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Experimental Study of Diffusive Tortuosity of Liquid-Saturated Consolidated Porous Media Zhaowen Li†,‡ and Mingzhe Dong*,§ Faculty of Engineering, UniVersity of Regina, Regina, Saskatchewan, S4S 0A2, Canada, and Department of Chemical and Petroleum Engineering, UniVersity of Calgary, 2500 UniVersity DriVe NW, Calgary, AB, T2N 1N4 Canada
Diffusive tortuosity factor is one of the key parameters in modeling solute diffusion in liquid-saturated porous media. However, the determination of diffusive tortuosity factor has to involve a diffusion process in liquidsaturated porous media, which was usually found complicated in laboratories. The incorrect use of diffusive tortuosity factor may cause significant errors in certain circumstances. This paper presents a method to evaluate the diffusive tortuosity factors for liquid-saturated consolidated porous media, i.e., sandstones, which are the typical porous media commonly encountered in contaminant transport in underground water and gas migration in liquid-saturated reservoirs. The proposed method applies two specific experiments to determine the diffusion coefficient in bulk liquid phase and the effective diffusion coefficient in liquid-saturated porous media, respectively. Diffusive tortuosity factor of the porous media is obtained by comparing the effective diffusion coefficient in porous media to the diffusion coefficient in bulk liquid. This study provides a procedure to evaluate the diffusive tortuosity factor for consolidated porous media and also the measured values of diffusive tortuosity factors for selected sandstone samples which can be used as input data for further studies. Another application of the proposed method is to determine the diffusion coefficient in bulk liquid phase for CO2 from the measured effective diffusion coefficients in porous media. Deff ) D/ε
Introduction Determination of solute effective diffusion coefficients in liquid-saturated porous media is essential to understand and evaluate contaminant transport, diffusive gas migration in a gas sequestration project, the solvent mixing process in a reservoir during an enhanced oil recovery process, and appraisal of diffusive losses in natural gas reservoirs in the geological time scale.1 To model these processes, effective diffusion coefficients in porous media, rather than diffusion coefficients in liquid phase, are being used to incorporate the tortuous characteristics of the pore-throat structures inside the porous media, which is usually referred to as tortuosity. If the diffusive tortuosity factors of the porous media are known, the commonly reported diffusion coefficient in homogeneous bulk liquids can be converted into the effective diffusion coefficient in porous media. The concepts of diffusion in liquid-saturated porous media and the relationship between the diffusion in porous media and that in the liquid phase have been discussed by many researchers.1-10 There are abundant data for diffusion coefficients in liquids, while the reported effective diffusion coefficient data in liquid-saturated porous media are inadequate, especially for gas diffusion in liquid-saturated porous media under high pressure and temperature conditions. This is partly due to the difficulties encountered when measuring gas diffusion coefficients in liquid-saturated porous media especially under high-pressure conditions. Therefore, the diffusive tortuosity factors, if it can be determined, will provide an important parameter to relate the available diffusion coefficients to that in porous media. The effective diffusion coefficients can be defined as * To whom correspondence should be addressed. Tel.: 1(403)-2107642. Fax: 1(403)-284-4852. E-mail:
[email protected]. † University of Regina. ‡ Currently with Husky Energy Inc., Calgary, Canada. § University of Calgary.
