Well test analysis on pressure of viscoelastic polymer solution with variable rheological parameters*

doi:10.4236/ns.2010.212161

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Vol.2, No.12, 1327-1332 (2010)

doi:10.4236/ns.2010.212161

Natural Science

Well test analysis on pressure of viscoelastic polymer

solution with variable rheological parameters*

Hongjun Yin#, Weili Yang, Siyuan Meng, Ming Cai

Key Laboratory of Enhanced Oil and Gas Recovery Ministry of Education; Northeast Petroleum University; Daqing, China;

#Corresponding Author: [email protected]

Received 27 September 2010; revised 28 October 2010; accepted 2 November 2010.

ABSTRACT

According to the behavior that the polymer so-

lution has both the characteristics of viscosity

and elastic properties, the transient flow mathe-

matical model considered the viscoelasticity of

the polymer solution has been established. The

model, in which the variation of the rheological

parameters during the seepage flow has been

also taken into consideration, has been solved

using finite-difference method. The type curves

have been plotted. The influence of some prop-

erties of polymer solution including the viscoc-

ity, the elastic properties and the rheological

parameters has been analyzed. Compared with

the curves of the power-law fluid, it is shown

that the pressure derivative curve considering

the elasticity of the polymer solution upwarps

less at the radial flow regime. Besides, it will

come down as the variation of the rheological

parameters, which is quite different from the

case regarding them as constants. Therefore, in

well test analysis on pressure of polymer solu-

tion, it’s nec essary to consider th e elasticit y and

the variation of the rheological parameters.

Keywords: Polymer Solution; Viscoelastic;

Relaxation Time; Rheological Parameters

1. INTRODUCTION

The polymer solution used in the oil field is a typical

kind of non-Newtonian fluid and its rheological property

in the porous media is very complicated because of the

effect of shear degradation, deconcentration, adsorption

and entrapment.

Many domestic and foreign scholars have studied its

rheological property and established several rheological

model. The initial researches on the flow behavior of the

non-Newtonian fluid are mainly focus on the power-law

fluid. In most well test analysis on pressure of polymer

solution, it is also assumed that the polymer solution is

pure viscous fluid, only the shear viscosity is considered

and the rheological parameters are always treated as

constants.

But a large number of experiments have demonstrated

that polymer solution has viscoelastic behavior and the

rheological parameters changes in the seepage flow

process. The viscosity of the polymer solution will de-

cline as the raising of the shear rate at a relatively lower

Darcy velocity. In the flow event, the viscosity is the

dominant influential factor and the elastic property can

be neglected. In addition, the rheological property of

polymer solution can be expressed by using pseudo-plastic

power-law model. However, once the Darcy velocity

exceeds the critical value，the viscosity will raise with

the increasing of the shear rate, the elastic effect will

enhance gradually and its influence will be too signifi-

cant to be ignored.

Reference [1] has declared that the effective viscosity

of polymer solution is composed of the individual con-

tributions of shear and strain viscosity. The elasticity

behavior and rheological property of polymer solution in

porous media under the conditions of reservoir flow rate

has been studied in [2]. A power-law fluid viscoelastic

semi-empirical model which could describe the viscoe-

lastic effect of polymer solution in porous media has

been developed in [3]. The researches on the viscoelas-

ticity have promoted the development of percolation

theory. Reference [4] has conducted numerical simula-

tion study for improvement of polymer flooding by vis-

coelastic effect. Reference [5] has studied the flow be-

havior of viscoelastic fluid, power law fluid and Newto-

nian fluid in pore throat by numerical method. Differ-

ence mathematical models of viscous-elastic polymer

solution have been established in [6] and [7] from dif-

*Project supported by the National Science Foundation of China (Grant

o: 50874023) and by

he Heilongjiang Provincial Science and Tech-

nology Plan Project (Grant No: GZ09A407) and by the Research Pro-

gram of Innovation Team of Science and Technology in Enhanced Oil

and Gas Recovery (Grant No: 2009td08).

H. J. Yin et al. / Natural Science 2 (2010) 1327-1332

1328



ferent angles. The influence of different factors on the

pressure of the formation near the injection well is ana-

lyzed too. But the variation of rheological behavior was

seldom taken into consideration and the consistency co-

efficient and the power-law index were always treated as

constants. Besides, the researches on the well test of

viscoelastic polymer solution were seldom.

