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Rheological Behavior of Foamy Oils† Patrice Abivin,‡,§ Isabelle Henaut,*,‡ Jean-Francois Argillier,‡ and Michel Moan§ IFP, 1 et 4 AVenue de Bois-Pre´au, 92852 Rueil-Malmaison, France, and Laboratoire de Rhe´ologie, UniVersite´ de Bretagne Occidentale, 6 aVenue Le Gorgeu 29238 Brest, France ReceiVed August 14, 2008. ReVised Manuscript ReceiVed October 12, 2008
When a reservoir is depleted, the lightest components (methane, ethane, etc.) can exsolve from the crude oil and create a gaseous phase. In conventional oils, the bubbles grow and coalesce quickly and the gas usually separates from the oil (slug flow). On the contrary, in heavy oils, bubbles are small, remain dispersed, and flow within the oil for a long time. This “foamy oil” phenomenon can drastically change the flow properties of the crude oil. This paper is devoted to the characterization of the heavy oil foaming behavior through a rheological study. Our objectives are to study the kinetics of bubble evolution in heavy oil and to measure the influence of the bubbles on the heavy oil viscosity. A new experimental method was developed on the basis of rheological measurements under pressure. Several heavy oils containing dissolved gas have been depleted inside the pressure cells of controlled stress rheometers to create foamy oils. Viscosity and viscoelastic properties have been continuously measured using respectively continuous and oscillatory tests from the nucleation to the disengagement of bubbles from oil. Results reveal that, under low shear rates, the presence of bubbles leads to an increase in heavy oil viscosity, as predicted by the hard sphere model or Taylor. A theoretical model describing the viscosity of foamy oil was then established. It takes into account both first-order kinetics of the appearance and release of bubbles in oil and a classic suspension model. Good agreement was obtained between experimental data and model predictions. Several tests reveal the strong influence of the shear rate on the foamy oil behavior and point out the major role of bubble deformation on the viscosity of foamy oils, as shown previously in other viscous materials, such as magmas and polymers. Under high shear rates, we suggest that the stabilization of the elongated bubbles in oil leads to the establishment of an anisotropic material, which can be seen as a sandwich-like structure. As a result, the viscosity appears lower in the direction of the flow.
Introduction Heavy oil rheological behavior appears as a key issue in heavy oil production and has been partly addressed in the past.1 These results on “pure” heavy oil are of primary importance, but progresses in determining the heavy oil flow properties during production and transport are still needed. Indeed, while oil is extracted, gas and water are also produced, leading to a much more complex mixture than the single crude oil. This paper focuses on the impact of bubbles on the flow properties of heavy oils. In a reservoir, a thermodynamic equilibrium is naturally established between the lightest hydrocarbons (methane, ethane, etc.) dissolved in the oil and the liquid phase. The amount of dissolved gas in the so-called “live oil” is proportional to the system pressure and given by Henry’s law (eq 1) [G] ) HP
(1)
where H is Henry’s constant, depending upon the chemical affinity of both fluids and the temperature,2 P is the pressure, †
Presented at the 9th International Conference on Petroleum Phase Behavior and Fouling. * To whom correspndence should be addressed. Tel: +33 1 4752 6386. E-mail: isabelle.henault.ifp.fr. ‡ IFP. Present address: Schlumberger, DBR Technology Center, 9450 17 Avenue, Edmonton, Alberta T6N 1M9, Canada. § Universite ´ de Bretagne Occidentale. (1) Pierre, C.; Barre, L.; Pina, A.; Moan, M. Composition and heavy oil rheology. Oil Gas Sci. Technol. 2004, 59 (5), 511–521. (2) Kuss, E. The visosity of gas/oil solutions at high pressure. High Temp.-High Pressures 1983, 15, 93–105.
