Among chemical enhanced oil recovery (EOR) methods, polymer flooding is the most widely used due to its simplicity and low cost. Polymers are used to increase viscosity of aqueous phase which improves mobility ratio and form a uniform flood front to displace oil  ,  . It has been as a technically and commercially proven technology and it is considered as one of the most mature chemical flooding methods. Polymer rheological behaviour through porous media affects sweep efficiency and injectivity during flooding process.
Many features of polymer flow through porous media and its impact on oil recovery are somehow known, but there are some technical challenges to be addressed. For instance, shear rate dependent polymer rheology in porous media. Polymer rheology is controlled through shear force exerted by wall of pores while flowing through porous media  . Viscoelastic polymers exhibit shear- thinning behavior during rotational rheometry experiments whereas, when flowing through porous media the same polymer solution depicts shear thickening behavior-at a certain medium/high shear rates    . Core flood experiments have confirmed apparent viscosity dependence on shear rate  ,  . Converging-diverging geometry due to pore size distribution cause polymer stretching and contraction, which results into shear thickening depending upon polymer relaxation time  ,  . Viscoelastic polymer behavior can be defined and is often analyzed by flow through media  ,  or by oscillatory rheology measurements     . Different studies focusing on shear thinning and thickening properties have been presented. For instance, comparing the pressure drop behaviour against interstitial velocity (porosity, ϕ x superficial velocity, u) as reported by Sobti et al.  , Heemskerk et al.  , Hincapie  , Kaur et al.  . Numerous models to calculate shear rate as function injection rates have been proposed and analyzed in the literature  ,    . In addition, shear rate experienced by a polymer while flowing through a capillary tube has been studied by Wei et al.  Furthermore, apparent viscosity in porous media (core plugs) has been calculated in literature using Darcy’s formula  .
This study focuses on the analysis of polymers based on single-phase core, sand-pack and capillary tube (CT) experiments coupled with their rheological behavior characterization, in order to evaluate and predict polymer behavior in porous media. Impact of polymer concentration, pre-shearing conditions and different type of polymers on polymer viscoelasticity in porous media is also emphasized considering various approaches. Elongation dominated excessive pressure drop is calculated by comparing total pressure drop happening in core flood and pressure drop accounting the rheometer viscosity. This investigation contributes to understand the viscoelastic flow dynamics of polymer in reservoir. Multiple approaches have been utilized to understand viscoelastic response of injected fluid, as viscoelasticity of fluid can recover more oil. Moreover this study endorses the existence and measurement of elongation dominated flow in porous media.
2. Material and Methods
A descriptive summary of materials and methods considered during this study can be seen from Figure 1.
2.1. Brine and Polymers
Brine with Total Dissolved Solids (TDS) of 4.0 g/l is used as make-up water for diluted polymer solutions. Brine composition is described in Table 1. Brine was filtered through 0.2 μm filter size by applying 2.0 Bar of Nitrogen gas pressure in order to avoid any undissolved component.
Figure 1. Descriptive flow chart of materials and methods adopted during this study.
Table 1. Brine compositions used during the discussed experiments.
Two synthetic high molecular weight hydrolyzed polyacrylamides (HPAM) polymers and one bio-polymer are used in this work. Polymer solutions are prepared at three concentrations 500 ppm, 1000 ppm and 1500 ppm. The preparation procedure used is the one adopted by Hincapie and Ganzer,  . A detailed description of polymers is provided in Table 2. Diluted polymer solutions are filtered using a membrane filter with a size of 1.2 µm (pre-sheared solutions) and 5 µm (non-sheared solutions). Purpose of pre-shearing of polymer solution is well documented by Hincapie,  , Hincapie and Ganzer,  . It attempts to evaluate polymer stability, injectivity and viscoelastic properties. Shearing is performed by injecting polymer solutions through shearing device with the principle of flow through small nozzles, proposed by Hincapie and Ganzer,  . The shearing is done by applying 30 bar pressure with the purpose of degrading the macromolecules of the polymer without affecting polymer’s yield viscosity. This approach is expected will reduce potential injectivity problems as pointed by Hincapie and Ganzer,  .
2.2. Porous Media
Bentheimer core plugs are used in this work, with an average length and diameter of 60 mm and 30 mm, respectively. Core plugs were dried by placing them in an oven at 60˚C for at least 5 days. Porosity was measured using a Micromeritics 1340 pycnometer and permeability using a Gas Permeameter. Brine was injected at five different injection rates (0.5. 1.0, 2.0. 5.0, and 10.0 ml/min) to measure brine permeability. Table 3 provides a summary of the core plugs.
Table 2. Characteristics of polymers used for the discussed experiments.
Table 3. Porous media characteristics used for core flooding/sand-pack.
