Determination of Diffusion Coefficients and Interface Mass-Transfer

Jul 10, 2008 - as CO2, in the crude oil plays a vital role in the oil recovery processes.4–7 Thus, it is ... are numerous dynamic and equilibrium in...
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Ind. Eng. Chem. Res. 2008, 47, 5447–5455

5447

Determination of Diffusion Coefficients and Interface Mass-Transfer Coefficients of the Crude Oil-CO2 System by Analysis of the Dynamic and Equilibrium Interfacial Tensions Daoyong Yang* and Yongan Gu Petroleum Technology Research Centre (PTRC), Petroleum Systems Engineering, Faculty of Engineering, UniVersity of Regina, Regina, Saskatchewan S4S 0A2, Canada

In this paper, a newly developed dynamic interfacial tension method has been applied to simultaneously determine the diffusion coefficients and interface mass-transfer coefficients of the crude oil-CO2 system at high pressures and a constant temperature. Experimentally, the dynamic and equilibrium interfacial tensions of the crude oil-CO2 system are measured by using the axisymmetric drop shape analysis (ADSA) technique for the pendant drop case. Theoretically, a mathematical model is formulated to obtain the time-dependent CO2 concentration distribution inside the pendant oil drop. Then, in terms of a predetermined calibration curve of the measured equilibrium interfacial tension versus the calculated equilibrium CO2 concentration in the crude oil, the dynamic interfacial tension at any time is calculated. Subsequently, an objective function is constructed to express the overall discrepancy between the numerically calculated and the experimentally measured dynamic interfacial tensions at different times. The CO2 diffusion coefficient and the mass-transfer Biot number are used as adjustable parameters and thus determined once the global minimum objective function is achieved. The diffusion coefficient, the mass-transfer Biot number, and the interface mass-transfer coefficient of CO2 mass transfer in a medium crude oil sample at P ) 0.1-5.0 MPa and T ) 27 °C are found to be 0.47-2.49 × 10-9 m2/s, 2.3-6.8, 0.88-8.41 × 10-5 m/s, respectively. 1. Introduction CO2 injection is considered as one of the most promising enhanced oil recovery (EOR) techniques for light and medium oils because it not only effectively enhances oil recovery due to dissolution of CO2 into the crude oil but also considerably reduces greenhouse gas emissions by sequestrating CO2 in a depleted oil reservoir. Recently, CO2 EOR has gained momentum in the oil and gas industry due to its potential for mitigating greenhouse gas emissions.1 It is estimated that about 80% of oil reservoirs worldwide might be suitable for CO2 injection based on the oil recovery criteria alone.2,3 Previous studies have already shown that molecular diffusion of an injected gas, such as CO2, in the crude oil plays a vital role in the oil recovery processes.4–7 Thus, it is essential that mass transfer of a crude oil-CO2 system be studied at the practical reservoir pressures and temperatures. Physically, three distinct stages are involved in the mass transfer process of an injected gas (e.g., CO2) in a crude oil. First, the injected gas moves toward the crude oil-gas interface, then penetrates the interface, and finally diffuses in the crude oil. The gas mass transfer in the crude oil causes the interfacial properties between the crude oil and the injected gas to change. Some previous studies have been conducted to determine the diffusion coefficients for different crude oil-gas systems. In these studies, the conventional methods involve compositional analysis of the crude oil-gas mixture samples taken at different times and locations during a diffusion test.8,9 On the other hand, the nonconventional methods measure the change of a property of the crude oil-gas system during the molecular diffusion process. Such a property can be gas volume,10,11 crude oil-gas interface position inside a capillary,12 gas pressure,13–16 shape,17 or volume18 of a pendant oil drop surrounded by a gas. * To whom correspondence should be addressed. Tel.: 1-306-3372660. Fax: 1-306-585-4855. E-mail: [email protected].

It is well-known that the gas mass transfer in the crude oil results in reduction of the interfacial tension between the crude oil and the injected gas. Also, the interfacial tension is an important parameter required in the reservoir simulation and field design of a gas injection operation. In the literature, there are numerous dynamic and equilibrium interfacial tension data available for various crude oil-CO2 systems at different pressures and temperatures.19–23 However, no studies have been conducted on the mass transfer in the crude oil-CO2 systems at high pressures and elevated temperatures by analysis of the measured dynamic and equilibrium interfacial tensions. In this paper, the dynamic interfacial tension method has been applied to study the mass transfer of a crude oil-CO2 system at high pressures and a constant temperature. In comparison with other experimental techniques, the dynamic interfacial tension method has the following advantages. First, both the gas diffusion coefficient and the interface mass-transfer coefficient can be determined simultaneously by analyzing the dynamic interfacial tension data measured in a single test at a constant pressure and a constant temperature. Second, a single dynamic interfacial tension measurement can be completed within 1 h, and only a small amount of the crude oil is needed. Finally, it is worthwhile to point out that the newly developed dynamic interfacial tension method can be used to study the mass transfer of any gases in the crude oil as long as there is an appreciable reduction of the dynamic interfacial tension. 2. Dynamic Interfacial Tension Method The dynamic interfacial tension method determines the diffusion coefficient and interface mass-transfer coefficient simultaneously. In the experiment, the dynamic and equilibrium interfacial tensions of the crude oil-CO2 system are measured by using the axisymmetric drop shape analysis (ADSA) technique for the pendant drop case. Theoretically, a mathematical model is formulated to study the mass transfer in the crude

10.1021/ie800053d CCC: $40.75  2008 American Chemical Society Published on Web 07/10/2008

