N. Zamani* (CIPR, Uni Research), R. Kaufmann (CIPR, Uni Research), T. Skauge (CIPR, Uni Research) & A. Skauge (CIPR, Uni Research)

Transcription

1 A18 Pore Scale Modelling of Polymer Flow N. Zamani* (CIPR, Uni Research), R. Kaufmann (CIPR, Uni Research), T. Skauge (CIPR, Uni Research) & A. Skauge (CIPR, Uni Research) SUMMARY Polymer flooding as an EOR method that has boomed in the last decade as oil prices have been rising, and new and larger polymer flood projects are being realized. Some milestones and examples are the largescale viscoelastic polymer flood implementation at the Daqing field in China, polymer injection in Marmul field, Oman, and the Dalia offshore polymer project. Polymer flood is a mature EOR technology, but as the reservoir targets get more diverse and the field conditions harsher, the current understanding of polymer flood is stretched to its limits. In order to explain viscous fingering in inter-mediate to heavy oil reservoirs and viscoelastic mobilization of residual oil there is a need for a better understanding of polymer flow mechanisms on the pore scale. Pore scale polymer flow characterization is very complex and involves several flow phenomena like; adsorption, viscous fingering, depleted layers, hydrodynamic retention, bridging/flow-induced adsorption, viscoelastic effects, in-accessible pore volume and more. In this study we have developed a Navier-Stokes model to analyse polymer flow and to compare against Newtonian fluids. The aim has been to identify the key parameters for polymer displacement. Examples of obtained results are that the depletion layer plays a major role in study of rheological properties. Increased depleted layer thickness lead to lower velocity at the centre of the pore and more slip effect near the pore wall. When a higher degree of shear thickening is included a larger drag on fluids in side channels will occur, this is consistent with oil mobilisation and lowering of residual oil saturation.

2 Introduction Polymer injection is one of the enhanced oil recovery methods which have been widely used in recent years (i.e. Daqing field in China, Marmul in Oman, Dalia out of Angola, etc.). The main reason for use of polymer is adverse mobility ratio with corresponding poor sweep. Mobility is defined as the ratio between permeability of the domain to the fluid and viscosity of fluid (k/μ). In addition to the macroscopic sweep several flow phenomena influence the process efficiency, such as, adsorption, inaccessible pore volume (IPV), straining, log-jamming, depletion layer, viscoelastic properties, etc. Polymers are large and flexible molecules with physical flow properties dependent on polymer concentration and flow local rate (shear rate). Generally, the polymer solution viscosity is a function of shear rate. Transitions from low to high shear rate involves change from Newtonian viscosity to shear thinning, apparent Newtonian, and shear thickening at high rate. As the shear rate increases beyond the critical shear rate (inverse of polymer relaxation time), the interaction between polymer molecules decrease and polymer solution shows shear thinning properties. At higher shear rates, no further interaction exists and polymer viscosity reaches to its lowest value with apparent Newtonian behavior. Beyond this shear rate extensional flow becomes dominant and causes a sharp increase in viscosity (shear thickening). Viscosity increases until the strain is so high that the carbon-carbon binding is disrupted and causes reduction in viscosity. Results from experimental polymer studies and modeling of polymer flow on pore scale indicate that viscoelasticity of polymer solution is effecting microscopic displacement or more specific the level of residual oil saturation. The key literature about polymer viscoelasticity and effect on oil mobilization will be reviewed. Experimental polymer studies of synthetic polymers and biopolymers, Ranjbar et al. (1992) performed on Bentheimer and Berea sandstone cores using HPAM and Xanthan, found that both increased the oil recovery compared to water flooding, but the increase in recovery was higher for HPAM. They concluded that although pressure increases due to the increasing of viscosity affect the recovery, viscoelasticity of HPAM was the main reason for the higher oil recovery factor. These observations were later confirmed in a study by Yang et al (26) performing core experiments and showed that oil recovery is higher for HPAM than Xanthan and the difference was also ascribed to the viscoelastic properties of HPAM. The argument was supported and elaborated by observation that for Xanthan increasing polymer concentration resulted in an increase in oil recovery, but more quickly reached a plateau value, while HPAM gave higher increase in recovery which did not seem to level off. Due to the rheological differences between Xanthan and HPAM, the increase was ascribed to the increase in viscoelasticity with concentration for HPAM polymers. Another experimental study by Wang et al (2) investigated effect of three different fluids; water, Glycerin (Newtonian) and HPAM (non-newtonian) on oil recovery in the glass etched 2D model. They observed that HPAM showed the highest oil recovery factor and possible due to the viscoelasticity of HPAM. Xia et al (28) also performed micro model studies displacing oil by water, glycerin, glucose and HPAM, and as Wang et al (2) they claim that viscoelastic polymers can reduce residual oil saturation after water flooding and that the oil displacement efficiency is increasing with increasing viscoelasticity. Flooding under oil wet condition showed an increase in oil displacement with chemical additive concentration although the effect might also influenced of PV flooded as no comparison with single concentration displacement was performed. They claimed that the increased oil displacement of HPAM at the highest concentration compared to glycerin was due to the viscoelastic effects of HPAM solution. More mechanistic studies by Wang et al (21) analyzed flow into dead-end pores as a function of increasing elastic modulus. Trapped oil in dead-end pores was observed to be mobilized by the polymer with highest degree of elasticity. However, the shear viscosity curves shows that this is also

