zSPE 49266

Modeling Relative Permeability Effects in

G.A. Pope, W, Wu, G, Narayanaswamy, M, Delshad, M.

Department of Petroleum and Geosystems Engineering,

Copyright 1998, Scciety C4Petroiaurn E~inaara, IM

ThiS PEpSr WaE - for~mam at he 1~ SPE W T~”&l c~ @Exhibii held in New OrtaMs, Leuieima, 27-30 Septem& 19SS,

Thie ~ wee safscted for premM!m by an SPE Prcgram Committee MM revtw ofInfermatim @i in m abstrect ~ittad by the euthw(s), CMents of the papar, sepraaentd have not k ~ by the S~ty of Petroleum Eng- md are subject towrrecfim by the edw(s). W material, as presented, *S not ~s~~ reflect any ~tbof the Stiaty of Petrulaum Enginaam, its offiars, or members. Papers praamtad at SPEmeetings are subject 10 ~i rwiaw by Edtorial Cmmfitms of h Society of PetmlmEnginaara. Elactrmii ra~odtim, distriiti~, w s[cfaga of any part & this papar for mmmercial~ws witiut the written msant of tie Sotiety of Petroleum Engimra is @hi,pamlissti to ~ in print is res~~ad to ~ ab~~r~~ ~ not ~~~ tin ~ m;

illustrat~s may Nt b W. The abstrad must main ccmspkums ~ledgmant of *and by Wm the papar was prasantad Write Libfarian, SPE, P.0, Box -, RiWaom TX7~ U.S.A, fax 01-S72-9S2-%

AbstractMany gas condensate wells show a significant decrease inproductivity once the pressure falls below the dew pointpressure. ~is significantly dec~ases the productivity of veryhigh rate gas wells in many of the world’s largest hydrocarbonreservoirs. A widely accepted cause of this decrease inproductivity index (PI) is the decrease in gas relativepermeability due to build up of condensate in the nearwellbore region. Predictions of well inflow performance requireaccurate models for the gas relative permeability as a functionof interracial tension (IFT) between the gas and condensatephases (actually it should be trapping number rather ,thanIFT). Since these relative permeabilities depend on fluidcomposition and pressure as well as condensate and watersaturations, a model is essential for both interpretation of labdata and for predictive field simulations. The key to such a gascondensate relative permeability model is the dependence cfthe critical condensate saturation on the capillary number or itsgeneralization called the trapping number. We have madecarefil comparisons of such data with a capillary trappingmodel and have found that we can match these data with asimple two-parameter model, We then developed a general

Societv of PetroleumEnoineers

Gas-Condensate

Sharma, P. Wang

Reservoirs

The University of Texas at Austin

scheme for computing the gas and condensate relativepermeabilities as a finction of the capillary trapping model andwith only data at low trapping number (high IFT) as inputand have found good agreement with the experimental data inthe literature. We then used this model and typical parametersfor gas condensates in a compositional simulation study of asingle well to better understand the PI behavior of the well andthe significance of the condensate buildup.

IntroductionAfidick et al.’ and Barnum et al. 2 have reported field datawhich show that under some conditions significant loss &well productivity can occur in some gas wells due to newwellbore condensate accumulation. As pointed out by Boomet al.,3even for lean fluids with very low condensate dropout,high condensate saturations may build up as many porevolumes of gas pass through the near wellbore region. As thecondensate saturation increases, the gas relative permeabilitydecreases and thus the productivity of the well decreases. Thegas relative permeability is a function of the interracial tensionbetween the gas and condensate among other variables. Forthis reason, several laboratory studies3-14have been reported onthe measurement of relative permeabilities of gas condensatefluids as a finction of interracial tension. These studies show asignificant increase in the relative permeability of the gas asthe interracial tension between the gas and condensatedecreases. The relative permeabilities of the gas andcondensate have often been modeled directly as an empiricalfi.mction of the interracial tension. 15 However, it has beenknown since at least 194916that the relative permeabilities ingeneral actually depend on the ratio of forces on the trappedphase, which can be expressed as either a capillary number or

