Effects of Operational Parameters on Diffusion Coefficients of CO2 in a

Article Cite This: Ind. Eng. Chem. Res. 2017, 56, 12799-12810

pubs.acs.org/IECR

Effects of Operational Parameters on Diffusion Coefficients of CO2 in a Carbonated Water−Oil System Guanli Shu, Mingzhe Dong,* Hassan Hassanzadeh, and Shengnan Chen Department of Chemical and Petroleum Engineering, Schulich School of Engineering, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4 S Supporting Information *

ABSTRACT: Diffusion of carbon dioxide (CO2) in a carbonated water−oil system is of great importance for proper design of CO2based enhanced oil recovery (EOR) processes. We study the effects of operational parameters such as saturation pressure, temperature, and phase volumes on diffusion coefficients of CO2 in a carbonated water−oil system. Results show that diffusion coefficients of CO2 in both phases are susceptible to saturation pressure. The greater the saturation pressure, the larger the diffusion coefficients. At a given saturation pressure, diffusion coefficients of CO2 in two phases increase by increasing temperature. Values of the coefficients determined at 40 °C are about twice those determined at 20 °C. The equilibration of the system was found to be much faster at the higher temperature. The results indicate that the predicted diffusion coefficients are insensitive to phase volumes, indicating applicability of the determined diffusion coefficients to simulate the mass transfer in large-scale reservoirs.

1. INTRODUCTION Mass transfer driven by molecular diffusion is an essential mechanism in many areas, such as petroleum engineering, soil science, chemical engineering, and biotechnology.1 Particularly in petroleum engineering, mass transfer is a key mechanism for determining the performance of solvent-based processes for oil recovery.2,3 Understanding the mass transfer processes is highly important for designing and optimizing oil recovery processes.4 Accordingly, it is imperative to accurately estimate molecular diffusion coefficients for the purpose of describing the mass transfer processes carried out at reservoir conditions.5−10 Generally speaking, there are two experimental methodologies (direct and indirect) to determine the diffusion coefficient. A direct method is carried out by the composition analysis of fluid samples extracted from a diffusion cell at various times by gas chromatography or other analytical devices.10,11 Nevertheless, this method is highly expensive and is prone to errors resulted from the disturbance that takes place throughout the withdrawal of a fluid sample.12 Indirect methods, including pressure pulse method3,5,9 and NMR,13 measure the changes of properties (for instance, pressure, volume, and interface movement) and then obtain the diffusion coefficients by means of analytical and numerical models or graphical methods.3,7 Considering the ease and affordability, indirect means have been broadly applied, and among which the pressure decay method is the most extensively used one. Pressure variations are recorded through the experiment and then used to estimate diffusion coefficients using a forward analytical or numerical model. © 2017 American Chemical Society

Over the past decades, the diffusion coefficients of CO2 in the aqueous phase and oleic phase have been studied separately. Experimental results of the diffusivities of CO2 in diluted water or brine have been published extensively at elevated pressures and temperatures.14−22 The diffusion coefficients were increased as a function of temperature, but there was no consistent conclusion on the effect of pressure change. Besides, a number of semiempirical correlations, in terms of either the kinetic theory of Chapman−Enskog or the hydrodynamic theory of Stokes−Einstein, have been proposed.23−28 For the investigation of the diffusion coefficient of CO2 in the oleic phase, most of the previous studies focused on heavy oil or bitumen.2,3,11,29−33 However, only a small proportion of the literature addressed the diffusion coefficient of CO2 in light oil.7,8 In order to recover the trapped residual oil from water-wet reservoirs, a new methodology of injecting carbonated water (CW) as a preflush before CO2 flood was developed to overcome the water blocking effect.34 In this method, CO2 is first dissolved in the water phase and then transferred into the oil phase by means of molecular diffusion through the injection of carbonated water. To accurately investigate the mass transfer processes, two diffusion coefficients are involved, i.e., the diffusion coefficients of CO2 in the water phase and oil phase. Received: Revised: Accepted: Published: 12799

June 21, 2017 October 9, 2017 October 17, 2017 October 17, 2017 DOI: 10.1021/acs.iecr.7b02546 Ind. Eng. Chem. Res. 2017, 56, 12799−12810

Article

Industrial & Engineering Chemistry Research

Figure 1. (a) Schematic diagram of the experimental setup. (b) Schematic of carbonated water−crude oil system in a closed diffusion cell. Reprinted from ref 35. Copyright 2017 American Chemical Society.

experiments were conducted and the diffusion coefficients of CO2 in the water phase and oil phase are determined at various experimental conditions. The effects of the operational parameters, such as saturation pressure, temperature, and changes of phase volumes, on the diffusion coefficients are discussed. Meanwhile, the impacts of these operational parameters on pressure, densities of water and oil phases, concentration distribution, interface displacement, and mass transfer of CO2 transferred in the oil phase are also investigated.

