 s
t = 440.000 st = 460.000 s
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 34
Conclusion
• Two models and schemes adapted to low Mach Numbercompositional 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 schemeLarger 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 colddeep sea production
- Complex networks
Mathematical and Numerical Aspects of Low Mach Number Flows - June 2004 35
The Girassol ( West Africa) production network