Mathematical Modeling ofSteam-Assisted Gravity Drainage

Serhat Akin, Middle East Technical U.

SummaryA mathematical model for gravity drainage in heavy-oil reservoirsand tar sands during steam injection in linear geometry is pro-posed. The mathematical model is based on the experimental ob-servations that the steam-zone shape is an inverted triangle withthe vertex fixed at the bottom production well. Both temperatureand asphaltene content dependence on the viscosity of the drainedheavy oil are considered. The developed model has been validatedwith experimental data presented in the literature. The heavy-oilproduction rate conforms well to previously published data cov-ering a wide range of heavy oils and sands for gravity drainage.

IntroductionGravity drainage of heavy oils is of considerable interest to the oilindustry. Because heavy oils are very viscous and, thus, almostimmobile, a recovery mechanism is required that lowers the vis-cosity of the material to the point at which it can flow easily to aproduction well. Conventional thermal processes, such as cyclicsteam injection and steam-assisted gravity drainage (SAGD), arebased on thermal viscosity reduction.1 Cyclic steam injection in-corporates a drive enhancement from thermal expansion. On theother hand, SAGD is based on horizontal wells and maximizingthe use of gravity forces.2 In the ideal SAGD process, a growingsteam chamber forms around the horizontal injector, and steamflows continuously to the perimeter of the chamber, where it con-denses and heats the surrounding oil. Effective initial heating ofthe cold oil is important for the formation of the steam chamber ingravity-drainage processes.3 Heat is transferred by conduction, byconvection, and by the latent heat of steam. The heated oil drainsto a horizontal production well located at the base of the reservoirjust below the injection well.

Based on the aforementioned concepts, Butler et al.4 derivedEq. 1 assuming that the steam pressure is constant in the steamchamber, that only steam flows in the steam chamber, that oilsaturation is residual, and that heat transfer ahead of the steamchamber to cold oil is only by conduction. One physical analogy ofthis process is that of a reservoir in which an electric heatingelement is placed horizontally above a parallel horizontal produc-ing well.

q = L�2��Sokg�h

m�s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)

In this equation, L is the length of the well, � is porosity, �So

is the difference between initial oil saturation and residual oilsaturation, k is the effective permeability for the flow of oil, g isthe acceleration caused by gravity, � is the thermal diffusivity, h isthe reservoir height, m is a constant between 3 and 5,5,6 and �s isthe kinematic viscosity of oil at steam temperature. The tempera-ture profile is assumed to be time-independent and to decline ex-ponentially with distance from the interface.

� T − TR

Ts − TR� = exp�−

U�

� �. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2)

In this equation, U is the local velocity of the steam-zoneinterface, and � is the coordinate perpendicular to the steam/oilinterface. There are three major consequences of this theory. First,steam-chamber growth is necessary for oil production, and oilproduction occurs so long as steam is injected. Second, oil rateincreases as the steam temperature increases. Third, at a givensteam temperature, the oil with the lowest viscosity (usually thehighest °API gravity) exhibits the greatest production response.

One major problem with the aforementioned model was thatthe observed experimental oil rates reported in the literature wereon the same order as those predicted by Butler’s model, butslightly lower. Butler1 associated the deviation of the observed rateto the factors that were not recognized in the derivation of theequation, such as a change in the effective steam-chamber height(i.e., it becomes lower than the reservoir height), because of deple-tion and because some heat is used to cause the lateral transfer ofthe draining oil to the fixed well.

Later, Reiss7 proposed the use of an empirical dimensionlesstemperature coefficient, a�0.4 (see Eq. 3) and the maximum ve-locity (rather than the local interface velocity) that led to a morerealistic representation of the experimental data reported in theliterature. He reported successful matches with some literature data.

q = L���Sokg�h

2amvs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3)

As can be seen, both of the aforementioned models do notinclude the effects of steam distillation and, hence, asphaltenedeposition. Because deasphalting causes drastic viscosity reduc-tion, the production rate is expected to increase with asphaltenedeposition. This paper reports a modification and enhancement toexisting gravity-drainage models by incorporating asphaltene-content-dependent viscosity. First, several viscosity models forincluding the effects of asphaltene deposition will be discussed.Then, these models will be coupled with gravity-drainage theory.Finally, the developed gravity-drainage models that consider as-phaltene deposition will be tested with the experimental data re-ported in the literature.

