Energy & Fuels 2006, 20, 2509-2517
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Measurements of Molecular Diffusion Coefficients of Carbon Dioxide, Methane, and Propane in Heavy Oil under Reservoir Conditions Asok Kumar Tharanivasan,† Chaodong Yang,‡ and Yongan Gu*,§ Department of Chemical and Petroleum Engineering, Schulich School of Engineering, UniVersity of Calgary, Calgary, Alberta T2N 1N4, Canada, Computer Modelling Group (CMG) Ltd., Calgary, Alberta T2L 2A6, Canada, and Petroleum Technology Research Centre, Faculty of Engineering, UniVersity of Regina, Regina, Saskatchewan S4S 0A2, Canada ReceiVed February 22, 2006. ReVised Manuscript ReceiVed August 18, 2006
In this paper, the so-called pressure decay method is applied to measure the molecular diffusivities of carbon dioxide, methane, and propane in heavy oil. In the experiment, a gaseous solvent is made in contact with a heavy oil, and thereby, the pressure in the solvent phase versus time data are accurately measured inside a closed high-pressure diffusion cell at a constant temperature while the solvent gradually dissolves into the heavy oil. In terms of the conservation law of mass and the equation of state for a real gas, the pressure in the solvent phase is calculated from the analytical solution to the diffusion equation for such a diffusion process. The equilibrium, quasi-equilibrium, and nonequilibrium boundary conditions are applied at the heavy oilsolvent interface, respectively. The solvent diffusivity in heavy oil is determined by finding the best match of the numerically calculated pressures with the experimentally measured data. It is found that the nonequilibrium boundary condition is the most applicable at the heavy oil-CO2 interface at small diffusion times. In addition, the determined diffusivity for the heavy oil-CH4 system is insensitive to the interface boundary condition. The mass transfer across the heavy oil-C3H8 interface is best described by applying the quasi-equilibrium boundary condition. In particular, a new strategy is adopted to find the equilibrium pressure for each heavy oil-solvent system from its measured solubility versus pressure data. Thus, the diffusion coefficient of each solvent in heavy oil can be determined by measuring the pressure decay in a short duration.
Introduction Western Canada has abundant reserves of heavy oil and bitumen. Effective and economical recovery of these reserves requires significant reduction of their viscosity. Among several oil recovery methods of interest, solvent-based processes, such as vapor extraction (VAPEX) and cyclic solvent injection processes, have shown great potential for enhancing the heavy oil recovery. In general, the solvent-based processes involve the dissolution of solvent into heavy oil by molecular diffusion and convective dispersion.1,2 The solvent gases can be carbon dioxide, flue gas, and light hydrocarbon gases, such as natural gas, methane, ethane, propane, and butane. During the VAPEX process, the vaporized solvents gradually dissolve into heavy oil to significantly reduce its viscosity. Consequently, the solvent-saturated heavy oil is mobile enough to drain down by gravity into the horizontal production well, which is placed below the horizontal injection well. Field design and reservoir simulation of solvent-based oil recovery processes need to determine the rate and amount of a solvent to be injected so as to reach the desired heavy oil viscosity reduction. This requires * To whom correspondence should be addressed. Tel.: 1-306-585-4630. Fax: 1-306-585-4855. E-mail:
[email protected]. † University of Calgary. ‡ Computer Modelling Group (CMG) Ltd.. § University of Regina. (1) Grogan, A. T.; Pinczewski, W. V. The role of molecular diffusion process in tertiary CO2 flooding. J. Pet. Technol. 1987, 39 (5), 591-602. (2) Yazdani J., A.; Maini, B. B. Effect of drainage height and grain size on the convective dispersion in the Vapex process: Experimental study. Presented at 2004 SPE/DOE Symposium on Improved Oil Recovery, Tulsa, OK, April 17-21, 2004; Paper SPE 89409.
