Scientia Iranica C (2014) 21(3), 763{771

Sharif University of TechnologyScientia Iranica

Transactions C: Chemistry and Chemical Engineeringwww.scientiairanica.com

Investigating the role of ultrasonic wave on two-phaserelative permeability in a free gravity drainage process

B. Keshavarzia,b, R. Karimia,b, I. Naja�a,b, M.H. Ghazanfaria,b and C. Ghotbia,b�

a. Ultrasonic Research Group, Sharif University of Technology, Tehran, Iran.b. Department of Chemical and Petroleum Engineering, Sharif University of Technology, Tehran, Iran.

Received 1 January 2013; received in revised form 18 August 2013; accepted 21 October 2013

KEYWORDSUltrasonic wave;Free gravity drainage;Relative permeability;Hagoort methodology;Experimental.

Abstract. In this work, the process of free gravity drainage under the in uence ofultrasonic waves was investigated. A glass bead pack porous medium was used to performfree fall gravity drainage experiments. The tests were performed in the presence andabsence of ultrasonic waves, and the data of recovery were recorded versus time under bothconditions. The wetting phase relative permeability curves were obtained using the dataof recovery versus time, based on the Hagoort backward methodology. Subsequently, usingthe wetting phase relative permeability curve, the relative permeability of non-wettingphases were calculated by performing history matching to the experimental productiondata. The results revealed that ultrasound considerably increases the recovery factor of thefree gravity drainage process. It was also observed that the relative permeability of bothwetting and non-wetting phases increases under exposure to ultrasonic waves. The resultsof this work can be helpful in better understanding/evaluating the behavior of the relativepermeability curves of both wetting and non-wetting phases during a free gravity drainageprocess under exposure to ultrasonic waves.c 2014 Sharif University of Technology. All rights reserved.

1. Introduction

Application of ultrasonic wave technology in enhancing uid ow in porous media has been investigated experi-mentally by many researchers. Duhon and Campbel [1]conducted a series of ood tests to characterize thebehavior of the system with and without addition ofsonic energy. The results of their work serve to showthat the addition of sonic energy increases displacemente�ciency. Chen [2], as well as Fairbanks and Chen [3],reported an increase in oil percolation rate throughporous medium due to wave radiation. Cherskiy etal. [4] described a sharp increase in the permeabilityof core samples saturated with fresh water in thepresence of an acoustic �eld. Neretin and Yudin [5]observed an increase in the rate of oil displacement by

*. Corresponding author. Tel.: +98 21 66165424E-mail address: ghotbi@sharif.edu (C. Ghotbi)

water through loose sand under ultrasound. Beresnevand Johnson [6] provided a comprehensive review ofmethods using elastic wave stimulation of oil produc-tion, including both ultrasonic and seismic methods,and mentioned that the application of elastic wavesimproves the permeability and production rate in mostcases. Numerical and experimental results of Aartset al. [7] showed that, by increasing ultrasonic power,the velocity of uid inside the porous media increases.Hamida and Babadagli [8] observed that ultrasonicwaves can enhance capillary imbibition oil recovery, de-pending on the uid, sound intensity, sound frequency,and the nature of the matrix. Naderi and Babadagli [9]also revealed the positive e�ect of ultrasonic energy onoil recovery for di�erent wettability rocks. However,to the best of our knowledge, no attempt has beenmade to experimentally investigate ow enhancementduring the gravity drainage process in the presence ofultrasonic radiation.

764 B. Keshavarzi et al./Scientia Iranica, Transactions C: Chemistry and ... 21 (2014) 763{771

Gravity drainage plays an important role in oilproduction from natural fractured reservoirs. Thisprocess will be highlighted, especially in cases wherea gas cap (initially or secondary) is present in directcontact with the oil zone [10]. In some of Iran's giantfractured reservoirs, where oil has been produced forseveral decades and the reservoir has been subject toboth water-drive from the bottom region and a gravity-drainage mechanism from the top region, materialbalance calculations indicate higher oil recoveries inthe gravity drainage areas than the water displacementregions [11]. Since the relative permeability of presentphases is the main parameter that has to be evaluatedto complete the description of the gravity drainage pro-cess [12,13], in order to investigate the process of grav-ity drainage under the e�ect of an ultrasonic wave, it iscritical to evaluate the wetting/non-wetting phase rela-tive permeability curves under exposure to ultrasound.