Viscosity of Oil/Asphaltene SuspensionsThe key point in this analysis is that as the asphaltene content ofthe oil increases, the viscosity of it also increases (i.e., an increasein oil/solvent mixture viscosity with increasing solvent concentra-tions). Most of the published models that aim to relate compositionto viscosity require the knowledge of critical parameters of mix-ture components.8 Unfortunately, when the crude contains heavyfractions such as resins and asphaltenes, these parameters cannotbe evaluated or have any physical meaning. For low concentrationsof asphaltene, Pfeiffer9 proposed the use of Einstein’s formula forcalculating the viscosity of benzene/asphaltene suspension:

�r = �1 + 2.5�asph�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4)

Here, �r is the relative viscosity (ratio of dynamic viscosities ofthe mixture and the pure, nonprecipitating solvent), and �asph isthe volume fraction of asphaltene suspended in benzene. Becausethe aforementioned formula underestimated the viscosities,Pfeiffer proposed Eiler’s formula for high concentrationsof asphaltene:

�r = �1 + 2.5�asph� �1 − 3.5�asph�. . . . . . . . . . . . . . . . . . . . . . . . (5)

Copyright © 2005 Society of Petroleum Engineers

This paper (SPE 86963) was first presented at the 2004 SPE International Thermal Op-erations and Heavy Oil Symposium and Western Regional Meeting, Bakersfield, California,16–18 March, and revised for publication. Original manuscript received for review 22 June2004. Revised manuscript received 8 April 2005. Paper peer approved 25 June 2005.

372 October 2005 SPE Reservoir Evaluation & Engineering

Derived for suspensions containing undeformable sphericalparticles with equal radii, this equation overestimated the mea-sured viscosities of asphaltene solutions.

A comprehensive study that related asphaltene content to vis-cosity of crude oil was conducted by Werner et al.10 The proposedmodel (W3BH) was based on a mixing rule that takes into accountthe compositional information and the model of Kanti et al.11 fordependence on pressure and temperature.

ln � ��p, T�

��p0, T0�� = c�1

T−

1

T0� + Eln � D + p

D + p0�. . . . . . . . . . . . . (6)

Here, p stands for pressure, T stands for temperature, and sub-script 0 refers to a reference state. Parameters c, D, and E can beobtained as a function of the viscosity, �, in the reference condi-tions using the following equations. It was shown by Werneret al.10 that the higher the temperature (i.e., >100°C), the closer Eis to 1. For SAGD operations, because reservoir temperature isusually high, E can be assumed to be equal to 1.

c = 16.07x02 + 634x0 + 1190.3. . . . . . . . . . . . . . . . . . . . . . . . . . . (7)

x0 = ln ��p0, T0�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8)

D = 60.9 exp�0.192x�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9)

x = ln � �p0, T�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10)

Four major fractions were used: gas (CO2 and hydrocarbongases), C6–C20 (light and intermediate hydrocarbons), C20+ [satu-rates, aromatics, and resins (SAR)], and asphaltenes. The mixingrule is given as follows:

ln � = 1 ln �1 + 2 ln �2 + 3 ln �3 + 4 ln �4 + 1212

+ 1313 + 1414 + 2323 + 2424 + 3434

+ 123123 + 124124 + 134134 + 234234

+ 12341234 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11)

To calculate the viscosity of given oil, the composition of the fluidfirst needs to be constructed. After that, Eq. 11 is used with theparameters given in Table 1 to obtain a reference viscosity at 35MPa, 100°C. The viscosity is then corrected with Eq. 6 for thedesired temperature and pressure. Note that if the composition ofthe fluid is not known, but a viscosity reference is available at agiven temperature and pressure, Eq. 6 can be used directly tocompute the viscosity at the desired temperature and pressure.When the pressure interval is small (for an upper limit of 70 MPa)and the temperature is elevated, as in the case of SAGD, E may betaken as 1. Moreover, if the reference temperature at which theviscosity is evaluated is taken as the steam-chamber temperature,then Eq. 6 can be written as follows:

