3564
Ind. Eng. Chem. Res. 1999, 38, 3564-3571
An Empirical Investigation of Polymer Flow in Porous Media Ali A. Garrouch* and Ridha B. Gharbi Petroleum Engineering Department, Kuwait University, P.O. Box 5969, Safat 13131, Kuwait
Steady-state flow experimental data have been analyzed for two commonly used polymers representing two generic classes, polysaccharides (Xanflood), and partially hydrolyzed polyacrylamides (Pusher-700), flowing inside bead packs and Berea sandstone. Oscillatory flow measurements have been used to compute the polymer solution’s longest relaxation time (θf1), which is referred to as the characteristic relaxation time in this paper. Steady-state flow experimental data for the two polymers combined with measured polymer viscous properties have been converted to average shear stress-shear rate data inside porous media. An average power-law exponent (n j ) is therefore obtained for the polymer flow inside the porous medium. Using θf1, n j , rock permeability (k), porosity (φ), and fluid flow velocity (u), a viscoelasticity number (Nv) is calculated and found to strongly correlate with the pressure gradient inside porous media. This correlation is the basis for defining a viscoelastic model for polymer flow, analogous to Darcy’s law. The proposed model asserts a nonlinear relationship between fluid velocity and pressure gradient. It accounts for polymer elasticity and for pore geometry changes due to molecular adsorption and mechanical entrapment. Introduction Polymers are used in a variety of petroleum engineering applications. In enhanced oil recovery, polymers are added to water to increase its viscosity and decrease its relative mobility. This decrease in water mobility improves significantly the volumetric sweep efficiency and the efficiency of the water flood.1,2 Even after decades of research, the mechanism of polymer flow in porous media remains vaguely understood. At low Reynold’s numbers (NRe < 1), Newtonian fluid flow in porous media obeys Darcy’s law, which is given by
u)
k ∆P µ L
(1)
In this notation u is the Darcy velocity, k is the porous media permeability, µ is the fluid viscosity, ∆P is the pressure drop, and L is the core length. For this type of flow, the fluid velocity is proportional to the pressure gradient (∆P/L). The constant of proportionality is the mobility, or the rock permeability divided by the fluid viscosity, which is constant for Newtonian fluids. Darcy’s law for fluid flow in porous media is valid for fluids behaving as a continuum with constant viscosity and for a constant pore geometry. The flow of polymers in porous media deviates from these assumptions because: (i) polymers’ viscosity is shear rate dependent, (ii) polymer molecules’ length is comparable to the pore throat length, a fact that enhances certain elastic phenomena, and (iii) molecular adsorption and mechanical entrapment make the geometry of the porous medium variable. Consequently, the use of Darcy’s law for polymer flow in porous media has serious flaws. The traditional approach to modeling polymer flow in porous media has been to use Darcy’s law with an effective viscosity, at a fixed shear rate, substituted for the Newtonian viscosity. This procedure corrects for the shear-rate dependence of viscosity but does not account for the noncontinuum and elastic flow properties.3,4
van Poollen and Jargon,5 and Willhite and Uhl6 showed by a simple empirical model the nonlinear relationship between the pressure drop and the flow rate for non-Newtonian fluid flow. This relation was expressed as
Qm+1 L Am+1
∆P ) F
(2)
In this notation, A is the cross-sectional area, Q is the volumetric flow rate, m is a power-law constant, and F is an empirical constant that has been correlated to the rock permeability, porosity, and tortuosity. A similar, more explicit, nonlinear dependence can also be obtained by extending the tube flow rate equation based on the viscous power-law model to the flow in a bundle of capillary tubes. The resulting model is as follows:
Q)
{
(3n+1)/n
1/n
φπ 21/n
r fp dr ∫0∞(3n+1)/n
∫0 πr2fp dr ∞
}
{ }{ } A m1/n
∆P L
1/n
(3)
Here, m and n are the power-law constants of the polymer, fp is the pore size distribution function of the rock, and r is a pore radius. This model clearly asserts two important findings achieved by van Poollen and Jargon5 and by Willhite and Uhl.6 The first is that the flow rate indeed relates nonlinearly to the pressure drop. The second is that the constant of proportionality depends on both the fluid properties (m and n) and on the rock properties such as the pore size distribution and the porosity. The above models fall short, however, in accounting for the elastic phenomena manifested by polymer flow in porous media. Because the length of polymer molecules is comparable to the length of pore throats and pore bodies in the porous media, these molecules compress as they hit the pore walls. Then, they stretch, or relax, to resume the flow under the influence of the flow potential. The characteristic relaxation time is
