An inverse solution methodology for estimating the

ngineering 53 (2006) 189–202www.elsevier.com/locate/petrol

Journal of Petroleum Science and E

An inverse solution methodology for estimating the diffusioncoefficient of gases in Athabasca bitumen from pressure-decay data

Hussain Sheikha, Anil K. Mehrotra, Mehran Pooladi-Darvish ⁎

Department of Chemical and Petroleum Engineering, University of Calgary, Calgary, Alberta, Canada T2N 1N4

Received 3 March 2006; received in revised form 6 June 2006; accepted 7 June 2006

Abstract

An inverse solution methodology is developed for the estimation of diffusion coefficient of gases in highly viscous, oil-sandsbitumens from isothermal, pressure-decay measurements. The approach involves modeling the rate of change in pressure using thediffusion equation for the liquid phase coupled with a mass balance equation for the gas phase. The inverse solution framework isutilized to arrive at two graphical techniques for estimating the diffusion coefficient. Both techniques involve the determination ofthe slope of a straight line resulting from plotting the experimental data in accordance with the developed model. An advantage ofthe proposed techniques is that the diffusion coefficient is estimated directly, i.e. without making it an adjustable parameter. Thenovelty of the proposed method is in its simplicity as well as its ability to isolate portions of the pressure-decay data affected byexperimental fluctuations. The effect of the initial pressure on the predicted diffusion coefficient and pressure-decay profile wasalso investigated. The diffusion coefficients of CO2, CH4, C2H6 and N2 in Athabasca bitumen at 50–90 °C and about 8 MPa wereestimated and compared with literature values.© 2006 Elsevier B.V. All rights reserved.

Keywords: Diffusion coefficient; Gas–bitumen mixtures; Inverse solution; Pressure-decay data; Mathematical modeling; Mass transfer; Interfaceboundary condition

1. Introduction

There is a growing need for the oil and gas industry toimprove the recovery of bitumens and heavy oils,particularly in view of declining reserves of conven-tional oils. The high viscosity of Alberta's bitumens andheavy oils poses a challenge in the in-situ recoveryprocesses. The mobility and recovery of bitumens andheavy oils can be improved by lowering their viscosity,which could be accomplished by injecting a diluent,

⁎ Corresponding author. Tel.: +1 403 220 8779; fax: +1 403 2844852.

E-mail address: [email protected] (M. Pooladi-Darvish).

0920-4105/$ - see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.petrol.2006.06.003

such as gases or light hydrocarbons (Butler and Mokrys,1991). For example, carbon dioxide with a relativelyhigh solubility in heavy oils and bitumens at typicalreservoir conditions causes a significant viscosityreduction (Mehrotra and Svrcek, 1982).

Molecular diffusion plays an important role inbitumen recovery processes as well as in many otherreservoir engineering applications. The Vapex process,for example, is an enhanced oil recovery method thatinvolves injecting light hydrocarbons to decrease the oilviscosity and enhance the flow. The oil production ratefrom the Vapex process is controlled by moleculardiffusion and dispersion (Butler and Jiang, 2000).

An accurate value of the diffusion coefficient ofgases in bitumens is essential for calculating the rate of

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http://dx.doi.org/10.1016/j.petrol.2006.06.003

190 H. Sheikha et al. / Journal of Petroleum Science and Engineering 53 (2006) 189–202

gas dissolution in bitumens and heavy oils. The gas-solubility, on the other hand, is a thermodynamicproperty that defines the extent to which a gas (orlight hydrocarbons) will dissolve in crude oils atreservoir conditions. Thus, the diffusion coefficient, atransport property, controls the rate of mass transfer ofgases in bitumens and heavy oils, whereas the solubilitydetermines the amount of gas that will dissolveeventually in the liquid phase.

Pomeroy et al. (1933) estimated the diffusioncoefficient and the solubility of lighter hydrocarbonsin quiescent liquids. They concluded that diffusioncoefficient is not affected by pressure or concentrationof the methane gas in solution up to a pressure of 300 psi(∼2 MPa). Reamer and Sage (1956) reported that anincrease in pressure at lower temperatures results in anincrease in the diffusion coefficient; however, attemperatures above approximately 27 °C, an increasein pressure resulted in a significant decrease in thediffusion coefficient. Reamer et al. (1956) calculated thediffusion coefficient of methane in hydrocarbon mix-tures by assuming a resistance at the interface, whichwas taken to be proportional to the mass transfer rate.Grogan and Pinczewski (1987) measured the diffusioncoefficient of carbon dioxide in oil and the oil shieldedby water by tracking the interface movement andcomparing it with the results from a numerical model.Renner (1988) determined the diffusion coefficient bymeasuring the volume of dissolved gas in the liquidphase with time at constant pressure. The experimentaldata were confirmed by the predicted straight-linebehavior after an initial period, termed the incubationperiod, during which the dissolution of gas into liquid atthe interface established the boundary condition.