and [G] is the amount of dissolved gas. While the live oil is extracted, it is depressurized during its transport from the reservoir to the wellhead. The equilibrium given by Henry’s law is then broken; the live oil is supersaturated; and the system tends to get rid of the dissolved gas in excess. This leads to the nucleation and growth of bubbles within the oil and thus to modifications in its composition and its flow properties. In the case of light oils, bubbles coalesce very quickly and a slug flow can appear out of the well.3,4 On the opposite, in the case of heavy oils, bubbles remain dispersed and flow within the oil, which gives to the produced oil the aspect of a “chocolate mousse”. This phenomenon is called “foamy oil phenomenon”,5,6 despite the fact that it is quite far from a conventional dry foam. The gas volume fraction is indeed usually comprised between 5 and 40%,7 and a better designation would thus probably be “bubbly oil”. This phenomenon appears to be linked to higher production rates than expected by reservoir modeling.5,8 In Canada, these high production rates seem to also be linked to (3) Taitel, Y. Stability of severe slugging. Int. J. Multiphase Flow 1986, 12 (2), 203–217. (4) Henriot, V.; Duret, E.; Heintze, E.; Courbot, A. Multiphase production control: Application to slug flow. Oil Gas Sci. Technol. 2002, 57 (1), 87–98. (5) Smith, G. E. Fluid flow and sand production in heavy oil reservoirs under solution-gas drive. SPE Prod. Eng. 1988, 3 (2), 169–180. (6) Maini, B. B.; Sarma, H. K.; George, A. E. Significance of foamy oil behaviour in primary production of heavy oils. J. Can. Pet. Technol. 1993, 32 (9). (7) Bauget, F. Production d’huiles lourdes par de´pressurisation: Etudes des interfaces huiles-air et mode´lisation du proce´de´. Ph.D. Thesis, University Paris-Sud, Orsay, France, 2002.
10.1021/ef8006646 CCC: $40.75 2009 American Chemical Society Published on Web 12/11/2008
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the production of sand and the formation of wormholes in the reservoir, enhancing the drainage process, but it appears that, in Venezuela, the foamy oil behavior is the most important parameter, with the production of sand being very low.9 Furthermore, developing the foamy oil behavior (by modifying the pumping process for instance) is a technique used to enhance heavy oil recovery in Albania.10 To understand why the foamy oil phenomenon is linked to higher production rates, several theories have been explored. One of them considers that the bubbles that are trapped in the reservoir maintain a high pressure in the porous medium for a longer time, which enhances the oil recovery.11 The bubble formation can also be seen as a “natural” gas lift process that improves oil rise in the well. Other theories suggest that the live oil viscosity is lower with the presence of bubbles, thus improving its transport in the well and the reservoir. For instance, a theory suggests that asphaltenes migrate to the surface of bubbles, which has been confirmed later in a toluene + asphaltene/methane interface,12 and the resulted impoverishment of the matrix in asphaltenes decreases the foamy oil viscosity.13 However, different experimental studies on foamy oil viscosity predict either an increase14-16 or a decrease17,18 in the live oil viscosity with the presence of bubbles. The influence of bubbles on the heavy oil flow properties is thus still debated. This study addresses this issue in bulk. In a previous paper, we studied the kinetics of gas exsolution in different oils.19 This one investigates the influence of bubbles on flow properties of heavy oils under one main aspect: the influence of the shear rate on the viscosity of foamy oils. It focuses on the flow properties of heavy oils in wells and pipelines by studying the flow behavior of foamy oil in bulk. Materials and Methods The density of the Canadian heavy crude oil mainly used in this study is 0.982 at 25 °C; its viscosity is 9 Pa s at 28.5 °C; it is (8) Huerta, M.; Otero, C.; Rico, A.; Jimenez, I.; de Mirabal, M.; Rojas, G. Understanding foamy oil mechanisms for heavy oil reservoirs during primary production. SPE Tech. Pap. 36749, 1996. (9) de Mirabal, M.; Rojas, G.; Gordillo, R.; Rodriguez, H.; Huerta, M. Impact of foamy oil mechanism on the hamaca oil reserves. SPE Tech. Pap. 36140, 1996. (10) Weatherhill, B. D.; Seto, A. C.; Gupta, S. K.; Cobo, L. Cold heavy oil production at Patos Marinza, Albania. Presented at SPE International Thermal Operations and Heavy Oil Symposium Calgary, Alberta, Canada, 2005. (11) Renard, G.; Nauroy, J.-F.; Deruyter, C.; Moulu, J.-C.; Sarda, J.-P.; Le Romancer, J.-F. Production froide des huiles visqueuses. Oil Gas Sci. Technol. 2000, 55, 35–66. (12) Bauget, F.; Langevin, D.; Lenormand, R. Dynamic surface properties of asphaltenes and resins at the oil-air interface. J. Colloid Interface Sci. 2001, 239 (2), 501–508. (13) Claridge, E. L.; Prats, M. A. Proposed model and mechanism for anomalous foamy heavy oil behaviour. SPE Tech. Pap. 29243, 1995. (14) Bora, R. Cold production of heavy oils: An experimental investigation of foamy oil flow in porous media. Ph.D. Thesis, University of Calgary, Alberta, Canada, 1998. (15) Fisher, D. G.; Espidel, J.; Huerta, M.; Randall, L.; Goldman, J. Use of magnetic resonance imaging as a tool for the study of foamy oil behaviour for an extra-heavy crude oil. Transp. Porous Media 1999, 35, 189–204. (16) Goodarzi, N.; Bryan, J.; Mai, A.; Kantzas, A. Heavy-oil fluid testing with conventional and novel techniques. Presented at SPE International Thermal Operations and Heavy Oil Symposium, Calgary, Alberta, Canada, 2005. (17) Islam, M. R.; Chakma, A. Mechanics of bubble flow in heavy oil reservoirs. SPE Tech. Pap. 20070, 1990. (18) Albartamani, N. S. Experimental studies on “foamy oil” phenomena. Ph.D. Thesis, University of Alberta, Edmonton, Alberta, Canada, 2000. (19) Abivin, P.; He´naut, I.; Argillier, J.-F.; Moan, M. Experimental study of foamy oil behavior; preliminary development of a viscosity model. Pet. Sci. Technol. 2008, 26 (13), 1545–1558.