10.1021/ie990121b CCC: $18.00 © 1999 American Chemical Society Published on Web 07/20/1999
Ind. Eng. Chem. Res., Vol. 38, No. 9, 1999 3565
sometimes comparable to the transient time defined as the time these molecules take to exit from a pore throat or a pore body. One can expect major elastic effects when the polymer molecule characteristic relaxation time is comparable to its transient time. In many instances these effects are felt as an additional pressure drop or an additional resistance to flow. A thorough summary on the subject is given by Savins.7 In this review paper, Savins7 highlights an interesting theoretical model expressed in a nondimensional form in terms of the friction factor (f), the Reynold’s number (Re), and the Deborah number (NDEB):
f Re ) [1 + ζ(NDEB)2]
(4)
The Deborah number is a measure of polymer elasticity. This number is expressed as a function of polymer concentration, molecular weight, viscosity at zero shear rate, the fluid superficial velocity, and a characteristic grain size diameter of the rock.7 On the basis of a perturbation analysis of flow of a spring-type of viscoelastic fluid characterized by a single relaxation time, Savins7 showed that, for a porous medium, the purely viscous pressure drop must be multiplied by a factor of [1 + ζ(NDEB)2] to obtain the pressure drop in the presence of elastic effects. The parameter ζ is a constant. A great number of published data, however, do not agree with the rate of increase in pressure loss predicted by this model.7 The objective of this research is to develop an empirical model for polymer flow in porous media. The model accounts for both viscous and elastic effects. Steadystate experiments for polymer flow in porous media are the means for defining the nonlinear dependence between the fluid velocity and the pressure gradient for horizontal flow. This dependence is expressed in terms of fundamental rock and fluid properties. Experimental Equipment and Procedures Data used in this analysis are extracted from three categories of experiments: measurement of polymer solution properties, measurements of petrograhic parameters for porous media, and flow experiments of the polymer solutions through porous media. Preparation of Polymer Solutions. Two types of water-soluble polymers were used to study the viscoelasticity effects on flow behavior in porous media. These are xanthan gum (Kelko’s Xanflood, hereafter referred to as Xanflood), and partially hydrolyzed polyacrylamide (Dow’s Pusher-700, hereafter referred to as Pusher-700). The polymers all came in powder form, and aqueous solutions were prepared in different ways depending upon the polymer type. Pusher-700 polymer solutions were made directly at the desired polymer and salt concentrations with a magnetic stirrer. The rate of addition of the polymer was slow enough for the particles to disperse in water without lump formation, but not so slow that the solution thickens before all the solid was added. From 8 to 24 h was generally required for complete hydration. The Xanflood solution was prepared by first dissolving sodium bicarbonate and an enzyme in distilled water. Then the xanthan gum powder was added and the mixture sheared in a high speed blender for 10 min. This provided a concentrated solution containing 1% by weight which was then blended with brine containing a biocide to obtain the desired polymer and sodium
chloride concentrations. The final composition was 1000 ppm xanthan gum, 10 000 ppm NaCl, 100 ppm NaHCO3, 4 ppm enzyme (Alcalase P1.5), and 120 ppm biocide (Dowicide B). Rheological Measurements. The bulk fluid property characterized is the viscosity as a function of shear rate ranging from approximately 100 000 s-1 at the injector wellbore to as low as 0.01 s-1 at some distance into the reservoir. This study included most of this range. The Contraves L-S 30 rheometer was used with various bob-and-cup arrangements for measuring shear stress-shear rate profiles.8 The instrument is valuable since ultralow shear stress measurements are possible for shear rates as small as 0.0174 s-1. The viscoelasticity was measured using an oscillatory tube flow system in which the frequency of oscillation and the amplitude are independently controlled. The apparatus and theory of measurement have been described by Thurston and Pope.9 In this method, the shear stress at the tube wall is determined from the pressure gradient while the shear rate at the tube wall is determined from the volume flow. The sinusoidal electrical signals representing the instantaneous shear stress and shear rate are analyzed to obtain the magnitudes and phase difference between the fundamental components, free of harmonics and noise. Porous Media Description. Two types of permeable media were used. These are glass-bead-packed columns (experiments A-D) and Berea sandstone cores (experiments E and F). Glass beads