3 the polymer with the highest shear viscosity. The elastic and shear forces can therefore not be discriminated in these experiments. The local wetting properties will influence the trapped oil mobilization and Wang et al (21b) proposed that oil trapped in oil-wet dead ends is mobilized by drag forces. The velocity gradient and viscosity near the wall is higher and causes higher drag force on oil film or in the dead-end pore. The central concept was that the drag forces exerted by the viscoelastic fluids may partially or completely mobilize residual oil after water flooding. Huh and Pope (28) performed core flooding using Berea and Antolini sandstone cores, the authors found that polymer flood cannot mobilize residual oil after water flooding at water-wet conditions. The suggested recovery mechanism was credited to elasticity of the polymer solution prevents the breakage of oil ganglia into smaller droplets and may therefore produce oil that otherwise would have been trapped. Also Jiang et al (28) performed polymer floods in water-wet and oil-wet cores and derived a correlation between displacement efficiency, capillary number and elasticity. There are several other results reported about viscoelastic effect of polymers in the literature, we would like to mention; Han et al. (1995),Yang et al. (26), Xia et al. (28), and Wang et al. (211). One of the most interesting consequences of the fluid viscoelastic effect or extensional viscosity is a strong increase flow resistance (pressure drop) in porous medium. To study polymer flow in porous media in addition to a model for the viscoelastic properties, a detailed description of the porous media is also necessary. The simplest and most controllable local models are pore network model and contraction-expansion models. More realistic and detailed porous media model currently limits mathematical, chemical and physical framework for describing the complex interactions between the polymer, brine and rock surface. Interaction between polymer solution and walls play a major role in microscopic study of polymer flow. Polymer molecules may either adsorb to the wall surface by the attractive forces or repel from the wall which is known as depleted layer. Adsorption reduce pore dynamic radius and increase the resistance to flow which alters the mobility ratio. In this paper the effect of different parameters has been modeled in an attempt to provide an overview of the impact of some phenomena such as adsorption, depletion layer and viscoelastic properties on polymer flow through various porous geometries such as contraction-expansion, dead-end and pore doublets by use of a Navier-Stokes model. Theory Polymer solution is known as non-newtonian fluid and in rheology non-newtonian fluids can be divided into three categories: time dependent, time-independent and viscoelastic fluids. These categories are not sharply defined and complex fluid can exhibit combined properties of these categories. One of the critical parameter in evaluating rheology of a fluid is observation time. Generally, the elastic response is faster than viscous, therefore at short time a fluid can exhibit elastic, but at longer time show viscous properties. Different models have been proposed for different categories such as the Carreau model or Power law for time independent category and Upper Convected Maxwel or Oldroyd-B for viscoelastic model. The models vary in the number of parameters required, the shear regions described and the degree of physical relevance incorporated in the parameters. All non-newtonian polymer fluids show viscoelastic properties. In rheology theory, a viscoelastic fluid is a fluid which shows both viscous and elastic properties, so the response of fluid viscosity to the applied stress will be both viscous and elastic: v + e (1) The viscoelastic fluids may store and release energy due to dualistic nature of having both viscous and elastic properties. If the flow rate is suddenly increases in a core which is flooding by a Newtonian

