Academic Journal of Science and Technology 
ISSN: 2771-3032 | Vol. 6, No. 1, 2023 

 

49 

Overview of Oil-water Two-phase Flow Pressure Drop 
Theory 
Shuaihu Jia, Chenyu Zhang 

College of Petroleum Engineering, Xi’an Shiyou University, Xi’an 710065, China 

 
Abstract: This is a review of the existing pressure drop theories for dispersed and annular flow in oil and water phases in 
straight wells is presented. The flow classification criteria, pressure drop calculation methods are outlined. For the dispersed 
flow, the homogeneous flow model and the two-fluid model are introduced. The key to the pressure drop of the homogeneous 
flow is the calculation of the frictional resistance, and the two-fluid model focuses on the drag force between the two phases. 
And for the core-annular flow, the two-fluid model is used to focus on the interference between the two-phase interface. 

Keywords: Oil-water two-phase flow, Pressure drop calculation, Homogeneous flow model, Two-fluid model. 
 

1. Introduction 
The research work of oil-water two-phase flow law 

originated from the study of drag reduction transport of thick 
oil in the petroleum industry in the early twentieth century. in 
1904, Isaacs and speed proposed a patent for water ring 
transport of thick oil. in 1950, Clark and Shapiro patented the 
injection of water containing emulsion breakers into thick 
oil.Charles et al. and Russell et al. found that adding water to 
oil could reduce the pressure drop. 

In 1961, G.W. GOVIER conducted oil-water two-phase 
experiments in a vertical pipe. He concluded that although the 
slip velocity equation proposed by Hlore and Wilde was 
written for a gas-liquid system, a similar equation was 
applicable to two systems of unmixed liquids, and he regarded 
liquids as high-density bodies in his experiments. Oil-water 
two-phase experiments were conducted using three different 
viscosities of oil to investigate the effect of oil phase viscosity 
on water holding rate, flow regime and pressure drop, and it 
was found that the viscosity of the mixture had little effect on 
the holding rate. 

In 1966, Charles and Lilleleht applied the Locldlan-
Maninelli formula for predicting the pressure drop in gas-
liquid two-phase flow to the prediction of the pressure drop 
in oil-water two-phase flow, and found that the prediction was 
not satisfactory. In 1980, Theissing attributed this deviation to 
the relatively small density difference between oil and water, 
and he modified the Maninelli equation to derive an empirical 
equation suitable for the pressure drop of oil-water and gas-
liquid two-phase flows. In 1994, Sapelberg and Mewes used 
the Locldlan-Maninelli method to predict the frictional 
pressure drop for oil-water two-phase flows with different 
pipe diameters. They found that the prediction results for 
small pipe diameters were similar to those of Charles 1966, 
while the method could not be used to predict the frictional 
pressure drop for various oil-water two-phase flow types for 
large pipe diameters. 

In 1998, Neima Brauner [1] proposed to use the 
homogeneous flow model to calculate the pressure drop of 
oil-water mixtures, and also summarized the equations of 
Zuber-Findlay (1965) and Harmathy (1960), et al. to propose 
a complete method for calculating oil-water two-phase flow, 
which divides the flow regime according to the holding rate 
and calculates the pressure drop of stratified and dispersed 

flow, and the method was widely accepted. 
In 2001 Antonio C. Bannwart [2] et al. proposed a 

phenomenological model to predict the pressure drop during 
oil-water annular flow through a vertical pipeline, which 
considered the effect of turbulence in the annular fluid and 
wavy interface, and also considered the effect of buoyancy on 
the vertical system proposed a method for calculating the 
pressure drop in oil-water two-phase core-annular flow. 

