Modeling Non-Equilibrium Mass Transfer Effects for a...

SPE 39746

Modeling Non-Equilibrium Mass Transfer EffectsWei-Jr. Wu, Peng Wang, Mojdeh Delshad, Chong Wang, Gary A.

Department of Petroleum and Geosystems Engineering, The

ABSTRACTTo assess the effect of non-equilibrium mass transfer on the

productivity ofa single weI1 producing horn a gas condensatefiel~ a model incorporating non-equilibrium mass transfm* was implemented into an equation-of-state @OS)compositional reservoir simulator developed at The Universityof Texas at Austin.

A comlation from the literature was used to account for theeffect of variables smh as gas velocity and difision coefficientson the mass -fer coefficient. However, no mass transferdata were available for gas condensates, so a sensitivity studyon the mass transfer coefficient was performed. Severalsimulations have been performed to evaluate the effm of thenon-equilibrium mass transfer on the flow behavior in theregion near the welIbore, The results tim these runs w=comparet with those obtained under the local equilibriumassumption. Such comparisons reveal that non-equilibriumphase behavior lead to a reduction in the condensate saturationin the region near the wellbore. The mole fractions for lightand heavy components in the oil phase are noticeably different.In the high veIocity layers, these ditinces become moresignificant. In generaI, non-equilibrium eti lead to slowerreductions in well productivity due to the fict that condensatedropout was reduced near the wellbore.

INTRODUCTIONMost reservoir simulation studies to date assume local

equilibrium between the fluid phases. For high rate gas wells,the residence time for fluids in the gridblocks near wellbore is

Society of Petroleum E~eers

for a Gas CondensatePope and Mukul M. Sharma

University of Texas at Austin

exmcted to be of the order of seconds.

Field

This is unlikely tofivide sufficient time for the fluids to reach equilibrium witheach other. Some laboratory experiments have supported thishypothesis (Burger et al., 1996).

The purpose of this work is to assess the effa of the non-equilibrium mass transfer on the productivity of a single wellproducing tim a gas condensate field. To do so, we haveimplemented a model dealing with non-equilibrium masstransfer into the equation-of-state (EOS) compositionalreservoir simulator UTCOMP, developed at The University cfTexas at Austin. This model is based upon data andcorrelation, published by Wilkins et al. (1995). To ourknowledge, there are no data on mass transfer coefficients underhigh temperature and high-pressure gas condensate reservoirconditions. Therefor, we performed a sensitivity study on themass transfer coefficient to assess the large uncertainty in it.Nghiem et a/. (1997) recently presented a simulation study cfdry gas displacing a light oil with mass transfer limitationsand a constant mass Wfer coefflcien~ but no comparisonswith data were made. Burger and Mohan~ (1997) studied theeffect of diffision on gas displacing oil, but each gridblock wasassumed to be at local equilibrium, so this is not non-equilibrium mass transfer in the local sense.

Several simulations have been performed to evaluate theeff- of the non-equilibrium mass transfer on the flowbehavior in the region near the wellbore. The dependence cfrelative permeability and residual saturations on the capillarynumber was not used so that we could assess just this oneeffect, but in future simulations the combined eff~ should beincluded since separate studies have shown that capillarynumber eff~ ca be very large under these conditions. Theresults from these simulations we~ compared with thoseobtained under the local equilibrium assumption.

THE UTCOMP SIMULATORThe UTCOMP simulator is a three-dimensional, EOS

compositional reservoir simulator (Chang et al., 1990). Theformulation follows the one by Acs et a/. (1985) with somechanges. The solution scheme is analogous to IMPEC(implicit pressure and explicit composition). A higher-orderfinite-difference method with the total variation diminishing

203

2 W.-J. WU, P. WANG, M. DELSHAD, C. WANG, G. A. POPE AND M. SHARMA

(~] scheme as well as the conventional one-point upstreamweighting method is avaiIable to discretize the partial-ti~ component-mass-balance equation. Three-phaseflash cal-n is also implemented so that UTCOMP iscapable of modeling four-phase flow. Both Peng-Robinson(PR) EOS (Peng and Robinson, 1976) and the modifiedRedlich-Kwong by Turek et al. (1984) can be used for phase-behavior calculations. Relative permeability can be treated as afimction of interracial tension and velocity through thecapillary number. Tracers, surfactant, foam and polymer etican also be modeled with UTCOMP.