4 fluid, the pressure drop will increase rapidly, but in viscoelastic fluid, although it will exhibit the same final pressure drop as Newtonian fluid, firstly it makes a higher pressure drop and decay exponentially toward a steady state value. Some polymer fluids, such as HPAM, exhibit strong viscoelastic effects such as shear thinning, shear thickening, time dependent viscosity. So far no significant model is proposed which can described all the viscoelastic properties at various flow conditions. A viscoelastic polymer solution shows different properties depending on the flow regimes and response of fluid inside porous medium in comparison with the rheometers results may differ due to subjecting to several contraction-expansions and several other phenomena such as adsorption. Modelling of viscoelastic properties of polymer solution in porous media is very complicated and numerically difficult to implement due to the numerical instabilities. Bai et al (211) theoretically studied effect of viscoelasticity of polymer solution on microscopic oil displacement. In his study, they separated effect of viscosity and elasticity and for each case calculated the critical oil droplet radius for being mobilized. They concluded that the fluid elastic properties will exert extra pressure which in case that the viscous pressure cannot exceed the capillary pressure the extra pressure which is provided by elastic properties can be added to the viscous pressure and defeat the capillary pressure. Based on their study the pressure drop due to the viscosity of fluid can be calculated as follows in which F v is viscous force: (2) They used power law model for simplicity and final pressure drop due to the fluid viscosity is: (3) (4) Where μ e is the elastic viscosity. By applying equation (1), total pressure drop is the summation of viscous and elastic pressure drop: (5) To mobilize the trapped oil in the configuration of Figure 1, the pressure drop should exceed the capillary pressure. (6) Figure 1 Trapped oil model in a pore throat [Bai et al, 211]

5 We can conclude that the critical oil droplet radius under elastic condition is smaller than viscous case. So the trapped oil droplet in jamming effect case can be enhanced easier by viscoelastic fluid than just viscous fluid. Bai et al (211) also studied the effect of viscoelasticity of polymer solution while passing through a sudden expansion channel (Figure 2). He declared that as polymer solution is passing through a capillary to a sudden expansion part of pore, it will show extrude swelling behavior due to normal stress difference which can mobilize residual oil. Figure 2 schematically shown the trapped oil inside a dead end and parameters which use for viscoelastic effect calculations [Bai et al, 211] Normal stress difference can be calculated as follows: N p = (7) (8) Beside the viscoelastic effect of polymer solution, extensional viscosity can happen while polymer molecules are passing through porous medium which can be also evaluated as the viscoelastic effect. Viscoelastic effects are the property of the fluid regardless of the medium and the extensional viscosity is directly related to porous medium and of course fluid properties. Extensional viscosity is mainly reported in the polymer flow in porous media. This phenomenon has been reported in several experimental results and is a debatable subject which is due to the difficulty of measuring of the extensional flow separately. The flow path of the polymer molecules inside the porous media is very tortuous. This causes the polymer molecules to be accelerated and decelerated. So the polymer molecules are not just sheared but also stretched. Beyond a critical shear rate, the viscosity increases abruptly. The reason for extensional flow is not very well understood, but two main theories, the coilstretch theory and the transient network theory, are developed for explaining of this phenomenon. In the coil-stretch theory (De Gennes, 1974) the sharp increase in flow resistance occurs as the shear rate exceeds a critical value where the polymer coil starts to un-coil and stretch. The increase continues until the polymer is fully stretched out. Thereafter, chain scission leads to a reduction in molecular weight, and thereby a reduction in apparent viscosity. In transient network theory (Odell et al, 1988) extensional flow is due to the formation of transient networks of polymer molecules. These occur because polymers are forced together and the entanglement time is shorter than the disentanglement time. The large increase in viscosity arises as these transient networks are stretched in the flow field. In extensional flow two dimensionless parameters are defined and widely used, one is Deborah number which is the ratio between the characteristic time of fluid and characteristic time of porous media, and directly correlate to the stretching rate and is more relevant to the porous media. When Deborah number is small, fluid will show viscous behavior, otherwise it will be elastic.

6 The other parameter is Weissenberg number which is defined as the product of the fluid relaxation time and the characteristic strain rate and directly correlates to the shear rate and is more related to the fluid flow regimes: (1) For the evaluation of extensional viscosity Trouton introduced a ratio between extensional and shear viscosity and declared that for Newtonian fluid, at low strain rate this ratio is constant and equal to three. (11) Calculation of extensional viscosity in some cases such as flow through porous media or high Trouton numbers (1 4 or 1 5 ) is important. Generally, techniques which are being used for measuring of extensional flow can be divided into two different types: flow-through and stagnation point. Some examples of these methods and new ones are: Lubricated planar stagnation die flows, filamentstretching techniques, on-line viscometry, Magnetic Resonance Imaging (MRI), Ultrasound Doppler Velocimetry and Laser Doppler Anemometer (LDA). The onset of extensional viscosity depends on several factors such as pore size distribution, tortuosity, brine salinity, polymer concentration and molecular weight. Modeling of extensional viscosity is very difficult due to its steep increase in viscosity which leads to numerical instabilities. Several models have been proposed to include extensional flow and provide more complete model for wider range of shear rate. Delshad et al (28) extended the Carreau model to include the term including extensional viscosity: 2 is the second time constant, n 2 is the second power law exponent and: μ w is water viscosity, C is polymer concentration in PPM, A 11, A 22 and C SEP Sp are the experimental parameters. At very high shear rates, the polymer may degrade due to large shear forces on the polymer chain. Shear degradation is not included in the models presented above. Stavland et al. (21) introduced the following formula to include the shear degradation: (9) (12) (13) (14) These two equations ((13) and (14)) are both extensions of the Carreau model. In the former model the maximum viscosity is used to scale the viscosity increase in the shear thickening regime, while the latter model utilizes the difference between initial and solvent viscosity as the scaling factor. In polymer flow, the interaction between pore wall (restrictions) and polymer molecules plays an important role. Most polymers used in EOR are ionic which cause attractive or repulsive interaction with pore surfaces. Another polymer pore surface interaction is depleted layer. This phenomenon is reported by several authors such as Aakura and Oosawa (1954), Chauveteau and Zaitoun (1981), Auvray (1981), De Gennes (1979), Chauveteau (1986), Sorbie (1991) in both adsorbing and nonadsorbing cases. Although the exact mechanism is debated, there is a general agreement that this effect arises due to entropic exclusion of polymer molecules from the wall of porous medium or rheometers. The molecules in some polymer solutions like Xanthan are rod like molecules and near

