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Rheology and Permeability of Crosslinked Polyacrylamide Gel
Journal of Colloid and Interface Science 240, 601–607 (2001) doi:10.1006/jcis.2001.7633, available online at http://www.idealibrary.com on
Rheology and Permeability of Crosslinked Polyacrylamide Gel Carlos A. Grattoni, Hamed H. Al-Sharji, Canghu Yang, Ann H. Muggeridge, and Robert W. Zimmerman1 T. H. Huxley School of Environment, Earth Sciences and Engineering, Imperial College of Science, Technology and Medicine, London SW7 2BP, United Kingdom Received December 12, 2000; accepted April 13, 2001; published online July 12, 2001
Gels produced by crosslinking polyacrylamide solutions with chromium (III) have been characterized by dynamic rheology studies. To vary the gel strength, different polymer concentrations were used, while keeping the temperature, salinity, and crosslinker concentration constant. Both the loss and storage moduli increased with the polymer concentration for this gel system. The storage modulus at the end of the gelation was used to characterize the gel strength. Steady-state water flow experiments through gel-filled capillary tubes were performed, with the aim of linking the gel strength and flow behavior. The permeability was found to be a function of the water flow rate (velocity) and polymer concentration. Two parameters were used to characterize the flow behavior, intrinsic gel permeability and elasticity index, which are each functions of the polymer concentration. However, only one parameter is needed to fully identify the flow and rheological gel properties, as the elasticity index and storage modulus are linked by a power-law relationship. The loss modulus and intrinsic permeability are correlated with the storage modulus and elasticity index, respectively. A theoretical model for this behavior linking both gel properties based on the dual domain structure was used to demonstrate that the flow and rheological behavior of the gel are indeed related and that the gel strength controls the water permeability. Implications for prediction of flow of water through gels emplaced in a porous medium are discussed. °C 2001 Academic Press Key Words: polyacrylamide; gels; rheology; water flow; permeability; elasticity index.
INTRODUCTION
Polymer gels are viscoelastic materials, formed by a reaction between a polymer solution and a cross-linking agent, e.g., polyacrylamide crosslinked with chromium acetate. During the gelation process the gelant undergoes a “phase transition” from a liquid to a viscoelastic solid as the crosslinking reaction takes place. During this process a three-dimensional network is formed that gives the gel unique physical properties (1). Although there is no generally accepted method for determining gel strength, dynamic rheological measurements are becoming more widely used to evaluate the gel’s viscoelastic properties (2–4). 1 To whom correspondence should be addressed. E-mail: r.w.zimmerman @ic.ac.uk.
Polymer/crosslinker gels are used in a number of applications, including the control of excess water production during oil and gas recovery (5). In this case polymer and crosslinker solutions are mixed at the surface and injected into the reservoir rock through a production well. This solution is designed to react and form a gel within the pore space of the rock at reservoir temperatures. An annulus of viscous gel solution is then formed around the production well. The resulting gel affects the flow of both water and oil. The bases of a successful gel treatment are (a) the formation of a stable gel within the pore space of the rock before oil production is resumed and (b) the reduction of water permeability in the near well-bore region compared with the reduction of oil permeability, known as disproportionate permeability reduction or DPR. There are numerous papers in the literature discussing possible mechanisms for DPR (6–10). Recent work by Al-Sharji et al. (11) has shown that water flows through the gel as if through a porous medium, but the gel permeability to water varies as a power-law function of fluid velocity. This velocity effect has also been observed by several other workers in this field (7, 12). Al-Sharji et al. (11) concluded that the gel permeability to water (and oil) depends upon, among other things, the elastic properties of the gel. The purpose of this paper is to link the rheological properties of a fully developed gel to its velocity-dependent transport behavior. Different polymer concentrations were used to vary the gel strength while keeping the temperature, salinity, and crosslinker concentration constant. The gelation process was monitored with a dynamic rheometer that allows the determination of the complex shear modulus under lowfrequency oscillating shear deformation. The loss and storage moduli are functions of the polymer concentration for our particular gel system. The storage modulus at the end of the gelation is used here to characterize the gel strength. The velocity-dependent water permeability of the resulting polymer gels, at different concentrations, was investigated by flowing water at various flow rates through gel-filled capillary tubes. This velocity-dependent behavior is characterized in terms of the intrinsic gel permeability and elasticity index, which are also functions of the polymer concentration. Finally, a power-law relationship between the elasticity index and storage modulus was found, demonstrating that the flow and rheological behavior of the gel are indeed related. A theoretical model for
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C 2001 by Academic Press Copyright ° All rights of reproduction in any form reserved.
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this behavior based on the dual domain structure (dilute–dense domains) is presented. MATERIALS AND PREPARATION
The polymer used in this study is a polyacrylamide (Alcoflood 254S), which was provided by Allied Colloids Ltd. in powder form. According to the manufacturer’s data sheet this polymer has a molecular weight of 340,000 and an anionic content of less than 5%. Solutions of 1.8, 2.0, 2.5, and 3.0% (by weight) of polyacrylamide were prepared by slow addition of the polymer powder, as received, to brine in a vortex created by magnetic stirring. After the complete dissolution of the polymer, 200 ppm of crosslinker (chromium acetate from Lancashire Chemical Works) was added while stirring. As all the gelant solutions contain a high polymer to crosslinker ratio, 90 to 150 w/w (13), an excess of polymer (uncrosslinked) will exist in the gel. The brine consisted of 2% NaCl (w/w) dissolved in distilled water, giving a viscosity of 1.0 × 10−3 Pa s and a density of 1014 kg/m3 at 25◦ C. This brine was also used during the flow experiments. GELATION AND RHEOLOGICAL PROPERTIES
Method An initial estimate of gelation time and gel strength was determined by visual observation using the bottle test (13). The test consists of placing a test tube, partially filled with polymer solution and crosslinker, in a water bath and slightly tilting the tube at different times. The time at which the gelant inside the tube can no longer flow is taken as the time at which the gel has formed. For a polymer solution of 1.8% at 70 ± 0.5◦ C, a gel was formed at approximately 8 h with a strength of code G (moderate deformable gel) according to the classification scheme of Sydansk (13). No further changes were observed after about 12 h. For solutions with higher polymer concentrations, the gel was formed at earlier times and produced stronger gels. The temperature and gelant composition were selected as representative of those encountered in oil and gas reservoirs. The gelation process was then studied more quantitatively using a Paar Physica rheometer UDS 200, with a Z3 DIN attachment (concentric cylinders). This instrument applies a small amplitude, constant frequency, and oscillatory shear to the gelling system and measures the elastic and viscous responses as a function of time. The elastic response is characterized by the storage modulus (G 0 ), which quantifies the ability of the material to store elastic energy that can eventually be recovered. The viscous response is characterized by the loss modulus (G 00 ), which quantifies the amount of mechanical energy that is transformed into heat (i.e., energy lost) due to viscous forces. The ratio G 00 /G 0 is the tangent of the phase angle between stress and strain during oscillatory shear. Further details on dynamic oscillatory testing can be found elsewhere (2–3). Before starting each test, the system was heated to a temperature of 70◦ C and was maintained at this temperature throughout
FIG. 1. Evolution of the storage modulus during the gelation process at 69.8 ± 0.2◦ C for different polymer concentrations.
