Journal of Petroleum Science and Engineering 182 (2019) 106263
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Journal of Petroleum Science and Engineering
journal homepage: www.elsevier.com/locate/petrol
A critical review of concept and methods related to accessible pore volume
during polymer-enhanced oil recovery
T
Saeed Akbaria, Syed Mohammad Mahmoodb,∗, Negar Hadian Nasra, Sameer Al-Hajria,
Maziyar Sabetc
a
Petroleum Engineering Department, Universiti Teknologi PETRONAS, Seri Iskandar, Tronoh 32610, Malaysia
Shale Gas Research Group (SGRG), Institute of Hydrocarbon Recovery, Petroleum Engineering Department, Universiti Teknologi PETRONAS, Seri Iskandar, Tronoh
32610, Malaysia
c
Petroleum and Chemical Engineering, Universiti Teknologi Brunei, BE1410, Brunei
b
A R T I C LE I N FO
A B S T R A C T
Keywords:
Inaccessible pore volume (IPV)
Polymer flooding
Single-slug experiment
Double-slug experiment
Excluded pore volume (EPV)
Tracer
Polymer flooding is an enhanced oil recovery (EOR) technique that improves frontal stability problems associated with waterflooding. It has two primary mechanisms, the reduction of mobility ratio and improvement of
conformance in heterogeneous reservoirs with a high coefficient of permeability variation (V). The former
mechanism is more pertinent to heavy oil whereas later is to light oil reservoir. It is imperative to estimate the
adsorption/retention losses for designing a slug that maintains integrity until the target destination. The optimum polymer slug is determined using reservoir simulators such as CMG & ECLIPSE, which require adsorption
loss data per unit of the accessible pore volume. Unlike water, polymer molecules are not able to penetrate the
entire pore volume due to size and wall exclusion. The fraction of total pore volume, which is not available for
polymer invasion is called IPV (Inaccessible Pore Volume). This paper reviews and critically evaluates IPV whose
determination is essential for finding polymer losses to the accessible pore volume. The five methods available
for IPV determination fall into either tracer-laden or tracer-free category. For tracer-laden, the analysis is based
on separation area between the tracer and polymer profiles or breakthrough time gap. It was observed that
estimation based on the separation between the tracer profile of the first injected polymer slug and a polymer
profile of the second injected slug is likely to provide the most accurate IPV value. It is hoped that this critical
review could guide researchers in selecting a suitable IPV estimation method for their polymer flooding studies
depending upon the complexity & time required and the availability of equipment in their laboratory.
1. An introduction to polymer losses in porous media
The primary recovery from oil reservoirs due to pressure depletion
is generally very low and ranges from 10 to 20% (Dembicki, 2017). A
secondary technique, such as water injection into oil reservoirs, is
commonly employed to enhance oil recovery by 15–55% (Lake, 2010).
Best recovery with piston-like displacement is expected when the mobility of injected water (kw/μw) is about the same or higher than the
mobility of reservoir oil (ko/μo).
Otherwise, the flood front may form fingering or encounter other
destabilizing phenomena, as shown in Fig. 1(a). The instabilities could
be worse in layered or fractured reservoirs. In polymer flooding, injected water viscosity is increased with a polymer to control mobility
and thereby improve sweep efficiency (Akbari et al., 2017; Green and
Willhite, 1998; Kaminsky et al., 2007; Seright, 2016; Zhang et al.,
2010). An idealized polymer flood is shown in Fig. 1(b).
Whereas polymer flooding is an established technology, it encounters polymer retention in the reservoir, causing a considerable
decrease in the effectiveness of oil displacement. The polymer is retained in the porous media because of the following reasons (Chen
et al., 2016; Choi et al., 2014; Dominguez and Willhite, 1977;
Farajzadeh et al., 2016; Huh et al., 1990; Sheng, 2011; Sorbie, 1991;
Szabo, 1975; Zhang and Seright, 2015);
a. Polymer Adsorption: Some polymer molecules are adsorbed on the
rock surfaces to which they come in contact.
b. Mechanical Entrapment: Some polymer molecules entering the
pores may not move further due to their hydrodynamic size because
∗
Corresponding author.
E-mail addresses: akbari.sa70@gmail.com (S. Akbari), mohammad.mahmood@utp.edu.my (S.M. Mahmood), negar.hadian@gmail.com (N.H. Nasr),
ensamyo87@gmail.com (S. Al-Hajri), maziyar.sabet@utb.edu.bn (M. Sabet).
https://doi.org/10.1016/j.petrol.2019.106263
Received 9 November 2018; Received in revised form 2 July 2019; Accepted 11 July 2019
Available online 12 July 2019
0920-4105/ © 2019 Elsevier B.V. All rights reserved.
Journal of Petroleum Science and Engineering 182 (2019) 106263
S. Akbari, et al.
Abbreviations
EPV
IPV*
IPV
PV
BPV
APV
RRF
Mw
Cp* (pv )
Excluded Pore Volume due to wall exclusion effect
Inaccessible pore volume due to only size exclusion effect
Inaccessible pore volume due to the combined effect of
size (IPV*) and the wall (EPV) exclusion effects
Pore volume
bulk pore volume
Accessible pore volume
Residual resistance factor
Cs* (pv )
PVct *= 0.5
PVcp*= 0.5
Cp
Ĉp
ap
bp
∇pwa
Nomenclature
ϕ
ϕeff
Ci
Cio
Original porosity
Effective porosity
Effluent concentration collected at the core outlet
Injected concentration into a core
∇pwb
Molecular weight
normalized concentrations of the polymer as a function of
injected pore volume
normalized concentrations of the tracer as a function of
injected pore volume
Injected pore volumes of tracer when the normalized
concentrations in the effluent reach 0.5
Injected pore volumes of the polymer when the normalized concentrations in the effluent reach 0.5
Injected polymer concentration
Adsorbed polymer concentration
Maximum mono-layer coverage capacity in Eq. (7)
The tuning parameter used in Eq. (7)
The pressure drop due to the flow of brine after the
polymer slug is injected
The pressure drop due to the flow of brine before polymer
injection
Fig. 1. Comparison of water transportation in (a) water flooding and (b) polymer flooding.