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Figure 1. Scheme of the rheometer pressure cell.
Newtonian on a wide range of shear rates; and its asphaltene content is roughly 11% [determined by saturates, aromatics, resins, and asphaltenes (SARA) analysis in heptane]. We use two controlledstress rheometers: Anton Paar Physica MCR 501 and TA Instruments AR2000. Both are fitted with a pressure cell (up to 100 bar, 107 Pa). A schematic diagram of these experimental devices is presented in Figure 1. Magnetic transmission is used between the rheometer motor and the geometry. Temperature is controlled by a Peltier system. Pressure is applied to the sample by gas (nitrogen or methane) and controlled by a pressure-regulating valve. The most appropriate geometries are a helix or a vane to avoid any slipping effect on the geometry and to provide a good mixing process to dissolve gas in oil. The apparent viscosity is calculated from these particular devices using the Couette analogy,20 and the results are checked on different internal standards on both rheometers. The results from both rheometers are also compared to one another and found similar. We use both continuous and oscillatory measurements. The basic principle of an oscillatory rheometer is to induce a sinusoidal shear deformation in the sample and to measure the resultant stress response. It is usually used to determine the viscoelastic properties of the material, but these measurements also provide the complex viscosity. The latter is equivalent to a “continuous” viscosity under very low applied strain deformations that are sufficiently small not to affect the material properties and not disturb the bubble growth and rise. A X-ray scanner was also used to visualize the bubbles in the pressure cell of the rheometers. The experiments are divided into two steps. First, the live oil is prepared in the pressure cell of the rheometer, recombining dead oil and gas. The oil is maintained under a constant pressure of methane and sheared (100 s-1). Some gas dissolves in the oil until the system reaches the equilibrium given by Henry’s law. Mixing is very important during the process: the better the mixing, the faster the system reaches the thermodynamic equilibrium, i.e., saturation. Using the helix and the vane devices, this process is several hours long and is almost impossible with a classic Couette geometry. During this process, the viscosity is continuously measured and drops with the dissolution of gas. The viscosity after the mixing process is stable, even after several hours without any shear applied on the sample. This shows that the drop in the viscosity is only due to dissolved gas and not the presence of entrained bubbles during the mixing process. Therefore, the live oil is considered as saturated when the viscosity is stabilized. The gas dissolution conditions were the same for all of the samples: 30 bar (3 × 106 Pa) in methane or 90 bar (9 × 106 Pa) in nitrogen, both corresponding to approximately the same amount of dissolved gas in oil. The latter has been measured using PVT cells and approximately corresponds to a gas oil ratio (GOR, ratio of the gas volume to the oil volume at room pressure and 15 °C) of 6. During the second step of the experiments, the live oil is depressurized from the saturation pressure to atmosphere. The viscosity of the system is continuously measured under a constant (20) Aït-Kadi, A.; Marchal, P.; Choplin, L.; Chrissemant, A.-S.; Bousmina, M. Quantitative analysis of mixer-type rheometers using the couette analogy. Can. J. Chem. Eng. 2002, 80 (6), 1166–1174.