Riazi (1996) developed a model to estimate the gasdiffusion coefficient from the gas–liquid interfaceposition or the pressure in a constant-volume cell. Inthis method, the velocity of the interface movement aswell as the rate of pressure change in the cell dependedon the rate of diffusion. Zhang et al. (2000) measuredthe decrease in the pressure of a constant volume of gasas it diffused into the bitumen. Their approach is asimplified version of the method by Riazi (1996), whichdid not require the measurement of the interface positionwith time. They coupled the diffusion coefficient withthe gas mass balance equation to match the gasabsorption data using the diffusion coefficient as anadjustable parameter; however, the effect of bitumenswelling was neglected and the gas compressibilityfactor was assumed to be constant.

Civan and Rasmussen (2001, 2002, 2006) developeda mathematical model for the experimental set-up of

Zhang et al. (2000) to estimate the gas diffusioncoefficient. They presented two models: equilibriumand non-equilibrium. The non-equilibrium modelaccounted for the delay in gas transport in the liquidphase. Civan and Rasmussen (2006) reported differentvalues of diffusion coefficients for the same data ofRiazi (1996) and Zhang et al. (2000). They concludedthat the accuracy of the available models is limited bythe inherent simplifying assumptions. Tharanivasan etal. (2004) employed the approach of Zhang et al. (2000)and found that the diffusion coefficient is sensitive to theinterface boundary condition. They recommended thatan appropriate boundary condition should be used foreach specific gas, i.e. equilibrium, quasi-equilibrium ornon-equilibrium. Yang and Gu (2003) presented anexperimental method along with a computationalscheme for determining the diffusion coefficient ofmethane and carbon dioxide in a heavy oil using thedynamic pendant drop shape analysis.

Upreti and Mehrotra (2000, 2002) developed anumerical technique to obtain the diffusion coefficientas a function of gas concentration. They obtained thediffusivity values from transient data obtained from thepressure-decay of a constant volume of gas as itdissolved in a layer of bitumen. The diffusion coeffi-cients of methane, ethane, carbon dioxide, and nitrogenin bitumens were provided at 25–90 °C at pressures up to8 MPa.

Recently, Sheikha et al. (2005) presented techniquesfor the estimation of the diffusion coefficient of gases inbitumens. Their graphical methods isolated the early andlate time periods that should be excluded in the analysisfor obtaining the diffusion coefficient. They estimatedthe diffusion coefficient from the data of Upreti andMehrotra (2000, 2002) at 4 MPa, which were within therange of literature values.

Creux et al. (2005) conducted two experimental teststo measure the diffusion coefficient of methane in heavyoils. In the first test, they estimated the diffusioncoefficient from the measurement of the pressure dropin a PVT cell. In the second test, they estimated thediffusion coefficient from the concentration variation ofmarkedmethanemolecules with 13C. The 13CH4 gas wasplaced on one side of a plane sheet and a methane–carbon dioxide mixture on the other, and a high-pressuresampler linked to an analyzer to measure the isotoperatio.

One of the shortcomings of the available approachesin the literature is their inability to differentiate betweenvarious stages in the gas diffusion process. This can leadto errors in the estimated diffusion coefficients obtainedfrom the experimental data. The effects of temperature

191H. Sheikha et al. / Journal of Petroleum Science and Engineering 53 (2006) 189–202

fluctuations, convection (for light oils) or incubationperiod, for example, are inadvertently included in theestimated value of the diffusion coefficient. Riazi (1996)emphasized the importance of checking the existence ofnatural convection in each phase in the vertical cell.

This paper presents a methodology that can differ-entiate between the various stages in the diffusionprocess. The method is used for the estimation ofdiffusion coefficient at higher pressures from theisothermal pressure-decay measurements by Upretiand Mehrotra (2000, 2002) at 8 MPa. It includes theeffects of initial pressure on the reproduced pressureprofile and the late time on the diffusion process. Twographical methods are developed for estimating thediffusion coefficient.

2. The forward problem

A mathematical model for the system underinvestigation is needed to predict the measurement. Itshould be based on a sound theoretical framework thatlinks the model parameters to the measured parameters.This prediction of observations, given the values of theparameters defining the model, constitutes the normalproblem or, in the terminology of the inverse problemapproach, the forward problem.

There are two parameters that control the gasdiffusion process, such as that in the experiments ofUpreti and Mehrotra (2000, 2002). One is the gassolubility, which establishes the concentration of gasdissolved at the gas–bitumen interface. The other is thediffusion coefficient, which controls the rate of masstransfer within the liquid (bitumen) phase.

For the estimation of diffusion coefficient, ananalytical solution is preferable for the gas pressure,which can then be inverted to find the diffusioncoefficient. An analytical solution is obtained by makinga few simplifying assumptions, including a non-reactivesystem, isothermal conditions, a constant diffusioncoefficient, constant gas compressibility factor, non-volatile bitumen, negligible bitumen swelling, equili-brium at the gas–bitumen interface, and the applicabilityof Henry's law. These assumptions were examined forgas–bitumen systems and the magnitude of theassociated errors was evaluated (Sheikha et al., 2005).

The two models for predicting the pressure of the gasphase are presented in the following section. These are:finite-acting model and the infinite-acting model. Thediffusion coefficient is obtained from the infinite-actingmodel. The finite-acting model is used to determine thetime at which the infinite-acting model ceases toadequately represent the system.

