Vol.2, No.12, 1327-1332 (2010)
doi:10.4236/ns.2010.212161
Copyright © 2010 SciRes. Openly accessible at http:// www. scirp.org/journal/NS/
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:
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
N
o: 50874023) and by
t
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
Copyright © 2010 SciRes. Openly accessible at http://www.scirp.org/journal/NS/
1328

w
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:


1
v
nr
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:



w
w
rr
HrHr e
 
 (2)
  
w
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:
31
21
2'
nv
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:
2
q
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.



1
31
1
21 2π
2'
nr qr
nr h
CK




 (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:



w1
w
31
1
21 2π
2'
nr qr
nr h
CK




 (7)
where n(rw) is the power-law index of polymer solution
in the bottom hole, dimensionless.
Letting



w
s
w
31
1
21 2π
2'
nr q
Fnr h
CK 

 (8)
The shearing rate can be eventually simplified as:
1
s
F
r 

 (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].
ef
2v

 

(10)
where, θf is the relaxation time, s.

H. J. Yin et al. / Natural Science 2 (2010) 1327-1332
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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.



  
w
ww ww
1
afsw
11
s
12 rr
nrr rnrr r
FrH re
Fr





 
 (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:
L
a
1C rp p
rrμrK


 



 t

(13)
Initial condition:
i
0 t
p
p (14)
Inner boundary condition:
w
aw
2π
rr
K
p
μrr




q
h
(15)
Outer boundary condition:
e
0
rr
p
r


 (16)
Define the dimensionless variable as follows:
Dimensionless radius
D
w
r
rr  (17)
Dimensionless pressure

D
*
2πKh
pp
qB 


i
p (18)
Dimensionless time
D*2
tw
K
t
tCr

 (19)
Where μ* is the characteristic viscosity, that is the ap-
parent viscosity of the polymer solution at the bottom of
the wellbore.
 
w
w
1
*
aw w
nr nr
rr Ar Cr



 
w
(20)
where


w1
ws
nr
AHrF   (21)


w
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:
D
D
DDe
*
DD
D
DD aDD
D0
*
D
D1
aD
D
D
1
0
1
0
t
r
rr
pp
r
rrr t
p
p
rr
p
r








 
















(23)
The difference equation at the point (i, j) can be estab-
lished by using Implicit Difference Method.

111
11
1, 2,,1
jjj
iiii iii
apb pcpdiN



 


00,1, 2,,1
i
pi N




10
,0
ii
pp di





11
10,
jj
ii
pp iN



where
D Di
22 2
Di Di
2
i
i
r
arx rx
 




Di
22
Di
21
i
j
bt
rx
 



DD
22 2
DD
'
2
i
r
crx rx





D
1
j
ii
dp
t








 
ww
ww
1
1()
ww
01
ww
11
AC
1
AC
w w
nr nr
xx
nr nr
ww
nr nr
ee
rr
nr nr
d
rr


 





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.

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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
11
.. ...
11
pd
pd abc
pd abc
pp






 










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
1.
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
0
1
2
3
23456789
lg(t
D
)
lg (p
D
), lg(p
D
'·t
D
)
10
lg(p
D
)
lg(p
D
'· t
D
)
Figure 1. The type curve of viscoelastic polymer solution.
0
1
2
3
23456789
lg( tD)
lg (pD),lg(pD'·tD)
10
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
Copyright © 2010 SciRes. Openly accessible at http://www.scirp.org/journal/NS/
1331331
0
1
2
3
4
23456789
lg(t
D
)
lg(p
D
),lg(p
D
'·t
D
)
10
H
(r
w
)=0.06Pa·s
n
H
(r
w
)=0.04Pa·s
n
H
(r
w
)=0.02Pa·s
n
Figure 3. The influence of consistency coefficient at the well-
bore.
0
1
2
3
4
23456789
lg( t
D
)
lg (p
D
),lg(p
D
'·t
D
)
10
n(r
w
)=0.5
n(r
w
)=0.3
n(r
w
)=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
0
1
2
3
23456789
lg( t
D
)
lg (p
D
),lg(p
D
'·t
D
)
10
α=-0.002
α=-0.006
α=-0.010
Figure 5. The influence of the variation factor of consistency
coefficient.
-1
0
1
2
3
4
2345678910111
lg( tD)
lg (pD),lg(pD'·tD)
2
β
=0.0005
β
=0.0010
β
=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
Copyright © 2010 SciRes. http://www.scirp.org/journal/NS/
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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