the diffusivity and solubility of a solvent in a heavy oil sample under the practical reservoir conditions. Few diffusivity data of heavy oil-solvent systems are available in the literature, though there are numerous solubility data. In the past, several experimental methods have been developed to determine the diffusion coefficient of a gas in heavy oil or bitumen.3-7 Most conventional methods require compositional analyses, which are expensive, time-consuming, and prone to large experimental errors.8 To eliminate this requirement, Riazi9 applied the so-called pressure decay method to measure the diffusion coefficient of methane in n-pentane. This nonintrusive method is based on the fact that the pressure in the gas phase decays as the molecular diffusion of the gas in the liquid proceeds in a closed diffusion cell. Later, the pressure (3) Das, S. K.; Butler, R. M. Diffusion coefficients of propane and butane in Peace River bitumen. Can. J. Chem. Eng. 1996, 74 (6), 985-992. (4) Fu, B. C. H.; Phillips, C. R. New technique for determination of diffusivities of volatile hydrocarbons in semisolid bitumen. Fuel 1979, 58 (8), 557-560. (5) Schmidt, T.; Leshchyshyn, T. H.; Puttagunta, V. R. Diffusivity of carbon dioxide into Athabasca bitumen. Presented at Annual Technical Meeting of the Petroleum Society of CIM, Calgary, AB, Canada, June 6-9, 1982; Paper 82-33-100. (6) Wen, Y.; Bryan, J.; Kantzas, A. Estimation of diffusion coefficients in bitumen solvent mixtures as derived from low field NMR spectra. J. Can. Pet. Technol. 2005, 44 (4), 22-28. (7) 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 (1), 48-57. (8) Upreti, S. R.; Mehrotra, A. K. Diffusivity of CO2, CH4, C2H6, and N2 in Athabasca bitumen. Can. J. Chem. Eng. 2002, 80 (1), 116-125. (9) Riazi, M. R. A new method for experimental measurement of diffusion coefficients in reservoir fluids. J. Pet. Sci. Eng. 1996, 14 (3-4), 235-250.
10.1021/ef060080d CCC: $33.50 © 2006 American Chemical Society Published on Web 10/07/2006
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decay method was applied to study a total of six specific crude oil-solvent systems and bitumen-solvent systems, and some significant improvements were made accordingly.8,10-12 However, there is no consensus on what boundary condition should be applied at a given heavy oil-solvent interface. For example, the heavy oil at the interface can be assumed to be saturated with the solvent under the so-called equilibrium pressure at all times,11 which is termed the equilibrium boundary condition (BC). On the other hand, Riazi9 and Upreti and Mehrotra8,10 assumed that the solvent concentration at the interface is equal to the saturation concentration, i.e., solubility, which varies with the existing pressure in the solvent phase during the measurement. This interface boundary condition is called the quasiequilibrium BC. Recently, Civan and Rasmussen13 applied the nonequilibrium BC at the heavy oil-solvent interface, which considers the interfacial resistance to the mass transfer across the interface. Tharanivasan et al.14 studied the above-mentioned three BCs by using the experimental data available in the literature.11 They have concluded that the nonequilibrium BC is more applicable to the heavy oil-carbon dioxide system, whereas the quasi-equilibrium or equilibrium BC is more suitable for the heavy oil-methane system. In this paper, some technical efforts are made to accurately measure the pressure decay data for the heavy oil-carbon dioxide, heavy oil-methane, and heavy oil-propane systems. The above three BCs are applied to these three heavy oilsolvent systems. For each BC, the diffusion coefficient is determined by finding the minimum objective function, which represents the minimum average pressure difference between the theoretically calculated and experimentally measured pressures at different times. The most suitable BC for each heavy oil-solvent system is found by comparing the minimum objective functions for the three BCs. Also, the effects of these three rather different BCs on the determined diffusion coefficient are studied. In particular, an alternative strategy is developed to determine the diffusivity of a solvent in heavy oil with the measured decaying pressures in a short period if the solubility data of the heavy oil-solvent system are known. Theory Mass Transfer Model. A schematic diagram of a heavy oilsolvent system inside a closed diffusion cell is shown in Figure 1, where the heavy oil-solvent interface is located at x ) L and the bottom of the diffusion cell is located at x ) 0. The diffusion process is governed by the unsteady one-dimensional diffusion equation,15
∂c ∂2c )D 2 ∂t ∂x
(1)
where c is the solvent concentration in heavy oil; x is the distance from the bottom of the diffusion cell; t is time; and D is the diffusion coefficient of the solvent in heavy oil, assuming that it is constant throughout the diffusion process.1,3,7,9,11-13 (10) Upreti, S. R.; Mehrotra, A. K. Experimental measurement of gas diffusivity in bitumen: Results for carbon dioxide. Ind. Eng. Chem. Res. 2000, 39 (4), 1080-1087. (11) Zhang, Y. P.; Hyndman, C. L.; Maini, B. B. Measurement of gas diffusivity in heavy oils. J. Pet. Sci. Eng. 2000, 25 (1-2), 37-47. (12) Sheikha, H.; Pooladi-Darvish, M.; Mehrotra, A. K. Development of graphical methods for estimating the diffusivity coefficient of gases in bitumen from pressure-decay data. Energy Fuels 2005, 19 (5), 2041-2049. (13) Civan, F.; Rasmussen, M. L. Analysis and interpretation of gas diffusion in quiescent reservoir, drilling and completion fluids: Equilibrium vs nonequilibrium models. Presented at SPE Annual Technical Conference and Exhibition, Denver, CO, October 5-8, 2003; Paper SPE 84072.