7 the wall cannot rotate freely consequently are forced to the center of the pore or forced to be stretched along the flow. Experimental results demonstrate that the polymer molecules in solution migrate away from solid boundaries (Sorbie 1991). This can also be due to the electric charges of the polymer molecules and the wall which may repel the molecules from the wall surface. This phenomenon exhibits significant impact on rheological behavior of polymer solutions inside porous media. Depletion layer will cause the polymer concentration profile to decrease gradually near the wall. Different configurations are considered for this phenomenon, such as two fluid model, linear model, etc. In two-fluid model two parts can be considered along the cross section of the channel which in both sections constant concentration will be considered. The concentration near the wall is less than bulk concentration and the concentration in the center of the channel is slightly more than the bulk concentration to provide the mass conservation. An analytical equation for the calculation of effective viscosity in a capillary tube at low shear rates was derived by Chauveteau (1982) based on the calculated viscosity at the center and near the wall. (15) In equation (15), is the depletion thickness, R 2 is the pore radius, μ 2 is viscosity near the wall, and μ 1 is the bulk viscosity. Sorbie (199) also proposed analytical solution for the calculation of apparent viscosity in linear model, which is more complicated and included more parameters. In linear model the concentration in the center of channel is constant and the concentration toward the wall decreases linearly. Due to the origin of the depleted layer, the thickness is in the order of the polymer molecular size. The depleted layer thickness is observed to be constant for dilute solutions, but is reduced with increasing concentration above the critical overlap concentration (Chauveteau et al. 1984, Omari et al. 1989). As a consequence of the reduction in polymer concentration toward the wall, the viscosity of the fluid reduces and leads to higher velocity which is known as apparent slippage along the wall. The presence of the depleted layer is responsible for the higher flow rate than expected at a given value of the wall shear stress. Because of the ratio of depleted layer thickness to pore radius, slip effects have a largest influence in low permeable rock where the average pore size is lower. The velocity profile is decreasing towards the pore wall due to frictional forces. As the friction is largest close to the wall it is also where the shear stress is highest. One of the interesting phenomena which can happen for polymer solution when depletion layer exists is earlier breakthrough of polymer molecules, which is known as velocity enhancement. It is one factor that often is included in the collective term of Inaccessible Pore Volume (IPV). Velocity enhancement can be calculated from the following formula (Sorbie 199): (16) (17) In the above formulation, C(r) is concentration profile and V z (r) is velocity profile. Another well-known phenomenon which is due to the interaction between polymer molecules and wall is adsorption which is fundamentally important in EOR. By adsorption, the permeability of the domain will decrease which can help to reduce the mobility ratio. The amount of permeability decrease depends on polymer type, the amount of adsorption, size of polymer molecules, polymer molecular weight, fluid velocity, pore diameter, nature and structure of the polymer molecules and rock surface.