2.1. The infinite-acting model

As the gas diffuses into the bitumen, its penetrationdepth increases with time. At early times, the diffusedgas behaves as though it were diffusing into an infinitecell. This assumption is valid as long as the diffusing gasdoes not reach the cell bottom. Pomeroy et al. (1933)investigated the validity of this assumption. They foundthat this assumption introduces an error of 0.25% whenthe liquid contains less than half of the gas required toreach full saturation, and an error of 4.7% when the gassaturation is 70%.

With the assumptions stated previously, the diffusionof gas into the bitumen can be modeled by Fick's secondlaw of diffusion:

DA2CAz2

¼ ACAt

ð1Þ

For the gas-free bitumen samples used in pressure-decayexperiments of Upreti and Mehrotra (2000, 2002), theinitial condition is:

C ¼ 0; zz0; t ¼ 0 ð2Þ

The interface boundary condition must satisfy tworequirements. First, the gas–bitumen interface is alwaysunder equilibrium at t>0. Secondly, the mass of thediffused gas is equal to the difference between the initialmass of the gas and the mass of the remaining gas phasein the pressure cell. Based on these two criteria, theinterface boundary condition is:

DAACAz

��z¼0

¼ VMZRT

dPdt

; z ¼ 0; t > 0 ð3Þ

The concentration of the gas in the bitumen can beexpressed in term of pressure by using Henry's law(Felder and Rousseau, 1986), as follows:

P ¼ KhC ð4ÞBy combining Eqs.(3) and (4), the interface boundarycondition takes the following form:

DAAC

Az

��z¼0

¼ VMKh

ZRT

dC

dt; z ¼ 0; t > 0 ð5Þ

During early times of a pressure-decay experiment,before the closed boundary at the cell bottom begins toaffect the rate of mass transfer between the gas and thebitumen, an infinite-acting boundary condition can beassumed:

C ¼ 0; zYl; t > 0 ð6Þ


The mathematical formulation, given by Eqs. (1), (2),(5) and (6) was solved using the method of Laplacetransform and the analytical solution in terms of the gasphase (cell) pressure is (Sheikha et al., 2005):

P tð Þ ¼ PiexpZRT

ffiffiffiffiD

p

LMKh

ffiffit

p� �2

erfcZRT

ffiffiffiffiD

p

LMKh

ffiffit

p� �ð7Þ

The above relationship is the forward solution forthe gas phase pressure as a function of time, whichrequires values of all other variables including thediffusion coefficient. An inverse solution method forobtaining the diffusion coefficient from Eq. (7) will bederived later. Eq. (7) suggests that pressure in the gaszone is a function of the square root of the diffusioncoefficient divided by Henry's constant. Therefore, theanalysis of the pressure-decay data will result inffiffiffiffiD

p=Kh. This is an important observation, which

suggests that the infinite-acting data do not containenough information to allow independent estimation ofboth the Henry's constant and the diffusion coefficient(Sheikha et al., 2005). Henry's constants for CO2–bitumen, CH4–bitumen and N2–bitumen mixtures wereobtained from the gas-solubility correlations forAthabasca bitumen (Mehrotra and Svrcek, 1982). Asimilar expression was obtained by fitting the experi-mental data for the solubility of ethane in Athabascabitumen, reported by Mehrotra and Svrcek (1985), asfollows.

lnRs ¼ 3:1995� 3:9356P0:5

þ 271610T2

ð8Þ

where Rs=volumetric solubility (m3/m3), P=pressure(MPa), and T=temperature (K).

2.2. The finite-acting model

As mentioned previously, the finite-acting modelwas developed to determine the time beyond whichthe assumption of the infinite-acting boundarycondition would not be valid. In the finite-actingmodel, the second boundary condition at the cellbottom is:

ACAz

¼ 0; z ¼ H ; t > 0; ð9Þ

The analytical solution of the finite-acting model,consisting of Eqs. (1), (2), (5) and (9), in the Laplace

domain is given by the following equation (Sheikha etal., 2005):

P sð Þ ¼c1Pi 1þ exp �2H

ffiffiffiffis

D

r� ��

c1sþffiffiffiffisD

r� �þ c1s�

ffiffiffiffisD

r� �exp �2H

ffiffiffiffisD

r� �

ð10Þwhere s denotes the Laplace transform variable and c1=(LMKh) / (DZRT). Numerical solutions for the finite-acting model were obtained using the Stehfest (1970)algorithm.

3. The inverse problem

The inverse problem involves using the results ofactual observations to determine the optimum values ofthe parameters characterizing the system under investi-gation. Generally speaking, the inverse problem maylead to non-unique results. Different values of the modelparameters may be consistent with the data but, forexample, the measurement of gas pressure in a cell maynot be sufficient to determine the height of bitumenlayer in the cell. Also, determining the values of allmodel parameters may require exploring a largeparameter space.