��p, T� = ��p0, T0� � D + p

D + p0�. . . . . . . . . . . . . . . . . . . . . . . . . . (12)

SAGD ModelThe following derivation follows the derivation presented byReiss.7 During steam injection from a horizontal well, the steamzone is expected to grow laterally as oil drains along the oil/steaminterface to the horizontal production well. The steam-zone crosssection resembles an inverted triangle (Fig. 1) as the steam zonewidens near the overburden formation. Assuming that the tempera-ture at the surface of this zone is uniform and constant, heat con-

duction into solid moving at a fixed velocity U can be obtainedfrom the following equation7:

� T − TR

Ts − TR� =

1

2� erfc � � − Ut

2��t�

+exp �U�

� � erfc � � + Ut

2��t�� . . . . . . . . (13)

Initially, the temperature that is equal to reservoir temperature,TR, jumps to steam-injection temperature, Ts, when the movementstarts. As the oil drains away, matter is lost, and for long times, Eq.13 can be approximated by Eq. 2.1 The variation of viscosity withtemperature depends on the properties of the particular oil in thereservoir. One arbitrary form of viscosity/temperature function thatcorresponds reasonably well ahead of the steam front can be ex-pressed by the following equation1:

�s

�o= � T − TR

Ts − TR�m

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (14)

The above form of the viscosity/temperature relationship makesmathematical manipulation quite easy. Moreover, this mathemati-cal function makes the viscosity of oil at reservoir temperatureinfinite. To calculate m, Butler1 proposed the use of the followingequation, which defined m as a function of the viscosity/temperature characteristics of the oil, the steam temperature, andthe reservoir temperature:

m =��s �TR

Ts �1

�−

1

�s� dT

T − TR�−1

. . . . . . . . . . . . . . . . . . . . . . (15)

Single-phase oil-production rate by SAGD along the steam/oilinterface obeys Darcy’s law. If qo is the oil-drainage rate along theinterface per unit length along the horizontal well, the oil flow canbe expressed as

dqo��� =ko��

�d�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (16)

At the interface, the flow potential can be approximated by thefollowing equation:

�� = � g sin � ≈ og sin �. . . . . . . . . . . . . . . . . . . . . . . . . . . . (17)

Substituting Eqs. 17 and 14 into Eq. 16 and then integrating theresulting equation from � equals zero, and infinity results in anoil-drainage rate along one side of the steam zone:

qo =kog� sin �

vomU. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (18)

Fig. 1—Schematic drawing of the steam-zone cross section.

373October 2005 SPE Reservoir Evaluation & Engineering

Considering the material balance at the interface of the steam zone(Fig. 1), the oil-drainage rate along one face of the steam zone perunit length along the horizontal well can be written as

qo =d

dt ���Sohws

2 �. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (19)

The time derivative of the width of the steam zone can be obtainedfrom the interface velocity as follows:

dws

dt=

U

sin �. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (20)

Substituting the time derivative of the width of the steam zone inEq. 19 and eliminating U and ws yields the oil-drainage rate

qo =���Sokoghws

2�sm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (21)

When integrated over time, the above equation yields the cumu-lative oil production per unit length along the horizontal well.

Qo =���Sokoghws

2�smt. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (22)

One major difference of the above derivation is that Eq. 22 doesnot include any empirical constants, as opposed to other solutionspresented in the literature.1,7

To include the steam-distillation and deasphaltation effects ofthe crude oil into the SAGD model presented above, the followingprocedure was used:

• Use an asphaltene-deposition model or experimental obser-vations. Use cubic splines if necessary.

• Calculate the viscosity at any time with Eq. 11.• Use Eq. 6 or Eq. 12 to correct for temperature and pressure.• Use Eq. 22 to calculate the cumulative production.

Results and DiscussionTo test and evaluate the proposed model, two different experimentaldata, presented by Canbolat et al.12 and Butler et al.,4 were used.