643

2. - G.A. POPE, W. WU, G. NARAYANASWAMY,M. DELSHAD,M. SHARMA,P. WANG SPE492&

Bond number. This has been recognized in recent years to betrue for gas condensate relative permeabilities.8”0 Below wedficriie a model for correlating relative permeabilities as afunction of a new trapping number which is more general thanany model in the literature but is still easy to use, requires aminimum of experimental data, and has been found to wewell with a variety of experimental data. This model is ageneralization of the approach first presented by Delshad etal.’7 As shown below, this relative permeability model agreeswell with gas condensate data from the literature. We thenshow simulations of a gas condensate weIl to illustrate theeffect of condensate buildup on the well productivity with andwithout the capillary number effect.

Model DescriptionThe fundamental probIem with condensate buildup in thereservoir is that capilla~ forces can retain the condensate in thepores unless the forces displacing the condensate exceed thecapillary forces. To the degree that the pressure fomes in thedisplacing gas phase and the buoyancy force on the condensateexceed the capillary force on the condensate, the condensatesaturation will be reduced and the gas relative permeabilityincreased. This problem is analogous to the trapping of oil bywater dur”mga waterflood in which case it is loosely referred toas residual oil saturation and has been studied for decades.BrownelI and Katz’b and others recognized early on that theresidual oil saturation should be a finction of the ratio ofviscous to interracial fomes and defined a capill~ number tocapture this ratio. Since then many variations on the definitionhave been published’’”’”with some of the most common oneswritten in terms of the velocity of the displacing fluid, whichcan be done by using Darcy’s law to replace the pressuregradient with velocity. However, it is the fome on the trappedfluid that is most fi.mdamentaland so we prefer the followingdefinition:

(1)

where the gradient of the flow potential is given byvat, = VPIJ – gpt~VD. Although the distinction is not

usuaIIy made, one should designate the displacing phase ?’and the displaced phase / in any such definition, In somecases, buoyancy forces can contribute significantly to the totalforce on the trapped phase, To quantib this effw~ the Bondnumber was introduced and also takes diffe~nt forms in theliterature.zOOne such definition is as follows:

_ (2)

For special cases such as vertical flow, the fome vectors =collinear and one can just add the scalar values of the viscousand buoyancy forces and correlate the residual oil saturationwith the this sum, or in some cases one force is negligiblecompared to the other fome and just the capillary number orBond number can be used by themselves. This is the casewith most laboratory studies including the recent ones byBoom et al.3’8and Henderson et al.’0 However, in general theforces on the trapped phase are not collinear in reservoir flowand the vector sum must be used. In fact, the viscous forces aremost ofien nearly horizontal and the buoyancy force is vertical,so a scalar sum is completely wrong. A generalization of thecapillary and Bond numbers was derived by Jin2’ and calledthe trapping number. The trapping number for phase fdisplaced by phase P is defined as follows:

. . . .. . (3)

This definition does not explicitly account for the veryimportant effects of spreading and wetting on the trapping of aresidual phase. However, it does an excellent job of correlatingthe residual saturation of both a wide variety of phases ofdiffering wettabilities and a wide variety of rock types, so it isvery useful for modeling a particular set of rock and fluidconditions if a few basic data have been measured to establishthe correlation.

The residual saturation is modeled based on the trappingnumber as shown below.

In the above equation, S/r is the residual saturation for phase ?

and S/ is the saturation of phase /. The superscripts high and

low refw to high and low trapping numbers. s~~gh i5

typically zero. The trapping parameters T/ and T/”are obtained

by fitting residual saturation data for phase /. ~~ is typically

1.0, however, some of the condensate data from the literaturecan be fit somewhat better by using t~ as a second fitting

644

SPE49266 MODELING REIATIVE PERMEABILl~ EFFECTS IN GAS-CONDENSATERESERVOIRS 3,.. .-.

parameter. Mass transfer can reduce the value of S/ to values

less than S/r which is the reason why the minimum is taken

in Eq. (4). For example, dry gas flowing by residualcondensate at a fixed pressure can strip the lighter componentstim the condensate and reduce its saturation.