(1)
where Deff is the gas effective diffusion coefficient in a liquidsaturated porous medium, D is the diffusion coefficient of gas in the infinite dilute liquid phase, and ε is the diffusive tortuosity factor of the liquid-saturated porous medium. It should be mentioned that the bulk rock volume-based concentration is required when using the Deff defined in eq 1 for calculating the diffusive flux. There are other forms of definition for Deff with the porosity of the porous medium involved, which requires the liquid phase-based concentration when it is used in the diffusive flux equation.1,5,6 More details have been given in the previous study.10 Given that the diffusion coefficient of a solute in a liquid and the diffusive tortuosity factor of a porous medium are available, the effective diffusion coefficient can be calculated by eq 1. Many attempts have been reported to obtain the diffusive tortuosity factors using a different method for liquidsaturated porous media.4,6,8 The tortuosity factors obtained from fluid flow can be significantly different from the diffusive tortuosity factors of porous media.4 It is believed that the tortuosity measured from electrical conductivity may agree with the diffusive tortuosity for liquid-saturated porous media considering the similarities between electrical conduction and molecular diffusion in liquid-saturated porous media. However, more studies in such an area are needed. Garrouch et al. compared the tortuosity measured from electrical conductivity and gas molecular diffusion in porous media and demonstrated the consistency between the tortuosities obtained from these two methods.8 However, only the gas-gas diffusion in dry porous media was determined in their diffusion measurements. More studies have been conducted recently on measurements of gas effective diffusion coefficients in liquid-saturated consolidated and unconsolidated porous media.9-11 With the measured effective diffusion coefficients Deff in liquid-saturated porous media, the tortuosity factors can be calculated from eq 1 if the diffusion coefficients D for the same fluid pairs are
10.1021/ie901765d 2010 American Chemical Society Published on Web 06/03/2010
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following assumptions apply: (1) the solution is infinitely dilute; hence, the diffusion coefficient D is a constant; (2) no densityinduced convection occurs in the diffusion process since CH4 dissolution does not cause the increase of the oil phase density; (3) the swelling of the oil phase due to the dissolution of methane is negligible; (4) the gas concentration at the liquid surface co is constant; and (5) gas and oil are immiscible. As shown in Figure 1, before the gas reaches the bottom of the diffusion cell, the oil column can be considered as a semiinfinite media. The diffusion process is described by Fick’s second law in one dimension:12 ∂c ∂2c )D 2 ∂t ∂z
(3)
The initial and boundary conditions are Figure 1. Schematic of gas diffusing into a semi-infinite oil column in a diffusion cell.
available.9,10 Both Deff and D have to be measured under the same pressure and temperature conditions for the same fluid pairs. The previous works have reported effective diffusion coefficients for CO2 in brine- and oil-saturated consolidated porous media.9,10 This provides a potential to calculate the diffusive tortuosity factors if the diffusion coefficients are available for the same fluid pairs of CO2/water and CO2/oil. However, most of the reported diffusion coefficients for CO2 in liquids are lack of reliability mainly due to the density induced natural convection. In addition, the oil or water sample used by different researchers in the literature are different from those used in previous studies.9,10 The focus of this paper is to further illustrate how to determine the diffusive tortuosity factors for liquid-saturated consolidated porous media by measuring both Deff and D using the same fluid pair. The fluid pair of CH4/oil was chosen for this study because there is no concern of a density-induced convection effect when the CH4 diffusion coefficient is determined by using the pressure-volumetemperature (PVT) method. The diffusive tortuosity factors for selected rock samples are calculated using eq 1. In addition, a demonstration is also given to convert the CO2 effective diffusion coefficients measured with porous media to that in bulk liquid phase. Experimental Section The proposed procedure to determine diffusive tortuosity factors involves three steps: (1) measure the diffusion coefficient of a gas in a bulk liquid phase, D, (2) measure the effective diffusion coefficient of the gas in oil-saturated porous media, Deff, and (3) calculate the diffusive tortuosity factor using eq 1 which can be modified into the following form: ε ) D/Deff