NON-EQUILIBRIUM MASS TRANSFER MODELme non-equilibrium mass transfer model of Wilkins et al.

(1995) is as follows:

Jiao =ko,(c:q-c, ) (1)

where Ji is the mass flux; a. is the specific interracial area; kO,

is the mass transfer coefficient of a component i; and C’iis thecone-on of component i in the flowing phase. The-citit ko, was evaluated using packed bed experiments.The superscript eq means the corresponding property isevaluated at equilibrium.

When the mole fiction of each species in the gas is usedin pIace of the mass concentration, Eq. (1) becomes:

axig~= %,(X: - X,g) (2)

where x,g is the mole fiction of component i in the gas

phase; and x,~ is the mole fraction of component i in the gas

phase at equilibrium.The coefficient ko, in Eq. (2) is the mass transfer coefficient

for component i in the gas phase and is computed using thefollowing correlation by Wilkins et al, (1995):

(3)

where m = 0.38 in l/cml 82 ;d50 is the mean grain size in cm;Vg is the gas interstitial velocity in cm/see; and Dl,g is the

~sion cceficient of component i in the gas phase incmzlsec.

The constant m is treated as an input to the UTCOMPsimulator. A sensitivity study was petiormed by varying thevalue of m. The mass transfer coefficient, ko,, has units &l/see. For the calculation of the mean grain size, d50, theCarman-Komny correlation (Carman, 1937) was used:

d“[3)d(l - #)2 0’550=

$(4)

In Eq. (4), k is the rock permeability in cmz and @ is the rockporosity. The computed values of d50 in each gridblockinstead of the average values are used in this work. Theconstant c in Eq. (4) is assumed to be 300 (Wilkins et al.,1995).

Using the finite difference scheme, Eq. (2) becomes:

~n+livg

,.[(x[;y+’-xfg] (5,n +Atko ~‘Xl+g

Using Eq (5), the mole fraction of each component can becalculated at a new time step. Then, the EOS parameters areupdated using the new composition. The phase properties alsoneed to be recalculated.

UTCOMP SOLUTION PROCEDURE FOR THE NON-EQUILIBRIUM MASS TRANSFER OPTION

The IMPEC nature and the volume-baIanced formulation ofthe UTCOMP simulator make extensions of its capability notas complicated as other simulators. The procedure weimplemented was to use Eq.(5) to correct the equilibriumphase composition. The calculation at one timestep may koutlined as follows:1.

2.

3.

4.

5.

6.

7.

compute coefficients of pressure equation and solvepressure equation implicitly;compute total number of moles fm each component ineach gridblock using component mass balance equation;perform equilibrium calculation (flash) to determinenumber and saturation of phases and composition for eachseparate phase in each gridblock using new pressure;account for non-equilibrium mass transfer effects using Eq,(5) for gas phase;R-compute oil-phase saturation and composition using amolar balance calculation;update rock and phase properties and phase relativepermeability using new phase saturation and composition;go to next timestep.

It cfi be seen that Steps 4 and 5 are added for this option,

FLUID DESCRIPTIONThe PR EOS is used to compute equilibrium mass ~fer

for hydrocarbon components and phase volumes. Indeveloping the model for the gas condensate examined, weemployed the procedure suggested by Wang et aI., 1997.which includes the following steps:. select a fluid sample representative of the actual gas

condensate where experimental PVT data is available;. describe the heavy end of the fluid sample using as many

as 44 different methods (combinations of different methodsin each step of the characterization procedure);

. choose a base fluid description that can best predictexperimental liquid dropout data of a constant-volumedepletion (CVD) process for water-free sample;

. petiorm free-tuning of critical properties of the heavypseudocomponents of the base fluid description to matchmeasured CVD data of water-bee sample;

204

MODELING NON-EQUILIBRIUM MASS TRANSFER EFFECTS FOR A GAS CONDENSATE FIELD 3

. test predictive capability of the model by applying it topredict phase behavior of both water-~ and water-containing samples.