Quartz sand with grain size range of 63 - 80 µm screened was selected for sand-pack experiments. Average length and diameter of sand packs was same as those of core plugs. Porosity of sand-packs is measured through mass balance and volume balance method while permeability calculation is done using Darcy’s law by brine injection. Apart from that, stainless steel capillary tube (CT) of 0.5 mm ID and 3 m length is used to investigate impact of no converging-di- verging geometry on polymer solution.
3. Experimental Procedure
Figure 2 presents a descriptive and detailed sequence of the key experimental procedures adopted throughout this study.
3.1. Single-Phase Core Flooding
Figure 3 provides a schematic representation of the core-flood setup  . It consists of a two-syringe injection pump connected in a multi-flow arrangement (infusion/withdrawal) and controlled by irreversible check valves. The core plug is placed in a core holder that receives confining pressure, applied using a positive displacement hand pump. A rubber sleeve is used to secure the core plug inside the core holder. We used a pressure transmitter to gather pressure data. The confining pressure was kept constant at 30bar to guarantee radial flow. Once calibrated, the pressure transducer measured the pressure difference between the holder’s inlet and exit ports, with the data recorded on a computer.
3.2. Single-Phase Sand Packs Injection
Single-phase sand pack experiments were performed with an essentially similar setup as the one shown in Figure 3. The experiment started with the preparation of the sand; placing it into the cell and previously weighing the dry mass of
Figure 2. Descriptive flow chart of key experimental procedures adopted in this study.
Figure 3. Sketch of core flood experimental setup. It is connected via PTFE Teflon tubing and stainless steel pipes to the syringe pump and a differential pressure gauge.
the sand pack. The sand pack was subsequently saturated with CO2 prior to brine injection. This was to remove air bubbles and thus avoid air bubbles effect. After sufficient brine had been injected into the sand pack, permeability was determined. The procedures undertaken to perform the sand pack experiments were similar to those described for core plug experiments.
3.3. Single-Phase Capillary Tube (CT) Injection
A CT (capillary tube) with an internal diameter of 0.5 mm and length 2.5 m was shaped into a helical structure as shown in Figure 4. Results from this approach are compared with results of core flooding and sand-pack injection. Experimental components e.g. Pressure measurement device and injection pump are the same as discussed in previous sections.
4. Results and Discussions
4.1. Measurement of Pressure Drop across the Sand Packs and Cores
Viscoelastic phenomenon of polymer can be observed by drawing a similar scheme to that proposed by Heemskerk et al.  , Hincapie, R.,  , Tahir et al.,  ; where the measured pressure drop during polymer flooding experiments is plotted against interstitial velocity.
Heemskerk’s work described shear thinning behavior for a slope below 45˚, shear thickening for a slope greater than 45˚, and a Newtonian response for a slope of precisely 45˚. The results presented here are in exceptionally good agreement with those reported by Heemskerk et al.  , Hincapie,  , Tahir
Figure 4. Stainless steel capillary tube (CT) setup. Designedto compare apparent viscosity in core flood and sand-pack flood (impact of converging-diverging geometry).
et al.,  . Figure 5 illustrates the following clearly observed behaviors of HPAM polymer solutions (pre-sheared and non-sheared): a characteristic slope for a Newtonian response at 45˚, shear thinning below 45˚, and shear thickening behavior above 45˚.
4.2. Pressure Drop Differentiation by Shear and Elongation
A typical plot to represent the pressure response is one where pressure response is related to injection flow rates or interstitial velocity   . The overall approach taken in this work is reported in Figure 6(a) for the Bentheimer core plug and in Figure 6(b) for a sand pack experiment.
The plot considers the total pressure (∆PT) obtained during the core flooding measurement and the pressures associated with shear (∆PSh), determined using Darcy’s law. These results enable determination of the pressure drop assumed to be by elongation (∆PElo). It would appear from Figure 6 that the calculated pressure drop only matched the experimental data at low interstitial velocity.
where,∆PT = Total pressure obtained during the core flooding∆PS = Determined using Darcy equation and Carreau-Yasuda flow curve fitted to rheometer data∆PE = Pressure drop by elongation = ∆PT − ∆PS.4.3. Evaluation of Apparent Viscosity over Shear Rates for Sand Packs and CoresTypical plots to compare the matching between bulk viscosity and apparent
Figure 5. Measured pressure drop as a function of interstitial velocity for an HPAM solution through a Bentheimer core plug. The data plotted is for the HPAM Flopaam 6035 S, non-sheared (a) and pre-sheared (b) at different concentrations in a 4 g/l TDS brine solvent. Pressure data plotted is the one recorded during the experiment by flooding polymer solutions for different injection rates.