Even though the topic of the diffusion coefficient of CO2 has been studied extensively, rarely addressed is the method of simultaneous determination of two coefficients in an immiscible liquid−liquid system, much less for a carbonated water−oil system. The primary difficulties of determining these coefficients are the presence of several unknowns and the mutual effect on the water phase and oil phase during the CO2 mass transfer. Accordingly, some researchers still applied the diffusion coefficients from the empirical correlations or experimental results from the CO2−water and CO2−oil systems solely. The inaccurate estimation produced unreal predictions of the mass transfer process. To deal with this problem, recently, we have developed an analytical model and designed an experimental method to determine the diffusivities of CO2 in the miscible liquid−liquid system.35 Our study35 revealed that the diffusion coefficients of CO2 in water and oil phases are independent of initial pressure, which proved the assumption for the model. In addition to the initial pressure, other operational parameters such as saturation pressure, temperature, and phase volumes are worthy of evaluation, which are subject of the current study. In this work, based on the analytical model and simplified experimental method proposed in our previous work,35 new

2. EXPERIMENTAL PROCEDURES AND THEORY 2.1. Experimental Procedures. The schematic diagram in Figure 1a demonstrates the experimental setup for investigating the diffusion mechanism of carbon dioxide in a carbonated water−oil system. Pressure change is recorded from the diffusion cell in which crude oil lies on the top and carbonated water at the bottom. The cell is immersed in the water bath where a desired experimental temperature could be kept constant. The properties of crude oil, preparation method, and procedure of carbonated water (CW), and the specifications of instruments, are elaborated in the Supporting Information. Following a gas leakage test, the diffusion cell and carbonated 12800

DOI: 10.1021/acs.iecr.7b02546 Ind. Eng. Chem. Res. 2017, 56, 12799−12810

Article

Industrial & Engineering Chemistry Research Table 1. Summary of Results of Diffusion Coefficients Obtained under Different Operation Conditions interface concn of CO2 at 10 h (10−3 mol/cm3)

exptl press. (MPa)

3

init phase vol (cm ) test no.

temp (°C)

water

oil

total

saturation press. (MPa)

init

final

init concn of CO2 in water phase (10−3 mol/cm3)

water

oil

1 2 3 4 5 6 7

20 20 20 20 40 40 40

80 80 80 80 80 100 45

70 70 70 70 70 50 105

150 150 150 150 150 150 150

16.00 17.93 19.31 20.68 17.93 17.93 17.93

16.56 18.75 19.98 21.08 18.59 18.79 18.52

18.08 22.83 21.97 22.55 21.82 22.05 19.69

1.3343 1.3475 1.3545 1.3617 1.0835 1.0836 1.0835

0.6500 0.7000 0.7500 0.8000 0.5000 0.5000 0.5000

2.1678 2.3200 2.4738 2.6293 2.0836 2.0814 2.0862

water cylinder were placed into the water bath and filled with distilled water. Lines connecting the crude oil vessel and the diffusion cell were vacuumed afterward. After introducing crude oil into the diffusion cell, the system temperature was set to the water bath temperature at a certain value, leaving the system for 24 h to ensure the test fluid equilibrated to the desired temperature. As the pressurization of the diffusion cell by crude oil to match the CW cylinder’s pressure, CW was injected through the bottom port of the cell using a syringe pump. When a designed volume of CW was transferred, the valve connecting the cell and the crude oil transfer vessel was closed, as was the valve between the diffusion cell and the CW cylinder. At this point, the designed amount of oil remained at the top and the CW remained at the bottom of the diffusion cell. The pressure recording was started immediately, with the amplitude of the pressure change recorded every minute, and was continued until no substantial change in pressure was observed. The state of the carbonated water and crude oil in the closed diffusion cell is shown in Figure 1b. 2.2. Theory. According to the schematic of the carbonated water−oil system in a closed cell, a theoretical model is developed, taking the following assumptions into consideration: (1) the mass transfer system is at an isothermal condition without any chemical reaction; (2) the interface between the water phase and oil phase is considered to remain at thermodynamic equilibrium; (3) diffusion coefficients of CO2 in the water phase and oil phase are assumed constant; (4) no natural convection is induced during the mixing process; (5) mass transfer happens between the two semi-infinite regions; (6) two phase interface moves as the diffusion continues. The detailed explanation of the assumptions can be found in the Supporting Information. By adopting Fick’s law, two governing equations are established based on the assumption that the phase densities are weak functions of the CO2 concentration. ∂Co(x , t ) ∂ 2Co(x , t ) = Do , ∂t ∂x 2

∂Cw(x , t ) ∂ 2Cw(x , t ) = Dw , ∂t ∂x 2

s (t ) < x < ∞

−∞ < x < s(t )

diffusion coeff (cm2/s) Dw 1.00 1.60 2.25 3.20 3.50 3.49 3.50

× × × × × × ×

Do 10−5 10−5 10−5 10−5 10−5 10−5 10−5

1.02 1.27 1.37 1.48 2.77 2.77 2.76

× × × × × × ×

10−6 10−6 10−6 10−6 10−6 10−6 10−6

phases at the interface (C*o , C*w ), and a partition coefficient (kpc, an average correlation between the partition coefficient and pressure has been presented by Shu et al.35), two equations describing the concentrations of CO2 in the water phase and oil phase are proposed as follows: C w (x , t ) = C i −

⎡ ⎛ ⎞⎤ C i − Cw* ⎢1 + erf⎜⎜ x ⎟⎟⎥ ⎛ ⎞⎢ ⎝ 2 Dw t ⎠⎥⎦ 1 + erf⎜λ Do ⎟ ⎣ ⎝ Dw ⎠ (3)

