New Experimental Method for Measuring Gas Diffusivity in HeavyOil by the Dynamic Pendant Drop Volume Analysis (DPDVA)

Chaodong Yang and Yongan Gu*Petroleum Technology Research Centre (PTRC), Faculty of Engineering, University of Regina, Regina,Saskatchewan S4S 0A2, Canada

This paper presents a new experimental method and its computational scheme for measuringgas diffusivity in heavy oil at high pressures and elevated temperatures by the dynamic pendantdrop volume analysis (DPDVA). In the experiment, a see-through windowed high-pressure cellis first filled with a test gas at a prespecified pressure and temperature. Then, a heavy oil sampleis introduced by using a syringe pump to form a pendant drop inside the pressure cell. Due tothe oil swelling effect, the subsequent dissolution of the gas into the pendant oil drop causes itsvolume to increase until the saturation state is reached. The sequential digital images of thedynamic pendant oil drop are acquired and analyzed by applying computer-aided imageacquisition and processing techniques to measure the oil drop volumes at different times. Amass-transfer model is developed theoretically to describe the diffusion process of the gas intothe pendant heavy oil drop. This model is numerically solved by applying the semidiscreteGalerkin finite element method. The volume of the dynamic pendant oil drop is calculated fromthe numerically predicted transient gas concentration distribution inside the pendant oil drop.The gas diffusivity in heavy oil and the swelling factor of gas-saturated heavy oil are, thus,determined by finding the best fit of the theoretically calculated volumes of the dynamic pendantoil drop to the experimentally measured data. This novel experimental technique is applied tomeasure CO2 diffusivities in a heavy oil sample and the swelling factors of a CO2-saturatedheavy oil at P ) 2, 3, 4, 5, and 6 MPa and T ) 23.9 C.

1. Introduction

To meet the growing demand for petroleum products,the oil industry needs to develop new technologies forexploiting the tremendous heavy oil and bitumen re-sources worldwide. These resources are approximatelyeight trillion barrels of oil in place throughout the world,a major portion of which is in Canada and Venezuela.1,2The key to successful recovery of heavy oil and bitumenis to dramatically reduce the oil viscosity, which istypically on the order of thousands to millions of centi-poises under the reservoir pressure and temperatureconditions. It is well-known that the injection of somegases, such as methane, ethane, propane, butane, andcarbon dioxide, into heavy oil reservoirs can significantlyreduce oil viscosity.3-5 Several solvent-based oil recoveryprocesses have been proposed to recover heavy oil andbitumen by injecting hydrocarbon and/or nonhydrocar-bon gases into oil reservoirs, such as vapor extraction(VAPEX),1,6 cyclic solvent injection,7 and CO2 immiscibleflooding.4 Previous studies have already shown thatmolecular diffusion of the injected gas in heavy oil playsa vital role in the solvent-based oil recovery processes.8-10Thus, gas diffusivity in heavy oil under the actualreservoir conditions becomes an important parameterfor the reservoir simulation and field design of solvent-based heavy oil recovery processes.

In the literature, there are several experimentalmethods for measuring the diffusivity of a gas in heavyoil. These experimental methods can be roughly catego-rized into conventional and nonconventional methods.Conventional methods involve composition analysis ofliquid samples taken from the gas-heavy oil mixture

at different times and locations during a diffusiontest.11-13 These methods are expensive, intrusive andtime-consuming, especially if the diffusion test is con-ducted at high pressures. In addition, compositionanalysis of a gas-heavy oil mixture is prone to largeexperimental error. Nonconventional methods measurethe change of a property of the gas-heavy oil systemduring the molecular diffusion process. This propertycan be gas volume,14 gas-oil interface position inside acapillary,15 laser light intensity of the gas-heavy oilmixture,16 gas pressure,17 or shape of a pendant heavyoil drop surrounded by a gaseous solvent.18 In particu-lar, the decaying gas pressure is measured while themolecular diffusion of the gas into heavy oil proceedswithin a closed, high-pressure diffusion cell. This isreferred to as the pressure decay method, which hasbeen applied to measure the diffusivities of methane,ethane, propane, nitrogen, and carbon dioxide in crudeoils17,19-23 and methane in water.24 This nonintrusivemethod requires a relatively simple experimental setup.Also, determination of the diffusivity from the measureddecaying pressure is quite straightforward. However,the pressure decay method is time-consuming, andgenerally, it takes several days or weeks to complete asingle diffusivity measurement. Moreover, the effect ofpressure on the diffusion coefficient cannot be examinedby using this experimental method because the gaspressure keeps decaying during the diffusivity measure-ment.