In 2007 Arun K. Jana [3] et al. conducted oil-water two-
phase experiments using kerosene and classified the flow 
patterns into four types: dispersed bubble flow, bubble flow, 
churning flow, and core-annular flow. And a suitable 
theoretical model was used to analyze the liquid holding rate 
and pressure drop of each flow pattern. The analysis shows 
that the homogeneous model is applicable to the dispersed 
bubbly flow, while the drift flux model can better predict the 
bubbly and churning-turbulent flow patterns. On the other 
hand, the separation flow model (N. Brauner, 1991) was used 
to accurately predict the pressure drop in the core circulation. 
However, this experiment only considered the effect of flow 
velocity and did not take into account the effect of antiphase 
and the effect of temperature on fluid viscosity. 

In 2018 Mohammad J. Hamidia et al. chose water and 
kerosene with a viscosity of 1.49 mPa-s and a density of 780 
kg/m3 as unmixed-phase fluids and used high-speed 
photography to identify the flow pattern. The pressure drop 
and heat transfer coefficient (HTC) of oil-water two-phase 
flow in a horizontal tube were experimentally measured to 
establish a new correlation to predict the HTCs of STMI flow 
pattern, D w/o flow pattern and o/w flow pattern. 

In 2021, Yang Hengqi et al. conducted a thick oil-water at 
high temperature and pressure experiment to derive and 
modify a two-phase pressure drop calculation model for thick 
oil-water in straight wells using Einstein's suspended 
viscosity equation. 

From the research, it can be seen that the calculation factors 
affecting the pressure drop in the well are as follows: 

1. the division of fluid flow patterns in the tubing; 
2. calculation of mixture viscosity; 
3. calculation of the fluid holding rate. 



 

50 

2. Oil-water Two-phase Veryical Flow 
Pattern Classification 

The determination of the flow pattern is extremely 
important during the two-phase flow analysis. This is mainly 
because all design parameters (e.g., pressure drop, fluid 
holding rate, heat and mass transfer coefficients, retention 
time distribution, and chemical reaction rates) are strongly 
dependent on the flow pattern under consideration. For a 
given two-phase flow system, the flow pattern is mainly 
determined by a combination of wellbore characteristics (wall 
roughness, wellbore ID and wellbore inclination), media 
characteristics (liquid density, liquid viscosity and interfacial 
surface tension) and operating conditions (liquid flow rate, 
system pressure “p”, system temperature T). 

2.1. Dispersion Flow Flow Pattern Judgment
In 1997, Flores et al. conducted 121 sets of oil-water two-

phase flow experiments in a vertical tube with an inner 
diameter of 50.8 mm using a conductivity probe, and defined 
two categories and six types of oil-water two-phase flow 
patterns, among which the flow patterns with water as the 
continuous phase include Dispersion o/w, Very Fine  
Dispersion o/w), transitional flow type of water-in-oil (o/w 
Churn); the flow type of oil in continuous phase includes 
transitional flow type of oil-in-water (w/o Churn), dispersion 
of water-in-oil (Dispersion w/o) and very fine dispersion of 
water-in-oil (Very Fine Dispersion w/o). 

When the water is continuous phase, most of the discrete 
oil droplets are diffusely distributed in the water, which shows 
the dispersion of water-in-oil droplet flow; when the flow 
velocity of water phase is high (more than 0.9 m/s), the size 
and distribution state of oil droplets become very regular, and 
the slip phenomenon of two phases can be ignored, which is 
the fine dispersion of water-in-oil droplet flow; as the oil 
volume increases, the oil content increases, and the 
concentration of oil droplets increases to make them coalesce 
Merge phenomenon to form a large block of oil, at this time 
for the water-in-oil transition flow type. When the oil content 
increases again, the oil phase dominates, i.e. oil is the 
continuous phase and water is the dispersed phase, and the 
transitional flow pattern of water-in-oil transforms into the 
transitional flow pattern of oil-in-water, and gradually 
transitions into the dispersed flow of water-in-oil until the fine 
dispersed flow of water-in-oil with the further increase of the 
flow rate of oil phase. It is generally accepted by the 
researchers. 