provided a uniform unconsolidated pack that was free of clays. Two different sizes of glass beads were used in this study, namely 100-110 µm (for experiments A-C) and 250-300 µm (for experiment D). The glass beads were packed in a 9/16 in. i.d. stainless steel pipe that was one-foot long. The column was composed of a piece of stainless steel pipe and two end caps. A detailed description of the glass-bead columns used to generate data for experiments A-D is given by Yuan.10 Multiple pressure taps were installed on these columns to observe the fluid entrance and exit end-effects. These pressure taps were also useful for monitoring the pressure distribution to determine if a fully developed flow was present inside the packed column or if any plugging occurred in the column. Berea core samples were used to generate data for experiments E and F. Berea provided a complex form of porous medium since it is a consolidated rock with some clay material present. A detailed description for the Berea core holder setup and procedures used to generate experiments E and F is provided by Jones8 and Wreath.11 Two-foot-long Berea core samples were cut with a 2 in.2 cross-sectional area. Nylon screening material of 500 µm opening size was placed between the end piece and the core face to insure an adequate initial fluid contact area which was uniform for the entire core face. Fittings were positioned on the top of the core, 6 in. from each core face, providing two pressure taps 12 in. apart. Tables 1 and 2 give a summary of the petrographic properties of each porous medium and the corresponding polymer solution used. Pumping System and Flooding Procedures. A constant flow rate feed system was employed to conduct the flow experiments through the packed columns and the Berea core samples. The flow system consisted of a Zenith pump for controlling the flow rate, a solution reservoir, a packed column with uniform glass beads
3566 Ind. Eng. Chem. Res., Vol. 38, No. 9, 1999 Table 1. Listing of All Experiments Analyzed experiment
solution
pack
A
1000 ppm Xanflood in 1% NaCl brine 1000 ppm Pusher-700 in 1% NaCl brine 1000 ppm Pusher-700 in 0.1% NaCl brine 1000 ppm Pusher-700 in 1% NaCl brine 1000 ppm Xanflood in 1% NaCl brine 1000 ppm Pusher-700 in 0.1% NaCl brine
100-110 µm beads
B C D E F
100-110 µm beads 100-110 µm beads 250-300 µm beads Berea sandstone Berea sandstone
Table 2. A Petrographic Summary of the Experimental Data Analyzed porosity permeability porosity permeability experiment (%) (md) experiment (%) (md) A B C
39.0 39.0 39.3
4000 4200 3600
D E F
37.4 19.7 22.0
37900 312 530
or Berea as the porous medium, a system of pressure transducer-carrier demodulator, and a chart recorder for pressure drop measurements, as well as a fraction collector to collect the effluent samples automatically. All the solutions were filtered and degassed before they were introduced into the solution reservoir. For all porous media samples, an initial brine flood was conducted before any polymer solutions were pushed into the medium to determine the absolute permeability. Brine with 1% NaCl concentrations by weight was made and filtered through 0.45 µm Millipore filter paper with the application of the Fann Filter Press under 40 psi pressure. It was then fed into the solution reservoir after degassing. After all the air bubbles in the tubing and fitting were purged out, the flow experiment started. Pressure drops were measured from pressure taps using pressure transducers. Three pressure drop values were read for each flow rate. The volumetric flow rate was determined after steady flow was established, which could be indicated by the stabilized pressure drop values. After the initial brine flood, polymer solutions were filtered through 1.2 µm Millipore filter paper under 40 psi and then were pushed through the porous medium to study the viscoelastic nature of the solutions. The porous medium which was flooded with 1% NaCl brine solution was then flushed with polymer solution at a medium flow rate. The volumetric flow rate was arbitrarily selected to be 1.7 cm3/min. This flush was conducted until a steady pressure drop was established and the viscosity of the effluent was measured to be identical to the injected polymer solution viscosity. This ensured that the packed column was indeed fully saturated with the polymer solution. Polymer flooding was then conducted from low rates to high rates while measuring the pressure drops and flow rates. The effluents were collected and viscosities were measured to see whether shear degradation occurred during the flow. The time span for each experiment was kept to a minimum to maintain the freshness of the polymer solutions. After the polymer flood had finished, the brine solution was run again to obtain the flushed permeability. The brine solution was run at a relatively high flow rate (25 cm3/min) until about 25-30 pore volumes were injected and a steady pressure drop had been established.