More recently, some state-of-the-art techniques havebeen applied to estimate the diffusion coefficients ofhydrocarbon solvents in heavy oil and bitumen, suchas magnetic resonance imaging (MRI),25-26 low fieldnuclear magnetic resonance (NMR) spectra,27 and X-raycomputer-assisted tomography (CAT) scanning.28 Thesetechniques require extremely expensive and highly

* To whom correspondence should be addressed. Tel.: (306)585-4630. Fax: (306) 585-4855. E-mail: peter.gu@uregina.ca.

4474 Ind. Eng. Chem. Res. 2005, 44, 4474-4483

10.1021/ie0501430 CCC: $30.25 2005 American Chemical SocietyPublished on Web 05/07/2005

sophisticated apparatuses, which have to be operatedby specially trained technicians. Also, the experimentalmodel should be made of nonmetallic materials. Thelatter requirement makes it difficult, if not impossible,to apply these techniques to measuring gas diffusivityin heavy oil at high pressures.

This paper presents a new experimental method andits computational scheme for measuring gas diffusivityin heavy oil under the practical reservoir conditions bythe dynamic pendant drop volume analysis (DPDVA).In the experiment, a see-through windowed high-pressure cell is first filled with a test gas at a prespeci-fied pressure and temperature, and then a heavy oilsample is introduced by using a syringe pump to forma pendant oil drop inside the pressure cell. The subse-quent diffusion of the gas into the pendant oil dropcauses its volume to increase until it is completelysaturated with the gas. The precise digital images ofthe dynamic pendant oil drop are acquired sequentiallyand stored as image files in a computer. Then, thesedigital oil drop images are analyzed and processedaccurately to determine the measured volumes of thedynamic pendant drop by applying computer digitalimage analysis and processing techniques. The increasein the volume of the dynamic pendant oil drop isattributed to the well-known oil swelling effect.10,29,30A mass-transfer model is developed theoretically todescribe the molecular diffusion process of the gas intothe pendant heavy oil drop. The finite element methodis applied to numerically solve the mass-transfer prob-lem. Furthermore, the volumes of the dynamic pendantoil drop at different times can be calculated if the gasdiffusivity and oil swelling factor are known. An objec-tive function is mathematically constructed to expressthe discrepancy between the theoretically calculatedvolumes of the dynamic pendant oil drop and theexperimentally measured data. The gas diffusivity inheavy oil and the oil swelling factor are used asadjustable parameters and, thus, are determined oncethe objective function is minimized. In this study, thenewly developed DPDVA method, together with itscomputational scheme, is applied to measure the diffu-sion coefficients of CO2 in a heavy oil sample at differentpressures of P ) 2, 3, 4, 5, and 6 MPa and a constanttemperature of T ) 23.9 C. The measured CO2 diffu-sivities and oil swelling factors are consistent with thosereported in the literature for similar gas-heavy oilsystems.

2. Theory2.1. Mass-Transfer Model. In the DPDVA method,

an axisymmetric pendant oil drop is formed from thetip of a cylindrical stainless steel syringe needle insidea high-pressure cell that is filled with the test gas asillustrated in Figure 1a. The outer radius of the syringeneedle is rn, the wall thickness of the needle is n, andthe height of the syringe needle is hn. The heavy oilphase inside the pendant drop, together with the heavyoil phase inside the syringe needle, is chosen as thecomputational domain, which is denoted by . Theboundaries formed between the syringe needle and theheavy oil, including the cutting plane at the top of thesyringe needle, are expressed by n. It should be notedthat the cutting plane is chosen far away from the tipof the syringe needle so that gas diffusion cannot reachit during the diffusion test. The surface of the pendantoil drop, i.e., the gas-heavy oil interface, is representedby int.

The heavy oil used in the experiment is nonvolatile.Therefore, only the mass transfer of gas into thependant heavy oil drop needs to be considered. Duringthe mass-transfer process, the interfacial tension be-tween the heavy oil and gas is gradually reduced andthe volume of the pendant oil drop slowly increasesbecause of the oil swelling effect.10,29,30 Thus, both theshape and volume of the pendant oil drop change duringthe diffusion test.18 This becomes an extremely compli-cated moving-boundary mass-transfer problem. Becauseof the slow molecular diffusion and the highly viscousheavy oil, however, the shape and volume of thedynamic pendant oil drop change rather slowly. In thisstudy, the effects of the shape and volume changes ofthe dynamic pendant oil drop on the mass transfer ofthe gas into the heavy oil phase are neglected. Thependant oil drop is assumed to remain unchanged inthe mass-transfer modeling. Accordingly, once the pen-dant heavy oil drop is formed, the pendant oil dropacquired at the beginning of a diffusion test is used todetermine the computational domain .