Based on the above procedure, a fluid sample that shows arelatively high liquid dropout (around 1.50A)at a temperature

of 353°F was chosen as the reference fluid. Among the 44 fluiddescriptions tried, the one using1. the exponential finction to split CT+ tition into 45

pseudocomponents;2. the Gaussian quadrature method to lump the split

components into two heavy groups;3. the TWU correlation (Twu, 1984) to compute the critical

properties and acenm-c factors of each pseudocomponent;and

4. two pseudocomponents to represent components betweenethane and hexane,

was found to reasonably predict experimental dewpointp~ssure and CVD liquid dropout data of the mf-ce fluid.Afier free-tuning, a 6-component PR EOS model for water-tifluid was developed, whose composition and properties heach component are listed in Table 1. Figure 1 shows theliquid dropout curve obtained from this model, which is insatisfactory agreement with experimental data. Additionally,this model can● estimate satisfactorily the Z-fictor of both dry and wet

samples, which are shown in Figures 2 and 3respectively; and

. predict acceptable shape of pressure-temperature phaseenvelop (Figure 4).

It should be added that models developed ti-omother fluid-description procedures predicted an un-closed shape of thephase envelop. Figure 5 shows a typical prediction for thephase envelop using such procedures.

It was found that the volume-shift parametem w notnecessarily needed in order to match the PVT data for this gascondensate.

DIFFUSION COEFFICIENTSThe difiion coefficients for each component in the gas

phase were calculated using the correlation of Wilke and Lee(1955). For the components in the liquid phase, the diffusioncoefficient correlation of Wilke and Chang ( 1955) was used. Adetailed description of these correlations is given in AppendixA. Some component physical properties required by thesecorrelations are taken from the reference book of Reid et al.(1987). Table 2 presents the diffision coefficients used in oursimulations, which are similar to the values used in the studyof Burger and Mohanty (1997).

RESERVOIR DESCRIPTION AND SIMULATIONSPECIFICATIONS

The simulation input was based partly on the descriptionand data for the Arun field given in the paper by Afidick et al.(1994). In order to estimate non-equilibrium e- on wellproductivity, radial flow into a single well was simulatedusing a two-dimensional x-z cross section with an angle of 36°(a pie shape as shown in Fig. 6). The simulation grid has 8layers in the vertical direction with the highest permeabilitylayer at the top (layer 1) and the lowest permeability at the

bottom (layer 8). The horizontal permeability for each layer islisted in Table 3. The grid has nineteen gridblocks in the xdirection that vary between 1 fi near the well and 500 R farthestfrom the well.

The reservoir temperature is 335 ‘F. me initial pressure is4100 psi, which is in the retrograde condensate region. Thecritical condensate saturation was 0.25. The residualsaturations for water and gas were 0.30 and 0.375,respectively.

Corey’s model was used to compute relative permeability.The parameters for this model were obtained by matching theexperimental data from Henderson et al. (1997). The endpointrelative permeability for gas and oil were 0.53 and 0.05 andthe exponents for gas and oil were 2.3 and 5.8, respectively.These values correspond to low capillary number conditionsand were not varied during these simulations.

The production rate for the entire well is 44 x 106SCF/day. In our simulations, only one tenth of this rate wasspecified because we were simulating a pie shape with only 36degrees. Further, we assumed that the well penetrates theentire eight layers. The production rate is allocated to eachlayer based on the phase mobility (a ratio of the phase relativepermeability to its viscosity) of the gridblock containing thewell. An open boundary with a constant pressure of 4100 psiis used for the outer boundary of the reservoir, which allowsthe fluid having the initial composition to flow into thesimulation domain.

Simulations were performed using the mass transfercoefficient proposed by Wilkins et al. (1995). This correlationwas developed based on experimental data under low pressureand temperature conditions. The variables in the correlationinclude gas-phase interstitial velocity, difision coefficient foreach component and mean grain size of the sand.

