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Mass Transfer of CO2 in a Carbonated Water−Oil System at High Pressures Guanli Shu, Mingzhe Dong,* Shengnan Chen, and Hassan Hassanzadeh 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: In this paper, CO2 diffusion coefficients in a carbonate water−oil system are determined by measuring the pressure buildup in the closed water−oil system experimentally and modeling the pressure change mathematically. The mathematical method of investigating one-dimensional, time-dependent heat conduction in a composite medium is adopted to solve the mass transfer problem between two liquid phases. The model is combined with well-designed trial-and-error method to determine diffusion coefficients of CO2 in both water and oil phases at the same time. The model considers a moving interface between carbonated water and oil as well as variations of interface concentrations of CO2 in these two phases, which more effectively conforms to reality. Results show that the pressure buildup during the diffusion process resulted from the increased density and swelling of the oil phase. The diffusion coefficient of CO2 in the water phase plays a major role in the interphase mass transfer process.

1. INTRODUCTION Water blocking may become a severe effect in water-wet reservoirs during carbon dioxide (CO2) flooding. As a consequence of water blocking, during the processes of conventional CO2 injection, continuous/slug CO2 injection,1 simultaneous water and gas injection (SWAG),2 and water alternating gas injection (WAG),3,4 a considerable amount of the residual oil is hindered from being contacted by the injected CO2. Such a continuous water phase, which is the waterblocking effect, greatly reduces the microscopic displacement efficiency.5−8 To eliminate the water-blocking effect in CO2 injection, active carbonated water is applied as a preflush prior to a CO2 flood.9 For a CO2−crude oil system, the mass transfer of CO2 into the oil phase results in oil swelling, viscosity reduction, and density increase;10−15 for the carbonated water system, the exsolution of CO2 decreases the density of the water phase.16 Peksa et al.17 experimentally studied CO2−crude oil mass transfer processes in a high pressure glass tube with a designed dead-end volume. In their work, the crude oil was used to represent residual oil that was trapped in dead end pores after waterflood. High pressure tests were conducted to compare the diffusion rates and the effect of water barriers in response to different injection schemes. With the presence of a water barrier, results from the visual observations showed that the injection of carbonated water, followed by CO2 flood, achieved the optimal tertiary oil recovery. If only carbonated water flood were applied, however, there was still a considerable amount of oil left unrecovered. Although the dynamics of the molecular diffusion processes were evident, a model or methodology for estimating the molecular diffusion coefficients of CO2 in the © XXXX American Chemical Society

water and oil phases was not reported in their work nor has it been reported in other past literature. Accurate estimation of molecular diffusion coefficients to describe the mass transfer processes conducted at reservoir conditions is of great significance.18−23 In general, methods of experimentally determined molecular diffusion coefficients can be categorized into direct and indirect methods. A direct method refers to analyzing the composition of fluid samples extracted from a diffusion cell at various times by gas chromatography or by other analytical devices.23−25 However, a direct method is highly expensive and is susceptible to generating experimental errors arising from the disturbance occurring during the extraction of a fluid sample.26 Indirect methods, such as the pressure pulse method19,22,27,28 and NMR,29 measure the changes of properties (for instance, pressure, volume, and interface movement), and then obtain the diffusion coefficients through analytical and numerical models or through graphical methods.27,30,31 In view of the convenience and inexpensiveness, indirect methods have been widely used; the pressure decay method is applied most extensively. Pressure changes are recorded during the experiment and then used to estimate diffusion coefficients through a mathematical model.32,33 In addition, various mathematical models based on pressure decay methodology have been proposed to investigate the mass transfer of CO2 into brine34 or crude oil.27,28,32,35,36 In most cases, CO2 was in a gaseous or in a pure liquid state under Received: Revised: Accepted: Published: A

September 26, 2016 December 5, 2016 December 9, 2016 December 9, 2016 DOI: 10.1021/acs.iecr.6b03729 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research

water was prepared with a high pressure transfer vessel and a syringe pump. The transfer vessel was first filled with the prepared water with 3.0 wt % salinity and then connected to a high pressure CO2 cylinder. Next, the vessel was connected to a syringe pump at the set pressure. To achieve phase equilibrium of the mixture of CO2 and brine water, the transfer vessel was placed in an air bath, set at room temperature, and rocked for 2 h with the syringe pump connected, followed by 24 h at equilibrium to ensure full saturation of the carbonated water.42 The solubilities of the CO2 in brine were calculated43,44 for different pressures and temperatures; the results are shown in Figure 1. From this figure, it is apparent that, as is typical with gases, the solubility of CO2 in the water phase decreases with temperature and increases with pressure.

various experimental conditions. The simultaneous estimation of the diffusion coefficients of CO2 for a carbonated water−oil system (or liquid−liquid system) has not been investigated in previous studies. On the basis of the preceding research,31,37−39 Riazi et al.40 have developed a pore level mathematical model to simulate the dynamic process of oil swelling. Two scenarios were envisioned: oil trapped in dead end pores was contacted directly with carbonated water, or it was contacted indirectly with the CO2 source separated by a water layer. Through sensitivity analyses, they compared the effects of parameters (such as CO2 density, initial equilibrium concentration of CO2, partition coefficient, initial oil volume) on oil swelling. In their study, however, solutions were not gained from an analytical model but instead from a numerical model. In addition, key parameters, such as diffusion coefficients and partition coefficient, were obtained using empirical correlation or from data reported in the literature. Besides, some researches consider the effect of diffusion coefficient on enhanced oil recovery. Zhang et al.27 found that the dissolution of carbon dioxide in oil could significantly affect the efficiency of CO2 EOR by reducing the effect of viscous fingering, decelerating gas breakthrough, and improving oil production rate. To directly investigate this effect, Alavian41 carried out reservoir simulation and found that the increase of diffusion coefficients accelerated the oil recovery rate and the decrease of the coefficients slowed down the oil recovery rate; however, the final oil recoveries were not changed so much. In this paper, a mass transfer model and an experimental method have been developed for simultaneously determining the diffusion coefficients of carbon dioxide in water and oil phases under the consideration of changes of the interface concentrations of CO2 in those phases. Given the complexity of determining diffusion coefficients, a trial-and-error procedure was designed and two convergence criteria were set to obtain the optimal values. The evolution of carbon dioxide concentration profiles during mass transfer was simulated with the estimated coefficients. Additionally, the effects of initial pressure and initial concentration of CO2 on the diffusion coefficients have been discussed, as well as variations of the oil density and concentration along the diffusion direction, and the interface displacement and mass of CO2 transferred.