Establishing the correlation of residual saturations with thetrapping number (or special cases of it as appropriate) is thefmt and most fimdamental step in correlating relativepermeability data as a fiction of interracial tension. Althoughwe have always found Eq. (4) to be adequate and convenientfor this purpose for a wide variety of data sets including gascondensates, a table could also be used to represent thedecrease in residual saturations with increasing trappingnumber, or for that matter some other simple fiction that fitsthe data, In cases such as gas condensates there are threeresidual phases (gas, condensate, and water) and thiscorrelation has been found to apply to all three phases.

The next step is to correlate the endpoint relativepermeability of each phase, which increases in a ve~predictable way as the trapping number increases and can becorrelated using the following equation.

where S~r is the residual saturation of the conjugate phase e.g.condensate is the conjugate phase for gas. This equation hasalso been found to-provide a good correlation of a wide varietyof data. The final step is to calcdate the relative permeabilityof each phase / as a fiction of saturation. One approach tothis problem is to assume a simple finction such as a Corey-type relative permeability fwction,” This then requirescorrelating the Corey exponent with trapping number.Equation (5) written in terms of the exponent rather than theendpoint can be used for this pu~ose.22 However, not allrelative permeability data can be fit with a Corey-type model,so we have generalized our approach by using the followingequation:

log krt = log k:/ + log S/

()low

krtlog — – log 5/

k;t

+ l+ Ty(NTf )’p

(6)

where kr~ is the relative permeability and k~l is the endpoint

relative permeability for a given trapping number ando low

saturation. k~w and krl are the relative permeability

and endpoint relative permeability at a low trapping numberfor phase !.

The normalized saturations ( S1 ) in the above equation aredefined as

1=1

where np is the number of phases present, S! isand S/r is the residual saturation for phasecalculated using equation (4).

“m

the saturationg, which are

Only the baseline relative permeability curve of each phaseat low trapping number corresponding to the usual laboratorymeasurements and the residual saturations as a finction of thetrapping number are needed in this approach. Figures 1 and 2show the computed relative permeability of gas and condensatecalculated for a wide range of trapping numbers using just twoparameters. Although this model captures the general trends inthe data vety well, there are still some issues such ashysteresis which have not been adequately investigated for gascondensates. However, it should be a great improvement overthe traditional approach used in compositional reservoirsimulators that are used to model gas condensate reservoirs.

Comparisons With Experimental DataAs pointed out above, the best starting point for understhdingand modeling relative permeability data as a finction ofinterracial tension is the relationship between the residualsaturations and trapping number (or its special cases &capillary number or Bond number when appropriate to theexperimental conditions). For this reason, we first show anexample of normalized residual saturations versus trappingnumber in Figure 3. The residual saturations were normalizedby dividing them by the low trapping number pIateau values.As seen from these data, there is a very large difference betweenthe nonwetting and wetting phase data. A much largertrapping number is required to decrease the residual saturationfor the wetting phase than for the nonwetting phase. This istypical of all of the data in the literature for all types of phasesand rocks (e.g. see the review in Delshad22).We selected thesedata from the many examples in the literature to make thepointin a

that even widely different phases have similar behaviorgiven rock if their nettability is the same. The

645

4 G.A. POPE, W, WU, G. NARAYANASWAMY,M. DELSHAD,M. SHARMA,P. WANG SPE49266

nonwetting phases in Fig. 3 are the gas and oil. The gas datatim Henderson et al.’0are for the equilibrium gas in a binarymixture of methane and n-butane intended to rep~sent a gascondensate fluid, The oil data from De1shadz2are for theequilibrium oiI for a mixture of decane, brine, isobutanol andsodium sulfonate under three-phase conditions. The wettingphases in Fig. 3 are the aqueous and microemulsion phases.The aqueous data tim Boom et al, 3’8are for the equilibriumaqueous phase in a ternary mixture of water, n-heptane andisopropyl alcohol. The microemulsion data horn Delshad2zmfor the equilibrium microemulsion, The condensate data *Henderson et al.’0 appear to be of intermediate nettability(between the gas and water), which emphasizes the importanceincluding all three phases in the experiments.