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Figure 2. Apparent complex viscosity ratio of a Venezuelan heavy oil saturated in nitrogen (90 bar) and depleted instantaneously to room pressure at t ) 180 s. (A) X-ray visualization of a foamy oil in the pressure cell of the rheometer. (B) Zoom-in on the first minutes of the experiment.
shear rate or an oscillatory test during and after the depletion. Figure 2 shows an example of a heavy oil (from Venezuela) saturated in nitrogen at 90 bar (9 × 106 Pa) and 40 °C depressurized instantaneously from the saturation pressure to room pressure. We use here an oscillatory test (2% strain, 5 Hz). The presence of bubbles during the experiments was checked by X-ray visualizations, as one can see in Figure 2A. To do so, the pressure cell of the rheometer was depressurized at rest (no rheological measurement) in the X-Ray scanner. The viscosity ratio plotted here versus time (t) is the ratio of the apparent complex viscosity of the mixture at t to the complex viscosity of the dead oil. Its evolution with time can be described as follows: (i) the viscosity ratio starts below 1 because the dissolved gas acts as a diluent and reduces the apparent viscosity of the live oil; (ii) the viscosity ratio shall equal 1 at the end of the experiment, when the live oil is at equilibrium at room pressure, i.e., when the oil has become a dead oil again and has removed all of the dissolved gas in excess; (iii) in between, the viscosity can remain below 1 or rise to a higher value, depending upon the viscosity of the oil.19 As one can see, in Figure 2, it remains below 1.
Results and Discussion Influence of the Bubbles on Foamy Oil Viscosity. The evolution of the viscosity ratio with time after the depressurization (Figure 2) can be described in three phases: (1) A drastic decrease in viscosity consecutive to the depressurization. This is due to the pressure dependence of the live oil viscosity. Indeed, any fluid viscosity is more or less pressure-dependent,21 as well as are crude oils22 and especially heavy oils because of their complex ramified structures.23 (2) A sharp increase in the viscosity. This is due to the release of gas from the live oil and, possibly, to the presence of bubbles. We know that a lot of bubbles are present in oil (see the X-ray visualization in Figure 2A), but we still need to determine their influence on the foam apparent viscosity. This will be addressed later in this paper. (3) A slow increase in the viscosity during several hours because of the same reasons: the impoverishment of the matrix in dissolved gas and, possibly, the presence of bubbles. (21) Glasstone, S.; Laidler, K. J.; Eyring, H. The Theory of Rate Processes: The Kinetics of Chemical Reactions, Viscosity, Diffusion and Electrochemical Phenomena; McGraw-Hill: New York, 1941. (22) Kuss, E. Extreme values of the pressure coefficient of viscosity. Angew. Chem., Int. Ed. Engl. 1965, 4 (11), 944–950.
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As one can see, foamy oil kinetics is very slow: the equilibrium is reached only several hours after the depressurization. This kinetics mainly depends upon the viscosity of the crude oil.19 The second important point here is that, in this case, the viscosity ratio is below 1 during the whole experiment. It becomes then challenging to understand the influence of the bubbles besides other parameters, such as the amount of dissolved gas in the live oil. Indeed, it has been observed through X-ray measurements that the nucleation seems to be progressive and not instantaneous, which fits with some models of the literature.7,24,25 Therefore, during the experiment, there is still a certain amount of dissolved gas that tends to diminish the apparent viscosity of the oil matrix, which can screen the role of the bubbles. To better appreciate the foamy oil phenomenon, a simple viscosity model has been proposed.19 It takes into account the different major parameters that control the viscosity of foamy oils. These main parameters are the pressure, the dissolved gas content, and the presence of bubbles. Therefore, this model is based on the sharing against time of gas as (i) dissolved gas, (ii) gas in bubbles, and (iii) disengaged gas and their different impact on the live oil viscosity. We designed by nO, the amount of gas in the oil phase, nB, the amount of gas in bubbles, and nG, the amount of gas released from oil. Influence of Pressure. The underlying foundations of the viscosity based on the Eyring theory state that the pressure dependence of the viscosity is exponential.21 A piezodependence constant κ is defined as follows: d log(η) (2) dp where η is the viscosity and p is the pressure. κ was experimentally measured for the different oils used in this study. Influence of the Dissolved Gas Content. To take into account the dissolved gas content on the viscosity of the oil, we used a model commonly used in dilution: κ)