Fig. 2 provides a schematic diagram of polymer retention mechanisms in a porous medium, as listed above. A comprehensive explanation
of different polymer retention mechanisms is provided by Sorbie
(1991). Several methods have been proposed to measure polymer retention (Al-Hajri et al., 2018; Dawson and Lantz, 1972; Dominguez and
Willhite, 1977; Huh et al., 1990; Szabo, 1975; Zhang and Seright,
2013).
the open pore throats for exit are too narrow to allow further propagation and therefore are entrapped.
c. Hydrodynamic Retention: Some polymer molecules enter in areas
(e.g., corners or dead ends) where they are shielded from the driving
forces due to laying outside the flow streamlines. Therefore, they
seize to flow and get temporarily entrapped until the streamlines are
re-distributed. This phenomenon is often referred to as flow-induced
hydrodynamic retention.
2. Inaccessible pore volume (IPV)
It is important to emphasize that deep in the reservoir, which is the
focus for reservoir performance; the polymer retention is mainly due to
adsorption. The mechanical entrapment and hydrodynamic retention
are significant at high flow rate conditions that are encountered only
near the injection well.
Most fluids, including water and tracer, can pass through entire pore
volume of a porous medium. On the contrary, there often is a fraction of
pore volume into which polymer molecules may not be able to invade
because of their large molecular size and flexible structure. This
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for reservoir simulation. It is, therefore, necessary to measure IPV that
later needs to be deducted from the total pore volume to compute the
accessible pore volume.
The determination of IPV by core flooding tests is not a straightforward process. The polymer adsorption decelerates polymer propagation, whereas IPV accelerates it. The coexistence of these two opposing factors complicates the interpretation of effluent polymer
profiles (AlSofi et al., 2017b; Clemens et al., 2016; Zhang and Seright,
2013). Fig. 3 is an idealized diagram of normalized polymer concentration (Ci/Cio, effluent concentration/injected concentration) profile during the injection of one pore volume slug of a polymer solution
containing tracer followed by brine injection.
Assuming there is no diffusion/dispersion between polymer slug
and leading or trailing brine, the four possible scenarios are depicted in
Fig. 3. Polymer profile is shown in black, whereas the tracer profile is
shown in red. Since the tracer is not affected by adsorption or IPV, it
always breaks through at 1 PV and cuts off at 2 PV. The piston-like
profile shown in this figure is possible only in case of homogenous cores
where the injected phase has lower mobility than the displaced fluid,
and the flow rate is low enough to support a stable front. A comparison
of polymer profile viz a viz’ the tracer profile provides a powerful diagnostic tool to assess the degree of adsorption and IPV.
In Fig. 3 (a), the polymer and tracer profiles are coinciding, indicating that there is no adsorption and no inaccessible pore volume.
Fig. 3 (b) is showing polymer breaks through at 0.75 pore volume and
cuts off at 1.75 PV (both 25% prematurely compared to tracer) implying the presence of 25% inaccessible pore volume but no adsorption.
The tracer profile in Fig. 3 (c) is showing polymer breakthrough at
1.2 PV (20% delayed) compared to tracer indicating 20% adsorption.
The polymer cutoff is at 1 PV coinciding with tracer because there is no
IPV. Section (d) of Fig. 3 illustrates the polymer cutoff at 1.75 PV (25%
prematurely) because of IPV alone since the back edge is not affected by
adsorption. The breakthrough time is seen as 0.95 PV (5% prematurely)
because it should have been advanced by 25% due to IPV and should
have been delayed by 20% due to adsorption, thus a net advancement
of the front by 5%.
In summary, the polymer bank appears earlier than expected when
IPV is present due to the quantity of the pore volume that is not accessible to the polymer. Thus the entire breakout curve is shifted forward (earlier). In contrast, adsorption postpones the front edge without
having an effect on the back edge, decreasing the size of the bank.
When both effects are combined, a bank that is shifted forward and is
Fig. 2. Schematic diagram of polymer retention mechanisms in a porous
medium. Adapted from (Huh et al., 1990).
volume, which is not accessible to polymer molecules, is called inaccessible pore volume (IPV).
Many polymer flooding studies have reported that polymer molecules are transported at a higher velocity than the non-adsorbing inert
tracers after the retention is completed (Dawson and Lantz, 1972;
Zaitoun and Kohler, 1987). They attributed this difference of post-adsorption velocities to the presence of IPV. A similar conclusion was also
drawn by other researchers (Bartelds et al., 1997; Unsal et al., 2018).
Another supporting evidence came from the coreflood experiments
of Stavland et al. (2010) wherein the calculated apparent viscosity was
lower than what would have been expected from bulk rheology at low
injection rates. Since their polymer was shear-thinning, they attributed
the lower than anticipated effective viscosity to the possibility of a
higher than anticipated effective velocity through porous media. This
observation was suggested to be due to the existence of a part of the
pore volume that was not available to polymer flow.
3. IPV and its importance for polymer loss estimation
Polymer adsorption is frequently expressed in lab studies as microgram (μg) of polymer per gram of rock. However, since the adsorption can take place only in the accessible pore volumes in porous
media, it is customary to report adsorption (or retention) as the adsorbed polymer mass per unit of accessible pore volume ( μ gram of polymer )
accessible PV
Fig. 3. Idealized polymer (black) and tracer (red) breakout curves for a bank size of 1 pore volume in different conditions; a) No adsorption and no inaccessible pore
volume, b) No adsorption and 25% inaccessible pore volume, c) 20% pore volume adsorption and no inaccessible pore volume, d) 20% pore volume adsorption and
25% inaccessible pore volume. Adopted from (Dawson and Lantz, 1972). (For interpretation of the references to colour in this figure legend, the reader is referred to
the Web version of this article.)