More examples of wetting phase data for several differentporous media are shown in Fig. 4. The corresponding data hthe nonwetting phase we shown in Fig. 5. These dataemphasize the strong dependence on the rock as well as thenettability of the phases. The strong conclusion is that onemust measure the residual saturations for the wetting state androck of interest to get usefil results that can be accuratelyapplied to a particular reservoir state, In particular, if there mthree phases in the reservoir such as there are with gascondensates, then to ensure the correct wetting and spreadingstate in the rock, three phases need to be in the laboratory mreeven if one of the phases such as the brine is always at residualsaturation. There are too many other similar examples in theliterature to review here, but many other data can be found inStegemeier,23 Morrow and Chatzis24 and Delshad22 amongothers. Stegemeier23provides an excellent theoretical treatmentas well.

All of the data shown in Figs. 3 to 5 wem fit using justone parameter T/ for each phase / and the value of thisparameter is shown on each figure, Next we show thecomparisons with endpoint relative permeabilities using thesesame values of T/. Figure 6 shows the endpoint relativepermeability of the gas phase as a function of trapping numberfor the methane/n-butane binary mixture tim both Hartmanand Cullick7 and Henderson et al,’0 Figure 7 shows theendpoint relative permeability fm various liquid phases andporous media as a finction of the trapping number. The valuesvary significantly due to the differing rocks and for the samerock such as Berea sandstone due to the differing nettability.However, the general trend of increasing endpoint relativepermeability with increasing trapping number is consistentand clear and agrees with that previously reported by Delshadet al’7 for widely different fluids.

The curve calculated tim Eq. (5) of the model is shownfor comparison with these data. In all of these cases, theendpoint relative permeability appears to approach 1,0 at a

sufficiently high trapping number. This high trapping numbervalue is sometimes mf=d to as the miscible value, butstrictly speaking it is still an immiscible value even if theintetiacial tension is ultralow. As shown in Delshad,22 theinterracial tension can be high and the trapping number sti 11made high enough in the laboratory by increasing the pressuregradient to make the endpoint approach one, so it is not theintefiaciaI tension that matters per se, but rather the trappingnumber, The endpoint relative permeability was not alwaysmeasured by these investigators at a sufficiently low value &trapping number to determine its value directly, so its valuehad to be estimated by fitting the available data at intermediatetrapping numbers. This value is often refwed to as theimmiscible value of the endpoint, but clearly this is not ausefil designation. Its value depends on the value of thetrapping number. In some cases, a plateau value of theendpoint at some sufficiently low trapping number isobserved, but this is not always the case, especially bheterogeneous rocks and any phase that is wetting or evenmixed wet. For these and other fundamental reasons, thedesignations of miscible and immiscible values are not corrector usefil, nor is meaningful to describe these data in terms &high and low interracial tension, but rather only in terms ofhigh or low trapping number.

Figures 8 to 11 show comparisons between the modelcurves and several sets of relative permeability data for gas andcondensate fluids. No new parameters were introduced and yetall of the trends in the data are captured reasonably well. Thecapillary numbers shown on Figs. 8 and 9 correspond to thedefinition used by Henderson et al.,’” which is based uponvelocity rather than potential gradient. Since their experimentswere done with their capillary number held constant, it mademore sense to plot these relative permeability data using theirdefinition. Under their experimental conditions, buoyancy wasnegligible, so our model could be used with either definitionof capillary number. In general, however, the trapping numbershould be used.

It is important to note that very fw parameters are neededin this model and that it goes to all of the correct limitsobserved not just for these data but for other literature data hvarious cores, fluids and conditions. Our modeling effortsshow that for reasonable prediction of relative permeability atvarious trapping numbers, the measurement of endpointrelative permeabilities at diffmt trapping numbers is moreimportant than the measurement of relative permeabilities atvarious saturations at different trapping numbers. It is our hopethat fiture experimental studies will show more emphasis onmeasuring endpoint relative permeabilities.