the test. The gelant solution was introduced into the cone-bob attachment and covered with mineral oil to prevent evaporation. The gelling system was then subjected to an oscillatory shear stress, with shear amplitude of 0.04 Pa at a frequency of 0.1 Hz, at constant temperature (69.8 ± 0.2◦ C) for up to 14 h. After initial tests the frequency and amplitude values were selected so that the gel structure would not be destroyed during the experiment. Results The changes in the storage and the loss modulus with time were used to follow the formation of the three-dimensional gel network structure for different polymer concentrations (see Figs. 1 and 2). At short times G 00 was higher than G 0 , and the system behaved as a viscous liquid. The time at which the loss and storage moduli were equal was used as an indication of the beginning of gelation. Comparison of Figs. 1 and 2 shows that the loss modulus remained low throughout the gelation process, while the storage modulus was initially low and increased sharply after the gelation started. A point was eventually reached after which no significant increase in G 0 was observed with time (∼ 400 min). At this point the gel structure was fully developed and the gel had achieved its maximum strength. Hence, it
FIG. 2. Evolution of the loss modulus during the gelation process at 69.8 ± 0.2◦ C for different polymer concentrations.
RHEOLOGY AND PERMEABILITY OF POLYACRYLAMIDE GEL
TABLE 1 Rheological Properties at the End of Gelation HPAM Storage Loss concn, C modulus, G 0 modulus, G 00 (%) (Pa) (Pa) 1.8 2.0 2.5 3.0
1.85 6.51 47.5 91.2
0.140 0.329 1.56 2.66
Sydansk code (nonflowing gels) G G–H H H–I
Moderately deformable Moderately–slightly deformable Slightly deformable Slightly deformable–rigid
seems logical to use the value of G 0 as an indicator of the gel strength of a mature gel. The values of G 0 and G 00 used henceforth correspond to fully developed gels at ∼ 70◦ C and are the averages over the last 60 min of measurement. The temperature reduction from gelation to ambient produces a moderate increase in both G 0 and G 00 while keeping its ratio almost constant. The properties of the gels as determined from the bottle test and the rheometer are summarized in Table 1. The gel’s storage and loss moduli both increased with initial polymer concentration (Fig. 3), although the amount of crosslinker used remained constant. It can be seen that there is a good correlation between G 0 and G 00 and the Sydansk code. As there is a linear relationship between storage and loss modulus, only one of them is needed to fully characterize the gel’s rheological properties. FLOW OF WATER THROUGH GELS
Method Steady-state flow experiments, consisting of water flowing through stationary gels, were then conducted to determine the gel’s permeability to water as a function of initial polymer concentration and velocity. In this case we were interested in the gel’s properties for the purposes of water shut off, so we chose to perform these experiments in gel-impregnated square glass capillary tubes. This tube shape resembles a typical single pore bounded by four rock grains, which is known as the grain bound-
FIG. 3. Maximum storage modulus and loss modulus, at the end of gelation, as functions of the polymer concentration (with C in percentage by weight).
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ary pore shape. To obtain a meaningful value of gel permeability, square cross-section capillary tubes of 0.1 × 0.1 × 7.0 cm were used. It should be noted that a measurement of the transmissivity of a gel-filled capillary tube would, in general, reflect the viscous resistance of both the gel and the channel walls. However, Al-Sharji et al. (11) demonstrated experimentally that, for a channel of this size, the effect of the channel walls is negligible, and the measured permeability accurately reflects that of the gel. Yang et al. (14) used a mathematical model based on a modified Brinkman equation and confirmed analytically that the wall effects are negligible when the ratio of the “permeability” of the empty tube to the permeability of the gel is greater than 100. The flow experiments were performed at room temperature. A syringe pump was used to induce a constant flow rate. The pressure was monitored and recorded using a pressure transducer connected to the inlet of the capillary tube, while the outlet of the model was kept at atmospheric pressure. The following preparation sequence was used for the flow experiment. First, the model was saturated with brine and the absolute permeability was measured. Then, 2 ml of gelant was injected at a constant flow rate of 5.6 × 10−4 cm3 /s. After cleaning the ports with brine, the model was shut in and placed into a water bath at 70 ± 0.5◦ C for 24 h to allow gelation. Finally, after gelation, dyed brine was injected at various flow rates. Results The results from the flow experiments are shown in Fig. 4. It can be seen that there is a nonlinear relationship between permeability and normalized velocity, v/vo (here vo is taken as 1 × 10−3 cm/s). When the gel’s permeability was analyzed it was found that it was velocity dependent, according to a power law that holds for velocities over several orders of magnitude (Fig. 4), k = ko (v/vo )b ,
[1]
where v = Q/A is the superficial velocity, vo is an arbitrary reference velocity, k is the gel permeability, ko is the gel
FIG. 4. Gel permeability as a function of the dimensionless water velocity for different polymer concentrations. The reference velocity vo is 0.001 cm/s.