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and electrolytes, concentrations, saturation, molecular weight (Mw),
and velocity (Sorbie, 1991). Lower permeability is generally associated
with a higher IPV* occupying up to 30% or more of the total pore space
(Lake, 1989).
A threshold Mw exists for any given pore-throat size or rock permeability above which the polymer molecules are not able to propagate
and can be retained upstream the pore throat. For polymer molecules to
enter a pore, the ratio of the pore-throat radius and the gyration radius
of the polymer, known as Jamming ratio, should be equal to or greater
than 5 when polymer is adsorbed on pore walls (Dan-dan et al., 2014;
Sheng et al., 2015; Wang et al., 2002). However, when there is no
adsorption, Jamming Ratio should be 3 to avoid polymer entrapping at
pore throat (Cozic et al., 2009).
It is interesting to note that there may be some reservoir rocks
having no (or minimal) IPV* allowing polymer molecules to pass
through easily. Such rocks generally have high permeability and are
composed of well-sorted (uniform) large grains (approaching 1-mm
diameter). Another characteristic of such rocks having low IPV* is the
lack of cementing and post-deposition diagenetic (chemical) changes in
the rocks.
smaller is produced (Dawson and Lantz, 1972).
Fig. 3, along with the accompanying discussion, show the importance of determining IPV for quantifying polymer retention (Li et al.,
2014). These corrections are highly significant, especially in low retention core-flood systems (Fletcher et al., 1991). For the simulation of
enhanced oil recovery by polymer flooding process, accurate knowledge of IPV is critical for precise determination of polymer breakthrough times (Liu et al., 2018; Lotsch et al., 1985; Zhao et al., 2017).
The effect of IPV is accounted for by reducing the original porosity (ϕ )
to the effective porosity (ϕeff ) as shown in Eq. (1) (Camilleri et al.,
1987).
ϕeff = (1 − IPV ) ϕ
(1)
For polymer flooding, many consider the presence of IPV in the
reservoir to be advantageous from the economic point of view. It accelerates polymer propagation into the formation, counteracts the diminishing effects of polymer retention, and also decreases polymer
losses due to adsorption. This can be especially true if small pores are
filled with irreducible or connate water and not the hydrocarbons (AlShalabi, 2018).
The presence of IPV may be disadvantageous if a polymer slug is
injected after a surfactant slug. The acceleration of polymer due to IPV
may cause the polymer from the chase polymer bank to penetrate the
surfactant bank and create a lower mobility polymer phase that compromises mobility control (Kolodziej, 1988; Trushenski et al., 1974).
Also, the polymer flooding benefit is muted in case of large IPV if the
small pores contain oil which will remain unrecovered because the
polymer won't be able to access them (Delamaide, 2014).
4.2. Wall exclusion effect
In the laminar flow of fluids through a capillary tube, the fluid
transports in layers of concentric cylinders that are moving in the direction of flow. The velocity of these layers in vertical direction varies
in a way that the central layer has the maximum velocity, and the ones
close to the wall have zero velocity. Fig. 5 shows the velocity distribution during laminar flow in a circular tube (parabolic Poiseuille
velocity profile (Sutera and Skalak, 1993)).
Since the porous media are often modeled as a collection of interconnected converging-diverging capillaries, polymer molecules tend to
flow in the center of the pores because of the highest velocity.
Fig. 6 shows the existence and movement of polymer molecules
through a vertical cross section of a capillary tube. The movement is
restricted in the surface boundary layer (a depleted layer having lower
polymer concentration) due to the polymer/rock interaction. The molecules in the center, on the other hand, are flowing/rotating freely. The
molecules near the wall tend to move towards the higher entropy in the
center (entropic exclusion) depleting polymer molecules from the pore
volume near the pore wall. The depletion layer effect has been firstly
investigated by Chauveteau (1982). This depleted layer is called Excluded Pore Volume (EPV) (Duda et al., 1981; Zaitoun and Chauveteau,
1998). A similar effect is observed for non-Poiseuille (i.e., non-Newtonian) flow conditions.
EPV was considered by Teeuw and Hesselink (1980) as the leading
cause of the velocity enhancement effect. Accordingly, Sorbie (1991)
highlighted that the impact of EPV is case-sensitive and is more
dominant for larger pore sizes in which IPV* would be negligible. On
the other hand, IPV* is more dominant in cores with smaller pore sizes.
Nonetheless, it is important to note that EPV is always present, and its
effect cannot be neglected.
4. Various mechanism contributing to the IPV
As discussed earlier, a significant consequence of the presence of
IPV is the velocity enhancement for the polymer molecules compared to
the tracer one. The mechanisms that lead to IPV and therefore, velocity
enhancement in porous media are discussed below.
4.1. Size exclusion effect
Reservoir rocks are composed of grains of various sizes due to the
sedimentation process and other geological processes. Consequently,
the pores sizes between those grains also vary widely, some of which
may be too small to allow large polymer molecules to penetrate through
(Adepoju et al., 2017; Chen et al., 2008; Clemens et al., 2011; de Melo
et al., 2002; Delamaide, 2014; Delamaide et al., 2016; Ferreira and
Moreno, 2016; Hatzignatiou et al., 2013; Mezzomo et al., 2002;
Pancharoen et al., 2010; Qing and Fulin, 2011; Rodriguez Manrique
et al., 2014; Sharma et al., 2011; Thiele et al., 2010; Wang et al., 2013;
Zhang and Seright, 2013). As a result, a fraction of the total pore volume in porous media is inaccessible to the polymer, whereas water and
tracers could pass through (Fig. 4).