8 From the Poiseuille law the thickness of adsorbed layer can be calculated as follows: = 1 ( / ) / (18) r is the pore radius, and are the flow rates due to a given pressure drop measured before and after the adsorption. Model We have used the Navier-Stokes equation to describe momentum conservation in the fluid flow. Finite Difference Model (FDM) on staggered grid is used for solving the Navier-Stokes equation. The general form of the Navier Stokes for the non-newtonian fluid is as follows: (19) Where p is pressure, μ is viscosity which is not constant, u is the velocity vector which for the 2D cases it includes two components: the horizontal velocity (u x ) and vertical velocity (u y ), Re is incompressible. (2) In equation (19), viscosity is considered as shear dependent variable. For the dependency of viscosity on shear rate a Carreau model is applied. Although the Carreau model includes four parameters and is numerically difficult to implement, this model is able to describe the changes in viscosity at very low and high shear rates, which are the interesting regions at reservoir conditions. Where μ,μ and are positive values, n (,2], 2 is the second time constant and n 2 is the second power law exponent. Based on the value of n, the fluid behavior is known as either shear thinning (n<1), Newtonian (n=1) or shear thickening (n>1). The boundary conditions are summarized in Figure 3. For the solid boundaries no-slip condition is considered. The velocity in the entire domain except for the inlet is initially set to zero. Detailed information about the discretization procedure and the numerical method is explained in Griebel et al. (1997). The model is validated by using bench mark results for the Newtonian fluid and also experimental results of Non-Newtonian fluids. (21)

9 Solid Boundaries: =, =, = Inlet: =..., = ( ), = Outlet: = = Solid Boundaries: =, =, = Figure 3 Boundary Conditions Numerical end effect is one source which can cause errors in the calculation of pressure drop. This phenomenon may occur when the flow is not fully developed and may lead to an overestimation of the pressure drop. Fully developed velocity profile is known for Newtonian and Power Law model and can be defined at the inlet as the velocity profile, but for the Carreau model the velocity profile should be calculated by numerical methods. In the following simulation (when the inlet and outlet have the same diameter), the inlet velocity profile will be replaced by the calculated velocity profile at the outlet of the domain at every fifty iterations. This is shown in Figure Fully developed pressure drop non-fully developed pressure drop Pressure [Psi] Number of Cells along the channel Figure 4 Effect of fully developed velocity profile on pressure drop profile along a single channel. (1 psi = bar) Results and Discussions Residual oil can be present in various forms based on wettability, pore structure, fluid properties and displacement situations. The oil elements can extend from a fraction of a pore to large multiple pore clusters. In this part we consider three types of microscopically trapped oil: a) Oil trapped in dead ends b) Oil films which exists in oil wet pores c) Oil droplets trapped in small channels due to capillary forces Here we present results of the influence of Newtonian and Non-Newtonian (shear thinning and shear thickening) fluids on trapped oil in various simplified pore geometries such as single channel, pore doublet and dead end.

10 Dead-end Pore In this part, we consider one phase flow of three different fluid types (shear thinning, shear thickening and Newtonian), compare different flow regimes and evaluate the potential of each fluid type for mobilizing of possible trapped oil in a dead end configuration. The simplified geometry of the deadend pore is demonstrated in Figure 5. The elastic properties (normal stresses) of the fluid are neglected and the main focus is on the viscous properties which are mainly shear dependent. The size of the dead end part is defined by the aspect ratio which is the ratio between the depth and the width of the dead-end. As the size of dead end pores varies inside the real porous medium, two different aspect ratios are considered, one for the narrow dead end (2:1) and the other one is representing wider pores (1:1). The parameter which is studied is the ability of each fluid type to sweep the dead end. Sweep ratio is defined as the ratio between the radius of the velocity isoline equal to 1% of the inlet velocity and the depth of the dead end. The velocity isolines for different fluid types are demonstrated in Figure 7. The results of sweep ratio are summarized in Figure 6 which shows that in the case of neglecting the elastic terms, just by increasing the power index (changing the fluid type from shear thinning to shear thickening) the fluid will be more diverted toward the side channel (higher swept area) which is consistent with the analytical calculation and causes higher drag force. The latter implies an enhanced potential for mobilizing oil trapped in the dead end Y X Figure 5 simplified dead-end pore geometry used in the simulation..9.8 Aspect Ratio = 1:1 Aspect Ratio = 2:1.7.6 Sweep Ratio Power Index,'n' Figure 6 Sweep ratios vs. power index for two aspect ratios

11 1.8 1 m/sec m/sec Y.4 6 Y X (a) X (b) m/sec m/sec Y Y X (c) X (d) 2 Figure 7 Velocity isoline distribution in a dead-end model, (a) extreme shear thinning, n=.1, (b) shear thinning, n=.5, (c) Newtonian, n=1, (d) Shear thickening, n=1.5. Oil Film Simplified geometry of a single channel is shown in Figure 8. In Figure 9 the velocity profile for different fluid types have been demonstrated along the vertical cross section of a single channel. The velocity gradient near the wall is higher than in the center of the channel. Given that the flow rate is sufficiently high to reach the extensional viscosity regime, the viscosity will be higher for a shear thickening fluid than for a shear thinning and a Newtonian fluid close to the wall. Consequently, the drag force exhibited on an oil film coating the wall surface will be larger. As an example, a stagnant oil droplet is considered in a single channel and the drag force on this stagnant oil droplet is calculated for shear thinning and shear thickening fluids. The results are presented in Figure 1. The drag force on the stagnant oil droplet is larger for shear thickening case than for the shear thinning case. It may therefore be possible that more oil is mobilized from the oil film for shear thickening fluids due to the larger drag force exhibited on oil film by the polymer..2.4 Y X Figure 8 Single channel model which is used in the simulation