Several simulations using the UTCOMP simulator withthe above mentioned specifications have been performed.Since no gas condensate experimental data are available tocompare with, simulations were conducted based on thevariation of the constant m in Eq. (3). We used 0.38, 0.038,and 38 for this constant.

RESULTS AND DISCUSSIONFigures 7 and 8 show the oil saturation profile in layer 1

(high permeability layer) afier 10 and 30 days. Non-equilibrium mass tisfer leads to a reduction of thecondensate saturation in the region near the wellbore comparedto local equilibrium. The saturation difference at 30 daysranged betsveen 4°/0 to 33°/0 for the case with m=0.38. Thisreduction in the condensate saturation increases as the non-equilibrium mass transfer coefficient decreases. For the casewith m=38, the simulation results are almost the same as thelocal equilibrium results. Similar results were also observedfor layers 3 and 8 (Figs. 9 and 10), which Iave relatively Towpermeability.

The diffemces in the pressure profiles (Figs. 1I and 12)and well productivities are not significant largely because thecondensate saturation was too low to flow in all cases (noteagain that a fixed residual condensate saturation of 0.25 wasused in these three simulations). For simplicity, the effect &capillary number was not included in these simulations. In

205


other simulations, we have found significant reductions incondensate saturation near the well due to the very highcapillary numbers near the well.

Figure 13 shows the velocity as a fiction of distance inlayers 1 and 8 afier a period of 30 days, Figure 14 illustratesthe mass transfer coefficients for the lightest component,methane, and the heaviest component, C 11’, corresponding tothe velocity profiles plotted in Figure 13 and the diffusionc~fficient listed in Table 2, The lighter components achieveequilibrium relatively faster than the heavier components. Thenonequilibrium calculations include both the mass tmnsfercoefficient and the magnitide of the driving force due to themole fiction ~erence between local equilibrium and flowingmole fictions. These calculations are coupled since thedriving force is a fbnction of pressure and thus the phasebehavior. Figures 15 and 16 show a significant effi on thecomponent concentration in the condensate phase but not inthe gas phase for all the components except for the heaviestcomponent of C 1l+. The term deviation in these figures meansthe ratio of the diffe~nce in component mole tiction betweenequilibrium and nonequilibrium cases to the component molehction tim the equilibrium case. An increase inconcentration of heavy components in the oil phase near thewellbore causes an increase in the specific gravity andviscosity of the oleic phase. This can reduce thetransmissibili~ of the oleic phase, which also helps explainthe results in Figures 15 and 16. In the gas phase, the molefraction of the heaviest component is 1.8x10-3compared to the1.3x10-3 of the equilibrium case. Thus, the effti ofcomponent deviation in the gas-phase properties would not belarge.

To fiuther study these phenomena, a simulation with drygas injection from the outer bound~ was performed. Theinjection well was kept at a constant bottomhole pressure cf4100 psi. The composition of the injected gas is listed inTable 4. Figures 17 and 18 show the saturation and pressuredistributions of the oil phase for the cases with and withoutnonequilibrium mass transfer, Similar trends are observedcompared to the previous results with the open boundary.These results indicated that the effii of nonequilibrium masstransfer on the productivity index is not significant. Figures19 and 20 show the effii of non-equilibrium mass transfer onthe component distribution.

SUMMARY AND CONCLUSIONSA simulation study of the effi of non-equilibrium mass

transfer on the well productivity of a gas condensate well hasbeen performed using an EOS compositional reservoirsimulator that includes a correlation of the mass transfercoefficient taken from the experimental literature. From thesesimulation results, we observed the following:

“the non-equilibrium phase behavior leads to a reduction inthe condensate saturation in the region near the wellbore

“the mole fictions for both the light and heavycomponents in the oil phase are noticeably different when themass transfer is not instantaneous, In the high velocity layers,the differences are more significant

These results suggest that non-equilibrium mass lransfermight be important under some conditions of very high flow

rate wells and thus merits Wer research. There is a need tomeasure mass transfer coefficients for gas condensates to eitherverifi the available mass transfer coefficient correlation ordevelop a new correlation. Also, the effa of capillarynumber and difisive flux should be included in titure studiesand more understanding of how mass transfer eff~ mightinteract with formation heterogeneity and non-Darcy flow inhigh rate gas condensate wells is needed.