5448 Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008

Figure 1. (a) Digital image of a well-deformed pendant oil drop surrounded by CO2 at the beginning at P ) 1.63 MPa and T ) 27 °C and (b) schematic of an axisymmetric pendant oil drop in CO2 phase depicted in the cylindrical coordinate system (r, z). Table 1. Compositional Analysis Results of the Weyburn Crude Oil carbon number

mole %

carbon number

mole %

C3 i-C4 n-C4 i-C5 n-C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17

0.50 0.28 1.30 1.13 1.55 3.14 7.87 7.69 6.45 7.01 4.57 4.66 3.92 4.38 3.72 3.27 3.25

C18 C19 C20 C21 C22 C23 C24 C25 C26 C27 C28 C29 C30 C31+ total C1 toC6 C7+

2.79 2.54 2.84 2.47 1.58 1.67 1.47 1.63 1.49 1.18 1.16 1.12 1.02 12.35 100.00 7.90 92.10

oil-CO2 system. The mass transfer model is solved numerically by applying the semidiscrete Galerkin finite element method to obtain the time-dependent CO2 concentration distribution inside the pendant oil drop. Then, in terms of a predetermined calibration curve of the measured equilibrium interfacial tension versus the calculated equilibrium CO2 concentration in the crude oil, the dynamic interfacial tension at any time is calculated. Subsequently, an objective function is constructed to express the overall discrepancy between the numerically calculated and the experimentally measured dynamic interfacial tensions at different times. The CO2 diffusion coefficient and the masstransfer Biot number are used as adjustable parameters and thus determined once the global minimum objective function is achieved. A brief description of the dynamic interfacial tension method is provided below, and its technical details can be found elsewhere.23,24 2.1. Mass Transfer Model. Figure 1a shows a digital image of a well-deformed pendant oil drop surrounded by CO2 at the beginning at P ) 1.63 MPa and T ) 27 °C. Figure 1b depicts a schematic of an axisymmetric pendant oil drop surrounded by CO2. The inner radius, wall thickness, and height of the syringe needle are expressed by rn, εn, and hn, respectively. The physical domain occupied by the pendant oil drop, including the oil phase inside the syringe needle, is denoted by ω. The boundaries formed between the walls of the syringe needle and the crude oil as well as the cutting plane at the top of the syringe needle are denoted by φn. Here, the cutting plane is chosen far

above the tip of the syringe needle so that CO2 diffusion cannot reach it at the end of each diffusion test. The crude oil-CO2 interface is represented by φint. In the Cartesian coordinate system, the CO2 molecular diffusion in the pendant oil drop seems to be a three-dimensional mass transfer problem. Nevertheless, it is more convenient to choose the cylindrical coordinate system as shown in Figure 1b, noting the axisymmetry of the pendant oil drop.25,26 Then CO2 diffusion in the crude oil becomes an unsteady two-dimensional mass-transfer problem. Several previous studies have found that the CO2 diffusion can lead to the interfacial tension reduction, the oil-swelling effect, and the light-ends extraction, particularly at high pressures.21–23 These three facts make the mass transfer become an extremely complicated moving boundary problem. On the other hand, the total mass transfer flux of CO2 into the pendant oil drop is proportional to its surface area. For the crude oil-CO2 system, the diffusion process proceeds so slowly that the oilswelling effect and the surface area change are rather slow. Thus, in this study, the effects of the pendant oil drop shape and volume changes on the diffusion process are neglected. It is assumed that the diffusion process occurs as if the pendant oil drop remains unchanged once it is formed. In comparison with gas mass transfer in a pure liquid, the mass transfer of CO2 in a crude oil becomes rather complicated because the crude oil is a multicomponent mixture. A reasonable method is to treat the multicomponent crude oil as one pseudocomponent since it is extremely difficult, if not impossible, to study the mass transfer of CO2 in each component. Therefore, the diffusion equation for describing CO2 concentration distribution inside the pendant oil drop, including the oil phase inside the syringe needle, c(r, z, t), can be expressed as27

[

]

∂c 1 ∂ ∂c ∂2c )D r + 2 , ∂t r ∂r ∂r ∂z

( )

(r, z) ∈ ω,

t>0

(1)

where D is the CO2 diffusion coefficient in the crude oil, assuming that it is a constant during each diffusion test and independent of CO2 concentration in this study. At the beginning of the mass-transfer process, there is no CO2 dissolution inside the pendant oil drop, and thus the initial condition (IC) for eq 1 becomes (r, z) ∈ ω

c(r, z, t)|t)0 ) 0,

(2)

Because the walls of the syringe needle are impermeable to CO2 and the cutting plane can be treated as an impermeable boundary, the corresponding nonpenetrating boundary condition (BC) is applied at the walls of the syringe needle and the cutting plane:

( ∂c∂r n + ∂c∂z n )

D

r

z

(r, z)∈φn

)0

(3a)

where nr and nz are the direction cosines, that is, r and z components of an outward unit vector normal to the boundaries formed between the syringe needle and the crude oil or the cutting plane. In this study, the interfacial resistance to CO2 mass transfer across the crude oil-CO2 interface is considered. Thus, the following nonequilibrium BC is applied at the interface:15

( ∂c∂r n + ∂c∂z n )

D

r

z

(r, z)∈φint

) k[ceq - c(r, z, t)|(r, z)∈φint]

(3b)

where nr and nz are the direction cosines, that is, r and z components of an outward unit vector normal to the crude oil-CO2 interface; k is the interface mass-transfer coefficient;

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Figure 2. (a) Schematic of the axisymmetric drop shape analysis (ADSA) system for the pendant drop case and (b) block diagram of the experimental setup used to measure the dynamic and equilibrium interfacial tensions of the crude oil-CO2 system under reservoir conditions.