In Canbolat et al.’s scaled experiments, crushed limestone(mesh size 0.5 to 1.4 mm) was used to create a medium. Thelimestone/water slurry was mixed with crude oil to yield oil andwater saturations of 75 and 25%. Initial saturations were heldconstant throughout the experiments. The oil was a 12.4°API vis-cous crude oil from the Bati Raman field. Fig. 2 gives the oilviscosity at the ambient-pressure-vs.-temperature relationship.Viscosity was roughly 600 cp at the initial experimental tempera-ture. Produced oil and water were measured together with injectionand production pressures. Additionally, the asphaltene content ofproduced oil was measured by titration experiments following themethod proposed by Kokal et al.13

The proposed model was tested using two different experi-ments in which the asphaltene content of the oil was reported(Fig. 3). In these experiments, the injector and producer well spac-ings were different. The asphaltene content of the initial oil inplace is roughly 35% by weight. Then, the asphaltene content ofthe produced oil experiences a drop upon initiation of productionin both cases. The decrease in asphaltene content indicates a partialupgrading of the produced oil through deposition of asphaltenes onthe limestone matrix. As the steam chamber develops, the asphalt-ene content increases somewhat but remains somewhat below theinitial value. This shows that adsorption or precipitation of asphalt-enes continues for a period of time. As the growth of the steamchamber continues, the asphaltene content of the produced oilincreases, indicating mobilization of asphaltenes. These asphalt-enes may be redissolved in the hot oil or mobilized as solid pre-cipitate. The time for asphaltene content to reach a minimum,peak, and begin to decline again varies with the well separation.

Using the experimental parameters reported in Table 2, a pro-posed SAGD model was tested. First, the asphaltene content of theproduced oil was represented by cubic splines. Because initialcomposition of this oil was not reported by Canbolat et al.,12 theviscosity of the crude at the steam temperature was used as astarting point. By using the reported asphaltene content as a con-straint, the composition of the oil was changed until a match wasobtained. During this process, as dead oil was used during theexperiments, a gaseous component fraction was taken as zero. Achange in asphaltene content was distributed among the SAR andlight and intermediate fractions equally. The viscosity is then cal-culated using Eq. 11 and corrected for the reported experimentalpressures (316 and 350 KPa for experiments 1 and 2). Then, thecumulative production is calculated using Eq. 22. The resultingproduction curve is given in Fig. 4. As can be seen, the productioncurve follows the trend of asphaltene content closely and shows agood match with the experimental production data. This indicatesthat the proposed model properly addresses the steam-distillationand deasphalting effects of the SAGD process.

For comparison purposes, empirical viscosity models were alsoevaluated. Because the asphaltene content of the crude oil waspretty high, Eiler’s model was selected. Eq. 5 was used to obtain

Fig. 2—Viscosity/temperature relationship for crude oil (afterCanbolat et al.12).

Fig. 3—Asphaltene content of the produced oil.

374 October 2005 SPE Reservoir Evaluation & Engineering

viscosity values, and Eq. 22 was used to compute the cumulativeoil produced. The model successfully predicts the experimentalbehavior up to 220 minutes before the experimental data start todeviate from the theoretical data (Fig. 4). At this point, asphaltenesprobably mobilize and redissolve in the hot oil or mobilize assolids precipitate. These effects could not be captured by Eiler’sformula. Moreover, it has been shown previously that Eiler’s for-mula underestimates the viscosities. Thus, it was concluded thatthe W3BH model is the better of the viscosity models used in theproposed SAGD model. This is expected because the W3BHmodel considers compositional dependency. Moreover, from amathematical point of view, the W3BH model is a higher-ordermodel compared to the simple Eiler formula.

The models proposed by Reiss7 and Butler and Stephens5 werealso used to model the same experimental data. Remember thatboth of these models rely on empirical constants to match experi-mental data. Similar to the use of Eiler’s formula in the proposedmodel, these models can represent the experimental data up to acertain point at which the asphaltene composition does not changemuch. As the steam-distillation effects start to dominate and com-positional changes start to occur, these models fail to represent theexperimental data (Fig. 4). It can be concluded that the modelproposed by Reiss7 is the better of the two models compared here.