In the Cartesian coordinate system, the moleculardiffusion of the gas inside the pendant heavy oil dropis a three-dimensional mass-transfer problem. Notingthe axisymmetry of the pendant drop,31,32 however, itis more convenient to choose the cylindrical coordinatesystem (r, z), where r denotes the radial coordinate andz stands for the axial coordinate as shown in Figure 1a.The origin of the cylindrical coordinate system is locatedat the apex of the pendant oil drop. Thus, the dissolutionof gas into the pendant heavy oil drop becomes anunsteady two-dimensional diffusion problem in thechosen coordinate system. The diffusion equation fordescribing the gas concentration distribution inside thecomputational domain can be expressed as33

where c is the molar concentration of the gas in heavyoil, i.e., the number of moles of gas dissolved per unit

Figure 1. (a) Schematic diagram of an axisymmetric pendant oildrop surrounded by a gas depicted in the cylindrical coordinatesystem (r, z); (b) triangular mesh used in the numerical computa-tion. Here, a mesh of 2,935 elements and 1,470 nodes is chosenfor graphical reason.

@c@t

) D[1r @@r(r @c@r) + @2c@z2], (r,z) 2 , t > 0 (1)

Ind. Eng. Chem. Res., Vol. 44, No. 12, 2005 4475

volume of heavy oil, and D is the diffusion coefficient ofthe gas in heavy oil.

Initially, the gas concentration inside the pendant oildrop is zero. Thus, the initial condition (IC) for eq 1 isgiven by

Regarding the boundary condition (BC) at the gas-heavy oil interface, the heavy oil at the interface isassumed to be saturated with the gas at all times.14,17,19,20Physically, this assumption means that there is nointerfacial resistance to the mass transfer of the gasacross the interface. The corresponding equilibriumboundary condition at the gas-heavy oil interface canbe written as

where csat is the molar concentration of the gas in thegas-saturated heavy oil at the experimental pressureand temperature. More specifically, in this study, csatis defined as the number of moles of gas dissolved perunit volume of heavy oil at the saturation pressure andtemperature. The boundary condition expressed in eq3 is classified as the first kind of boundary condition,which is also called the Dirichlet boundary condition.

The syringe needle is impermeable to the gas, andthe cutting plane can also be treated as an impermeableboundary. Thus, the nonpenetrating boundary conditioncan be applied at the boundaries n

where nr and nz are the direction cosines, i.e., the r andz components of the outward unit vectors normal to theboundary. This BC belongs to the second kind ofboundary condition, which is also called the Neumannboundary condition.

To solve the gas diffusion equation eq 1 subject to theIC in eq 2, the equilibrium BC in eq 3, and thenonpenetrating BC in eq 4, it is more convenient tonondimensionalize all the equations by introducing thefollowing dimensionless variables:

Here, C is the dimensionless gas concentration in heavyoil, R and Z are the dimensionless radial and axialcoordinates, respectively, and is the dimensionlesstime. With these dimensionless variables, eqs 1-4become

where nR and nZ are the direction cosines, i.e., the Rand Z components of the outward unit vectors normalto the boundary.

Equation 6 subject to eqs 7-9 is solved numericallyby applying the semidiscrete Galerkin finite elementmethod.34 In this numerical method, the discretizationconsists of the following two steps: spatial discretizationand temporal discretization. In the spatial discretiza-tion, the space variables are discretized by using theGalerkin method. This transforms the partial dif-ferential equation, i.e., eq 6, into a system of time-dependent ordinary differential equations (ODEs). Inthe temporal discretization, the resultant system ofODEs is then solved by using the -method.34 Thedetailed mathematical formulations of the semidiscreteGalerkin finite element method for the same diffusionequation are given elsewhere.35 Figure 1b shows atriangular mesh for the numerical simulation. As shownin this figure, a finer mesh is used near the gas-heavyoil interface, whereas a coarser mesh is adopted nearthe center of the pendant oil drop so as to determine amore accurate gas concentration distribution near theinterface and achieve a higher overall computationalefficiency as well. For graphical reasons, Figure 1bshows a triangular mesh of 2,935 elements and 1,470nodes. The numerical results presented in this paperare obtained from various triangular meshes withapproximately 7,000 elements and 3,500 nodes. Typicaldimensional mesh sizes for elements near the gas-heavy oil interface are approximately 30 m.

