Model of Relative Permeability Curves of Viscoelastic Polymer Flooding CAO Renyi

Physical and Numerical Simulation of Geotechnical Engineering
4th ISSUE, Sep. 2011
Model of Relative Permeability Curves of Viscoelastic
Polymer Flooding
CAO Renyi1, CHENG Linsong1, HOU Jun2
1. School of Petroleum Engineering, China University of Petroleum at Beijing
2. Sinopec International Exploration and Production Corporation
ABSTRACT: Due to the viscoelastic behavior of polymer solution, its flowing characteristic is
different to the Newton and viscosity fluid in porous media, so that there is essentially difference of
relative permeability between the polymer flooding and water flooding. Through the core
displacement experiment, this article characterizes the relationship between the endpoint of relative
permeability curve and pore throat ratio and permeability, which confirmed the viscoelasticity of
polymer flooding could enhance the displacement efficiency. Besides, the article improved the
measurement of relative permeability experiment, and also characterizes features of relative
permeability of polymer flood. On the basis of interpolation of polymer fluid‘s concentration, a
mathematical model of relative permeability curve is established and added in the numerical module
with polymer flooding simulator. The results show that this module can reflect the viscoelastic
mechanism of the polymer flooding and the predetermination is consistent with the field experience.
KEYWORDS: Polymer flooding, Viscoelasticity, Relative permeability curve, Mathematical model,
Numerical simulation
1 INTRODUCTION
The HPAM is used as the solvent in polymer flooding
now in China EOR practice (Wang et al. 2000; Han et
al.2006; Liu et al. 2008; Zhou et al. 2007, 2008; Wang et al.
2008). This fluid is a kind of viscoelastic rheological
material while flowing through the porous media that will
become shear thinning, be absorbed and detented and so on
(Heemskerk et al.1984, Wang et al. 2000, and Seright et al.
2009). The viscoelastic fluid is different to the common
non-Newtonian fluid, so that the displacement mechanism
of polymer solution is different to water or other pure
viscous flooding (Wang et al. 2000, and Zhang, 2004). So
that there is essentially difference of relative permeability
between the polymer flooding and water flooding. Relative
permeability curve is the synthetical description of the
flowing process of oil and gas, so it‘s very essential in the
process of reservoir engineering and numerical simulation.
With the maturity of polymer flooding technology and field
practice, the measurement and characterization of the
relative permeability curves of the polymer solution
flooding have gained more and more attention. Its accuracy
has a direct impact on the feasibility evaluation of chemical
flooding, the prediction of effects and the reservoir
management.
Many researchers also suggest that polymer flooding
may reduce the relative permeability of water phase
(Barreau et al., 1999; Grattoni et al., 2004; Zheng et
al.,2000). Recent study shows that the polymer flooding
residual oil saturation is lower than the waterflood residual
oil saturation, when polymer flooding is employed right
after primary production or at early stages of waterflood
(Huh and Pope, 2008; Kamaraj et al, 2011). The rheology
of polymer solution leads to a large difference of relative
permeability features between polymer drive and water
© ST. PLUM-BLOSSOM PRESS PTY LTD
drive, and also bring a huge difficulty when measuring the
relative permeability curves and handling the data
(Schneider and Owens, 1982; Lei and Xu, 1994; Shi and Li,
2001; Chen et al. 2005). During the calculation process of
classical numerical simulation and reservoir engineering,
the oil and gas relative permeability curves are still used
without considering the variable features of endpoint and
trend resulted by the polymer flooding (Pan, 2008; Chen et
al. 2011). In the process of conventional numerical
simulation, a permeability decrease coefficient is given in
the relative permeability curves of water flooding, which
indicates the permeability decrease and polymer absorption
(Barreau et al, 1997; Ogunberu and Asghari, 2004; Vasquez
and Miranda, 2010). However this method may lead to big
errors and bad forecast results (Chen, 2011).
Aimed the viscoelastic rheology and physical and
chemical reaction of polymer solution flowing in porous
media, we designed a series of core experiments and
established the relationship between relative permeability
features and the core and parameters of polymer.
Subsequently we built a mathematical model and a numeric
simulation module of the relative permeability curves for
EOR simulation software.
2 EXPERIMENTS AND CHARACTERIZATION
OF THE RESIDUAL OIL SATURATION OF
POLYMER FLOODING
The viscoelasticity of polymer flooding can enhance the
displacement efficiency (Wang et al. 2000, and Zhang,
2004). Based on a large number of experiments and
theories, many scholars have confirmed that the higher the
viscoelasticity, the better the displacement efficiency. The
effect of polymer‘s viscoelasticity varies with the different
types of throats and interfacial properties of porous media,
Model of Relative Permeability Curves of Viscoelastic Polymer Flooding
DOI: 10. 5503/J. PNSGE. 2011.04.015
