Coupling heat and mass transfer for a gas mixture–heavy oil system at high pressures and elevated temperatures

International Journal of Heat and Mass Transfer 74 (2014) 173–184

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Coupling heat and mass transfer for a gas mixture–heavy oil system at high pressures and elevated temperatures Huijuan Sun, Huazhou Li 1, Daoyong Yang ⇑ Petroleum Systems Engineering, Faculty of Engineering and Applied Science, University of Regina, Regina S4S 0A2, Canada

a r t i c l e

i n f o

Article history: Received 29 October 2013 Received in revised form 3 March 2014 Accepted 3 March 2014 Available online 2 April 2014 Keywords: Heat transfer Mass transfer Gas mixture Heavy oil Diffusion coefficient Swelling effect

a b s t r a c t A generalized methodology has been developed to couple heat and mass transfer of a gas–heavy oil system and a gas mixture–heavy oil system at high pressures and elevated temperatures. Theoretically, the Peng–Robinson equation of state (PR EOS) incorporating with a one-way heat and mass transfer model has been developed to couple heat and mass transfer from either a hot gas or a hot gas mixture into heavy oil. Experimentally, diffusion tests have been conducted with a PVT setup for a hot CO2–heavy oil system and a hot C3H8–CO2–heavy oil system under a constant pressure, respectively. Both the gas-phase volume and liquid-phase swelling effect are simultaneously recorded during the measurement. The gas chromatography method is employed to measure compositions of the C3H8–CO2 mixture at the beginning of the diffusion measurement. The heat transfer is found to proceed faster than mass transfer, leading to that the thermal equilibrium is achieved more quickly than mass equilibrium. The heavy oil expands rapidly at the initial stage of the coupled heat and mass transfer, and then swells gradually in the subsequent stage of mass transfer. The diffusion coefficient is determined by minimizing the discrepancy between the measured and calculated swelling factors of heavy oil during the diffusion tests. Adding C3H8 to CO2 stream is found to not only improve mass diffusion, but also accelerate the heat diffusion and consequently an enhanced swelling effect of heavy oil. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Adding alkane solvent(s) (e.g., C3H8 or n-C4H10) into CO2 stream has been found to be a promising method for enhancing heavy oil recovery through an immiscible process [1–3]. Meanwhile, the hot solvent injection technique has attracted considerable attentions since it is particularly effective for improving heavy oil recovery due to the incremental benefits of heat transfer together with mass transfer from solvent(s) into heavy oil. It has been well-recognized that a steam–solvent process enhances oil mobility and thus a higher oil production rate compared to the steam-only process [4,5]. Hot solvent-enriched CO2 flooding might be able to significantly further improve heavy oil recovery by combining advantages of both enriched CO2 flooding process and hot solvent injection process. Three distinct benefits arise from the hot solvent-enriched CO2 flooding process. Firstly, the existence of CO2 in the injected gas mixture is able to maintain a high reservoir pressure, which is critically important for developing heavy oil reservoirs [6]. Secondly, the solvent-enriched CO2 stream has a much ⇑ Corresponding author. Tel.: +1 306 337 2660; fax: +1 306 585 4855. 1

E-mail address: [email protected] (D. Yang). Now with the University of Alberta.

http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.03.004 0017-9310/Ó 2014 Elsevier Ltd. All rights reserved.

higher solubility in heavy oil compared to pure CO2, resulting in enhanced viscosity reduction and swelling effect of heavy oil [7]. Thirdly, the heat transfer from gas mixture to heavy oil contributes to the further reduction of heavy oil viscosity [8]. Therefore, it is of fundamental and practical importance to study the coupled heat and mass transfer of solvent–CO2–heavy oil system so that the underlying mechanisms associated with dissolution of hot solvent(s)–CO2 mixture into heavy oil can be better understood and quantified. Numerous efforts have been made to study mass transfer from either a gas [9–12] or a gas mixture [13–15] into heavy oil. Fick’s diffusion law is generally employed to describe the mass transfer process, while the volume change of heavy oil is usually considered due to the dissolution of a gas solvent. As a critical parameter in the diffusion equation, diffusion coefficient is generally determined by matching a theoretically calculated parameter to its experimentally measured value. The diffusion coefficient of a solvent has been assumed to be a constant [11,12,14,15], a function of the concentration of solvent in heavy oil [10,16] or viscosity of heavy oil [17,18], and even a multi-parameter equation associated with pressure, temperature, and viscosity of liquid [19], respectively. Since the convective mixing due to a high initial mass transfer rate together with surface tension-driven instability plays an important

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Nomenclature A a ac b ci ci,sat cp,g, cp,l Di H H0 H⁄ MWi m mg,i Nexp n ng nc O(SF) p pc R SE Sm ST SFX SF ical SF iexp SG Tc Tg, Tl Tl,0 Tr

cross-sectional area of the PVT cell, cm2 attraction parameter in the PR EOS model, kPa m3/kmol factor in correlation of attraction parameter in the PR EOS model van der Waals volume, m3/kmol concentration of ith solvent in heavy oil, mol/m3 saturated concentration of ith solvent in heavy oil, mol/m3 heat capacity of gas and liquid phase, J/(kg K) diffusion coefficient of ith solvent, m2/s updated height of heavy oil, cm initial height of heavy oil, cm updated height of heavy oil at last time step t  Dt, cm molecular weight of ith component, g/mol the mth node along the diffusion direction mass of ith component in gas phase, kg number of data obtained from the experiment the nth time step during the numerical calculation number of gas component number of components in the diffusion system objective function pressure, kPa critical pressure, kPa universal gas constant, kPa m3/(K kmol) energy input to the gas phase from the environment, W/m3 mass loss due to the mass transfer from gas phase to liquid phase, kg/s energy loss due to the heat transfer from gas phase to liquid phase, W/m3 swelling factor calculated swelling factor of the ith data point measured swelling factor of the ith data point specific gravity critical temperature, K temperature of gas and liquid phase, K initial temperature of liquid phase, K reduced temperature, K

