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Engineering, Technology & Applied Science Research Vol. 6, No. 6, 2016, 1258-1263 1258  
  

www.etasr.com Suarez-Dominguez et al.: Homogeneous and Stratified Liquid-Liquid Flow Effect of a Viscosity Reducer 
 

Homogeneous and Stratified Liquid-Liquid Flow 
Effect of a Viscosity Reducer:  

I. Comparison in parallel plates for heavy crude  
 

E. J. Suarez-Dominguez E. F. Izquierdo-Kulich A. Rodríguez-Valdez 
Laboratorio de Mecanica y Materiales 
FADU Universidad Autonoma de 

Tamaulipas 
Tampico, Tamaulipas, Mexico 
Edgardo.suarez@uat.edu.mx 

Departamento de Quimica-
Fisica  

F. Q. Universidad de la Habana 
La Habana, Cuba 
elenaik@fq.uh.cu 

Facultad de Ingenieria  
Universidad Nacional Autonoma de 

Mexico 
Ciudad de Mexico, Mexico 
arodriguezv@ii.unam.mx 

F. Solorio-Ordaz A. E. Chavez-Castellanos A. Palacio-Perez 
Facultad de Ingenieria 

Universidad Nacional Autonoma de 
Mexico 

Ciudad de Mexico, Mexico 
fjso@unam.mx 

Facultad de Ingenieria 
Universidad Nacional Autonoma 

de Mexico 
Ciudad de Mexico, Mexico 

aechavez@unam.mx 

Facultad de Ingenieria 
Universidad Nacional Autonoma de 

Mexico 
Ciudad de Mexico, Mexico 

apalaciop@ii.unam.mx  
 

 

Abstract—Production of heavy crude oil in Mexico, and 
worldwide, is increasing which has led to the application of 
different methods to reduce viscosity or to enhance 
transport through stratified flow to continue using the 
existing infrastructures. In this context, injecting a 
viscosity improver that does not mix completely with the 
crude, establishes a liquid-liquid stratified flow. On the 
basis of a parallel plates model, comparing the increase of 
flow that occurs in the one-phase case which assumes a 
complete mixture between the crude and the viscosity 
improver against another stratified liquid-liquid (no 
mixing between the oil and compared improver); it was 
found that in both cases there is a flow increase for the 
same pressure drop with a maximum for the case in which 
the flow improver is between the plates and the crude. 

Keywords- stratified flow; velocity profile; heavy oil   

I. INTRODUCTION  
Transporting heavy oil has gained significant interest due to 

high viscosity and operational problems such as high pumping 
costs, heavy fractions deposition and other phase changes [1-3]. 
The study of velocity profiles is very important because these 
effects are directly related to the pumping power and transport 
costs [4], as well as the design of monitoring pipeline [5]. 
Although frequently emulsions are used to improve pipeline 
transport [6, 7] and viscosity reducers are injected [8], few 
studies have examined the effects related to the shape of the 
injected fluid and dosing of products in the crude oil. Most of 
the time the research focuses on pump dimensioning [9, 10] 

limiting the capacity to use the same infrastructure for heavier 
fluids. 

It is important to consider the mechanism that determines 
the transport of a compound injected to a fluid to reduce its 
viscosity.  This can determine the effectiveness of its action 
and interaction effects that are set at the molecular level and 
that modify the rheological properties. [11-14]. In this regard, it 
has been found that the injection of a liquid into the inner 
periphery of a pipeline can substantially improve the flow of 
transported fluid [15, 16]. Various numerical methods [18, 19] 
and theoretical simulation models have been proposed to 
explain this effect [17]. Studies have also been carried out 
focusing on the involvement of surface tension exerted by the 
flow improvers and leading to an oil phase dispersion increase 
and the consequent influence in viscosity [20-22]. However, 
there are not papers reporting the effects of injection of other 
liquids on flow enhancement, as in the case of highly viscous 
fluids such as heavy and extra heavy crude, that generally 
maintain a laminar flow. A liquid-liquid stratified flow can be 
established if mechanical mixing between the oil and the 
chemical is not presented, as when oil extraction is performed 
by the reservoir natural pressure, or the flow improver is 
injected at a later point or away from the pump discharge.  

The aim of this work is to model theoretically the behavior 
of a system consisting of crude oil and flow improver moving 
between two parallel plates considering three different 
conditions: i) complete mixing between the two fluids, so that 
we have a single-phase system of lower viscosity, ii) no mixing 
between the crude oil and the flow improver, establishing a 
stratified system with only one interface between them iii) 



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www.etasr.com Suarez-Dominguez et al.: Homogeneous and Stratified Liquid-Liquid Flow Effect of a Viscosity Reducer 
 

establishing a stratified flow where one of the fluids is in the 
center and the other in the duct walls, creating two interphases. 