Harsan and Kabir et al. used the volume fraction of the 
dispersed phase 0.25o 0.25o  to determine the transition from 
small spherical bubbles to large oil bubbles and segment plugs 
earlier, and Lihua. Huang et al. used the ratio δ of the particle 
size of Sauter's average droplet to the pipe to classify the flow 
pattern: 

 
Table 1. Reference range for determining the flow pattern 

for δ 
Flow 

pattern 
Dispersion 
Flow (DF) 

Bubble flow 
(BF) 

Stretch flow 
(SF) 

Range of δ >0.27 0.27~0.50 0.50~1.00 
 
The maximum droplet size dmax is used to estimate the 

Sauter mean diameter of the droplet: 
 

3 2
32 max/d /i i i i dn d n d d kdmax /d k/ dmax /       (1) 

 
Azzopardi and Hewitt et al (1997). suggested a range of 

values for Kd from 1.5 to 3. 
2.1.1. Calculation of the maximum droplet size 

(1) Laminar flow 
For the case where the viscosity of the viscous continuous 

phase is much larger than that of the dispersed phase, Taylor 
(1934) and Arivos (1978) proposed that elongated droplet 
rupture in axisymmetric strain motion can be used to estimate 
droplet size, and the maximum droplet size can be calculated 
using the following equation, which considers the maximum 
droplet size to be related to the surface tension of the 
continuous phase liquid:  

 
1/6 1/6

max 0.296 0.074 ; 1c c d

c d c m d c

d
D D U

1
DD dddddd

(2) 

 
 is the average shear rate of the fluid 4 /mU D4 /mU D/m μ is 

the viscosity of each phase of the fluid, and the subscripts c, 
d refer to the continuous and dispersed phases, respectively. 

(2) Turbulence 
In 1955 Kolmogorov and Hinze proposed that the size of 

the droplet depends on thecritical Weber number: 
 

       max
crit [1 ( )]dWe C F On         (3) 

 
For a dilute dispersion flow, taking into account the wall 

friction coefficient of the tube wall, Kubie and Gardner 1977 
proposed that: 

 
0.6 0.4max

max max0.55 ; 0.1c ko
o

dd We f d D
D maxk dmaxmaxmax (4) 

 
£k is the Kolmogorov microscale and 0.1D represents the 

inertial subscale (the length scale containing the energy of the 
vortex). N. Branner presented the limitations of the Kubie et 
al. model, supplemented the model for dense dispersions, and 
proposed the H-model for calculating the maximum droplet 
diameter as follows: 

 
0.6 0.08max 1.88 c c

o

d We Re
D

          (5) 

 
0.6 0.4

0.6 0.08
max 7.61 1

1 1
d d d

H c c
d c d

d C We Remaxdmax 7 67.6 H cCH c
0 6C 0 60 60.6667.67.6

0.6 0.40.6

d d d1d d d1 d d1111
1

1
1d c d111

1   (6) 

 

max max maxo
d Max d dmax maxmaxd Max d dmax mmaxmax            (7) 

 

HCHCH  is an adjustable constant, (1)HC O(1)HC OH , the model is 

applicable to the case of 0.7
max1.82 0.1cRe dmax 0.1maxdmmaxm   or 

Rec>2100, For max 0.1dmax 0.1dm  , N. Branner proposed the K 



 

51 

model: 
 

1 0.2
max max0

30 ; 0.1c cd We Re d1 0 2
max max0

0.1maxc cd We Re d1 0 2
max 30 ;1 0.2

m        (8) 

 

1 0.2
max 174

1
d

K c c
d

d C We Remaxdmax 177417 ddd

d

      (9) 

 

max max maxo
d Max d dmax maxmaxd Max d dmax mmaxmax       (10) 

 
According to Hinze (1955), if the dispersed phase viscosity 

is large, then the effect of the dispersed phase viscosity μd on 
the maximum particle size dmax must be considered, and the 
effect of the dispersed phase viscosity is expressed by the 
Ohnesorge number, and the On number is increased 

0.6[1 ( )]F On  at the right end of both the k model and the 
H model when the On number is not negligible 

If the viscosity of the dispersed phase is much greater than 
that of the continuous phase, consider the viscosity of the 
dispersed phase (o/w) Paul and Sleicher suggest using: 

