10.Transient

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10 Transient two-phase flow

Objectives: Quantitative insights into transient mechanisms and phenomena

Until now we have assumed the same mass flow in and out and constant mass flow along the

pipe. For given gas and liquid flow out, can we find the gas and liquid flow, speed, density

and pressure gradient along the pipe, and so estimate the pressure drop and flow capacity.

But if the inflow is changed, it takes time before the mass flow again becomes stationary.

During this period, the outflow is quite different from before, and also quite different from the

outflow after the transient period is over. In the transient period the liquid in the pipe may

flow and fill the inlet separators at the process facility. In a gas lift system, transient

fluctuations affect the inflow so that the changes may be sustained and reinforced so that thesystem becomes dynamically unstable, Asheim/1988 /. Thus, we can not ignore transient

effects.

To calculate transient two-phase flow, we must include continuity equations for gas and

liquid. In the flow equation, we must include the acceleration during time. This is done in

numerical simulators widely used in the oil industry, Bendiksen oa/1991 /. But since numeric

simulators are large and their source code often confidential, we will here focus on analytical

methods. This enables simple estimates and provide insight into transient mechanisms that

also may enable more efficient use of numerical simulators.

10.1 Pressure waves

10.1.1 Wave velocityThe relationship between pressure, velocity and density can be expressed by the Bernoulli

equation: 0dpdpvdvdp f h . If mass flow is constant: vAm and density depends

on the pressure we get the acceleration term: d vvdv 2 so that the pressure gradient is

dpd v1

dxdpdxdp

dx

dp2

f h

(10-1)

We see that if the denominator in (10-1) is zero, the pressure gradient approaches infinity.

This implies a critical speed related to the density and pressure

dp / d

1v*

(10-2)

This velocity is the largest that can occur in a pipes of constant cross section. It also quantifies

how fast a change in pressure propagates. If the outflow from a well increases, the velocity

will first increase at the top of the well. It will then propagate down the pipe as a pressure

wave at velocity given above.



Equation (10-2) is valid for all compressible fluids (gas, liquid, and mixtures), but the

density/pressure relation varies. For single-phase gas, this can be estimated reasonably

accurately with the adiabatic state equation. For two-phase flow density changes can be

related to the compressibility of the mixture: TP TP TP cdpd . Compressibility must

correspond to the average for gas and liquid

l l g g TP yc ycc (10-3)

Average density is: l l g g TP y y . Putting this into (10-2) the wave speed in

homogeneous two-phase mixtures may be estimated

l l l TP TP

*

y1 y

p

c

1v

(10-4)

The approximation indicated above is obtained by neglecting liquid compressibility: cl = 0,and assuming isothermal compressibility and constant z-factor to express gas compressibility:

p1c g , and neglecting the gas density. In total this assumes that the gas provides the

compressibility of the mixture and the liquid the mass, which is reasonably accurate at modest

pressure.

Figure 10.1 shows the critical speed calculated from (10-4), for a mixture of gas and liquid

with specific densities g = 0.7 , and: l = 1, temperature 288 K, z = 0.9, and: cl = 10-9. We see

that at low pressure and liquid fraction around: 0.5, the wave speed is close to 30 m/s.

Figure 10.1 Speed of pressure waves in two phase mixture

While the wave velocities in pure gases and liquids are reasonably constant, the velocity in

mixtures depends on the fraction. Mixtures can thus behave quite differently from the pure

substances they are made up of. The sometimes dramatic effects of powder snow avalanches

may be attributed to this: If the avalanche approaches the critical velocity of the snow/air



mixture, it will have little ability to flow around obstacles, but to hit with full weight.

10.1.2 Pressure Change Speed change propagates as a pressure wave propagates along the pipe. Figure 10.2 illustrates

a control volume that encloses the pressure wave and follows this

Ahead of the control volume a fluid mixture of density:1TP

enters with speed: 1m

* vv . In

the rear a fluid mixture leaves with density:2TP and speed: 1m

*vv . A momentum balance

then gives for when pressure drop:

2

1m

*

1TP

2

2m

*

2TP a vvvv p

. Density changescan be estimated as: a1TP TP 1TP 2TP pc . This gives the pressure drop estimate

1m2m

*

1TP

1TP TP

1m2m

*

1TP a vvv2

c1

vvv2 p

(10-5)

10.2 Wave equation

Changes of speed and pressure will then propagate along the pipe with speed v*, so that that

changes at distance: x along the pipe, will occur at time: t = L / v*. This and similar

phenomena, we can express the general wave equation

t v x f z f k (10-6)

The equation describes a form: f (z) moves with speed: vk . The form can be a single wave, or

periodic fluctuations and may change with time and place, but must be recognizable. Figure

10.3 illustrates a wave front, shock: y z f . We may imagine a tsunami approaching land

with speed: vk = 10 m / s, the front initially being : xo. Five seconds later: t-to = 5s, the front

have reached: m50t t v x x ok o , that is 50 meter closer to shore.