ACKNOWLEDGMENTSWe thank Kathy Hartman, Ravi Vaidya, Mary Coles andMyung Hwang of Mobil Exploration and ProductionTechnical Center (Meptec) for usefil discussions and tidingthis research.

NOMENCLATUREspecific interracial area, cmzconcentration of solute in the flowing phase, volumefictiondifision coefficient of component i in gas phase in

cm2/secmean grain size, cm

mass flux, g/cm2/secrock permeability, cmzmass transfer coefficient, l/seegas interstitial velocity, cm/sec

mole fiction of component i in gas phase

mole fiction of component i in gas phase at

equilibrium.

206


REFERENCEAcs, G., Doleschall, S., and Farkas, E.: “General PurposeCompositional Model,” Sot, Pet. Eng. J,, Vol. 25, No. 4(1985) 543-553.

Afidick, D., Kaczorowski, N.J., and Bette, S.: “ProductionPerformance of a Retrograde Gas Reservoh A Case Study cfthe Arun Field,” paper SPE 28749 presented at the SPE AsiaPacific Oil and Gas Conference, Melbourne, Australia, Nov. 7-10,1994.

Burger, J.E. and Mohanty, K.K,: “Mass Transfer hBypassed Zones During Gas Injection,” SPE ReservoirEngineering, 124-130 (May 1997).

Burger, J.E., Spingate, G.S,, and Mohanty, K.K.:“Experiments on Bypassing During GasFloods inHeterogeneous Porous Media,” SPE Reservoir Engineering,109 (May 1996).

Carman, P.C., “Fluid Flow ~rough a granular bed”Transactions of tie Institution of Chemical Engineering,London, Vol. 15, 150-167 (1937).

Chang, ”Y.B., G.A. Pope, and K, Sepehrnoori: “A Higher-Order Finhe Difference Compositional Simulator;’ J, Pet. Sci.Eng., 5 (1990) 35-50.

Henderson, G.D., Danesh, D,H., Tehrani, S.1, Pantling, J.and A1-Shaidi, S.: “Gas Condensate Recovery Project, ”Progress Report August-December 96, Heriot-WattUniversity, UK, 1997.

Nghiem, L.X., and Sarnmon, P.H., “A non-EquilibriumEquation-of-State Compositional Simulator,” paper SPE37980 presented at the 1997 SPE reservoir SimulationSymposium, Dallas TX, June 8-11.

Peng, D.-Y. and Robinson, D.B.: “A New Two-ConstantEquation of State,” Ind. Eng. Chem, Fundam., Vol. 15, No. 1(1976) 59-64.

Reid, R.C., Prausnitz, J.M. and Poling, B.E. The propertiesof Gases and Liquid, Forth edition, McGraw-Hill Inc., NewYork (1987).

Turek, E.A., Metcalfe, R,S,, Yarborough, L., and Robinson,R.L.: “Phase Equilibria in C02-MulticomponentHydrocrubon Systems: Experimental Data and An ImprovedPrediction Technique,” Sot. Pet. Eng. J., Vol. 24, No, 3(1984) 308-324.

Twu, C.H.: “An Internationally Consistent Correlation hPredicting the Critical Properties and Molecuair Weights &Petroleum and Coal-Tar-Liquids,” Fluid Phase Equilibria, 16(1984) 137.

Wang, P., Pope, G.A. and Sepehrnoori, K. “Development cfEquation of State for Near Critical Gas Condensates:’Submitted to Ind. & Eng. Chem. Research ( 1997).

Wilke, C.R.: “Difiional Properties of MulticomponentGases,” Chem. Eng. Progress, 95 (1955).

Wiike, C.R. and Chang, P.: AZCHE J. 1:264 (1955)

Wllke, C.R. and Lee, C.Y.: Ind. Eng. Chem., 47:1253 (1955)

Wilkins, M.D., L.M. Abriola, and K.D. Pennell: “AnExperimental Investigation of Rate-limited Nonaqueous PhaseLiquid Volatilization in Unsaturated Porous Media: SteadyState Mass Transfer,” water resources research, vol. 31, no. 9,2159-2172 (Sept. 1995).