Figure 1. Schematic of the heavy oil-solvent system inside a closed diffusion cell.
The assumption of constant diffusion coefficient in the solventsaturated heavy oil is reasonable, because the solvent concentration in heavy oil is generally low under the test conditions. In the above diffusion equation, the natural convection is not taken into account because the density of the solvent-enriched oil phase near the heavy oil-solvent interface is lower than that of the heavy oil at the bottom of the diffusion cell. In addition, the oil swelling effect is neglected and only one-way diffusion of the pure solvent in heavy oil is considered, because the latter is nonvolatile under the test conditions. The nonvolatility of heavy oil is verified by undertaking compositional analysis of the solvent phase after the diffusion test. The overall diffusion process is also assumed to be isothermal. At the beginning of the diffusion test, the heavy oil contains no solvent. Thus, the initial condition (IC) is given by
c(x,t)|t)0 ) 0
0exeL
(2)
For the impermeable rigid boundary at the bottom of the diffusion cell, the Neumann boundary condition can be applied:16
|
∂c )0 ∂x x)0
t>0
(3)
The equilibrium boundary condition assumes that the heavy oil-solvent interface is saturated with the solvent under the socalled equilibrium pressure at all times.11 The corresponding Dirichlet boundary condition at the interface is written as
c(x,t)|x)L ) csat(Peq)
t>0
(4a)
Here, Peq is the equilibrium pressure in the solvent phase when the heavy oil is completely saturated with the solvent, and csat(Peq) is the solvent saturation concentration or solubility of the solvent in the heavy oil under the equilibrium pressure Peq. The equilibrium BC indicates that the solvent concentration at the interface remains constant throughout the diffusion test. Therefore, it is a suitable boundary condition only if the pressure decay is small. In the quasi-equilibrium boundary condition,8-10 the heavy oil-solvent interface is assumed to be saturated with the solvent at the existing pressure in the solvent phase. Hence, the (14) Tharanivasan, A. K.; Yang, C.; Gu, Y. Comparison of three different interface mass transfer models used in the experimental measurement of solvent diffusivity in heavy oil. J. Pet. Sci. Eng. 2004, 44 (3-4), 269282. (15) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena, second ed.; John Wiley & Sons: New York, 2002. (16) Crank, J. The Mathematics of Diffusion, second ed.; Clarendon Press: Oxford, U.K., 1975.
Molecular Diffusion Coefficients of CO2, Methane, & Propane ∞
respective Dirichlet boundary condition is expressed as
c(x,t)|x)L ) csat[P(t)]
C(X,τ) )
t>0
|
∂c ) k[csat(Peq) - c(x,t)|x)L] ∂x x)L
t>0
(4c)
where k is the mass-transfer coefficient at the heavy oil-solvent interface and 1/k represents the interfacial resistance to the mass transfer across the interface. It is worthwhile to point out that the nonequilibrium BC expressed by the above equation approaches the equilibrium BC in eq 4a if the mass-transfer coefficient is sufficiently large. Similar to the equilibrium BC, the nonequilibrium BC is more applicable when the pressure decay is smaller. Equations 1-4c can be nondimensionalized by introducing the following dimensionless variables and parameter:
C)
t kL c x , X ) , τ ) 2 , kD ) L D csat(Peq) L /D
(5)
Here, τ is the dimensionless time, which is also called the masstransfer Fourier number; and kD is referred to as the masstransfer Biot number. If the equilibrium BC in eq 4a is applied at the heavy oilsolvent interface, the analytical solution to the diffusion equation in eq 1 subject to eqs 2 and 3 can be expressed in the following dimensionless form:16
∞
C(X,τ) ) 1 - 2
∑(-1)n n)0
cos
[
(2n + 1)π X 2
(2n + 1)π
]
e-[(2n+1)π/2] τ (6) 2
2 To obtain a semianalytical solution to the diffusion equation in eq 1 subject to eqs 2, 3, and 4b, it is assumed that the solvent saturation concentration in heavy oil is proportional to the existing pressure in the solvent phase and that the measured decaying pressures can be explicitly expressed as an exponential function of time,11
P(t) ) m1 e-t/k1 + m2 e-t/k2 + Peq
{
(4b)
where P(t) is the existing pressure in the solvent phase, which decays with time during the diffusion test until the equilibrium pressure Peq is reached. According to the nonequilibrium boundary condition,13 the solvent mass-transfer flux across the interface is proportional to the difference between the solvent saturation concentration under the equilibrium pressure and the existing solvent concentration at the interface. The corresponding Robin boundary condition is given by
D
Energy & Fuels, Vol. 20, No. 6, 2006 2511
(7)
where the specific values of m1, m2, k1, and k2 for a given heavy oil-solvent system can be found by using the nonlinear regression. With the above two assumptions, the semianalytical solution to eq 1 can be derived from the general solution given by Carslaw and Jaeger,17 (17) Carslaw, H. S.; Jaeger, J. C. Conduction of Heat in Solids; Clarendon Press: Oxford, U.K., 1959.