To properly account for the lack of accessibility in all pores, the bulk
pore volume may be divided into two sections as follows:
BPV = APV + IPV *
(2)
where BPV is the bulk pore volume composed of all pores, APV is the
fraction of pore volume through which polymer can pass through, and
IPV* is the fraction of pore volume through which polymer cannot pass
through.
The effect of the IPV* is that the polymer molecule progresses
through the porous media as if this pore volume does not exist, allowing
faster propagation of polymer concentration compared to water
(Dawson and Lantz, 1972).
The IPV* strongly depends on the characteristics of porous media
(such as permeability, porosity, and pore size distribution) and the
hydrodynamic size of the polymer in the solution (Alishaeva and Entov,
1983; Sheng, 2011). The latter can be affected by the types of polymer
Fig. 4. Schematic diagram of inaccessible pore volume due to size exclusion
(IPV*) in porous media.
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In addition, some cores, especially of lower permeability ones, may
also have inaccessible pore volume (IPV*) into which polymer may not
be able to enter due to the small size of throats compared to polymer
molecule size. Whereas EPV exists in all cores, IPV* may be absent in
some very high permeability cores. IPV* expedites the polymer breakthrough as compared to the tracer since it has to invade a pore volume
that has been reduced by IPV*. No such expedition of breakthrough is
observed in tracer since it can invade the entire pore volume due to its
smaller sizes. The existence of IPV* can be thought of as reducing the
effective area of the core through which polymer has to travel, thus
increasing the flow velocity.
In practicality, however, it is very difficult to distinguish IPV* and
EPV experimentally and often unnecessary to measure IPV* and EPV
separately. Therefore, IPV has often been reported as a combination of
both wall exclusion and the size exclusion effect in several published
works (Poellitzer et al., 2009; Verma et al., 2009). Since this may cause
confusion in our discussion in the coming section, we will also refer to
the combined effect as inaccessible pore volume (IPV) as defined below:
Fig. 5. Velocity profile of liquid flow through a capillary under laminar conditions.
IPV = IPV *+ EPV
(3)
6. IPV determination with and without tracers
IPV can be determined by polymer flooding in porous media using
any of the five experimental methods. These five methods fall into two
categories; the ones using tracer and the other without tracer. Further
explanation of these methods is provided below.
6.1. IPV determination with tracers
Fig. 6. Exclusion of molecules near the wall from the free-flowing molecules in
the center region. Reproduced from (Sorbie, 1991). This figure has been firstly
proposed by Chauveteau and Zaitoun (1981) for rodlike polymer molecules like
xanthan gum.
The methods using tracers are the ones most commonly used for IPV
determination in which an inert tracer (non-retainable) is usually mixed
with the polymer solution. As shown in Fig. 3, the breakthrough and
cut-off times will not be affected by adsorption or IPV since the selected
tracer is not retained/adsorbed in porous media, thus providing a clear
picture of front advancement. The methods using tracers described in
the following sections have gradually evolved to overcome the uncertainties in adsorption and desorption behavior.
5. IPV as summation of size and wall exclusion effects
As mentioned earlier, there is a depleted layer around grains of
porous media having lower polymer concentration and restricted
movement of the polymer molecules. The depletion layer effect which is
due to steric repulsion between the pore wall and macromolecules takes
place in all cores, including high permeability ones. In the presence of
EPV, polymer molecules encounter entropic driving forces that bring
them towards the center of flow where the velocity is highest, thus
expediting flow velocity. Contrarily, tracer molecules do not experience
an entropic driving force because of their low molecular weight and
rigid structure; therefore, travel with normally expected velocity as if
there was no EPV.
6.1.1. Dawson & Lantz Method (based on the injection of a single tracerladen polymer slug)
First introduced by Dawson and Lantz (1972), this method of IPV
estimation in porous media was later utilized by other researchers
(AlSofi et al., 2017a, 2017b, 2018; Huh et al., 2009; Najafiazar et al.,
2016). In this technique, the porous medium is initially saturated with
brine, then several pore volumes of the tracer-laden polymer solution of
known concentrations are injected. Effluent samples are collected at
Fig. 7. Inaccessible pore volume determination using single slug experiment: (a) in an ideal polymer (black) and tracer (red) breakout curves and (b) in a typical areal
flood, adapted from (Dawson and Lantz, 1972). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this
article.)
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S. Akbari, et al.
unfavorable mobility ratio, this method lacks accuracy because of the
long tailing of the back front.
In addition to avoiding the viscous fingering, it is vital that no adsorbed or entrapped polymer is released during the washout brine injection to obtain accurate results from this method. This is because if
the adsorbed or mechanically trapped components get released during
washout brine flooding, the area under the polymer curve may be larger
than the tracer area. In this case, the IPV may become negative and lose
its original physical meaning (Najafiazar et al., 2016), except that a
negative IPV may indicate the release of retained matter. Fig. 8 shows a
core flood case where trapped particles were released during the
washout brine flooding and produced a much larger area under the
cellulose curve than normally expected.
Previous studies (Manichand and Seright, 2014; Zhang, 2013) have
observed the release of mechanically entrapped and hydrodynamically
retained polymer as a result of washout brine flooding for an extended
duration, indicating that polymer retention due to these phenomena is
reversible.
regular intervals from the other end for concentration measurements.
Ideally, the injection should continue until the normalized polymer
concentration of the effluent becomes 1 (i.e., the injected and produced
concentrations are equal). The injection is often terminated after only
2–3 PV since meeting this criterion may take a long time.
The next phase involves washing out the polymer that has not been
irreversibly-adsorbed by injecting brine of same salinity as was used for
initial saturation of the core and the preparation of polymer solution.
The brine injection should continue until the washout is complete as
evidenced by pressure stabilization and the absence of any measurable
polymer or tracer in the effluent. Practically, due to time constraint, the
brine injection is stopped after more or less two pore volumes.