12 Shear Thinning, n=.1 Shear Thinning, n=.5 Newtonian Shear Thickening, n=1.5 Velocity [m/sec] Number of cells along the vertical cross section Figure 9 Velocity Profile at the outlet for different fluid types; Extreme shear thinning (n=.1), moderate shear thinning (n=.5), Newtonian (n=1) and shear thickening (n=1.5). Figure 1 Drag Force distributions around an obstacle (Oil droplet), for (a) shear thinning and (b) shear thickening fluid. Pore Doublet model (a) A simplified configuration of a pore doublet model is shown in Figure 11. We consider various radius ratios to evaluate the behavior of different types of fluid. Oil can be trapped in this configuration due to capillary forces. By polymer flooding, polymer can either block a channel and diverts the flow or providing higher pressure drop which can overcome the capillary pressure. In this part three different pore radius ratios (larger channel to smaller one) are considered, 3:1, 3:2, 2:1. The ratio of fluid that is passing through small channel compared to the larger channel as a function of the Carreau power index which are the representatives of fluid type ranging from extreme shear thinning fluid (n=.1) to extreme shear thickening case, (n=1.9) are summarized in Table 1. The relative size of the pore channels is the dominating factor that gives the larger impact on fluid flow. For each given pore channel ratio, a clear increase in flow in the smaller channel is observed for increasing shear thickening behavior. This model therefore predicts a diversion of flow from the larger to the smaller pores for a shear thickening polymer. Diversion of fluid flow to smaller pores may mobilize capillary trapped oil and/or improve the microscopic sweep efficiency in comparison with the shear thinning and Newtonian fluids. (b)

13 1.8.6 Y.4.2 Figure 11 Simplified pore doublet geometry which is used in the simulation. n R 2 /R 1 = 3: R 2 /R 1 = 2: R 2 /R 1 = 3: Table 1 Flow rate ratio (small channel/large channel) vs. power index for three channel ratio sizes. Extensional viscosity X In order to study the extensional viscosity a modified Carreau model proposed by Delshad et al (28) has been used. Effect of pore geometry, polymer concentration, and salinity has been studied by Chauveteau (1982). Extensional viscosity will happen when the extensional flow become dominant. Chauveteau declared that the product of Deborah number and shear rate is constant for the transition where extensional viscosity starts. In this part based on Chauveteau s experimental data, we have studied the effect of pore geometry on the onset of dilatant (extensional flow) behavior. With this goal contraction-expansion geometry (Figure 12) is considered with the following properties: L is the length of pore throat, L exp is the length of pore body, R is radius of pore body and r is the radius of pore throat (b) (a) Figure 12 (a) single contraction-expansion model, (b) geometrical parameters. Based on Chauveteau s experiments, we define the base case of this study so that the contraction ratio is 5 and L/r = 1. In this part, effect of L/r, contraction ratio and number of contraction-expansions on onset of dilatant behavior will be studied and the results have been compared by the experimental results. For calculation of the apparent viscosity, first pressure drop along the model in the case of Newtonian fluid is calculated ( P New ) and for various inlet velocities, the ratio between the pressure drop along the model at the current case and the Newtonian pressure drop will be calculated ( P/ P New ). The ratio is called apparent viscosity. For the shear rate variation, the maximum wall shear rate will be defined as representative of the shear rate. Length of pore throats plays a major role on the onset of dilatant behavior. As this length increases, Deborah number will increase and dilatant behavior happened at higher shear rates (Table 2).

14 Theoretically as the L/r tends to infinity the fluid does not show dilatant behavior. These results are consistent with the experimental results. L/R Onset Shear rate (1/sec) Table 2 onset of dilatant behavior for various L/R On the other hand when the contraction ratio increases, the velocity inside pore throat section will increase and decrease the Deborah number. Consequently the onset of dilatant behavior happens at lower shear rates (Table 3). These results are also consistent with experimental results. Contraction ratio Onset Shear rate (1/sec) Table 3 onset of dilatant behavior for various contraction ratios. Another parameter which can be interesting to study is the effect of number of contractionexpansions. The flow inside multiple contraction expansions can be important as the experimental results have shown build up in viscosity as the polymer solution is passing through more contractions that are known as viscosity hysteresis. In the other words, while polymer solution is passing through several contraction-expansion channels, for the first contraction-expansion channel initial stress is zero which is non-zero for the succeeding contraction sections. This is also called the memory effect. In this part the flow inside five successive contraction expansion channels is studied. The results with five contractions provide the same results as the one contraction model. The modeling results are not consistent with experimental results. The difference may be related to the viscosity model. The Carreau model does not include any time dependent parameter (memory effect) and just consider the local shear rates. As it has been demonstrated in Figure 13, the viscosity pattern is repeating in different contraction-expansions.