Figure 1. Shear stress versus shear rate profile for experiment A (100-110 µm beads pack flooded with 1000 ppm Xanflood in 1% NaCl brine solution).
Results and Discussion Characterizing Polymer Viscous Effects. The experimental data of flow rate versus pressure drop was converted to shear stress versus shear rate data inside the porous medium. The apparent viscosity of polymer solution inside the porous medium was calculated using Darcy’s law as
µapp )
kfA∆P qL
(5)
Here kf is the permeability of the flushed porous medium. It was measured after the brine flood following the polymer flood. The porous medium shear rate was calculated using the Blacke-Kozeny model8,10 as
γ˘ p )
[3n4n+ 1]
n/(n-1)
4u
x8kfφ
(6)
The exponent n is the power-law parameter determined from the viscous flow measurements obtained using the LS-30 rheometer by plotting the shear stress-shear rate profile. The parameter n is the slope from this log-log plot. The corresponding shear stress inside the given porous medium was therefore calculated using
τp ) µappγ˘ p
(7)
Figure 1 shows the shear stress-shear rate profile for a typical Xanflood solution flowing inside a bead pack. The Xanflood solution appears to fit reasonably well the power-law model given by
j γ˘ pnj τp ) m
(8)
implying insignificant elastic effects. The exponent n j is an average power-law constant inside the porous medium. The molecules of Xanflood are described as being relatively stiff in the literature.12 Their flow behavior is governed by their shear viscosity. The results obtained in this study confirm these observations. The Pusher-700 solutions showed some deviation from the viscous power-law model given by eq 8. Figure 2 shows the shear stress-shear rate profile for experiment B, which is run on a similar sand pack to that of experiment A, but with Pusher-700 solution instead. The shear-rate magnitude increased by approximately 2 orders of magnitude, as compared with values for experiment A, because of the shear-thickening behavior of Pusher-700. The deviation of the shear stress-shear rate profile from a straight line in a log-log plot
Ind. Eng. Chem. Res., Vol. 38, No. 9, 1999 3567
Figure 2. Shear stress versus shear rate profile for experiment B (100-110 µm beads pack flooded with 1000 ppm Pusher-700 in 1% NaCl brine solution).
Figure 3. Deborah number versus flow rate profile for experiments A and B (100-110 µm bead packs flooded with 1000 ppm Xanflood in 1% NaCl brine and 1000 ppm Pusher-700 in 1% NaCl brine, respectively).
confirms the literature observations that classify Pusher700 as a viscoelastic polymer.9,12 Characterizing Polymer Elastic Effects. The viscoelasticity was measured using an oscillatory tube flow system in which the frequency of oscillation and the amplitude were independently controlled. The complex coefficient of viscosity η* was obtained from the complex ratio of shear stress τ to the shear rate γ˘ ,
η* )
τ ) ηv - iηE γ˘
(9)
where ηv is the energy dissipative viscous component and ηE is the nondissipative elastic energy storage component.9 Both ηv and ηE are functions of the angular frequency (ω) and the shear rate (γ˘ ) at which the measurement is made. The experimental data were fitted to the classic Rouse model.13 In this model, the polymer molecule is represented as a flexible chain of N beads connected by N - 1 elastic springs. When immersed in a solvent fluid, such a model molecule has N - 1 relaxation times which are related to the storage and loss moduli as follows:
θfj
N-1
ηv ) ηs + RT
∑ j)1
1 + (θfjω)2
(10)
and N-1
ηE ) RT
∑ j)1
ωθfj2 1 + (θfjω)2
(11)
The longest relaxation time, θf1, was used for the correlation with flow data through cores and packs. By fitting the predictions of the model for ηv and ηE to the experimental data, an estimate for θf1 was obtained. This characteristic relaxation time was used to calculate the Deborah number defined as7
NDEB )
θf1u Dpeffφ
(12)
where Dpeff is the effective porous medium diameter8,10 and is given by
Dpeff )
x
150(1 - φ)2k φ3
(13)
Figure 4. Comparing the Deborah number for experiments B and C (100-110 µm bead packs flooded with Pusher-700 in 1% NaCl and with Pusher-700 in 0.1% NaCl brine solution, respectively).