Figure 2 shows the calculated respective dimension-less gas concentration contours inside the pendant oildrop and the syringe needle at ) 0.1, 0.2, and 0.4. Inthe numerical calculation, the following values areused: initial drop volume of V0 ) 5.764 mm3, syringeneedle radius of rn ) 0.79 mm, syringe needle wallthickness of n ) 0.59 mm, and syringe needle heightof hn ) 2.4 mm. The pendant drop profile is taken fromthe measured initial pendant heavy oil drop for theCO2-heavy oil system at P ) 4 MPa and T ) 23.9 C.It can be seen from Figure 2 that, at the early stage ofthe diffusion process ( ) 0.1), there is a large gasconcentration gradient near the interface of the pendantoil drop. As the gas gradually dissolves into the heavyoil, the gas concentration gradient becomes smaller andsmaller ( ) 0.2 and 0.4). More and more gas diffusesinto the center of the pendant oil drop and enters thesyringe needle. This diffusion process proceeds until thependant oil drop is completely saturated with the gas.

2.2. Dynamic Pendant Drop Volume. For a givenpendant oil drop, the dimensionless gas concentrationdistribution C(R,Z,) inside the pendant oil drop isnumerically calculated by solving the above-formulatedmass-transfer model at any dimensionless time .Recalling the relations between the dimensionless anddimensional variables defined in eq 5, the number ofmoles of gas dissolved in the pendant heavy oil drop andinside the syringe needle at any time t, n(t), can befound:

Meanwhile, the volume of the dynamic pendant oil dropat any time t is equal to the summation of the initialheavy oil drop volume and the volume change causedby the gas dissolved in the heavy oil

c(r,z,t)jt)0 ) 0, (r,z) 2 (2)

c(r,z,t) ) csat, (r,z) 2 int, t > 0 (3)

D(@c@r nr + @c@z nz) ) 0, (r,z) 2 n, t > 0 (4)

C ) ccsat

, R ) rrn

, Z ) zrn

, ) trn

2/D(5)

@C@

) 1R

@

@R(R @C@R) + @2C

@Z2, (R,Z) 2 , > 0 (6)

C(R,Z,)j)0 ) 0, (R,Z) 2 (7)

C(R,Z,) ) 1, (R,Z) 2 int, > 0 (8)@C@R

nR +@C@Z

nZ ) 0, (R,Z) 2 n, > 0 (9)

n(t) ) csatrn3 ss(R,Z)2 C(R,Z, Dtrn2)R dR dZ (10)

Vc(t) ) V0 + vgasMgasn(t). (11)

4476 Ind. Eng. Chem. Res., Vol. 44, No. 12, 2005

where Vc(t) is the calculated volume of the dynamicpendant drop at any time t; V0 is the measured initialvolume of the dynamic pendant oil drop at t ) 0; vgasrepresents the volume change per unit mass of gasdissolved in the gas-heavy oil mixture, which is aconstant for a given gas-heavy oil system at constantpressure and temperature; and Mgas is the molecularweight of the gas.

In phase behavior studies, the oil swelling factor isoften used to quantitatively describe the oil swellingeffect when a gas dissolves into heavy oil.4,13,36,37 In thispaper, the symbol ksw is used to represent the oilswelling factor, which is defined as the ratio of thevolume of the gas-saturated heavy oil to the volume ofthe initial clean heavy oil without any gas dissolution.Similar to vgas defined in eq 11, the oil swelling factorksw is a property of the gas-heavy oil system of interest,which depends on the saturation pressure and temper-ature only. On the basis of the above definition, the oilswelling factor ksw at the saturation state can be relatedto vgas used in eq 11 by

where csat is the number of moles of gas dissolved perunit volume of heavy oil at the saturation pressure andtemperature. Substituting n(t) in eq 10 and vgas in eq12 into eq 11 yields

Therefore, once the dimensionless gas concentrationdistribution C(R,Z,) inside the computational domain is numerically computed for a pendant heavy oil drop

with the measured initial pendant oil drop volume V0at t ) 0, eq 13 can be readily used to calculate thevolume of the dynamic pendant oil drop at any time t ifD and ksw are known.