Figure 10.3: Wave propagation



Periodic waves usually expressed as: t kxcos A y , or: t kxi Ae y

, where denotes

the angular frequency and k is the wave number. The speed then corresponds to the ratio

between angular frequency and wave number: k vk . Periodic waves are thus consistent

with the general wave equation (6.10).

10.3 Kinematic waves

Kinematic waves denote waves that occur related to concentration. It often called continuity

waves. Such wave phenomena can occur in many contexts.

An entertaining example is traffic jams: Most people have probably experienced that heavy

traffic seldom flows smoothly and that the congestions can form and dissolve for no apparent

reason. If one considers the flow of cars, it will obviously be small at low car density, but also

small at high car density. (Drivers will slow down when the distance to the car in front

becomes uncomfortably short). The flow thus depends on density. Speed adjustment from asingle driver will change the local density and the change can propagate backward to form

congestions.

10.3.1 Kinematic waves in two-phase flow In two-phase flow, changes in liquid fraction may propagate along the pipe. Since mass can

not disappear, such changes must satisfy the continuity equation

0 x

v

t

mmTP

(10-7)

The sum of gas and liquid fraction is unity, so it is sufficient to look at the liquid fraction. The

gas fraction will follow as: y g = 1-yl . When we neglect the changes of density, the continuity

equation can be written

0 x

yv

t

y l k

l

(10-8)

mvl

sl k

dy

dvv (10-9)

Equation (10-8) is in accordance with the general wave equation (10-1). We see this byexpressing a change in liquid fraction as a wave: t v x y y z f k l l and verify that this

fulfills (10-8). Thus fraction changes may propagate as waves, with speed: l sl k dydvv .

10.3.2 Fluid flow and wave speedWe have previously assumed that the gas speed can be related to the fluid speed and slip

parameters: ,.v ,C ,vv ool g and calculated stationary liquid fraction: ,...v ,C ,v ,v y oo sl sg l . If the

liquid fraction and the total flow is known, we can find liquid flow similarly:

,.v ,C , y ,vv ool m sl . Assuming the linearization of the slip relationship: ol o g vvC v we get



l l o

2

l ol oml m sl

y y1C

yv yvv y ,vv

(10-10)

When the liquid fraction is: yl and the total flow is: vm, the flow liquid may thus be estimated.

The flow of gas : vsg = vm-vsl. Wave speed is found then by differentiating (10-10)

2l l o

2

l ool ooooml mk

y y1C

yv1C yvC 2C vv y ,vv

(10-11)

10.3.3 Change of waveform

Figure 10.4 illustrates a fractionation step: yl = .018 to .07, the overall speed: vm = 3.58m / s

and the wave speed calculated according to (10-11); (other parameters are: vo = 1.2m / s, Co =

1.2) . Wave speed depends on the concentration, so that the fraction yl = 0.07 flows faster than

yl = 0.018. Fraction changes will thus flow rarify along the pipe.

Figur 10.4 Rarifying

Figure 10.5 is calculated for the same conditions as 10.4, except that the fraction initially

increases. The wave velocities then cause the fraction profile to concentrate along the pipe.

But since the initial profile is already sharply, further concentration will be physically

unstable (something analogous to the chain collision on the highway). The initial profile will

be maintained as a shock.

Figure 10.5 Concentrating kinematic wave

If the initial fraction of the increase was gradual, the profile would gradually concentratealong the pipe. If the pipe was long, a shock would eventually formed.



10.4.3 Shock Mathematically we approximate an abrupt change in concentration by a shock. Figure 10.6

shows a shock enclosed in a control volume. We disregard the details inside the control

volume, but require that mass and may be also momentum is conserved through the shock

Figure 10.6 Shock

If the shock moves with speed vf , as illustrated above, and we let the control volume follow

the shock, the fluid flow into the control volume becomes: 2 ,l 2 ,l f i yvvQ , and fluid flow

out 1 ,l 1 ,l f u yvvQ . Liquid speed depends superficial velocity and fraction: l sl l yvv . To

maintain mass balance, the inlet and outlet streams should be equal: Qu = Qi. This gives the

shock speed

1 ,l 2 ,l

1 , sl 2 , sl

f y y

vvv

(10-12)

Similar momentum balance will provide an estimate of the pressure drop across the shock.

Since the shock speed usually will not deviate much from the average flow speed, the

momentum change will be small and may usually be neglected.

References

1973 Whitham, G. B.: Linear and nonlinear waves

John Wiley & sons, NY, 1973

1988 Asheim, H.: "Criteria for Gas-Lift Stability".

J. of Petroleum Technology, Nov. 1988, pp. 1452-1456

1991 Bendiksen, K.H., Malnes, D., Moe, R., Nuland, S.:

“The Dynamic Two-Fluid Model OLGA: Theory and Application”

SPE Prod. Engr., May 1991

1998 Asheim, H., and Grødal, E.:

"Holdup propagation predicted by steady-state drift flux models",

Int. Journal of Multiphase Flow, vol. 24, no. 5, August 1998, 757

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10.Transient