15 Figure 13 Viscosity surface plots for 136 ppm HPAM for the inlet velocity range 6-18 mm/s, for five contraction-expansions. Depletion Layer and Adsorption In this part the effect of depletion layer and adsorption on the apparent viscosity and flow field are studied. In the depletion layer the concentration profile decreases toward the pore wall. Here two-fluid model was considered for the concentration profile. In the model two regions will be considered along the vertical cross section of the channel. From the wall to a distance of (depletion thickness) is considered as the wall concentration (C w ) which is lower than the bulk concentration and between the two regions (depleted layers) are considered the bulk concentration that is slightly higher than initial concentration due to mass conservation. The Carreau parameters such as μ,nand depend on the local concentration. The dependencies of the mentioned parameters on polymer concentration have been defined by several researchers (i.e. Sorbie 1991). The equations which are used for that are as follows: (22) (23)

16 (24) μ w is the water viscosity, A 11,A 22, P 11 and P 22 are constant coefficients and C is the polymer concentration. Velocity profiles for different values of depletion layer are demonstrated in Figure 14. As it has been demonstrated the velocity decreases at the center that causes higher velocity near the wall also known as slip effect. This will also cause higher shear rates near the wall. (25) Velocity (mm/s) Number of cells along the vertical cross section Figure 14 Effect of depleted layer thickness on velocity profile. The effect of depletion layer thickness on rheology is studied. In the case of depletion layer, maximum wall shear rate cannot be considered as the representative shear rate. So the average shear rate along the vertical cross section of the channel will be calculated as the shear rate (equation (26)). (26) The fitted Carreau parameters are summarized in Table 4, as the depletion layer thickness increases, the apparent viscosity at low shear rate decreases and also n (slope of the curve in the power law region) and (the onset of shear thinning) increase. By increasing the depletion layer thickness, the velocity enhancement will increase and causes earlier polymer breakthrough.

17 Relative Depletion layer Thickness /R Velocity enhancement Fitted Carreau Parameters μ (cp) n Table 4 The effect of Two-Fluid model for depletion layer on different flow parameters. For Concentration = 1 PPM. Another phenomenon studied here is the effect of adsorption on polymer flow, flow resistance and apparent viscosity. The same approach as depletion layer is considered here. The cross section of the channel is divided into two subsections but unlike the depletion layer, in the section near the wall apparent viscosity is relatively high. The model is considered to be a monolayer and semi impermeable domain, which fluid flow can exist, but due to high viscosity and low permeability, will be diverted toward the pore center (Figure 15). Figure 15 schematically shown the proposed model for the adsorption [adapted from Gramain and Myard, 198]. In Figure 16 the effect of adsorption thickness on velocity profile has been demonstrated. As the relative adsorption thickness increases the shear rate near the wall increase. Meanwhile by increasing the adsorption thickness the resistance to flow in a single channel will increase and causes higher pressure drop along the channel (Figure 17). Also effect of adsorption thickness on apparent viscosity for two different shear rates is also studied. Adsorption shows higher impact on apparent viscosity in lower shear rates (Figure 18).

18 Velocity [m/sec] No adsorption Rel. Ads. Thick =.6 Rel. Ads. Thick =.1 Rel. Ads. Thick =.16 Rel. Ads. Thick =.2 Rel. Ads. Thick = Number of cells along the vertical cross section Figure 16 Effect of adsorption on velocity profile along the vertical cross section No adsorption Rel. Ads. Thick =.6 Rel. Ads. Thick =.1 Rel. Ads. Thick =.16 Rel. Ads. Thick =.2 Rel. Ads. Thick =.3 Pressure Number of Cells along the vertical cross section Figure 17 Effect of adsorption on pressure drop 2 High shear rate = 6(1/s) Low shear rate =.3(1/s) Apparent viscosity [mpa] Relative adsroption thickness Figure 18 effect of adsorbed layer thickness on apparent viscosity at two different shear rates