and there still isn‘t any accurate method to quantitatively
characterize the increase scope of recovery ratio. So we
measure the endpoint value of the relative permeability of
cores with different throat and permeability, and probe the
displacement effect of polymer fluid‗s viscoelasticity.
measurement system. The cores used in the experiment are
natural cores, through the mercury injection experiment and
cast thinning and scanning electron microscopy, we
measured the pore structure parameters of nine cores with
different permeabilities (Table 1). The water used in the
experiment is formation water with a salinity of 4120 mg/L.
The oil is a mixture of crude oil and kerosene, and its
viscosity is 6.278 mPa·
s. The polymer is anti-salt
polyacrylamide with a molecular weight of 18 million.
2.1 Equipment and conditions of experiments
The displacement equipment consists of displacement
system, casing pressure system and calculation and
Table 1 Parameters of experimental cores
Core
numbers
Porosity
volume
(ml)
Permeability
(10-3μm2)
Porosity
(%)
Effective
radius
（μm）
Coordinate
number
Pore
throat
ratio
Grading
factor
DJT-1
8.41
58.9
28.1
1.3
1.7
78.9
6.22
DJT-2
8.24
70.6
23.9
1.5
2.4
62.8
3.97
DJT-3
11.51
80.4
30.4
1.5
2.4
62.8
3.97
DJT-4
10.94
145.2
29.3
2.0
1.9
51.5
3.57
DJT-5
12.12
255.6
32.6
2.5
2.1
40.2
2.63
DJT-6
10.20
279.9
31.7
2.7
2.1
40.2
2.63
DJT-7
8.78
424.3
27.8
3.5
1.7
26.2
1.38
DJT-8
10.32
528.4
28.6
3.8
1.7
26.2
1.38
DJT-9
10.53
724.1
30.1
4.4
2.7
7.1
0.64
2.2 Process and results
saturation is higher than cores with high permeability,
which means that there is more residual oil left in the pores
with the shape of films, islands and dead-end. So the
polymer could effect the more viscoelastic flooding in low
permeability cores, and more residual oil is droved by
polymer solution.
We can figure out from figure 1 and figure 2 that there is
a good linear relationship between △ Sor and permeability
and pore throat ratio, which can be expressed by equation 1
and equation 2:
(1)
Sor  a1  b1
(2)
Sor  a2 K  b2
聚驱与水驱残余油饱和度差（%）
聚驱与水驱残余油饱和度差（%）
First, injected water until the water content is 98%, and
then injected 0.4 PV polymer solution with concentration of
1200 mg/L, and then continued to flood the core with water
until there was no oil coming out. Figure 1 and figure 2 are
the curves showing the relationships between the core
permeability and the differential residual oil saturation of
polymer flooding and water flooding (△ Sor), which can be
fitted and exhibited by linear expressions. There exists an
inverse relation between the core permeability and pore
throat ratio. The curves also show that the lower the
permeability or the bigger the pore throat ratio, the higher
the residual oil saturation difference (△ Sor). This is
because that the pore and throat relation of the cores with
lower permeability is more complex, so their residual oil
10.0
9.5
y = -0.0036x + 9.3679
9.0
2
R = 0.9509
8.5
8.0
7.5
7.0
6.5
6.0
0
200
400
600
-3
800
y = 0.0363x + 6.7214
9.5
2
R = 0.9169
9.0
8.5
8.0
7.5
7.0
6.5
6.0
0
20
40
60
80
100
孔喉比
Pore
throat ratio
2
-3
渗透率（10 μm ）
Permeability(10
μm2)
Fig 1 The relationship between K and △ Sor
10.0
Fig 2 The relationship between pore throat ratio and △ Sor
96
Physical and Numerical Simulation of Geotechnical Engineering
4th ISSUE, Sep. 2011
Two cores were chosen to measure the relative
permeability curves, and the basic data of cores and fluids
are as follows (Table 2).
3 EXPERIMENTS AND CHARACTERIZATION
OF RELATIVE PERMEABILITY CURVES
Table 2 Basic data of cores and fluids
Core
number
Porosity
(%)
Permeability
(10-3μm2)
Residual oil
saturation of water
flooding
(%)
Residual oil
saturation of
polymer flooding
(%)
Viscosity of
simulated oil
(mPa·
s)
Viscosity of
polymer
(mPa·
s)
c-1
23.9
0.21
44
45
6.9
14.1
c-2
27.1
1.3
30
32
6.9
17.3
Normally we use the non-steady method (Shi and Li,
2001; Chen et al. 2005) (―JBN‖ method) to deal with the
data of polymer flooding‘s relative permeability. Due to the
difficulty of calculating the produced fluid in the core‘s
outlet of polymer drive, we need to use differential calculus,
which exists huge difficulties whiling handling it in
practice. Oil and water are always recovered in the form of
emulsion, which will result in big errors when using the
automatic measuring system. So we used a difference
quotient to replace the derivation, and calculated it after the
oil and water being standed for a while or by adding
demulsifer. So this method can increase the accuracy of
calculation, which makes the measured curves more close
to the real condition. The formulas of the modified ―JBN‖
method are:
  1     1   f (S )
K ro ( S w )  f o ( S w )      /       o w
I
 Vt     IVt  
K rw ( S w )  K ro ( S w )
fo (Sw )  qo / q
(5)
f w  1  f o ( Sw )
I  po / pt
(6)
(7)
The results of C-1 and C-2 cores‘ relative permeability
experiments are shown as figure 3 and figure 4. In the
semi-logarithmic coordinates, we used the dimensionless
water saturation Sw* as the right angle horizontal axis, and
the relative permeability Kro, Krw as the logarithmic
ordinate, and we can see from the matching relation that
there is a good linear relationship between Sw* and Kro,
Krw, which fits in the following expression:
(8)
lg K ro  cS w*  d
lg K rw  eSw*  f
While: S *  S w  S wr
w
1  S wr  Sor
（3）
1
(9)
1
0.1
0.1
Kro
Kro
Kro,Krw
ef f w ( S w ) (4)