Similar results were obtained for the case in which the injec-tion/production well spacing was smaller (10 cm) when comparedto the previous experiment’s (15 cm), as reported by Canbolatet al.12 Using the aforementioned composition-selection methodand the data presented in Table 2, an attempt was carried out tomatch the experimental data. Because the compositional change ofthe crude oil was more pronounced compared to the first experi-ment, the mismatch was greater, especially at around 150 minutes(Fig. 5). The overall behavior of the mathematical model matchedexperimental behavior, but the match was not as good as in the

previous case. Similar to the previous case, the use of Eiler’sequation in the SAGD model resulted in smooth changes. Thistime, however, the match was good. The models proposed byReiss7 and Butler and Stephens5 showed a similar performance;they both described the experimental data up to a certain extent.

The last set of example data is from Butler et al.4 In thisexperiment, the oil saturation at the beginning of the experimentwas 100% (Table 2), and the permeability of the system wassomewhat smaller than the previous experiments. Once again, be-cause the composition of the bitumen was not reported, the oilviscosity at the steam temperature was used as a starting point.Because the asphaltene-deposition data were not available, the datagiven by Canbolat et al.12 were used. By doing so, we assumedthat a similar steam-distillation and asphaltene-deposition mecha-nism was expected. Because the proposed SAGD model success-fully represented the experimental data (Fig. 6), it was concludedthat the aforementioned asphaltene-deposition behavior is to beexpected in SAGD operations. Eiler’s formula also can be used forthis case. Other analytical models, however, overestimated thecumulative production at late times. Similar observations werereported by Butler1 and Reiss.7 Butler1 attributed such deviationsto factors not recognized in the derivation of the model. (Forexample, with time the effective height of the steam chamberbecomes lower than h because of depletion and because some ofthe heat is used to cause lateral transfer of the draining fluid to thefixed well). He then used a constant equal to 1.5 instead of 2.0within the square root in Eq. 1. Likewise, Reiss used a constantequal to 0.4 to attain a match. But, as shown by Canbolat et al.,12

the steam-zone size does not change at late times. The time re-quired to reach a constant size, however, changes as a function ofinjection/production spacing and the presence of noncondensablegases in the system. Thus, it can be concluded that at late times,rather than the steam-zone size and lateral heat transfer, steam-distillation and asphaltene-deposition effects dominate the SAGDprocess.

ConclusionsA mathematical model for gravity drainage in heavy-oil reservoirsand tar sands during steam injection in linear geometry is pro-posed. Temperature, pressure, and asphaltene-content dependenceon the viscosity of the drained heavy oil is considered by means ofa compositional viscosity model. Unlike the previous models, theproposed model successfully predicts cumulative oil production byeffectively modeling steam-distillation and asphaltene-depositioneffects. It was observed that rather than the change in steam-zoneheight and lateral transfer of the draining fluid to the fixed wellsteam-distillation and asphaltene-deposition effects are more dom-inant in the SAGD process.

Nomenclaturea � empirical constant (0.4)c � parameter that determines the variation of viscosity as

a function of temperature

Fig. 5—Comparison of oil-production models with data from theCanbolat et al.12 Experiment 2.

Fig. 6—Comparison of oil-production models with data fromButler et al.4

Fig. 4—Comparison of oil-production models with data from theCanbolat et al.12 Experiment 1.