 
0.5 0.72

max 1 0.7c c c c d cU d U UC   (11) 

 
Where C=38~43 

2.1.2. Core-annular flow  
In 1996 Joseph et al. proposed the use of the effective 

viscosity method to determine the presence of a central 
annular flow: 

 

1, eff 2, eff  0.5          (12) 

 
μeffis the effective viscosity of the fluid, ε is the volume 

number of the core, and the subscripts "1" and "2" are the core 
and annular phases 

If the case of perfect core-annular flow is considered (two-
phase fluid is assumed to be free of any interference) the 
following equation can be used: 

 

2 2
1 2 2 2

2

0.0005  for 2000J DJ D (13) 

 
ρ2 is the density of the annular fluid and J2 is the apparent 

flow velocity of the fluid. 
In 2001 Bannwart applied the Eötvos number to 

supplement the criterion for the appearance of annular flow 
based on Branner's idea: 

 
2

8
4
gD

 

3. Pressure drop theory 
3.1. Homogeneous flow model 

Homogeneous flow models usually treat the two-phase 
mixture as a homogeneous single-phase fluid, and the 
frictional pressure drop is a key element in the accuracy of the 
total pressure drop calculation. 

For a fully expanded dispersive flow in a vertical pipe and 
neglecting acceleration gradients, the total pressure gradient 
including gravitational and frictional pressure gradients can 
be calculated as (Brauner, 1998): 

 

tp g f

dp dp dp
dz dz dz

          (14) 

 
where the "±" symbols correspond to vertical upward or 

downward flow, respectively.  
The gravitational pressure gradient is: 

 

M
g

dp g
dz

            (15) 

 
Frictional pressure gradients are: 

 
2

2 M
tp M

f

Udp f
dz D

         (16) 

 
where ftp is the friction coefficient, ρM is the density of the 

mixture, UM is the flow rate of the mixture, D is the diameter 
of the pipe 

The density of the mixture is determined by the weighted 
average of the volume fractions: 

 

M o o w w              (17) 

 
ε is the volume fraction, and the subscripts o and w 

represent the oil and water phases, respectively 
The friction coefficient ftp is calculated according to the 

Blasius equation: 
 

Re n
tp mf C                (18) 

 
For turbulent flow C=0.079, n=0.25 and laminar flow C=16, 

n=1. 
If the effective viscosity of the mixture is known, the 

Reynolds number of the mixture Rem is: 
 

Re M M
m

e

DU
             (19) 

 
In 1998 Flores et al. proposed a method to calculate the 

two-phase friction coefficient by considering the difference 
between the water-dominated and oil-dominated flows. The 
constants C and n in the above equation will be determined 
from experimental data by redefining Rem: 

 



 

52 

Re M M
m

c

U D
             (20) 

 
where the subscripts “M” and “c” refer to the mixture and 

the continuous phase. 
3.1.1. Effective viscosity model 

Most current calculations of oil-water flow use the Dukler 
volume factor weighting method to calculate viscosity: 

M o o w w

Oil-water two-phase flow differs from gas-liquid two-
phase flow in that the viscosity of the dispersed phase in oil-
water two-phase flow plays a large role in transmitting shear, 
and the deformation of the droplet near the wall (by shear) 
thus affects the flow of the droplet in the center of the pipe. 
Some models have been proposed such as Brinkman (1952) 
and Roscoe (1952) model: 

 
2.51e

d
c

            (21) 

 
Pal model (2001): 

 
1.5

1
3

1
3

2 5
9
8 12 5

e d

c c de

c d
d

c

K

K
(22) 

 
3.1.2. Calculation of the holding rate 

Due to the different densities of the two-phase fluids, the 
accuracy of the calculation of the holding rate directly affects 
the calculation of the pressure drop. The two mainstream 
methods for calculating the holding rate are the homogeneous 
model and the drift flow model. 