c ) 0 (0 e z e zo, t ) 0)
(4)
c ) co (z ) zo, t > 0)
(5)
and
where c is the gas concentration in oil at time t and distance x and co is the gas concentration at the gas/oil interface under the test pressure. The solution of eq 3 is given as12 c ) co[1 - erf(ζ)]
(6)
where ζ)
z0 - z
(7)
√4Dt
and erf(ζ) )
2 √π
∫
ζ -s2
0
e
ds
(8)
The mass conservation for the diffusion process, i.e., the gas lost in the gas phase equals the amount of gas diffused into the liquid phase, is given as -
∂c V dP ) DA ZRT ∂z
|
dt
(9)
z)z0
where V is the volume of the gas phase, R is the universal gas constant, T is the test temperature, Z is the gas compressibility factor, and A is the cross-sectional area of the diffusion cell (or the contact area of gas/liquid). Rearranging eq 9 and integrating it with respect to time t over the period from to to t and the corresponding pressure change from Po to P yields
(2)
∆P ) kp(√t - √to)
(10)
∆P ) Po - P
(11)
where Since the experimental procedure to measure Deff has been given in the previous studies,9,10 the following section is mainly focused on how to measure the diffusion coefficient D of CH4 in the bulk oil phase. Theoretical. In this study, the diffusion coefficient of CH4 in bulk oil is determined by using the traditional PVT method. A schematic of gas diffusing into an oil column contained in a diffusion cell is shown in Figure 1, where h is the height of the gas column and z0 is the height of the liquid column. For the gas/liquid system (CH4/hexadecane) used in this study, the
kp )
2ZRTco h
Dπ
(12)
Note that to is the initial time when diffusion occurs and Po is the initial pressure at time to. According to eq 10, plotting the experimental data of the pressure drop versus the square root of time yields a straight line which crosses the origin at to
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Figure 2. Schematic of the experimental setup for CH4 diffusion coefficient measurement.
) 0. Thus, the diffusion coefficient is determined from eq 12 with the slope kp obtained from eq 10. Apparatus. Figure 2 shows a schematic diagram of the experimental apparatus. It mainly consists of a diffusion cell, two high-pressure piston cylinders, two high-pressure positivedisplacement pumps (hand pump), and a data acquisition system. The high-pressure diffusion cell with a diameter of 6.3 cm and depth of 22 cm was used to contain the test gas/liquid system. Two high-pressure cylinders (500 cm3) were used to contain high pressure CH4 and oil. Both the diffusion cell and the high-pressure cylinders were placed in a water bath, where the temperature was maintained within (0.1 °C of the desired value. Two high-pressure hand pumps were used to pressurize gas and oil samples and to transfer them into the diffusion cell at a desired pressure. The pressure of the diffusion cell was monitored by using the same Heise digital pressure gauge (Dresser Instrument) with an accuracy of (0.025% of the fullscale span (34.5 MPa) and a resolution of 1 kPa. Procedure. Before each test, the entire system was tested for leakage at approximately 10 MPa. The pressure in the gas cylinder was then adjusted to the desired pressure and the diffusion cell was completely filled with the oil sample. The diffusion cell was connected to the oil sample cylinder but separated from the gas cylinder. Both the diffusion cell and the gas cylinder were put into the water bath for more than 24 h prior to each measurement to ensure the temperature of the test sample was at the desired temperature. After that, the following procedure was applied to measure the gas diffusion coefficient: (1) Adjust the pressure in the diffusion cell and the gas cylinder to ensure that they are as close as possible. Vacuum all the lines connecting the gas cylinder, the pressure gauge, and the diffusion cell. (2) Open the valve connecting the diffusion cell and the gas cylinder. Use two hand pumps to transfer a desired volume of CH4 into the diffusion cell and, at the same time, draw the same volume of oil to the oil sample cylinder. During the gas transferring process, the pressure of the system was maintained constant. Record the total volume of oil extracted from the diffusion cell, which is used to determine the height of the gas column inside the diffusion cell. (3) After the designed volume of gas was transferred into the diffusion cell, separate the diffusion cell from the gas and oil cylinder (by closing the valves to the gas cylinder and to the oil cylinder) and immediately start recording the cell pressure (pressure decay measurement).