ηO ) ηDO exp(-KnO)
(3)
where ηO is the viscosity of the oil phase (Pa s), ηDO is the viscosity of the dead oil (Pa s), and nO (mol) is the dissolved gas content in the oil phase. K was chosen to fit the experiments. Influence of the Presence of Bubbles. To quantify the role of the presence of bubbles on the viscosity of the foam, we used a classic Batchelor model,26 considering that the bubbles remain spherical and neglecting the circulation of fluid in them because of a relatively strong interfacial film.7,27-29 This model only depends upon the gas volume fraction and not the size distribution of bubbles, which is very difficult to assess in our experiments. Using this model, the viscosity is given by ηLO ≈ (1 + 2.5Φ + 6.2Φ2)ηO
(4)
(23) Kioupis, L. I.; Maginn, E. J. Impact of molecular architecture on the high pressure rheology of hydrocarbon fluids. J. Phys. Chem. B 2000, 104, 7774–7783. (24) Firoozabadi, A.; Ottesen, B.; Mikkelsen, M. Measurements of supersaturation and critical gas saturation. SPE Form. EVal. 1993, 7 (4). (25) Arora, P.; Kovscek, A. R. Mechanistic modeling of solution-gas drive in viscous oils. J. Pet. Technol. 2001, 53 (6). (26) Batchelor, G. K. The effect of Brownian motion on the bulk stress in a suspension of spherical particles. J. Fluid Mech. 1977, 93 (1), 97–117. (27) Sheng, J. J. Foamy oil flow in porous media. Ph.D. Thesis, University of Alberta, Edmonton, Alberta, Canada, 1997. (28) Zaki, N. N.; Poindexter, M. K.; Kilpatrick, P. K. Factors contributing to petroleum foaming. 2. Synthetic crude oil systems. Energy Fuels 2002, 16, 711–717. (29) Poindexter, M. K.; Zaki, N. N.; Kilpatrick, P. K.; Marsh, S. C.; Emmons, D. H. Factors contributing to petroleum foaming. 1. Crude oil systems. Energy Fuels 2002, 16, 700–710.
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Figure 3. (Left) Apparent complex (2%, 5 Hz) viscosity ratio of the foamy Canadian heavy oil. The oil was saturated at 90 bar in nitrogen and depressurized from 90 bar to room pressure instantaneously at t ) 180 s. Comparison of the experimental data and the model. (Right) Comparison of the model and the experimental data in the case of a Brazilian oil in the same conditions. The overshoot shows that the viscosity of the foam is higher than the viscosity of the dead oil.
Figure 4. Comparison of the calculated relative complex viscosity and the calculated complex viscosity ratio for the Canadian heavy oil depressurized in the same conditions as on the left side of Figure 3.
where Φ is the gas volume fraction and depends upon nB and ηLO is the live oil viscosity. Kinetics of Gas Exsolution. To model the progressive nucleation, we considered first-order equations. The gas exsolution process is divided into two different steps: the nucleation and the disengagement of bubbles from oil. Evolution of nO, nB, and nG are given below: dnB 1 ) nO dt τB
(5)
dnG 1 ) nB dt τG
(6)
τB, corresponding to a characteristic time constant for the nucleation of bubbles and τG, a time constant for the liberation of bubbles from the oil phase, are both adjusting parameters. The modeling results are compared to the experimental data in the case of the Canadian heavy oil and a Brazilian heavy oil, less viscous (Figure 3). The principal objective of this model is to characterize and study the kinetics of gas exsolution in different types of oil.19 However, this model also gives information about the influence of the bubbles on the viscosity of foamy oil through nB and using the Batchelor equation. On Figure 4, the relative complex viscosity (ratio of the foamy oil complex viscosity to the complex viscosity of the live oil matrix)
Figure 5. Foamy oil viscosity ratio under different constant shear rates (10, 50, and 100 s-1) and under an oscillatory test (2%, 5 Hz). The Canadian heavy oil is saturated in methane (30 bars) and depleted instantaneously from 30 bar to room pressure at t ) 180 s.
