Paper No 7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Investigation on the Mlultiphase Fluid Flow Regime and Pressure
Distribution in Wellbores
Hao Yu and Yueshe Wang
State Key Labortary of Multiphase Flow in Power Engineering, Xi'an Jiaotong University,
Xi'an, 710049, China
hao.yu~,stu.xjtu.edu.cn and wangys~,mail.xjtu.edu.cn
Keywords: multiphase flow in wellbores, driftflux models, transient flow
Abstract
It is particularly crucial to investigation on the multiphase fluid flow regime and its pressure distribution in wellbores for the
practical oil and gas exploitation and development. Driftflux modeling techniques are commonly used to represent
multiphase flow in wellbores. Unlike other mechanistic models, driftflux models are continuous, differentiable and relatively
fast to compute, which apply two basic parameters: the profile parameter Co and dirft velocity V, .Then the holdup can be
calculated with the superficial velocity. So they are very suitable to be used in the description the characteristics of flow
regime for oil reservoir. In this paper, based on the fundamental laws of conservation of mass, momentum and energy fluid
mechanics, a hydraulic transient model on multi component fluid in a wellbore is proposed, which ignores the mass transfer
between different phase interfaces. By assuming an appropriate transportation boundary, setting an pertinent initial value and
effective algebraic correction, the calculation methods, discrete schemes and difference equations suit for wellbore fluids are
put forward. On the basis of the proposed models, profiles of pressure fluctuation, velocity variation of each phase and
holdup distribution along with time in the wellbores are illustrated. Compared with the other mechanistic models, the more
effective formulas can come out, which have the important theoretical and practical significance for the oil and gas
exploitation and development. It is shown that the driftflux model is well suited for use in reservoir simulators. The
predictions of the pressure distribution and gas holdup along with depth of the wellbores is more accurately using this model.
Introduction
The complicatedcomponent multiphase flow is usually
encountered in the oil and gas field development.
Therefore, it is very useful to model the hydrodynamic
characteristics, such as pressure drop and flow regime, of
the multiphase flows along the wellbores for the oil and
gas exploitation and production investment control. Due to
uncertainty of spacetime and interaction between
individual phases, it is extremely difficult to analysis its
flow characteristics along the wellbore. Especially,
Simulation model describing the behaviors of multiphase
flow under hightemperature and highpressure is of
significance to understand the flow mechanism of wellbore
multicomponent fluid. Any innovated technologies
following the models will have overwhelming potential
economic benefit.
Generally speaking, independent or concerted effects of a
lot of factors make modeling the multiphase flow much
more difficult. What method is adopted influence the
precision of simulation directly. So far, uniform fluid ,
separate fluid and drift flow model in multiphase theory
are adopted. The homogeneous models supposed that flow
rates of gas and liquid are identical with the advantages of
being relatively simple, continuous and differentiable. So
they are well suited for the use in reservoir simulators. But
this model neglects the interaction between fluid phases
(i.e.,the fluid phases all move at the same velocity ). So,
the error is more apparent and they are not appropriate for
use in reservoir simulators because they fail to capture the
complex relationship between the in situ volume fraction.
The two fluid models just show the conservation equations
in quality, momentum and energy of gas and liquid phase
respectively, and furthermore they really consider the
interaction between liquid and gas phases. As a matter of
fact, the two fluid models are the most precise and reliable
model theoretically, but their mathematical models are
rather difficult to solve and the regulation of the juncture
between the two phases is not fully understood yet, so it is
difficult to get the wholly precise solution. The driftflux
model (DFM) however, is constructed on describing the
bubbles' distribution and two structural parameters of
relative slip of gas and liquid. They have the same
characteristics with homogeneous flow models and easy to
solve, meanwhile they could depict local features of the
twophase flow. When adopted in simulating multiphase
flow in the wellbores, it will have the advantages of
continuity, differentiability and high calculation speed,
which together made this model more applicable. As a
result, they are a good choice for use in reservior
simulators.