(4) After a pressure decay measurement, carefully release the pressure in the diffusion cell. Vacuum the oil sample in the diffusion cell to remove the dissolved gas. Then, inject the oil into the diffusion cell from the oil sample cylinder to completely refill the cell and prepare for the test at the next pressure level. Materials. CH4 was chosen as the gas and an oil sample, n-hexadecane (92+ %), was chosen as the liquid. Methane (CH4) used in the experiment was supplied by Praxair (Regina, Canada) with a purity of 99.9%. n-Hexadecane (92+ %) (Alfa Aesar) was used as the oil sample. The molecular weight of the n-hexadecane was 226.45 g/mol and the density was 0.7733 g/cm3 at 18 °C and 0.7596 g/cm3 at 40 °C.13 The viscosity of the oil was measured to be 3.35 mPa s at 20 °C and 2.14 mPa s at 40 °C. A total of 19 core samples cut from 8 long Berea (Ber) and Benthiemer (Ben) sandstone plugs (about 30 cm each) were used as the porous media for effective diffusion coefficient measurements. These are high-quality sandstones and are composed of mainly of quartz sand cemented together by silica. The dimensions and properties of the 19 core samples are listed in Table 1. For the first five core plugs, Ben-1, Ben-2, Ber-1, Ber-2, and Ber-3, each of them was cut into two samples with different lengths. The longer ones were about 20 cm and the short ones were about 10 cm. The longer samples were used in the previous study to measure the effective diffusion coefficient of CO2 in brine-saturated porous media,9 while the shorter ones were used to determine the diffusive tortuosity factors using CH4 in this study. The remaining three core plugs Ber-4, Ber5, and Ber-6 were cut into three samples of about equal length (10 cm). Among the three cuts from each plug, two were used for CO2/oil effective diffusion coefficient measurements, which have been reported,10 and one cut was used to determine the diffusive tortuosity factors with CH4 in this study. Once the diffusive tortuosity factor is determined through the CH4-oil fluid pair, the measured CO2 effective diffusion coefficients in brine- or oil-saturated porous media can also be converted to diffusion coefficients in bulk liquid phase using eq 1. Results and Discussion Diffusion Coefficient of CH4 in Bulk Oil. Five measurements were conducted to determine the diffusion coefficient of CH4 in the test oil sample (hexadecane) at a pressure range of ap-
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Table 1. Dimension, Porosity, and Permeability of Core Samples Used for Different Gas/Liquid Systems
a
core pluga
diameter (cm)
permeability (md)
porosity (%)
test sample
length (cm)
test system
Ben-1
5.17
1,819
21.73
Ben-2
5.16
1,527
22.53
Ber-1
5.123
119
17.08
Ber-2
5.008
227
18.92
Ber-3
5.142
80
17.5
Ber-4
5.148
263
18.85
Ber-5
5.046
163
18.2
Ber-6
5.147
160
18.32
Ben-1-1 Ben-1-2 Ben-2-1 Ben-2-2 Ber-1-1 Ber-1-2 Ber-2-1 Ber-2-2 Ber-3-1 Ber-3-2 Ber-4-1 Ber-4-2 Ber-4-3 Ber-5-1 Ber-5-2 Ber-5-3 Ber-6-1 Ber-6-2 Ber-6-3
19.83 10.1 19.97 9.4 20.05 9.39 19.83 9.42 20.1 9.29 9.9 10.25 9.87 10.148 9.74 10.3 10.21 9.98 9.66
CO2/brine9 CH4/oil CO2/brine9 CH4/oil CO2/brine9 CH4/oil CO2/brine9 CH4/oil CO2/brine9 CH4/oil CO2/oil10 CO2/oil10 CH4/oil CO2/oil10 CO2/oil10 CH4/oil CO2/oil10 CO2/oil10 CH4/oil
Ber, Berea rock sample. Ben, Benthiemer rock sample. Ber-4, -5, -6 were numbered as Ber-1, -2, and -3 in the previous study.10
Figure 3. Experimental results for measurement of DCH4/O-1 at a temperature of 40 °C: (a) pressure vs diffusion time (pressure decay curve) and (b) pressure drop vs square root of time.
proximately 2-10 MPa. Figure 3 shows the measured smallpressure decay curve (a) and the pressure drop as a function of the square root of time (b) for measurement of DCH4/O-1. It is shown in Figure 3b that a linear relationship exists between the pressure drop and the square root of time except in the curvature at the very early stage. This early stage curvature was also found in the experiments by the previous researchers.14-17 Similar to the procedures taken by them, the early stage curvature was disregarded and the slope of the straight section of the plot was used to calculate the diffusion coefficient using eq 12.