646

sPE- MODELING RELATIVEPERMEABILITYEFFECTS IN GAS-CONDENSATERESERVOIRS 5

Numerical SimulationsWe used the relative permeability curves of Figs. 1 and 2 toinvestigate the effect of trapping number on the productivity &a single weII in a gas condensate reservoir. The equation-of-state (EOS) compositional reservoir simulator UTCOMP wasused in this study.zs The fluid description and phase behavioris the same as given in Wu et al.26 using the approach ofWang et aI.z7 to accurately model the behavior of a gascondensate fluid using six hydrocarbon components. Forsimplicity, a layered-permeability description was used for thisinitial simulation study. Both the reservoir description andphase behavior are similar to those of the Arun field studied byAfidick et al.’ However, these simulations are not meant toapply to the Arun field, but rather were done simply toillustrate the trends in the PI with trapping number. Asystematic simulation study of the Arun field including ahistory match of PI can be found in Narayanaswamy.n

Description of Simulation DataThe simulation domain used was a two-dimensional verticalcross-section (x-z) with a fm shape at an angle of 36° (Figure12), The simulation grid has 8 layers (Table 1) with thehighest permeability layer at the top (90 md) and the lowestpermeabili~ layer at the bottom (1.5 red). Nineteengridblocks were used in the x direction with a variablegridblock size of 1 to 500 fi and with the smaller gridblockslocated near the6wellbore, The well is producing at a constantrate of 4.4 x 10 SCF/D (corresponds to one-tenth of the fillwell rate) and it penetrates all 8 layers. A constant pressure(4100 psia) boundary condition was applied to the outerboundary of the fan-shaped reservoir, which allows fluid withthe initial composition to flow into the simulation domain.The reservoir temperature and pressure are 335 ‘F and 4100psi% so the initial fluid composition is in the two-phaseregion.

Discussion of ResultsWe performed simulations with and without the eff@ &trapping number on the relative permeability to demonstrateits significance on condensate saturation and productivityindex (PI). Figures 13 and 14 illustrate the distribution cfcondensate with and without the ~ of trapping number,Figure 13 shows that the condensate saturation goes tiough amaximum in the high permeability layers with distance hthe well. This is because the trapping number results in lowvalues of condensate very close to the well in the highpermeability layers, then the condensate saturation increases asthe trapping number decreases with increasing distance b

the well, and it finally decreases again due to the increase inthe pressure. The maximum condensate saturation occurstier away from welIbore in the top (high permeability)layers. Figure 14 shows the condensate saturation in the nearwellbore region for the case without trapping number modeledis high near the well and then decreases away from the well inall layers.

Figure 15 illustrates the range of the trapping numbersencountered in both the top and bottom layers for thesimulation as a finction of distance from the wellbore. In thetop layer, trapping numbers are above the critical trappingnumber (about 10-6 from Fig. 3) for up to the 100 R from thewelIbore. In the bottom layer that has the lowest permeability,no increase in relative permeability due to trapping number* will be seen because the trapping number is less thanthe critical trapping number.

Figures 16 shows the condensate saturation in layer 1 as afiction of distance horn the well with and without trappingnumber modeled and for two values of endpoint gas relativepermeability. The variations in condensate saturation are moresignificant with the lower endpoint gas relative permeability.The use of 0.2 as the endpoint gas relative permeabilityerdarges the range of relative permeabilities and hence moresignificant effects of the trapping number are observed.

The normalized well productivity index, with and withouttrapping number and for two different gas endpoints, is shownin Fig. 17 as a fi,mctionof the average reservoir pressure. Theproductivity index (PI) in this study was computed using thefollowing equation:

PI= QP ave – Pwf

(8)

where Q is total production rate in MMSCF/D, Pave is theaverage reservoir pressure in psia and Pwf is bottomholeflowing pressure from the top layer in psia. The ratio of the PIto the initial PI gives the normalized PI. The initialproductivity index was taken at time 0.01 days for each case.