In this paper, based on the fundamental laws of
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
and it gives the definition of drift velocity, which means
when the twophase flows at one mixture velocity, the gas
phase flows at the drift velocity relative to this mixture
velocity and also the fluid flows at the counter drifti
velocity in order to keep flowing continuity. It has been
utilized to solve many engineering problems involving
twophaseflow dynamics.In particular, its application to
forced convection system has been quite successful.
This model puts forward the concept of the drift velocity in
the conservation equations instead of relative velocity
between the phases and depicts gasfluid flowing state by
additive gas phase continuity equations. What's more, the
model takes the twophase fluid as just one homogeneous
fluid, and describes velocity difference by means of linear
relationship between each phase slip velocity and the
mixture velocity.
The driftflux models require many empriical parameters
(S.Livescu et al 2008). Most of the parameters used before
may not be directly applicable to wellbore flows. K.Aziz et
al reported a number of more precise parameters for
wellbores through the experiments.
Zuber and Findlay argues that it is necessary to consider
the slip between gas and fluid, void fraction in the
crosssectional area and the inhomogeneous distribution of
velocity, and they proposed the plwsical model of drift
flow in order to derive the void fraction formula in theory
(Zuber, and Findlay et al 1965). The velocity of gas phase
V, and the mixture velocity Vax could be joined together by
applying the parameter Co and the drift velocity of gas Vd
vgCoi ,tv (1)
Here Co is the distribution coefficient (or the profile
parameter), which describes the effect of the velocity and
concentration profiles (J.P. Schlegel et al 2009). The drift
velocity of the gas Vd meaSures partial relative velocity
between gas and liquid (S.Livescu et al 2008). Using these
paramters, in situ phase volume void fraction can be
calculated from the phase flow velocitvs.
The average mixture velocity Vax is the sum of the gas
and liquid superficial, specifically, it could be depicted as:
Paper No
conservation of mass, momentum and energy fluid
mechanics, a twdraulic transient model on
multicomponent fluid in a wellbore is proposed, which
ignores the mass transfer between different phase
interfaces. By assuming an appropriate transportation
boundary, setting an pertinent initial value and effective
algebraic correction, the calculation methods, discrete
schemes and difference equations suit for wellbore fluids
are put forward. On the basis of the proposed models.
profiles of pressure fluctuation, velocity variation of each
phase and holdup distribution along with time in the
wellbores are illustrated. Compared with the other
mechanistic models, the more effective formulas can come
out, which have the important theoretical and practical
significance for the oil and gas exploitation and
development.
Nomenclature
g gravitational acceleration(ms2)
P pressure (Nm2)
A cross sectional area (m )
V velocitv(ms')
V mixture velocitv(ms )
V, velocity of gas phase(ms')
VI velocity of liquid phase(ms')
Vd drift velocitv(ms')
V, superficial velocitv(ms')
z axial distance(m)
t time
Co distribution parameter
D diameter of the wellbore(m)
T temperature("C)
h enthalpy
H hold up of liquid phase dimensionlesss)
S wetted perimeter(m)
Greek letters
oc void fraction dimensionlesss)
p density (Kg/ m )
Deviation from vertical
Tc shear stress
VM = Sg + SL
al~ + HI 1
Subsripts
m nuxture
g gas
liquid
w wellbore
Here, as ,is the gas holdup and H1 is the liquid holdup.
With the parameters of Co, Vd and the converted velocity
of gas and liquid, we could calculate the void fraction a ,
and liquid holdup H1. That is to say, how the precision of
the predicted as, depends on the value of Co and Vd
(Nassos. and Bank 1967) (T.Hibiki et al 2002)
Driftflux model
The driftflux model is also called mixture model or drift
model. It is one of the most practical and accurate models
proposed by zuber& Findlay aiming at the deviation of the
homogeneous flow model and two fluid model with the
fact (Zuber and Findlay et al 1965). It has since been
refined by many researchers (e.g., Ishii, Nassos and
Bankoff, Wallis, K.Aziz et al. ). It is the kind of model
which focuses on the lwpothesis of thermodynamic
equilibrium and constructs on the twophase velocity field,
Paper No
velocity profile
concentration profile
local relative velocity
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
8 8 2 dP
at 1H I VIA) + ( lHI V A) + Al 8: (6)
SZ1ls1 + Zis. pl g HIA
The first item in the left side is the variable quantity with
time of the momentum in the control volume, the second
item is the changing quantity of the momentum for the
different qualities of gas and liquid inside and outside the
control volume, the third item is the pressure impulse. The
first item in the right side is the friction impulse between
gasliquid phase and the pipe wall, second item is friction
impulse between gas and liquid and the last item is the
momentum of gravity along the flowing direction.