o f o ( S w )
Krw
0.01
Kro
Krw
0.01
0.001
0
0.05
0.1
Sw
0.001
0.15
0
0.02
0.04
*
0.06
0.08
0.1
0.12
0.14
Sw *
Figure 3 The relative permeability curve of C-1
Figure 4 The relative permeability curve of C-2
and (9) established the mathematical model of viscoelastic
polymer flooding.
(1) The constitutional equation of water relative
permeability:
o
(10)
K rw  K rw
( S w* )n1 K ro  K roo (1  Sw* )n 2
4 THE MATHEMATICAL MODEL OF RELATIVE
PERMEABILITY CURVES
In the numerical simulator, there are two ways of
endowing relative permeability, one is to give the maximal
and minimal endpoint value of the relative permeability
function, and another way is to give several saturations and
correspondent relative permeability table or function to
calculate the endpoint value (Liao, 1999). This article used
the first method, and according to the expressions (2), (8)
(2) Endpoint values affected by polymer‘s viscoelasticity
The relationship between the permeability and residual oil
saturation of polymer with a high elasticity (  P
follows:
Sor high  a high K  bhigh
97
(11)
high
) is as
(cPlow):
The relationship between the permeability and residual
oil saturation of polymer with a low elasticity (  P
low
lg K rolow  clow S w*  d low
) is as
lg K rw
follows:
Sor low  alow K  blow
(12)
low
e S  f
low
*
w
low
(17)
(18)
So the parameters of a certain concentration(cP):
  
low
So the parameters related to the relationship between the
permeability and residual oil saturation of polymer drive
cP  cP low
 high  low 
cP high  cP low
with a characteristic time of certain elastic (  P ):
parameter c, d, e or f
  p P
 high
a  a low   high
a  a low 
low 