Figure 3. Sequential digital images of a well-deformed pendant crude oil drop surrounded by CO2 at different times at P ) 1.63 MPa and T ) 27 °C.

and ceq is the maximum equilibrium CO2 concentration (i.e., CO2 solubility) in the crude oil at the prespecified pressure and temperature. Equations 1–3b can be nondimensionalized by introducing the following four dimensionless variables and one dimensionless parameter: C)

krn r z Dt c , R ) , Z ) , τ ) 2 , kD ) ceq rn rn D r

(4)

n

Here, C is the dimensionless CO2 concentration; R and Z are the dimensionless radial and axial coordinates, respectively; τ is the dimensionless time; and kD is the so-called mass-transfer Biot number. Physically, the mass-transfer Biot number, kD, represents the ratio of the bulk resistance to the mass transfer due to the molecular diffusion, rn/D, to the interfacial resistance to the interface mass transfer, 1/k.15,28 After eq 1 together with its IC in eq 2 and BCs in eqs 3a and 3b has been nondimenionalized, the semidiscrete Galerkin finite

element method is applied to numerically solve the above dimensionless equations. To obtain more accurate CO2 concentration distribution near the interface and higher overall computational efficiency, a finer mesh is chosen in the region close to the crude oil-CO2 interface or near the boundaries, whereas a coarser mesh is used near the center of the pendant oil drop. While the detailed mathematical formulations can be found elsewhere,23 the detailed numerical procedure is summarized as follows. (1) The sequential digital images of the pendant oil drop surrounded by the test gas are acquired and the dynamic interfacial tensions, γm(tmi), i ) 0, 1, 2,..., N, are measured. (2) The first digital image of the pendant oil drop at t ) 0 is used to define the physical domain ω for the mass transfer modeling, as shown in Figure 1b. (3) At a given pressure and temperature, 11 trial points of mass transfer Biot number are chosen as kD, i, i ) 0, 1, 2,..., 10. (4) For a given kD, i, the mass transfer model is solved numerically by applying the semidiscrete Galerkin finite element method to obtain the dimensionless gas concentration distribution C(R, Z, τ) inside the computational domain Ω.23,24 (5) The lower and upper limits of the diffusion coefficient D are chosen as [D0, D10] for a known kD, i. (6) The initial interval of uncertainty for D is subdivided into ten equally spaced subintervals with 11 points Dj, j ) 0, 1, 2,..., 10. (7) For a chosen Dj, the dimensionless time τ is converted into the real time t. With the predetermined calibration curve of the measured equilibrium interfacial tension versus the

5450 Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008

Figure 4. (a) Triangular mesh of 2820 elements and 1529 nodes for a pendant oil drop at P ) 1.63 MPa and T ) 27 °C, where the drop volume V ) 9.229 mm3, the inner syringe needle radius rn ) 0.20 mm, the syringe needle wall thickness εn ) 0.59 mm, the syringe needle height hn ) 3.50 mm; (b) calculated dimensionless CO2 concentration contours at τ ) 0.8 (t ) 36 s, D ) 0.89 × 10-9 m2/s) and kD ) 2.3; and (c) calculated dimensionless CO2 concentration contours at τ ) 4.0 (t ) 180 s, D ) 0.89 × 10-9 m2/s) and kD ) 2.3. Table 2. Measured Diffusion Coefficient D, Mass-Transfer Biot Number kD, Interface Mass-Transfer Coefficient k, and Global Minimum Objective Function Emin of the Crude Oil-CO2 System at Different Pressures and T ) 27 °C

Figure 5. Measured dynamic interfacial tensions (symbols) and the calculated dynamic interfacial tensions (lines). The diffusion coefficients and the mass-transfer Biot numbers used in calculating the dynamic interfacial tensions are taken from Table 2 for the crude oil-CO2 system under five different pressures at T ) 27 °C.

calculated equilibrium gas concentration, for each pair of E(Dj, kD, i), j ) 0, 1, 2,..., 10, the dynamic interfacial tension at any time t is calculated by using linear interpolation of the calibration curve. With the measured and calculated dynamic interfacial tensions, the corresponding objective function E(Dj, kD, i), j ) 0, 1, 2,..., 10, is determined from eq 5.

P (MPa)

D (10-9 m2/s)

kD

k (10-5 m/s)

Emin (%)

0.32 1.63 2.65 3.22 4.39

0.47 0.89 1.23 1.99 2.49

3.8 2.3 6.8 4.8 6.8

0.88 1.00 4.16 4.72 8.41

0.86 1.77 0.88 0.74 0.84

8) By comparing E(Dj, kD, i), j ) 0, 1, 2,..., 10, the local minimum objective function E(Dn, kD, i), 1 e n e 9, is found. Then the new interval of uncertainty for D becomes [Dn-1, Dn+1]. (9) If the new interval of uncertainty [Dn-1, Dn+1] is already small enough to satisfy a prespecified accuracy for the diffusion coefficient determination, Dn is the local optimum value for the given kD, i. Otherwise, the new interval of uncertainty [Dn-1, Dn+1] is considered as the initial interval of uncertainty for D and Steps 6-9 are repeated until the prespecified termination criterion is satisfied. (10) Steps 4-9 are repeated for all kD, i, i ) 0, 1, 2,..., 10. By comparing E(Dn, kD, i), i ) 0, 1, 2,..., 10, 1 e n e 9, the global minimum objective function E(Dn, kD, l), 1 e l e 9, is found, where Dn and kD, l are the global optimum values of the diffusion coefficient and the mass transfer Biot number. Therefore, Dn and kD, l are the true diffusion coefficient and the true mass transfer Biot number, respectively. Accordingly, the numerical optimization procedure is terminated. In this paper,

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Figure 6. (a) Measured equilibrium interfacial tension and the calculated equilibrium CO2 concentration (i.e., CO2 solubility) determined by using an existing correlation29 for the crude oil (30°API) tested in this study versus the equilibrium pressure at T ) 27 °C and (b) measured equilibrium interfacial tension versus the calculated equilibrium CO2 concentration29 for the crude oil-CO2 system at T ) 27 °C.