The above procedure were applied to obtain the diffusion coefficient for all the measurements, DCH4/O-1 through DCH4/ O-5. The results and the parameters used in the calculation of the diffusion coefficient are summarized in Table 2. The plots for all the experimental data have been given elsewhere.18 The average pressures in Table 2 are the time-averaged pressures for each measurement. Both the compressibility factor, Z, and concentration, co, in eq 12 were determined based on the timeaveraged pressure in each measurement. The concentrations were determined based on the experimental measurements with more details being given previously.18 The height of the oil column, h, was calculated from the recorded volume divided by the cross-sectional area of the diffusion cell. As shown in Table 2, the measured diffusion coefficient ranges approximately from 2 × 10-5 to 3 × 10-5 cm2/s under the test conditions. The diffusion coefficient is plotted against the average pressure of the measurement in Figure 4, which shows that the diffusion coefficient decreases slightly with increasing pressure. A similar dependence of the CH4 diffusion coefficient on pressure was observed by previous researchers for different CH4/liquid hydrocarbon systems.14,19 A linear fit of the diffusion coefficient and pressure, as shown in Figure 4, was later used to calculate the diffusion coefficient of CH4 in oil under different pressures for back-calculating the diffusive tortuosity factors. For all the measurements listed in Table 2, the maximum pressure drop is approximately 8% of the test pressure. The maximum change of the diffusion coefficient based on the measured results is less than 1.5% in one pressure decay measurement. This justifies the assumption that the CH4 concentration at the CH4/oil interface and the diffusion coefficient are constant in each of the small-pressure decay measurement. With the measured diffusion coefficients, further calculations were made to examine whether CH4 could reach the bottom of the diffusion cell during a diffusion measurement. For a given diffusion coefficient of 3 × 10-5 cm2/s (which is greater than the actual value in the measurement) and a liquid column of 15 cm (which is shorter than the actual value in the measurement), the calculation shows that it takes more than 50 h for the gas to reach the bottom of the cell. Since all the CH4 diffusion measurements were finished within about 2 h, the semi-infinite condition assumption holds in all the measurements. Attempts were made to compare the measured results in this study with the reported results of CH4 diffusion coefficients in
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Table 2. Summary of the Measurements of CH4 Diffusion Coefficient in Bulk Oil at 40 °C
a
pressure (kPa) measurement no.
initial
final
average
h (cm)
co (10-3 mol/cm3)
Z
kp (kPa/min1/2)
D (10-5 cm2/s)
DCH4/O-1 DCH4/O-2 DCH4/O-3 DCH4/O-4 DCH4/O-5
2 124 4 207 6 343 8 337 9 918
1 956 3 919 6 035 8 017 9 624
2 017 4 025 6 148 8 137 9 736
1.925 1.636 2.534 3.208 4.299
0.308 0.615 0.939 1.243 1.488
0.965 0.931 0.8995 0.876 0.861
18.31 40.23 37.14 36.49 29.72
2.72 2.55 2.4 2.23 1.92
a
D, diffusion coefficient measurement; -CH4/O, CH4/oil system.