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FIG. 5. Effect of a velocity cycle on gel permeability for a gel with a polymer concentration of 1.8%. The reference velocity vo is 0.001 cm/s.
permeability at velocity vo (hereafter referred to as the intrinsic permeability), and b is an exponent that lies between 0 and 1. This velocity-dependent behavior has been noted previously by Al Sharji et al. (11), among others. The permeability behavior does not show hysteresis when the velocity is increased and then decreased (Fig. 5), demonstrating that the velocity dependence is not produced by gel breakdown at higher velocities. Table 2 summarizes the results from Fig. 4 in terms of the gel’s intrinsic permeability ko and the power law exponent b. It can be seen that there is a decrease in the intrinsic permeability and in the exponent b as the concentration of the polymer increases. DISCUSSION
Velocity Dependence of Permeability Velocity-dependent (power-law) behavior of flow properties is usually observed when a non-Newtonian fluid, such as a polymer solution, flows through a rigid porous medium. However, as the saline water used in our studies is Newtonian, the power-law behavior can only be attributed to the gel properties. Typically, the gel consists of a three-dimensional crosslinked macromolecular polymer network with the interstitial space filled with the solvent, e.g., water (15). Normally the gel holds this solvent in place through its “interaction forces,” which also give the gel its solidity. In our flow experiments the imposed TABLE 2 Summary of Flow Properties as a Function of Polymer Concentration HPAM concn, C (%)
Intrinsic permeability, ko (m2 )
Elasticity index, b
1.8 2.0 2.5 3.0
0.0134 × 10−12 0.0059 × 10−12 0.0048 × 10−12 0.0044 × 10−12
0.806 0.797 0.693 0.622
FIG. 6. Flow elasticity index as a function of polymer concentration (with C in percentage by weight).
pressure gradient is greater than these interaction forces and so the water flows through the polymer network as if through a porous media. If the interaction forces were only holding the water in place and not giving the gel its solidity then the gel permeability would be independent of flow rate and the exponent b in Eq. [1] would be zero. However we observe that gel permeability increases with flow rate. It thus seems reasonable to suppose that when we enable the water to flow by overcoming the gel’s interaction forces, we are also deforming the gel in some way that is proportional to the imposed pressure gradient. From Table 2 and Fig. 6 we see that the exponent b decreases as the concentration of the polymer solution used to make the gel increases, and from Table 1 and Fig. 3 we see that the storage modulus of the gels increases with initial polymer concentration. In other words, the stiffer the gel, the lower the exponent b. Henceforth, we shall refer to b as an elasticity index and assume that it is related in some way to the storage modulus of the gel. Several researchers (1, 16, 17) have studied water flow through gels. Weiss and Silberberg (16) investigated the influence of polymer concentration on the viscoelasticity, heterogeneity, and permeability of polymer gels. They did not observe a velocity dependence of the gel permeability. Tokita and Tanaka (1) also claimed a linear relationship between pressure drop and water velocity, indicating that the water permeability for a given gel is constant. Nevertheless, in the present and some previous studies (11, 12) it was found that the permeability is velocity dependent. The discrepancy between the various results may be due to the type of gel and/or rigidity of the gels. Tokita and Tanaka (1) used higher polymer concentrations 5–8 wt%, while Weiss and Silberberg (16) used gels with storage moduli (2–3 × 103 Pa) almost 100 times larger than the values obtained for our system. For such rigid gels, it is likely that water flows through them as if through a rigid porous medium. Thus, the velocity dependence of gel permeability will vanish, resulting in a constant permeability. Additionally, the range of velocities used by both groups of workers was very narrow, 2–6 × 10−5 cm/sec. Within this range, it might be difficult to observe the variation of permeability with velocity.
RHEOLOGY AND PERMEABILITY OF POLYACRYLAMIDE GEL
Effect of Polymer Concentration on Intrinsic Gel Permeability Tokita and Tanaka (1) found a power-law relationship between the gel permeability and the correlation length, ξ , ko = αξ 2 .
[2]
They also related the gel permeability and polymer concentration, C, by ko = βC −1.5 ,
[3]
where α and β are arbitrary constants and ξ is defined as the average distance between neighboring contact points of the polymers. Our gel system also shows a similar power-law relationship between the gel’s intrinsic permeability and the polymer concentration (Fig. 7), but with an exponent of −1.85. The correlation length for our gel system is of the order of 10−7 m, which is comparable with the estimated size of the polymer molecule. The change in gel permeability and other rheological properties with polymer concentration may be explained as follows. Gel network structures are, in general, highly heterogeneous (1, 16, 18, 19). The reason for the heterogeneity of gels is mostly related to the crosslinking process. This results in a lattice-like structure consisting of two domains: the dilute domain (freedraining space) and the dense domain (nondraining space). This was first observed by Tanaka et al. (20), using small-angle neutron scattering. This dual domain structure explains the controls on gel permeability. The dense domains are centered on the crosslinked regions (crosslink clustering), where the polymers are bound together, and extend outward by entanglement of the free polymer onto the crosslinked regions. They are much less permeable than the dilute domains. As a result the gel permeability is controlled by the correlation length of the three-dimensional structure, i.e., the distance between the dense domains or crosslinked regions (15). If we assume that, even at the lowest polymer concentra-
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tions, all the crosslinker reacts with the polymer solution to form a gel, then an increase in polymer concentration, at a constant crosslinker concentration, would mainly cause an increase in the number of free polymer coils. Such free polymers will preferentially fill the dilute domains of the gel lattice, blocking or reducing the size of flow pathways, resulting in a decrease in gel permeability. Conversely, if some of the crosslinker has not been used up at the lowest initial polymer concentrations then as the polymer concentration is increased more crosslinked regions will form. This will also result in a less permeable structure as polymer concentration increases. Cohen and Christ (21) found that adsorbed polymer molecules in a porous media deform under flow-induced shear stress. Saphiannikova et al. (22) reported that “polymer brushes” can actually collapse under shear due to solvent flow. In our case, although the structure is more complex, as the free polymer molecules are retained by the crosslinked structure rather than adsorbed onto a surface, we suggest they can deform in a similar way. As the solvent velocity increases, more free polymer chains are forced (compressed) into the crosslinked regions. This increases the effective size of the pathways through the dilute domains, and hence the gel permeability increases as a function of velocity. This suggests that the compression process created by the solvent flow should be related to the elasticity of polymer chains and the gel structure. Let us first consider the system where all the crosslinker has been used up, even at the lowest initial polymer concentration. In this case, as the free polymer concentration increases, the crosslinked regions increase their relative volume due to entanglement. This effect reduces the volume of the dilute domains and, hence, the overall gel permeability (for a given flow rate). In the alternative case, the effect of increasing the polymer concentration is to increase the number of crosslinked regions while keeping the concentration of free polymer molecules in the dilute domains more or less constant. The increase in the number of crosslinked regions results in small and fewer dilute regions, and again we see a reduction in the gel permeability, for a given velocity, as the initial polymer concentration increases. In both cases, as the total volume remains more or less constant, an increase in the polymer concentration decreases the ratio of volumes between dense and dilute domains. We characterize this behavior by the elasticity index b defined in the preceding section. A plot of elasticity index vs. polymer concentration is shown in Fig. 6. It can be seen that there is a strong inverse relationship between them, showing that when polymer concentration increases the elasticity index decreases. Effect of Polymer Concentration on Rheological Properties
FIG. 7. Intrinsic gel permeability as a function of polymer concentration (with C in percentage by weight).