D and kD are chosen from 0.01-10 × 10-9 m2/s and in the range of 0.3-8.8, respectively. The termination criterion for D is |Dn+1 - Dn-1| e 0.005 × 10-9 m2/s. 2.2. Objective Function and Its Optimization. Once a pendant oil drop is formed inside the pressure cell prefilled with CO2, the latter gradually diffuses in the former. The CO2 molecular diffusion in the oil drop results in the interfacial tension reduction of the crude oil-CO2 system. In the experiment, the sequential digital images of the pendant oil drop surrounded by CO2 are acquired and the dynamic interfacial tensions, γm(tmi), i ) 1, 2,..., N, and, finally, the equilibrium interfacial tension, γeq(P), are measured at each equilibrium pressure and a constant temperature. Meanwhile, the equilibrium CO2 concentration, ceq(P), in the crude oil is correlated to the equilibrium pressure by using an existing correlation.29 Therefore, the measured equilibrium interfacial tension under a prespecified pressure is related to the calculated equilibrium CO2 concentration under the same equilibrium pressure. This relation is termed the calibration curve of the measured equilibrium interfacial tension versus the calculated equilibrium CO2 concentration. The first digital image of the dynamic pendant drop at t ) 0 is used to define the physical domain ω for the mass transfer modeling under each pressure, as shown in Figure 1b. Then, the mathematical model is solved numerically by applying the semidiscrete Galerkin finite element method to obtain the dimensionless CO2 concentration distribution, C(R, Z, τ), inside

Figure 7. (a) Variations of the objective function E(D, kD) with the diffusion coefficient D at different mass-transfer Biot numbers kD for the crude oil-CO2 system at P ) 0.32 MPa and T ) 27 °C and (b) variations of the objective function E(D, kD) with the diffusion coefficient D at the global optimum mass-transfer Biot number kD for the crude oil-CO2 system under different pressures at T ) 27 °C.

Figure 8. Effects of the equilibrium CO2 concentration on the measured diffusion coefficient D and the interface mass-transfer coefficient k for the crude oil-CO2 system at T ) 27 °C.

the computational domain Ω and the average dimensionless CO2 concentration at the crude oil-CO2 interface, Cint(τ). The average dimensionless CO2 concentration at the interface can be transformed into the dimensional transient interface CO2 concentration, cint(t), at any time. In this study, it is assumed that the relation between the to-be-calculated dynamic interfacial tension and the predicted transient interface CO2 concentration follows the calibration curve between the measured equilibrium

5452 Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008 Table 3. Comparison of the measured diffusion coefficients and interface mass-transfer coefficients in different crude oil-CO2 systems at different pressures and temperatures

a

crude oil

P (MPa)

T (°C)

µ (mPa · s)

D (10-9 m2/s)

k (10-5 m/s)

Weyburn oila Stock tank oil11 Ontario oil17 Maljamar oil12 Lloydminster heavy oil18 Athabasca bitumen16

0.1-5.0 15 2.9 5.2 2.0-6.0 4

27 66 25 25 23.9 25-90

13.0 @ 27 °C 290 @ 25 °C 43.8 @ 27 °C 3 @ 23 °C 23 000 @ 23.9 °C 767 @ 80 °C

0.47-2.49 3 1.14 2 0.20-0.55 0.16-0.47

0.88-8.41 5.7

Represents the crude oil used and the experimental data obtained in this study.

interfacial tension, γeq(P), and the calculated equilibrium CO2 concentration, ceq(P).30 Thus, the dynamic interfacial tension, γc(t), can be calculated from the predicted transient interface CO2 concentration, cint(t), at any time. An objective function is constructed to express the overall discrepancy between the theoretically calculated dynamic interfacial tensions and the experimentally measured dynamic interfacial tensions. Let tmi and γm(tmi), i ) 1, 2,..., N, be a set of times and the experimentally measured dynamic interfacial tensions, and tci and γc(tci), choosing tci ) tmi, i ) 1, 2,..., N, be another set of times and the theoretically calculated dynamic interfacial tensions. Thus, the objective function E is defined as E)

 [

N γm(tmi) - γc(tci) 1 N i)1 γm(tmi)

∑

]

2

× 100%

(5)

It can be seen from eq 5 that the above-defined objective function is the root-mean-squared relative error between the calculated and measured dynamic interfacial tensions. As described previously, the calculated dynamic interfacial tension, γc(t), is determined from the predicted transient CO2 concentration at the interface, i.e., cint(t), by using the established calibration curve. Mathematically, the predicted transient interface CO2 concentration depends on two unknown parameters, the diffusion coefficient, D, and the mass-transfer Biot number, kD. Therefore, the objective function is dependent on D and kD, once the experimentally measured dynamic interfacial tensions, γm(tmi), i ) 1, 2,..., N, are obtained E ) E(D, kD)

(6)