is compared to the complex viscosity ratio (ratio of the foam complex viscosity to the constant dead oil complex viscosity) in the case of the Canadian heavy oil (same as on the left-hand side of Figure 3). As one can see, the viscosity ratio remains below 1 because of the strong influence of the dissolved gas in the matrix during all of the experiments. However, the relative viscosity shows the strong impact that the bubbles have on the foamy oil viscosity. It appears indeed that the relative viscosity goes up to 1.4 at its maximum. Therefore, the bubbles cannot be neglected in the foamy oil viscosity evaluation. Moreover, the quality of fits in Figure 3 tends to show that the assumption of the hard sphere model is a good choice in this case. Under an oscillatory test implying a very low strain, the bubbles contribute to an increase in the foamy oil viscosity. The same experiments were carried out under different constant shear rates, instead of oscillatory measurements. The aim is to study the foamy oil behavior and the foamy viscosity under shear. Figure 5 depicts the evolution of the Canadian heavy oil viscosity saturated in methane (30 bar, 28.5 °C) under shear rates of 10, 50, and 100 s-1. We checked that the dead oil is Newtonian on this range of shear rates. The complex viscosity ratio is also plotted for comparison (oscillatory regime). Two observations have to be pointed out. First, it appears that the higher the shear rate, the lower the viscosity ratio. Second, the viscosity seems somehow stabilized under a constant shear
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rate. Actually, there is an increasing trend of the viscosity for all of the samples if you look at it for a longer time. This trend is more pronounced at lower shear rates. However, it takes a really long time (several days) to be able to see this trend. Finally, it should also be noted that the kinetics model used previously cannot match these new experimental results, showing that the assumptions used (the use of hard sphere model, for instance) cannot be used in these new experimental conditions. The drop of the apparent viscosity of a bubbly liquid below the viscosity of the liquid itself has already been observed in other viscous materials, such as magmas or melts of glass and polymers.30-35 These studies have shown that this behavior is due to the bubble elongation under shear. This explains why the hard sphere model cannot describe the viscosity of foamy oils under shear. As a result, several studies use the capillary number, Ca, to determine the impact of bubbles on the viscosity of a bubbly liquid. The capillary number is the ratio of the viscous forces, which tends to deform the bubble, to the surface tension, which restores its spherical shape. It is written as follows: η0aγ˙ Ca ) σ
(7)
where η0 is the viscosity of the continuous phase (Pa s), γ˙ is the shear rate (s-1), σ is the interfacial tension between gas and oil (taken at 31 mN/m after measurements of an interface air/ Canadian oil) and a is the bubble radius. For high capillary numbers, bubbles are elongated and the bubbly liquid relative apparent viscosity drops below 1, and when Ca , 1, bubbles remain spherical and contribute to an increase in the material apparent viscosity.32 To experimentally check this theory on heavy oils in our conditions, the visualization of the bubble shape during the experiments is required. Because the heavy oil is completely opaque and the X-ray experiments could not be performed under a constant shear rate, we used a transparent silicon oil, the viscosity of which is the same as that of the Canadian heavy oil in our conditions of temperature. We checked that this oil exhibits the same behavior as the Canadian heavy oil regarding the shear rate (decreasing viscosity with the shear rate), even if the amount of dissolved gas (GOR ) 12 for a saturation at 30 bars and 28.5 °C) and the surface tension (20 mN/m) are different. Experiments on the silicon oil have also been conducted using the multipass rheometer (MPR).36 The MPR is a pressurized capillary rheometer fitted with a window through which the sample can be visualized during the experiments. It has been developed in the Chemical Engineering Department of the University of Cambridge, U.K. Results are presented in Figure (30) Manga, M.; Loewenberg, M. Viscosity of magmas containing highly deformable bubbles. J. Volcanol. Geotherm. Res. 2001, 105, 19–24. (31) Llewellin, E. W.; Mader, H. M.; Wilson, S. D. R. The rheology of a bubbly liquid. Proc. R. Soc. A 2002, 548 (20), 987–1016. (32) Rust, A. C.; Manga, M. Effects of bubble deformation on the viscosity of dilute suspensions. J. Non-Newtonian Fluid Mech. 2002, 104, 53–63. (33) Stein, D. J.; Spera, F. J. Shear viscosity of rhyolite vapor emulsions at magmatic temperatures by concentric cylinder rheometry. J. Volcanol. Goetherm. Res. 2002, 113, 243–258. (34) Bagdassarov, N.; Pinkerton, H. Transient phenomena in vesicular lava flows based on laboratory experiments with analogue materials. J. Volcanol. Geotherm. Res. 2003, 302, 1–22. (35) Thompson, M. J.; Pearson, J. R. A.; Mackley, M. R. The effect of droplet extension on the rheology of emulsions of water in alkyd resin. J. Rheol. 2001, 45 (6), 1341–1358. (36) Mackley, M. R.; Marshall, R. T. J.; Smeulders, J. B. A. F. The multipass rheometer. J. Rheol. 1995, 39 (6), 1293–1309.
Figure 6. Pressure drop versus flow rate measured for the silicon oil and the bubbly silicon oil (foam) in the MPR. The error bars take into account the compressibility of the bubbly oil.