1. Homogeneous model (ignoring the slip between the two 
liquid phases): 

The in-situ holding rate is determined by the input volume 
flow rate of the two liquids 

 

; 1 ;ds cs
d c d m ds cs

m m

U U U U U
U U

ds c1d m; 1 ;1U Uds c1 UmU Uc d;; 1 ;ds ; 1; 1 ;U  (23) 

 
where U is the input volume flow rate and the subscripts d, 

c, ds, cs, and m refer to the volume fraction of the dispersed 
phase, the volume fraction of the continuous phase, the 
volume flow rate of the dispersed phase, the volume flow rate 
of the continuous phase, and the total flow rate of the mixture, 
respectively. 

2. Drift flow model Zuber-Findlay (1965): 
 

sw
w m e

w

U C U U
Y

             (24) 

 
 

e w1 NU U Y              (25) 

The drift flow model mainly considers the slip 
phenomenon due to the density difference between the two 
phases of the liquid. 

Where:Usw is the apparent velocity of water phase; Cw and 
N are the phase distribution parameters and particle size index, 
respectively; U∞ is the limiting rise velocity of water droplets 
in oil continuum medium.Co is a distribution parameter which 
explains the droplet velocity and concentration distribution. 
For a uniform droplet concentration, Co = 1, Co > 1 when the 
droplet tends to flow in the center, and Co < 1 when the droplet 
concentration is higher near the wall. the value of N depends 
mainly on the droplet size. hassan and Kabir (1990) and and 
Flores et al. (1997) suggest that for large droplets (of the order 
of tube diameter), N ≈ 0, while for liquid-liquid dispersions, 
N = 1.5 ~ 2.5. 

Harmathy (1960) proposed a method to calculate the 
limiting rise velocity of droplets: 

 
1/4

2

| |1.53
c

gu             (26) 

 
where σ is the surface tension of the fluid. Δρ is the density 

difference between the two phases, and ρc is the density of the 
continuous phase. g is the acceleration of gravity. 

Drift flow models applicable to specific flow types have 
been proposed as follows: 

 
1.Nicolas model (suitable for dispersion flow): 

 
1so

m w
w

, 0.5
1

NU U U Y N
Y

      (27) 

 
 

2.Flores model (suitable for disturbance flow): 
 

2.5s0
m w

w

1.5so
m w

w

1.038 0.142 D o/w
1

1.1045 0.142 CF o/w
1

U U Y
Y

U U Y
Y

   (28) 

 
3.1.3. Two-fluid model 

The two-fluid model was originally proposed by Taitel and 
Dukler for the stratified flow pattern, based on the assumption 
of separate flow of the two phases and the presence of shear 
stress at the phase interface; the two-fluid model for o/w 
dispersed flow was proposed by Poesio et al. based on the 
two-fluid model for stratified flow: 

 

, / 0 sin 0o D o w o
dpA AF A g
dz

    (29) 

 

, / sin 0W W W D o w W W
dpA S AF A g
dz

(30) 

 
Where A and SW denote the cross-sectional area of the pipe 



 

53 

and the water wall wetting perimeter, respectively, τw is the 
wall shear stress of water, and θ is the inclination angle of the 
pipe. 

The above two equations are combined to eliminate the 
pressure drop, so that the o/w dispersion flow model can be 
introduced: 

 

0 , /sin 0w w W w D o wA g S F A   (31) 

 
Similarly a two-fluid model for w/o dispersive flow can be 

introduced: 
 

, / sin 0W D w o W W
dpA AF A g
dz

     (32) 

 

, / sin 0O O O D w o O O
dpA S AF A g
dz

 (33) 

 
Eliminate the pressure drop to obtain the w/o dispersion 

flow model: 
 

0 0 0 0 , /0(1 ) sin (1 ) 0W D wA g S F A (34) 

 
Assuming that the dispersed phase is not in contact with the 

tube wall: 
 

0WS S D                   (35) 

 
Assuming that FD is the drag force of the phase of the 

continuous relatively spherical dispersed phase: 
 