The effect of the dual domain structure on the gel’s rheological properties is as follows. The highly crosslinked regions at the center of the dense domains will probably control the gel’s elastic response or its storage modulus. In contrast the dilute domains, which contain mainly fluid, will have a predominantly
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viscous response. However, as mentioned earlier, an increase in concentration will affect both domains and as a consequence both the elastic and the viscous responses will be affected. If we assume that some of the crosslinker has not been used up at the lowest initial polymer concentrations, then, as the polymer concentration is increased, more crosslinked regions will form, resulting in a higher storage modulus, while the loss modulus should remain constant. Conversely, if all of the crosslinker has reacted with the polymer solution and we have an excess of initial polymer concentration, then an increase in polymer concentration will result in more free polymer coils in the dilute domain and induce a more viscous structure with a larger loss component (G 00 ). In this case the crosslinked polymer backbone has little freedom of movement, resulting in less flexing and twisting of the polymer, and we would expect the storage modulus to be more or less constant as the initial polymer concentration increases. However, the overlapping of the free polymer molecules will increase with polymer concentration, resulting in further entangling with the crosslinked network. The polymer chains cannot then slide over each other, thus resulting in an increase in the storage modulus. The relationship between the storage modulus and the polymer concentration for our gel system is shown in Fig. 3. It can be seen that the storage modulus increases with the initial polymer concentration and that this behavior is well described by a power-law relationship. As there is also a power-law relationship between polymer concentration and gel permeability, it is to be expected that a similar relationship should exist between storage modulus and gel intrinsic permeability. This hypothesis is confirmed in Fig. 8. Our storage modulus exponent is very different from that measured by Takebe et al. (23) using dynamic light scattering and mechanical studies. However, the values of storage modulus reported here are similar to those presented by Kakadjian et al. (3) and Broseta et al. (4) for similar gel systems and compositions. Thus, the discrepancy in exponent may be due to the type and rigidity of the gels used in the various studies.
FIG. 9. Flow elasticity index as a function of storage modulus.
To link gel strength and flow behavior, a relationship between storage modulus and elasticity index is needed. Figure 9 shows that there is a slow decrease in elasticity index with an increase in storage modulus, which can also be represented by a power law. This can be explained as follows. The larger the storage modulus, the stiffer the gel, and hence it more closely resembles a rigid porous medium. The permeability of a rigid porous medium does not of course depend upon velocity, so the elasticity index is zero. The stiffness of the gel increases with polymer concentration, because the free polymer coils become increasingly entangled with the crosslinked polymer. This entanglement reduces the flexibility of the backbone and increases the gel stiffness. In summary, the gel structure, or distribution of dense and dilute domains, controls the solvent flow and the intrinsic gel permeability. The same structure determines the gel rheological behavior. For a given polymer and crosslinker concentration, an increase in flow velocity compresses the free polymer chains into the dense domains. A similar effect occurs when mechanical forces deform the gel. An increase in polymer concentration increases the number of free polymer coils, resulting in more entanglement, with dense domains reducing the size of the dilute domains and hence reducing the permeability and increasing the storage modulus of the gel. CONCLUSIONS
FIG. 8. Intrinsic gel permeability as a function of storage modulus.
The rheological properties of crosslinked, polyacrylamide gel have been measured and related to the permeability of the gel to water, over a range of polymer concentrations. These properties, and their behavior as a function of polymer concentration, have been interpreted in terms of the dual domain structure first identified by Tanaka et al. (20). The gel system examined always had an excess of polymer to crosslinker, resulting in significant quantities of free polymer coils existing within the gel structure after gelation was complete. The storage and loss moduli of the gel were measured during the gelation process by applying a low-amplitude,
RHEOLOGY AND PERMEABILITY OF POLYACRYLAMIDE GEL
constant-frequency oscillatory shear to the gelling system. The values of these parameters were consistent with results obtained using the Sydansk bottle test. Both the storage and the loss moduli obtained at the end of gelation increase with polymer concentration, according to a power-law relationship. The gel’s permeability to water was found to increase with flow velocity and decrease with polymer concentration. Again, these behaviors can be modeled with a power-law expression. In particular, the velocity-dependent behavior of the gel permeability for a given polymer concentration can be represented in terms of a so-called intrinsic gel permeability and an elasticity index. Both of these parameters were found to be functions of polymer concentration and, hence, of gel storage modulus. Our results imply that the velocity-dependent behavior of the gel’s permeability to water is due to its elasticity. We propose that its intrinsic permeability is controlled by the distribution of dense and dilute domains within the gel structure, while its velocitydependent behavior is governed by movement and deformation of the free polymer molecules existing within the dilute domains. The relationships among polymer concentration, storage modulus, gel intrinsic permeability, and elasticity index obtained in this study, combined with the theoretical modeling results of Yang et al. (14), have the potential to enable engineers to predict water permeability behavior directly from rheological measurements. This would allow engineers to select the most appropriate gelant formulation for a given reservoir water shut off treatment using data obtained from standard rheology studies, rather than from extensive core tests as is done at present. ACKNOWLEDGMENTS H. Al-Sharji’s work was supported by Petroleum Development Oman, and C. Yang’s work was supported by a CVCP Overseas Research Students Award. The authors also thank Paul Luckham of Imperial College for allowing the use of the Paar Physica rheometer.