The inverse solution methodology finds manyapplications in engineering. In well testing, for example,the reservoir permeability can be estimated by measur-ing the pressure change in the well-bore. The governingequation in the propagation of pressure is the diffusivityequation of single-phase fluid flow in the radialdirection. Techniques have been developed for identify-ing a portion of the data that follows the ideal forwardsolution (Mattar, 1999). Likewise, the dispersion andthermal conductivity coefficients can be estimated fromthe concentration and the temperature measurements,respectively.

In this study, an inverse solution technique ispresented for the determination of diffusion coefficientby analyzing the measured gas pressure values withtime. A graphical approach is developed that isanalogous to that for the determination of reservoirpermeability in well-testing (Mattar, 1999). In thegraphical technique used in well-testing, the pressure,or its derivative, is plotted on certain coordinates and thepermeability is found from the resulting plot. Theadvantage of graphical techniques is that they allow theidentification of portions of the data that are affected byprocesses that are not included in the forward model.For the diffusion problem, if the data at early times were

Table 1Summary of gas–bitumen pressure-decay experiments at about 8 MPaby Upreti and Mehrotra (2000, 2002)

Pressure-decayexperiment

Gas Temperature(°C)

Pi

(MPa)Pf

(MPa)Duration(h)

H(m)

Run 1 CH4 50 7.95 7.46 85 0.0112Run 2 CH4 90 7.99 7.38 28 0.0112Run 3 C2H6 75 7.77 6.74 27 0.0107Run 4 C2H6 90 8.27 7.40 12 0.0103Run 5 N2 50 7.84 7.75 32 0.0101Run 6 N2 75 7.90 7.74 30 0.0109Run 7 N2 90 7.90 7.74 20 0.0115Run 8 CO2 50 7.68 6.96 45 0.0106Run 9 CO2 75 7.82 6.57 37 0.0116Run 10 CO2 90 7.65 6.84 14 0.0100


affected by pressure changes or if the later-time datawere affected by the closed boundary or convection, thegraphical representation would allow the identificationof such data for their exclusion from calculations. Twosuch graphical techniques are presented in the followingsection.

The data from ten pressure-decay tests of Upreti andMehrotra (2000, 2002) were subjected to the inversesolution methodology derived from Eq. (7). These tests,referred to as Runs 1–10, involved the diffusion ofmethane, ethane, nitrogen, and carbon dioxide intobitumen at a constant temperature (i.e. approximately50, 75 or 90 °C). The experimental conditions of theseten pressure-decay runs are summarized in Table 1. Thediffusion coefficients were obtained for these tests andcompared with published values.

3.1. Graphical Method A

This method is a modification of Method I presentedin our previous study (Sheikha et al., 2005). The non-linearity in Eq. (7) did not allow an explicit expressionfor the diffusion coefficient. The magnitudes of boththe complementary error function term and theexponential term in Eq. (7) were compared, whichshowed that the exponential term remained close tounity (Sheikha et al., 2005). Thus, in developingGraphical Method A, the exponential term in Eq. (7)was ignored altogether. Eq. (7) was rearranged into thefollowing expression:

erfc�1 PðtÞPi

� �¼ ZRT

ffiffiffiffiD

p

LMKh

ffiffit

p ð11Þ

Note that Eq. (11) gives a straight line on a graph oferfc−1{P(t)/Pi} versus

ffiffit

p, from which the diffusion

coefficient could be obtained from the slope of thisstraight line. The slope, denoted as b, is equal toZRT

ffiffiffiffiD

p=LMKh. With b determined graphically, the

diffusion coefficient can be obtained from the followingexpression.

D ¼ bLMKh

ZRT

� �2

ð12Þ

Obtaining the diffusion coefficient by means ofMethod A is somewhat similar to Method I of Sheikha etal. (2005), except Method A involves using the slope ofthe resulting straight line for finding the diffusioncoefficient instead of the derivative. For estimating thediffusion coefficient, Method A is more straightforwardthan Method I of Sheikha et al. (2005) because it avoidsthe need for data-smoothening due to amplifiedfluctuations introduced in the derivative method.However, the end of the infinite-acting period in MethodAwas not as sharp as it was in Method I of Sheikha et al.(2005).

3.2. Graphical Method B

In this new method, the exponential term in Eq. (7) isnot ignored. Instead, the complementary error functionis approximated using the following expression (Abra-mowitz and Stegun, 1970):

erfcð/Þ ¼ ða3b3 þ a2b2 þ a1bÞ � expð/�2Þ

þ eð/Þ ð13Þ

where,

b ¼ 1a4/þ 1

ð14Þ

with a1=0.3480242, a2=−0.0958798, a3=0.7478556,a4=0.47047 and |ε(ϕ)|≤2.50×10−5.

Next, the following substitution is used to simplifyEq. (7).

/ ¼ ZRTffiffiffiffiD

p

LMKh

ffiffit

p ð15Þ

By substituting Eq. (15) into Eq. (7), we obtain:

PðtÞ ¼ Pi � expð/Þ2erfcð/Þ ð16ÞThen, by substituting Eq. (13) into Eq. (16), thefollowing expression was obtained.

PðtÞ ¼ Pi � a3b3 þ a2b

2 þ a1b� ð17Þ


The substitution of Eq. (14) into Eq. (17) yielded thefollowing expression.