3
4

d c d c
D c D

d

U U U U
F C

d
       (36) 

 
where CD is the drag coefficient, Ud is the velocity of the 

dispersed phase, Uc is the velocity of the continuous phase, 
and dd is the effective diameter of the dispersed phase droplet. 
The drag coefficients are: 

 
0.68724 1 0.15 800

0.44 Re 800
D

Re Re
C Re

800
(37)

 
where the droplet Reynolds number: 

 

c d c d

m

U U d
Re             (38) 

 
Mixture viscosity using Einstein's suspension viscosity 

formula:
 

c d1 2.5m H             (39) 

 

3.2. Core-annular air flow pressure drop 
calculation method 

At present, for the calculation of annular air pressure drop, 
the most used is the two-fluid model. This model takes into 
account the interface interference between the phase-phase 
and phase-tube wall, and the error of pressure drop pre is 
relatively small. 
3.2.1. Perfect Core-annual Flow 

Taitel and Dukler first proposed a perfect core-annular flow 
two-fluid model for interfacial smoothing, and the pressure 
drop calculation model was obtained after eliminating the 
two-phase pressure drop phase: 

 
2 1 2 12 2

f 4 2 2 4 2

(1 )[1 (1 )] (1 )128 128
(1 )1 (1 ) 1 (

 
1 ) 1

g m gQ Q
D m m D (40) 

 
where μ ρ Q,g are viscosity, density, flow rate, and 

gravitational acceleration, respectively. ε is the volume 
number of the core, and the subscripts "1" and "2" are the core 
and annular phases. 2 1/m  

ε is calculated by the following equation: 
 

i

i i

9 5
24 2 2

1 i,0 2 ref 
3 ln(1 ) (1 ) 0

2 1

i
i

i

nn n
nn nJ s J V (41) 

 
 

i

i
i

1
21 2

2 zz 1 2 i 2
ref i z

2 2z 1 2

| |
n

nn g DDg
V a g D

g
(42) 

 
gz is the axial component of gravitational acceleration, ai 

and ni take the values of: 
 

Re 2000 : 16 1
2000 : 0.079 0.25

k k k

k k k

a n
Re a n

  (43) 

 
3.2.2. Considering wave characteristics and circulating 

air turbulence effects 
2001 Bannwart predicts the pressure drop during oil-water 

annular flow through a vertical pipe by means of a 
phenomenological model that takes into account the effect of 
turbulence in the annular fluid and wavy interface, and also 
the effect of buoyancy on the vertical system, where the 
pressure drop gradient can be expressed as: 

2
m m

f 2 1
m

(1 )
2

n
JD Jb C g

D
(44) 

 
The core volume fraction ε is calculated by the following 

equation: 
 

2
1 i,0 2 ref(1 ) (1 ) 0mJ s J cV    (45) 

 
where c, m are adjustable parameters that can be adjusted 

experimentally to suit the wave velocity data, and Bai (1995) 
determined that the optimal data: 



 

54 

i,o 1.5 0.02 2ms c         (46) 

 
The constants b, n, c are the parameters to be adjusted in 

the experiment, n=0.25 for turbulent flow, n=16 for laminar 
flow (turbulent and laminar flow in smooth pipe walls), 
Bannwart recommends taking n=0.25, b=0.257, C=0.159 

Core-annulal flow mixture density: 
 

m 1 2(1 )            (47) 

 
Core-annular flow mixture viscosity: 

 

m 1 2 2

1 1 1
        (48) 

 
The two-fluid model and the uniform flow model are based 

on different assumptions, from the assumptions, the uniform 
flow model is suitable for oil-water two-phase flow where one 
phase is continuous, which is applicable to the dispersed flow 
type, while the two-fluid model is suitable for the case where 
there is a clear interface between the two phases and both are 
in a continuous state. They compared the pressure drop data 
predicted by the uniform flow model and the two-fluid model 
in an inclined pipe and found that the uniform flow model 
could predict the inverse phase point while the two-fluid 
model could not, but the two-fluid model was more accurate 
in terms of the accuracy of the pressure drop prediction. 

References 
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