Figure 1. Solubilities of CO2 in the water phase at different pressures and temperatures.

2.2. Experimental Setup. The primary methods of determining diffusion coefficients based upon the pressure decay method can be categorized into three types and have been summarized by Haugen and Firoozabadi.45 The first type is a fixed cell volume with pressure maintained constant by supplementing gas from the top of the cell; diffusion coefficients are inferred from the amount of gas consumed as a function of time. The second type is a closed cell of fixed volume with changing pressure during the experiment; diffusion coefficients are determined according to the recorded pressure and/or liquid volume change. The third type is designed to maintain a constant pressure, but with a changeable height of the diffusion cell; diffusion coefficients are determined from the changes of phase volumes. In this study, the second type is applied and a schematic diagram of the experimental setup is illustrated in Figure 2. It primarily comprises a diffusion cell, a carbonated water (CW) cylinder, and two transfer vessels (i.e., a CW vessel and a crude oil vessel). The diffusion cell is a double-ended blind cylinder, supplied by Swagelok, with an inner diameter of 3.58 cm and a height of 15 cm. It has a maximum operating pressure of 30 MPa, and the operating temperature ranges between −50 and +50 °C. The mass transfer process takes place in the diffusion cell. The diffusion cell and the CW transfer vessel are placed in the water bath (Core-Parmer Model 12111-11), wherein the temperature is maintained within ±0.01 °C of the desired value. Two syringe pumps (Teledyne ISCO 500D, the flow range is 0.001−204 mL/min, the flow accuracy is 0.5% of set point) are applied to transfer the liquids (crude oil or CW) from the transfer vessels to the diffusion cell. A highly sensitive digital

2. EXPERIMENTAL SECTION 2.1. Materials. Experiments were conducted to study the mass transfer between the crude oil and the carbonated water. In this study, a light oil sample from the Bakken formation in Canada was used. Physical properties of the light oil are listed in Table 1. Carbon dioxide (CO2) and nitrogen (N2), each with purities of 99.99%, were supplied by Praxair. The carbonated Table 1. Physical Properties of Bakken Crude Oil property viscosity, cP

density, kg/m3

total acid number, (mg of KOH)/g molecular weight, g/mol

value at at at at at at

15 20 30 15 20 30

°C °C °C °C °C °C

2.54 2.17 1.65 805.0 801.2 793.1 0.40 162 B

DOI: 10.1021/acs.iecr.6b03729 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Figure 2. Schematic diagram of the experimental setup.

pressure gauge, supplied by Heise, with a maximum measuring pressure of 34.5 MPa and accuracy of 0.1−0.025% full scale, is used to record pressure variations and is connected to a computer for data collection. The pressure change during the test is recorded with a digital acquisition system. Given that the tests are conducted at high pressure, a backpressure regulator (BPR) is applied to set and precisely control the upstream pressure. A graduated tube is used to collect the produced liquid from the BPR. To evacuate the lines connected to the diffusion cell, a vacuum pump (Welch Model 1376) is employed. An additional transfer vessel was applied to inject pressurized nitrogen (N2) into the BPR dome to set the back pressure. 2.3. Experimental Procedure. Before the experiment was performed, a gas leakage test was conducted for the experimental setup. The system was pressurized to 10 MPa by injecting N2. The pressure was recorded for 48 h to check for leakage using a liquid leak detector. When the system was leak-free, the following experimental procedure was applied: (1) The diffusion cell and CW cylinder were placed into the water bath and filled with distilled water. (2) The lines connecting the crude oil vessel and the diffusion cell were vacuumed. (3) Crude oil was introduced into the diffusion cell, the temperature was set to the water bath temperature at a certain value, and the system was left for 48 h to ensure that the test fluid equilibrated to the desired temperature. (4) The diffusion cell was pressurized with crude oil to match the pressure of the CW cylinder. (5) The CW was injected into the diffusion cell from the bottom port of the diffusion cell using the syringe pump. (6) After a designed volume of CW was transferred into the diffusion cell, the valve connecting the diffusion 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. (7) 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.

Figure 3. Schematic of carbonated water−crude oil system in a closed diffusion cell.

Initially, carbon dioxide is merely dissolved in carbonated water with a known concentration. At time zero, the water and oil phases are brought into contact, without mixing, along the interface of x = 0. 3.2. Assumptions. The following assumptions are made for the developed theoretical model: (1) Because the diffusion cell is immersed in a water bath wherein the temperature can be controlled within ±0.01 °C of the desired value, the mass transfer system is at an isothermal condition with no chemical reaction. (2) The interface between the water phase and oil phase is considered to remain at thermodynamic equilibrium. (3) Diffusion coefficients of carbon dioxide in the water phase and the oil phase are assumed to be constant. Although diffusivities are affected by pressure and temperature, in this research, the values of diffusivities do not change significantly when a small pressure range is used in the determination of diffusion coefficients. (4) No natural convection is induced by mixing during the diffusion process. The oil phase, which is initially free of

3. THEORY 3.1. Model. A schematic of the carbonated water−crude oil system in a closed diffusion cell is illustrated in Figure 3. C


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the corresponding diffusion coefficients of CO2 in the oil and water phases, respectively, in cm2/s. At time zero, CO2 is entirely dissolved in the water phase. The following two initial conditions are applied:

CO2, is placed on the top of the carbonated water phase. The dissolved CO2 in the water diffuses upward due to the existing concentration gradient between the two phases. As a result of the dissolution of CO2, the density of the oil phase slightly increases from bottom to top, but it is still lighter than the water phase. Therefore, densityinduced convection also will not emerge. Regarding the water phase, the exsolution of CO2 causes the density of the water phase to slightly decrease from top to bottom. As a result, the water phase also remains stable and density-driven convection will not appear. (5) Given the diffusion time scale is rather long, the carbonated water and oil phases are considered as two semi-infinite regions before the CO2 diffusion front has reached the top and bottom boundaries. (6) Due to oil swelling, the two-phase interface is considered to move as the diffusion continues. This is a reality because the dissolution and exsolution of CO2 cause the volume changes of both water and oil phases. Moving boundary problems have been discussed widely in different areas, such as oxidation of alloys,46 spontaneous emulsification,47 absorption by the liquid of a single component from a mixture of gas,48 and brass diffusion couples with multilayers,49 among others. Danckwerts48 presented a general solution for the moving interface problem in unsteady-state heat conduction or in diffusion within two-phase regions. In this section, the mass transfer of CO2 from the water phase into the oil phase, with a moving interface, is studied. Solutions are obtained by applying Danckwerts’s method, but with different boundary conditions. The differential equation describing the mass transfer process can be obtained using Fick’s law. The problem of interest is illustrated in Figure 4 with the moving interface of x = s(t).

Co(x ,0) = 0

(3)

Cw(x ,0) = Ci

(4)

where Ci is the initial concentration of CO2 in the water phase. In addition, at the early stage when the CO2 has not yet diffused sufficiently to reach the boundaries, the following boundary conditions are applied: Co( +∞,t ) = 0

(5)

Cw( −∞,t ) = Ci

(6)

Subject to the aforementioned initial and boundary conditions, the general solutions for eqs 1 and 2 can be written as48,50,51 ⎛ x ⎞ ⎟⎟ Co(x ,t ) = A1 + A 2 erf⎜⎜ ⎝ 2 Dot ⎠

(7)

and ⎛ x ⎞ ⎟⎟ Cw(x ,t ) = B1 + B2 erf⎜⎜ ⎝ 2 Dwt ⎠

(8)

where A1, A2, B1, and B2 are constants to be determined, and 2 erf(z) is the error function in the form of erf(z) = 2 ∫ z0e−t dt. π

Combining eqs 5 and 7 gives A 2 = −A1

(9)

Likewise, using eqs 6 and 8 yields B2 = B1 − Ci

(10)

According to the preceding assumption that equilibrium exists between the phases at the moving interface (x = s(t)) and using eqs 7−10, one can write

Assuming that the phase densities are weak functions of the CO2 concentration, the governing equations are

∂Cw(x ,t ) ∂ 2Cw(x ,t ) = Dw ∂t ∂x 2

⎛ s (t ) ⎞ ⎟⎟ = Cw* Cw(s(t ),t ) = B1 + (B1 − Ci) erf⎜⎜ ⎝ 2 Dw t ⎠

(12)

Co*

A2 = −

2

s (t ) < x < ∞

(11)

where C*o and C*w are the concentrations of CO2 in the oil and water phases at the interface, respectively. In addition, there is a relationship between the two interface concentrations, which is that C*o = kpcC*w , where kpc is the partition coefficient of CO2 in the two phases at the interface. A particular description of kpc will be given in the following section. From eqs 9 and 11, we can obtain

Figure 4. Two semi-infinite regions of CO2 mass transfer from the water phase into the oil phase with moving interface.

∂Co(x ,t ) ∂ Co(x ,t ) = Do ∂t ∂x 2

⎡ ⎛ s (t ) ⎞ ⎤ ⎟⎟⎥ = Co* Co(s(t ),t ) = A1⎢1 − erf⎜⎜ ⎢⎣ ⎝ 2 Dot ⎠⎥⎦

( ) s(t ) 2 Dot

(13)

( )

(14)

1 − erf

(1)

and −∞ < x < s(t )

(2)

A1 =

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 are

Co* 1 − erf

D

s(t ) 2 Dot


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diffusion coefficients, pressure turns out to be the most important experimental parameter, indicating its dominant effect on the mass transfer of carbon dioxide from the carbonated water into the oil. The mass transfer (or diffusion) experiment is performed within a closed cell. After the cell was evacuated, the test oil is introduced from the top of the cell. While achieving the desired experimental pressure by pressurizing the oil, the carbonated water is injected from the bottom to a desired volume. When the two phases are brought into contact, pressure measurement is started. Figure 6a shows the pressure curve for a typical diffusion test (Test 1) for a carbonated water−oil system. It can be seen that the pressure increases with time during the mass transfer process. The ascending rate of pressure is higher at the initial stage and levels off during the late stage. 4.2. Estimation of Diffusion Coefficients. The detailed steps of estimating diffusion coefficients are set forth in the following trial-and-error procedure: (1) For each test, determine the initial parameters, such as the initial concentration of CO2 in the water phase Ci, the initial total volume Vini, the initial phase volumes of the water (Vw,ini) and oil (Vo,ini), where Vini = Vw,ini + Vo,ini, the initial masses of oil, water, and CO2, the experimental temperature, and the initial pressure. Table 2 lists the parameters used to estimate the diffusion coefficients for Test 1.

As a result, the concentration of CO2 in the oil phase can be expressed as ⎡ ⎛ ⎞⎤ Co* ⎢1 − erf⎜⎜ x ⎟⎟⎥ 1 − erf(λ) ⎢⎣ ⎝ 2 Dot ⎠⎥⎦

Co(x ,t ) =

(15)

where λ = s(t )/(2 Dot ). Using eqs 10 and 12 yields Cw* − Ci

B2 =

1 + erf

( ) s(t ) 2 Dw t

(16)

and Cw* − Ci

B1 = Ci +

1 + erf

( ) s(t ) 2 Dw t

(17)

Thus, the concentration of CO2 in the water phase is Cw(x ,t ) = Ci −

⎡ ⎛ ⎞⎤ Ci − Cw* ⎢1 + erf⎜⎜ x ⎟⎟⎥ ⎛ ⎞⎢ ⎝ 2 Dw t ⎠⎥⎦ 1 + erf⎜λ Do ⎟ ⎣ ⎝ Dw ⎠ (18)

where λ =

s(t ) 2 Dot

and thus

s(t ) 2 Dw t

=λ

Do Dw

.