The normalized concentration is plotted vs. pore volume injected at
test conclusion, and a careful comparison is made between the polymer
and tracer profiles. The IPV is determined by the areal difference at the
trailing (back) edge of the two curves (the shaded areas in Fig. 7).
A typical normalized concentration profile of polymer and salt
tracer is shown in Fig. 7. Fig. 7 (a) shows an ideal case, where no dispersion or retention mechanism is present other than polymer adsorption. Fig. 7 (b) shows a real example involving dispersion as evidenced
by the “S” shape of the breakthrough profiles (Perkins and Johnston,
1963; Rodriguez Manrique et al., 2014) and also mechanical entrapment, which is the primary mechanism for the slow recovery of polymers after breakthrough (Farajzadeh et al., 2016; Huh et al., 1990).
For estimating IPV, this method intentionally avoids using the
leading edges (unshaded area in Fig. 7), i.e., the tracer and polymer
profiles after breakthroughs. This is because polymer adsorption decelerates polymer front advancement whereas IPV accelerates it, thus
partially or completely offsetting the lag between polymer and tracer
profiles. Therefore, this method estimates IPV by only using the difference between the tracer and the polymer profiles of the trailing edges
(i.e., depletions or shaded area in Fig. 7).
It is important to consider the impact of viscous fingering on front
profiles. During the polymer injection phase, a more viscous fluid
(polymer) displaces a less viscous fluid (brine). Thus the frontal interface between the two fluids is stable, i.e., there is no viscous fingering.
Therefore, the leading edges of the two profiles are not influenced by
instability. In contrast, the situation is reversed during the washout
phase when brine is displacing polymer and during which viscous fingering could occur distorting the trailing edge profiles. Since this
method uses trailing edge profiles for analysis, it is critical to ascertain
that displacement front during the washout period is stable by avoiding
high flow rates and gravity-aided core orientation. Otherwise, the
tracer profile at cutoff may appear earlier, and the polymer appears
later than expected due to faster diffusion (mixing) of the tracer component as compared to the polymer. Therefore, in the case of
6.1.2. Lotsch et al. Method (based on injection of double tracer-laden
polymer slugs)
First proposed by Lotsch et al. (1985), this method differs from the
Dawson & Lantz Method in that a second cycle of polymer injection and
brine washout sequence is repeated in the same manner as the first
cycle of the Dawson & Lantz method described above. Other researchers
have also endorsed this method (Mezzomo et al., 2002; Poellitzer et al.,
2009; Wassmuth et al., 2007). It must be emphasized though that the
adsorption in the first cycle should be complete before starting the
second cycle for obtaining accurate results by this method.
IPV is assessed using only the breakthrough front part of the effluent
curves during the second polymer slug injection (shaded areas in
Fig. 9). The back edges (cutoff) of the first cycle profiles that were used
in Dawson & Lantz Method are ignored in this method to avoid the
problems and uncertainties associated with viscous fingering and extended production of low concentration fluids (Manichand and Seright,
2014). After sample collection and concentration measurement during
the second cycle, the tracer and polymer profiles are analyzed to determine IPV by one of the two proposed strategies: areal difference and
breakthrough time difference.
6.1.2.1. IPV determination based on areal differences. For the areal
difference strategy, the IPV is determined by finding the separation
area between tracer and polymer profiles of the front edge of the second
slug, as shown in shaded areas of Fig. 9. An idealized diagram is
depicted in Fig. 9 (a) in which the IPV approximation is demonstrated
for an ideal case with the absence of dispersion, whereas Fig. 9 (b)
Fig. 8. Breakthrough curve for the sandpack flood with 0.5 wt % nanofluid cellulose nanocrystals. Adapted from (Aadland et al., 2018).
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Fig. 9. Inaccessible pore volume determination using double slug experiment in (a) an ideal polymer (black) and tracer (red) breakout curves and in (b) a typical
areal flood, adapted from (Zhao et al., 2017). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this
article.)
where Cp* (pv ) and Cs* (pv ) are normalized concentrations of polymer and
tracer as a function of injected pore volume, respectively. The limits of
the integral (20.13 and 29.79) are the pore volumes at the start and end
of the separation. The area between the curves can be graphically
computed using one of several software such as OriginPro by OriginLab
or Prism by GraphPad.
The majority of the researchers (Hughes et al., 1990; Lund et al.,
shows a real example.
Equation (4) can be used to obtain IPV from Fig. 9 (b) front edge
profiles (Zhao et al., 2017):
29.79
IPV =
∫ (Cp* (pv) − Cs* (pv)) dPV = 25.8%
20.13
(4)
Fig. 10. Tracer front shift during two successive Scleroglucan polymer slugs. Adapted from (Fournier et al., 2018).
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2010; Poellitzer et al., 2009) prefer to use the difference in pore
volumes corresponding to 0.5 normalized concentration (Ci*) between
polymer front in the second cycle and tracer front in the first cycle.
Fig. 12, in which the second cycle is also plotted starting from PV = 0
as the starting point, clarifies this concept. A horizontal line is drawn
through the polymer front profile of 2nd cycle up to the tracer front
profile of the 1st cycle. The pore volume values corresponding to the
0.5 normalized concentration of the two curves are read by drawing
vertical lines at points of intersection between the horizontal line and
the corresponding profiles. IPV is considered as the difference between
these pore volumes as mathematically expressed in Eq. (5).
1992; Manichand and Seright, 2014; Osterloh and Law, 1998; Wan and
Seright, 2016; Zhao et al., 2017) prefer to consider the areal difference
between tracer and polymer front edges of the second injection cycle
(shaded area of Fig. 9 (b)).
An alternative approach preferred by some researchers such as
Mukherjee et al. (2018) is to plot the second cycles on the same plot as
the first cycle but start counting the pore volume injected from zero so
that the second cycle data overlaps the first cycle as shown in Fig. 10.