Co(x , t ) =

where λ =

⎡ ⎛ ⎞⎤ Co* ⎢1 − erf⎜⎜ x ⎟⎟⎥ 1 − erf(λ) ⎢⎣ ⎝ 2 Dot ⎠⎥⎦

s(t ) 2 Dot

and thus

s(t ) 2 Dw t

(4)

= λ Do /Dw . At the moving

interface (x = s(t)), the following equation is derived in terms of the mass conservation: −

Dw Do

2 Co* C i − Cw* e−λ2(Do / Dw ) + e −λ ⎛ Do ⎞ 1 − erf(λ) ⎟ 1 + erf⎜λ ⎝ Dw ⎠

= (Co* − Cw*)λ π

(5)

The detailed derivation process is shown in the Supporting Information. To determine the diffusion coefficients of CO2 in the water phase and oil phase, collecting experimental pressure data from the diffusion test is the first step. After obtaining the pressure at a selected time, a well-designed trial-and-error procedure is applied. Initial guesses are made for interface concentrations and diffusion coefficients. The concentration profiles of CO2 in two phases can be plotted based on the guessed values by applying eqs 3−5. The densities and volume changes of two phases are calculated in accordance with the initial conditions and correlations of densities of the water and oil phases, and the correlation of the partition coefficient. Two convergence criteria are used to obtain the optimal results; one is to compare the difference of the volume change of the two phases and the other is to compare the difference of the diffusion fluxes at the interface for the two phases. The detailed trial-and-error procedure has been described by Shu et al.35

(1)

(2)

where Co(x,t) and Cw(x,t) are the concentrations of CO2 in the oil and water phases, respectively, in mol/cm3. Do and Dw and are the corresponding diffusion coefficients of CO2 in the oil and water phases, respectively, in cm2/s. After assuming the initial concentration of CO2 in the water phase (Ci) and two boundary conditions (Co(+∞,t) = 0, Cw(−∞,t) = Ci), concentrations of CO2 in the oil and water

3. RESULTS AND DISCUSSION The schematic of the problem, the material used in the experiments, the experimental setup, and the experimental procedure have been detailed previously35 and are avoided here for the sake of brevity (for a better understanding, Supporting 12801


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Figure 2. Measured pressure data of CO2 mass transfer in a carbonated water−oil system versus time for four tests conducted at different saturation pressures and the experimental temperature of 20 °C: (a) test 1, (b) test 2, (c) test 3, and (d) test 4.

Figure 3. Calculated density changes of water and oil phases versus distance from the initial interface x = 0 at different times for four tests: (a) test 1, (b) test 2, (c) test 3, and (d) test 4.

and the oil phase. Results of diffusion coefficients from seven tests, carried out at different operational conditions, are summarized in Table 1. From Table 1, it is of significance that the diffusion coefficients are greatly relevant to the saturation pressure and temperature. The difference of phase volumes seems to be less influential. The detailed model and procedure for estimation of the diffusion coefficients reported

Information is provided). In this work, we used the same methodology described in our previous work to study the effect of operational parameters on the estimation of the molecular diffusion coefficient of CO2 in water and oil. As such, the same crude oil is used throughout all tests. In this work, experimental studies were performed to analyze the effects of saturation pressure, temperature, and phase volumes on the diffusion coefficients of CO2 in the water phase 12802


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Figure 4. Calculated concentration profiles of CO2 in water and oil phases at six times (5, 10, 15, 20, 25, 30 h) for four tests: (a) test 1, (b) test 2, (c) test 3, and (d) test 4.

20 °C. From Figure 2, it is noticeable that the system pressure increases with time and gradually reaches a plateau. Within these four tests, test 2 is chosen as a special case with a considerably longer experimental time (more than 500 h) to examine the actual duration needed to reach the equilibrium for this carbonated water−oil system. The result reveals that the mass transfer of CO2 from the water phase to the oil phase is a long time process. Hence, a shorter time (around 70 h) is selected in this study for other tests. The pressure buildup during the CO2 mass transfer process can be attributed to two main factors: density increase and swelling of the oil phase. This density increase tendency has been proven by previous literature and described in the work conducted by Shu et al.35 In their study, under the pressure of 20 MPa and temperature of 20 °C, the density of the oil/CO2 mixture increases from 0.8138 to 0.8184 g/cm3 when the concentration of CO2 is increased from 0 to 0.5366 × 10−3 mol/cm3. With the continuous increase of CO2 concentration in the oil phase, the oil/CO2 mixture density continuously increases. Shu et al.35 also presents a correlation equation for the density of the oil/ CO2 mixture under different pressure and temperature. Figure 3 demonstrates the calculated density changes of the water phase and oil phase as a function of distance at different times for four tests. From Figure 3, a similar trend can be observed. With the diffusion of CO2 from the carbonated water phase into the oil phase, the density of the water phase decreases. By contrast, the density of the oil phase increases. This is because of the declining concentration of CO2 in the water phase and the incremental concentration of CO2 in the oil phase. Figure 4 shows the calculated concentration profiles of CO2 in the water phase and oil phase versus time for four tests. As listed in Table 1, the different saturation pressures bring about different initial concentrations of CO2 in the water phase;