2.3. Objective Function and its Minimization.The dynamic pendant drop volume analysis (DPDVA)method determines the gas diffusivity in heavy oil andits swelling factor simultaneously. The strategy em-ployed is first to define an objective function whichexpresses the discrepancy between the theoreticallycalculated and experimentally measured volumes of thedynamic pendant drop at different times. Then, the gasdiffusivity and the swelling factor are determined oncethe objective function is minimized.

Let Vm(t) be the measured volume of the dynamicpendant oil drop and Vc(t) be the calculated volume ofthe dynamic pendant oil drop at any time t, 0 e t e tm,where tm is the total duration of the diffusion measure-ment. It is noted that, in the measurement of Vm(t), timezero (t ) 0) is defined when the first digital image ofthe dynamic pendant drop of heavy oil is taken. Becauseof the high viscosity of heavy oil and the small innerdiameter of the syringe needle, it usually takes ap-proximately tf ) 8 s to completely form a pendant heavyoil drop at the tip of the syringe needle. Once thependant oil drop is completely formed, its first digitalimage is taken and the corresponding drop volume isdenoted by V0. Therefore, the gas dissolved in thependant heavy oil drop at any time t after the first dropimage is taken can be found from eq 10

where tf is the time needed to completely form the

Figure 2. Calculated dimensionless gas concentration contours inside the pendant oil drop and the syringe needle at (a) ) 0.1; (b) )0.2; and (c) ) 0.4 (drop #1 of CO2-heavy oil system at P ) 4 MPa and T ) 23.9 C, initial drop volume V0 ) 5.764 mm3, syringe needleradius rn ) 0.79 mm, syringe needle wall thickness n ) 0.59 mm, syringe needle height hn ) 2.4 mm).

vgas )ksw - 1Mgascsat

(12)

Vc(t) )

V0 + (ksw - 1)rn3 ss(R,Z)2 C(R,Z,Dtrn2)R dR dZ (13) n(t + tf) - n(tf) ) csatrn3 ss(R,Z)2{C[R,Z,D(t + tf)rn2 ] - C(R,Z,Dtfrn2)}R dR dZ (14)

Ind. Eng. Chem. Res., Vol. 44, No. 12, 2005 4477

pendant drop of heavy oil and n(tf) represents thenumber of moles of gas dissolved in the pendant heavyoil drop during this period. More precisely, the abovechange of the number of moles of gas dissolved from tfto t + tf should be used in the numerical calculation ofthe volume of the dynamic pendant oil drop at any timet. Accordingly, eq 13 can be rewritten as

In this study, the following objective function E isdefined to describe the discrepancy between the theo-retically calculated oil drop volume Vc(t) and the ex-perimentally measured oil drop volume Vm(t), 0 e t etm:

Mathematically, the objective function E is equal to theroot-mean-squared relative error between the theoreti-cally calculated and experimentally measured volumesof the dynamic pendant drop. Physically, the minimumobjective function corresponds to the best fit of thetheoretically calculated volumes to the experimentallymeasured volumes of the pendant oil drop. Once thevolumes of the dynamic pendant drop at different timesare measured, the objective function depends on thetheoretically calculated volumes only. Thus, it can beseen from eq 15 that, in this case, the objective functionE defined in eq 16 is solely dependent on the two to-be-determined parameters, the gas diffusivity D and theoil swelling factor ksw, i.e.,

Therefore, D and ksw can be used as the adjustableparameters to minimize the objective function E. Oncethe minimum objective function Emin is found, thecorresponding values of D and ksw are the measured gasdiffusivity and oil swelling factor, respectively.

It is noticed from eq 15 that, in the minimization ofthe objective function E(D,ksw), the calculated volumeVc(t) is a linear function of the oil swelling factor ksw.Therefore, for a given guessing value of D, the corre-sponding local optimum value of ksw at which theobjective function E(D,ksw) is minimized can be foundanalytically. Substituting eq 15 into eq 16 and dif-ferentiating the objective function E(D,ksw) with respectto ksw yields

where

When the partial derivative in eq 18 is equal to zero,the oil swelling factor ksw is determined as

On the other hand, eq 15 shows that the calculatedoil drop volume Vc(t) at any time t is a complicatedfunction of the gas diffusivity D. Thus, the minimizationof the objective function E(D,ksw) with respect to D hasto be carried out numerically. In this study, the one-dimensional sequential search method38 is applied tofind the optimum gas diffusivity D that corresponds tothe minimum objective function. This search methodrequires that the objective function E(D,ksw) is a uni-modal function of D, i.e., there is only one minimum overthe search interval of D for the local optimum kswdetermined from eq 20. This requirement is satisfiedbecause, physically, a pair of the true gas diffusivity Dand the true swelling factor ksw exists for a given gas-heavy oil system at constant pressure and temperature.The detailed numerical procedure for determining thegas diffusivity D and the oil swelling factor ksw simul-taneously is described as follows:

(1) Sequential digital images of the dynamic pendantoil drop surrounded by a gas are acquired and its volumeVm(t), 0 e t e tm, is, thus, measured. Obviously, Vm(t)jt)0) V0.