The IPV is then calculated as the difference in area between the secondcycle polymer front-edge profile (curve a) and the first-cycle tracer
breakthrough curves (curve c).
Whether to use the tracer front profile of the first cycle or the second
cycle is still debatable. Some researchers (Ferreira and Moreno, 2017;
Fournier et al., 2018) prefer to use the second tracer front in their IPV
calculation, not the first one. Sometimes, the first and second tracer
fronts overlap at a normalized concentration of 0.5, allowing either
tracer front to be used. However, in some cases, such as the one shown
in Fig. 10, the first and second tracer front positions are not close to
each other. When the gap is significant, the decision as to which of the
tracer front profile to use for the IPV calculation should be made
carefully.
In almost all two-cycle experiments, the second tracer front breakthroughs earlier than the first one (compare curves b & c in Fig. 11)
because the polymer adsorption in the first cycle renders some pore
volume inaccessible to the tracer in the second cycle, thus reducing the
available pore volume for the tracer. Therefore, the area between the
first and second tracer profiles measures the pore volume which has
been rendered inaccessible to the brine (tracer) (Hughes et al., 1990;
Lund et al., 1992; Zaitoun and Kohler, 1987).
It is better to calculate IPV as the areal difference between the
second polymer front (curve a), and the first tracer front (curve c) since
the IPV of only the polymer component needs to be calculated that
should not include the IPV of the brine.
Moreover, taking Eq. (1) into account as the main equation for
computing the effective porosity in reservoir simulators, it is clear that
IPV is the primary cause of the difference between effective porosity
(ϕeff ) and the original porosity (ϕ ). While the polymer's effective porosity (pore volume at polymer break through time/bulk volume) can be
represented by the second polymer front, the original porosity (pore
volume at tracer break through time/bulk volume) will be better represented by the first tracer front at first cycle, rather than the tracer
front appeared in the second cycle.
IPV = (PVct*= 0.5 − PVcp*= 0.5) × 100
(5)
where PVct*= 0.5 and PVcp*= 0.5 are the injected pore volumes of tracer and
polymer, respectively, when the normalized concentrations in the
effluent reaches 0.5.
The theory of diffuse percolation states that when a water-saturated
core is flooded by a solution at an injected apparent velocity of V, the
front having a normalized concentration (Ci*) of 0.5 also moves at V.
Therefore, an accurate breakthrough time is th moment that this concentration reaches the outlet (Li et al., 2016). The cumulative injected
PV can be considered as the dimensionless breakthrough time of
polymer front.
Like in the earlier method based on the areal difference, a decision
has to be made regarding which of the tracer front profile to use for the
IPV calculation. The decision is easier and less significant if the first and
second tracer fronts have a small gap, or they overlap entirely. If the
gap is significant, however, a choice has to be made carefully after
giving due consideration.
6.1.2.3. Mechanical entrapment: IPV method selection. When mechanical
entrapment is negligible, both methods (areal and breakthrough-time
difference) may yield similar values for IPV (Lund et al., 1992). The
results diverge as the role of mechanical entrapment in the polymer
retention process increases with decreasing permeability, a condition
which is associated with high molecular weight polymer injection into
low permeability porous media.
An example of the difference in IPV determination by the two
methods when mechanical entrapment is occurring is shown in Fig. 13.
Without mechanical entrapment, the normalized polymer concentration (curve b) is zero until breakthrough and jump and stays to unity
after that. With mechanical entrapment, on the other hand, the
breakthrough is slightly delayed (curve c) and the concentration at a
breakthrough time jumps from zero to only around 0.5. After that, the
concentration gradually increases until eventually reaching unity after
many pore volumes of injection.
6.1.2.2. IPV determination based on breakthrough time difference. Several
researchers (Al-Hashmi et al., 2016; Al-Maamari et al., 2015; Divers
et al., 2018; Dupuis et al., 2017; Gaillard et al., 2014; Pancharoen et al.,
Fig. 11. Propagation of two successive xanthan slugs through a shaly sand pack. Adapted from (Zaitoun and Kohler, 1987).
8
Journal of Petroleum Science and Engineering 182 (2019) 106263
S. Akbari, et al.
Fig. 12. Inaccessible pore volume determination using a double slug experiment at 0.5 normalized concentration in a typical areal flood. Adapted from (Gaillard
et al., 2014).
Therefore, the areal difference method can be suggested only for a
polymer flooding in which there is no mechanical entrapment suspected. Else, the breakthrough time difference method should be used,
which can provide better accuracy for the one in which the mechanical
entrapment is suspected. It should be mentioned that if mechanical
entrapment occurs during coreflood experiment, the measurement of
polymer adsorption on the rock cannot be determined with a high degree of accuracy using the current methods.
As can be seen from Fig. 13, the areal difference between tracer
profile (curve a) and the polymer profile (curve b) without entrapment
is different and much smaller than the tracer profile (curve a) and the
polymer profile (curve c) with entrapment.
Another observation to be made from Fig. 13 is that the effect of
mechanical entrapment on the polymer effluent profile can be observed
only after recovering of 50% of injected polymer concentration (curve
c) in most cases. The mechanical entrapment of polymer can be a
lengthy process leading to a slow polymer recovery in the effluent
(Farajzadeh et al., 2016). As shown in Fig. 13, there is no polymer recovery up to Cp* = 0.5 (curve c) and a slow recovery thereafter generating a large areal difference between tracer (curve a) and polymer
(curve c) profiles which is not related to the IPV, and which may cause
incorrect determination of IPV.
6.1.2.4. Highlighting the statistical differences between different
methods. In a little known but an interesting study by Lotsch et al.