role at the early stage of diffusion [10], diffusion coefficient may be underestimated at the initial stage of the mass transfer process if it is assumed to be constant. Consequently, it may be more reasonable to assume that diffusion coefficient is dependent on the solvent concentration and/or viscosity of the liquid, though its accuracy relies on the forms of representation functions. A hot solvent such as methane, ethane, propane, butane, and their combinations have been tested to examine their effectiveness for enhancing heavy oil recovery. Experimentally, effects of temperature, gravity, permeability, pore size, asphaltene precipitation, pressure, and solvent type on heavy oil recovery have been extensively studied [5,8,20–24]. Theoretically, thermal reservoir models have been developed for numerically simulating steam injection and other thermal processes [25–27]. The swelling of heavy oil contributes to an increase in mobile oil saturation as well as an increase in relative permeability of heavy oil, and hence heavy oil recovery is improved [28,29]. Nonetheless, few attempts have been made to develop a generalized framework for accurately not only quantifying the swelling effect, but also determining the diffusion coefficient due to the coupled heat and mass transfer. In this study, a generalized methodology has been developed to couple heat and mass transfer from a gas and its mixture to heavy oil at a constant pressure. Theoretically, the Peng–Robinson Equation of State (PR EOS) [30] incorporating a coupled heat and mass transfer model has been developed to describe the heat and mass

t V Vcorrected V1 V2 xi, xj y yg ZRA

time, s molar volume, m3/kmol corrected molar volume, m3/kmol molar volume of heavy oil at test pressure and initial temperature, m3/kmol molar volume of solvent-diluted heavy oil at test pressure and temperature, m3/kmol composition of the ith and jth component in liquid phase, respectively, mole fraction coordinate direction of liquid phase, cm coordinate direction of gas phase, cm Rackett parameter

Greek symbols a alpha function in PR EOS model, constant in Eq. (12) b constant in Eq. (12) Dt time interval Dy space interval d BIP matrix k thermal conductivity, W/(m K) l dynamic viscosity, cP m kinematic viscosity, m2/s q density, kg/m3 s dimensionless time / volume fraction x acentric factor Subscripts 0 initial state cal calculated exp experimental g gas l liquid o heavy oil s solvent sat saturated

transfer from a gas or a gas mixture to heavy oil at high pressure and elevated temperatures. Experimentally, PVT tests are conducted to measure the decaying volume of gas phase and the swelling volume of liquid phase during the heat and mass transfer for a hot CO2–heavy oil system and a hot C3H8–CO2–heavy oil system under a constant pressure, respectively. The diffusion coefficient of the gas or gas mixture in heavy oil is determined once the discrepancy between the measured and calculated swelling factor of heavy oil during the diffusion measurement has been minimized. Also, the temperature profile and concentration profile can be correspondingly determined. 2. Mathematical formulations 2.1. PR EOS model Due to its wide application in the petroleum and chemical industries, the PR EOS is chosen as the equation of state for describing phase behavior of solvent(s)–CO2–heavy oil systems. The PR EOS model can be described as [30,31],

p¼

RT a  V  b VðV þ bÞ þ bðV  bÞ

a ¼ ac aðT r ; xÞ

ð1Þ ð2aÞ

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ac ¼

0:457235R2 T 2c pc

ð2bÞ

0:0777969RT c b¼ pc

ð2cÞ

where p is pressure, pc is critical pressure, R is universal gas constant, T is temperature, Tc is critical temperature, Tr is reduced temperature, V is molar volume, x is acentric factor, and a(Tr, x) is the so-called alpha function. To more accurately model the phase behavior of the solvents– CO2–heavy oil systems, the following modified alpha function [32] is used in the PR EOS model, (

a ¼ exp ð0:13280  0:05052x þ 0:25948x2 Þð1  T r Þ h

pffiffiffiffiffi i2 þ 0:81769 ln 1 þ ð0:31355 þ 1:86745x  0:52604x2 Þð1  T r Þ

)

ð3Þ

For a mixture system, the van der Waals mixing rule is applied to calculate a and b, nc X nc X pffiffiffiffiffiffiffiffi a¼ xi xj ð1  dij Þ ai aj

ð4aÞ

i¼1 j¼1

b¼

nc X xj b j

ð4bÞ

2.2. Mass and energy conservation Fig. 1 shows a schematic of the coupled heat and mass transfer between gas solvents and heavy oil in a closed PVT cell. Three reasonable assumptions are made as follows: (1) one-dimensional and one-way heat and mass transfer [10,11]; (2) the equilibrium boundary condition exists at the gas–liquid interface at a constant pressure [10,34]; and (3) the gas solvent exists in liquid state when dissolved in the heavy oil [2]. The mathematical model includes the conservation equations for gas phase and liquid phase as well as the PR EOS model used to perform the two-phase flash calculation and solve the molar volume of liquid phase as a function of time. 2.2.1. Gas phase 2.2.1.1. Mass conservation. The equation for mass conservation of gas phase obeys the following form,

@

Png

i¼1 mg;i

@t

 ¼ Sm

ð8Þ

where mg,i is mass of the ith component in gas phase, ng is number of the gas components, t is time, and Sm is the mass loss due to the mass transfer from gas phase to liquid phase, i.e., heavy oil. Once the composition of gas is obtained from Eq. (8), the PR EOS model can be performed to calculate the volume of gas phase as a function of time.

j¼1

where nc is number of the components in the system, xi and xj are the molar fraction of ith and jth component in the mixture, respectively, and dij is binary interaction parameter (BIP) between the ith component and the jth component. The BIP matrix for the C3H8–CO2–heavy oil system used in this study is shown in Table 1, where d13 and d23 can be calculated by [2],

d13 ¼ 0:5462T=T c  0:4596SG  0:0238x þ 0:7523

ð5aÞ

d23 ¼ 0:4560T=T c þ 0:1817

ð5bÞ

where Tc, SG and x are the critical temperature, specific gravity and acentric factor of heavy oil, respectively. The experimental phase behavior data for CO2–heavy oil system and C3H8–heavy oil system have been used for obtaining the BIP correlations, which are also validated with the experimentally measured swelling factor [7]. A volume-translation strategy is then employed to correct the molar volume obtained from Eq. (1) [31],