The expression for the conservation of mass at the moving interface (x = s(t)) is Dw

∂C ∂Cw d s (t ) − Do o = (Co* − Cw*) dt ∂x ∂x

Table 2. Parameters Used To Determine Diffusion Coefficients for Test 1

(19)

Substituting eqs 15 and 18 into eq 19 and considering the boundary conditions yields −

Dw Do

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

= (Co* − Cw*)λ π

(20)

After the value of λ was determined from the above transcendental equation by using the bisection method, the interface movement distance s(t) can be obtained as a function of time, as well as the concentrations of CO2 in the oil and water phases through eqs 15 and 18. The model described here forms the forward model for predicting the diffusion coefficients of CO2 in the oil and water phases.

parameters

value

initial concentration of CO2 in water phase, 10−3 mol/cm3 initial volumes water phase, cm3 oil phase, cm3 total, cm3 initial masses in the carbonated water−oil system oil, g water (or brine), g CO2, g total, g experimental temperature, °C initial pressure, MPa

1.3343 80 000 70 000 150 000 56.83 78.34 4.70 139.87 20 16.56

(2) According to the experimental pressure data, the pressure at t = 10 h is obtained. There are two reasons for choosing 10 h to determine the diffusion coefficients. One is to avoid the pressure fluctuation at the very outset of the mass transfer process. A second reason is that the diffusion process has entered into a pure diffusion process at 10 h. Accordingly, the pressure change that occurs then is only related to the density and volume changes of the two phases. Figure 7 shows the measured pressure change curve versus the square root of time for Test 1. The solid line is the actual experimental pressure data. The dashed line shows the deviation of the actual experimental pressure change compared with the linear change. At around 10 h, the pressure falls on a straight line. (3) Assume initial estimates for the interface concentration of CO2 in the water phase (Cw*) and the diffusion coefficients (Do and Dw) as the first guess. The interface concentration of CO2 in the oil phase can be calculated

4. METHODOLOGY OF DETERMINING DIFFUSION COEFFICIENTS FOR A CARBONATED WATER−OIL SYSTEM The developed model described in the previous section is used to estimate the diffusion coefficients of CO2 in the carbonated water and in the oil. No direct method is available to obtain these coefficients. Therefore, in this study, the proposed solution methodology makes use of the pressure changes obtained experimentally at constant temperature. The diffusion coefficients at a certain time are estimated throughout a welldesigned trial-and-error procedure combined with the aforementioned established analytical model. 4.1. Experimental Data of Pressure Change during Interphase Mass Transfer. In the course of determining E


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j+1 n j+1 the total volume Vj+1 o (Vo = ∑1Voi ) of the oil phase at the selected time. (7) Follow the same procedure for the water phase (Vjwi = j+1 Vjw/n) to determine the water phase volume Vj+1 w (Vw = n j+1 ∑1Vwi ). Here, the density correlation of the CO2 + brine solution is applied (eq S3, its derivation process is shown in the Supporting Information). j j+1 j j+1 j j (8) Compare Vj+1 o and Vo, Vwi and Vw. If |Vo − Vo|/Vo × j+1 j j 100 < 0.01% and |Vw − Vw|/Vw × 100 < 0.01% are satisfied, then calculate the phase volume changes (ΔVj+1 o j+1 j+1 j+1 j+1 and ΔVj+1 w ), where ΔVo = |Vo − Vo,ini| and ΔVw = |Vw − Vw,ini|. At the same time, calculate the diffusion fluxes in the water and oil phases at the interface using eqs 15 and 18. Results have been shown in eqs 21 and 22.

on the basis of the correlation of the partition coefficient (eq S5, shown in the Supporting Information), Co* = kpcCw*. (4) Input the estimated interface concentrations and diffusion coefficients into eq 20 to solve the value of λ using the bisection method. (5) Then, apply the obtained value of λ and the initial conditions to generate the concentration profiles for a chosen time (e.g., t = 10 h) using eqs 15 and 18. The phase volumes (Vjo and Vjw) can be determined on the basis of the interface position, s(t); (6) The next step is to equally divide the volume of the oil phase into n elements (Figure 5); thus, the volume for

Jo = Jw =

Co* Do /πt e−(x /2 1 + erf(λ) Ci − Cw* 1 + erf(λ Do /Dw )

Dot )2

(21)

Dw /πt e−(x /2

Dw t )2

(22)

At the two-phase interface, the mass of CO 2 transferred out of the water phase shall be equal to that of the CO2 entered into the oil phase. The trial-anderror procedure terminates when the convergence j+1 3 j+1 j+1 criteria (|ΔVj+1 o − ΔVw | < 0.03 cm and |Jo − Jw | < 10−9 mol/(cm2·s)) are satisfied. Thus, the current estimates of the diffusion coefficients can be considered to be the converged values; otherwise, repeat steps 3−7. (9) In general, after satisfying the two criteria mentioned above, the values of the interface concentrations and diffusion coefficients should be considered to be the final results. In this work, however, more than one set of values may satisfy the two criteria. The final step should be employed to reach the optimal results: that is, to compare the differences of the diffusion fluxes calculated from a series of C*w and choose the set of values with a minimum difference to be the final results. Additionally, to promote a fast convergence, a good initial estimate is needed (a common criterion for iterative procedures, particularly nonlinear ones). The interface concentrations and diffusion coefficients may vary over wide ranges. Therefore, the ranges should be narrowed using the experimental data from the literature. For the interface concentrations, because they are far away from the equilibrium concentration at 10 h, the initial guess of the interface concentration of CO2 in the water phase (Cw*) can be selected at around 50% of the initial concentration and enlarged to a range of 30−70% later. Moreover, in the trial-and-error