375October 2005 SPE Reservoir Evaluation & Engineering

D, E � parameters that determine the variation of viscosity asa function of pressure

g � acceleration caused by gravityh � thickness (reservoir)k � permeabilityL � length of horizontal wellm � dimensionless viscosity exponentp � pressureq � volumetric rateQ � cumulative oil productionS � saturation, dimensionlesst � time

T � temperatureU � local velocity of steam-zone interfacew � steam-zone half-width� � thermal diffusivity� � coordinate horizontal to steam/oil interface� � relative viscosity� � potentialij � binary interaction coefficient between fraction i and

fraction jijk � ternary interaction coefficientijkl � quaternary interaction coefficient

� � dynamic viscosity� � kinematic viscosity� � coordinate perpendicular to steam/oil interface� � porosity� � asphaltene content � mass fraction

Subscripts and Superscriptsi � injectoro � oilr � relativeR � reservoirs � steam0 � reference state

1,2,3,4 � fraction

References1. Butler, R.M.: Thermal Recovery of Oil and Bitumen, Prentice-Hall,

Englewood Cliffs, New Jersey (1991).2. Butler, R.M.: “SAGD comes of age,” J. Cdn. Pet. Tech. (1998) 37, No.

7, 9.3. Elliot, K.E. and Kovscek, A.R.: “A Numerical Analysis of the Single-

Well Steam Assisted Gravity Drainage (SW-SAGD) Process,” Pet. Sci.Tech. (2001) 19, No. 7–8, 733.

4. Butler, R.M., McNab, G.S., and Lo, H.Y.: “Theoretical studies on thegravity drainage of heavy oil during in-situ steam heating,” Cdn. J.Chem. Eng. (1981) 59, No. 4, 455.

5. Butler, R.M. and Stephens, D.J.: “The gravity drainage of steam-heatedheavy oil to parallel horizontal wells,” J. Cdn. Pet. Tech. (1981) 20, No.2, 90.

6. Butler, R.M.: “New interpretation of the meaning of the exponent ‘m’in the gravity drainage theory for continuously steamed wells,” AOSTRAJ. of Res. (1985) 2, No. 1, 67.

7. Reiss, J.C.: “A steam–assisted gravity drainage model for tar sands:linear geometry,” J. Cdn. Pet. Tech. (1992) 31, No. 10, 14.

8. Pedersen, K.S. and Fredenslund, A.: “Viscosity of crude oils,” Chem.Eng. Sci. (1987) 42, 182.

9. Pfeiffer, J.P.: The Properties of Asphaltic Bitumen, Elsevier PublishingCo., New York City (1950).

10. Werner, A. et al.: “A new viscosity model for petroleum fluids withhigh asphaltenes content,” Fluid Phase Equilibria (1998) 147, 319.

11. Kanti, M. et al.: “Viscosity of liquid hydrocarbons, mixtures and pe-troleum cuts, as a function of pressure and temperature,” J. Phys.Chem. (1989) 93, 3860.

12. Canbolat S., Akin, S., and Kovscek, A.R.: “A Study of Steam-AssistedGravity Drainage Performance in the Presence of NoncondensableGases,” paper SPE 75130 presented at the 2002 SPE/DOE ImprovedOil Recovery Symposium, Tulsa, 13–17 April.

13. Kokal, S.L. et al.: “Measurement and correlation of asphaltene pre-cipitation from heavy oils by gas injection,” J. Cdn. Pet. Tech. (1992)31, No. 4, 24.

SI Metric Conversion Factors°API 141.5/(131.5+°API) � g/cm3

bar × 1.0* E+05 � Pacp × 1.0* E–03 � Pa�sft × 3.048* E–01 � m

ft2 × 9.290 304* E–02 � m2

°F (°F – 32)/1.8 � °Cin. × 2.54* E+00 � cm

in.3 × 1.638 706 E+01 � cm3

*Conversion factor is exact.

Serhat Akin is an associate professor of petroleum and naturalgas engineering at the Middle East Technical U. (METU), An-kara, Turkey, where he has served on the faculty since 1999.Before joining METU, Akin was a post-doctoral research affiliatein the Petroleum Engineering Dept. at Stanford U. He has pub-lished more than 70 technical articles, reports, and conferenceproceedings. His research interests include computerized to-mography applications, image processing, enhanced oil re-covery, and reservoir and geothermal engineering. Akin holdsa PhD degree in petroleum and natural gas engineering fromMETU. He has served on the SPEREE review committee since1999 and received the Outstanding Technical Editor Award in2001, 2002, and 2004.

376 October 2005 SPE Reservoir Evaluation & Engineering