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REFERENCES 1. Tokita, M., and Tanaka, T., J. Chem. Phys. 95, 4613 (1991). 2. Winter, H., and Chambon, F., J. Rheol. 30, 367 (1986). 3. Kakadjian, S., Rauseo, O., and Mejias, F., SPE Paper 50751, 1999. 4. Broseta, D., Marquer, O., Zaitoun, A., Baylock, P., and Fery, J. J., SPE Res. Eval. 3, 204 (2000). 5. Dalrymple, E. D., Eoff, L., Reddy, B. R., and Botermans, C. W., in “Proceedings, 11th Oil Field Chemistry Symposium, Norway,” 2000. 6. Dawe, R. A., and Zhang, Y., J. Pet. Sci. Eng. 12, 113 (1994). 7. Liang, J. T., and Seright, R. S., SPE Prod. Facil. 12, 225 (1997). 8. Liang, J. T., and Seright, R. S., Soc. Pet. Eng. 59344 (2000). 9. Barreau, P., Lasseux, D., Bertin, H., Gl´enat, P., and Zaitoun, A., Pet. Geosci. 5, 201 (1999). 10. Nilsson, S., Stavland, A., and Jonsbr˚aten, H. C., SPE Paper 39635, 1998. 11. Al-Sharji, H. H., Grattoni, C. A., Dawe, R. A., and Zimmerman, R. W., SPE Paper 56738, 1999. 12. Yan, Z., McCool, C. S., Geen, D. W., and Willhite, G. P., SPE Paper 50753, 1999. 13. Sydansk, R. D., SPE Res. Eval. 5, 346 (1990). 14. Yang, C., Grattoni, C. A., Muggeridge, A. H., and Zimmerman, R. W., in “7th EAGE Eur. Conf. Math. Oil Recovery, Italy,” 2000. 15. Zhang, X., Hu, Z., and Li, Y., Polymer 39, 2783 (1998). 16. Weiss, N., and Silberberg, A., Br. Polym. J. 6, 144 (1977). 17. Allain, C., and Amiel, C., Phys. Rev. Lett. 56, 1501 (1986). 18. Tokita, M., and Tanaka, T., Science 253, 1121 (1991). 19. Matsuo, E. S., Orkisz, M., Sun, S. T., Li, Y., and Tanaka, T., Macromolecules 27, 6791 (1994). 20. Tanaka, T., Hocker, L. O., and Benedek, G. B., J. Chem. Phys. 59, 5151 (1973). 21. Cohen, Y., and Christ, F. R., SPE Res. Eng. 1, 113 (1986). 22. Saphiannikova, M. G., Pryamitsyn, V. A., and Cosgrove, T., Macromolecules 31, 6662 (1998). 23. Takebe, T., Nawa, K., Suehiro, S., and Hashimoto, T., Chem. Phys. 91, 4360 (1989).
Rheology and Permeability of Crosslinked Polyacrylamide Gel Carlos A. Grattoni, Hamed H. Al-Sharji, Canghu Yang, Ann H. Muggeridge, and Robert W. Zimmerman1 T. H. Huxley School of Environment, Earth Sciences and Engineering, Imperial College of Science, Technology and Medicine, London SW7 2BP, United Kingdom Received December 12, 2000; accepted April 13, 2001; published online July 12, 2001
Gels produced by crosslinking polyacrylamide solutions with chromium (III) have been characterized by dynamic rheology studies. To vary the gel strength, different polymer concentrations were used, while keeping the temperature, salinity, and crosslinker concentration constant. Both the loss and storage moduli increased with the polymer concentration for this gel system. The storage modulus at the end of the gelation was used to characterize the gel strength. Steady-state water flow experiments through gel-filled capillary tubes were performed, with the aim of linking the gel strength and flow behavior. The permeability was found to be a function of the water flow rate (velocity) and polymer concentration. Two parameters were used to characterize the flow behavior, intrinsic gel permeability and elasticity index, which are each functions of the polymer concentration. However, only one parameter is needed to fully identify the flow and rheological gel properties, as the elasticity index and storage modulus are linked by a power-law relationship. The loss modulus and intrinsic permeability are correlated with the storage modulus and elasticity index, respectively. A theoretical model for this behavior linking both gel properties based on the dual domain structure was used to demonstrate that the flow and rheological behavior of the gel are indeed related and that the gel strength controls the water permeability. Implications for prediction of flow of water through gels emplaced in a porous medium are discussed. °C 2001 Academic Press Key Words: polyacrylamide; gels; rheology; water flow; permeability; elasticity index.
INTRODUCTION
Polymer gels are viscoelastic materials, formed by a reaction between a polymer solution and a cross-linking agent, e.g., polyacrylamide crosslinked with chromium acetate. During the gelation process the gelant undergoes a “phase transition” from a liquid to a viscoelastic solid as the crosslinking reaction takes place. During this process a three-dimensional network is formed that gives the gel unique physical properties (1). Although there is no generally accepted method for determining gel strength, dynamic rheological measurements are becoming more widely used to evaluate the gel’s viscoelastic properties (2–4). 1 To whom correspondence should be addressed. E-mail: r.w.zimmerman @ic.ac.uk.
Polymer/crosslinker gels are used in a number of applications, including the control of excess water production during oil and gas recovery (5). In this case polymer and crosslinker solutions are mixed at the surface and injected into the reservoir rock through a production well. This solution is designed to react and form a gel within the pore space of the rock at reservoir temperatures. An annulus of viscous gel solution is then formed around the production well. The resulting gel affects the flow of both water and oil. The bases of a successful gel treatment are (a) the formation of a stable gel within the pore space of the rock before oil production is resumed and (b) the reduction of water permeability in the near well-bore region compared with the reduction of oil permeability, known as disproportionate permeability reduction or DPR. There are numerous papers in the literature discussing possible mechanisms for DPR (6–10). Recent work by Al-Sharji et al. (11) has shown that water flows through the gel as if through a porous medium, but the gel permeability to water varies as a power-law function of fluid velocity. This velocity effect has also been observed by several other workers in this field (7, 12). Al-Sharji et al. (11) concluded that the gel permeability to water (and oil) depends upon, among other things, the elastic properties of the gel. The purpose of this paper is to link the rheological properties of a fully developed gel to its velocity-dependent transport behavior. Different polymer concentrations were used to vary the gel strength while keeping the temperature, salinity, and crosslinker concentration constant. The gelation process was monitored with a dynamic rheometer that allows the determination of the complex shear modulus under lowfrequency oscillating shear deformation. The loss and storage moduli are functions of the polymer concentration for our particular gel system. The storage modulus at the end of the gelation is used here to characterize the gel strength. The velocity-dependent water permeability of the resulting polymer gels, at different concentrations, was investigated by flowing water at various flow rates through gel-filled capillary tubes. This velocity-dependent behavior is characterized in terms of the intrinsic gel permeability and elasticity index, which are also functions of the polymer concentration. Finally, a power-law relationship between the elasticity index and storage modulus was found, demonstrating that the flow and rheological behavior of the gel are indeed related. A theoretical model for
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C 2001 by Academic Press Copyright ° All rights of reproduction in any form reserved.