PðtÞPi

¼ a3

ða4/þ 1Þ3 þa2

ða4/þ 1Þ2 þa1

ða4/þ 1Þ ð18Þ

The above equation was rearranged into the followingsimplified form:

a/3 þ b/2 þ c/þ d ¼ 0 ð19Þwhere,

aua34PðtÞPi

� �; bu

3a24PðtÞPi

� a1a24

� �;

cu3a4PðtÞ

Pi� a2a4 � 2a1a4

� �; and

duPðtÞPi

� a3 � a2 � a1

� �:

Note that Eq. (19) is a cubic equation, which wasfound to have one real root and two imaginary roots(which were ignored). After obtaining the real root ateach measured pressure, the results were plotted onCartesian coordinates according to Eq. (15), i.e. with

ffiffit

pon the abscissa and ϕ on the ordinate. The slope of theresulting straight line, b was equated to ZRT

ffiffiffiffiD

p=LMKh,

and the diffusion coefficient was obtained from Eq. (12).Method B is the new inverse-solution method forestimating the diffusion coefficient.

Fig. 1. Estimation of the diffusion coefficient from the early-time d

4. Results and discussion

4.1. Estimation of diffusion coefficient from the inversesolution

In this section, Graphical Methods A and B areapplied to obtain the diffusion coefficient of methane,ethane, nitrogen and carbon dioxide in Athabascabitumen. The data from the ten pressure-decay tests ofUpreti and Mehrotra (2000, 2002) were subjected to theinverse solution methodology derived from Eq. (7). Thediffusion coefficient values were obtained from theportion of data that conformed to the infinite-actingbehavior.

In Graphical Method A, the diffusion coefficient wasobtained by plotting the experimental data in accordancewith Eq. (11). Fig. 1 presents the experimental data forRun 1, which was for the methane–bitumen system at50 °C. Note that the data corresponding to both theearly-time and late-time periods show departures fromthe straight line; hence, these were excluded incalculating the slope of the straight line.

The infinite-acting time was found by plotting thedifference between the pressures for the finite-actingand infinite-acting periods, as shown in Fig. 2. Thetime for the end of the infinite-acting behavior in Fig.2 corresponds to that in Fig. 1. For the results in Fig.2, it was found that the bitumen phase containedabout 50% of the gas needed to reach saturation afterabout 42 h. The slope of the straight line in Fig. 1was used to calculate the diffusion coefficient fromEq. (12).

ata using Graphical Method A for Run 1 with CH4 at 50 °C.

Fig. 2. Determining the end of the infinite-acting behavior from the difference between the predicted pressure-decay profiles for finite-acting andinfinite-acting models for Run 1.


The same data of Run 1 were analyzed usingGraphical Method B for obtaining the diffusioncoefficient. The calculations involved finding thefunction ϕ and then plotting the experimental dataaccording to Eq. (15). The results are shown in Fig. 3.Similar to Graphical Method A, an importantcharacteristic of this method was to identify theportion of the data that did not follow the mathema-tical model. After determining the value of slope b,Eq. (12) was used to obtain the diffusion coefficient.As shown in Table 2, Methods A and B gave about

Fig. 3. Estimation of the diffusion coefficient from the early-time d

the same diffusion coefficient of 1.3×10−10 and1.4×10−10 m2/s, respectively.

The early time experimental data for Runs 5–7 areplotted in Fig. 4 according to Graphical Method A. Notethat the experimental data deviate from the straight lineat early times. For all runs, the pressure data for t<1 hwere excluded from calculating the slope of the straightline. As a further validation of the results of thesecalculations, the diffusion coefficients for Run 1 ob-tained from Graphical Methods A and B were used toreproduce the experimental pressure-decay profile. As

ata using Graphical Method B for Run 1 with CH4 at 50 °C.

Table 2Comparison of estimated diffusion coefficients for CH4, C2H6, N2 and CO2 in Athabasca bitumen; data of Upreti and Mehrotra (2000, 2002)

Pressure–decayexperiment

Gas Temperature,(°C)

Diffusion coefficient (D), m2/s×1010

This study at Pi≈8 MPa Sheikha et al. (2005) at Pi≈4 MPa Upreti and Mehrotra (2002)

Graphical MethodA

Graphical MethodB

Graphical MethodI

Graphical MethodII

AtPi≈4 MPa

AtPi≈8 MPa

Run 1 CH4 50 1.3 1.4 n.a. n.a. n.a. 1.5Run 2 CH4 90 5.3 5.8 5.5 7.3 4.3 8.6Run 3 C2H6 75 1.3 1.6 n.a. n.a. 4.2 4.9Run 4 C2H6 90 2.5 3.0 n.a. n.a. 6.0 6.9Run 5 N2 50 1.0 1.0 n.a. n.a. 0.5 1.7Run 6 N2 75 3.3 3.4 2.5 2.6 2.3 4.7Run 7 N2 90 5.0 5.2 4.0 2.8 5.0 7.5Run 8 CO2 50 1.0 1.1 n.a. n.a. 2.4 4.0Run 9 CO2 75 4.3 5.3 5.1 4.6 3.7 7.5Run 10 CO2 90 6.4 8.7 7.9 7.6 4.3 9.3


shown in Fig. 5, a good match was obtained between theexperimental and predicted pressure-decay profiles withboth values of the diffusion coefficient.