Appendix A: Calculation of the diffusion coefficients

Wilke and Lee ( 1955) correlation:

~ ~.03 -(0.981M$2)10-3 T’I’AB=

P~~2~~~D

WhereT = temperature, kMA, MB = molecular weights of A and B, g/mole

MAB = 2[(1/MA)+(l/MB)]-1P = pressure, bar

c= 1.18V:’3

where Vb = liquid molar volume, cm3/mole

A c E G‘D ‘~+~(T ) exflDT ) + exp( FT ) + exp(HT )

* *

with T* =Tl(<ik)m

where A= 1.06036 B = 0.15610C = 0.19300 D = 0.47635E = 1.03587 F = 1.52996G = 1.76474H=3189411(</k)m =1.15Tb

Tb = the normal boiling point (at 1 atm), k

The Wilke-Chang Correlation

7.4x10 -8(M; )1’2TD, =

pov:b

(1955) :

207


with

Vbi= 0.285Vj ’048

Whm ~0 = oil phase viscosity,

Vti= VOILUUe of COmpOi’IeI’It i at Tb,Vc = critical volume

Table~. Description of the Gas Condensate.

Comp. z Tc Pc Vc 0) Parachor

nme ‘??0 K atm cm3/mol

C02 16.18 304.2 72.8 93,90 0.225 49.0

c1 70.98 190.6 45.4 99.20 0.008 71.0

C2.3 7.90 330.5 45.7 169.64 0.119 130.0

C4.6 2.60 453.8 34.9 298.58 0.226 230.0

C9 2.00 606.0 26.0 490.35 0.359 327.0

C22 0.34 869.7 14.9 1058.94 0.788 761.0

Table 2. me Difision Coefficients (cm2/see) for EachComponent

Component Difiion coefficient Difiion coefficientin oil phase (x104) in gas phase (x104)

C02 5.38 11,8

methane 6.23 15.0

ethane 4.30 9.17

propane 3.22 5.75

n-butane 2.78 4,461 I

n-pentane I 2.49 3.47 I

Table 3. Reservoir Permeability and Porosity.

hyer Thickness Porosity ermeability(f) (’??) (red)

1 10 0.300 902 10 0.250 753 30 0.214 504 50 0.220 285 100 0.209 126 50 0.219 177 150 0.127 2.68 370 0.120 1,5

Table 4. Injection Fluid Compositions

Component Mole fraction

C02 0.16551

methane 0.72624

C2.3 0.08072

C3.4 0.02584

PIUS1 0.011687

1 I

n-undecane 0.879 1.53 I

208


2.0 I 1 I I

0.00 1000 2000 3000 4000

“5000

Pressure (psi)

Figure 1. The liquid dropout curve using the 6-component PR EOS (T=352’F),

1.4 ~ ............................................. ................

~no Exp

1.3 . ... ........

Cal

1.2 . ............,..,,.{,,,............. ....... .....................

t

o.9t’’’’i’’’’ i”’”4000 4500 5000 5500 6000 6500 7000 7500

Pressure (psi)

Figure 2. Z-factor from the 6-component PR EOS.

..—

W. -J. WU, P. WANG, M. DELSHAD, C. WANG, G. A. POPE AND M. SHARMA

..l. l, ., ., .!..,...

7;o Exp . ... . ..

Cal ‘

/

..L0.95 ........... .......

0.90 ‘

..........

...... ...................

/

/

<

0...... .

.

4000 4500 5000 5500 6000 6500 7000 7500

Pressure (psi)

Figure 3. Z-factor for a wet fluid sample from the 6-component PR EOS.

r400 j--- -

350 ““ ““”””’”’j’“ ~~~-----

I300 ‘“’

~ ~50 ,,,,,,,,,,,,,,

g~ 200 . ,,,

~150 “ L/“,,,

. . . ..... ..,,,,,,,,.,.;,+ Reservoil

..............

.~.gg. ....+....\ ....... :.iI

,.0985..* .......

I

I1.. . . . . . .. . . . . .