∑(-1)n(2n + 1)π cos n)0 4κ1R
[ [
(2n + 1) π κ1 - 4 4κ2β 2 2
(2n + 1) π κ2 - 4 2 2
[
]
(2n + 1)π X × 2
] ]
e-τ/k1 - e-(2n+1) π /4τ + 2 2
e-τ/κ2 - e-(2n+1) π /4τ + 2 2
4 (2n + 1) π
2 2
[
1 - e-(2n+1) π /4τ 2 2
]}
(8)
where
R)
m2 k1 k2 m1 , β ) , κ1 ) 2 , κ 2 ) 2 Peq Peq L /D L /D
are dimensionless parameters. For the nonequilibrium BC in eq 4c at the heavy oil-solvent interface, the specific solution to the diffusion equation in eq 1 was given by Walas,18 ∞
C(X,τ) ) 1 - 2
∑ n)1(λ
sin λn n
+ sin λn cos λn)
cos(λnX) e-λn τ (9) 2
where λn, n ) 1, 2, 3, ..., represent the eigen values, which are the positive roots of the characteristic equation, tan λn ) kD/λn. Calculated Pressure Decay Curve. The existing pressure Pcal(t) in the solvent phase at any time can be calculated by using an equation of state (EOS) for a real gas.
Pcal(t)Vsolvent ) Z(Pcal(t),T)Ng(t)RT ) Z(Pcal(t),T)[Ni - No(t)]RT (10a) where Vsolvent is the volume of the solvent phase in the diffusion cell; Z(Pcal(t),T) is the Z-factor19 at Pcal(t) and T; Ng(t) represents the number of moles of solvent remaining in the gaseous solvent phase; R is the universal gas constant; T is the absolute temperature; No(t) is the number of moles of solvent dissolved into heavy oil at any time, which is obtained by numerically integrating the transient solvent concentration distribution in the heavy oil; and Ni represents the initial total number of moles of solvent in the closed diffusion cell, which can be calculated from the prespecified initial pressure Pi by applying the EOS for the solvent phase, eq 10a, at the beginning of the diffusion test,
PiVsolvent ) Z(Pi,T)NiRT
(10b)
where Z(Pi,T) is the Z-factor at Pi and T. Determination of Equilibrium Pressure Peq. In the abovedescribed mass-transfer model, the equilibrium pressure Peq is required to determine the solvent saturation concentration csat(Peq) in heavy oil. Ideally, the equilibrium pressure can be measured when the heavy oil is completely saturated with the solvent. In practice, such a slow diffusion process may take an unreasonably long time to reach the equilibrium state. Alternatively, in this study, the equilibrium pressure Peq is calculated (18) Walas, S. M. Modeling with Differential Equations in Chemical Engineering; Butterworth-Heinemann: Boston, MA, 1991. (19) Lee, B. K.; Kesler, M. G. A generalized thermodynamic correlation based on three-parameter corresponding states. AIChE J. 1975, 21 (3), 510527.