Therefore, in this paper, the relationship between elas-

tic viscosity and shear viscosity has been insisted in this

paper. In addition, the variation of the rheological pa-

rameters in the flow event has been taken into considera-

tion. What’s more, the expression of the apparent viscos-

ity of polymer solution which considered the viscoelas-

ticity and the changing parameters of polymer solution

has been developed. Then the non-dimensional radial

instable flow mathematical model has been established

and solved using finite-difference methods. The pressure

of different formation points near the bottom of the well

at different time has been calculated. Then the well test

analysis curves have been plotted. In the end, the influ-

ence of different factors on the curves has been ana-

lyzed.

2. APPARENT VISCOCITY OF POLYMER

SOLUTION

2.1. The Shear Viscosity

The viscosity of the power-law fluid can be described

as:



Hrr





 (1)

where r is the radius away from the wellbore, m; γ is the

shear rate, s-1; H (r) is the consistency coefficient,

mPa·s n; n (r) is the power-law index of polymer solution,

dimensionless.

Reference [8] has studied the variation of rheological

behavior through experiments and developed the basic

models about the variation of rheological parameters

along the seepage flow direction. In this paper it means

that the consistency coefficient and the power-law index

are changing along the radial direction.

For the well where the polymer solution is injected

into, the variation of rheological parameters can be ex-

pressed as:





HrHr e





 (2)



 

nrnrr r



 (3)

where rw is the radius of the wellbore, m; α is the varia-

tion factor of the consistency coefficient, dimensionless;

β is the variation factor of the power-law index, dimen-

sionless.

According to the researches in [9], the relationship

between the shearing rate and the seeping rate is as fol-

lows:

γnCK





 (4)

where v is the flow velocity through porous medium, m/s;

K is the permeability of the formation, μm2; c’ is the

factor related to the tortuosity of capillary, 2.08～2.50;

Φ is the porosity of the formation, dimensionless.

The flow velocity through porous medium can be ap-

proximately expressed by using the following linear re-

lation:

vrh

 (5)

where q is the injection rate of the polymer solution,

m3/d; h is the reservoir thickness, m.

Then the expression of shearing rate considering the

variation of rheological parameters can be derived.







21 2π

nr qr

nr h









 (6)

The power-law index changes with the radius. How-

ever, the resulting changes of the shear rate make little

sense to the viscosity of the fluid at the same point. As a

result, the variation of the power-law index can be ig-

nored when calculating the shearing rate. Then the

shearing rate can turn to the expression as follows:







21 2π

nr qr

nr h









 (7)

where n(rw) is the power-law index of polymer solution

in the bottom hole, dimensionless.

Letting







21 2π

nr q

Fnr h





 (8)

The shearing rate can be eventually simplified as:





 (9)

And the final expression of the shear viscosity of the

polymer solution considering the variation of theological

parameters can be reached.

2.2. The Elastic Viscosity

The relationship between elastic viscosity and shear

viscosity has been obtained in [10].



 



(10)

where, θf is the relaxation time, s.

H. J. Yin et al. / Natural Science 2 (2010) 1327-1332

1321329

2.3. The Apparent Viscosity

There are both the shearing deformation and the elas-

tic deformation when the polymer solution seeping in the

porous medium because of the continuous contraction

and spreading of the runner. As a result, the apparent

viscosity is composed of the shear viscosity (μv) and the

elastic viscosity (μe).





aev f

12 v



 

 (11)

According to the expressions of the shear rate and the

shear viscosity, the final expression of the apparent vis-

cosity can be obtained.



  

ww ww

afsw

12 rr

nrr rnrr r

FrH re









 

 (12)

3. MATHETICAL MODEL AND ITS

SOLUTION

The mathematical model for transient flow of viscoe-

lastic polymer solution is derived with the following

equations.

The partial differential equation for transient flow of

viscoelastic polymer solution:

rp p

rrμrK





 







 t



(13)

Initial condition:

p (14)

Inner boundary condition:

2π

μrr









(15)

Outer boundary condition:

r





 (16)

Define the dimensionless variable as follows:

Dimensionless radius

 (17)

Dimensionless pressure



2πKh







(18)

Dimensionless time

D*2

tCr



 (19)

Where μ* is the characteristic viscosity, that is the ap-

parent viscosity of the polymer solution at the bottom of

the wellbore.