In this study, the impact of ultrasonic waveson the relative permeabilities of both wetting andnon-wetting phases in a drainage process is investi-gated through a series of drainage experiments. Eachexperiment is done in the presence and absence ofultrasonic wave radiation. Then, using the productiondata in each experiment, the wetting phase relativepermeability curve is obtained through the HagoortBackward Method [14,15]. Afterwards, the gas relativepermeability is calculated using history matching to theproduction data. Finally, the relative permeabilitiesof the wetting and the non-wetting phases are plottedand compared in the presence and absence of ultrasonicradiation.

2. Experiments

2.1. Experimental setup and uidsThe experimental setup includes a cylinder made ofPlexiglas with a length of 30 cm, an outer diameterof 4 cm and an inner diameter of 3 cm, a graduatedcylinder to measure the amount of extracted liquidand an ultrasonic wave generator with an averagee�ective output power of 80 watts, which produceswaves with 22 kHz frequency. A schematic view of theexperimental setup is shown in Figure 1. The averagesize of the beads was 150-180 �m. The porosity ofthe medium was calculated by measuring the volumeof uid entering the cell at each experiment. The valueof porosity was 37 � 0:5%. The absolute permeabilityof the bead pack was also calculated by injecting uidinto the cell with a speci�c pressure gradient throughthe cell and measuring the ow rate. The amount ofpermeability was obtained to be 34� 1 Darcy. Air wasused as the non-wetting phase, and water, kerosene,crude oil A and crude oil B (in di�erent experiments)were the wetting phases. The properties of the wetting uids can be seen in Table 1.

Figure 1. The experimental setup used in this study.

Table 1. Physical properties of wetting uids understandard condition.

Fluid Density(kg/m3)

Viscosity(cp)

Asp.content

Dis. water 1000 1 -Kerosene 739 0.9 -Crude oil A 860 6.5 0.3%Crude oil B 920 83 5%

Before beginning each experiment, the glass beadswere packed in the Plexiglas cylinder and the apparatuswas assembled. Unused dry glass beads were usedin each experiment to avoid any error according tochanges in wettability. The cell was �rstly vacuumedto a desired pressure and then fully saturated with theliquid sample. Afterwards, the cell was placed in avertical position, the faces of the cell were opened, andthe liquid produced was measured versus time. Theexperiments were conducted under ambient laboratorytemperature (25�C) under two di�erent conditions;the presence and absence of ultrasonic waves, andthe results were used for determination of relativepermeability curves.

3. Relative permeability determination

Firstly, the Hagoort [15] backward methodology wasused and wetting phase relative permeability curveswere drawn using the data of recovery versus time.

B. Keshavarzi et al./Scientia Iranica, Transactions C: Chemistry and ... 21 (2014) 763{771 765

Then, the results of wetting phase relative permeabilityand production data were used and relative permeabil-ity curves for the non-wetting phase were determinedthrough history matching to the experimental produc-tion data.

3.1. Wetting phase relative permeabilitydetermination through Hagoortmethodology

By considering the case of free gravity drainage attimes beyond breakthrough time, one can consider gasvelocity to be negligible [16]. Hagoort [15], by applyingthe Buckley-Leveret equation, assuming in�nite gasmobility and negligible capillary pressure, proposed abackward methodology for the calculation of relativepermeability based on Corey-type equations [15,17,18].Derivation of the backward methodology is describedin Appendix A. Due to Hagoort, after the time ofbreakthrough, the amount of normalized oil productioncan be calculated with Eq. (1):

Np = 1��

1� 1n

��1

nk0rotD

� 1n�1

; (1)

where n is the Corey equation exponent and k0ro is the

Corey constant. According to this equation, the plot of1�Np versus tD in a log-log paper results in a straightline, and the value of the Corey exponent (n) can becalculated from the slope of the straight line, as follows:

dln(1�Np)dlntD

= �(1

n� 1): (2)

Also, the amount of k0ro can be obtained from Eq. (1),

when tD = 1. Derivations of Eqs. (1) and (2) aredescribed in Appendix A. Now, by having Corey con-stants (n and k0

ro), the values of relative permeabilityfor the wetting phase after the time of breakthrough isdetermined. But, to obtain the relative permeabilitiesof the wetting phase before the time of breakthrough,an extrapolation of the values of relative permeabilityafter breakthrough time has to be undertaken [14].This process is illustrated by an example in the resultsand discussion section.