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Figure 2. Schematic diagram of the experimental setup used to measure the diffusion coefficient of the heavy oil-solvent system.
from the measured solubility versus pressure data for the same heavy oil-solvent system. At the equilibrium state, the EOS for the solvent phase, eq 10a, can be rewritten as
PeqVsolvent ) Z(Peq,T)NeqRT )
[
Z(Peq,T) Ni -
]
χ(Peq,T) moil RT (10c) 100 Msolvent
where Z(Peq,T) is the Z-factor and χ(Peq,T) is the solubility of the solvent in heavy oil in weight percentage at Peq and T; Neq denotes the number of moles of solvent remaining in the gaseous solvent phase at the equilibrium pressure Peq; moil is the mass of heavy oil inside the closed diffusion cell; and Msolvent is the molecular weight of solvent tested. In the experiment, Vsolvent, Msolvent, moil, and T are known. In this study, χ(Peq,T) at Peq and T is found by interpolating the measured solubility versus pressure data for the given heavy oil-solvent system. Numerical Optimization. After the pressure versus time curve, i.e., Pcal(t), is calculated from eq 10a for each of the three BCs, the true diffusion coefficient D can be determined by finding the best match between the calculated pressure-time data Pcal(t) and the measured pressure-time data Pexp(t). In this study, the so-called history-matching technique is used to minimize an objective function ∆Pave, which is defined as20
∆Pave )
x
m
|Pcal(ti) - Pexp(ti)|2 ∑ i)1 m
(11)
Here, Pcal(ti) are the calculated pressures and Pexp(ti) are the measured pressures at any time ti, i ) 1, 2, 3, ...., m. Physically, this objective function represents the average pressure difference between the theoretically calculated and experimentally measured pressures. The detailed numerical optimization procedure can be found elsewhere.14,20 Experimental Section Materials. The heavy oil sample is collected from the Lloydminster area, Canada. The density and viscosity of heavy
oil are 988 kg/m3 and 20 267 mPa.s at 1 atm and 23.9 °C, respectively. Carbon dioxide, methane, and propane have the purity of 99.99%, 99.97%, and 99.5%, respectively. Nitrogen is used for drying and leakage testing, and its purity is equal to 99.998%. These gases are purchased from Praxair, U.S.A.. Apparatus. Figure 2 shows a schematic diagram of the experimental setup used to measure the decaying pressures as the solvent gradually dissolves into heavy oil inside a closed diffusion cell at a constant temperature. The apparatus comprises a stainless steel cylindrical diffusion cell (CYL-0250-10-NP316-2, DBR, Canada) of 5.4 cm inner diameter and 16.0 cm length. The maximum operating pressure and temperature of the diffusion cell are 69 MPa and 200 °C, respectively. The top port of the diffusion cell is connected to the solvent supply cylinder through a needle valve (SS-0VS2, Swagelok, U.S.A.), whereas the bottom port is connected to a heavy oil transfer cylinder. The heavy oil sample is introduced into the diffusion cell by using a positive displacement pump (PMP-500-1-10HB-316-MO-CO, DBR, Canada). The bottom port is also connected to a needle valve for evacuating the diffusion cell before the experiment or draining the heavy oil sample after the experiment. In addition to the high-pressure diffusion cell, another major component in the experimental setup is a high-precision Digiquartz pressure transducer (46KR-101, Paroscientific, Inc., U.S.A.). This pressure transducer is used to measure the absolute pressure in the solvent phase inside the closed diffusion cell with an accuracy of 0.01% at its full scale of 41 MPa. It is carefully tested and accurately calibrated by the manufacturer prior to shipping. On-line data acquisition and real-time pressure display are made available through an interface board together with the pertinent software, which are purchased from the manufacturer. The pressure transducer also has a built-in function to automatically display and record the temperature of the solvent phase in a personal computer. To maintain a constant temperature condition throughout the experiment, the whole experimental setup is placed inside an (20) Tharanivasan, A. K. Measurements of Molecular Diffusion Coefficients of Carbon Dioxide, Methane and Propane in Heavy Oil under Reservoir Conditions. Master Thesis, Petroleum Systems Engineering, Faculty of Engineering, University of Regina, Canada, 2004.