 

aw w

nr nr

rr Ar Cr







 

(20)

where



AHrF 

 (21)



fws

2nr

CHrF



 (22)

According to the dimensionless variable above, the

seeping model of the polymer solution considering the

variation of the rheological parameters was established:

DDe

DD aDD

rrr t









































(23)

The difference equation at the point (i, j) can be estab-

lished by using Implicit Difference Method.



111

1, 2,,1

jjj

iiii iii

apb pcpdiN







 





00,1, 2,,1

pi N









pp di











10,

pp iN







where

22 2

Di Di

arx rx

















22 2

crx rx











 

1()

nr nr















And Δx = ln(re/rw)/N, which is the grid spacing; re is

the radius of the external boundary, m; rw is the radius of

the wellbore, m; i is the number of the node; N is the

grid number; pe is the supply boundary pressure, Pa.

H. J. Yin et al. / Natural Science 2 (2010) 1327-1332

1330







Then the tridiagonal coefficient matrix equation of the

dimensionless mathematical model was built up as:

D0 0

D1 1

111

D2 2

222

DN e

...

abc





































The equation has been solved. The pressure and its

derivative value at the bottom of the wellbore at different

time have been calculated.

4. TYPE CURFE OF VISCOELASTIC

POLYMER SOLUTION

According to the solution of the mathematical model,

the type curve has been drawn, as may be seen in Figure

As shown in Figu re 1, when considering the variation

of the rheological parameters and the elasticity of the

polymer solution, the characteristic of the type curve is

as follows:

The pressure curve and the pressure derivative curve

both change along the straight line with the slope of 1 at

the pure wellbore storage phase.

In transition section, the pressure curve flattens and

the pressure derivative curve appears to be a transporta-

tion hump.

The pressure derivative curve goes up after the transi-

tion regime. This variation is a comprehensive action of

adsorption, shear, and elastic deformation.

The pressure derivative curve goes down once the in-

fluence of the changing rheological parameters become

obviously.

When the effects of the closed outer boundary play a

role, the pressure curve and the pressure derivative curve

both go up.

5. ANALYS IS OF INFLUENTIAL

FACTORS

Some parameters which could influence the charac-

teristics of the type curve have been analyzed. These

parameters include the relaxation time, the variation

factor of the consistency coefficient and the variation

factor of the power-law index, as well as the consistency

coefficient and the power-law index of the viscoelastic

polymer solutions.

The influence of relaxation time is analyzed on the

basis of Figure 2. As the relaxation time increases, the

pressure and pressure derivative values increase after the

pure wellbore storage phase, the “hump” of the pressure

derivative curve in transition section increases and the

23456789

lg(t

)

lg (p

), lg(p

'·t

)

lg(p

)

lg(p

'· t

)

Figure 1. The type curve of viscoelastic polymer solution.

23456789

lg( tD)

lg (pD),lg(pD'·tD)

θf=0.010s

θf=0.005s

θf=0s

Figure 2. The influence of the relaxation time.

pressure derivative curve upwarps less at the radial flow

regime. What’s more, compared with the curves simply

considered the polymer solution as power-law fluid, it is

shown that the pressure derivative curve upturns less at

the radial flow regime considering the elasticity of the

polymer solution. The larger the relaxation time is, the

elastic property of the polymer solution is stronger and

the greater the energy is required, then the larger the

bottom hole pressure is and the faster it changes. There-

fore, it’s necessary to consider the elasticity in well test

analysis on pressure of viscoelastic polymer solution.

The influence of the consistency coefficient at the

wellbore is analyzed on the basis of Figure 3. It is shown

that the consistency coefficient of the polymer solution

at the wellbore has no significant effect on the go-up of

the pressure derivative curve. However, as the consis-

tency coefficient at the wellbore increase, the apparent

viscosity of the polymer at the wellbore will increase,

the pressure and its derivative value of the radial flow

regime will increase.

The influence of the power-law index at the wellbore

is analyzed on the basis of Figure 4. The power-law index

at the wellbore mainly influences the radial flow regime.

As the power-law index at the wellbore decreases, the

upturned level of the pressure derivative curve increases.