V corr ¼ V 

nc X xj V j

2.2.1.2. Energy conservation. The energy conservation of gas phase can be expressed by,

@ðqg cp;g T g Þ @ ¼ @t @yg

@T g kg @yg

ð6Þ

  RT c ð0:29441  Z RA Þ V  ¼ 0:40768 Pc

þ ST þ SE

ð9Þ

where cp,g, qg, and kg are heat capacity, density, and thermal conductivity of the gas phase, respectively. SE is the energy input to the gas phase, ST is the energy loss due to the heat transfer from gas phase to liquid phase, Tg is temperature of the gas phase and yg is the distance from the gas–liquid interface of the diffusion cell (see Fig. 1). In this study, the energy loss ST can be compensated by the energy input SE, leading to ST þ SE ¼ 0. In addition, the temperature of gas phase can be assumed uniform due to the fast heat con@T duction in gas phase, resulting in @ygg ¼ 0. Therefore, Eq. (9) can be simplified as,

Gas p, Tg

j¼1

where Vcorr is the corrected molar volume and V j is a volume correction of the jth component. The volume correction of a pure component is given by [31],

!

Heat and mass transfer yg

ð7Þ

where ZRA is Rackett parameter. In particular, the Rackett parameter of a heavy oil sample is relative with its molar weight and specific density [33].

y=H at t y=H0 at t0 Heavy oil Tl

y Heat transfer T=Tg

Table 1 BIP matrix for C3H8–CO2–heavy oil system used in the PR EOS model. Component

CO2

C3H8

Heavy oil

CO2 C3H8 Heavy oil

0 0.13500 d31

0.13500 0 d32

d13 d23 0

o Fig. 1. Schematic of a solvent(s)–heavy oil system in a closed diffusion cell for heat and mass transfer.

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@ðqg cp;g T g Þ ¼0 @t

ð10Þ

  @ci @ @ci Di ¼ @y @t @y

ð11Þ

where ci is concentration of the ith solvent component in heavy oil in mol/m3, Di is mass diffusion coefficient of the ith solvent, and y is the distance from the bottom of diffusion cell (see Fig. 1). During heat and mass transfer between the gas mixture and heavy oil, temperature in the liquid phase increases until temperature reaches its equilibrium value, while its viscosity reduces. According to Hayduk and Cheng [17], the diffusion coefficient shows an exponential relationship with viscosity of the gas–diluted heavy oil,

D ¼ alb l

ð12Þ

where a and b are to-be-determined constants which are dependent on the gas solvent, ll is viscosity of the gas–diluted heavy oil. The viscosity can be determined by using the Lobe’s mixing rule [35],

v l ¼ /s v s expð/o ao Þ þ /o v o expð/s as Þ

ð13Þ

where

as ¼ 1:7 ln



vo vs





vo vs

  0:5 vo þ 1:3 ln

vs

ð14aÞ

ð14bÞ

2.2.2.2. Energy conservation. Heat transfer in the liquid bulk phase due to conduction from the gas–liquid interface to the bulk liquid phase can be formulated as,

  @T l @ @T l ¼ kl @y @t @y

t¼0

0 6 y 6 H0 ;

ð15Þ

where cp,l, kl and ql are heat capacity, thermal conductivity and density of the liquid phase, respectively, and Tl is temperature of the liquid. 2.2.3. Initial and boundary conditions Since no solvent and energy is transported into heavy oil at the beginning of the diffusion process, the initial conditions of liquid phase can be described by,

ð16aÞ

t¼0

ð16bÞ

where H0 is initial height of heavy oil in the diffusion cell, and Tl,0 is initial temperature of heavy oil. The Neumann boundary condition for mass diffusion is applied at the bottom of the diffusion cell,

@ci ¼ 0; @y

y ¼ 0;

t>0

ð17aÞ

Since the equilibrium boundary condition is assumed, the corresponding Dirichlet boundary condition for mass diffusion at the gas–liquid interface is given by,

ci ¼ ci;sat ;

y ¼ H;

t>0

ð17bÞ

where ci,sat is saturation concentration of the ith component at the gas–liquid interface at the existing temperature and pressure, H is the expanded height of the liquid phase, which can be sequentially updated in the diffusion process. Meanwhile, effect of liquid-phase swelling on diffusion process can be quantified by changing the boundary conditions during the dynamic evolvement. Given the total compositions of solvent components and heavy oil, saturation concentration of each solvent component at the boundary can be determined by performing two-phase flash calculations [36]. In this study, the two-phase flash calculations are performed with the volume-translated PR EOS model. Since temperature in the PVT chamber and gas-phase is constant, the temperature can be considered constant at both the bottom of diffusion cell and gas–liquid interface, which is the socalled the first type of boundary condition,

T l ¼ T g ; y ¼ H and 0;



where vl, vs and vo are kinematic viscosity of the liquid mixture, solvent and heavy oil, respectively. The kinematic viscosity vl is related to the dynamic viscosity ll by v l ¼ lql . /s and /o are the volume fracl tions of solvent and heavy oil in the liquid phase, respectively. If a solvent mixture diffuses into heavy oil, it is essential that its viscosity be calculated prior to calculating the viscosity of the solvent-diluted heavy oil. As for the solvent mixture, vs and vo in Eq. (13) are the small and large kinematic viscosity of the two solvents, respectively. As for the heavy oil tested in this study, the Lobe’s mixing rule is experimentally found to be superior to predict viscosity of the solvent mixture than other mixing rules [15]. After the temperature equilibrium achieves, only the mass transfer proceeds. In this stage, the diffusion coefficient is assumed constant, which can be determined by combining the experimental data with the calculated results.