/

I

/. . . . . . . . . ...2.....

:emperatuje J

11..........,,.,,/jk!-100 ..................

50 ‘ ~~~~~~~~~

o150 225 300 375 450 525 600

Temperature (k)

Figure 4. The phase envelop from the 6-component EOS.


(4.4 MM5&/D)

t

200 250 300 350 400 450 500 550Temperature (K)

Figure 5. Phase envelope from a model of tuning

the binary inter~ction coefficients.

flowing boundary

(4100 psi)

T7s0 ft

I

Figure 6. Schematic of grids used in the simulation study

211

10 W. -J. WU, P. WANG, M. DELSHAD, C. WANG, G. A. POPE AND M. SHARMA

0.35 .— Equilibrium

'''"""''""''"""'''''' ''"'~''"''""`'''""'"'"""'"'"''""''"'"''"""'""""`'""=``''`````---- y;n-;q;;pm=.

— -- Non-Equilibrium............. .. . . .. .. ................ . .. .......(m = 0.038)

0.20 :1 .j .../..................................................... ............................. ... .................._

. !1

0.15 :'~""t~'"''""""''''' '''''''''"'""''"`""'```"`"''"'"""'"."```-```'``"-""""-"""""+---`-"-""-""--"""""""---""""""""""-------1

............... . . .. ....................... ... .........................-

Distance from producer (ft)

!00

Figure 7. Effect of non-equilibrium mass transfer on oi saturation in layer

(Time = 10 days)

0.35— Equilibrium

--”””””’”’”””/;’”””’’”’”’’----- --- N’:n-;q;.ibriurn=.

..................................................................... — -- Non-Equilibrium

~ \\(m = 0.038)

0.20: “’”’ ;“” \ ‘“’’”/”””’”’” ““””’””””’”””’””””””’’”””~:

II0.15 ““””\l””’!\ .

........................ .....................................................I

——


Figure 8. Effect of non-equilibrium mass transfer on oil saturation in layer 1(Time =30 days)

.

212


0.35

0.30

0.10

0.05

0.00

1“””\....

‘1\l... ............‘\\l..‘\\\.......... .

--”\

.............

..... .................................

— Equilibrium

- -- Non-Equilibrium(m= 0.38)

— -- Non-Equilibrium(m= 0.038)

.. .. . . . {.,,,. .,, .,,.,;.,..,,.... ... .........-

.: ........ . ...{....... . .... . .......!.. .. .......... . . ... ......

) 40 60 80 1Distance from producer (ft)

Figure 9. Effect of non-equilibrium mass transfer on oil saturation in layer 3(Time =30 days)

0.35 ;

. .........................................

0.00 ~ -10 2

— Equilibrium.................. .. ..........

I-- ~- Non-Equilibrium(m= 0.38)

I....——......................—- Non-Equilibriun~

(m= 0.038)

() 30 40Distance from producer (ft)

Figure 10. Effect of non-equilibrium mass transfer on oil saturation(Time =30 days)

in layer 8

213

12 W. -J. WLl, P. WANG, M. DELSHAD, C. WANG, G. A. POPE AND M. SHARMA

4100

4000n

3700

................................................................. . . ................... ........... ... .

~*.............................4.... . . . . . . . . . . . . . . . . . . . . . . . . ...+... . . . . . . . . . . . . . . . . . . . . . . . .

I I—--Nol~-Equilibrium(m= 0.038)

36000 20 40 60 80 100

Distance (ft)Figure 11. Effect of non-equilibrium mass transfer on presure distribution

in layer 1 (Time = 30 days) -

4000 ~. ..................................................................................................

3900

3850 ...z ~~- ;

1{ :

—Equilibrium

3800 ~~~~~~~~~~~~~.........................................- -- “ y:::q:;ym.

!3750 ............................................J........................- - “ “:~:q;;;ym.. .3700 J

{,

o 20 40 60 80 1

Figure 12. Effect of non-equilibrium mass transfer on presure distributionin layer 8 (Time = 30 days)

214


103

102

10’

10°

~ol

. : .—=——

—


Figure 13. Gas phase interstitial velocity profiles at 30 days

....~..m= o+38........\...................j.....~....{....j...{...~... ................{............~........{......{..........\...~

:: :1:0 ~ ~ ~~~::loo ~ ~ ~~::looo. . .