It means that the fluid is Newtonian fluid when the value

of the power-law index is 1. The more the power-law

index at the wellbore deviated from the value of 1, the

more obviously of the non-Newtonian flow characteris-

tic is, the greater the flow resistance is and the more ob-

viously the pressure derivative curve goes up.

H. J. Yin et al. / Natural Science 2 (2010) 1327-1332

1331331

23456789

lg(t

)

lg(p

),lg(p

'·t

)

)=0.06Pa·s

)=0.04Pa·s

)=0.02Pa·s

Figure 3. The influence of consistency coefficient at the well-

bore.

23456789

lg( t

)

lg (p

),lg(p

'·t

)

n(r

)=0.5

n(r

)=0.3

n(r

)=0.1

Figure 4. The influence of power-law index at the wellbore.

The influence of variation factor of the consistency

coefficient is analyzed on the basis of Figure 5. As the

variation factor of the consistency coefficient increases,

the radial flow regime disappear earlier, the pressure

derivative curve goes down earlier and deeper instead of

going on turning up. It is mainly for the following rea-

sons: The smaller the variation factor of the consistency

coefficient is, the faster the consistency coefficient

changes and the faster the apparent viscosity of polymer

solution decreases. In the same period of flowing time,

the smaller the apparent viscosity of the polymer solu-

tion is, the smaller the flow resistance is. As a result, the

pressure derivative curve goes down earlier, the recessed

part is wider and the radial flow period stopped earlier.

The influence of variation factor of the power-law in-

dex is analyzed on the basis of Figure 6. As the variation

factor of the power-law index increases, pressure and

pressure derivative values are smaller in the same period

of flowing time, the pressure derivative curve upwarps

less obviously at the radial flow regime and the radial

flow period become shorter. What’s more, the Concave

appears earlier and deeper. It is mainly for the following

reasons. The greater the variation factor of the

power-law index is, the faster the power-law index

changes, the sooner the apparent viscosity of polymer

solution decreases, then the smaller the apparent viscos-

ity is and the lower the flow resistance is. It is the reasons

23456789

lg( t

)

lg (p

),lg(p

'·t

)

α=-0.002

α=-0.006

α=-0.010

Figure 5. The influence of the variation factor of consistency

coefficient.

-1

2345678910111

lg( tD)

lg (pD),lg(pD'·tD)

=0.0005

=0.0010

=0.0015

Figure 6. The influence of variation factor of the power-law

index.

above make the pressure derivative curve falls earlier.

6. CONCLUSIONS

According to the behavior that the polymer solution

has viscoelasticity and its rheological parameters are not

constants along the seepage flow direction, the transient

flow mathematical model has been established. It has

been found that the type curve is different from the

curves without considering the viscoelasticity and the

rheological parameters’ variation.

Compared with the curves simply considered the

polymer solution as power-law fluid, the pressure curve

of viscoelastic polymer solution is higher and the pres-

sure derivative curve upturns less in the radial flow re-

gime.

The greater the relaxation time is, the greater the elas-

tic viscosity of polymer solution is and then the greater

the seepage resistance the fluid encountered is. It means

that the energy required in the seepage flow is higher. So

the injection pressure required is higher and the pressure

at the other point of the formation is higher.

The smaller the power-law index of polymer solution in

the bottom hole is, the more seriously the non-Newtonian

behavior is. Hence, at the same rate, the apparent viscos-

ity of the polymer solution near the bottom hole is

smaller. This will make the pressure changes more slow-

H. J. Yin et al. / Natural Science 2 (2010) 1327-1332

1332

ly and make the pressure derivative curve upturns more

at the radial flow regime. The consistency coefficient of

polymer solution in the bottom hole has no significant

influence on the upturned degree of the pressure deriva-

tive curve. It mainly has impact on the value of the

pressure and the pressure derivative. The larger the con-

sistency coefficient of polymer solution in the bottom

hole is, the larger the value of the pressure and the pres-

sure derivative is.

Openly accessible at

Because of the variation of the rheological parameters,

the apparent viscosity of polymer solution reduces con-

tinuously in the flow process, which makes the pressure

derivative curve declines after the upturned section in

the radial flow and make a concave section appears be-

fore the upturned section caused by the closed outer

boundary. The more seriously the rheological parameters

changes, the more obviously the concave will be.

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