ql cp;l

0  y  H0 ;

T l ¼ T l;0 ;

2.2.2. Liquid phase 2.2.2.1. Mass conservation. Based on the Fick’s second law, mass transfer from the gas–liquid interface to liquid phase can be expressed by,

ao ¼ 0:27 ln

ci ¼ 0;

t>0

ð18Þ

The mass loss due to the mass transfer between the gas and liquid phases in Eq. (8) can be determined by,

Dt  S m ¼

Z ng  A X MW i t

V corr

i¼1

H

0

 cti dy 

Z ng  X MW i tþDt

HðtþDtÞ

A V corr

i¼1

0

Dt ctþ dy i



ð19Þ

where MWi is molar weight of the ith gas solvent component and ng is number of the gas solvent components. 2.3. Discretization method Fig. 2 shows the schematic of domain discretization, where concentration or temperature Cm,n at Node #P in the center of Volume #I is taken to be its characteristic mean value. Parameters m and n are the nodal number in y direction and the number of time step, respectively. Other two nodes in the volumes adjacent to Volume #I are Nodes #W and #E, respectively. Accordingly, w represents the interface between two adjacent Volumes #I and II, and e represents the interface between Volumes #I and III. The diffusion equations, i.e., Eqs. (11) and (15), can then be discretized by an explicit scheme. Taking Eq. (15) for an example, the integration of unsteady term in time interval [t, t + Dt] is,

Z

w

Z

e

tþDt

ql cp;l

t

@T l dydt ¼ ðql cp;l ÞP  ðT P  T 0P ÞDy @t

ð20Þ

where TP is temperature of Volume #I at time t þ Dt and T 0P is temperature of Volume #I at time t. Integrating the diffusion term, we have,

Z t

tþDt

Z e

w

" #   @ @T l T 0E  T 0P T 0P  T 0W dydt ¼ ke kl  kw Dt @y @y ðdyÞe ðdyÞw

ð21Þ

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2.4. Determination of diffusion coefficients

W

Cm-1,n

The coupled heat and mass transfer process can be solved by performing the computational procedure as follows:

II

δy

w I

P

Cm,n

e

( δ y )e (δ y )e

∆y

−

+

III E

Cm+1,n

Fig. 2. Schematic of one-dimensional domain discretization.

where T 0W and T 0E are temperatures at time t of Volumes #II and III, respectively. Dy is the length of one control volume in y direction, while dy is the distance between two adjacent nodes in y direction. kw and ke are equivalent thermal conductivity at Interfaces # w and #e, respectively, which can be calculated by the harmonic mean [36],

ðdyÞe ðdyÞe ðdyÞeþ ¼ þ ke kP kE

(1) Perform two-phase flash calculation to determine the boundary concentration of each solvent component in Eq. (17b); (2) Determine a time interval Dt and an initial space interval Dy for the discrete equations; (3) Assume two coefficients a and b in Eq. (12) during the stage of heat and mass transfer and the constant diffusion coefficient during the subsequent stage of mass transfer. It should be noted that the diffusion coefficient calculated by Eq. (12) is varied along the y direction since there exists a temperature gradient inside the liquid phase; (4) Start the iteration at time t and use the volume-translated PR EOS model to calculate the molar volume of the liquid phase for each small volume with Eqs. (1), (2a)–(2c), (3), (4a), (4b), (5a), (5b), (6), (7). Calculate properties of the liquid mixture by using correlations in Appendix A together with Eq. (13); (5) The swelling factor of heavy oil due to heat and mass transfer is calculated at time t with integral equation similar to the definitions in the literature [7,38],

ð22Þ SF X ¼

where ðdyÞe is distance between Interface #e and Node #P, and ðdyÞeþ is distance between Interface #e and Node #E (see Fig. 2). By rearranging Eqs. (15), (20), and (21), T P at time t þ Dt has the following form,

aP T P ¼ aW T 0W þ aE T 0E þ ðaP  aW  aE ÞT 0P

ð23Þ

where

aP ¼

ðql cp;l ÞP Dy Dt

aW ¼

aE ¼

ð24aÞ

kw ðdyÞw

ð24bÞ

ke ðdyÞe

ð24cÞ

Similarly, Eq. (11) can be discretized as,

bP cP ¼ bW c0W þ bE c0E þ ðbP  bW  bE Þc0P

ð25Þ

where

bP ¼

Dy Dt

bW ¼

bE ¼

ð26aÞ

Dw ðdyÞw

ð26bÞ

De ðdyÞe

ð26cÞ

where diffusion coefficient D is a function of liquid viscosity in the stage of heat and mass transfer as shown in Eq. (12) and subsequently a constant in the stage of mass transfer. The viscosity at the interface can also be obtained by applying the harmonic mean [37],

ðdyÞe

le

¼

ðdyÞe

lP

þ

ðdyÞeþ

lE

ð27Þ

1 H0

Z

H

y¼0

V2  V1

1 dy ng X xi 1

ð28Þ

i¼1

where H⁄ is the expanded liquid-phase height at the last time step t  Dt, V1 is molar volume of heavy oil at the test pressure and its initial temperature, V2 is molar volume of hot gas–diluted heavy oil at the same pressure and existing temperature, and xi is mole fraction of the ith solvent component in heavy oil. Meanwhile, the liquid-phase height is correspondingly updated for the next time step at time t + Dt, and a new space interval is also updated by keeping the same total number of nodes; (6) Solve the solvent concentrations and temperature profiles inside the liquid phase at the time t with Eqs. (11) and (15), respectively; (7) Calculate the mass loss in the gas phase formulated in Eq. (19) and then substitute it into Eq. (8) to update the mass of gas phase and then the gas volume; (8) Start a new iteration at t ¼ t þ Dt to repeat Steps #4–7 by using the updated data until the end of the diffusion test; (9) The objective function O(SF) is firstly evaluated with the assumed diffusion coefficients. The objective function can be expressed by,

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 u Nexp u 1 X SF iexp  SF ical t OðSFÞ ¼ Nexp i¼1 SF iexp

ð29Þ

where Nexp is the number of data obtained from the experiment. SF ical and SF iexp are the calculated and measured swelling factor of the ith data point, respectively. If OðSFÞ < 103 , the iteration terminates and the resulted diffusion coefficients are considered to be the actual ones; otherwise, further iteration will be initiated by repeating Steps #3–9; and (10) Once the diffusion coefficients at both stages are determined, repeat Steps #2–9 to find a reasonable time interval Dt and initial space interval Dy which are able to satisfy the computational accuracy (note: the space interval is changed at every time step due to the increased liquid-phase height).