Figure 6. Measured and calculated pressure drop as a function of interstitial velocity for a HPAM solution through a Bentheimer Core (a) and Sand-pack (b). The data plotted is for the HPAM Flopaam 6035 S, non-sheared at 1500 ppm in a 4 g/l TDS brine solvent. Measured data plotted is the one recorded during the experiment by flooding polymer solutions for different injection rates. Calculated pressure is the one obtained by Darcy equation and the Carreau-Yasuda flow curve fitted to rheometer data.
viscosity are shown in Figure 7. It is a common practice to present the data in such a way because it enables demonstrating the differences observed during the core flooding to data obtained in the rheometer. Figure 7(a) presents a comparison of the different approaches in the case of a hydrolyzed polyacrylamide (HPAM) solution. Approach 1 is based upon equation used for shear rate calculation proposed by Cannella et al.,  , the approach 2 is taken from the equation proposed by Christopher and Middleman,  , the approach 3 is taken from Lee,  and the approach 4 is evaluated from the equation presented by Sheng,  . A summary of the shear rate correction coefficients is given in Table 4. Moreover, Figure 7(b) depicts the case for a biopolymer. Note that it is important to point out the differences of both plots. There was a tendency for the viscosity of the HPAM to increase at a certain apparent shear rate. This tendency
Table 4. Shear rate coefficients for solutions.
Figure 7. Apparent flow curves obtained by relating the bulk viscosity to core flooding. The data shown belong to a Hengfloc 63026 1500 ppm non-sheared in a 4 g/l TDS brine solvent (a) and a biopolymer Actigum CS6 1500 ppm non-sheared in a 4 g/l TDS brine solvent (b).
was not observed for the biopolymer. It is evident that these results demonstrate flow thickening or flow thinning in the porous media. This is pointed as a characteristic of flow in porous materials that is not observed in rheological measurements at similar rates. Similar observations have been presented previously by Cannella et al.,  , Hejri et al.,  , Teeuw and Hesselink,  . The results presented here are in good agreement with those reported in the literature.
Shear rates and apparent viscosity calculations using two different type of porous media (Bentheimer core and Quartz sand-pack) and CT are shown in Figure 8 for Hengfloc 63026 polymer solution. From Figure 8 it can be observed that the apparent viscosity and shear rates are calculated using the approach 1. As can be seen the core flooding and sand-pack data depicted an increased apparent viscosity due to the increment in pressure. This phenomenon has been previously observed as an onset. This is assumed to be related to stronger viscoelastic properties at higher shear rates. Unlike core flooding and sand packs, for the case of the CT experiments the only shear thinning behavior is observed. This is due to: 1) non-existence of pore size distribution (contrac- tions-expansions) in CT and 2) it is indeed a pipe flow (expected behavior). One
Figure 8. Apparent flow curves obtained by relating the bulk viscosity to core flooding, sand-pack and capillary tube injection. The data shown belongs to a Hengfloc 63026 1500 ppm non-sheared in a 4 g/l TDS brine solvent. Approach 1 is used for apparent shear rate calculation.
can argue that the apparent viscosity in sand-pack experiments is higher compared to the core flooding. This is explained by the variation in pore size distribution due to the sand grain size, as well as the differences in permeability and porosity, due to the low compaction.
Comparing shearing conditions of solutions given in Figure 9(a), sheared polymer showed lower bulk and apparent viscosity as related to non-sheared solutions over low shear rate values (1 s−1). This was expected behavior as mechanical shearing break-downs the molecular chains of polymer to avoid injectivity problems. But at higher shear rates (100 s−1), rheometer bulk viscosity for shear and non-sheared solution was almost same. At these higher shear rates, apparent viscosity for sheared solution was higher than non-sheared solution. This is because sheared polymers have less viscous forces as compare to non-sheared polymers. Thus less viscous forces will result more ease in flow and will result to dominance of turbulent pressure drop. This excessive pressure drop will contribute excessive apparent viscosity. But contribution of turbulence pressure drop at high injection rates need to be calculated  . Also in Figure 9(b) is the comparison of Flopaam 6035 S and Hengfloc 63026 1000 ppm non-sheared in a 4 g/l TDS brine solvent. Flopaam 6035S showed higher bulk and apparent viscosity as compare to Hengfloc 63026 solution. Reason could be difference in molecular structure and molecular weight of polymers (Table 2). Due to higher bulk and apparent viscosity, Flopaam 6035 S also showed injectivity issues as compare to Hengfloc 63026 in sand-packs.
Figure 9. Apparent flow curves obtained by relating the bulk viscosity to core flooding. The data shown on (a) belong to a Hengfloc 63026 1500 ppm non-sheared/sheared in a 4 g/l TDS brine solvent. The data shown on (b) belong to a Hengfloc 63026 1000 ppm non-sheared and Flopaam 6035 S 1000 ppm non-sheared in a 4 g/l TDS brine solvent. Approach 1 is used for apparent shear rate calculation.