4.2. Effect of initial pressure

In the forward model, Eq. (7), P(t) is not only afunction of the diffusion coefficient but also the initialpressure, Pi. The diffusion coefficient determines therate of gas dissolution in the bitumen, which with timedecreases the pressure. On the other hand, the initialpressure relates to the time for the commencement of thediffusion process. For some of the runs, it was observedthat the predicted diffusion coefficient, although correct,did not provide a satisfactory match of the experimental

Fig. 4. Deviation of the experimental data at early-times fro

pressure-decay profile. Fig. 6 shows such a comparisonfor Run 2, where the difference between the predictedand experimental pressure increases with time.

Both Eqs. (11) and (15) represent straight lines, withthe intercept equal to zero. Calculations for Runs 2 and5, for example, gave best-fit lines with the intercept notbeing zero. The initial pressure, in pressure-decayexperiments, would ideally relate to the time at whichthe desired initial pressure is established and thediffusion process is commenced. However, the diffu-sion process would commence while the cell is filledwith the gas, i.e. it would commence earlier thanachieving the desired initial pressure. In this sense, itwould be difficult to obtain the true value of the initialpressure at t=0. Likewise, for diffusion calculations

m the developed model for Runs 5, 6 and 7 with N2.

Fig. 5. Data and predictions for the pressure-decay profiles for Run 1 with CH4 at 50 °C using the estimated diffusion coefficients from GraphicalMethods A and B.


related to the pressure-decay experiment, the actualpressure at t=0 can be less than the experimentallyrecorded value of Pi.

In order to obtain an estimate of the correct value ofPi, the intercepts of the straight lines in Figs. 1 and 3were obtained. For Graphical Method A, the value of P(0) at t=0 was obtained from the intercept, erfc−1(P(0)/Pi). For Graphical Method B, the following procedurewas used to estimate P(0). The intercept for the best-fitstraight line was obtained. The value of ϕ was thenobtained from this intercept and substituted into Eq. (18)to obtain P(0).

Fig. 6. Effect of initial pressure on the predicted press

A sensitivity analysis was undertaken to investigatethe effect of Pi on the slope of the straight line, b, andconsequently the diffusion coefficient. The results ofcalculations for a 0.5% increase and decrease in Pi ondata of Run 5 are shown in Fig. 7. The results indicatedthat the slope of the straight line does not change as aresult of changes in Pi. As shown in Fig. 8, for a changein Pi of −0.38%, which was small enough to make theintercept equal to zero, the slope of the straight line didnot change. This small change in Pi also provided aconsiderable improvement in the match of the pressure-decay profile, as shown in Fig. 6.

ure-decay profile for Run 2 with CH4 at 90 °C.

Fig. 7. Effect of ±0.5% variations in the initial pressure on the slope of the straight line.


4.3. Predicted diffusion coefficients

The various assumptions and approximationsinvolved in deriving the mathematical models forestimating the diffusion coefficient from the experimen-tal data introduce errors in reported values of the diffusioncoefficients (Riazi, 2005). As a result, there are no uniquevalues and the reported values can vary slightly.

The diffusion coefficient values obtained for all tenruns with the four gases are given in Table 2. The resultsin Table 2 suggest that the diffusion coefficient increaseswith temperature for all gas–bitumen mixtures, which isin agreement with the trends reported by Upreti and

Fig. 8. Predicted pressure-decay profiles using the experimental a

Mehrotra (2000, 2002). Table 2 also presents acomparison of the results obtained in this study withthose of Sheikha et al. (2005) and Upreti and Mehrotra(2000, 2002). For the same pressure-decay data at P-i≈8 MPa, the diffusion coefficients from this study aresomewhat lower than those reported by Upreti andMehrotra (2000, 2002). Also, in Table 2, the diffusioncoefficients at 8 MPa are generally higher than thosereported at 4 MPa by Sheikha et al. (2005), and thisobservation is also supported by the results of Upreti andMehrotra (2000, 2002).

In Fig. 9, the diffusion coefficients from bothgraphical methods are in agreement with those reported

nd modified initial pressure for Run 2 with CH4 at 90 °C.

Fig. 9. Comparison of diffusion coefficients predicted from Graphical Methods A and B with literature data.


by other investigators (Upreti andMehrotra, 2000, 2002;Schmidt, 1989). Note that Schmidt (1989) reported thediffusion coefficient of methane in bitumen at 50 °C overa range of 0.75×10−10 to 4.0×10−10 m2/s, and thelowest and the highest values have been plotted in Fig. 9.

The reproduced pressure-decay profiles for all gasesat 50 °C, 75 °C and 90 °C are compared with the data ofUpreti and Mehrotra (2000, 2002) in Figs. 10, 11, and12, respectively. For all cases, the estimated values ofthe diffusion coefficient provided a satisfactory match ofthe experimental pressure-decay profiles.