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The time interval and initial space intervals are determined by using the following method. When the diffusion process has been solved by using the respective time interval Dt1 and initial space interval Dy1, the new time interval and initial space interval are increased to be 1.5 times of their original values to initiate a new computational iteration. For example, three time intervals Dt1, Dt2 (Dt2 = 1.5Dt1) and Dt3 (Dt3 = 1.5Dt2), and three initial space intervals Dy1, Dy2 (Dy2 = 1.5Dy1) and Dy3 (Dy3 = 1.5Dy2) are assumed to be used in solving the equations, while SF1, SF2 and SF3 are the corresponding swelling factors. If |SF2  SF1| < 103 and |SF2  SF3| P 103, the results obtained by applying Dt2 and Dy2 can be considered independent of the discretization method. Meanwhile, Dt2 and Dy2 can be the reasonable intervals for both the computational accuracy and efficiency. 3. Experimental 3.1. Materials Dissolution of both pure hot CO2 and hot CO2–C3H8 mixture in heavy oil are measured and used to validate the proposed mathematical model. The heavy oil sample used in this study is collected from the Lloydminster area in Saskatchewan, Canada. The reservoir temperature and pressure are 294.15 K and 3500 kPa, respectively. The molecular weight and specific gravity of the oil sample are 482 g/mol and 0.9997, respectively. The simulated distillation method is used to measure the composition of oil sample (see Table 2). The purities of CO2 and C3H8 (Praxair, Canada) are 99.998 mol% and 99.99 wt%, respectively. The heat capacity, density, viscosity and thermal conductivity of heavy oil are listed in Appendix A, while correlations for predicting the viscosity of liquid CO2 and C3H8 can be found elsewhere [7,39]. The viscosity of the solvent diluted-heavy oil can be obtained by using the aforementioned mixing rule, i.e., Eqs. 13, 14a, 14b. The

heat capacity and thermal conductivity of the solvent diluted-heavy oil can be calculated by using the correlations in Appendix A as long as the molecular weight and specific gravity of diluted-heavy oil are updated during the dissolution of solvent mixture. The density of gas mixture can be predicted by using the volume-translated PR EOS model. 3.2. Experimental setup Fig. 3 shows the schematic of PVT apparatus used in this study. The mercury-free DBR PVT system (PVT-0150-100-200-316-155, DBR, Canada) is applied to study the diffusion process of hot CO2 and hot C3H8–CO2 mixture in a heavy oil, respectively. The PVT system mainly consists of a transparent high-pressure PVT cell with an inner diameter of 3.177 cm and a length of 20.320 cm. The PVT cell is used to contain high-pressure and high-temperature sample fluids and permits a maximum operating pressure and temperature of 69000 kPa and 473.15 K, respectively. There is a floating piston inside the PVT cell to isolate the sample fluids from the hydraulic oil. At the end of PVT cell containing the hydraulic oil, a section of tubing is added to maintain a constant pressure inside the PVT cell throughout the measurements by using a high-precision syringe pump (500HP, Teledyne ISCO Inc., USA). The fluid height can be recorded by using a video-based digital cathetometer with a measurement precision of ±0.002 cm. Two pressure gauges are used in the PVT system for more accurately reading the cell pressure. One is Heise pressure gauge, and the other connected to the inlet tubing of the PVT cell is a high-precision gauge (Model No.: 2089, Ashcroft, USA) with a full scale of 6895 kPa and an accuracy of 0.05%. The transfer cylinder used to contain the gas is placed vertically inside the air-bath to ensure that only vapor-state gas will be released upwards from the gas cylinder. During the measurements, temperature of the system except heavy oil is maintained constant in the air-bath, while its pressure is kept constant by using the syringe pump. The heavy

Table 2 Compositional analysis result of the Lloydminster heavy oil sample.

a

Carbon no.

wt%

mol%

Assumed MWa, g/mol

Carbon no.

wt%

mol%

Assumed MWa, g/mol

C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17 C18 C19 C20 C21 C22 C23 C24 C25 C26 C27 C28 C29 C30

0 0 0 0 0 0 0 0 0.786 0.844 1.037 1.267 1.817 1.950 2.300 2.143 2.286 2.238 2.048 1.857 2.071 1.329 1.743 1.571 1.714 1.600 1.583 1.650 1.452 1.281

0 0 0 0 0 0 0 0 2.73 2.64 2.96 3.32 4.39 4.38 4.83 4.22 4.24 3.92 3.40 2.93 3.11 1.91 2.39 2.07 2.17 1.95 1.85 1.86 1.58 1.35

16 30 44 58 72 86 100 114 128 142 156 170 184 198 212 226 240 254 268 282 296 310 324 338 352 366 380 394 408 422

C31 C32 C33 C34 C35 C36 C37 C38 C39 C40 C41 C42 C43 C44 C45 C46 C47 C48 C49 C50 C51 C52 C53 C54 C55 C56 C57 C58 C59 C60+

1.390 1.329 0.981 1.083 1.307 1.343 0.917 0.900 1.583 1.600 0.750 0.830 1.380 0.929 0.929 0.700 0.833 0.792 0.792 0.733 0.750 0.717 0.717 0.683 0.650 0.650 0.667 0.667 0.667 36.164

1.42 1.31 0.94 1.01 1.18 1.18 0.78 0.75 1.29 1.27 0.58 0.63 1.02 0.67 0.65 0.48 0.56 0.53 0.51 0.46 0.47 0.44 0.43 0.40 0.37 0.37 0.37 0.36 0.36 15.01

436 450 464 478 492 506 520 534 548 562 576 590 604 618 632 646 660 674 688 702 716 730 744 758 772 786 800 814 828 1072

Note: the assumed MWs are provided by the Saskatchewan Research Council (SRC).