Figure 4. Diffusion coefficient of CH4 in hexadecane as a function of pressure at 40 °C.
similar hydrocarbon liquids. The diffusion coefficient of CH4 in a gasoline fraction at 1-2 MPa and 30 °C was approximately 5 × 10-5 cm2/s as measured by Pomeroy et al.20 Hill et al. recommended a correlation to calculate the diffusion coefficient based on their experimental data of CH4 in different hydrocarbon liquids.21 This correlation yields a diffusion coefficient of 3.4 × 10-5 cm2/s for CH4 in a hexadecane, which is slightly higher than the values measured in this study. The diffusion coefficient of CH4 in a decane at 38 °C, as measured by Reamer et al. decreased from 5.9 × 10-5 to 1.7 × 10-5 cm2/s as pressure increased from 3.5 to 30 MPa.14 The CH4 diffusion coefficient in decane is significantly higher than that in hexadecane from both this study and Hill et al.’s correlation.21 This might be attributed to the higher viscosity of hexadecane than decane. Despite the discrepancies of the results for different systems and slightly different test conditions, all the results show a trend of decrease in the diffusion coefficient with increasing pressure. CH4 Effective Diffusion Coefficient in Oil-Saturated Rock Samples and Diffusive Tortuosity Factors. Eight measurements in total were conducted for CH4 diffusion in oilsaturated porous rocks as indicated in Table 1. All the measurements of CH4 effective diffusion coefficient in this study were conducted at 40 °C and approximately 3.5-4 MPa. As an example, the experimental and theoretical pressure drop, ∆P, vs square root of time, t, of the measurement of EDCH4/O-1 is shown in Figure 5. The detailed procedures to obtain the corrected experimental data and predicted curves shown in the figure and complete plots for all eight measurements have been reported elsewhere.18 Table 3 summarizes the results of the eight measurements of the CH4 effective diffusion coefficient. With the measured values of Deff in Table 3 and D in Figure 4, the diffusive tortuosity factor ε was determined for each core sample. The maximum pressure change in the pressure decay measurements as listed in Table 3 is less than 6% of the average pressure. The maximum uncertainty due to the concentration
Figure 5. Experimental and theoretical ∆P vs t1/2 plots for measurements EDCH4/O-1 at 40 °C.
and compressibility change is estimated to be less than 7%. On the basis of the pressure dependence of the CH4 diffusion coefficient, the relative change of the effective diffusion coefficient in a pressure decay measurement is estimated to be less than 0.8%. Unlike the CO2/oil system, the dissolution of CH4 in the oil phase causes a decrease in the oil phase density, which can also induce natural convection when CH4 diffuses into the oil-saturated porous column under certain conditions. For CH4/oil-saturated porous rock samples, the absolute values of the Rayleigh numbers range from 0.2 to 4.72. The largest Rayleigh numbers correspond to the highest permeability rock samples used in the measurements, Ben-1-1 and Ben-1-2. The effect of natural convection on diffusion for such low Rayleigh numbers has been examined to be negligible.22 As shown in Table 3, the diffusive tortuosity factors obtained for the eight sandstone plugs are within the range of 2-5. Large tortuosity values, as observed by Cussler,4 were not seen in this study. This is due to the difference of the microscopic structures of the porous media used in this study and that used by Cussler.4 Nevertheless, it is seen that the same type of Berea cores may have significantly different tortuosity factors, while different types of sandstones (Berea and Benthimer) may have very similar tortuosity factors despite the fact that huge differences exist in porosities and permeabilities. Similar observations were also found by Garrough et al.8 Their measurements showed that, for some sandstone samples with three orders of difference in permeability, the measured diffusive tortuosities for gas in dry porous media can be rather close.8 Since tortuosity was introduced to account for the sinuous diffusion path of the solute, it is probably more dependent on the sinuousness of the pore structures, the sand grain size and arrangements, and the cementing between sand grains rather than the most accessible petrophysical properties, such as porosity and permeability. The conductive tortuosity factors may bear certain similarities with the diffusive tortuosity factors, but further studies in this area are needed.