elsewhere.35 The detailed analyses of the new experiments are illustrated in the following sections. 3.1. Effect of Saturation Pressure on Diffusion Coefficients. Tests 1−4, shown in Table 1, are carried out to investigate the effect of saturation pressure on diffusion coefficients of CO2 in the water phase and oil phase. Saturation pressure is the pressure for a corresponding saturation temperature at which the brine is fully saturated with carbon dioxide. Different saturation pressures lead to varied initial concentrations of CO2 in the carbonated water phase. For one thing, comparing different initial concentrations may shed light on the impact on diffusion coefficients, which are used to confirm the accuracy of values of coefficients determined in previous literature and the feasibility of applying these coefficients in other systems. For another thing, the expression of concentration of CO2 in the water phase, rather than the degree of carbonization, is applied to directly clarify the impact of initial concentration on the diffusion coefficients. From Table 1, it can be seen that the saturation pressures for four tests are in the range between 16.00 and 20.68 MPa. The corresponding initial concentrations of carbon dioxide in water varies from 1.3343 × 10−3 to 1.3617 × 10−3 mol/cm3. It can be concluded that the greater the saturation pressure, the larger the initial concentration. Meanwhile, the variation on the interface concentration can also be observed. Since the initial pressures for four tests are distinct, the corresponding experimental pressures at 10 h are different. As explained by Shu et al.,35 the change of pressure inside a closed isothermal cell is only induced by the mass transfer of CO2 in the carbonated water−oil system. Figure 2 illustrates the measured pressure data of the CO2 mass transfer in a carbonated water− oil system as a function of time for four tests at different saturation pressures and the same experimental temperature of 12803


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corresponding time scale is between 16.98 and 17.66 MPa. The volume of the water phase is reduced from 79.896 to 79.701 cm3, and the volume of the oil phase is increased from 70.132 to 70.313 cm3. The relative error of the total volume and the difference of the phase volumes are quite small (around 0.02 cm3); however, the difference of diffusion fluxes changes substantially. As the time increases from 10 to 30 h, the difference keeps increasing. Similar trends can also be observed in the other three tests, which indicate the inappropriateness of applying the same diffusion coefficients over a longer time (more than 30 h). For test 2, the pressure changes from 18.95 to 19.69 MPa in the range 5−30 h; the phase volume changes from 79.857 to 79.626 cm3 for the water phase and from 70.172 to 70.396 cm3 for the oil phase. For test 3, the system pressure varies from 20.47 to 21.27 MPa and the phase volume changes from 79.849 to 79.597 cm3 for the water phase and from 70.177 to 70.420 cm3 for the oil phase. For test 4, the system pressure varies from 21.59 to 22.22 MPa and the phase volume changes from 79.816 to 79.547 cm3 for the water phase and from 70.197 to 70.480 cm3 for the oil phase. As indicated from Figure 4, it can be seen that the interface of the water phase and oil phase moves gradually toward the water phase as a result of the mass transfer of CO2 from the carbonated water phase into the oil phase. The comparison of interface displacements of tests 1−4 is shown in Figure 6. The interface displacement versus time, plotted in a log−log graph, results in a straight line, which reveals the diffusive nature of the mass transfer process. The interface displacement is calculated based upon the phase volume change of the oil phase divided by the cross-sectional area (A = 10 cm2). The phase volume changes for tests 1−4 are shown in Table 2. It is noticeable from Figure 6 that the greater the diffusion coefficient, the faster the interface displacement. The diffusion coefficient of CO2 in the oil phase for test 4 is the greatest within four tests; thus it has the fastest interface displacement. By contrast, the diffusion coefficient of CO2 for test 1 is the smallest, resulting in the lowest displacement. The four tests exhibit linear relationships between the interface displacement and time, which demonstrates the nature of the mass transfer process. Figure 7 illustrates the mass change of CO2 in the oil phase during the mass transfer. The mass of CO2 dissolved in the oil phase as a function of time in a log−log graph also results in a straight line, which again manifests the diffusive nature of the mass transfer process and indicates that the greater the diffusion coefficient, the faster the mass of CO2 transferred in the oil phase. According to Fick’s law, regression equations for the four tests shown in Figure 7 can be used to verify the diffusion coefficients.

therefore, different interface concentrations of CO2 in the two phases are observed. At 10 h, test 4 has the greatest interface concentrations of CO2: 0.8000 × 10−3 mol/cm3 in the water phase and 2.6293 × 10−3 mol/cm3 in the oil phase. Test 1 has the lowest values of interface concentrations of CO2: 0.6500 × 10−3 mol/cm3 in the water phase and 2.1678 × 10−3 mol/cm3 in the oil phase. From Figure 4, it can be seen that there is a concentration difference between interface concentrations of CO2 in the water phase and in the oil phase, which is resulted from the partition coefficient of CO2, which is defined as the ratio of the concentration of CO2 in the water phase to that in the oil phase. Previous study conducted by Shu et al.35 demonstrates that the partition coefficient is a function of pressure at a given temperature, and it is less affected by the initial CO2 concentration. Another point from Figure 4 is that, following the mass transfer of CO2 in the water phase close to the two-phase interface, the concentration of CO2 in the water declines gradually which causes the decreased density. This decline will continue until the completion of mass transfer. In addition, Figure 4 also indicates the phase volume changes which can be seen through the interface displacement. This displacement explains the phenomenon of oil swelling, owing to the mass transfer of CO2 Diffusion coefficients determined at 10 h for four tests under different saturation pressures have been listed in Table 1. The results reveal that diffusion coefficients of CO2 in both phases increase by increasing the saturation pressure. Test 4 has the greatest diffusion coefficients of CO2: 3.20 × 10−5 cm2/s in the water phase and 1.48 × 10−6 cm2/s in the oil phase. To better understand this change, the values of diffusion coefficients as a function of the saturation pressure are plotted in Figure 5.