Using different approaches for estimating the diffu-sion coefficient can lead to somewhat different results

Fig. 10. Comparison of experimental and predicted

even for the same set of data. However, the reportedresults are generally within the same order of magni-tude. Schmidt (1989) reported the diffusion coefficientof methane in bitumen, at the same conditions ofpressure and temperature, as a range of values. Thediffusion coefficient of methane in water reported bySchas (1998) is 1.4×10−9 m2/s at 25 °C. However, theestimated diffusion coefficients at the same conditionsobtained by Civan and Rasmussen (2002) were4.2×10−9 m2/s. Yang and Gu (2003) used a dynamicpendant drop shape analysis to report a diffusioncoefficient of 1.14×10−9 m2/s for CO2 diffusing inbitumen at 25 °C, which is an order-of-magnitude higher

pressure-decay profiles for all gases at 50 °C.

Fig. 11. Comparison of experimental and predicted pressure-decay profiles for all gases at 75 °C.


than that reported by Upreti and Mehrotra (2002).Ignoring the swelling of bitumen phase leads an errorthat gets magnified particularly in the higher-pressureexperiments, and consequently yields a lower estimatefor diffusion coefficient. The approach presented byUpreti and Mehrotra (2000, 2002) takes the bitumenswelling into consideration.

4.4. Comments on the applicability of the proposedmethods

The graphical methods developed in this study canserve to identify the end of the infinite-acting behavioras well as for finding the diffusion coefficient from that

Fig. 12. Comparison of experimental and predicted

portion of the data for which the infinite-actingassumption is valid. As shown, if the pressure-decaydata are affected by experimental fluctuations or theclosed boundary at the bottom of the cell, the data didnot fall on the straight line and were not used forobtaining the diffusion coefficient. That is, only that partof data that followed the infinite-acting behavior wasused in the analysis.

The predicted pressure-decay profiles with thediffusion coefficients estimated from the two graphicalmethods are very similar to the extent that one methodcannot be judged to be preferable over the other. Thepredicted diffusion coefficients from both graphicalmethods compared well for all gas–bitumen systems.

pressure-decay profiles for all gases at 90 °C.


Another advantage of the proposed methods is that therewas no need to take the derivative of the pressureequation, which can amplify the fluctuations associatedwith the pressure-decay data.

5. Conclusions

Two graphical methods were developed for theestimation of diffusion coefficient of gases in bitumenfrom pressure-decay measurements. These methodswere based on the inverse solution approach for solvingthe mathematical model. It was found that the early-timedata do not adequately follow the developed mathema-tical model; hence, these data were excluded from thecalculation of the diffusion coefficient. Moreover, due toexperimental uncertainties, the measured initial pressuremay not agree with the forward solution. An optimizedmethod was proposed for adjusting the initial pressure,which was successful in reproducing the experimentalpressure-decay profiles.

The estimated values of diffusion coefficient forgas–bitumen mixtures, involving four gases over atemperature range of 50–90 °C at an initial pressureof approximately 8 MPa, varied from about 1×10−10

to 9×10−10 m2/s. These values were shown to be insatisfactory agreement with those reported in theliterature.

NomenclatureA cross-sectional area of the pressure cell (m2)b slope of the straight line in Graphical Method

A and BC concentration of gas in bitumen (kg/m3)c1 (LMKh) / (DZRT)D diffusion coefficient (m2/s)Kh Henry's constant (Pa m3/kg)H thickness of bitumen layer in the pressure cell

(m)L height of gas zone in the pressure cell (L=

0.03−H) (m)M molar mass of gas (kg/kmol)md mass of gas diffused in bitumen (kg)P pressure (Pa)P(s) pressure term in the Laplace domainPi initial pressure (Pa)R universal gas constant; R=8314 J kmol−1 K−1

Rs volume-basis solubility of gas in bitumen (m3

(at 0 °C and 101.3 kPa)/m3)s Laplace transform variablet time (s)T temperature (°C or K)V volume of gas (m3)

z depth (m)Z gas compressibility factor

Subscriptsf final valuei initial value

Greek symbolsϕ function defined by Eq. (15)

Acknowledgment

Financial support from the Natural Sciences andEngineering Research Council of Canada (NSERC), theConsortium for Heavy Oil Research by UniversityScientists (CHORUS), and the Department of Chemicaland Petroleum Engineering at the University of Calgaryis gratefully acknowledged. We thank Dr. S.R. Upreti(presently at Ryerson University, Toronto, Canada) forproviding the pressure-decay data used in this study.

References

Abramowitz, M., Stegun, I.A., 1970. Handbook of mathematicalfunctions with formulas, graphs, and mathematical tables. Natl.Bur. Stand., Appl. Math. 55 (Ninth printing).

Butler, R.M., Jiang, Q., 2000. Improved recovery of heavy oil byVapex with widely spaced horizontal injectors and producers.J. Can. Pet. Technol. 39 (1), 48–56.

Butler, R.M., Mokrys, I.J., 1991. A new process (Vapex) forrecovering heavy oils using hot water and hydrocarbon vapor.J. Can. Pet. Technol. 30 (1), 97–106.