In the optimization scheme, D and kD are used as the adjustable parameters to find the best fit of the theoretically calculated dynamic interfacial tensions to the experimentally measured data. Once the global minimum objective function is achieved, the corresponding values of D and kD are considered as the true diffusion coefficient and the true mass-transfer Biot number, respectively. The detailed numerical procedure for simultaneously determining the diffusion coefficient and interface masstransfer coefficient can be found elsewhere.23,24 3. Experimental Section 3.1. Materials. A crude oil sample is collected from the Weyburn Oilfield in Saskatchewan, Canada. The density and viscosity of the Weyburn crude oil are 0.877 g/cm3 and 13.0 mPa · s at the atmospheric pressure and 27 °C, respectively. The compositional analysis results for the Weyburn crude oil are given in Table 1. The crude oil sample is filtered through filter papers to remove fine solids prior to the experiment. The purities of carbon dioxide (Praxair, U.S.A.) and nitrogen (Praxair, U.S.A.) used are 99.99% and 99.998%, respectively. 3.2. Apparatus. In this study, the ADSA technique for the pendant drop case is used to measure the dynamic and

equilibrium interfacial tensions of the crude oil-CO2 system at different pressures and T ) 27 °C. A schematic of the ADSA system for the pendant drop case used in this study is shown in Figure 2a. The major component of this system is a see-through windowed high-pressure cell (IFT-10, Temco, U.S.A.), which has a chamber volume of 41.5 cm3. In the ADSA system, the see-through windowed high-pressure cell is placed between a light source and an MZ6 microscope camera (Leica, Germany). The entire experimental setup is mounted on a vibration-free table (RS 4000, Newport, U.S.A.). A Dell desktop computer is used to acquire the sequential digital images of the dynamic pendant oil drop and perform the subsequent drop image analysis, digitization, and computation. The temperature during the measurement is maintained by wrapping the pressure cell with two heating tapes (HT95504 × 1, Electrothermal, U.S.A.), which are connected to a stepless temperature controller (CN45515, Thermolyne, U.S.A.). The maximum operating pressure and temperature of this pressure cell are 69 MPa and 177 °C, respectively. In the experiment, the maximum uncertainties of the pressure and temperature measurements are equal to 0.02 kPa and 0.2 °C, respectively. Figure 2b shows a block diagram of the experimental setup used to measure the dynamic and equilibrium interfacial tensions of the crude oil-CO2 system under reservoir conditions. This setup consists of the ADSA system, control panel, and fluidshandling system. Mounted on the vibration-free table, the control panel has eight needle valves that are used to control the flow rate and direction. In the fluids-handling system, three cylinders are used to pressurize the pressure cell with CO2, supply the crude oil, and clean the pressure cell with the toluene and acetone, respectively. There are a total of six ports around the cylindrical pressure cell, as shown in Figure 2b. In this study, the top port is used to introduce the crude oil and form a pendant oil drop inside the pressure cell, and the bottom port is served as a draining outlet. Among the other four ports on the side, one is for pressure measurement using a digital pressure gauge (DTG-6000, 3D Instruments, U.S.A.), one is for temperature measurement using a thermocouple (JMQSS-125U-6, Omega, U.S.A.) and a temperature display (MDSS 41-T-A, Omega, U.S.A.), one is for the installation of a rupture valve (P-7019, Oseco, U.S.A.), and the last one is for the injection of CO2. 3.3. Experimental Procedure. Prior to the first experiment, the whole system is tested for leakage with deionized water up to 10 MPa and then cleaned with acetone, flushed with nitrogen, and finally evacuated. The pressure cell is then pressurized with CO2 to a prespecified pressure by using a manual positive displacement pump (PMP-500-1-10-HB, DBR, Canada). After CO2 is injected, it usually takes 30-60 min for the pressure and temperature inside the pressure cell to reach their stable values. The general experimental procedure for the dynamic and equilibrium interfacial tension measurements of the crude oil-CO2 system is described below. An oil drop is introduced

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from the crude oil cylinder whose pressure is maintained at 0.1-0.5 MPa higher than that of CO2 phase inside the pressure cell. The pendant oil drop is formed by using a specially designed high-pressure syringe delivery system at the tip of the stainless steel needle, which is installed at the top of the pressure cell. After the pendant oil drop is formed in the CO2 phase, its digital image is well focused, acquired sequentially and stored automatically in the computer memory. For each digital oil drop image, a standard grid image is used to calibrate the drop image and correct possible optical distortion. Then the ADSA program for the pendant drop case is executed to determine the dynamic and equilibrium interfacial tensions of the pendant oil drop and the pendant drop profile as well. It should be noted that the accuracy of the interfacial tension measurement is ( 0.05 dyne/ cm. The interfacial tension measurement is repeated for at least three pendant oil drops to ensure satisfactory repeatability at each specified pressure and temperature. After each test, the pressure cell and the whole tubing system are thoroughly cleaned with toluene and acetone and then flushed with nitrogen and finally vacuumed. In this study, the pressure range of P ) 0.1-5.0 MPa and a constant temperature of T ) 27 °C are chosen to study the mass transfer of CO2 in the crude oil. 4. Results and Discussion 4.1. Pendant Oil Drop Images and CO2 Concentration Distribution. Sequential digital images of the dynamic pendant oil drop surrounded by CO2 are acquired, and the dynamic and equilibrium interfacial tensions are measured at P ) 0.1-5.0 MPa and T ) 27 °C. For example, Figure 3 shows four sequential digital images of a well-deformed pendant oil drop in the CO2 phase inside the pressure cell at P ) 1.63 MPa and T ) 27 °C. It is found that the dynamic interfacial tension is reduced with time due to dissolution of CO2 into the oil phase. The first digital image of the pendant oil drop at t ) 0 is used to define the physical domain ω for the mass transfer modeling at P ) 1.63 MPa and T ) 27 °C. Then the mass transfer model is solved numerically by applying the semidiscrete Galerkin finite element method to obtain the dimensionless CO2 concentration distribution C(R, Z, τ) inside the computational domain Ω. Figure 4a shows a triangular mesh for the numerical simulation. As shown in this figure, a finer mesh is chosen near the crude oil-CO2 interface, whereas a coarser mesh is used near the center of the pendant oil drop. For graphical reasons, Figure 4a shows a triangular mesh of 2820 elements and 1529 nodes. The numerical results presented in this paper are obtained from finer triangular meshes of approximately 4200 elements and 2300 nodes. Typical dimensional mesh sizes for the elements near the crude oil-CO2 interface are approximately 40 µm. Figures 4b and c show the calculated dimensionless CO2 concentration contours inside a pendant oil drop at P ) 1.63 MPa and T ) 27 °C under the nonequilibrium BC (kD ) 2.3) and with the diffusion coefficient D ) 0.89 × 10-9 m2/s at τ ) 0.8 and τ ) 4.0, respectively. At τ ) 0.8 (t ) 36 s), the average dimensionless CO2 concentration at the crude oil-CO2 interface is equal to Cint ) 0.78, whereas the average dimensionless CO2 concentration inside the pendant oil drop is found to be Cave ) 0.27. At τ ) 4.0 (t ) 180 s), Cint ) 0.93 and Cave ) 0.64. It can be seen from Figure 4b that, at the early stage of the diffusion process (τ ) 0.8, t ) 36 s), there is a larger CO2 concentration gradient near the crude oil-CO2 interface. As CO2 gradually dissolves into the crude oil, for example, at τ ) 4.0 (t ) 180 s), the CO2 concentration gradient near the interface