(2) The first digital image of the dynamic pendant oildrop at t ) 0 is used to define the computational domain for the mass-transfer modeling, as shown in Figure1.

(3) The mass-transfer equation eq 6 subject to eqs 7-9is solved by applying the semidiscrete Galerkin finiteelement method to obtain the dimensionless gas con-centration distribution C(R,Z,) inside the computa-tional domain .

(4) The initial interval of uncertainty for the gasdiffusivity D is chosen as [D0, D10].

(5) The interval of uncertainty [D0, D10] is subdividedinto 10 equally spaced subintervals with 11 nodal pointsDi, i ) 0, 1, 2, , 10.

(6) For a given gas diffusivity Di, i ) 0, 1, 2, , 10,the corresponding local optimum oil swelling factor ksw,i,i ) 0, 1, 2, , 10 is determined from eq 20, at whichthe local minimum objective function E(Di,ksw,i) islocated.

(7) For each pair of (Di,ksw,i), i ) 0, 1, 2, , 10, thevolume Vc(t) of the dynamic pendant drop at any timet is calculated from eq 15, and the correspondingobjective function E(Di,ksw,i), i ) 0, 1, 2, , 10 isdetermined from eq 16.

(8) By comparing E(Di,ksw,i), i ) 0, 1, 2, , 10, theminimum objective function E(Dn,ksw,n), 1 e n e 9, isfound. Then, the new interval of uncertainty for the gasdiffusivity D becomes [Dn-1, Dn+1].

Vc(t) ) V0 + (ksw - 1)rn3 ss(R,Z)2{C[R,Z,D(t + tf)rn2 ] - C(R,Z,Dtfrn2)}R dR dZ (15)

E ) x 1tms0tm [Vm(t) - Vc(t)Vm(t) ]2 dt 100% (16)

E ) E(D,ksw) (17)

@E(D,ksw)@ksw

)

1tm{(ksw - 1)s0tm Q2(t)Vm2(t) dt - s0tm [Vm(t) - V0]Q(t)Vm2(t) dt}

x 1tm s0tm [Vm(t) - V0 - (ksw - 1)Q(t)Vm(t) ]2 dt(18)

Q(t) ) rn3 ss(R,Z)2 {C[R,Z,D(t + tf)rn2 ] -

C(R,Z,Dtfrn2)}R dR dZ (19)

ksw ) 1 +

s0tm [Vm(t) - V0]Q(t)Vm2(t) dts0tm Q

2(t)

Vm2(t)

dt

(20)

4478 Ind. Eng. Chem. Res., Vol. 44, No. 12, 2005

(9) If the new interval of uncertainty [Dn-1, Dn+1] isalready small enough to satisfy a prespecified accuracyfor the gas diffusivity determination, Dn and ksw,n arethe global optimum values of the gas diffusivity and oilswelling factor, which correspond to the global minimumobjective function E(Dn,ksw,n). Therefore, Dn and ksw,n arethe measured gas diffusivity and oil swelling factor,respectively. The minimization procedure is, thus, ter-minated. Otherwise, the above-described numericalsearch procedure after step 5 is repeated for the newinterval of uncertainty [Dn-1, Dn+1] until the prespecifiedaccuracy for the gas diffusivity D is satisfied.

3. Experimental Section

3.1. Materials. Carbon dioxide is purchased fromPraxair Canada with a stated purity of 99.99% (instru-ment grade). The heavy oil sample is collected from theLloydminster area, Canada. Prior to usage, the heavyoil sample is cleaned to remove water, fine sandparticles, and any solution gas. The density of thecleaned heavy oil sample is 988 kg/m3, and its viscosityis 23 000 mPas at 23.9 C. The asphaltene content (n-heptane insoluble) of the heavy oil is 11.5 wt %.