Selecting a Viscoelasticity Number. For two similar sand packs, one flooded with Xanflood (experiment A) and the other flooded with Pusher-700 (experiment B), the Deborah number-flow rate profile appears to be identical (Figure 3). This is not surprising since a slight difference in the characteristic relaxation time for these two polymer solutions does not cause a significant difference in NDEB, according to eq 12. The surprising element though is the fact that xanthan material, which is described in the literature12 as long rodlike molecule with little flexibility, has a relaxation time comparable to Pusher-700, which is a partially hydrolyzed polyacrylamide made of flexible and elastic chain-structured molecules. This might be explained by the sensitivity of Pusher-700 to solution salinity and hardness. Salinity causes the molecular chain to collapse, which results in a much smaller random coil structure with a reduced relaxation time. A fully extended partially hydrolyzed polyacrylamide molecule in a good solvent would exist as a filament of 7-25 Å in diameter and exceeds 10 µm in length. The molecular length of xanthan is estimated to be 2-10 µm. The diameter of a single-chain xanthan backbone is 20 Å.12 In a saline solution (1%) the length and width of the xanthan molecule makes up for its elasticity as compared to the narrow partially hydrolyzed molecule whose length has been reduced. This is confirmed in Figure 4, where a comparison is made between the Deborah number of two Pusher-700 solutions of different salinities (1% and 0.1% NaCl). The lower salinity solution appears to be more elastic than the higher salinity solution. This result indicates that the Deborah number may not be an adequate parameter for characterizing the viscoelasticity of polymer flow in porous media since an elastic polymer (Pusher-700) appears to behave similarly to a viscous polymer (Xanflood) in porous media because of the sensitivity of Pusher-700 to solution salinity. This suggests the
3568 Ind. Eng. Chem. Res., Vol. 38, No. 9, 1999
Figure 5. Viscoelasticity number versus flow rate for experiments A and B (100-110 µm bead packs flooded with 1000 ppm Xanflood in 1% NaCl brine and 1000 ppm Pusher-700 in 1% NaCl brine, respectively).
Figure 6. Flow rate term versus pressure gradient term for experiment D (250-300 µm beads pack flooded with 1000 ppm Pusher-700 in 1% NaCl brine solution).
introduction of a group number (Nv), which we refer to as the viscoelasticity number in this paper, and it is given by14
Nv )
xkφ θf1unj -1
(14)
The viscoelasticity number becomes dimensionless when n j takes a value of 2 and carries units of velocity when n j equals 1. It also increases with increasing n j values as long as the fluid velocity (u) is maintained for less than 1 m/s, which is always the case for polymer flow in porous media. Figure 5 shows a plot of the viscoelasticity number versus flow rate for two similar packs (experiments A and B) but for two displacing polymers (Pusher-700 for experiment B and Xanflood for experiment A). Here the distinction between the two polymers is crystal clear both in magnitude and in profile. The shear-thickening Pusher-700 appears to have a negative slope, unlike the shear-thinning Xanflood, despite the solution high salinity. The viscoelasticity of the Pusher-700 is clearly indicated by the magnitude of Nv at any flow rate compared to Xanflood. Modeling Polymer Flow. As shown in Figure 6, a log-log plot of the flow rate term (Qm j 1/nj /Aφ) versus the 1/n j pressure gradient term (∆P/L) , for Pusher-700 solution flooded in a bead pack, did not result in a straight line. This suggests that the Pusher-700 floods did not follow the flow rate-pressure drop relationship indicated by the capillary tube model given by eq 3. This is not surprising since the capillary-tube model was derived for viscous but not elastic polymers. On the other hand, Xanflood in both bead packs and Berea appeared to mimic the capillary-tube model reasonably well (Figures 7 and 8). One expects Xanflood to follow the
Figure 7. Flow rate term versus pressure gradient term for experiment A (100-110 µm beads pack flooded with 1000 ppm Xanflood in 1% NaCl brine solution).