After adding the each phase equations, we can rewrite the
momentum equation of the twophase mixture as follows:
d 22
(t Ra V + plHIVl)_( ga V +t plg' HI~'l
=(p a +plHI A
Figure.1: Profile and local slip mechanisms in the
driftflux model
So far, through semiempirical method the relatively
precise value of Co and Vd can be calculated if a is
adopted. The more precise parameters which fit to the
wellbore flows are reported in detail by K.Aziz et al
(2003).
Numerical Scheme
For the instability of flow between liquid and gas, the
following consumption should be constructed:
First, the flow of gas and liquid is transient in the wellbore,
and they conforms the onedimensional flow along the
wellbore:
Second, the gas and liquid flow at their average velocity in
the control volume (Takashi Hibik et al 2003);
Then, there is no mass transfer between the gas and liquid.
Under these three basic consumption, the continuity
equations, the momentum equations and the energy
equations can be proposed according to the fundamental
regulations of hydrodynamics and the whole heat transfer
process in the wellbore (M. Ishii and K. Mishima 1984).
The gasphase continuity equations:
The energy conservation equation is thus:
12 2
d( 4p agF(h + + gz) + 4pl~HI hl z)
S2 2
+A(p a (u+ ) PNIU+ pl)
+K~r D(T TO )= 0
(8)
The auxiliary equation of the gasphase state equation is
thus:
p = p(P, T)
A formulation that combines the two mechanisms is thus:
g= pay 4 7 (10)
( p A4a r ) + ( p a )= 0
dz g g g dt & &
The liquidphase continuity equation:
The gasphase momentum equation:
a a 2
(pa u 4 )+(p a i 4)+ 4
at g g g az ggg g
= 7 s 7.s. p g a .
g g I;g g
The liquidphase momentum equation:
The flow equations above are nonlinear partial differential
equations, and they are tough to solve the problems, so we
can not get the more precise numerical solutions (Wenquan
Tao 2001). The equations are solved by the conservation
(4) laws in this paper.
It is common to use the analysis of eigenvalues and the
implicit difference method to get the numerical
implementation. Generally in the literature, explicit
method is often adopted in the dynamic calculation for its
simple program and relatively fast to compute, but it is
(5) often limited to Courant conditions. And also, the improper
selection of space, time step and initial value will lead to
unsteady of the results. While there really exists large
amount of calculation and time consuming if implicit
difference method is used, it could also avoid the
constraint of time step while using analysis of eigenvalues.
This research employs formal storing parameters of
alternating network, the specific method of storing is to
" I I I
0 500 1000 1500 2000 2500 3000
Depth (m)
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
test in the bottom of the well. So we usually get the
prediction pressure distribution with the pressure of the
outlet.
,e  lo~s
10 15 20 25 30 35 40
Test Pressure(MPa)
Figure 2: Predicted pressure distribution using driftflux
model for wellbores
We predict the pressure distribution of 14 wells. Then we
make a comparison with the actual test pressure of the
wells bottom.
From the figure above, we can see that the driftflux model
above mentioned is practical to the pressure prediction of
the well bottom. The maximum error is about 10% which
is situable for the actual oil and gas exploitation.
Paper No
conserve the velocity of gas and liquid phases on the
boundary of the network, and store the pressure,
temperature and gas holdup in the centre of the grid, this
will get the difference of adjacent velocity component in
order to avoid the waveform distribution of the pressure .
We calculate the value of velocity, holdup and pressure in
nonboundary point using the methods of laxfriedrichs
and Mac Cormack. Also we get the value in boundary point
with the auxiliary equations and the slip velocity.
The equations of flow may be written as
where u is the independent variable.