becomes much smaller. More and more CO2 diffuses toward the center of the pendant oil drop, and some even enters the oil phase inside the syringe needle. This mass transfer process continues until the entire pendant oil drop is completely saturated with CO2. 4.2. Measured Dynamic Interfacial Tensions. The measured dynamic interfacial tensions of the crude oil-CO2 system as a function of time under five different pressures at T ) 27 °C are shown in Figure 5. The measured dynamic interfacial tensions reach their equilibrium values after at least 300 s under different pressures. It can also be seen from this figure that the dynamic interfacial tension is lower at a higher pressure. This is because the solubility of CO2 in the crude oil is higher at a higher pressure. Furthermore, the dynamic interfacial tension reaches its equilibrium value more quickly at a higher pressure. Due to rather slow CO2 diffusion inside the pendant oil drop, as shown in Figures 4b and c, it is obvious that CO2 reaches its equilibrium concentration at the interface more quickly than it does inside the pendant oil drop far away from the interface. This means that the equilibrium interfacial tension is achieved at the interface before the pendant oil drop is completely saturated with CO2. 4.3. Measured Equilibrium Interfacial Tensions. At the end of each dynamic interfacial tension measurement, there always exists a constant interfacial tension, which is referred to as the equilibrium interfacial tension. The measured equilibrium interfacial tensions of the crude oil-CO2 system at five different pressures and T ) 27 °C are plotted in Figure 6a. It is seen from this figure that the equilibrium interfacial tension decreases as the equilibrium pressure increases. This is because the equilibrium CO2 concentration (i.e., CO2 solubility) in the crude oil is higher at a higher pressure. In addition, by using an existing correlation in the literature,29 the equilibrium CO2 concentration is correlated to the equilibrium pressure, which is also shown in Figure 6a. This correlation provides an excellent approximation for the equilibrium CO2 concentration in the Weyburn crude oil under pressures up to 8.0 MPa at T ) 59 °C.31 It can be seen from this figure that the calculated equilibrium CO2 concentration increases quickly with the equilibrium pressure in the pressure range tested in this study. With the results shown in Figure 6a, the relation between the measured equilibrium interfacial tension and the calculated equilibrium CO2 concentration in the crude oil at T ) 27 °C can be obtained and is shown in Figure 6b. This figure clearly shows that the equilibrium interfacial tension decreases as the equilibrium CO2 concentration increases. Physically, the equilibrium interfacial tension reduction with pressure is due to increased equilibrium CO2 concentration in the crude oil at a higher pressure. In this study, Figure 6b is used as the calibration curve of the measured equilibrium interfacial tension versus the calculated equilibrium CO2 concentration of the crude oil-CO2 system at T ) 27 °C. 4.4. Diffusion Coefficient and Mass-Transfer Biot Number. With the measured dynamic interfacial tension data, the objective function E(D, kD) in eq 6 can be minimized to determine the diffusion coefficient D and the mass-transfer Biot number kD of CO2 in the crude oil at each pressure. The objective function, E(D, kD), is minimized by following the numerical procedure described in the theoretical section. For example, Figure 7a shows variations of the objective function E(D, kD) with the diffusion coefficient D at different mass-transfer Biot numbers kD for the crude oil-CO2 system at P ) 0.32 MPa and T ) 27 °C. It is clearly seen from this figure that for a given mass-transfer Biot number, there is a local optimum value