Figure 8. Flow rate term versus pressure gradient term for experiment E (Berea sandstone flooded with 1000 ppm Xanflood in 1% NaCl brine solution).
capillary-tube model since the polymer has insignificant elasticity, and since the bundle-of-tubes model is a close approximation for bead packs, and these facts are in line with the original assumptions made in deriving the capillary-tube model given by eq 3. The results of Xanflood flow in Berea (Figure 8) make a little bit of surprise though since Berea sandstone’s pore geometry is far more complex than the simple geometry assumed in developing the capillary-tube model. These results may suggest that the flow of a viscous polymer with insignificant elasticity inside bead packs or Berea sandstone adheres to the following nonlinear relationship between flow rate and pressure gradient:
Q 1/nj ∆P m j )κ Aφ L
1/n j
[( ) ]
(15)
Here κ and are two empirical constants that depend on the porous medium geometry and the polymer rheological properties. Both κ and are obtained from a log-log plot of the flow rate term versus the pressure gradient term. A statistical summary of the values of these constants, and the corresponding regression coefficients for each experiment are given in Table 3. Data from the six experiments investigated appeared to fit very well in a log-log plot of viscoelasticity number versus pressure gradient. Examples of these responses for both Berea and bead packs flooded with Xanflood and Pusher-700 are displayed in Figures 9-12. A viscoelatic model is therefore suggested as
(∆PL)
Nv ) R
β
(16)
In this notation β is the slope of a log-log plot of Nv versus pressure gradient. The empirical constant R is the intercept at a value of (∆P/L) equal to 1 from the same log-log plot. By using the definition of the group
Ind. Eng. Chem. Res., Vol. 38, No. 9, 1999 3569 Table 3. Statistical Summary of Regression Parameters R
experiment
β 10-6
5.130 × 7.261 × 10-3 3.005 × 10-4 5.171 × 10-3 3.790 × 10-6 4.442 × 10-4
A B C D E F
κ
rR-β2
0.4514 -0.2954 -0.0523 -0.1975 0.5725 -0.01023
10-3
1.1367 × 6.3600 × 10-2 3.1410 × 10-1 3.0892 × 10-3 3.2970 × 10-6 2.7072 × 10-6
0.9947 0.9810 0.9715 0.9831 0.9963 1.0000
Figure 9. Viscoelasticity number versus pressure gradient for experiment A (100-110 µm beads pack flooded with 1000 ppm Xanflood in 1% NaCl brine solution).
rκ-2
0.9955 0.9836 0.9915 0.9880 1.561 1.033
0.9953 0.8437 0.8906 0.8524 0.9989 1.0000
Figure 12. Viscoelasticity number versus pressure gradient for experiment F (Berea sandstone flooded with 1000 ppm Pusher700 in 0.1% NaCl brine solution). Table 4. Rheological Data Summary of all Experiments Analyzed
Figure 10. Viscoelasticity number versus pressure gradient for experiment E (Berea sandstone flooded with 1000 ppm Xanflood in 1% NaCl brine solution).
Figure 11. Viscoelasticity number versus pressure gradient for experiment C (100-110 µm beads pack flooded with 1000 ppm Pusher-700 in 0.1% NaCl brine solution).