3.2. Non-wetting phase relative permeabilitydetermination

In this part, a free gravity drainage process is simu-lated. The relative permeability data of the wettingphase that has been calculated in the previous sectionis used. A Corey type relative permeability functionis considered for the gas relative permeability andthe constants of this function are determined usinga history matching of the simulation results to theproduction data. The main assumptions here are asfollows:

a) Capillary pressure at the interface is assumed con-stant in the whole porous media.

b) The Darcy ow equation and its conditions arevalid.

c) The ow is assumed to be piston like and one-dimensional, while the lateral faces are sealed.

For a two phase gas and oil system the followingequations are used:

Qo = � ko�o

(A)(�o2 � �ozH � z ); (3)

Qg = � kg�g

(A)(�gz � �g1

z); (4)

where Qo and Qg are Darcy ow rates; ko and kgare e�ective permeabilities; �o and �g are viscosities;�o and �g are the hydraulic potentials of oil andgas, respectively; z is the distance of the oil and gasinterface from the top of the column; A is the crosssectional area; and H is the height of the column. Theschematic of the gravity drainage process is depictedin Figure 2. After integrating and using the conceptof capillary pressure (pc = �gz � �oz), the followingequation could be generated:

�g1 � �o2 � Pc + ��g(H � z) =Q�R2

��gkgz

+�oko

(H � z)�; (5)

in which pc is capillary pressure and subscripts 1 and2 are introduced in Figure 2; �� is the di�erence be-tween oil and gas (wetting and non-wetting) densities;constant R is the radius of the column; and g is thegravity acceleration. As the assumption is a piston-like displacement, both Qo and Qg are consideredto be equal and are shown by Q in Eq. (5). By

Figure 2. Schematic of gas oil drainage process.

766 B. Keshavarzi et al./Scientia Iranica, Transactions C: Chemistry and ... 21 (2014) 763{771

considering negligible capillary pressure at the exitface (�g1 � �o2 = 0) [19], by replacing Q

�R2 with u,by de�ning Mobility ratio, M , as ( kg�g )=( ko�o ) and byinvolving Eq. (5) the following equation is obtained:

uDarcy =kg(�Pc + ��g(H � z))�g[z(1�M) +MH]

: (6)

The actual velocity is equal to the derivative of theinterface location to time (uactual = dz

dt ). So, accordingto the de�nition of actual ow rate the followingequation is obtained:

uact =uDarcy

� =dzdt; (7)

in which t is time in seconds and � is the porosity ofporous media [19]. Combining Eqs. (6) and (7), thelocation of the interface between two phases with timewill be described in this form:

dzdt

=kg(�Pc + ��g(H � z))��g [z(1�M) +MH]

: (8)

De�ning the amount of recovery as Np = zH , the

following equation can be derived:

dNpdt

=kg(�PcH + ��g(1�Np))H � �g[Np(1�M) +M ]

: (9)

To predict the relative permeabilities of gas, a freegravity drainage process is simulated by solving Eq. (9)

using the Runge-Kutta method of order 4, knowingthat Np is equal to zero at the initial time. Thevalues of the wetting phase relative permeability fromHagoort backward methodology [15] are used, togetherwith estimates for the relative permeability of thenon-wetting phase. The gas relative permeabilityparameters are changed through an optimization tech-nique until a reasonable match between the simulationcalculated recoveries and experimental production datais achieved.

4. Results and discussion

The results of free fall gravity drainage experimentscan be seen in Figure 3. The relative error for the oilrecovery data plotted in this �gure is in the range of2-4%.

Radiation of the ultrasonic wave could highlyincrease the amount of ultimate recovery, as well asincreasing the rate of drainage recovery. Figure 3(a)demonstrates that kerosene ultimate recovery has in-creased from 51% to 62%. In the case of crude sampleA (Figure 3(b)) and water (Figure 3(c)), the values of�nal recovery were increased from 35% and 33% to 49%and 45%, respectively.

