Two models for the simulation of multiphase flows in oil and gas pipelines

8/10/2019 Two models for the simulation of multiphase flows in oil and gas pipelines

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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]


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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


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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


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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 ...


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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


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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


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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


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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


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Two models for oil and gas pipelines

I - Pipe and fluid representations

II - Drift Flux model

III - No pressure wave model


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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


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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)


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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


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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|


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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


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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


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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)|


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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


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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


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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


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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


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Change in the flow direction at the oulet: vL


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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


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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


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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


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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


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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


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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


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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


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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


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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


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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


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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)


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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


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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


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The Girassol ( West Africa) production network

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Two models for the simulation of multiphase flows in oil and gas pipelines

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