H. Sun et al. / International Journal of Heat and Mass Transfer 74 (2014) 173–184

179

Fig. 3. Schematic of PVT experimental system for studying dissolution of hot solvent(s) into heavy oil.

oil housed in another transfer cylinder is placed outside the airbath and kept at a lower environment temperature to allow the subsequent heat transfer from the gas to heavy oil in the PVT cell. The uncertainties of measuring pressure, volume, and temperature and determining the mixture composition are ±4 kPa, ±1.59  102 cm3, ±0.1 K, and ±0.3 wt%, respectively. 3.3. Experimental procedure One pure CO2–heavy oil system (Feed #1) and one C3H8–CO2– heavy oil systems (Feed #2) are tested. Table 3 shows the overall compositions of the scenarios and the physical properties of CO2, C3H8 and heavy oil are tabulated in Table 4. The tests are performed at the same pressure of 5500 kPa. The experimental procedure for Feed #2 is briefly described as follows. Before the fluids are introduced to the PVT cell, C3H8–CO2 mixture is pre-mixed inside the gas transfer cylinder. Before the mixing, the transfer cylinder is placed vertically. C3H8 is first injected into the transfer cylinder, leading to a certain pressure. CO2 is subsequently discharged into the cylinder, and the pressure of mixture

increases. The C3H8–CO2 mixture then stays for 12 h to achieve homogeneity. Meanwhile, the PVT cell and tubing are thoroughly cleaned with kerosene and then vacuumed. The floating piston is placed closer to the upper section of the PVT cell as shown in Fig. 3. After 12 h, the pre-mixed C3H8–CO2 mixture is discharged from the top valve of the transfer cylinder to the bottom of the PVT cell and a 10 cm3 sampler (Swagelok, Canada, 1000 psi) simultaneously. The inlet valve at the bottom of PVT cell is then closed, while the sampler is removed for gas composition analysis, followed by maintaining a constant pressure, i.e., 5500 kPa, of the PVT cell by the syringe pump. The air-bath temperature is set to the desired test temperature of 331.15 K for at least 12 h, while another transfer cylinder housed with heavy oil is placed at the environment with a lower temperature of 295.65 K for 12 h. After the temperature of PVT cell filled with the C3H8–CO2 mixture reaches its stable value, a certain amount of heavy oil sample is introduced into the bottom of PVT cell quickly and then the inlet valve at the bottom of PVT cell is closed. When the diffusion starts, both the gas-phase volume and liquid-phase volume are monitored simultaneously by tracking the upper interface between

Table 3 Initial compositions and two-phase flash calculations for Feeds #1 and 2. Feed

Initial composition mole fraction

Two-phase flash mole fraction Gas phase

Liquid phase

Liquid-phase saturation concentration mole fraction

#1

xCO2 ¼ 0:7821 xoil ¼ 0:2179

xCO2 ¼ 1:0000 xoil ¼ 0:0000

xCO2 ¼ 0:4206 xoil ¼ 0:5794

cCO2 ;sat ¼ 0:4206 coil;sat ¼ 0:5794

#2

xCO2 ¼ 0:6291 xC3 H8 ¼ 0:1597 xoil ¼ 0:2112

xCO2 ¼ 0:8664 xC3 H8 ¼ 0:1336 xoil ¼ 0:0000

xCO2 ¼ 0:3693 xC3 H8 ¼ 0:1882 xoil ¼ 0:4425

cCO2 ;sat ¼ 0:3693 cC3 H8 ;sat ¼ 0:1882 coil;sat ¼ 0:4425

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heat and mass transfer and the subsequent stage of mass transfer is,

Table 4 Physical properties of CO2, C3H8 and Lloydminster heavy oil. Component CO2 C3H8 Heavy oil

pc, kPa 7378 4246 1265

Tc, K 304.14 369.80 933.66

x 0.2238 0.1520 1.0288

(

ZRA 0.2736 0.2763 0.2622

piston and gas as well as the lower interface between gas and heavy oil in a continuous manner. During the dissolution of hot gas mixture into the heavy oil, the liquid phase expands, while the gas phase contracts, and thus the temperature difference between the gas and liquid phases decreases. The volume recordings are terminated after 45 h. Considering that the C3H8–CO2–heavy oil system cannot achieve equilibrium after 45 h, the magnetic stirrer at the bottom of PVT cell starts functioning to force the mass equilibrium between the gas and liquid phases. After the volumes of both gas and liquid phases do not change for 12 h, it can be considered to reach the equilibrium and their volumes are recorded. Then the PVT cell is reversed upside down so that the mixture is at the top and the hydraulic oil is at the bottom of the PVT cell. Another gas sampler is connected to the top of PVT to sample the residual C3H8–CO2 mixture for compositional analysis until the heavy oil falls down and the gas rises up. A high-precision gas chromatography (GC) analyzer (Model No.: 6890 N, Agilent) is used to analyze the compositions of the two gas samples. Table 5 shows the initial and equilibrium compositions of Feed #2. The experimental procedure for Feed #1 is similar to that for Feed #2 except that the pre-mixing step and compositional analysis can be avoided because the composition of pure gas can be obtained from its height in the PVT cell together with the PR EOS model. In addition, the air-bath temperature is kept at 330.95 K. The equilibrium concentration of pure CO2 in heavy oil cannot be obtained in this study due to the fact that the viscosity of CO2-diluted heavy oil is still higher than the minimum viscosity under which the magnetic stirrer can function properly.