(1985), the IPV values were determined by each of the four possible
methods described above associated with the use of tracers under both
oil-free and residual oil saturation conditions. The eight results were
Fig. 13. Effect of mechanical entrapment on the concentration profile. Adapted from (Farajzadeh et al., 2016).
9
Journal of Petroleum Science and Engineering 182 (2019) 106263
S. Akbari, et al.
then averaged to obtain the final IPV. The tests were performed on
relatively high permeability (1600 md ≤ kw ≤ 2000 md) Bentheimer
sandstone cores of normal porosity (22–24%) using two biopolymers
(Scleroglucan and Xanthan).
Fig. 14 is a hypothetical diagram presented to show the concept of
their approach. The values in this figure do not correspond to the data
presented in Table 1 though the reference pointers such as A1 correspond to the one in the table and are useful in understanding the
concept. A schematic diagram of the test sequence is shown in Fig. 14.
The first injected slug had a higher polymer concentration than the
second slug to overcome the adsorption losses completely. The polymer
concentration in the second slug injection is lower since the role of
adsorption has been eliminated by the previous cycle, and the polymer
transportation phenomenon is dominant. Injection history is also shown
at the end of the figure.
Table 1, extracted from Lotsch et al.'s (1985) work, shows the results
of IPV determinations using the four different methods for each of the
oil-free and at residual oil conditions and the overall average of the
eight results. The first cycle of the test was used for the analysis of one
method (Dawson & Lantz), and the second cycle was used in the remaining three methods. The front profiles in the second cycle were
analyzed by two methods based on Lotsch et al. approach (areal &
breakthrough time differences). The last method was Dawson & Lantz
back-edge profile areal difference in cycle 2.
As evidenced by Table 1, different methods generated different IPV
values. The results are likely to be more reliable since having residual
oil in one of the tests makes the conditions closer to the hydrocarbon
reservoirs for which the tests were conducted in the first place. It is
noteworthy to mention that incorrect approximation of IPV will result
in an incorrect calculation of polymer losses in the porous media,
thereby leading to an inaccurate assessment of project profitability and
economic losses.
Finding the average of various methods is complicated and timeconsuming because it requires IPV to be computed from all existing
approaches. It is not clear if such an average will provide more accurate
results under all conditions, though reliability may be slightly improved. For example, if mechanical entrapment is present, then the
three methods based on the separation areas of tracer and polymer
profiles will be affected and will introduce error in the average IPV.
widely and successfully, there may arise a situation where doping with
tracer may not be the best option. Three methods that do not require
doping with tracers are explained below.
6.2.1. Pore volume at 0.5 of the normalized concentration of second
polymer slug
Several researchers (Idahosa et al., 2016; Rodriguez Manrique et al.,
2014) have applied this method of dual slug polymer injection without
a tracer for quick estimation of IPV. The technique requires an estimation of the effective pore volume of the core before polymer slug
injection (Seright et al., 2011).
The core is fully saturated with brine, and a polymer solution prepared in the same brine is injected until steady-state conditions are
reached. The purpose of the first polymer slug is to let the polymer
occupy all the adsorption sites. The brine is then injected to wash out all
non-retained polymer from the core until reaching to a steady-state
condition again. In the second cycle, the polymer is re-injected and
continued until reaching a steady-state condition. The count for the
cumulative pore volume injection for the second cycle starts from
PV = 0. Thus, the PV data is plotted, starting from the origin. The pore
volume injection of the second slug when the normalized concentration
reached 0.5 is noted (PVcp*= 0.5) and subtracted from 1 to yield IPV as per
Eq. (6) (Li et al., 2016):
IPV = 1 − PVcp*= 0.5
(6)
6.2. IPV determination without tracers
Fig. 15 is an example of a test on an artificial homogenous core. The
point Ci/Ci0 = 0.5 (y-axis) corresponds to an injected pore volume of
0.75 (x-axis). Therefore, the computed IPV equals 0.25 (i.e. 1–0.75).
Since the method does not include tracer injection, it might be associated with some risks such as having fractures in the porous medium.
The polymer breakthrough will happen earlier than expected because of
fracture. Thus the measured IPV will be erroneously higher. The fracture problem would have been identified if the tracer was also injected
and its concentration measured since the tracer breakthrough would
occur before one pore volume injection. Without fracture, the breakthrough would have been expected at one pore volume of injection.
Therefore, it is suggested that the homogeneity of the porous media be
independently verified using methods such as X-ray CT Scanning to
exclude the possibility of fracture or other abnormalities before using
this method. The existence of large dead volumes in the coreflood is
also a great factor of uncertainty for this method.
IPV determination methods using tracers require measurement of
tracer concentrations in the effluent samples along with the measurement of polymer concentrations. Whereas tracers have been used
6.2.2. By Simulating Polymer Floods with numerical simulators
In CMG (Computer Modelling Group), ECLIPSE, and other numerical simulators having polymer flooding options, there are keywords
Fig. 14. Principle of tracer test. Adapted from (Lotsch et al., 1985).
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S. Akbari, et al.
Table 1
Determination of inaccessible pore volume (IPV). Adopted from (Lotsch et al., 1985).a
Xanthan S/PF-4
Oil-Free Conditions
Polymer concentration (g/L)
The shift of PV between tracer & polymer front end profiles at
normalized concentration of 0.5
IPV by profile separation area at the front end (A2)
IPV by profile separation area at the backend of cycle 1 (A1)
IPV by profile separation area at the backend of cycle 2 (A3)
Mean Value
a
With 30% Residual Oil
Cycle 1
Cycle 2
Cycle 1
Cycle 2
2.5
–
1
8
2.5
–
1
10
–
11
9
–
10
–
10
–
10
10
IPV = (8 + 10+9 + 10+11 + 10+10 + 10)/8 = 10 Accuracy ± 2 (% PV) Standard Deviation = 0.9
Values do not correspond to Fig. 14 which is shown as a typical example though the reference pointers such as A1 etc. can still be used to understand the concept.
simulation method is thus not favorable if the aim is only to find the
IPV.