4.4. Apparent Viscosity Related to Steady Shear Viscosity
The flow behaviour of an HPAM in a rheometer experiment compared to that in a core flooding experiment is commonly expressed as shown in Figure 10. Since the purpose in this work was to describe that onset or shear thickening, a different approach was chosen. Howe et al.,  proposed a plotting approach from observations in microchannel that seem to be useful, as expressing the behaviour observed in core plugs. The suggested evaluation is based on plotting the normalized data obtained by taking the apparent viscosity (measured during core experiments) divided by bulk viscosity (rheometer) against the apparent shear rate. As reported by Howe et al.,  , this manner of plotting is equivalent to plotting the measured pressure during core experiments divided by behaviour expected according to Darcy’s equation.
The results obtained for Flopaam 6035 S non-sheared and at different concentrations, are depicted in Figure 10(a). It is evident from the plot that at a certain shear rate the onset of elasticity occurs. This is consistent with results obtained by Howe et al.,  and clearly indicates non-dependence of this onset on polymer concentration. This is contrary to the generally accepted assumption often made by the authors in petroleum related literature. For instance, it is assumed that elasticity in porous media depends strongly on the linear viscoelasticity (the one observed in small amplitude oscillatory SAOS measurements). As reported by Hincapie et al.,  , SAOS can provide an insight only on whether a polymer might be elastic or inelastic, but it fails to predict the behaviour under flow. A similar trend was observed for pre-sheared solutions Figure 10(b).
Figure 10. Ratio between apparent viscosities measured in Bentheimer core plugs and Bulk viscosity by fitting the Carreau-Ya- suda model plotted against the apparent shear rate corrected by rheometer data. The data plotted is for the HPAM Flopaam 6035 S, non-sheared (a) and pre-sheared (b) in a 4 g/l TDS brine solvent.
Comparing Figure 10(a) and Figure 10(b) shows that an early onset seemed to occur for the non-sheared solutions. Quantitatively the ratio of apparent viscosity and bulk viscosity above 50% was found to be around 30 s−1, where the onset started for the non-sheared solutions and around 70 s−1 for the pre-sheared solutions. This was determined by observing the deviation of the ratio from the unity.
Additional support can be given to the results obtained in this work by considering the Weissenberg number. The Weissenberg number is described as a measure of the degree of nonlinearity or that degree to which normal stresses manifest during the flow  . For simplicity it can be considered as the product of the relaxation time (of the fluid) and the characteristic shear rate of the flow. In other words relaxation times obtained in Small Amplitude Oscillatory Shear (SAOS) multiplied by the apparent shear rate determined for each measurement. It can be observed in Figure 11 that the onset corresponded to the elastic properties of the polymer solutions. It could be inferred then, as reported by Hincapie,  , Howe et al.,  that flow instabilities are related to elastic turbulence.
Figure 11. Ratio between apparent viscosity measured in Bentheimer core plugs and bulk viscosity by fitting the Carreau-Yasuda model plotted against the apparent viscosity corrected by rheometer data as well as Weissenberg number. The data plotted is for the HPAM Flopaam 6035 S, non-sheared at different concentrations in a 4 g/l TDS brine solvent.
5. Summary and Conclusion
A detailed investigation of polymer viscoelasticity is performed and summarized using various approaches suggested by researchers from rheometer to its response in porous media. When comparing data obtained for core and sand pack experiments, there was a tendency for the viscosity of HPAM to increase at a certain apparent shear rate. This was not observed for the biopolymers. These results demonstrate flow thickening in the porous media. Further these observations were noted as a characteristic of flow in porous materials that was not observed in rheological measurements and CT injection at similar rates. Further evaluations on the shear thickening behaviour were achieved by plotting the normalized data against the apparent shear rate. The normalized data was obtained by dividing the apparent viscosity (measured during core experiments) by bulk viscosity (rheometer). It is evident from the plot that at a certain shear rate, the onset of elasticity is clearly seen. It is clear that this onset is independent of polymer concentration. Additional support to the elastic onset evaluation obtained in this work was given by considering the Weissenberg number. The following were clearly observed in polymer flooding in porous media: a characteristic slope for Newtonian behaviour equal to 45˚, shear thinning behaviour below to 45˚ and shear thickening behaviour above 45˚. As expected, a concentration dependence related to the pressure was observed. Based on this pressure differentiation, it was evident that the pressure drop occurring in the sand pack and core experiments were not only due to shear deformation, but also due to elongational deformation. Therefore to estimate the elongational pressure drop additional parameters need to be integrated.
The authors are thankful to the Institute of Petroleum Engineering (ITE) from Clausthal University of Technology in Germany, for providing the necessary facilities and financial support for the investigation.The authors also acknowledge the support given by the Open Access Publishing Fund of Clausthal University of Technology.
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