4. Results and discussion 4.1. Diffusion coefficient The boundary concentrations for Feeds #1 and #2 obtained from two-phase flash calculations are shown in Table 3. The diffusion coefficient of the dissolved CO2 in heavy oil determined at p = 5500 kPa and Tg = 295.65–330.95 K during the initial stage of

DCO2 ¼

1:5e  8l0:45 ; Tl < Tg l

ð30aÞ

2:1e  9; T l ¼ T g

It should be noted that each component in C3H8–CO2 gas mixture is assumed to have the same diffusion coefficient, which is the socalled ‘‘apparent diffusion coefficient’’ [14]. The apparent diffusion coefficient of C3H8–CO2–heavy oil system obtained at p = 5500 kPa and Tg = 295.65–331.15 K during the two stages is determined as,

(

DC3 H8 CO2 ¼

4:5e  9l0:60 ; Tl < Tg l

ð30bÞ

1:7e  9; T l ¼ T g

The correlation of diffusion coefficient proposed for the diffusion of pure C3H8 into heavy oil available at the p = 200–800 kPa and Tg = 297.05 K is [18],

DC3 H8 ¼ 2:56  109 l0:51 l

ð30cÞ

The diffusion coefficient calculated by the correlation obtained in this study is found to be much higher than that predicted by Eq. (30c) due to the much higher test pressure, i.e., 5500 kPa. 4.2. Swelling factor Fig. 4 depicts the measured and calculated swelling factors of heavy oil as a function of time for Feeds #1 and #2 at p = 5500 kPa. At the end of the diffusion test, the swelling factor for Feed #2 is calculated to be 1.103, which is greater than that for Feed #1, i.e., 1.045. This is attributed to the fact that addition of C3H8 into CO2 facilitates an enhanced swelling factor of heavy oil [7]. In addition, the swelling factor of heavy oil for Feed #2 is calculated to be 1.140 at the equilibrium state. By comparing with the swelling factors of heavy oil due to only mass transfer [15], the swelling factor increases much faster at the initial stage at the presence of heat transfer. Therefore, during the initial stage of heat and mass transfer, the swelling factors for both feeds increase rapidly mainly due to the increased temperature and reduced viscosity of heavy oil, while the swelling factors increase gradually at the subsequent stage of mass transfer mainly due to the increased concentration of gas solvent in heavy oil. Fig. 5 shows the measured and calculated decaying volume of gas phase as a function of time for Feeds #1 and #2, respectively. The mean relative errors between the measured and calculated gas-phase volume are found to be 0.84% and 0.65% for Feeds #1 and #2, respectively. Such a small deviation proves the viability of the developed mathematical model for describing the one-way

Table 5 Measured composition of C3H8–CO2 mixture at the beginning and equilibrium state of the diffusion test. System

C3H8–CO2 mixture

Rounded C3H8–CO2 composition

Component

H2 CO2 C2H4 C2H6 C3H6 C3H8 i-C4H10 n-C4H10 C6+ C5+ O2 N2 CH4 CO

Composition, mol% Beginning

Equilibrium

<0.01 79.77 <0.01 <0.01 <0.01 20.23 <0.01 <0.01 <0.01 <0.01 <0.01 <0.01 <0.01 <0.01 (20.2 mol% C3H8, 79.8 mol% CO2)

<0.01 85.26 <0.01 <0.01 <0.01 14.56 <0.01 <0.01 <0.01 <0.01 <0.01 0.18 <0.01 <0.01 (14.6 mol% C3H8, 85.4 mol% CO2)

H. Sun et al. / International Journal of Heat and Mass Transfer 74 (2014) 173–184

181

Fig. 4. Measured and calculated swelling factors of heavy oil for two feeds.

coupled heat and mass transfer from either a pure gas or gas mixture into heavy oil at high pressures and elevated temperatures. 4.3. Temperature profile Fig. 6 plots the temperature profiles inside the liquid phase for Feeds #1 and #2, respectively. As can be seen, the thermal equilibrium between gas and liquid phases achieves at about t = 40 min for the CO2–heavy oil system, while the liquid-phase temperature reaches the gas-phase temperature at approximately t = 7 min for the C3H8–CO2–heavy oil system. C3H8 is found not only to improve the mass transfer [15] but also to accelerate the heat transfer from the gas into heavy oil. Based on the fact that the heat transfer proceeds and achieves equilibrium faster than the mass transfer, the coupled heat and mass transfer process is divided into two stages, i.e., coupled heat and mass diffusion at the initial stage and mass diffusion at the subsequent stage. The time for reaching thermal equilibrium can be used as the criterion for dividing two stages. The heavy oil expands rapidly at the initial stage due to the heat diffusion from hot gas into heavy oil (see Fig. 4). Once achieving the thermal equilibrium, the mass diffusion dominates the process and the heavy oil swells slowly due to the sole contribution from mass transfer. 4.4. Concentration profile Fig. 7 depicts variations of the mean compositions with time in the liquid phase for Feeds #1 and #2, respectively. As can be seen from Fig. 7a, the measured and calculated compositions for Feed

Fig. 6. Calculated temperature profile in liquid phase for (a) Feed #1 and (b) Feed #2.