Figure 17 shows that the PI decreases rapidly ascondensate builds up in the reservoir, but that the effm issomewhat attenuated when the reduction in condensatesaturation at high values of trapping number is modeled. Thetrapping number effa is more significant for the case of lowendpoint gas relative permeability than for the higher value,Afier a certain period of production, the diffe~ce in theproductivity between these two runs becomes almostunchanged: the run with the trapping number effects remainsaround 35% higher productivity for the case with an endpointof 0.53. The productivity modeled with trapping number is

647

6 G.A. POPE, W. WU, G. NARAYANASWAMY,M. DELSHAD,M. SHARMA,P. WANG SPE 49266

approximately twice the productivity without trapping numberwhen using the low endpoint gas curve (0.2). Table 2 lists theproductivity index for the partial well in each layer with andwithout trapping number eti after 10 days of production.The layer-averaged pressure and bottomhole pressure at eachlayer were used for the Pave and Pwf in Eq. (8). It can be seen& Table 2 that the productivity index is more than twotimes greater in the top two layers when trapping numberefffi are modeled. The productivity index decreases as theformation permeability decreases. In the bottom two layers, nosignflcant eff- of the trapping number on the productivityindex were observed.

This simtiation study shows why relative permeabilityshould not be modeled based on IFT only. The fundamentalnature of changes in relative permeability has been shownusing the trapping number concept. The basic equationsclearly indicate that IFT is not the only factor affecting residualsaturations and the critical condensate saturation in particular.Hence, the crucial rate effi which contribute significantly inthe near wellbore region will not be accounted for by an IFTmodel. The reduction in condensate saturation in highpermeability layers near the wellbore will not be shown bysuch a model. Conditions in the near wellbore region for highpermeability layers are the most important factors affming thePI. Hence, relative permeability models based on IFT onlyhave a very poor chance of making accurate predictions of PI.Furthermore, it would be very diff]cult to model all of theseinteracting effects analytically since heterogeneity, pressure,phase behavior, interracial tension, trapping number andrelative permeability are all coupled in a complex andnonlinear way. The coupling with heterogeneity is strong evenwhen the reservoir description is very simple as in thisexample. We are currently using stochastically generatedpermeability fields to ~er elucidate this coupling for morerealistic reservoir descriptions.28 Assuming a uniformcondensate buildup close to the wellbore is not correct mddoes not lead to accurate predictions, nor is it reasonable toassurrre a donut shaped condensate bank near the wellbore.

In addition, for simplicity and clarity, we did not includenon-Darcy effects in this illustrative simulation, but for veryhigh rate gas wells, non-Darcy flow can be significant and isalso coupled with all of the above variables and should ktaken into account,zs-30

Summary And ConclusionsThe buildup of condensate close to gas condensate wells cansignificantly reduce the gas relative permeability and thus thePI of the well and must be accounted for with an accuraterelative permeability model. Although intetiacial tension can

be low and variable and does effd the gas and condensaterelative permeabilities, it is not correct or accurate to modelthe relative perrneabilities directly as a finction of intetiacialtension, but rather they should be modeled as a function ofthe combined effects of pressure gradient, buoyancy andcapillary forces. This requires a generalization of the classicaIcapillary number and Bond number into a trapping number.As shown in this paper, this trapping number can be used in ageneralized relative permeability model to correlate gascondensate data and then used in a simulator to predictchanges in PI due to changes in condensate saturation,

AcknowledgmentsWe thank Mobil for finding this research project andspecifically thank the members of the gas condensate team atMobil Exploration and Production Technical Center (Meptec):Kathy Hartman, Myung Hwang, Ravi Vaidya, Jim Dixon,Steve Weber and Mary Coles for usefil discussions andsuggestions during the course of this research. We also wishto thank Rene Frederiksen for his help with some of the modelcalculations.

References1.

2.

3.

4.

5.

6.

7.

8.