3.2. Apparatus. Figure 3 shows the schematic dia-gram of the experimental setup. The major componentof the setup is a see-through windowed high-pressurecell (IFT-10, Temco, U.S.). The pressure cell has achamber volume of 41.5 cm3 and can sustain pressuresup to 69 MPa. The maximum working temperature ofthe pressure cell is 177 C. The syringe needle used toform the pendant oil drop is made of stainless steel. Itsouter radius is rn ) 0.79 mm, and its wall thickness isn ) 0.59 mm. To specify the computational domain ,the height of the syringe needle is chosen as hn ) 2.4mm in this study. This height is used to ensure thatCO2 diffusion does not reach the cutting plane at theend of the diffusion test. Heavy oil is introduced from aheavy oil sample cylinder to the syringe needle by usinga programmable syringe pump (100DX, ISCO Inc.,U.S.). The pressure cell is wrapped with two heatingtapes (HT95504 1, Electrothermal, U.S.), and itstemperature is controlled by using a stepless tempera-ture controller (CN45515, Thermolyne, U.S.). The pres-

sure inside the pressure cell is measured by using adigital pressure gauge (DTG-6000, 3D Instruments,U.S.).

A light source and a diffuser are used to provideuniform illumination for the pressure cell. A microscopecamera (MZ6, Leica, Germany) with a resolution of 640 480 pixels is used to capture the sequential digitalimages of the dynamic pendant oil drop inside thepressure cell. The high-pressure cell is placed betweenthe light source and the microscope camera. The high-pressure cell, light source, diffuser, and microscopecamera are placed on a vibration-free table (RS4000,Newport, U.S.). The digital images of the dynamicpendant drop are acquired by using a digital framegrabber (Ultra II, Coreco Imaging, Canada) in taggedimage file format (TIFF). This optical system is able tograb the sequential digital images at the speed of threedigital images per second.

3.3. Procedure. Prior to each test, the high-pressurecell and the fluids handling system are thoroughlycleaned with kerosene and acetone, rinsed with carbondioxide several times, and finally evacuated to removeany traces of the cleaning reagents. Then CO2 isintroduced into the high-pressure cell from the CO2cylinder. After CO2 is injected, it usually takes about30 min for the pressure and temperature inside thepressure cell to reach their stable values. Then, a heavyoil sample is introduced by using the syringe pump toform a pendant drop at the tip of the syringe needle,which is installed at the top of the high-pressure cell.Once the pendant oil drop is formed, a digital imageacquisition program is executed to grab its sequentialimages during CO2 dissolution. The time intervals forthe sequential image acquisition are properly set so thatthe time intervals are smaller at the beginning andlarger when the pendant drop is almost saturated withCO2. These digital images are automatically stored inthe hard drive of the computer in TIFF format. For eachmeasured digital drop image, a high-precision calibra-tion grid is used to calibrate and correct possible opticaldistortions. A LIA (LIA Technologies Inc., Canada)computer image analysis and processing program forthe pendant drop case is executed to analyze thesedigital drop images. For each digital image of the

Figure 3. Schematic diagram of the experimental setup used for measuring the gas diffusivity in heavy oil and the oil swelling factorby the dynamic pendant drop volume analysis (DPDVA).

Ind. Eng. Chem. Res., Vol. 44, No. 12, 2005 4479

dynamic pendant oil drop, the output of this programis a set of coordinates of the pendant oil drop profileand its volume. After all the digital images of thedynamic pendant oil drop are processed, the experimen-tally measured volumes of the dynamic pendant oil dropat different times, Vm(t), 0 e t e tm, are, thus, obtained.

4. Results and Discussion

In this study, the developed DPDVA method isapplied to measure the diffusion coefficient and theswelling factor of a CO2-heavy oil system at fivedifferent pressures, P ) 2, 3, 4, 5, and 6 MPa, and aconstant temperature of 23.9 C. This temperature isthe same as that of the actual oil reservoir from whichthe heavy oil sample is collected and used in this study.Pressures are selected to cover the most practical casesof interest for solvent-based heavy oil recovery pro-cesses.