6. The pressure drop measured for the oil without any bubble is compared to the one obtained in the same conditions with the bubbly oil. It can be seen that the pressure drop is lower in the case of the foam. Pictures taken during the experiments also show that bubbles are more and more elongated as the flow rate increases. Bubbles (CO2) were added to the oil directly in the MPR at room pressure. Thus, the lower pressure drop measured in the case of foams is not due to the presence of dissolved gas in the matrix. These data support the assumption that the presence of elongated bubbles can lead to a decrease in the fluid apparent viscosity. To understand why the viscosity is stabilized at this low value, a pressurized reactor fitted with a window has been used to visualize the foam morphology after depressurization in function of the shear rate. The mineral oil is saturated in methane (30 bar) and depressurized to room pressure instantaneously, thus mimicking the experimental conditions in the rheometer. Pictures are presented in Figure 7. It can be easily seen that the bubbles are small, elongated (Figure 7A), and maintained much longer in the oil under shear than without mixing (Figure 7B). One can think that this is due to our stirring geometry used to dissolve the gas in the oil, because it highly disturbs the flow lines in the cup. However, we checked that bubbles are also maintained in oil under a simple shear, using a classic Couette geometry (Figure 7C). To explain the slow rise of bubbles under a shear rate, the friction coefficient can be evoked. It is indeed higher for ellipsoids37 than for spherical bubbles (Stokes relation). Thus, when a bubble is horizontally elongated, its rising velocity is much lower. Another explanation for the stability of the viscosity under shear can be suggested. Because it has been shown that the apparent viscosity is lower in the direction of the flow, one can imagine that the material behaves as an anisotropic sandwich-like material. Therefore, the apparent viscosity in the direction normal to the flow may be higher than expected, which slows down the bubble rise. A parallel can be established between this anisotropy and the shear-induced band structure in polymers. To support this theory of anisotropy, a homogenized shear viscosity in the direction of the flow can be calculated. For this purpose, we consider a periodic representative cell and apply an affine velocity field on the cell boundary. The averaged shear rates and shear stresses can then be (37) Perrin, R. Mouvement Brownien d’un ellipsoı¨de. II. Rotation libre et de´polarisation des fluorescences. Translation et diffusion de mole´cules ellipsoı¨dales. J. Phys. Radium 1936, 7, 1–11.
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Figure 7. Pictures taken 5 min (A) and 60 min (B) after the saturated silicon oil was depressurized instantaneously from saturation pressure (30 bar methane) to room pressure. Comparison of a depletion without mixing (left side) and under a constant shear rate (right side). (C) Trapped, elongated bubbles under an homogeneous shear rate (Couette geometry), 15 min after the depressurization. The black indicators in the pictures on the left and right are 5 mm long.
computed, leading to a homogenized shear viscosity. We use here the fact that, for small deformations, the nonlinear terms of the Green-Lagrange tensor can be neglected, and thus, the strain tensor has the same expression as in fluid mechanics. For this calculation, the finite element method was applied on two different configurations: the gas volume fraction is the same in both cases, but in one case, the bubbles are ellipsoids, with an axis ratio of 0.15, and in the other case, the axis ratio is 0.85; i.e., the bubbles are almost spherical. The Young modulus of the gas is chosen negligible compared to the “liquid phase”. As expected, the homogenized shear modulus is lower by 11% in the case of the elongated holes (ellipsoid axis ratio of 0.15) than for almost spherical holes (ellipsoid axis ratio of 0.85), a fact that confirms the experimental observations. In these simulations, we ignored the effects of surface tension and recognize that the analyzed configurations do not correspond exactly with the experimental observations, but in any case, they allowed us to identify the evolution tendency and explain, at least qualitatively, the rheological measurements. It has been shown in this section that the viscosity of a foamy oil highly depends upon the shear conditions. Indeed, the bubbles can lead to an increase in the viscosity if remaining spherical, while they can induce a drop in the apparent viscosity in the flow direction if the bubbles are elongated under high shear rates. Thus, the capillary number must be considered as an important characteristic of the studied system. In this study, the capillary number can be roughly evaluated considering an average radius diameter of 1 mm for the bubbles, according to our visualizations. Under shear rates from 10 to 100 s-1, the capillary number is comprised between 2 and 20. Under our oscillatory test, the capillary number can be evaluated at 0.1 at the maximum of deformation. These values are in accordance with Rust and Manga,32 who found a critical capillary number around 1. Demonstration of the Different Flow Regimes. These experimental results show that the presence of bubbles in heavy oil can