Figure 5. Changes of diffusion coefficients of CO2 in water and oil phases as a function of saturation pressure for tests 1−4 at 20 °C.

Figure 5 shows that linear relationships exist between the diffusion coefficients and the saturation pressures. Two regression equations provide two correlations for determining diffusion coefficients of CO2 in both phases at different saturation pressures and at 20 °C. For the oil phase, DoCO2 = 0.0972Psat − 0.5118 (R2 = 0.9790); for the water phase, DwCO2 = 0.4630Psat − 6.5436 (R2 = 0.9649). Figure 5 also manifests that the diffusion coefficient of CO2 in the water phase is higher than that in the oil phase. After determining the diffusion coefficients at 10 h, it is imperative to examine the effect of applying these coefficients throughout each test. In this section, five additional analysis times were selected (5, 15, 20, 25, 30 h). Results of the tests are listed in Table 2. For test 1, the examined pressure range for the

flux = DoA

∂Co ∂x

= Co*A x=0

Do πt

(6)

Integrating eq 6 yields the expression of the mass of CO2 transferred into the oil phase. Q = 2Co*A

Do 1/2 t π

(7)

where Q is the mass of CO2 transferred into the oil phase in grams; C*o is the interface concentration of CO2 in the oil phase in mol/cm3; A is the cross-sectional area in cm2; Do is the diffusion coefficient of CO2 in the oil phase in cm2/s; t is the diffusion time in seconds. 12804


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Table 2. Summary of Results for Tests 1−4 at Different Times and Saturation Pressures at the Experimental Temperature of 20 °C phase vol determined (cm3) time (h)

press. (MPa)

water

oil

5 10 15 20 25 30

16.98 17.19 17.33 17.47 17.59 17.66

79.896 79.839 79.794 79.762 79.726 79.701

70.132 70.183 70.223 70.256 70.285 70.313

5 10 15 20 25 30

18.95 19.16 19.32 19.46 19.58 19.69

79.857 79.790 79.744 79.698 79.664 79.626

70.172 70.235 70.283 70.324 70.360 70.392

5 10 15 20 25 30

20.47 20.68 20.84 20.99 21.14 21.27

79.849 79.776 79.718 79.676 79.631 79.597

70.177 70.247 70.301 70.346 70.384 70.420

5 10 15 20 25 30

21.59 21.79 21.93 22.05 22.15 22.22

79.816 79.736 79.680 79.625 79.584 79.547

70.197 70.277 70.339 70.390 70.436 70.480

total Test 150.028 150.022 150.017 150.018 150.011 150.014 Test 150.028 150.025 150.027 150.022 150.024 150.018 Test 150.025 150.023 150.019 150.021 150.015 150.017 Test 150.013 150.012 150.019 150.016 150.020 150.026

vol change (cm3) rel error of total vol (%) 1 (Dw = 1.00 0.02 0.02 0.01 0.01 0.01 0.01 2 (Dw = 1.60 0.02 0.02 0.02 0.02 0.02 0.01 3 (Dw = 2.25 0.02 0.02 0.01 0.01 0.01 0.01 4 (Dw = 3.20 0.01 0.01 0.01 0.01 0.01 0.02

−5

× 10

× 10−5

× 10−5

× 10−5

water 2

cm /s, 0.104 0.161 0.206 0.238 0.274 0.299 cm2/s, 0.143 0.210 0.256 0.302 0.336 0.374 cm2/s, 0.151 0.224 0.282 0.324 0.369 0.403 cm2/s, 0.184 0.264 0.320 0.375 0.416 0.453

oil Do = 1.02 0.132 0.183 0.223 0.256 0.285 0.313 Do = 1.27 0.172 0.235 0.283 0.324 0.360 0.392 Do = 1.37 0.177 0.247 0.301 0.346 0.384 0.420 Do = 1.48 0.197 0.277 0.339 0.390 0.436 0.480

diff of phase vol changes (cm3) × 10

−6

× 10−6

× 10−6

× 10−6

diff of diffusion fluxes (mol/(cm2·s))

2

cm /s) 0.028 0.022 0.017 0.018 0.011 0.014 cm2/s) 0.028 0.025 0.027 0.022 0.024 0.018 cm2/s) 0.025 0.023 0.019 0.021 0.015 0.017 cm2/s) 0.013 0.012 0.019 0.016 0.020 0.026

5.54 5.02 9.71 1.27 1.37 1.39

× × × × × ×

10−10 10−12 10−11 10−10 10−10 10−10

7.37 1.37 1.51 1.90 2.01 2.03

× × × × × ×

10−10 10−11 10−10 10−10 10−10 10−10

6.30 6.42 1.86 2.18 2.25 2.24

× × × × × ×

10−10 10−11 10−10 10−10 10−10 10−10

9.05 4.48 2.15 2.62 2.74 2.75

× × × × × ×

10−10 10−11 10−10 10−10 10−10 10−10

Figure 6. Interface displacements versus time in a log−log graph for tests 1−4.