Molecular Diffusion Coefficients of CO2, Methane, & Propane
air bath. A temperature controller (Standard-89000-00, ColeParmer Instrument Company, Canada) is used to control the heat supply from an electric heater. Temperature feedback to the temperature controller is obtained from a thermocouple (Type-T-08500-96, Cole-Parmer Instrument Company, Canada) inside the air bath. Two fans are used to circulate the air inside the air bath. Pressure Decay Measurement. Prior to the experiments, the diffusion cell and all the connections are pressurized with nitrogen and tested for leakage several times. The leakage test is conducted by monitoring the pressure in the pressurized cell and connections for at least 24 h. The following experimental procedure is implemented for recording the decaying pressures due to the molecular diffusion. First, the diffusion cell is cleaned and evacuated. The temperature controller, fans, and electric heater are turned on for the temperature maintenance. Then the diffusion cell is purged three times and filled with the test solvent to reach a certain pressure, which is estimated from the initial pressure in the solvent phase to be tested and the volume of heavy oil to be injected. After the diffusion cell is filled with the solvent, it is tested for any leakage for 24 h and its temperature is allowed to reach an equilibrium value. Finally, a certain amount of heavy oil sample is injected from the heavy oil transfer cylinder into the diffusion cell by using the positive displacement pump. As the solvent gradually dissolves into heavy oil, the pressure and temperature data inside the diffusion cell are acquired, displayed, and stored in the personal computer. These experimental data are recorded once every 2 min during each pressure decay measurement. The data acquisition process continues until the daily pressure decay is within the accuracy of the pressure transducer, i.e., ∼4 kPa/day. In this study, the pressure decay tests for three different solvents, CO2, CH4, and C3H8, are conducted at a constant temperature of T ) 23.9 °C. The percentage of heavy oil volume in the diffusion cell is equal to 27.2%, 16.4%, and 27.4% for the heavy oil-CO2, heavy oil-CH4, and heavy oil-C3H8 systems, respectively. After the completion of each test, gas chromatographic analysis of the remaining solvent phase is performed to determine whether there are any hydrocarbon gases that are extracted from the heavy oil sample. It has been found that there are no hydrocarbon gases (i.e., no light-ends extraction) in the heavy oil-CO2 system and that there are no other hydrocarbon gases in the heavy oil-CH4 and heavy oil-C3H8 systems. These findings support the assumption of nonvolatility of the heavy oil under the test conditions, i.e., there is only one-way molecular diffusion from the pure solvent phase to the heavy oil, which is made in the mass-transfer modeling. Solubility Measurement. The solubilities of carbon dioxide, methane, and propane in heavy oil are measured by saturating heavy oil with each solvent inside a see-through, windowed high-pressure cell (P/N 2329-800, Ruska Fluid Products, Chandler Engineering, U.S.A.). The detailed technical description of the experimental setup can be found elsewhere.20 The experimental procedure is briefly summarized below. First, after the pressure cell is thoroughly cleaned, it is filled with a test solvent to reach a prespecified initial pressure. The initial cell pressure P0 and the solvent volume V0 are recorded before the heavy oil sample is introduced. Then the heavy oil sample is injected slowly from the top port into the pressure cell by using a syringe pump at a constant flow rate of 0.005 cm3/min. The slow injection process of heavy oil takes place by forming pendant oil drops continuously at the tip of 1/8 in. stainless steel tubing inside the pressure cell. Formation of the pendant oil drops at such a slow rate accelerates the mass transfer of the
Energy & Fuels, Vol. 20, No. 6, 2006 2513
solvent in heavy oil by creating a large contact area. After the specified amount of heavy oil is injected into the pressure cell, the syringe pump is stopped and the pressure inside the pressure cell is recorded and monitored continuously. Eventually, the heavy oil is completely saturated with the solvent and no pressure change is observed within 2-3 days. The corresponding pressure in the solvent phase, Pf, and the solvent volume, Vf, are recorded. The conservation law of solvent mass in the closed pressure cell is applied to compute the solubility of the solvent in heavy oil,