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this behavior based on the dual domain structure (dilute–dense domains) is presented. MATERIALS AND PREPARATION
The polymer used in this study is a polyacrylamide (Alcoflood 254S), which was provided by Allied Colloids Ltd. in powder form. According to the manufacturer’s data sheet this polymer has a molecular weight of 340,000 and an anionic content of less than 5%. Solutions of 1.8, 2.0, 2.5, and 3.0% (by weight) of polyacrylamide were prepared by slow addition of the polymer powder, as received, to brine in a vortex created by magnetic stirring. After the complete dissolution of the polymer, 200 ppm of crosslinker (chromium acetate from Lancashire Chemical Works) was added while stirring. As all the gelant solutions contain a high polymer to crosslinker ratio, 90 to 150 w/w (13), an excess of polymer (uncrosslinked) will exist in the gel. The brine consisted of 2% NaCl (w/w) dissolved in distilled water, giving a viscosity of 1.0 × 10−3 Pa s and a density of 1014 kg/m3 at 25◦ C. This brine was also used during the flow experiments. GELATION AND RHEOLOGICAL PROPERTIES
Method An initial estimate of gelation time and gel strength was determined by visual observation using the bottle test (13). The test consists of placing a test tube, partially filled with polymer solution and crosslinker, in a water bath and slightly tilting the tube at different times. The time at which the gelant inside the tube can no longer flow is taken as the time at which the gel has formed. For a polymer solution of 1.8% at 70 ± 0.5◦ C, a gel was formed at approximately 8 h with a strength of code G (moderate deformable gel) according to the classification scheme of Sydansk (13). No further changes were observed after about 12 h. For solutions with higher polymer concentrations, the gel was formed at earlier times and produced stronger gels. The temperature and gelant composition were selected as representative of those encountered in oil and gas reservoirs. The gelation process was then studied more quantitatively using a Paar Physica rheometer UDS 200, with a Z3 DIN attachment (concentric cylinders). This instrument applies a small amplitude, constant frequency, and oscillatory shear to the gelling system and measures the elastic and viscous responses as a function of time. The elastic response is characterized by the storage modulus (G 0 ), which quantifies the ability of the material to store elastic energy that can eventually be recovered. The viscous response is characterized by the loss modulus (G 00 ), which quantifies the amount of mechanical energy that is transformed into heat (i.e., energy lost) due to viscous forces. The ratio G 00 /G 0 is the tangent of the phase angle between stress and strain during oscillatory shear. Further details on dynamic oscillatory testing can be found elsewhere (2–3). Before starting each test, the system was heated to a temperature of 70◦ C and was maintained at this temperature throughout
FIG. 1. Evolution of the storage modulus during the gelation process at 69.8 ± 0.2◦ C for different polymer concentrations.
the test. The gelant solution was introduced into the cone-bob attachment and covered with mineral oil to prevent evaporation. The gelling system was then subjected to an oscillatory shear stress, with shear amplitude of 0.04 Pa at a frequency of 0.1 Hz, at constant temperature (69.8 ± 0.2◦ C) for up to 14 h. After initial tests the frequency and amplitude values were selected so that the gel structure would not be destroyed during the experiment. Results The changes in the storage and the loss modulus with time were used to follow the formation of the three-dimensional gel network structure for different polymer concentrations (see Figs. 1 and 2). At short times G 00 was higher than G 0 , and the system behaved as a viscous liquid. The time at which the loss and storage moduli were equal was used as an indication of the beginning of gelation. Comparison of Figs. 1 and 2 shows that the loss modulus remained low throughout the gelation process, while the storage modulus was initially low and increased sharply after the gelation started. A point was eventually reached after which no significant increase in G 0 was observed with time (∼ 400 min). At this point the gel structure was fully developed and the gel had achieved its maximum strength. Hence, it
FIG. 2. Evolution of the loss modulus during the gelation process at 69.8 ± 0.2◦ C for different polymer concentrations.
RHEOLOGY AND PERMEABILITY OF POLYACRYLAMIDE GEL
TABLE 1 Rheological Properties at the End of Gelation HPAM Storage Loss concn, C modulus, G 0 modulus, G 00 (%) (Pa) (Pa) 1.8 2.0 2.5 3.0
1.85 6.51 47.5 91.2
0.140 0.329 1.56 2.66
Sydansk code (nonflowing gels) G G–H H H–I
Moderately deformable Moderately–slightly deformable Slightly deformable Slightly deformable–rigid
seems logical to use the value of G 0 as an indicator of the gel strength of a mature gel. The values of G 0 and G 00 used henceforth correspond to fully developed gels at ∼ 70◦ C and are the averages over the last 60 min of measurement. The temperature reduction from gelation to ambient produces a moderate increase in both G 0 and G 00 while keeping its ratio almost constant. The properties of the gels as determined from the bottle test and the rheometer are summarized in Table 1. The gel’s storage and loss moduli both increased with initial polymer concentration (Fig. 3), although the amount of crosslinker used remained constant. It can be seen that there is a good correlation between G 0 and G 00 and the Sydansk code. As there is a linear relationship between storage and loss modulus, only one of them is needed to fully characterize the gel’s rheological properties. FLOW OF WATER THROUGH GELS
Method Steady-state flow experiments, consisting of water flowing through stationary gels, were then conducted to determine the gel’s permeability to water as a function of initial polymer concentration and velocity. In this case we were interested in the gel’s properties for the purposes of water shut off, so we chose to perform these experiments in gel-impregnated square glass capillary tubes. This tube shape resembles a typical single pore bounded by four rock grains, which is known as the grain bound-
FIG. 3. Maximum storage modulus and loss modulus, at the end of gelation, as functions of the polymer concentration (with C in percentage by weight).
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ary pore shape. To obtain a meaningful value of gel permeability, square cross-section capillary tubes of 0.1 × 0.1 × 7.0 cm were used. It should be noted that a measurement of the transmissivity of a gel-filled capillary tube would, in general, reflect the viscous resistance of both the gel and the channel walls. However, Al-Sharji et al. (11) demonstrated experimentally that, for a channel of this size, the effect of the channel walls is negligible, and the measured permeability accurately reflects that of the gel. Yang et al. (14) used a mathematical model based on a modified Brinkman equation and confirmed analytically that the wall effects are negligible when the ratio of the “permeability” of the empty tube to the permeability of the gel is greater than 100. The flow experiments were performed at room temperature. A syringe pump was used to induce a constant flow rate. The pressure was monitored and recorded using a pressure transducer connected to the inlet of the capillary tube, while the outlet of the model was kept at atmospheric pressure. The following preparation sequence was used for the flow experiment. First, the model was saturated with brine and the absolute permeability was measured. Then, 2 ml of gelant was injected at a constant flow rate of 5.6 × 10−4 cm3 /s. After cleaning the ports with brine, the model was shut in and placed into a water bath at 70 ± 0.5◦ C for 24 h to allow gelation. Finally, after gelation, dyed brine was injected at various flow rates. Results The results from the flow experiments are shown in Fig. 4. It can be seen that there is a nonlinear relationship between permeability and normalized velocity, v/vo (here vo is taken as 1 × 10−3 cm/s). When the gel’s permeability was analyzed it was found that it was velocity dependent, according to a power law that holds for velocities over several orders of magnitude (Fig. 4), k = ko (v/vo )b ,
[1]
where v = Q/A is the superficial velocity, vo is an arbitrary reference velocity, k is the gel permeability, ko is the gel
FIG. 4. Gel permeability as a function of the dimensionless water velocity for different polymer concentrations. The reference velocity vo is 0.001 cm/s.