P
P


low





p
P
b  blow   high
b high  blow 
low 




P
P


5 THE MODULE AND
CALCULATION
WITH
SIMULATOR

is
(19)
low
While the relationship between
(13)
(12)
P
P  f  cp 
(14)
(3) The relative permeability of polymer drive:
The parameter of relative permeability is based on the
interpolating calculation of relative permeability with
different concentrations of polymer. And high and low
represent values of different concentrations respectively. So
the relative permeability of a known high concentration
(cPhigh):
(15)
lg K ro high  c high S w*  d high
lg K rw high  ehigh S w*  f high
THEORETICAL
NUMERICAL
On the basis of UTCHEM simulator, we modified the
processing module of viscoelastic polymer flooding‘s
relative permeability curve. We simulated a well group with
four injection wells and one production well. The grid is
10×10×1 while and the step is 20m×20m×5m. The
permeability is 300mD, the porosity is 0.25, the slug size is
0.5PV, and the mass fraction is 0.15%. The fluid parameters
are measured parameters of C-2. The results of simulation
are shown by figure 4 and figure 5. The recovery ratio,
while considering viscoelasticity of polymer, increases
4.3% when compared to that without considering
viscoelasticity, which coincides with the field experience.
and cP is as
follows:
(16)
The relative permeability of a known low concentration
100
100
含水率(%)
40
20
0
0
2
4
80
Recovery (%)
60
60
采收率(%)
80
Water Cut (%)
含水率(%)
Model of Relative Permeability Curves of Viscoelastic Polymer Flooding
DOI: 10. 5503/J. PNSGE. 2011.04.015
60
40
水驱
Water flooding
水驱
纯粘性聚驱
Viscous polymer flooding
纯粘性聚驱
粘弹性聚驱
Viscoelastic polymer
flooding
粘弹性聚驱
20
0
50
40
30
06
28
时间(a)
410
612
8 14
Time时间(a)
(years)
10
12
14
Figure 4 Water cut ratio of polymer flooding
Water
水驱
Viscous
纯粘性聚驱
polymer
Viscoelastic
粘弹性聚驱
polymer
Figure 5 Recovery ratio comparison of polymer flooding
concentration, but also relates to the microscopic pore
structure of the porous media. The experiments show that
the lower the permeability of cores the more complex the
pore and throat relation, the higher the residual oil
saturation after water flooding and more oil could be
displaced by viscoelastic polymer flooding, and also show
that the decrease of the residual oil saturation is inversely
proportional to permeability and proportional to pore throat
ratio, meanwhile, the curves display good linear
relationships.
(3) According to the improved non-steady relative
permeability experiments, we can conclude that the water
relative permeability of polymer flooding declines severely,
while the oil phase curve raises and the isotonic point
moves right and lowers as well. We also fit the function
6 CONCLUSIONS
Through the core-scale endpoint value experiments of
residual oil saturation of polymer drive and the improved
non-steady relative permeability experiments, together with
the modified description method of polymer drive‘s relative
permeability, we draw several conclusions as follows:
(1) The core-scale endpoint value experiment of residual
oil saturation shows that the polymer can decrease the
residual oil saturation of water flooding, which further
proves that polymer flooding can enhance the final
displacement efficiency.
(2) The mechanism about that polymer can decrease the
residual oil saturation not only relates to the polymer‘s
98
Physical and Numerical Simulation of Geotechnical Engineering
4th ISSUE, Sep. 2011
equation using the logarithmic coordinates.
(4) Based on the relative permeability‘s functional
equation of polymer flooding, we established the
mathematical model of polymer drive, and developed the
correspondent simulation module to calculate it
theoretically, and the results could be consistent with that
obtained through field practice.
[6].
[7].
EXPRESSIONS
[8].
Kro(Sw)—— oil phase relative permeability
[9].
μef —— Viscosity of polymer solution equivalent
μo——water viscosity
Sw—— water saturation
fo(Sw)—— oil content ratio
Swe—— water saturation at the outlet point
Vt ——dimensionless accumulative injection volume
[10].
[11].
(V/Vp)
——dimensionless
accumulative oil production (Vo/Vp)
Vot
[12].
Vp—— effective pore volume of core
Krw(Sw)——oil phase relative permeability
I—— flow capacity ratio
Swi——irreducible water saturation
△ pt——pressure difference at t
△ p0——initial pressure difference
λ——pore throat ratio
K——permeability
a1, a2, b1, b2—— fitting undetermined coefficient
△ Sor——the residual oil saturation difference of polymer
and water displacement
[13].
[14].
[15].
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