~~~=(U,~ ~ ''' )
w means the conservation parameters.
w = p a ,plHI paV + 'pHIV
We difine the variables as follows
(wi w ) +
2At 71J 2 1
(14)
(w . w. .)+
2At ; 1, 1,} 2
(15)
;, ~
7
.
Then the equations of flow can be obtained:
w"n + 1 n 7 qn
7, J i, J A 1
S+2 J
In ) tn
1 .
Figure 3: Predicted pressure distribution using driftflux
model for one gaswater well
Predictions pressure distribution of one gaswater well are
shown in Figure 3. From the figure above, we can get that
the pressure is higher with the increasing depth of the well.
The next is the predictions for as in a gaswater well.
We use the method of MacCormack to predict and correct.
And we can get the more precise results in the wellbores.
Results and Discussion
A novel numerical model for several wellborereservoirs
systems is presented. In order to evaluate the adaptability
of the driftflux model to the wellborereservoirs
simulators, the comparison of simulation results and
available data from oilfield are made. Then we analyze the
accuracy of this model, the multiphase fluid flow regime
and the optimized characteristic parameters for the
wellbores.
The research on pressure distribution in the wellbores is
very important for the oil and gas exploitation. For the high
pressure well, we can not get the data of the pressure with
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
(1) The driftflux model was well suited for use in
reservoir simulators because it was relatively simple
and it provided a continuous represention of
multiphase flow in wells. It could predict the pressure
distribution and flow regime more accurately. Most
error was about 10 percent or even less.
(2) The pressure distribution in wellbores was higher with
the increasing depth of the well. It could be very high
at the bottom of the well.
(3) The gas holdup was lower with the increasing depthof
the well. Under the same depth of the wellbore,the
gas holdup was affected by the gas water
ration(GWR).
In future work, we plan to consider the factor of the
temperature in the wellbores, and take into account the
affect of the temperature on the flow regime and the
pressure distribution. And how to choose the optimized
parameters for wellbores is also the focus of our research.
Acknowledgements
We wish to thank State Key Labortary of Multiphase Flow
in Power Endineering and Sinopec Research Institute of
Petroleum Engineering for supporting this work. The
authors also are grateful to National S&T Major Project for
providing funding for this work (2()(8ZX()5()(5()(61()HZ).
References
H.Shi, J.A.Holmes, L.J.Durlofsky, K.aziz, L.R.Diaz,
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Multiphase Flow in Wellbore, the SPE Annual Technical
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S.Livescu, L.J.Durlofsky, and K.Aziz, A Semianalytical
Thermal Multiphase Wellbore Flow Model for Use in
Reservoir Simulation, the 2008 SPE Annual Technical
Conference and Exhibition, Denver, September 2008
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076
I074
on
~072
av 
* *
*
sco io'oo s00 2000 25M0 3000
Depth (m)
Figure 4: Predicted gas holdup using driftflux model for
one oilgas well
From the calculated gas holdup, we can come to the
conclusions that the gas holdup is lower with the
increasing depth of the well. This case show that the flow
regime is churn flow in this wellbore. We can also predict
that the liquid is transported in the shape of the bubbles.
We then assume that the water production is constant, and
we just calculate the holdup of gas and flow regime with
the different gas water ration (GWS) respectively.
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own a
I \
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Deph (m)
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Reservoir
Figure 5: Predicted gas holdup using driftflux model for
different GWR
Figure 5 shows that the gas holdup is lower with the
increasing depth of the wellbore under the same gas water
ration (GWR) conditions. Under the same depth of the
wellbore, the gas holdup is more with the increasing gas
water ration (GWR). The flow regime is also changed with
the different gas water ration.
Conclusions
More accurately prediction the pressure distribution can
make us control and change the gas and oil exploitation
and develop the better technology. It is of great
significance in the oil and gas fields. In this paper, we uesd
the dirftflux experiment parameters for wellbores to
simulate the multiphase fluid flow regime and pressure
distribution in the wellbores. Then we calculated the gas
holdup of different gas water ration. The following
conclusions can be drawn from this work:
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Paper No 7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
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