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FIG. 5. Effect of a velocity cycle on gel permeability for a gel with a polymer concentration of 1.8%. The reference velocity vo is 0.001 cm/s.
permeability at velocity vo (hereafter referred to as the intrinsic permeability), and b is an exponent that lies between 0 and 1. This velocity-dependent behavior has been noted previously by Al Sharji et al. (11), among others. The permeability behavior does not show hysteresis when the velocity is increased and then decreased (Fig. 5), demonstrating that the velocity dependence is not produced by gel breakdown at higher velocities. Table 2 summarizes the results from Fig. 4 in terms of the gel’s intrinsic permeability ko and the power law exponent b. It can be seen that there is a decrease in the intrinsic permeability and in the exponent b as the concentration of the polymer increases. DISCUSSION
Velocity Dependence of Permeability Velocity-dependent (power-law) behavior of flow properties is usually observed when a non-Newtonian fluid, such as a polymer solution, flows through a rigid porous medium. However, as the saline water used in our studies is Newtonian, the power-law behavior can only be attributed to the gel properties. Typically, the gel consists of a three-dimensional crosslinked macromolecular polymer network with the interstitial space filled with the solvent, e.g., water (15). Normally the gel holds this solvent in place through its “interaction forces,” which also give the gel its solidity. In our flow experiments the imposed TABLE 2 Summary of Flow Properties as a Function of Polymer Concentration HPAM concn, C (%)
Intrinsic permeability, ko (m2 )
Elasticity index, b
1.8 2.0 2.5 3.0
0.0134 × 10−12 0.0059 × 10−12 0.0048 × 10−12 0.0044 × 10−12
0.806 0.797 0.693 0.622
FIG. 6. Flow elasticity index as a function of polymer concentration (with C in percentage by weight).
pressure gradient is greater than these interaction forces and so the water flows through the polymer network as if through a porous media. If the interaction forces were only holding the water in place and not giving the gel its solidity then the gel permeability would be independent of flow rate and the exponent b in Eq. [1] would be zero. However we observe that gel permeability increases with flow rate. It thus seems reasonable to suppose that when we enable the water to flow by overcoming the gel’s interaction forces, we are also deforming the gel in some way that is proportional to the imposed pressure gradient. From Table 2 and Fig. 6 we see that the exponent b decreases as the concentration of the polymer solution used to make the gel increases, and from Table 1 and Fig. 3 we see that the storage modulus of the gels increases with initial polymer concentration. In other words, the stiffer the gel, the lower the exponent b. Henceforth, we shall refer to b as an elasticity index and assume that it is related in some way to the storage modulus of the gel. Several researchers (1, 16, 17) have studied water flow through gels. Weiss and Silberberg (16) investigated the influence of polymer concentration on the viscoelasticity, heterogeneity, and permeability of polymer gels. They did not observe a velocity dependence of the gel permeability. Tokita and Tanaka (1) also claimed a linear relationship between pressure drop and water velocity, indicating that the water permeability for a given gel is constant. Nevertheless, in the present and some previous studies (11, 12) it was found that the permeability is velocity dependent. The discrepancy between the various results may be due to the type of gel and/or rigidity of the gels. Tokita and Tanaka (1) used higher polymer concentrations 5–8 wt%, while Weiss and Silberberg (16) used gels with storage moduli (2–3 × 103 Pa) almost 100 times larger than the values obtained for our system. For such rigid gels, it is likely that water flows through them as if through a rigid porous medium. Thus, the velocity dependence of gel permeability will vanish, resulting in a constant permeability. Additionally, the range of velocities used by both groups of workers was very narrow, 2–6 × 10−5 cm/sec. Within this range, it might be difficult to observe the variation of permeability with velocity.
RHEOLOGY AND PERMEABILITY OF POLYACRYLAMIDE GEL
Effect of Polymer Concentration on Intrinsic Gel Permeability Tokita and Tanaka (1) found a power-law relationship between the gel permeability and the correlation length, ξ , ko = αξ 2 .
[2]
They also related the gel permeability and polymer concentration, C, by ko = βC −1.5 ,
[3]
where α and β are arbitrary constants and ξ is defined as the average distance between neighboring contact points of the polymers. Our gel system also shows a similar power-law relationship between the gel’s intrinsic permeability and the polymer concentration (Fig. 7), but with an exponent of −1.85. The correlation length for our gel system is of the order of 10−7 m, which is comparable with the estimated size of the polymer molecule. The change in gel permeability and other rheological properties with polymer concentration may be explained as follows. Gel network structures are, in general, highly heterogeneous (1, 16, 18, 19). The reason for the heterogeneity of gels is mostly related to the crosslinking process. This results in a lattice-like structure consisting of two domains: the dilute domain (freedraining space) and the dense domain (nondraining space). This was first observed by Tanaka et al. (20), using small-angle neutron scattering. This dual domain structure explains the controls on gel permeability. The dense domains are centered on the crosslinked regions (crosslink clustering), where the polymers are bound together, and extend outward by entanglement of the free polymer onto the crosslinked regions. They are much less permeable than the dilute domains. As a result the gel permeability is controlled by the correlation length of the three-dimensional structure, i.e., the distance between the dense domains or crosslinked regions (15). If we assume that, even at the lowest polymer concentra-
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tions, all the crosslinker reacts with the polymer solution to form a gel, then an increase in polymer concentration, at a constant crosslinker concentration, would mainly cause an increase in the number of free polymer coils. Such free polymers will preferentially fill the dilute domains of the gel lattice, blocking or reducing the size of flow pathways, resulting in a decrease in gel permeability. Conversely, if some of the crosslinker has not been used up at the lowest initial polymer concentrations then as the polymer concentration is increased more crosslinked regions will form. This will also result in a less permeable structure as polymer concentration increases. Cohen and Christ (21) found that adsorbed polymer molecules in a porous media deform under flow-induced shear stress. Saphiannikova et al. (22) reported that “polymer brushes” can actually collapse under shear due to solvent flow. In our case, although the structure is more complex, as the free polymer molecules are retained by the crosslinked structure rather than adsorbed onto a surface, we suggest they can deform in a similar way. As the solvent velocity increases, more free polymer chains are forced (compressed) into the crosslinked regions. This increases the effective size of the pathways through the dilute domains, and hence the gel permeability increases as a function of velocity. This suggests that the compression process created by the solvent flow should be related to the elasticity of polymer chains and the gel structure. Let us first consider the system where all the crosslinker has been used up, even at the lowest initial polymer concentration. In this case, as the free polymer concentration increases, the crosslinked regions increase their relative volume due to entanglement. This effect reduces the volume of the dilute domains and, hence, the overall gel permeability (for a given flow rate). In the alternative case, the effect of increasing the polymer concentration is to increase the number of crosslinked regions while keeping the concentration of free polymer molecules in the dilute domains more or less constant. The increase in the number of crosslinked regions results in small and fewer dilute regions, and again we see a reduction in the gel permeability, for a given velocity, as the initial polymer concentration increases. In both cases, as the total volume remains more or less constant, an increase in the polymer concentration decreases the ratio of volumes between dense and dilute domains. We characterize this behavior by the elasticity index b defined in the preceding section. A plot of elasticity index vs. polymer concentration is shown in Fig. 6. It can be seen that there is a strong inverse relationship between them, showing that when polymer concentration increases the elasticity index decreases. Effect of Polymer Concentration on Rheological Properties
FIG. 7. Intrinsic gel permeability as a function of polymer concentration (with C in percentage by weight).