19 Conclusions In this study, effect of pore geometry on the onset of dilatant behavior, depleted layer and adsorption were studied. The effect of fluid type on fluid flow pattern was studied in various simplified pore geometries, i.e. pore dead-end, pore doublet and contraction-expansion channel. Pore geometry plays a major role on the occurrence of dilatant behavior. Effect of three pore geometrical parameters, L/R, contraction ratio and several contraction expansion channels were studied. As L/R increases, the residence time of fluid in pore throat will increase which cause the onset of dilatant behavior happens at higher shear rates. Another parameter which studied is the effect of contraction ratio. When the contraction ratio increase, the residence time of the fluid in contracted part of the pore will decrease and based on the Deborah number explanation, dilatant behavior happens at lower shear rates. Although based on experimental results several contraction-expansion channels do not have any impact on the onset of dilatant behavior, this should have impact on the intensity of dilatant behavior. The simulation model did not reproduce the experimental results from Chauveteau (1982) for increasing number of contraction-expansion channels. The simulations showed indifference in both the onset and intensity of dilatant behavior with increasing number of contraction-expansion channels. This indicates that for some configurations the polymer memory effects need to be included to match the experiments. In the depletion layer, as concentration toward the wall decreases, this will cause an apparent slip effect. The slip effect was reproduced in simulations and shows a decrease in the velocity profile along the cross section of the channel. Depleted layer excludes a fraction of the pore volume i.e. that near the pore wall and the higher concentration of polymer molecules will be located at the center of the channel which leads to early breakthrough of polymer molecules. Depletion layer also decreases apparent viscosity and increase the values of the Carreau parameters such as power index n and relaxation time. In contrast to depleted layer, adsorption will increase apparent viscosity and delay breakthrough of polymer. Shear thickening fluid shows more distribution of flow into the side channel in dead-end and narrow channel in the pore doublet model and may explain additional oil mobilization.

20 Nomenclature A 11, A 22, Experimental Constants p c Capillary Pressure P 11,P 22 C Polymer Concentration[PPM] Velocity enhancement De Deborah Number Strain Rate F v Viscous Force Fluid density[kg/m 3 ] N p Normal Stress Difference μ Fluid viscosity[cp] p Pressure [psi] μ Viscosity at zero shear rate[cp] Q Flow Rate[m 3 /sec] μ Viscosity at infinite shear rate[cp] Tr Trouton Number μ e Elastic Viscosity [cp] u Velocity vector (u,v) μ app Apparent Viscosity [cp] <v> Average Velocity [m/sec] μ w Water Viscosity [cp] We Weissenberg Number Shear rate [sec -1 ] Pressure Drop Interfacial tension [mn/m] e Elastic Pressure Drop 1 4 Relaxation time [sec] v Viscous Pressure Drop Stress Tensor Acknowledgement We gratefully acknowledge financial support from PETROMAKS program, Norwegian Research Council. References Aakura, S. and Oosawa,F. [1954] On Interaction between Two Bodies Immersed in a Solution of Macromolecules. J. Chemical Physics, Vol. 22(7), DOI: 1.163/ Auvray, L. [1981] Solutions of Rigid Macromolecules- Wall and Confinement Effects-Orientation by Flow, J. de Physique, Vol. 42(1), DOI: 1.151/jphys: Bai, Y., Zhang, X. and Zhao, G. [211] Theoretical analysis of microscopic oil displacement mechanism by viscoelastic polymer solution. Theoretical and Applied Mechanics Letters. Vol 1(2), DOI: 1.163/ Chauveteau G. and Zaitoun, A. [1981] Basic rheological behavior of xanthan polysaccharide solutions in porous media: Effects of pore size and polymer concentration. Enhanced Oil Recovery, Vol. 13, ED. Fayers, F.J., pp: ISBN: Chauveteau, G. [1982] Rodlike Polymer Solution Flow through Fine Pores: Influence of Pore Size on Rheological Behavior. J. of Rheology. Vol. 26 (2), DOI: / Chauveteau, G., Tirrell, M. and Omari, A. [1984] Concentration-Dependence of the Effective Viscosity of Polymer Solutions in Small Pores with Repulsive or Attractive Walls. J. Colloid Interface Science. Vol. 1, DOI: 1.116/ (84)941-7 Chauveteau G. [1986] Water Soluble Polymers-Beauty with Performance. Ed Glass, J. E., Vol. 213, De Gennes, P.G. [1974] Coil-Stretch transition of dilute flexible polymers under Ultrahigh Velocity gradients, J. Chemical Physics., Vol. 6(12), pp: DOI: 1.163/ Delshad, M., Kim, D.H., Magbagbeola, A., Huh, C., Pope,G.A. and Tarahhom, F., [28] Mechanistic Interpretation and Utilization of Viscoelastic Behavior of Polymer Solutions for

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