Figure 5. Selection of oil and water domain subintervals for determination of the diffusion coefficients.

each element is equal to Vjoi = Vjo/n (i = 1, 2, 3, ..., n; j = 1, 2, 3, ...). The subscript i denotes the location of each element, and the superscript j denotes the trial number. The optimal element n will be analyzed and the effect of number of subintervals in determination of diffusion coefficients is addressed in the Supporting Information Section 3. According to the concentration profile of CO2 in the oil phase, the average concentration for each element (Cjoi) can be calculated with the trapezoidal formula. This average concentration is used to determine the density of the mixture of CO2 and oil (ρjmix,oi) for each element on the basis of the density correlations from eq S4 (the detailed derivation process is shown in the Supporting Information). In addition, on the basis of the average concentration Cjoi, volume of each element Vjoi, and the molecular weight of CO2, the mass of CO2 j (mCO ) in each element can be obtained. Consequently, 2,i the mass of mixture of CO2 and oil (mjmix,oi) for each element can be determined. Following that, the volume j+1 j of each element Vj+1 oi can be calculated by Voi = mmix,i/ j ρmix,oi. Summing up the volumes of all elements obtains

Table 3. Summary of Initial and Interface Concentrations of CO2 in Three Tests at the Experimental Temperature of 20 °C concn of CO2 in water and oil phases at the interface at 10h(10−3 mol/cm3)

initial phase volume (cm3) test no.

water

oil

total

initial pressure (MPa)

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

pressure at 10 h (MPa)

water

oil

1 2 3

80 000 80 000 80 000

70 000 70 000 70 000

150 000 150 000 150 000

16.56 18.93 21.18

1.3343 1.3353 1.3365

17.19 19.32 21.68

0.6500 0.6900 0.7200

2.1678 2.2857 2.3672

F


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Figure 6. Measured pressure change data of CO2 diffusion from carbonated water to crude oil versus time at the experimental temperature of 20 °C for three tests: (a) Test 1, (b) Test 2, and (c) Test 3.

procedure, we first assume a fixed value for C*w and fix the value of the diffusion coefficient of CO2 in the water phase (Dw). We iterate on the diffusion coefficient in the oil phase (Do) until an optimal value is obtained and the convergence criteria are satisfied. Then, we change the value of Cw* and follow the same procedure.

Table 3 shows the calculated interface concentrations of CO2 in the water and oil phases for three tests at 10 h. It can be seen from this table that the tests were carried out under the same conditions, except for initial pressure, resulting in distinct experimental pressures at 10 h (17.19, 19.32, and 21.68 MPa, respectively). The interface concentrations of CO2 in the oil phase at 10 h vary (2.1678 × 10−3, 2.2857 × 10−3, and 2.3672 × 10−3 mol/cm3, respectively) because of the different pressures at 10 h. The values of concentrations of CO2 in the water phase at the interface are also different (0.6500 × 10−3, 0.6900 × 10−3, and 0.7200 × 10−3 mol/cm3, respectively), again because of the different pressures at 10 h, resulting in different partition coefficients: 3.335 for Test 1, 3.313 for Test 2, and 3.288 for Test 3 (the detailed calculation method can be found in the Supporting Information). As a result, it can be concluded that the initial pressure has only a slight impact on the interface concentrations. A higher initial pressure leads to a relatively higher interface concentration of CO2 in the oil phase. 5.3. Effects of Saturation and Initial Pressures on Diffusion Coefficients. In this section, the diffusion coefficients for the three diffusion tests are determined and the effects of saturation and initial pressures are discussed. To avoid the effect of pressure fluctuations at the beginning of each experiment, the acceptable pressure data points are selected from a later time when the pressure increase follows the expected linear trend when plotted versus the square root of time. This behavior is characteristic of pure diffusion, and it is depicted in Figure 7. The solid lines display the actual experimental pressures, and the dashed line shows the deviation of the actual experimental pressure change compared with the linear change. Except for Test 2, in which the initial fluctuation stage ends around 4 h, Tests 1 and 3 each have a very short fluctuation period. As a result, to avoid the influence of the

5. RESULTS AND DISCUSSION 5.1. Pressure Change during CO2 Mass Transfer. In this section, the results of three tests are presented to study the variation of pressure during the diffusion of CO2 from the carbonated water phase to the oil phase. These three tests were initially prepared at the same saturation pressure (Psat), at which point the water is fully saturated with carbon dioxide. Then, the CO2 saturated water was brought into contact with oil at different initial pressures by pressurizing the carbonated water solely. Thus, the initial concentrations of CO2 in water remained the same. Details of the tests are shown in Table 3. It can be seen that the initial concentrations of CO2 in the water phase are around 1.3353 × 10−3 mol/cm3. To maintain consistency, the initial phase volumes for the three tests were the same: 80 000 cm3 of water phase, 70 000 cm3 of oil phase, and therefore a total volume was 150 000 cm3, which is the design volume of the diffusion cell. Figure 6 illustrates the measured pressure data as a function of time for the carbonated water−crude oil system for the three tests, at a pressure range of approximately 16−22 MPa and at the experimental temperature of 20 °C. From these figures, it can be seen that the system pressure builds up with time and gradually plateaus at a later time. Two mechanisms are responsible for the pressure variations: the increased density of the oil phase due to the mass transfer of CO2, which tends to decrease the pressure, and the swelling of the oil phase, which tends to increase the pressure. The oil swelling, however, is shown to be the dominant mechanism. These conclusions will also be proved in the following sections. 5.2. Concentration of CO2 in the Water and Oil Phases at the Interface. During the study of diffusion processes, the interface concentrations of CO2 in the water and oil phases are of great significance and need to be determined. As pointed out, interface concentrations vary with time during the experiments. Past studies52,53 on the effect of interface concentrations on interface movement reveal that, the greater the difference of interface concentrations, the lesser the amount of interface movement. Considering the uncertainty caused by the difference, interface concentrations are determined simultaneously with diffusion coefficients within the trial-and-error procedure.