However, an unexpected result was observed forgravity drainage in the case with crude oil B duringexposure to the ultrasonic wave, which is an asphalteniccrude oil. As can be observed in Figure 3(d), thevalues of oil recovery have been decreased after expo-

Figure 3. Results of sonication on total recovery and rate of uids drainage: (a) Kerosene; (b) crude oil A; (c) water; and(d) crude oil B.

B. Keshavarzi et al./Scientia Iranica, Transactions C: Chemistry and ... 21 (2014) 763{771 767

sure to ultrasonic waves. According to the previousstudies on the e�ects of ultrasonic waves on asphaltenicoil properties [20], this observation can be describedas a result of the breakdown of asphaltene micellestructures in the oil and their dissolution in the uid,which increases the viscosity of the uid after longexposure to radiation. The increment in the viscosityof uid leads to slowing down the rate of drainagerecovery.

4.1. Wetting and non-wetting phase relativepermeability results

It was shown in the previous section that ultrasonicenergy could considerably enhance the recovery of adrainage process. Here, it is aimed to calculate thechanges in the relative permeability of both wettingand non-wetting phases in the drainage process underexposure to ultrasonic waves.

Here, the procedure for calculations of relativepermeability for the experiment, in which kerosene wasthe wetting phase, is presented as a sample of thecalculations. As depicted in Figure 4(a), breakthrough

occurs at tD = 0:15 for non-sonicated and at tD = 0:14for sonicated kerosene drainage, which shows a slightlysooner breakthrough under ultrasound.

To calculate the value of n, it is expected touse the experimental data after the breakthrough timeby crossing a line through the data, as shown inFigure 4(b). So, n is obtained according to Eq. (2), andthe value of Np acquired from the straight line at tD =1 is also used for determination of k0

ro through Eq. (1).The result of the backward methodology for calculationof kerosene relative permeability is demonstrated inFigure 4(c). n and k0

ro were found to be 5.926 and1.87 in the case of non-sonicated and 4.03 and 1.96 inthe case of sonicated kerosene, respectively. Values ofCorey's constants for each experiment could be seenin Table 2. Ultrasound decreased the value of n forall cases and increased k0

ro for water and crude oil,A. It should be noted that the Hagoort methodologycan only be applied to situations in which there are nochanges in the physical properties of the uid. So, wecannot apply this method to the case of crude oil Bbecause there are some variations in oil viscosity as the

Figure 4. Step by step calculations of Corey type equation constants for oil relative permeability.

Table 2. Constants of Corey's equation before and after wave application.

Experiment Fluid n(NUS)/n(US) k0ro(NUS)/k0

ro(US)1 Dis. water 5.926/4.03 1.87/1.962 Kerosene 9.47/5.98 1.58/1.383 Crude oil B 5.184/5.09 1.43/1.76

768 B. Keshavarzi et al./Scientia Iranica, Transactions C: Chemistry and ... 21 (2014) 763{771

result of exposure to ultrasonic waves. As mentionedearlier, it should be kept in mind that the valuesobtained for the constants only pertain to a limitedrange of saturation (i.e. saturations after breakthroughtime). To have a thorough curve of relative perme-ability versus saturation, we are supposed to makean extrapolation between these determined values ofrelative permeability and the value with kro equal to 1at SoD = 1. The interpolation part of the curve isdepicted in Figure 4(c).

Relative permeabilies of the non-wetting phase(air) in di�erent experiments were also determinedthrough the history matching of the results of solvingEq. (9) to the experimental production data (Fig-ure 4(d)). Relative permeability of the non-wettingphase was increased in all the experiments. Relativepermeabilities of both phases are plotted together inFigure 5. For each case, there is an increase foroil and gas relative permeabilities as the result ofultrasonic wave radiation. The amount of gas endpoint relative permeability has been enhanced andalso there is a reduction in the value of critical gassaturation.