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of the diffusion coefficient at which the local minimum objective function is achieved. For different mass-transfer Biot numbers, there is only one pair of the global optimum values of the diffusion coefficient D and the mass-transfer Biot number kD, at which the global minimum objective function is reached. At P ) 0.32 MPa and T ) 27 °C, for instance, the true diffusion coefficient and the true mass-transfer Biot number of the crude oil-CO2 system are found to be D ) 0.47 × 10-9 m2/s and kD ) 3.8, respectively. The corresponding global minimum objective function is equal to 0.86%. Figure 7b shows variations of the objective function E(D, kD) with CO2 diffusion coefficient D, which corresponds the global optimum mass-transfer Biot number at P ) 0.32, 1.63, 2.65, 3.22 and 4.39 MPa, respectively. It can be seen from this figure that the local minimum objective function is sensitive to CO2 diffusion coefficient D and that there is an obvious global minimum objective function in each curve (i.e., at each equilibrium pressure). Therefore, the diffusion coefficient D and the mass-transfer Biot number kD are the true values once the global minimum objective function is achieved. The diffusion coefficient is equal to 0.47-2.49 × 10-9 m2/s, and the mass-transfer Biot number is determined to be 2.3-6.8 at T ) 27 °C in the pressure range tested in this study. Physically, these relatively small mass-transfer Biot numbers indicate that there is appreciable interfacial resistance to the mass transfer of CO2 across the crude oil-CO2 interface, in comparison with the bulk resistance to the mass transfer of CO2 due to the molecular diffusion. The measured diffusion coefficient, the mass-transfer Biot number, the interface mass-transfer coefficient, and the global minimum objective function Emin for the crude oil-CO2 system are listed in Table 2 for P ) 0.32, 1.63, 2.65, 3.22, and 4.39 MPa and T ) 27 °C. With the measured diffusion coefficient and masstransfer Biot number given in Table 2, the dynamic interfacial tension at any time can be calculated from the predetermined calibration curve of the measured equilibrium interfacial tension versus the calculated equilibrium CO2 concentration (see Figure 6b). Such theoretically predicted dynamic interfacial tensions (lines) are compared with the experimentally measured data (symbols) under different pressures at T ) 27 °C and shown in Figure 5. It can be seen from this figure that there exists an excellent agreement between the theoretically calculated and experimentally measured dynamic interfacial tensions. In addition, Table 2 lists the minimum objective function, Emin(D, kD), that is, the root-mean-squared relative error between the calculated and measured dynamic interfacial tensions, which is equal to or less than 1.77% in this study. 4.5. Effects of Equilibrium CO2 Concentration on the Measured Diffusion Coefficient, Mass-Transfer Biot Number, and Interface Mass-Transfer Coefficient. Figure 8 shows the measured diffusion coefficient and interface mass-transfer coefficient as a function of the equilibrium CO2 concentration, i.e., solubility of CO2. It is seen from Table 2 and Figure 8 that both the diffusion coefficient and the interface mass-transfer coefficient increase with the equilibrium CO2 concentration due to a reduced viscosity of CO2-saturated crude oil at an increased pressure. In this case, CO2 can pass relatively easily through the crude oil-CO2 interface and diffuses in the bulk crude oil phase. Table 3 compares the measured diffusion coefficient D and interface mass-transfer coefficient k in this study with those published in the literature for similar crude oil-CO2 systems. It is seen from this table that the measured diffusion coefficients in this study (0.47-2.49 × 10-9 m2/s) are close to the literature data values of 2.0 × 10-9 m2/s for the Maljamar oil-CO2

system,12 1.14 × 10-9 m2/s for the Ontario oil-CO2 system,17 and 3 × 10-9 m2/s for a stock tank oil-CO2 system.11 However, they are much larger than CO2 diffusion coefficients in Lloydminster heavy oil (0.20-0.55 × 10-9 m2/s)18 of which viscosity is several orders of magnitude higher and that in Athabasca bitumen (0.16-0.47 × 10-9 m2/s)16 of which viscosity is also much higher. In addition, the measured interface mass-transfer coefficient in this study is in the range of k ) 0.88-8.41 × 10-5 m/s. In comparison with the interface masstransfer coefficient of k ) 5.7 × 10-5 m/s for the Ontario oil-CO2 system at P ) 2.9 MPa and T ) 25 °C,17 the measured interface mass-transfer coefficient (k ) 4.16 × 10-5 m/s) of the Weyburn oil-CO2 system at P ) 2.65 MPa and T ) 27 °C is in a similar range. 5. Conclusions The diffusion coefficients and interface mass-transfer coefficients of the crude oil-CO2 system at P ) 0.1-5.0 MPa and T ) 27 °C have been determined by applying the newly developed dynamic interfacial tension method. The diffusion coefficient, the mass-transfer Biot number, and the interface mass-transfer coefficient of the crude oil-CO2 system are found to be 0.47-2.49 × 10-9 m2/s, 2.3-6.8, and 0.88-8.41 × 10-5 m/s, respectively. The experimental results show that these three quantities of the crude oil-CO2 system increase with pressure due to a lower viscosity of CO2-saturated crude oil at a higher pressure. The measured CO2 diffusion coefficients and interface mass-transfer coefficients are close to the data published in the literature for similar crude oil-CO2 systems. Acknowledgment The authors acknowledge the Discovery Grants from the Natural Sciences and Engineering Research Council (NSERC) of Canada to D. Yang and Y. Gu, respectively. Literature Cited (1) Moritis, G. CO2 Injection Gains Momentum. Oil Gas J. 2006, 104, 37. (2) Taber, J. J.; Martin, F. D.; Seright, R. S. EOR Screening Criteria Revisited-Part 1: Introduction to Screening Criteria and Enhanced Recovery Field Projects. SPE ReserVoir Eng. 1997, 12, 189. (3) Taber, J. J.; Martin, F. D.; Seright, R. S. EOR Screening Criteria Revisited-Part 2: Applications and Impact of Oil Prices. SPE ReserVoir Eng. 1997, 12, 199. (4) Stalkup F. I., Jr. Miscible Displacement; Monograph Series;SPE: Richardson, TX, 1983, Vol. 8. (5) Boustani, A.; Maini, B. B. The Role of Diffusion and Convective Dispersion in Vapour Extraction Process. J. Can. Pet. Technol. 2001, 40, 68. (6) Bijeljic, B.; Muggeridge, A. H.; Blunt, M. J. Multicomponent Mass Transfer across Water Films during Hydrocarbon Gas Injection. Chem. Eng. Sci. 2003, 58, 2377. (7) Hoteit, H.; Firoozabadi, A. Numerical Modeling of Diffusion in Fractured Media for Gas Injection and Recycling Schemes. Paper SPE 103292 presented at the SPE Annual Technical Conference and Exhibition, San Antonio, TX, September 24-27, 2006. (8) Sigmund, P. M. Prediction of Molecular Diffusion at Reservoir Conditions, Part I. Measurement and Prediction of Binary Dense Gas Diffusion Coefficients. J. Can. Pet. Technol. 1976, 15, 48. (9) Nguyen, T. A.; Farouq Ali, S. M. Effect of Nitrogen on the Solubility and Diffusivity of Carbon Dioxide into Oil and Oil Recovery by the Immiscible WAG Process. J. Can. Pet. Technol. 1998, 37, 24. (10) Denoyelle, L.; Bardon, C. Diffusivity of Carbon Dioxide in Reservoir Fluids. Paper No. 114 presented at the Annual Meeting of Canadian Institute of Mining, Metallurgy and Petroleum, Ottawa, ON, April 15-19, 1984. (11) Renner, T. A. Measurement and Correlation of Diffusion Coefficients for CO2 and Rich Gas Applications. SPE ReserVoir Eng. 1988, 3, 517.

Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008 5455 (12) Grogan, A. T.; Pinczewski, V. W.; Ruskauff, G. J.; Orr Jr., F. M. Diffusion of CO2 at Reservoir Conditions: Models and Measurements. SPE ReserVoir Eng. 1988, 3, 93. (13) Riazi, M. R. New Method for Experimental Measurement of Diffusion Coefficients in Reservoir Fluids. J. Pet. Sci. Eng. 1996, 14, 235. (14) Zhang, Y. P.; Hyndman, C. L.; Maini, B. B. Measurement of Gas Diffusivity in Heavy Oils. J. Pet. Sci. Eng. 2000, 25, 37. (15) Civan, F.; Rasmussen, M. L. Determination of Gas Diffusion and Interface Mass-transfer coefficient for Quiescent Reservoir Fluids. SPE J. 2006, 11, 71. (16) Upreti, S. R.; Mehrotra, A. K. Experimental Measurement of Gas Diffusivity in Bitumen: Results for Carbon Dioxide. Ind. Eng. Chem. Res. 2000, 39, 1080. (17) Yang, C.; Gu, Y. A New Method for Measuring Solvent Diffusivity in Heavy Oil by Dynamic Pendant Drop Shape Analysis (DPDSA). SPE J. 2006, 11, 48. (18) Yang, C.; Gu, Y. New Experimental Method for Measuring Gas Diffusivity in Heavy Oil by the Dynamic Pendant Drop Volume Analysis (DPDVA). Ind. Eng. Chem. Res. 2005, 44, 4474. (19) Gasem, K. A. M.; Dickson, K. B.; Shaver, R. D., Jr. Experimental Phase Densities and Interfacial Tensions for CO2/Synthetic-Oil and a CO2/ Reservoir-Oil System. SPE ReserVoir Eng. 1993, 8, 170. (20) Yang, D.; Gu, Y. Interfacial Interactions of Crude Oil-Brine-CO2 Systems under Reservoir Conditions. Paper SPE 90198 presented at the SPE Annual Technical Conference and Exhibition, Houston, TX, September 26-29, 2004. (21) Yang, D.; Gu, Y. Interfacial Interactions between Crude Oil and CO2 under Reservoir Conditions. Pet. Sci. Technol. 2005, 19, 216. (22) Yang, D.; Tontiwachwuthikul, P.; Gu, Y. Interfacial Tensions of the Crude Oil + Reservoir Brine + CO2 Systems at Pressures up to 31 MPa and Temperatures of 27°C and 58°C. J. Chem. Eng. Data 2005, 50, 1242.

(23) Yang, D. Interfacial Interactions of the Crude Oil-Reservoir Brine-Reservoir Rock Systems with Dissolution of CO2 under Reservoir Conditions. Ph.D. Dissertation, University of Regina, Regina, Canada, 2005. (24) Yang, D.; Tontiwachwuthikul, P.; Gu, Y. Dynamic Interfacial Tension Method for Measuring the Gas Diffusion Coefficient and the Interface Mass Transfer Coefficient in a Liquid. Ind. Eng. Chem. Res. 2006, 45, 4999. (25) Rotenberg, Y.; Boruvka, L.; Neumann, A. W. Determination of Surface Tension and Contact Angle from the Shapes of Axisymmetric Fluid Interfaces. J. Colloid Interface Sci. 1983, 93, 169. (26) Cheng, P.; Li, D.; Boruvka, L.; Rotenberg, Y.; Neumann, A. W. Automation of Axisymmetric Drop Shape Analysis for Measurement of Interfacial Tensions and Contact Angles. Colloids Surf. 1990, 43, 151. (27) Ghez, R. Diffusion Phenomena: Cases and Studies; Kluwer Academic & Plenum Publishers: New York, 2001. (28) Saatdjian, E. Transport Phenomena: Equations and Numerical Solutions; John Wiley & Sons: Chichester, U.K., 2000. (29) Simon, R.; Graue, D. J. Generalized Correlations for Predicting Solubility, Swelling and Viscosity Behavior of CO2-Crude Oil Systems. J. Pet. Technol. 1965, 17, 102. (30) Hansen, R. S.; Wallace, T. C. The Kinetics of Adsorption of Organic Acids at the Water-Air Interface. J. Phys. Chem. 1959, 63, 1085. (31) Dong, M.; Huang, S.; Dyer, S. B.; Mourits, F. M. A Comparison of CO2 minimum Miscibility Pressure Determinations for Weyburn Crude Oil. J. Pet. Sci. Eng. 2001, 31, 13.

ReceiVed for reView January 13, 2008 ReVised manuscript receiVed May 6, 2008 Accepted May 19, 2008 IE800053D