χ(Pf,T) )
[
]
Msolvent P0V0 PfVf × 100 RTmoil Z0(P0,T) Zf(Pf,T)
(12)
where Z0(P0, T) and Zf(Pf, T) are the Z-factors at the initial cell pressure P0 and the final saturation pressure Pf at constant temperature T, respectively. In eq 12, the measured solvent solubility χ(Pf,T) at the final saturation pressure is expressed as the weight percentage of the heavy oil prior to solvent dissolution. Results and Discussion Solubility Data and Equilibrium Pressures. The measured solubility data for the heavy oil-CO2 system, heavy oil-CH4 system, and heavy oil-C3H8 system at T ) 23.9 °C are plotted in parts a-c of Figure 3, respectively. Linear correlations of the measured solubility versus pressure data are also shown in these figures. The large correlation coefficients indicate good linearity between the measured solubility and pressure in the pressure ranges tested. In the literature, this linearity is also reported for other heavy oil/bitumen-CO2 systems,21-23 heavy oil/bitumen-CH4 systems,24,25 and heavy oil/bitumen-C3H8 systems.26 Furthermore, comparison of the measured solubility data in parts a and b of Figure 3 shows that the solubility of carbon dioxide in the heavy oil is ∼1 order of magnitude higher than that of methane in the heavy oil. Therefore, much more carbon dioxide can dissolve into the heavy oil than methane under the same reservoir pressure and temperature conditions. On the other hand, it is also noted from Figure 3c that the solubility of propane in the heavy oil sample is quite high, even at a much lower pressure. In principle, solubility of a solvent in a heavy oil sample can be related to its critical temperature and normal boiling point. It has been stated that a gas with a higher critical temperature and a higher normal boiling point is more soluble in hydrocarbon liquids.27 The critical temperatures of carbon dioxide, methane, and propane are equal to 31.06, (21) Mehrotra, A. K.; Sarkar, M.; Svrcek, W. Y. Bitumen density and gas solubility predictions using the Peng-Robinson equation of state. AOSTRA J. Res. 1985, 1 (4), 215-229. (22) Mehrotra, A. K.; Svrcek, W. Y.; Helper, L. G.; Xu, Y.; Fu, C. T. Comments on solubility of carbon dioxide in tar sand bitumen: Experimental determination and modeling. Ind. Eng. Chem. Res. 1992, 31 (5), 14221423. (23) Quail, B.; Hill, G. A.; Jha, K. N. Correlations of viscosity, gas solubility, and density for Saskatchewan heavy oils. Ind. Eng. Chem. Res. 1988, 27 (3), 519-523. (24) Mehrotra, A. K.; Svrcek, W. Y. Viscosity, density and gas solubility data for oil sand bitumens. Part III: Wabasca bitumen saturated with N2, CO, CO2 and C2H6. AOSTRA J. Res. 1985, 2 (2), 83-93. (25) Svrcek, W. Y.; Mehrotra, A. K. Gas solubility, viscosity and density measurements for Athabasca bitumen. J. Can. Pet. Technol. 1982, 21 (4), 31-38. (26) Frauenfeld, T. W. J.; Kissel, G.; Zhou, S. PVT and viscosity measurements for Lloydminster-Aberfeldy and Cold Lake blended oil systems. Presented at SPE International Thermal Operations and Heavy Oil Symposium and International Horizontal Well Technology Conference, Calgary, AB, Canada, Nov 4-7, 2002; Paper SPE/Petroleum Society of CIM/CHOA 79018.
2514 Energy & Fuels, Vol. 20, No. 6, 2006
Figure 3. (a) Measured solubility of CO2 in heavy oil as a function of pressure at T ) 23.9 °C. (b) Measured solubility of CH4 in heavy oil as a function of pressure at T ) 23.9 °C. (c) Measured solubility of C3H8 in heavy oil as a function of pressure at T ) 23.9 °C.
-82.50, and 96.70 °C, respectively, and their respective normal boiling points are equal to -78.20, -161.49, and -42.04 °C.28 Thus, propane has the highest solubility in the heavy oil in comparison with carbon dioxide and methane. As a first step in the diffusivity determination, the equilibrium pressures Peq for the heavy oil-carbon dioxide, heavy oil(27) 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 (2), 24-31. (28) Whitson, C. H.; Brule, M. R. Phase BehaVior; Society of Petroleum Engineers: Richardson, TX, 2000.