number Nv (eq 14), the above equation can be rearranged, for n j different from 1, to give the following relationship between Darcy’s velocity (u) and the pressure gradient (∆P/L):
u)
[ ] [ ] xkφ Rθf1
1/(n j -1)
∆P L
β/(1-n j)
(17)
The model presented by eq 17 asserts the nonlinear relationship between fluid velocity and pressure gradient. It also captures both viscous and elastic effects of the polymer solutions, since it includes the characteristic relaxation time (θf1) and the power-law constant (n j ) that reflects the porous medium relationship be-
experiment
n j
m j × 1000
β/(1 - n j)
1/n j
A B C D E F
0.6902 1.3512 1.0558 1.2495 0.6400 1.0100
45.326 1.7260 19.992 2.9221 31.190 36.600
1.4570 0.8411 0.9408 0.7916 1.5903 1.0230
1.4488 0.7400 0.9471 0.8003 1.5625 0.99009
tween shear stress and shear rate. Adsorption and mechanical entrapment effects on the pore structure are implicitly accounted for through the use of the powerlaw constant n j . This is true since the flushed permeability kf is used to calculate the porous medium shear rate (eq 6). This data is ultimately used to generate n j from a plot of shear stress versus shear rate inside the porous medium. The model also accounts for the effects of the original microscopic structure and geometry of the porous medium through rock permeability (k) and porosity (φ). As shown in Table 4, it is interesting to note that the pressure gradient exponent (β/(1 - n j )) of the proposed viscoelastic model (eq 17) is approximately equal to the pressure gradient exponent (1/n j ) of the capillary tube model (eq 3). This implies that the term (β/(1 - n j )) of eq 17 makes up for the viscous part of the pressure drop needed for the polymer to flow. The model presented by eq 17 can, therefore, be rearranged to give an expression for the fluid velocity in terms of n j instead of β. A statistical summary of the R and β values and the corresponding regression coefficients for each experiment are given in Table 3. The intercept R appears to strongly correlate with n j . The correlation coefficient between R and n j is approximately 0.86. The deterministic relationship between R and the remaining fluid and porous media parameters is yet to be determined. However, the proposed model can still be applied by estimating R using the empirical correlation given by
R ) exp[ln a0 + n j ln a1 + m j ln a2 + φ ln a3 + k ln a4] (18) where a0, a1, a2, a3, and a4 are regression constants given in the Appendix. Here porosity is input as a fraction, and permeability is in mdarcy. The above
3570 Ind. Eng. Chem. Res., Vol. 38, No. 9, 1999
empirical correlation reproduces the experimental R values with an average absolute relative error of 2.7% and a standard deviation on this relative error of 2.8%.
m: m j: n:
Conclusions A steady-state flow experimental data set of two polymer solutions (Xanflood and Pusher-700) flowing in porous media (Berea sandstone and bead packs) has been analyzed. The analysis led to the development of an empirical viscoelastic model for polymer flow in porous media analogous to Darcy’s law. A nonlinear relationship between fluid velocity and pressure gradient is asserted though. The proposed model accounts for polymer elasticity by using the longest relaxation time (θf1) and accounts for polymer viscous properties by using an average porous medium power-law constant (n j ). It also accounts for porous medium structure by using porosity and permeability. The changing pore structure is accounted for in estimating n j since the in situ shear rate, used to generate n j , depends on the afterflush permeability (kf). Analysis of similar data, for a variety of polymers flowing inside heterogeneous rocks, is essential for confirming the empirical relationships between parameters R and β, the fluid rheological properties, and the petrographic properties. This confirmation may then generalize our results to other types of porous media. The Xanflood flow in both bead packs and Berea sandstone is found to mimic a nonlinear relationship between flow rate and pressure gradient similar to that predicted by the capillary-tube model. This implies the development of a second empirical model that may work only for flow of viscous polymers with insignificant elasticity. The study shows that the Deborah number may not be an adequate parameter to characterize the viscoelastic flow of polymers in porous media. A group number, Nv, called the viscoelasticity number, is found to be more adequate since it clearly distinguishes viscous from viscoelastic flow. Acknowledgment The authors would like to thank the Kuwait Foundation for the Advancement of Sciences (KFAS) for its financial support of project KFAS-94-8-2.