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Table 3. Summary of CH4 Effective Diffusion Coefficient Measurements at 40 °C pressure (kPa) measurement no.
sample
initial
final
average
V (cm3)
c0 (10-3 mol/cm3)
Z
Deff (10-6 cm2/s)
D (10-5 cm2/s)
ε
EDCH4/O-1 EDCH4/O-2 EDCH4/O-3 EDCH4/O-4 EDCH4/O-5 EDCH4/O-6 EDCH4/O-7 EDCH4/O-8
Ber-1-2 Ber-2-2 Ben-1-2 Ben-2-2 Ber-3-2 Ber-4-3 Ber-5-3 Ber-6-3
3 958 4 125 3 522 3 995 4 110 4 016 3 894 4 027
3 757 3 960 3 351 3 811 3 950 3 795 3 791 3 850
3 821 4 012 3 406 3 870 4 001 3 863 3 850 3 905
72.08 74.71 75.89 77.34 76.42 61.48 74.28 81.65
0.584 0.613 0.518 0.591 0.591 0.59 0.588 0.597
0.933 0.931 0.935 0.933 0.931 0.933 0.933 0.932
9.40 5.83 5.80 6.32 5.96 8.30 6.92 8.82
2.58 2.56 2.62 2.57 2.57 2.57 2.56 2.62
2.74 4.39 4.51 4.07 4.31 3.10 3.70 2.97
Table 4. Calculated CO2 Diffusion Coefficients in Bulk Brine at 59 °C measurement no.
sample
pressure (kPa)
Deff (10-6 cm2/s)a
ε
D (10-5 cm2/s)
EDCO2/W-1 EDCO2/W-2 EDCO2/W-3 EDCO2/W-4 EDCO2/W-5 EDCO2/W-6 EDCO2/W-7 EDCO2/W-8
Ber-1-1 Ber-2-1 Ben-1-1 Ben-2-1 Ber-3-1 Ber-3-1 Ber-3-1 Ber-3-1
4 303 4 300 4 314 4 294 2 436 4 238 5 096 7 340
7.19 4.66 4.73 4.70 3.12 4.45 5.50 6.30
2.74 4.39 4.52 4.07 4.31 4.31 4.31 4.31
1.97 2.05 2.14 1.91 1.34 1.92 2.37 2.72
a
Deff from ref 9.
Table 5. Calculated CO2 Diffusion Coefficients in Bulk Oil at 40 °C measurement no.
sample
pressure (kPa)
Deff (10-6 cm2/s)a
ε
D (10-5 cm2/s)
EDCO2/O-1 EDCO2/O-2 EDCO2/O-3 EDCO2/O-4 EDCO2/O-5 EDCO2/O-6
Ber-4-1 Ber-4-2 Ber-5-1 Ber-5-2 Ber-6-1 Ber-6-2
2 283 6 259 4 244 4 925 5 600 3 031
6.54 8.01 5.98 6.40 7.95 6.90
3.10 3.10 3.70 3.70 2.97 2.97
2.03 2.48 2.21 2.37 2.36 2.05
a
Deff from ref 10.