Figure 14. Mass transfer coefficient profiles in layer 1 at 30 days

215

i4 W. d. WU, P. WANG, M. DELSHAD, C, WANG, G. A. POPE AND M. SHARMA

Figure

1.50

1.00

1,

;;L—C02

mmn

----- c1,Sti...................................7...... . . . . . . . . . . ...+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

‘ \jj;~~ / —.. c

t“. ~ j \ : : ---c2-3

0.50 “~.f:i..”’.””””’’””’.”.”------------------------, /4-5

- ----” PIUS1* ●.=. . . “=----”-”P1US2

~.#~m#=wm~”wMwJwMw”& ”* Mwtiwu=u&vJ* i-””-

“==””3=’”=-””==.”==.’”=—. . ....d------- .- q---- ------- .-

"""""."""".""'"""""-""-."`"-"."---"--.---"-------------------------------------------J ii

-1.00 ;’’. --: - . - . : ----: ----:---: . -’---””--~=o.3

100 200 300 400 500 600 700 [ oDistance from well (ft)

5. Effect of non-equilibrium mass transfer on component distributionsin oil phase Layer 1, Time = 30 days)

0.35

0.30

0.25

0.20

0.15

0.10

0.05

0.00

-0.05 ~

.!● :

-.

5 10Distance from well (ft)

—C02----- c1

—.. c2-3

-- - C3-4

-=--- P1US1. . . . =P1US2

............. ....................

........................ .... ....

Figure 16. Effect of non-equilibrium mass transfer on component distributionsin gas phase (Layer 1, Time = 30 days)

216


-—

0.30

0.25

0.20

0.15

0.05

0.00 b- ”-;”-”——-—:———.

20

~ - — - Non-equilibrium,. ................. (m=O.38)

— Equilibrium\.........................7................. .................................. .. .

.............. .. . ............!.............——

-——- .—— —— ;-——-1 60 80 1

Distance from well (ft)

Figure 17. Effect of non-equilibrium mass transfer on oil saturation(Layer 1, Time= 60 days)

4200 -~x 1; j

4000 “................................. ................................. .............................. ................................................................!

8 _4 #—- —--—-J-—

,

3800 “ fl-............ ,..,,,,,,,.,.$.,.!............................. ................................ ................................. ............................../;

: ~:

: ~

3600 -., ............................ .................................................................. ...............................................................

3400 “

?.1...m

. ............................... ................................. ............................:

i32m *

;

10

-— --0 20 40 60 80 100

Figure 18. Effect of non-equilibrium mass transfer on pressure(Layer 1, Time= 60 days)

217

W. -J.WU, P. WANG, M. DELSHAO, C. WANG, G. A. POPE AND M. SHARMA

1.50

1.00

0.50

0.00

-0.50

-1.00

........ .. . . ............................................... .................... ..............

..... ............................................4...... . . . . . . . . . . . . . . . . . . . . . . . . . . . .

-+-_:_~., =..

--;---=-;- ----:-

— C02----- c1—-. c

2-3-— - C34

-=-’ -P1US1

“---”””=-PIUS2-~w .,

m=O.38

100 200 300 400 5k 6& 7&

—..—

0Distance from well (ft)

Figure 19. Effect of non-equilibrium mass transfer on com orient distributionfin oil for the gas injection case (Layer 1 at 10 ays)

0.25

-=-----“”” -?==---- . . . . . . . . . . . . .4 . . . ..~#.=~.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..- *- . . . .

0.00: -

—C02----. c1—-. c

2-3.- - C3-4

-=-= -PIUS1

--=”=”=--P1US2

m=O.38

u 3 15 2Distance f;~m well (ft)

Figure 20. Effect of non-equilibrium mass transfer on component distributionin gas for the gas injection case (Layer 1 at 10 days)

Modeling Non-Equilibrium Mass Transfer Effects for a...

Documents

Transcript of Modeling Non-Equilibrium Mass Transfer Effects for a...