One point that should be considered here isthat for all the experiments, except in the case ofasphaltenic crude oil, in which changes in the physicalproperties of the uid are observed, recovery of thedrainage process was increased, which is somehowin contrast with the fact that the gas breakthroughoccurs sooner under exposure to ultrasound. On theother hand, most of the recovery is drained after thetime of breakthrough. So, it should be mentionedthat ultrasound stimulates both the wetting and non-wetting phases, which leads to a quicker breakthrough,but, ultimately, the value of �nal recovery will beaugmented. It should be mentioned that it wouldbe very complicated if one were to consider an extraterm for the contribution of ultrasonic waves in thegravity drainage process. But, calculated values ofthe relative permeability of wetting and non-wettingphases discussed in this work, in the presence andabsence of ultrasonic waves, respectively, could revealthe enhancement e�ect of ultrasound on two-phasegravity drainage ow behavior.

5. Conclusions

In this paper, the process of free-fall gravity drainageunder the in uence of ultrasonic wave radiation wasstudied. The Hagoort (1980) backward methodologywas applied and the relative permeabilities of wettingphases (kerosene, water and crude oil A) were ob-tained from unsteady state displacement data. Then,the obtained wetting-phase relative permeability andproduction data were used for history matching todetermine the relative permeability of the non-wetting

Figure 5. Obtained relative permeability curves forwetting and non wetting phases: (a) Kerosene-air; (b)crude oil A-air; and c) water-air.

phase, air. Based on the obtained results in this work,the following conclusions can be drawn,

� This work illustrates successful application of theHagoort backward methodology for evaluating wet-ting phase relative permeability from unsteady dis-placement data of the free-fall gravity drainageprocess under the in uence of ultrasound waveradiation.

B. Keshavarzi et al./Scientia Iranica, Transactions C: Chemistry and ... 21 (2014) 763{771 769

� The ultrasonic wave increased the recovery of thegravity drainage process for non-asphaltenic uidsamples, while an adverse result was observed in thecase of asphaltenic crude oil. This could be justi�edas the result of increments in viscosity under longtime ultrasonic exposure.

� It was shown that wave radiation increases the rela-tive permeability of both wetting and non-wettingphases. It also increased the end point relativepermeability of the non-wetting phase and decreasedthe value of critical gas saturation.

� The results of this work can be helpful to betterunderstand the role of ultrasonic waves on therelative permeability curves of the two-phase gravitydrainage process.

Nomenclature

A Cross sectional area of porous media(L)

foD Dimensionless fractional ow functiong Gravity acceleration (L/T2)H Height of the porous media (L)J Leverett functionk Absolute permeability (L2)kr Relative permeabilityk0ro Corey equation constantM Mobility ration Corey equation exponentNcg Capillary to gravity forcesNgv Gravity to viscous forcesNp Fractional produced oil

PV Pore volume (L3)pc Capillary pressure (M/LT2)Q Flow rate (L3/T)qt Production rate (L3/T)R Radius of the sand pack (L)Swi Initial water saturationSorg Residual oil saturation after gas

injectionSoD Outlet normalized oil saturationSoDe Outlet normalized oil saturation at end

pointt Time (T)tD Dimensionless timeu Fluid velocity (L/T)uDarcy Darcy uid velocity (L/T)zD Dimensionless displacement

Greeks

� Viscosity (M/LT)

� Density (M/L3)

� Hydraulic potential (M/LT2)� Porosity� Interfacial tension (M/T2)

Subscripts

D Dimensionlessg Gaso Oil

References

1. Duhon, R. and Campbell, J. \The e�ect of ultrasonicenergy on ow through porous media", 2nd AnnualEastern Regional Meeting of SPE/AIME (Charleston,WV) (1965).

2. Chen, W. \In uence of ultrasonic energy upon the rateof ow of liquids through porous media", PhD Thesis,OSTI ID: 5442582, USA (1969).

3. Fairbanks, H. and Chen, W. \Ultrasonic accelerationof liquid ow through porous media", Chem. Eng.Prog., Symp. Ser. (United States), West Virginia Univ.(1971).

4. Cherskiy, N., Tsarev, V., Konovalov, V. and Kus-netsov, O. \The e�ect of ultrasound on permeability ofrocks to water", Transactions (Doklady) of the USSRAcademy of Sciences, Earth Science Section (1977).