CO2 Diffusion Coefficient in Bulk Brine and Oil. With the determined tortuosity factors ε listed in Table 3 and CO2 effective diffusion coefficient measured previously,9 CO2 diffusion coefficient in bulk brine, D, can be calculated by rearranging eq 1. Table 4 shows the calculated results of the diffusion coefficients. The CO2 diffusion coefficient in bulk brine exhibits an increasing trend with increasing pressure. This trend agrees with the results of the CO2 diffusion coefficient in bulk brine as reported by Renner16 and CO2 diffusion coefficients in bitumen as measured by Upreti and Mehrotra.19 However, the values of the CO2 diffusion coefficient in bulk brine measured in this study (1.34 × 10-5 to 2.72 × 10-5 cm2/s) are significantly lower than Renner’s results (3.07 × 10-5 to 7.35 × 10-5 cm2/s) despite the fact that the temperature in his measurements (38 °C) is lower than that in this study (59 °C). Diffusion coefficient should be lower at lower temperatures. One reason causing higher values in his experiment could be that natural convection still existed and contributed to the mass transfer process in the measurements. The natural convection effect on the measured diffusion coefficients in Table 4 is quantitatively examined to be negligible.22 Similar to the calculation of CO2 diffusion coefficient in the water phase, the CO2 diffusion coefficients in the oil phase are calculated and listed in Table 5. The diffusion coefficient of CO2 in the test oil sample falls in the range of 2 × 10-5 to 2.5 × 10-5 cm2/s. These results are in the same range as those reported by Grogen et al. (1.80 × 10-5 to 3.21 × 10-5 cm2/s) for CO2/hexadecane at a pressure range of 2.1-5.3 MPa and 25 °C.23 They are significantly lower than the results for CO2/ decane (1.97 × 10-5 to 5.05 × 10-5 cm2/s) measured by Renner at similar temperature and pressure ranges (1.5-5.86 MPa and
38 °C).16 Despite the discrepancies between the results from this study and those reported in the literature, all the measured values are in the range of approximately 2 × 10-5 to 5 × 10-5 cm2/s. The higher trend of the literature values might be ascribed to the contribution of natural convection, which was not examined in their studies even though they indicated that natural convection was negligible in their measurements.16,23 Also shown in Table 5, the CO2 diffusion coefficient in hexadecane increases slightly with increasing pressure, which agrees with the observations made by other researchers.16,23 The procedure described above also provides an alternative for measuring CO2 diffusion coefficients in bulk liquids, such as water and oil, where natural convection occurs as CO2 dissolves in these liquids which interferes with the diffusion process and causes the traditional method unsuitable for CO2 diffusion coefficient measurements. Conclusions and Recommendations (1) An experimental procedure is developed for the evaluation of the diffusive tortuosity factors of liquid-saturated porous media. This provides a tool to relate the diffusion coefficient measured in the bulk liquid phase to that in porous media. (2) The proposed method was tested with CH4/oil systems for different rock samples. The measured diffusive tortuosity factors for tested sandstone rock samples fall in the range between 2 and 5. These values can serve as reference data to convert the diffusion coefficient in the bulk liquid phase into that in porous media for modeling of solute diffusion in liquid-saturated sandstone formations. (3) With the measured diffusive tortuosity factors, the diffusion coefficients of CO2 in the liquid phase were calculated based on the measured diffusive tortuosity factors and the previously reported effective diffusion coefficients. This also provides an alternative for determining the CO2 diffusion coefficient in the bulk liquid phase to exclude the interference of density-induced natural convection in traditional PVT methods. Acknowledgment The financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Petroleum Technology Research Center (PTRC), Regina, Saskatchewan, is gratefully acknowledged. Nomenclature A ) cross-sectional area of diffusion cell, (cm2) c ) concentration of gas in liquid phase, (mol cm-3) co ) concentration of gas in liquid phase at gas/liquid interface, (mol cm-3) D ) diffusion coefficient of gas in liquid, (cm2 s-1) Deff ) gas effective diffusion coefficient in porous medium, (cm2 s-1)
Ind. Eng. Chem. Res., Vol. 49, No. 13, 2010 -2
g ) gravitational acceleration, (980 cm s ) h ) height of gas column, (cm) kp ) slope of ∆P vs t plot, (kPa/s1/2) n ) total mass flux, (mol cm-2 s-1) P ) pressure, (MPa) or (kPa) Po ) initial pressure of gas phase in diffusion measurement, (kPa) R ) universal gas constant, (8 314 kPa cm3 mol-1 K-1) t ) time, (s or min) to ) initial time of diffusion, (s or min) T ) temperature, (K) V ) volume of gas phase in diffusion cell, (cm3) z ) vertical distance of oil column, (cm) z0 ) height of oil column, (cm) Z ) gas compressibility factor Greek Symbols ∆P ) pressure drop in gas phase, (kPa) ε ) diffusive tortuosity of porous rock φ ) porosity of porous medium
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ReceiVed for reView November 7, 2009 ReVised manuscript receiVed May 18, 2010 Accepted May 20, 2010 IE901765D