II. MATHEMATICAL MODEL 
The mathematical model used to predict the velocity 

profiles and the total flow related with the volume fraction of 
the flow improver is based on the equations of continuity and 
change in cylindrical coordinates, in all cases considering i) 
Newtonian flow with constant density and viscosity, ii) 
isothermal system, iii) laminar flow regime and iv) steady state. 
Schemes corresponding to the cases to be considered are shown 
in Figures 1-3. 

The distance x between the plates is to be considered much 
less than the width Y, so that one can assume that the 
momentum transport occurs preferentially in the x direction, 
while the liquid moves in the direction z. To quantify the effect 
of the flow improver on total flow is determined by the 
relationship: 

12

1

Q
q=

Q
  and  

12 1 2Q =Q +Q            (1)

 
where the subscript 1 refers to the crude and the subscript 2 

refers to the flow improver. Q12 represents the homogenous 
mix of the two substances. Q1 represents the flow of oil; q 
allows to quantify the flow increase because of the flow 
improver injection, for the same pressure drop. 

 

 
Fig. 1.  Case A: single-phase flow consisting of the mixture of oil and the 
flow improver. 

 

 
Fig. 2.  Case B: no mixing between the oil crude and the flow improver, so 
that the system behaves as a two-phase stratified flow with a single interface 
between the fluids 

 

 
Fig. 3.  Case C: one of the fluids is injected in the center, such that 
stratified flow is developed with two interfaces between the fluids 

A. Case A 
When a complete mix is produced between crude oil and 

the flow improver, either via a mixing tank or an injection prior 
to pumping, blend viscosity decreases.  Many empirical 
equations exist to describe this effect. The ones proposed by 
Lederer: 

2
12 2

1

μ
lnμ =g ln +1

μ
      (2)                                                  

( )
2

2
2 2

σy
g =

σ 1-y +y

σ=0.5-0.004T

    (3)             

 
and Kendall and Monroe:

 
3

1

22
3

1

11
3

1

12  xx          (4)   
 

respectively, are classical results, which can be used for 
mix of oil and improver flows [23, 24]. In this paper, (2) is 
used in order to describe the system behavior when a complete 
mix is produced because of experimental results showed this is 
one of the higher correlation mixing rule [25]. 

From the considerations set, we obtained that the velocity 
profile established in this system is described by the partial 
differential equation: 

2

2

p d v
- =μ

z dx

¶

¶
         (5)                                                   

where p is pressure and v is the fluid velocity. Defining the 
dimensionless parameter γ:  

x
γ=

α
                       (6)                                                  

and: 

2 p
Φ=α

z

¶

¶
            (7)                                                   

equation (5) is written in the form: 
2

2

d v
-Φ=μ

dγ
        (8)                                                      

with the boundary conditions: 

( )
( )

v 0 =0

v 1 =0
             (9)                                                    

Differential equation (8) has an analytical solution, and 
considering (9), velocity profile is obtained: 

( )2
1 Φ

v= γ-γ
2 μ

       (10)                                                  

From the velocity profile, volumetric flow is obtained for 

the single-phase case ( )
Y 1

2

0 0

1 Φ
Q= γ-γ dγdy

2 μ

æ ö÷ç ÷ç ÷÷çè øò ò
 and 

therefore: 



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www.etasr.com Suarez-Dominguez et al.: Homogeneous and Stratified Liquid-Liquid Flow Effect of a Viscosity Reducer 
 

1 Φ
Q= Y

12 μ
       (11)                                                                                                                  

 
Taking into account equations (1), (2) y (11) yields: 

( )
( ) ( )( )

1

2 2

1
2 2

0.5-0.004T y μ
μ1-y 1- 0.5-0.004T +y

μ
q=

exp ln +1
æ ö÷ç ÷ç ÷ç ÷çè ø

    (12)                                                                       

B. Case B 
The system of differential equations describing spatial 

velocity profile of a stratified flow with a single interface 
between the fluids is given by: 

2

1

1 2

d v
-Φ=μ

dγ
 for 0<γ<α            (13) 

2

2

2 2

d v
-Φ=μ

dγ
 for α<γ<1          (14) 

Near of the wallsm it is considered that velocity is zero, 
therefore: 

( )
( )

1

2

v 0 =0

v 1 =0
  (15.a) 

The others boundary conditions are stablished taken into 
account the continuity of velocity and shear stress on interface 
α, in such way that: 

 
( ) ( )
( ) ( )

1 2

1 2

v α =v α

τ α =τ α
                    (15.b) 

where  is fluid shear tensor, defined: 
dv

τ=-μ
dx

                               (16) 

The differential equations (13) and (14) have, in this case, 
the solution: 

( )
1 22 16

2

1

γ1
μ2v γ =C +C γ- Φ    for 0<γ<α       (17)                                                     