The effect of the dual domain structure on the gel’s rheological properties is as follows. The highly crosslinked regions at the center of the dense domains will probably control the gel’s elastic response or its storage modulus. In contrast the dilute domains, which contain mainly fluid, will have a predominantly
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viscous response. However, as mentioned earlier, an increase in concentration will affect both domains and as a consequence both the elastic and the viscous responses will be affected. If we assume that some of the crosslinker has not been used up at the lowest initial polymer concentrations, then, as the polymer concentration is increased, more crosslinked regions will form, resulting in a higher storage modulus, while the loss modulus should remain constant. Conversely, if all of the crosslinker has reacted with the polymer solution and we have an excess of initial polymer concentration, then an increase in polymer concentration will result in more free polymer coils in the dilute domain and induce a more viscous structure with a larger loss component (G 00 ). In this case the crosslinked polymer backbone has little freedom of movement, resulting in less flexing and twisting of the polymer, and we would expect the storage modulus to be more or less constant as the initial polymer concentration increases. However, the overlapping of the free polymer molecules will increase with polymer concentration, resulting in further entangling with the crosslinked network. The polymer chains cannot then slide over each other, thus resulting in an increase in the storage modulus. The relationship between the storage modulus and the polymer concentration for our gel system is shown in Fig. 3. It can be seen that the storage modulus increases with the initial polymer concentration and that this behavior is well described by a power-law relationship. As there is also a power-law relationship between polymer concentration and gel permeability, it is to be expected that a similar relationship should exist between storage modulus and gel intrinsic permeability. This hypothesis is confirmed in Fig. 8. Our storage modulus exponent is very different from that measured by Takebe et al. (23) using dynamic light scattering and mechanical studies. However, the values of storage modulus reported here are similar to those presented by Kakadjian et al. (3) and Broseta et al. (4) for similar gel systems and compositions. Thus, the discrepancy in exponent may be due to the type and rigidity of the gels used in the various studies.
FIG. 9. Flow elasticity index as a function of storage modulus.
To link gel strength and flow behavior, a relationship between storage modulus and elasticity index is needed. Figure 9 shows that there is a slow decrease in elasticity index with an increase in storage modulus, which can also be represented by a power law. This can be explained as follows. The larger the storage modulus, the stiffer the gel, and hence it more closely resembles a rigid porous medium. The permeability of a rigid porous medium does not of course depend upon velocity, so the elasticity index is zero. The stiffness of the gel increases with polymer concentration, because the free polymer coils become increasingly entangled with the crosslinked polymer. This entanglement reduces the flexibility of the backbone and increases the gel stiffness. In summary, the gel structure, or distribution of dense and dilute domains, controls the solvent flow and the intrinsic gel permeability. The same structure determines the gel rheological behavior. For a given polymer and crosslinker concentration, an increase in flow velocity compresses the free polymer chains into the dense domains. A similar effect occurs when mechanical forces deform the gel. An increase in polymer concentration increases the number of free polymer coils, resulting in more entanglement, with dense domains reducing the size of the dilute domains and hence reducing the permeability and increasing the storage modulus of the gel. CONCLUSIONS
FIG. 8. Intrinsic gel permeability as a function of storage modulus.
The rheological properties of crosslinked, polyacrylamide gel have been measured and related to the permeability of the gel to water, over a range of polymer concentrations. These properties, and their behavior as a function of polymer concentration, have been interpreted in terms of the dual domain structure first identified by Tanaka et al. (20). The gel system examined always had an excess of polymer to crosslinker, resulting in significant quantities of free polymer coils existing within the gel structure after gelation was complete. The storage and loss moduli of the gel were measured during the gelation process by applying a low-amplitude,
RHEOLOGY AND PERMEABILITY OF POLYACRYLAMIDE GEL
constant-frequency oscillatory shear to the gelling system. The values of these parameters were consistent with results obtained using the Sydansk bottle test. Both the storage and the loss moduli obtained at the end of gelation increase with polymer concentration, according to a power-law relationship. The gel’s permeability to water was found to increase with flow velocity and decrease with polymer concentration. Again, these behaviors can be modeled with a power-law expression. In particular, the velocity-dependent behavior of the gel permeability for a given polymer concentration can be represented in terms of a so-called intrinsic gel permeability and an elasticity index. Both of these parameters were found to be functions of polymer concentration and, hence, of gel storage modulus. Our results imply that the velocity-dependent behavior of the gel’s permeability to water is due to its elasticity. We propose that its intrinsic permeability is controlled by the distribution of dense and dilute domains within the gel structure, while its velocitydependent behavior is governed by movement and deformation of the free polymer molecules existing within the dilute domains. The relationships among polymer concentration, storage modulus, gel intrinsic permeability, and elasticity index obtained in this study, combined with the theoretical modeling results of Yang et al. (14), have the potential to enable engineers to predict water permeability behavior directly from rheological measurements. This would allow engineers to select the most appropriate gelant formulation for a given reservoir water shut off treatment using data obtained from standard rheology studies, rather than from extensive core tests as is done at present. ACKNOWLEDGMENTS H. Al-Sharji’s work was supported by Petroleum Development Oman, and C. Yang’s work was supported by a CVCP Overseas Research Students Award. The authors also thank Paul Luckham of Imperial College for allowing the use of the Paar Physica rheometer.
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