10−5 cm2/s and Guo et al.37 found the diffusion coefficient of CO2 in the oil phase was 2.723 × 10−6 cm2/s at similar conditions. Therefore, the diffusion coefficients determined in this study are of the same order of magnitude. The lower values can be attributed to the lower initial concentration of CO2 in the water phase. This finding also demonstrates the feasibility of applying the determined diffusion coefficients in other binary or multicomponent systems. 3.2. Effect of Temperature on Diffusion Coefficients. Except for the effect of saturation pressure, temperature also plays an important role in determining the diffusion coefficients

The calculated diffusion coefficients (in the oil phase) for four tests are 1.16 × 10−6, 1.46 × 10−6, 1.55 × 10−6, and 1.70 × 10−6 cm2/s, respectively. Compared with the diffusion coefficients determined from the analytical calculations (see Table 1), the differences are small. This indication proves the accuracy of the determined diffusion coefficients. Another way to confirm the accuracy of determined values is to compare these values with those published in the previous literature. Even though the same experimental conditions applied in the current study have not been used yet, Cadogan et al.36 reported the diffusion coefficient of CO2 in the water phase was 2.233 × 12805


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Figure 7. Mass change of CO2 in the oil phase versus time in a log−log graph for tests 1−4.

accelerates the mass transfer process and fastens the completion of phase equilibria. Another phenomenon shown in Figure 8 is the slight fluctuation during the test. This fluctuation is caused due to the automatic temperature adjustment made by the water bath. It is not entirely sealed and water will vaporize in a slow pace. The internal mechanism of the water bath then will heat up the water to a slightly higher temperature (0.1−0.2 °C) than the set temperature and cool down to match the desired temperature. During the test, temperature-caused fluctuation will periodically happen and become more dramatic as the set temperature increases. This phenomenon can also be seen in Figure 13. In addition, the variation tendency of densities of the water phase and oil phase follows the same law. Figure 9 displays the

of CO2 in the water phase and oil phase. Hence, how much impact the temperature has on the diffusion coefficient is a question valuable to be discussed. In this section, data of tests 2 and 5 are used to investigate this effect. Test 2 was conducted at the experimental temperature of 20 °C, whereas test 5 was conducted at 40 °C. As listed in Table 1, at the same phase volumes and saturation pressure, the initial concentrations of CO2 are different: 1.3475 × 10−3 mol/cm3 for test 2 and 1.0835 × 10−3 mol/cm3 for test 5. This is because the increased temperature reduces the solubility of CO2 in water. Although the initial pressures for two tests are almost the same, the interface concentrations of CO2 in the water and oil phases at 10 h for test 5 are lower than that for test 2. The reason can be attributed to the greater diffusion coefficients. Compared with test 2, the diffusion coefficients of CO2 for test 5 increased more than twice: 3.50 × 10−5 cm2/s (water phase) and 2.77 × 10−6 cm2/s (oil phase), respectively. The increased diffusion coefficients alter the mass transfer of CO2 from the carbonated water phase to the oil phase, which is exhibited by the pressure change, the interface displacement, and the mass change of CO2 transferred into the oil phase. Figure 8 shows the measured pressure change during the CO2 mass transfer in a carbonated water−oil system as a

Figure 9. Calculated density changes of water and oil phases versus distance from the initial interface x = 0 at different time points for test 5.

calculated density changes in the water and oil phases as a function of distance at different times for test 5. Considering that the time needed to reach equilibrium is shorter than in test 2, five additional new analysis times (5, 7, 10, 12, and 15 h) are selected for test 5. Similarly, the mass transfer of CO2 from the water phase to the oil phase decreases the density of the water phase and increases the density of the oil phase. Density changes can also be explained by concentration changes, shown in Figure 10. Table 3 summarizes the specific changes of the system pressure and phase volumes from 5 to 15 h for test 5. Within the studied time range, the system pressure builds up from 19.11 to 19.55 MPa. The phase volumes change from

Figure 8. Measured pressure data of CO2 mass transfer in the carbonated water−oil system versus time for test 5 at the experimental temperature of 40 °C.

function of time for test 5. Likewise, the pressure builds up with time. Compared with test 2 (see Figure 2b), the system pressure tends to reach equilibrium in test 5 around 200 h; however, the pressure is still increasing after 500 h in test 2. Hence, it can be concluded that a greater temperature 12806


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Figure 10. Calculated concentration profiles of CO2 in water and oil phases at different times for test 5.

Figure 11. Interface displacements versus time in a log−log graph for tests 2 and 5.