5. Neretin, V. and Yudin, V. \Results of experimentalstudy of the in uence of acoustic treatment on perco-lation processes in saturated porous media", Topics inNonlinear Geophysics, pp. 132-137 (1981).

6. Beresnev, I.A. and Johnson, P.A. \Elastic-wave stim-ulation of oil production: A review of methods andresults", Geophysics, 6(59), pp. 1000-1017 (1994).

7. Aarts, A., Gijs, O., Bil, K. and Bot, E. \Enhancementof liquid ow through a porous medium by ultrasonicradiation", SPE Journal, 4(4), pp. 321-327 (1999).

8. Hamida, T. and Babadagli, T. \E�ect of ultrasonicwaves on the capillary imbibition recovery of oil", SPEAsia Paci�c Oil and Gas Conference and Exhibition,Jakarta, Indonesia (2005).

9. Naderi, K. and Babadagli, T. \E�ect of ultrasonicintensity and frequency on heavy-oil recovery fromdi�erent wettability rocks", International Thermal Op-erations and Heavy Oil Symposium, Calgary, Alberta,Canada (2008).

10. Zendehboudi, S., Chatzis, I., Sha�ei, A. and Dusseault,M.B. \Empirical modeling of gravity drainage in frac-tured porous media", Energy & Fuels, 3(25), pp. 1229-1241 (2011).

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11. Firoozabadi, A., Soroosh, H. and Hasanpour, G.\Drainage performance and capillary-pressure curveswith a new centrifuge", Journal of Petroleum Technol-ogy, 7(40), pp. 913-919 (1988).

12. Pedrera, B., Bertin, H., Hamon, G. and Augustin, A.\Wettability e�ect on oil relative permeability duringa gravity drainage", SPE Annual Technical Conferenceand Exhibition (2002).

13. Parvazdavani, M., Masihi, M. and Ghazanfari, M.\Gas-oil relative permeability at near miscible con-ditions: An experimental and modeling approach",Scientia Iranica, 3(20), pp. 626-636 (2012).

14. Rostami, B., Kharrat, R., Ghotbi, C. and Tabatabaie,S. \Gas-oil relative permeability and residual oil sat-uration as related to displacement instability anddimensionless numbers", Oil & Gas Science andTechnology-Revue de l'Institut Fran�cais du P�etrole,2(65), pp. 299-313 (2009).

15. Hagoort, J. \Oil recovery by gravity drainage", OldSPE Journal, 3(20), pp. 139-150 (1980).

16. Chatzis, L. and Ayalollahi, S. \Investigation of theGAIGI process in strati�ed porous media for therecovery of water ood residual oil", Technical Meet-ing/Petroleum Conference of the South SaskatchewanSection (1995).

17. Ayatollahi, S. Lashanizadegan, A. and Kazemi, H.,\Temperature e�ects on the oil relative permeabilityduring tertiary gas oil gravity drainage (GOGD)",Energy & Fuels, 3(19), pp. 977-983 (2005).

18. Corey, A., Rathjens, C., Henderson, J. and Wyllie,M. \Three-phase relative permeability", Journal ofPetroleum Technology, 11(8), pp. 63-65 (1956).

19. van Golf-Racht, T.D., Fundamentals of FracturedReservoir Engineering, Elsevier Science (1982).

20. Naja�, I., Mousavi, S., Ghazanfari, M., Ghotbi, C.,Ramazani, A., Kharrat, R. and Amani, M. \Quantify-ing the role of ultrasonic wave radiation on kinetics ofasphaltene aggregation in a toluene-pentane mixture",Petroleum Science and Technology, 9(29), pp. 966-974(2011).