Figure 7. Measured pressure data versus square root of time for the three tests. G


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Industrial & Engineering Chemistry Research initial fluctuations on the analysis, the pressure at 10 h is chosen to determine the diffusion coefficients. Calculated diffusion coefficients for the three tests are shown in Table 4. Because the initial concentration of CO 2

in the oil phase do not. As a result, at the same saturation pressure, it can be concluded that the initial pressure significantly affects the diffusion coefficients. This also explains the reason that it is imperative to determine the diffusion coefficients within a small pressure range. Additionally, there are some uncertainties while determining the diffusion coefficients, including the calculation of the densities of oil and water phases and the trial-and-error procedure. 5.4. Feasibility of Applying Diffusion Coefficients throughout the Test. After determining the diffusion coefficients of CO2 for the pressure chosen at 10 h, it is necessary to examine the feasibility of applying these determined diffusion coefficients throughout each test. This examination is necessary to validate the assumption of using the constant diffusion coefficients in the mathematical model within a small pressure range. If the differences between the phase volume change and diffusion fluxes of two phases at the interface are considerable, the assumption of the constant diffusion coefficients cannot be justified. Table 5 summarizes the pressures at different times, the phase volume differences, and the diffusion flux differences for the three tests. As shown in Table 5, the system pressure of Test 1 increases from the initial value of 16.56 to 17.66 MPa (after 30 h), as the process of the CO2 mass transfer occurs. In the meantime, the phase volumes of water and oil are also changing. The water phase shrinks during the test, the phase volume decreases from the initial value of 80 000 cm3 to 79 701 cm3 (at 30 h), and the shrinking ratio is 0.37%, where the shrinking ratio is defined as the percentage of difference of the initial water phase volume and volume at 30 h divided by the initial phase volume. On the contrary, the oil phase swells, and the phase volume increases from the initial value of 70 000 to 70 313 cm3 (at 30 h), with a swelling ratio of 0.45%, where the swelling ratio is defined as

Table 4. Diffusion Coefficients of CO2 in Water and Oil Phases Determined at 10 h for Three Tests at the Experimental Temperature of 20 °C experimental pressure (MPa) test no. 1 2 3

initial 16.56 18.93 21.18

at 10 h 17.19 19.32 21.68

diffusion coefficients (cm2/s)

final 18.08 20.18 22.42

Dw

Do −5

1.00 × 10 1.33 × 10−5 1.65 × 10−5

1.02 × 10−6 1.08 × 10−6 1.14 × 10−6

corresponds to the saturation pressure, which was the same in all tests, variations of the diffusion coefficients are associated with the different initial pressures. In this work, three initial pressures are considered: 16.56 MPa (Test 1), 18.93 MPa (Test 2), and 21.18 MPa (Test 3). It can be observed from Table 4 that the diffusion coefficients increase with increasing initial pressure from 1.00 × 10−5 to 1.65 × 10−5 cm2/s for the water phase and from 1.02 × 10−6 to 1.14 × 10−6 cm2/s for the oil phase. Although the same experimental conditions applied in this work have not been used yet, Cadogan et al.54 reported the diffusion coefficient of CO2 in the water phase was 2.233 × 10−5 cm2/s and Guo et al.55 found the diffusion coefficient of CO2 in the oil phase to be 2.723 × 10−5 cm2/s at similar conditions. Our diffusion coefficients are of the same order of magnitude. The lower values of diffusion coefficients can be attributed to the lower initial concentration of CO2 in the water phase. With respect to changes in oil phase, although the diffusivities of CO2 in the water phase alter significantly, those

Table 5. Summary of Results for Three Tests at Different Times Conducted at the Different Initial Pressures and at the Experimental Temperature of 20 °C volume change (cm3)

phase volume determined (cm3) time (hour)

pressure (MPa)

water

oil

total

relative error of total volume (%)

water −5

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

19.13 19.32 19.49 19.63 19.76 19.87

79 876 79 815 79 767 79 732 79 694 79 659

70 155 70 212 70 255 70 291 70 322 70 351

5 10 15 20 25 30

21.58 21.68 21.80 21.90 22.01 22.11

79 857 79 800 79 749 79 712 79 671 79 642

70 155 70 221 70 269 70 309 70 343 70 375

oil

difference of phase volume changes (cm3) −6

Test 1 (Dw = 1.00 × 10 cm /s, Do = 1.02 × 10 cm /s) 150 028 0.02 0.104 0.132 0.028 150 022 0.02 0.161 0.183 0.022 150 017 0.01 0.206 0.223 0.017 150 018 0.01 0.238 0.256 0.018 150 011 0.01 0.274 0.285 0.011 150 014 0.01 0.299 0.313 0.014 Test 2 (Dw = 1.33 × 10−5 cm2/s, Do = 1.08 × 10−6 cm2/s) 150 031 0.02 0.124 0.155 0.031 150 027 0.02 0.185 0.212 0.027 150 022 0.01 0.233 0.255 0.022 150 023 0.01 0.268 0.291 0.023 150 016 0.01 0.306 0.322 0.016 150 009 0.00 0.341 0.351 0.009 Test 3 (Dw = 1.65 × 10−5 cm2/s, Do = 1.14 × 10−6 cm2/s) 150 013 0.01 0.143 0.155 0.013 150 021 0.01 0.200 0.221 0.021 150 017 0.01 0.269 0.269 0.017 150 021 0.01 0.288 0.309 0.021 150 015 0.01 0.329 0.343 0.015 150 017 0.01 0.358 0.375 0.017 2