 ( )
2 26 15

2

2

1
2

γ
μv γ =C +C γ- Φ   for α<γ<1       (18) 

where: 

( )( )
( )

( )( )
( )

( ) ( )
( )

22

2

1 1 2

15

2 1 1 2

2

1 1 2

16

1 1 1 2

1 2

26

2 1 1 2

C =0

μ -α μ -μΦ
C =

2μ μ -α μ -μ

μ -α μ -μΦ
C =

2μ μ -α μ -μ

μ -μ α α-1Φ
C =

2μ μ -α μ -μ

                       (19) 

To find the  value that corresponds to the interphase 
position, the flow values are determined as a function of : 

( )( )
( )

22

1 1 2

1

1 1 1 2

μ -α μ -μ1 α 3
Q = YΦ -α

6 μ 2 μ -α μ -μ

æ ö÷ç ÷ç ÷ç ÷çè ø
     (20) 

( ) ( )
( )( )
( )( )

( ) ( )
( )( )

( )
2

1 1 2 1 2 2
2

2 1 1 2 1 1 2

μ -α μ -μ μ -μ α α-11 YΦ 3
Q = 1-α 1+α +3 - 1+α+α (21)

6 μ 2 μ -α μ -μ μ -α μ -μ

æ ö÷ç ÷ç ÷ç ÷ç ÷÷çè ø
 

and from defining the volume fraction of the injected flow 
improver y2 = Q2/Q12 relative to the total two-phase flow,  is 
cleared from the equation: 

( )
( )2 22 1 1 2 1 22

1 4 2 4 3 2
2 2 1 2 1 2 1 2

μ -2αμ +4αμ +α μ -α μy
=μ α-1

1-y α μ -μ α μ -2μ α μ +3μ α μ
     (22)

 

After determining the value of the position of the interface 
for a fraction of the injected liquid ratio, relationship between 
the total biphasic flow and single phase flow of crude oil is 
determined:

( )( )
( )

( )
( )

( )( )
( )( )

( ) ( )
( )( )

( )

2
2 1 1 2

1 1 1 2

2
1 1 2 1 2 2

2 1 1 2 1 1 2

μ -α μ -μα 3
q=2μ -α

μ 2 μ -α μ -μ

μ -α μ -μ μ -μ α α-11-α 3
+2μ 1+α +3 - 1+α+α (23)

μ 2 μ -α μ -μ μ -α μ -μ

æ ö÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷÷ç ÷çè ø
æ ö÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷÷ç ÷çè ø

      

C. Case C 
Following the same methodology as in Case B, but 

considering two interfaces among fluids, the velocity profile 
that is set for two-phase flow, where one fluid is in contact with 
both plates, (Figure 3) is obtained: 

( )
A 22 16

1
2

2

A

γ
μv γ =C +C γ- Φ  for 0<γ<α      (24) 

( )
A 22 16

2

A

1
2

γ
μv γ =C +C γ- Φ  for β<γ<1        (25) 

( )
B 26 15

2

B

1
2

γ
μv γ =C +C γ- Φ  for α<γ<β      (26) 

where: 

( )( )( )

( )

22

B A B

15

A B

16

A

A B

26

A B

C =0

μ + μ -μ α+βΦ
C =

2 μ μ

1 Φ
C =

2 μ

αβ μ -μΦ
C =-

2 μ μ

    (27) 

In this case there may be two possibilities. First is that the 
flow improver moves inside (in the centre) and the crude is in 
contact with both plates. In which case we obtain: 



Engineering, Technology & Applied Science Research Vol. 6, No. 6, 2016, 1258-1263 1261  
  

www.etasr.com Suarez-Dominguez et al.: Homogeneous and Stratified Liquid-Liquid Flow Effect of a Viscosity Reducer 
 

( ) ( ) ( )( )( )
( ) ( ) ( )( )

2 2

2 1 1 22

2 2
2

2

1
μ β-α 3μ α+β -2αβμ + μ -3μ α +βy

=
1-y β-1 2β+1 -α 2α-3

        (28) 

( ) ( ) ( )( )
( ) ( ) ( )( )( )

2 2

2 2

2 1 1 2

2

q= β-1 2β+1 -α 2α-3

1
+ β-α 3μ α+β -2αβμ + μ -3μ α +β
μ

       (29) 

second case is if oil flows through the inside and the flow 
improver is in contact with both plates: 

( ) ( ) ( )( )
( ) ( ) ( )( )( )

2 2

2
1 2 2

2 1 1 2 2

β-1 2β+1 -α 2α-3y
=μ

1-y β-α 3μ α+β - β +α 3μ -μ -2αβμ
        (30) 

( ) ( ) ( )( )( )

( ) ( ) ( )( )