#1 agree well with each other with the diffusion coefficient as obtained from Eq. (30a). The solvent composition in liquid phase for Feed #1 during the diffusion test cannot be experimentally measured, but they can be accurately determined on the basis of the measured gas-phase composition, gas-phase volume, and mass conservation in the closed system. The solvent compositions in gas phase for Feed #2 cannot be directly measured because the composition of each gas in the gas mixture cannot be measured during the diffusion test and only the equilibrium composition of each gas component can be measured. Fig. 7b shows that more gas solvents (C3H8 and CO2) for Feed #2 is dissolved in heavy oil than CO2 for Feed #1, leading to an enhanced swelling factor of heavy oil due to the heat and mass transfer of C3H8–CO2 mixture. Fig. 8 illustrates the composition profile of CO2 in liquid phase for Feed #1, while Fig. 9 shows the composition profiles of CO2 and C3H8 in liquid phase for Feed #2, respectively. The composition gradient inside the liquid is smaller for Feed #2 than that for Feed #1 due to the faster dissolution of C3H8–CO2 mixture in heavy oil. Also, Fig. 9 shows that the compositions of C3H8 and CO2 in heavy oil increase at the same rate since the same diffusion coefficient of the two components has been made. 5. Conclusions

Fig. 5. Measured and calculated gas-phase volume for two feeds.

A generalized technique has been developed to couple heat and mass transfer for the dissolution of a hot gas or gas mixture in heavy oil at high pressures and elevated temperatures. Theoretically, the volume-translated PR EOS model is applied to calculate the changing liquid-phase volume for realizing the moving boundary due to the swelling effect of heavy oil. A one-way diffusion model is formulated to describe the coupled heat and mass transfer

182

H. Sun et al. / International Journal of Heat and Mass Transfer 74 (2014) 173–184

Fig. 7. Compositions in liquid phase for (a) Feed #1 and (b) Feed #2.

Fig. 9. Composition profiles of (a) CO2 and (b) C3H8 in liquid phase for Feed #2.

between either a hot gas or a gas mixture and heavy oil. Experimentally, a PVT setup is employed to conduct the diffusion tests for a CO2–heavy oil system and a CO2–C3H8–heavy oil system, respectively. The thermal equilibrium is found to be achieved much faster than the mass equilibrium. The time of achieving thermal equilibrium can be used as a criterion to divide the heat and mass transfer into two stages, i.e., the initial stage of coupled heat and mass transfer and the subsequent stage of mass transfer. The diffusion coefficient is assumed to be a function of viscosity of the solventdiluted heavy oil at the stage of coupled heat and mass transfer while constant in the subsequent stage of mass transfer. Heat transfer plays the most important role in the initial stage, resulting

in a fast expansion of heavy oil due to the decreased viscosity. Mass transfer dominates the subsequent stage of the diffusion tests after thermal equilibrium has been achieved, though the swelling factor of heavy oil increases gradually because of the increased concentration of gas solvent in heavy oil. The presence of C3H8 in CO2 stream contributes to both faster heat diffusion and quicker mass diffusion. The dissolution of hot solvent–CO2 mixture in heavy oil is found to be an effective way to enhance the swelling of heavy oil. Acknowledgments The authors acknowledge a Discovery Grant and a Collaborative Research and Development (CRD) Grant from the Natural Sciences and Engineering Research Council (NSERC) of Canada to D. Yang and EHR Enhanced Hydrocarbon Recovery Inc. for financial support. Appendix A. Correlations of heat capacity, viscosity, density and thermal conductivity for heavy oil The heat capacity of heavy oil can be predicted by the following empirical formulas [40],

cp;o ¼ A1 ðB1 þ C 1 TÞ; 145K < T < 0:8T c

ðI:1Þ

where cp,o is heat capacity of heavy oil in kJ/(kg K) and T is temperature in K. The viscosity of heavy oil can be accurately predicted based on the experimentally measured viscosity data for the same oil [7],

log10 ½log10 ðlo Þ ¼ 3:7042log10 ðTÞ þ 9:7600 Fig. 8. Composition profile of CO2 in liquid phase for Feed #1.

where

lo is the viscosity of heavy oil in cP.

ðI:2Þ

H. Sun et al. / International Journal of Heat and Mass Transfer 74 (2014) 173–184 Table A1 Constants used in Appendix A [40]. Equation A1 B1 C1

A1 = 1.4651 + 0.2302Kw B1 = 0.306469  0.16734SG C1 = 0.001467  0.000551SG

Also, according to the experimental measurements, the density of heavy oil can be calculated by [7],

qo ðT; PÞ ¼

q0 ðT; P Þ 0

1  b ln

Bþ0:001P Bþ0:1



ðI:3Þ

where

q0 ðT; P0 Þ ¼ 784:0044 þ 1:7217T  3:3752  103 T 2

ðI:4aÞ

b ¼ ð6:1774 þ 0:0213TÞ

ðI:4bÞ

B ¼ 3:7614  104  1:8009  107 T 1 þ 2:0605  109 T 2

ðI:4cÞ

3

where qo is density of heavy oil in kg/m and P is pressure in kPa. The thermal conductivity of heavy oil can be predicted by [41],

    1 1 1 1 ¼ A2  1 þ B2 ¼ A2  1 þ B2 ko F RI F RI ðT 0 Þ=qðT 0 Þ qðTÞ

ðI:5aÞ

where FRI is the Lorentz–Lorenz function of refractive index. Since F RI =q ¼ 1=3 (i.e., the so-called ‘‘one-third rule’’) for 7 crude oils in a temperature range of 10–70 °C [42], Eq. (I.4a) can be simplified as,

    1 1 3 ¼ A2  1 þ B2 ¼ A2  1 þ B2 ko F RI qðTÞ

ðI:5bÞ

where qðTÞ and ko are density in g=cm3 and thermal conductivity in W=ðm  KÞ, respectively. The constants used in Appendix Aare tabulated in Table A1. In Table A1, Kw is Watson characterization factor, MW is molecular weight in g/mol and SG is specific gravity. The definition of Watson factor is [43], 1=3

Kw 

Tb SG

ðI:6Þ

where Tb is the normal boiling point in °R, which can be predicted by [43],

T b ¼ 1928:3  1:695  105 MW 0:03522 SG3:266  exp½4:922  103 MW  4:7685SG þ 3:462  103 MW  SG

ðI:7Þ

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