H

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

2

5.54 5.02 9.71 1.27 1.37 1.39

× × × × × ×

10−10 10−12 10−11 10−10 10−10 10−10

6.72 1.45 1.08 1.45 1.57 1.60

× × × × × ×

10−10 10−11 10−10 10−10 10−10 10−10

6.61 3.09 1.25 1.61 1.72 1.74

× × × × × ×

10−10 10−12 10−10 10−10 10−10 10−10


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Industrial & Engineering Chemistry Research the percentage of difference of the initial oil phase volume and volume at 30 h divided by the initial phase volume. Considering that the experiments were conducted in a closed cell, this is a good indication that the oil swelling may have caused the pressure buildup as a result of the mass transfer of CO2 from the water phase into the oil phase. When the same diffusion coefficients determined at 10 h are applied throughout the experiment (between 5 and 30 h), the relative errors in the total volume are negligibly small, sitting at around 0.01−0.02%. The differences of the phase volume changes and diffusion fluxes are within the acceptable convergence ranges (ΔV < 0.03 cm3, ΔJ < 10−9 mol/(cm2·s)). This finding supports the assumption of pressure independence of the diffusion coefficients within the studied time range. It is notable, however, that as time elapses (or as the pressure increases) the difference of diffusion fluxes also increases, indicating the potential for considerable errors after applying a higher pressure or after a longer time. Nevertheless, although the increased difference of diffusion fluxes at 5 h is an outlier to that trend, considering that it occurs at the initial stage of the mass transfer process, it is a negligible effect. For Test 2, the similar trend of pressure increase can be observed. The experimental pressure builds up from the initial pressure of 18.93 to 19.87 MPa (at 30 h). Meanwhile, the water phase shrinks from 80 000 cm3 to 79 659 cm3 with a shrinking ratio of 0.43%; the oil phase swells from 70 000 cm3 to 70 351 cm3 with a swelling ratio of 0.50%. Compared with Test 1 results, the shrinking and swelling ratios are both increased. From Table 5, small relative errors and differences of phase volume changes and diffusion fluxes are also found to back up the assumption of constant diffusion coefficients. The trend of ascending pressure for Test 3 is comparable to results of Tests 1 and 2. The pressure builds up from the initial 21.18 to 22.11 MPa (at 30 h) and, correspondingly, the phase volumes have changed. The water phase volume decreases from 80 000 cm3 to 79 642 cm3 (the shrinking ratio is 0.45%), and the oil phase volume increases from 70 000 cm3 to 70 375 cm3 (the swelling ratio is 0.54%). Similar to previous test results, the relative errors remain small. 5.5. Changes of Concentration and Density as the CO2 Mass Transfer. The variations of oil and water densities versus distance above and below the interface at different times (5, 10, 20, and 30 h) are illustrated in Figure 8 for the three tests. It demonstrates the trend of declining water density in response to the CO2 diffusion from the water phase into the oil phase. Following the mass transfer of CO2 in the water phase close to the two-phase interface, the concentration of CO2 in the water declines, causing the decreased density. This decline continues until the completion of the mass transfer. In the oil phase, the oil density increases due to the elevated concentration of CO2 which results in swelling. At the same time, these figures also indicate the changes of phase volumes. A detailed discussion of the effect of oil swelling will be addressed in the following section. Figure 9 shows the concentration profiles of CO2 during the process of diffusion for the three tests. As demonstrated, all concentration profiles show the same trend. Close to the interface, with the mass transfer of CO2 from the water phase to the oil phase, the CO2 concentration starts to decrease from the initial interface (x = 0) in the water phase and increase in the oil phase. Meanwhile, the dissolution of CO2 swells the oil phase and shrinks the water phase, which can be seen through the interface movement. However, considering the constraint of

Figure 8. Density changes of water and oil phases versus distance from the initial interface x = 0 at different times for three tests, (a) for Test 1, (b) for Test 2, and (c) for Test 3.

a closed cell, the interface movement is not significant. For Test 1, at 30 h, the interface moves toward the left (toward the water phase) by about 0.031 cm. This movement is calculated through subtracting the volume at 30 h by the initial phase volume of 70 000 cm3 and dividing the result by the crosssectional area of 10 cm2. The specific values have been shown in Table 5. The interface movements at 30 h of Tests 2 and 3 are about 0.035 and 0.038 cm, respectively. 5.6. Effects of Diffusion Coefficients on Interface Displacement and Mass of CO2 Transferred. A detailed comparison of the interface displacement toward the water phase is shown in Figure 10a. It can be seen that the interface between the water and oil phases moves gradually toward the water phase as a result of the mass transfer of CO2 from the I


Article


revealing the diffusive nature of the mass transfer process. In addition, according to Fick’s law of diffusion: J = DoA

∂Co ∂x

= Co*A x=0

Do πt

(23)

where J is flux, mol/cm2; Do is the diffusion coefficient of CO2 in the oil phase, cm2/s; A is the cross-sectional area, cm2; Co is the concentration of CO2 in the oil phase, mol/cm3; x is the distance, cm; and t is the elapsed time, s. Integrating the above equation yields the following expression for the total mass of CO2 transferred into the oil phase: Q = 2Co*A

Do 1/2 t π

(24)

where Q is the total moles of CO2 transferred into the oil phase. Accordingly, the diffusion coefficients of CO2 in the oil phase can be calculated on the basis of the intercept obtained from the regression function (shown in Figure 10b). The calculated diffusivities for the three tests are 1.16 × 10−6, 1.24 × 10−6, and 1.30 × 10−6 cm2/s, respectively. When they are compared with the diffusion coefficients determined from the analytical calculations (Table 4), the differences are indeed small. The uncertainties of determining CO2 diffusion coefficients in water and oil phases by analyzing the pressure curve include the calculation of the densities of oil and water phases and the trial-and-error procedure. The above comparison proves the accuracy of applying the proposed trial-anderror procedure and phase density calculations in determining the diffusion coefficients. 5.7. Examination of Uncertainties Resulted from Pressure Variation. As mentioned in the previous sections, concentration, densities, and diffusion coefficients are all dependent on pressure. Thus, it is indispensable to estimate the possible errors resulted from pressure variation during the experimental period of interest (30 h). Comparison with the time-average pressures of three tests yields relative errors of maximum pressure variations during the studied period of 6.36% for Test 1, 4.83% for Test 2, and 4.27% for Test 3. As the pressure changes, densities of the oil phase and water phase are correspondingly changed. The average maximum relative error of density of oil phase for three tests caused by CO2 concentration change at the two-phase interface is about 2.24%; for the water phase, this value is 0.51%. As a result, the uncertainties caused by pressure variation will not significantly affect the determined values of diffusion coefficients. Figure 9. Concentration profiles of CO2 in water and oil phases at six times for three tests: (a) Test 1, (b) Test 2, and (c) Test 3.