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methane, and heavy oil-propane systems are calculated from their measured solubility data at T ) 23.9 °C. This calculation is carried out by using eq 10c and the linear correlations of the measured solubility versus pressure data in Figure 3 parts a-c. The calculated equilibrium pressures and solubilities are shown in parts a-c of Figure 3 for the three heavy oil-solvent systems, respectively. With the determined equilibrium pressure for each heavy oil-solvent system, the number of moles of solvent remaining in the gaseous solvent phase, Neq, at the equilibrium pressure, Peq, is also computed from eq 10c. Then in terms of conservation of solvent mass in the closed diffusion cell, the solvent saturation concentration in heavy oil is equal to csat(Peq) ) (Ni - Neq)/Voil. It is worthwhile to note that, though in this study the pressure decay test for each heavy oil-solvent system lasts for a sufficiently long period, the termination pressures of Pt ) 3 530.0 kPa for the heavy oil-CO2 system, Pt ) 4 918.0 kPa for the heavy oil-CH4 system, and Pt ) 393.3 kPa for the heavy oil-C3H8 system are still slightly higher than their corresponding equilibrium pressures, Peq ) 3 480.7, 4 901.1, and 381.4 kPa, respectively. Measured Pressure Decay Curves. The measured pressure decay data for the heavy oil-carbon dioxide, heavy oilmethane, and heavy oil-propane systems at T ) 23.9 °C are plotted in parts a-c of Figure 4, respectively. These figures also include the best-fit curves of the experimental data at different diffusion times, t ) 5, 10, and 20 days, respectively. Here, the diffusion time is defined as the duration in which the measured experimental data are used to find the best-fit curve. In addition, the best-fit curve of the complete experimental data is also plotted at the respective termination time of 37.3 days for the heavy oil-CO2 system in Figure 4a, 23.8 days for the heavy oil-CH4 system in Figure 4b, and 23.5 days for the heavy oil-C3H8 system in Figure 4c. Here, eq 7 with the obtained respective equilibrium pressure, Peq, is used to find the best-fit curve of the measured pressure decay data for each heavy oilsolvent system at a different diffusion time. Parts a-c of Figure 4 show that the correlation coefficients for all the best-fit curves are close to unity. This fact indicates that the expression for pressure as an exponential function of time given in eq 7 is an excellent approximation. Heavy Oil-Carbon Dioxide System. The diffusion coefficients of carbon dioxide in heavy oil at four different diffusion times for each of the three different BCs are given in Table 1. It is noted from this table that, in general, the diffusivity slightly decreases with the diffusion time for each BC. The variations of the diffusivity with the diffusion time can be attributed to the changes of the heavy oil properties due to the gradual solvent dissolution.8 It is also found that the diffusion coefficients of carbon dioxide in the Lloydminster heavy oil tested in this study are larger than those of 0.12-0.22 × 10-9 m2/s in the Athabasca bitumen.8,10 This is because the Lloydminster heavy oil has a much lower viscosity than the Athabasca bitumen.29 When the minimum objective functions ∆Pave given in Table 1 for three different BCs at the diffusion time of t ) 5 days are compared, the nonequilibrium BC gives the best history matching of the calculated pressures with the measured data. Also, if the nonequilibrium BC is applied, the mass-transfer Biot number kD increases with the diffusion time. This means that the interfacial resistance to the mass transfer across the heavy oil-solvent interface, 1/k, is relatively important only at the early diffusion stage. At large diffusion times, t ) 10, 20, (29) Hayduk, W.; Cheng, S. C. Review of relation between diffusivity and solvent viscosity in dilute liquid solutions. Chem. Eng. Sci. 1971, 26 (5), 635-646.
Molecular Diffusion Coefficients of CO2, Methane, & Propane
Energy & Fuels, Vol. 20, No. 6, 2006 2515 Table 1. Diffusivities of CO2 in Heavy Oil at Different Diffusion Times at T ) 23.9 °C diffusion time (day)
boundary condition
D (10-9 m2/s)
5
equilibrium quasi-equilibrium nonequilibrium equilibrium quasi-equilibrium nonequilibrium equilibrium quasi-equilibrium nonequilibrium equilibrium quasi-equilibrium nonequilibrium
0.72 0.52 0.94 0.67 0.53 0.70 0.59 0.49 0.62 0.56 0.46 0.57
10 20 37.3
Figure 4. (a) Measured pressure decay data and the best-fit curves at different diffusion times for the heavy oil-CO2 system at T ) 23.9 °C. (b) Measured pressure decay data and the best-fit curves at different diffusion times for the heavy oil-CH4 system at T ) 23.9 °C. (c) Measured pressure decay data and the best-fit curves at different diffusion times for the heavy oil-C3H8 system at T ) 23.9 °C.
and 37.3 days, in terms of the minimum ∆Pave, nevertheless, the quasi-equilibrium BC gives the best history matching of the calculated pressures with the measured data in comparison with the other two BCs. This suggests that the solvent concentration at the heavy oil-CO2 interface varies with the existing pressure in the CO2 phase and satisfies the quasiequilibrium BC at large diffusion times.
kD
19.20 >80 >100 >100
∆Pave (kPa) 11.0 11.8 6.5 11.7 10.7 300
∆Pave (kPa) 23.6 16.1 23.7 28.8 22.0