Appendix A

In the presence of viscous, capillary and gravity forces,the following equation can present the amount of oilfractional ow:

@SoD@tD

+@foD@zD

= 0; (A.1)

where dimensionless time, outlet normalized oil satu-ration, dimensionless displacement and dimensionlessfractional ow function are presented as follows:

tD =��oggkt

�o'(1� Sorg � Swi) ; (A.2)

SoD =So � Sorg

1� Sorg � Siw ; (A.3)

zD =zL; (A.4)

foD =1Ngv

�gkrokrg�o + kro +Ncg @J

@SoD@SoD@zD

1 + �gkrokrg�o

; (A.5)

where Ngv, Ncg and J(SoD) could be obtained by thefollowing equations:

Ngv =��oggk�ou

; (A.6)

Ncg =�p'

k��oggl

; (A.7)

J(SoD) =�p'

kpc

: (A.8)

By considering Hagoort assumptions for capillary pres-sure and gas mobility, the fractional ow will be onlydependent on SoD:

foD = kro(SoD): (A.9)

So, the fractional ow equation could be written as:

@SoD@tD

+@kro@SoD

@SoD@zD

= 0: (A.10)

This equation has been solved by the characterizationmethod of the Buckley-Leverett solution:

zD(SoD) =@kro@SoD

tD(SoD): (A.11)

Before the gas break-through, the amount of enteredgas is equal to the amount of oil produced:

Np =R t

0 qtdtPV

=tDNgv

: (A.12)

After gas break-through:

Np=1�SoDZ0

(1� zSoD )dSoD = 1� SoDe + kro(SoDe)tD:(A.13)

zSoD is the vertical distance at the speci�c saturation,and SoDe is also the out ow normalized oil saturation,which can be expressed as:

SoDe =�

1nk0

rotD

� 1n�1

; (A.14)

where k0ro is considered to be a curve �tting param-

eter. By assuming a Corey-type function for relativepermeability, kro = k0

roSnoD; Np can be calculated as:

B. Keshavarzi et al./Scientia Iranica, Transactions C: Chemistry and ... 21 (2014) 763{771 771

Np = 1� SoDe + k0roS

0oDtD: (A.15)

At the out ow end, zD = 1

@kro@SoD

tD(SoD) = 1: (A.16)

So:

Np = 1��

1� 1n

��1

nk0rotD

� 1n�1

: (A.17)

The amount of k0ro can be obtained from this equation,

when tD = 1. The value of n is calculated from thefollowing equation:

dln(1�Np)dlntD

= ��

1n� 1

�: (A.18)

Biographies

Behnam Keshavarzi holds an MS degree inPetroleum Engineering from Sharif University of Tech-nology, Tehran, Iran. He has authored/co-authored anumber of publications in the area of ultrasonic appli-cations in the oil industry, asphaltene damage removaland uid percolation in fractured porous medium underelastic waves. His research interests include modelingasphaltene damage stimulation by ultrasonic wave, welltesting technology and minimum miscibility pressurecalculation methods.

Rahim Karimi has a BS degree in Petroleum En-gineering and an MS degree in Aerodynamics fromSharif University of Technology, Tehran, Iran. Hisresearch interests include drilling engineering and uid

dynamics, and he has authored/co-authored a numberof papers in these areas of expertise.

Iman Naja� is a PhD degree student of PetroleumEngineering at Sharif University of Technology,Tehran, Iran. His research interests include modelingthe ow induced by waves in petroleum reservoirs,heavy oil upgrading using ultrasonic wave technologiesand ultrasound assisted asphaltene removal, and he hasauthored/coauthored a number of publications in thisarea of expertise.

Mohammad Hossein Ghazanfari obtained a BSdegree from Isfahan University of Technology, Iran,and PhD and MS degrees, in Chemical Engineering,from Sharif University of Technology, Tehran, Iran,where he is currently Assistant Professor of Chemicaland Petroleum Engineering and head of the EnhancedOil Recovery Laboratory. His research interests in-clude modeling of transport in porous media, mi-cro/core scale EOR studies, and asphaltene precipita-tion/deposition in static/dynamic conditions.

Cyrus Ghotbi was born in 1956 in Tehran, Iran.He obtained his BS degree from the Department ofChemical Engineering at Sharif University of Technol-ogy, Tehran, Iran, in 1978, and MS and PhD degreesin Petroleum Engineering from the French Institute ofPetroleum (FIP), Paris, in 1981 and 1984, respectively.He is presently Professor of Chemical Engineering atSharif University of Technology, Tehran, Iran. Hisresearch interests include thermodynamics, separationprocesses, and petroleum characterization. ProfessorGhotbi is the author of about 50 papers published ininternational journals, and a bout 90 papers presentedat international and national conferences.