6. CONCLUSIONS In this paper, an analytical model and an experimental method are developed for determining CO2 diffusion coefficients in a carbonated water−oil system. A theoretical model has been developed by referencing the mathematical theory of heat transfer. In this model, a moving interface is considered and the interface concentrations of CO2 in water and oil phases are considered to be functions of time, which more effectively conforms to reality. Considering the complexity of determining the interface concentrations and diffusivities, a methodology based on the experimental pressure data and designed trial-anderror procedure is proposed. A detailed methodology has been developed to estimate the diffusion coefficients of a diffusing species within two liquids across a moving interface between the liquid phases, in our case, carbonated water and oil.

carbonated water phase into the oil phase. The interface displacement versus time plotted in a log−log graph results in straight lines for all three tests, which again reveals the diffusive nature of the mass transfer process. Although the three tests display similar interface movement trends, the Test 3 interface moves faster because the diffusion coefficients of Test 3 are greater than the other tests. This phenomenon can also be explained by the mass change of CO2 in the oil phase. Figure 10b shows the mass change of CO2 in the oil phase as proceeding from the mass transfer. A similar trend of the increase in mass of CO2 as time goes on can be seen in all three tests. The total mass of CO2 dissolved in the oil phase versus time in a log−log graph also results in a straight line, again J


Article


Figure 10. Results of mass change and interface displacement versus time in a log−log graph for three tests: (a) interface displacement and (b) mass change of CO2 in the oil phase.

An experimental method has been established to measure the pressure changes for a liquid−liquid system. Diffusion coefficients of CO2 in water and oil phases are determined simultaneously, based on the measured pressure data and on the analytical model developed. Results indicate that the pressure builds up as the dissolved CO2 in the carbonated water transfers to the oil phase. During the process of mass transfer, the CO2 concentration in the water phase decreases gradually and correspondingly increases gradually in the oil phase. The change of concentration causes a density increase and swelling of the oil phase, which explains the buildup of pressure. A change of initial concentration of CO2 in the water phase has essentially negligible effect on the diffusion coefficients. When the three tests are compared, the diffusivity of CO2 in the water phase is greater than that in the oil phase. By using the proposed experimental and theoretical methods, researchers can determine the actual diffusion coefficients in different conditions and directly apply determined values in conducting modeling for forecasting the potentiality of recovering the residual oil from water-wet reservoirs. Considering the deviation of estimated values calculated by using empirical equations, these values are more accurate. This approach can also be applied for simultaneous measurements of diffusion coefficient in other liquid−liquid systems.

Hassan Hassanzadeh: 0000-0002-3029-6530 Notes

The authors declare no competing financial interest.

■

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

■

*M. Dong. E-mail: [email protected].

NOMENCLATURE cross-sectional area of diffusion cell (cm2) CO2 concentration (mol/cm3) initial concentration of CO2 in water phase (mol/cm3) concentration of CO2 in oil phase (mol/cm3) concentration of CO2 in water phase (mol/cm3) interface concentration of CO2 in oil phase (mol/cm3) interface concentration of c CO2 in oil phase (mol/cm3) average concentration for each element (mol/cm3) diffusion coefficient of CO2 in oil phase (cm2/s) diffusion coefficient of CO2 in water phase (cm2/s) error function diffusion fluxes in the oil phase (mol/(cm2·s)) diffusion fluxes in the water phase (mol/(cm2·s)) partition coefficient, dimensionless total length (or height) of the mass transfer system (cm) mjCO2,i mass of CO2 (g) mjmix,oi mass of mixture of CO2 and oil (g) n number of elements P pressure (MPa) t time (s) Vjo volume of oil phase (cm3) j Vw volume of water phase (cm3) j Voi volume of oil phase at element i (cm3) Vjwi volume of water phase at element i (cm3) Vini initial volume of both phases (cm3) Vo,ini initial volume of oil phase (cm3) Vw,ini initial volume of water phase (cm3) x distance (cm)

ORCID

Greek Symbols

■

A C Ci Co Cw C*o C*w Cjoi Do Dw erf Jo Jw kpc L

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.6b03729. To simplify the trial-and-error procedure, three correlations were developed to calculate the densities of the water and oil phases and of the partition coefficient during the mass transfer process, which are shown as eqs S3−S5; the effects of the number of subintervals in determination of diffusion coefficients and the determination of optimal diffusion coefficients are also discussed to support and extend the content of this paper (PDF)

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AUTHOR INFORMATION

Corresponding Author

α

Guanli Shu: 0000-0002-5534-4324 K

slope of curve for brine without CO2 (g/(cm3·MPa)) DOI: 10.1021/acs.iecr.6b03729 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article

Industrial & Engineering Chemistry Research β ρ ρws ρp ρ0 ρom ρjmix,oi ΔVj+1 o ΔVj+1 w λ

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slope of density vs concentration (g/mol) density of liquid (g/cm3) density of CO2 + brine solution at any pressure (g/cm3) density of the water phase without dissolution of CO2 concentration at given pressure (g/cm3) intercept of curve for brine without CO2 (g/cm3) density of CO2 + crude oil mixture (g/cm3) density of the mixture of CO2 and oil (g/cm3) volume change of oil phase (cm3) volume change of water phase (cm3) constant solved from transcendental equation

Superscripts

j iteration cycle or trial number Subscripts

i location of each element

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