lead to two different flow regimes: (1) In the first one, the bubbles are spherical. The foamy oil behaves roughly as a dispersion of hard spheres and can be modeled this way. The foamy oil viscosity is then higher than the liquid phase itself, which makes its extraction from the reservoir even more complicated. (2) In the second regime, the bubbles start to deform under a sufficient shear rate. The whole material behaves then as an anisotropic material, in which the apparent viscosity decreases in the direction of the flow (the direction of the bubble elongation) and increases in the perpendicular direction of the flow. This increase in the viscosity in this direction delays the rise and release of the bubbles, which explains the stability of the measure under a constant shear rate. To “picture” these two flow regimes, two specific experimental tests can be carried out. The first one consists of depleting
the live oil under a constant shear rate (50 s-1) and, when the flow regime is stabilized, to follow it by an oscillatory measurement (2% strain, 5 Hz; Figure 8). Under the constant shear rate, the bubbles are trapped and elongated and the apparent viscosity ratio of the material in the flow direction is lower than 1. When this constant shear rate is stopped, the bubbles retrieve their spherical shape. They then contribute to an increase of the apparent viscosity of the material (circled overshoot in Figure 8). However, they are not trapped within the oil any more: they rise and are progressively released from the matrix. When the bubbles are gone, the viscosity ratio of the material is equal to 1. There is no gas, neither dissolved nor dispersed, in the oil any more; therefore, if a constant shear rate is then re-applied, the viscosity ratio remains equal to 1. The second experiment (Figure 9) consists of depleting slowly the live oil sample under a constant relatively high shear rate (100 s-1). The depletion rate is 1 bar/min. In the first few minutes, the viscosity increases because of the presence of tiny spherical bubbles. As long as the depletion continues, the bubbles grow for thermodynamic reasons. They become bigger and thus more deformable and start to elongate. At 10 bar (106 Pa), the flow regime switches. The viscosity decreases because of the presence of more and more elongated bubbles. The apparent viscosity ratio is then stabilized at a much lower value than 1. These two experiments clearly show how the two flow regimes described above can switch from one to another, depending upon the shear rate applied to the material. Conclusion The influence of bubbles on the foamy oil viscosity is highly dependent upon the shear conditions. Through an original experimental approach, we point out the competition between two different regimes: in the first one, the bubbles induce an increase in the foamy oil viscosity, and in the second one, the presence of bubbles lead to a decrease in the apparent viscosity of the material in the flow direction, thus facilitating oil production and transport. This difference is due to the deformability of the bubbles in the viscous oil. On one hand, under a low shear rate, the bubbles remain spherical. The foamy oil behaves as a hard-sphere dispersion, which leads to an increase of the relative viscosity. On the other hand, under a high shear rate, the bubbles are elongated. We believe that the foamy oil then behaves as an anisotropic material: its viscosity is reduced in the flow direction, which facilitates its transport, and increased in the normal direction, which contributes to maintain the bubbles dispersed within the oil. These two flow regimes can switch from one to another depending upon the mechanical load on the material, as we have shown in two specific experiments.
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Figure 8. In this first experiment, the foamy oil viscosity is first stabilized at a low value under a constant shear rate (“creep test”, at 50 s-1). Then, at t ) 16 800 s, the constant shear rate is stopped and an oscillatory measurement of the complex viscosity is carried out (“oscillatory test”, 2%, 5 Hz).
Figure 9. Second experiment: viscosity ratio under a constant shear rate (100 s-1) and a slow depletion rate from 30 bar (saturation pressure in methane) to room pressure on the Canadian heavy oil.
As explained in the Introduction, this paper focuses on the flow behavior of heavy oil in bulk, i.e., in wells and pipelines. At this point, we cannot extend these results to the flow in porous media or even in wormholes because the ratio of the bubble radius and the channel size is much lower than in our experimental conditions. In addition to this, these results are only experimental. To go further, a predictive model of the pressure drop in a pipe in function of the presence of bubbles and the flow conditions would constitute a major breakthrough for the oil industry. Moreover, the role of water droplets in extra-
heavy oils should also be investigated, because they may behave in a different way if they elongate under high shear rates. Acknowledgment. The authors thank Prof. Malcolm Mackley, Dr. Tri Tuladhar, and Dr. Nitin Nowjee from the University of Cambridge, U.K., Prof. Francisco Chinesta and Kevin Angibaud from LMSP, France, for their helpful collaboration in this work, and Schlumberger for supporting our participation in the Petrophase 2008 Conference. EF8006646