Special consideration is warranted to the polymer adsorption (retention) model when using this approach. Unlike IPV, which can be
determined experimentally, there is currently no direct method to obtain polymer adsorption/retention trend as a function of polymer concentration, which is required for simulating polymer flooding in the
porous media. The only way to obtain adsorption/retention trend is to
use it as a tuning parameter in the history matching process. In a
number of commercial simulators such as UTCHEM, CMG, and
ECLIPSE, polymer adsorption is demonstrated by the Langmuir isotherm model, which can be mathematically expressed as shown in Eq.
(7) (Goudarzi et al., 2013):
Cˆp =
ap (Cp − Cˆp)
1 + bp (Cp − Cˆp)
(7)
where Cp and Ĉp are the injected polymer concentration and adsorbed
polymer concentration, respectively. Cp - Ĉp is the equilibrium polymer
concentration in the rock. ap and bp are the tuning parameters that
allow ap to determine the maximum mono-layer coverage capacity.
During model validation and the history matching process, the tuning
parameters need to be adjusted to find the best match between observed results and simulation results.
If both IPV and adsorption/retention are determined by tuning the
simulator to match experimental history, the results may not be unique
since it is possible to have more than one pair of the tuned values that
could adequately match the historical data (Hatzignatiou et al., 2013).
This is because the effect of IPV and adsorption/retention on polymer
propagation are in the opposite direction, the former accelerates,
whereas the later decelerates. The combined effect, therefore, is
somewhere in between. In this case, the IPV may be estimated lower or
higher than the actual value, depending on the polymer adsorption/
Fig. 15. Dual slug polymer concentration profiles of an artificial homogenous
core with 0.1026 μm2 permeability. Adapted from (Li et al., 2016).
(e.g., PORFT keyword in the CMG and the first argument of the
PLYROCK keyword in the ECLIPSE) to define IPV. The simulator subtracts IPV from the total pore volume to determine the pore volume
accessible to polymer for each grid cell.
The IPV has been estimated by using it as a tuning parameter in the
simulation process of polymer flooding experimental results (AlSofi
et al., 2013; Hatzignatiou et al., 2013; Lüftenegger and Clemens, 2017).
IPV is found by selecting a value that provides the best match with the
experimental data (such as production profile and pressure drop). To
simulate a polymer flood accurately, a complete data set such as rock/
fluid properties is required that is time-consuming to acquire. This
Table 2
Pros and cons of various IPV determination methods.
IPV Determination Method
Pros
Cons
Stavland et al. Empirical.
Easiest method requiring only one cycle of
polymer flooding data and only RRF extracted
from experiments.
No tracer injection or tracer measurement
required thereby lower complexity of
experimental work.
only one cycle of polymer flooding.
Accuracy of the method not independently verified yet.
Pore Volume at 0.5 of the normalized
concentration of second polymer slug.
Dawson & Lantz Method (Based on the injection
of a single tracer-laden polymer slug).
Lotsch et al. Method (Based on the injection of
double tracer-laden polymer slugs).
By Simulating Polymer Floods with Numerical
Simulators.
Considered as the most reliable technique.
Since IPV is used as a tuning parameter only, thus
the experimental determination of IPV is not
required.
11
Not accurate if the porous medium has a fracture(s) or if the coreflood
system has high dead volume.
Lacks accuracy because of the long tailing of the back front if mobility
ratio is unfavorable.
Involves performing two cycles of tracer-laden polymer injection.
This method is considered the be the toughest. It requires the rock and
fluid properties data that is time-consuming. Then the history matching
process is itself a time-consuming process. The results may also not be
unique.
Journal of Petroleum Science and Engineering 182 (2019) 106263
S. Akbari, et al.
IPV determination without tracer.
retention value. In other words, the best breakthrough time match for
the observed effluent profile may be the result of a high (or low) estimation of adsorption and low (or high) estimation of the IPV. Therefore, this method of IPV estimation is risky and should be avoided if
possible. It is possible, though not verified, that if both the polymer
flooding and the brine flooding for washout are history-matched using
both tuning parameters, a unique pair of IPV and adsorption/retention
values may be obtained. It is hypothesized that on the basis that the
cutoff edge of the polymer slug will possibly not be affected by polymer
adsorption/retention.
•
6.2.3. Stavland et al. Empirical Model
Stavland et al. (2010) proposed a new method for IPV approximation based on the assumption that the porous medium is made of capillary tubes of varying diameters that obey the Poseuille Law. They
proposed Eq. (8) to approximate IPV assuming that the residual resistance factor (RRF) defined in Eq. (9) is known. This method has been
endorsed later by Hatzignatiou et al. (2013).
IPV = 1 −
RRF =
1
RRF
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(8)
∇pwa
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7. Comparing the advantages and disadvantages of each IPV
method
The discussion presented above is summarized in Table 2 to allow
for an informed decision before selecting an IPV method based on the
accuracy requirements.
8. Summary and concluding remarks
In a polymer flooding process, the inaccessible pore volume (IPV) is
a pore volume through which water/brine can propagate but not the
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IPV determination with tracer.
•
• Pore Volume at 0.5 of the normalized concentration of second polymer
slug: One should be sure that the porous medium is not fractured by
using methods such as X-ray CT Scanning.
• Stavland et al. Empirical Model: This method of IPV estimation
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subject of future works.
By Simulating Polymer Floods with Numerical Simulators: This method
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• Dawson & Lantz Method (Based on the injection of a single tracer-laden
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negligible, and that no adsorbed or entrapped polymer can be released.
Lotsch et al. Method (Based on injection of double tracer-laden polymer
slugs): Considered to be the best method provided that the first
tracer front is taken into account, rather than the tracer front of the
second slug.
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