4.1. Measured Dynamic Pendant Drop Volumes.At each prespecified pressure and the constant temper-ature, the pendant oil drop volume measurements arerepeated for at least three different oil drops to ensuresatisfactory repeatability. For example, three differentpendant oil drops (drop #1, drop #2, and drop #3) aremeasured at P ) 4 MPa and T ) 23.9 C with theirinitial drop volumes of V0 ) 5.764, 5.690, and 5.601mm3, respectively. The sequential digital images of thedynamic pendant oil drop (drop #1) are shown in Figure4 at six different times, t ) 0, 140, 300, 450, 720, and1200 s. This figure clearly shows that the volume of thedynamic pendant oil drop increases as CO2 graduallydissolves into the heavy oil phase. More specifically, in

comparison with its initial value V0 ) 5.764 mm3 at t )0, the pendant oil drop volume increases by 4.54%,6.26%, and 7.06% at t ) 300, 720, and 1200 s, respec-tively. The measured pendant oil drop volume versustime curves are shown in Figure 5 for the three differentpendant drops of heavy oil at P ) 4 MPa and T ) 23.9C. It can be seen from Figure 5 that the measured dropvolume increases due to the oil swelling effect as CO2dissolves into heavy oil. Also, the volume increase of thependant drop is faster at the beginning and becomesslower later. Finally, the volume of the dynamic pendantdrop gradually approaches a constant as the pendant

Figure 4. Sequential digital images of the dynamic pendant heavy oil drop surrounded by CO2 at P ) 4 MPa and T ) 23.9 C (drop #1,initial drop volume V0 ) 5.764 mm3, drop formation time tf ) 8 s).

Figure 5. Comparison between the measured and calculatedvolumes of the three dynamic pendant heavy oil drops surroundedby CO2 at P ) 4 MPa and T ) 23.9 C (drop formation time tf )8 s).

4480 Ind. Eng. Chem. Res., Vol. 44, No. 12, 2005

oil drop is completely saturated with CO2. This isbecause, according to Ficks law,33 the mass-transferrate of CO2 into the pendant oil drop is proportional tothe CO2 concentration gradient at the CO2-heavy oilinterface. During the diffusion process, the CO2 con-centration gradient is larger at the beginning andbecomes smaller as more CO2 dissolves into the pendantoil drop. When the pendant oil drop is completelysaturated with CO2, the gas concentration gradient isequal to zero and, thus, the volume of the pendant oildrop reaches its maximum value.

4.2. Determination of CO2 Diffusivity and OilSwelling Factor. With the measured volumes of thedynamic pendant drop of heavy oil at different times,the objective function defined in eq 16 can be minimizedto determine the CO2 diffusivity D in heavy oil and theoil swelling factor ksw simultaneously. As describedpreviously in the theoretical section, for a given gasdiffusivity D, the local optimum ksw that correspondsto the local minimum objective function E(D,ksw) can bedetermined from eq 20. The local minimum objectivefunction E(D,ksw) versus CO2 diffusivity D curves areshown in Figure 6 for the three different pendant drops(drop #1, drop #2, and drop #3) at P ) 4 MPa and T )23.9 C. It can be seen from this figure that there is adistinctive dip in each curve where the global minimumobjective function is located. Thus, the correspondingdiffusivity D and oil swelling factor ksw are the measuredCO2 diffusivity and oil swelling factor, respectively.

The measured CO2 diffusivities D and oil swellingfactors ksw are included in Figure 6 for the threedifferent pendant oil drops tested at P ) 4 MPa and T) 23.9 C. The measured results are independent of thependant oil drop volume and show satisfactory repeat-ability of the determined CO2 diffusivities and swellingfactors for different pendant oil drops. The numericalresults also show that the global minimum objectivefunction, i.e., the root-mean-squared relative errorbetween the calculated and measured volumes of thedynamic pendant oil drop, is

the measured CO2 diffusivity and oil swelling factorlisted in Table 1, the volumes of the dynamic pendantoil drops can be calculated from eq 15 at each givenpressure. Figure 7 shows the comparison of the calcu-lated relative pendant oil drop volumes, i.e., Vc(t)/V0,with the measured relative pendant oil drop volumes,i.e., Vm(t)/V0. It can be seen from this figure that thetheoretically calculated results are in good agreementwith the measured data. Table 1 also shows that theglobal minimum objective function E(D,ksw), i.e., theroot-mean-squared relative error between the calculatedand measured volumes of the dynamic pendant oil drop,is

made available to implement the DPDVA method. Ingeneral, the DPDVA method can be applied to study thediffusion process of a fluid into an immiscible liquid aslong as the dissolution of the former into the latter cancause an observable volume change in their mixture.

Acknowledgment

The authors acknowledge the discovery grant fromthe Natural Sciences and Engineering Research Council(NSERC) of Canada and the Innovation Fund from thePetroleum Technology Research Centre (PTRC) to Y.G.

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Received for review February 4, 2005Revised manuscript received March 31, 2005

Accepted April 13, 2005

IE0501430

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