2 21
1 1 2 2

2 1

2 21

2

μ 1
q= β-α 3μ α+β - β +α 3μ -μ -2αβμ

μ μ

μ
+ β-1 2β+1 -α 2α-3
μ

           (31)

 

III. RESULTS AND DISCUSSION 
The results obtained with the model were based on a crude 

with viscosity 1 = 35 Pa.s and a flow improver with viscosity 
2 = 0.007 Pa.s. To make the study, the velocity profile was 
analyzed considering Φ=1 for a flow improver volume fraction 
of 0.03. Results predicted by the model are shown in Figures 
4.a, 4.b, 4.c y 4.d. It is appreciated that when there is mixing 
between the two liquids (Case A) the speed of transport is 
increased significantly compared with that achieved in the 
absence of flow improver. If there is no mixing, we see a well-
defined interface between the two flows, where the speed 
reached in the liquid layer corresponding to the flow improver 
is significantly higher than the phase of crude oil (Case B), but 
is lower than that reached with complete mixing. When the 
flow improver is injected in the center (Case C.1), even when 
the speed is increased, this increase is less pronounced than the 
corresponding liquid-liquid stratified flow with a single 
interface, while a further increase occurs when the flow 
improver is in contact with both plates.  

As shown in Figure 5, the model predicts different behavior 
between the volume fraction and flow improvement is achieved 
for the same pressure drop depending on the form of injection. 
Thus very significant increases are achieved when the enhancer 
is in contact with the walls. This apparently is because it has 
significantly lower viscosity than crude, so that the shear stress 
on the wall of the plates decreases, which is manifested in 
increased flow for the same pressure drop. In correspondence 
with this, the smallest increase in flow is achieved for the case 
where the crude oil is in contact with both plates. However, this 
two interfaces model between fluids has two limiting factors: 
the first is that there is no explicit criteria established by the 
asymmetry of the same (distance from each of the interfaces), 
and it only has a flow ratio to determine the height of the 
interface, so that the asymmetry must be defined to make other 
predictions.   

The second limitation is that it is necessary to define under 
what conditions interfaces can be established because in 
horizontal ducts, the density difference between both flows 
favors the formation of a stratification. When Cases A and B 
are compared, it can be seen that in both cases there is an 
increased flow.  Although this increase is more marked when 
mixing, it is still advisable to inject the flow improver before 
pumping.  The results in terms of flow profiles are consistent 
with those obtained in [16, 26-27]. The maximum possible 
flow is the flow improver and oil crude substituted in (1), such 
that q=5000; it can be seen that the better effect is around 95%.  

 

  
                      

   

Fig. 4.  Velocity profiles predicted by the model. Case A: complete mix 
with the fraction of flow improver as parameter. Case B: stratified flow with a 
single interface. Case C: stratified flow with two symmetrical interfaces. 1 the 
oil crude is located in contact with both plates; 2. The flow improver is in 
contact with both plates 

IV. CONCLUSION 
The injection of a flow improver always causes an increase 

of fluid flow for the same pressure drop, where the increased 
magnitude order, from lowest to highest, is: stratified flow with 
two interphases, which is crude oil that is in contact with the 
walls; stratified flow with an interface between the two flows; 
complete mixing between the two fluids with the establishment 
of a single phase flow of lower viscosity and two interfaces 

Case A   Case B    

Case C.2   Case C.1   



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stratified flow where the flow improver is in contact with the 
plates. These results indicate that the way in which the 
injection of the flow improver and the existence of mechanical 
mixing with oil crude occur influences the overall enhancement 
effect. The model has as limitation that it cannot explicitly set 
the conditions under which the formation of two interfaces 
occurs, so that oil flows through the center, which may 
reasonably be expected if similar fluids densities are obtained. 
It is necessary to develop technologies that allow the injection 
to be made in the wall with subsequent increase in improved 
operating systems extraction and transportation of crude oil. 

 

0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.020 0.022 0.024 0.026 0.028 0.030
13.2

13.4

13.6

13.8

14.0

14.2

14.4

14.6

14.8

15.0

y

q

 

0.005 0.010 0.015 0.020 0.025 0.030

6.56

6.57

6.58

6.59

6.60

6.61

6.62

6.63

y

q

 

0.005 0.010 0.015 0.020 0.025 0.030

1.840

1.845

1.850

1.855

1.860

y

q

 

0.005 0.010 0.015 0.020 0.025 0.030

4756

4758

4760

4762

4764

4766

4768

y

q

 
Fig. 5.  Behavior of flow increase with respect to the volume fraction of 
improver 

ACKNOWLEDGMENT 

The authors would like to thank the Consejo Nacional de 
Ciencia y Tecnología (CONACYT) of Mexico, project 
number 166923 for the support and Prof. Lynda Kay Deckard 
for editing this paper. 

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Case A   

Case B   

Case C.1   

Case C.2   



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