3

79.789 to 79.642 cm for the water phase and from 70.221 to 70.377 cm3 for the oil phase. To better understand the effect of temperature on the interface displacement and mass of CO2 transferred in the oil phase, Figures 11 and 12 are applied. Apparently, due to the greater diffusion coefficient of CO2 in the oil phase, test 5 has a faster interface displacement and mass change of CO2. Likewise, a linear relationship exists for both the interface displacement and the mass, revealing the process still follows the diffusive nature of mass transfer at the increased temperature. According to Fick’s law, the regression equations in Figure 12 can be used to verify the diffusivities. The calculated diffusion coefficient of CO2 in the oil phase for test 5 is 3.10 × 10−6 cm2/s, which is close to the determined value of 2.77 × 10−6 cm2/s listed in Table 1. 3.3. Effect of Phase Volumes on Diffusion Coefficients. In general, the diffusion coefficient is independent of the phase volume change. Studies are rare, however, to verify this conclusion. Therefore, the objective of this section is to examine the effect of changed phase volumes on the diffusion coefficients of CO2 in the carbonated water phase and the oil phase. The primary results for tests 5, 6, and 7 at the changed phase volumes have been listed in Table 1. All tests are carried out at the same temperature and saturation pressure, leading to the same initial concentrations of CO2 in the water phase. Results from Table 1 indicate that the interface concentrations of CO2 at 10 h are the same regardless of the phase volumes. More notably, the diffusion coefficients determined from these three tests are changed insignificantly. This is a good indication of applying the determined diffusion coefficients regardless of the phase volumes within the carbonated water and oil system. The detailed comparisons are shown as follows. In addition to Figure 8 (test 5), Figure 13 demonstrates the measured pressure change of the CO2 mass transfer in the carbonated water−oil system as a function of time for tests 6

Figure 12. Mass change of CO2 in the oil phase as a function of time in a log−log graph for tests 2 and 5.

and 7. The same tendency of increased pressure with time also exists. The density changes of the water phase and oil phase conform to the same law during the CO2 mass transfer from the water phase to the oil phase, which have been plotted versus distance in Figure 14. As the phase volumes change, the configurations of the concentration profiles (Figure 15) appear to be different. However, the fundamental variation tendency does not vary. Table 4 summarizes the results of tests 6 and 7. Compared with test 5 (shown in Table 3), in the time range 5−15 h, the system pressure changes from 19.39 to 19.86 MPa for test 6 and from 18.95 to 19.17 MPa for test 7. Correspondingly for test 6, the phase volume changes from 99.782 to 99.624 cm3 for the water phase and from 50.227 to 50.387 cm3 for the oil phase. For test 7, the phase volume changes from 44.802 to

Table 3. Summary of Results for Test 5 at Different Times under the Experimental Temperature of 40 °C and Phase Volumes of 80 cm3 of Carbonated Water and 70 cm3 of Crude Oil (Dw = 3.50 × 10−5 cm2/s, Do = 2.77 × 10−6 cm2/s) phase vol determined (cm3)

vol change (cm3)

time (h)

press. (MPa)

water

oil

total

rel error of total vol (%)

water

oil

diff of phase vol changes (cm3)

5 7 10 12 15

19.11 19.22 19.36 19.44 19.55

79.789 79.754 79.702 79.677 79.642

70.221 70.261 70.310 70.338 70.377

150.010 150.015 150.012 150.015 150.019

0.01 0.01 0.01 0.01 0.01

0.211 0.246 0.298 0.323 0.358

0.221 0.261 0.310 0.338 0.377

0.010 0.015 0.012 0.015 0.019

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diff of diffusion fluxes (mol/(cm2·s)) 2.48 3.81 4.24 4.25 4.13

× × × × ×

10−10 10−10 10−10 10−10 10−10


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Figure 13. Measured pressure data of CO2 mass transfer in the carbonated water−oil system versus time for tests at different phase volumes and at 40 °C: (a) test 6 and (b) test 7.

Figure 14. Calculated density changes of water and oil phases versus distance from the initial interface x = 0 at different time points for (a) test 6 and (b) test 7.

Figure 15. Calculated concentration profiles of CO2 in water and oil phases at different times for (a) test 6 and (b) test 7.

44.661 cm3 for the water phase and from 105.215 to 105.382 cm3 for the oil phase. Comparisons of the interface displacement and mass change of CO2 in the oil phase for three tests have been plotted in Figures 16 and 17, respectively. It is apparent that these three tests exhibit the same interface displacement and mass change. This finding also proves that the diffusion coefficients are not changed with phase volumes. Furthermore, the diffusion coefficient of CO2 in the oil phase, calculated based on the regression equation from Figure 17, is about 2.90 × 10−6 cm2/s, which confirms the accuracy of the new analysis method used to find the diffusion coefficients under different experimental temperatures. 3.4. Examination of Uncertainty Resulted from Pressure Measurement and Pressure Variation. Considered parameters, such as concentration, densities, and diffusion coefficients, are dependent on pressure. The accuracy of the pressure gauge is first taken into consideration. In this work, a highly sensitive digital pressure gauge is supplied by Heise with a maximum measuring pressure of 34.5 MPa and accuracy of 0.1−0.025% full scale. The uncertainty resulted from instru-

ment itself is excluded. Then it is necessary to estimate the possible errors resulted from pressure variation during the experimental period of interest (30 h for tests 1−4 and 15 h for tests 5−7). Comparison with the time-average pressures of seven tests yields relative errors of maximum pressure variations during the studied period of 6.36% for test 1, 4.88% for test 2, 6.21% for test 3, 5.22% for test 4, 5.03% for test 5, 5.49% for test 6, and 3.43% for test 7. As a result, the uncertainty caused by pressure variation will not significantly affect the determined values of diffusion coefficients.

