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]


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

WaterLiquid


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


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 ...


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


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


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


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


Two models for oil and gas pipelines

• I - Pipe and fluid representations

• II - Drift Flux model

• III - No pressure wave model


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


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)


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


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|


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


• 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)


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 )|


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


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


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


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


• 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


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

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


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


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


“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


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


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


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


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


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


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


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)


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


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


The Girassol ( West Africa) production network

two models for the simulation of multiphase flows in oil and gas

Documents