Civan, F., Rasmussen, M.L., 2001. Accurate Measurement of GasDiffusivity in Oil and Brine Under Reservoir Conditions. SPEProduction and Operations Symposium, Oklahoma City, OK. SPEPaper, vol. 67319.

Civan, F., Rasmussen, M.L., 2002. Improved Measurement of GasDiffusivity for Miscible Gas Flooding under Non-equilibrium vs.Equilibrium conditions. SPE/DOE Improved Oil RecoverySymposium, Tulsa, OK. SPE Paper, vol. 75135.

Civan, F., Rasmussen, M.L., 2006. Determination of gas-diffusion andinterface-mass-transfer coefficients for quiescent reservoir liquids.SPE J. 11 (1), 71–79.

Creux, P.,Meyer, V., Graciaa, A., Cordelier, P.R., Franco, F.,Montel, F.,2005. Diffusivity in Heavy Oils. SPE International ThermalOperations and Heavy Oil Symposium, Calgary, AB. SPE Paper,vol. 97798.

Felder, R.M., Rousseau, R.W., 1986. Elementary Principles ofChemical Processes. Wiley, New York.

Grogan, A.T., Pinczewski, W.V., 1987. The role of molecular diffusionprocesses in tertiary CO2 flooding. J. Pet. Technol. 39 (5),591–602.

Mattar, L., 1999. Derivative analysis without type curves. J. Can. Pet.Technol. (special edition CD ROM), 38(13): paper 97–51 (6 pp).

Mehrotra, A.K., Svrcek, W.Y., 1982. Correlations for properties ofbitumen saturated with CO2, CH4 and N2, and experiments withcombustion gas mixtures. J. Can. Pet. Technol. 21 (6), 95–104.


Mehrotra, A.K., Svrcek, W.Y., 1985. Viscosity, density and gassolubility data for oil sand bitumens. Part I: Athabasca bitumensaturated with CO and C2H6. AOSTRA J. Res. 1 (4), 263–268.

Pomeroy, R.D., Lacey, W.N., Scudder, N.F., Stapp, F.P., 1933. Rate ofsolution of methane in quiescent liquid hydrocarbons. Ind. Eng.Chem. 25 (9), 1014–1019.

Reamer, H.H., Sage, B.H., 1956. Diffusion coefficients in hydrocarbonsystems: methane–n-butane–methane in liquid phase. Ind. Eng.Chem., Chem. Eng. Data Ser. 1 (1), 71–77.

Reamer, H.H., Opfell, J.B., Sage, B.H., 1956. Diffusion coefficient inhydrocarbon system methane–decane–methane in liquid phase.Ind. Eng. Chem. 48 (2), 275–281.

Renner, T.A., 1988. Measurement and correlation of diffusioncoefficients for CO2 and rich-gas applications. SPE Reserv. Eng.517–523 (May).

Riazi, M.R., 1996. A new method for experimental measurement ofdiffusion coefficients in reservoir fluids. J. Pet. Sci. Eng. 14 (3–4),235–250.

Riazi, M.R., 2005. Characterization and Properties of PetroleumFractions. ASTM International, West Conshohocken, PA.

Schas, W., 1998. The diffusional transport of methane in liquid Water:method and result of experimental investigation at elevatedpressure. J. Pet. Sci. Eng. 21 (3–4), 153–164.

Schmidt, T., 1989. Mass transfer by diffusion. In: Hepler, L.G., Hsi, C.(Eds.), AOSTRA Handbook on Oil Sands, Bitumens and HeavyOils. AOSTRA, Edmonton, Canada. Chapter 12.

Sheikha, H., Pooladi-Darvish, M., Mehrotra, A.K., 2005. Develop-ment of graphical methods for estimating the diffusivity coefficientof gases in bitumen from pressure-decay data. Energy Fuels 19 (5),2041–2049.

Stehfest, H., 1970. Algorithm 368: numerical inversion of Laplacetransforms. Commun. ACM 13 (1), 47–49.

Tharanivasan, A.K., Yang, C., Gu, Y., 2004. Comparison of threedifferent interface mass transfer models used in the experimentalmeasurement of solvent diffusivity in heavy oil. J. Pet. Sci. Eng. 44(3–4), 269–282.

Upreti, S.R., Mehrotra, A.K., 2000. Experimental measurement of gasdiffusivity in bitumen: results for carbon dioxide. Ind. Eng. Chem.Res. 39 (4), 1080–1087.

Upreti, S.R., Mehrotra, A.K., 2002. Diffusivity of CO2, CH4, C2H6

and N2 in Athabasca bitumen. Can. J. Chem. Eng. 80 (1), 116–125.Yang, C., Gu, Y., 2003. A new method for measuring solvent

diffusivity in heavy oil by dynamic pendant drop shape analysis(DPDSA). SPE Annual Technical Conference and Exhibition,Denver, CO. SPE Paper, vol. 84202.

Zhang, Y., Hyndman, C.L., Maini, B., 2000. Measurement of gasdiffusivity in heavy oils. J. Pet. Sci. Eng. 25 (1–2), 37–47.

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