4. CONCLUSIONS In this paper, diffusion coefficients of carbon dioxide in the water phase and oil phase were determined under different experimental conditions. The effects of saturation pressure, temperature, and phase volumes on diffusion coefficients were evaluated. Meanwhile, the impacts on changes of pressure, densities of water phase and oil phase, concentration distribution, interface displacement, and mass amount of CO2 transferred into the oil phase were also investigated. The conclusions are as follows: 12808


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Industrial & Engineering Chemistry Research Table 4. Summary of Results for Tests 6 and 7 at Different Times and Phase Volumes at 40 °C phase vol determined (cm3) time (h)

press. (MPa)

water

oil

5 7 10 12 15

19.39 19.47 19.63 19.73 19.86

99.782 99.747 99.694 99.668 99.624

50.227 50.267 50.318 50.348 50.387

5 7 10 12 15

18.95 18.99 19.05 19.14 19.17

44.802 44.767 44.724 44.694 44.661

105.215 105.257 105.311 105.338 105.382

total

vol change (cm3) relative error of total vol (%)

Test 6 (Dw 150.009 150.015 150.012 150.016 150.011 Test 7 (Dw 150.017 150.024 150.035 150.032 150.044

water

oil

diff of phase vol changes (cm3)

diff of diffusion fluxes (mol/(cm2·s))

= 3.49 × 10−5 cm2/s, Do = 2.77 × 10−6 cm2/s) 0.01 0.218 0.227 0.009 0.01 0.253 0.267 0.015 0.01 0.306 0.318 0.012 0.01 0.332 0.348 0.016 0.01 0.376 0.387 0.011 = 3.50 × 10−5 cm2/s, Do = 2.76 × 10−6 cm2/s) 0.01 0.198 0.215 0.017 0.02 0.233 0.257 0.024 0.02 0.276 0.311 0.035 0.02 0.306 0.338 0.032 0.03 0.339 0.382 0.044

2.55 3.81 4.22 4.21 4.09

× × × × ×

10−10 10−10 10−10 10−10 10−10

2.66 3.81 4.16 4.14 4.01

× × × × ×

10−10 10−10 10−10 10−10 10−10

for applying the determined diffusion coefficients to simulate the mass transfer in large-scale reservoirs. 4. In the process of the mass transfer of CO2, the system pressure builds up with time and levels at the late stage. The density of the water phase declines as the concentration of CO2 decreases; on the contrary, the density of the oil phase increases with an augmentation of the mass amount of CO2 transferred in the oil phase. The mass transfer of carbon dioxide from the carbonated water to oil phases also results in the interface displacement. By applying the correlations proposed in this work, researchers can determine the actual diffusion coefficients in different conditions and directly use these determined values for modeling and forecasting the feasibility and potentiality of extracting the residual oil through a carbonated water flooding process. Compared with values estimated by empirical equations, the determined diffusion coefficients are more reliable and accurate.

Figure 16. Interface displacements versus time in a log−log graph for tests 5−7.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.7b02546. Schematic of the problem, material used in the experiments, experimental setup, and experimental procedure (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected].

Figure 17. Mass change of CO2 in the oil phase as a function of time in a log−log graph for tests 5−7.

ORCID

Guanli Shu: 0000-0002-5534-4324 Mingzhe Dong: 0000-0001-5093-0116 Hassan Hassanzadeh: 0000-0002-3029-6530

1. Diffusion coefficients of CO2 in both phases are a function of saturation pressure. The greater the saturation pressure, the larger the diffusion coefficients. A linear relationship exists between saturation pressure and the diffusion coefficients. 2. Temperature plays an important role in diffusion coefficients. At a given saturation pressure, diffusion coefficients of CO2 in two phases increase with temperature. Diffusion coefficients determined at 40 °C are about 2 times greater than those determined at 20 °C. The increased temperature reduces the time required for phase equilibrium to be achieved. 3. At the varied volumes of water phase and oil phase, diffusion coefficients remain constant, indicating the potential

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors would like to thank the Petroleum Technology Research Centre (PTRC) in Regina, Saskatchewan, Canada, and the Natural Sciences and Engineering Research Council of Canada for their financial support of the project. The support provided by the China Scholarship Council (CSC, No.2011644003) is also acknowledged. 12809


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NOMENCLATURE Ci = initial concentration of CO2 in water phase (mol/cm3) Co = concentration of CO2 in oil phase (mol/cm3) Cw = concentration of CO2 in water phase (mol/cm3) C*o = interface concentration of CO2 in oil phase (mol/cm3) C*w = interface concentration of CO2 in water phase (mol/ cm3) Do = diffusion coefficient of CO2 in oil phase (cm2/s) Dw = diffusion coefficient of CO2 in water phase (cm2/s) erf = error function kpc = partition coefficient, dimensionless t = time (s) x = distance (cm)

Greek Symbol

λ = constant solved from transcendental equation

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Effects of Operational Parameters on Diffusion Coefficients of CO2 in a

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