N: n j: Nv: NDEB: r: Re: rR-β2: rκ-2: Q: u: Qm j 1/nj /Aφ: (∆P/L)1/nj :
fluid power-law constant from rheological measurements intercept from a log-log plot of shear stress versus shear rate of the rock fluid power-law constant from rheological measurements number of beads of a flexible chain that models a polymer molecule average power-law constant inside porous medium viscoelasticity number Deborah number pore radius Reynold’s number coefficient of determination for regression line given by R and β coefficient of determination for regression line given by κ and volumetric flow rate Darcy velocity flow rate term pressure gradient term
Greek Letters R:
intercept from a log-log plot of Nv versus pressure gradient slope from a log-log plot of Nv versus pressure gradient slope from a log-log plot of flow rate term versus pressure gradient term rock porosity complex shear rate average porous medium shear rate intercept from a log-log plot of flow rate term versus pressure gradient term Newtonian fluid viscosity apparent viscosity of polymer solution inside rock complex coefficient of viscosity the nondissipative elastic energy storage component of η* the energy dissipative viscous component of η* steady-flow viscosity complex shear stress average shear stress in porous medium relaxation time longest relaxation time angular frequency a constant
β: : φ: γ˘ : γ˘ p: κ: µ: µapp: η*: ηE : ηv: ηs: τ: τp: θf: θfj: ω: ζ:
Nomenclature A: a0: a1: a2: a3: a4: DPeff: ∆P: F: f: fp: k: kf : L:
cross sectional area regression constant for Appendix regression constant for Appendix regression constant for Appendix regression constant for Appendix regression constant for Appendix effective porous medium pressure drop empirical constant friction factor pore size distribution rock permeability flushed permeability length of core sample
eq 18 given in the eq 18 given in the eq 18 given in the eq 18 given in the
Appendix: Constants for the Empirical Correlation Given by Eq 18 a0 a1 a2 a3 a4
) ) ) ) )
1.06999 × 10-9 330072.2 2.13337 × 1011 0.019213 1.000024
eq 18 given in the
Literature Cited
diameter
(1) Lake, L. W. Enhanced Oil Recovery; Prentice Hall: Englewood Cliffs: New Jersey, 1989. (2) Xian-Qing, H.; Wei-Ying, W.; Ying, X. The Viscoelastic Behavior of HPAM Solutions in Porous Media and its Effects on Displacement Efficiency. SPE International Meeting on Petroleum Engineering, Beijing, November 1995; Paper 30013. (3) Pruess, W. K.; Witherspoon, P.A. Displacement of a Newtonian Fluid by a Non-Newtonian Fluid in a Porous Medium. Transport in Porous Medium 1991, 6, 115.
Ind. Eng. Chem. Res., Vol. 38, No. 9, 1999 3571 (4) Flew, S.; Sellin, R. H. J. Non-Newtonian Flow in Porous MediasA Laboratory Study of Polyacrylamide Solutions. J. NonNewtonian Fluid Mechanics. 1993, 47, 169. (5) van Poollen, H. K.; Jargon, J. R. Steady state and unsteady state flow of non-Newtonian fluids through porous media. In AIME Transactions: Dallas, TX, 1969; Vol. 246, p 80. (6) Willhite, G. P.; Uhl, J. T. Correlation of the mobility of biopolymer with polymer concentration and rock properties in sandstone. Polym. Sci. Eng. 1986, 55, 577. (7) Savins, J. G. Non-Newtonian Flow Through Porous Media. Ind. Eng. Chem. 1969, 61, 18. (8) Jones, K. The Rheology of Viscoelastic Fluids for Oil Recovery: A Study of Micellar/Polymer Fow Over a Wide Range of Shear Rates in Porous Media. M.S. Thesis, The University of Texas at Austin, Austin, TX, 1981. (9) Thurston, G. B.; Pope, G. Shear Rate Dependence of the Viscoelasticity of Polymer Solutions. J. Non-Newtonian Fluid Mechanics 1981, 9, 69. (10) Yuan, M. A Rheological Study of Polymer and Microemulsion in Porous Media. M.S. Thesis, The University of Texas at Austin, Austin, TX, 1981.
(11) Wreath, D. A Study of Polymerflooding and Residual Oil Saturation. M.S. Thesis, The University of Texas at Austin, Austin, TX, 1989. (12) Sorbie, K. Polymer-Improved Oil Recovery; Blackie and Son Ltd: London, 1991. (13) Heemskerk, J.; Rosmalen, R. J.; Holtslag, R. J.; Teeuw, D. Quantification of Viscoelastic Effects of Polyacrylamide Solutions. SPE/DOE Fourth Symposium on Enhanced Oil Recovery, Tulsa, OK, April 1984; Paper 12652. (14) Wreath, D.; Pope, G.; Sepehrnoori, K. Dependence of Polymer Apparent Viscosity on the Permeable Media and Flow Conditions. In Situ 1990, 14, 263.
Received for review February 18, 1999 Revised manuscript received May 28, 1999 Accepted June 1, 1999 IE990121B