Afidick, D., N.J. Kaczorowski, and S. Bette: “’ProductionPerformance of a Retrograde Gas: A Case Study,.of the Arunfield,’”paper SPE 28749 presented at the Asia Pacific Oil &Gas Conference, Melburne, Australia, Nov. 7-10, 1994.Barnum, R.S., F.P. Brinkman, T.W. Richardson and A.G.Spillette: “Gas Condensate Reservoir Behavior:Productivity and Recovery Reduction Due toCondensation,” paper SPE 30767 presented” at the SPEAnnual Technical Conference & Exhibition held in Dal Ias.Texas, Oct. 22-25, 1995.Boom, W., K. Wit, J.P.W. Zeelenberg, H.C. Weed% and J.G.Maas: “On the Use of Model Experiments for AssessingImproved Gas-Condensate Mobility Under Near-WellboreFlow Conditions,” paper SPE 36714 presented at the SPEAnnual Technical Conference and Exhibition held at Denver,Colorado, Oct. 6-9, 1996.Asar, H and L.L. Handy: “Influence of Interracial Tension onGas/Oil Relative Permeability in a Gas-Condensate System,”’SPERE (Feb. 1988) 257-264.Hanif, M.S. and J.K. Ali: “Relative Permeability and LowTension Fluid Flow in Gas Condensate Systems: paper SPE20917 presented at the European Petroleum Conference heldin The Hague, Netherlands, Oct. 22-24 1990.Bardon, C. and D.G. Longeron: “Influence of Very LowInterracial Tensions on Relative Permeability,” SPEJ (Oct.1980), 391-401.Hartman, K.J. and A.S. Cullick: “Oil Recovery by GasDisplacement at Low Interracial Tension,” J. Pet. Sci. andEng. IO (1994) 197-210.Boom, W., K. Wit, A.M. Schulte, S. Oedai, J.P.W. Zeelenberg.and J.G. Maas: “Experimental Evidence for ImprovedCondensate Mobility at Near-wellbore Flow Conditions,”paper SPE 30766 presented at the SPE Annual TechnicalConference and Exhibition, Dallas Texas, Oct. 22-25, 1995.

648

SPE 49266 MODELING REIATIVE PERMEABILITY EFFECTS IN GAS-CONDENSATE RESERVOIRS 7

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

Che~ H.L., T.G. Monger-McClure, S.D. Wilson: 26.“Determination of Relative Permeability and Recovery forNorth Sea Gas Condensate Reservoirs,” paper SPE 30769presented at the SPE AnnuaI Technical Conference &Exhibition, Dallas, Texas, October 22-25, 1995.Henderson, G.D., A. Danesh, D.H. Tehrani, S. A1-Shaidi, and 27.J.M. Peden: “Measurement and Correlation of GasCondensate Relative Permeability by the Steady-StateMetho~ SPU (Jan. 1996).Kalaydjian, F.J.M., B.J. Bourbiaux and J.M. Lombard: 28.“Predicting gas-condensate reservoir performance: how flowparameters are altered when approaching production wells,” 29.SPE 36715 presented at the SPE Annual TechnicalConference, Denver, Colorado, Oct. 6-91996.Ali, J.K., P.J. McGauley and C.J. Wilson: “The Effects ofHigh-Velocity Flow and PVT Changes Near the Wellbore onCondensate Well Performance; paper SPE 38923 presented 30.at the SPE Annual TechnicalConferenceand Exhibitionheldin San Antonio, Texas, October 5-8, 1997.Morel, D.C., A. Nectoux and J. Danquigny: “ExperimentalDetermination of the Mobility of Hydrocarbon Liquidsduring Gas Condensate Reservoir Depletion: Three ActualCases; paper SPE 38922 presented at the SPE AnnualTechnical Conference and Exhibition, San Antonio, Texas,October 5-8, 1997.McDougall, S. R., P.A. Salino and K. S. Sorbic: “The Effect ofInterracial Tension Upon Gas-Oil Relative PermeabilityMeasurements: Interpretation Using Pore-Scale Models,”paper SPE 38920 presented at the SPE Annual TechnicalConference and Exhibition, San Antonio, Texas, October 5-8,1997.Coats, K.H.: “An Equation of State Compositional Model,”SPEJ, 20 (1980) 363-376.Brownell, L.E. and D,L. Katz: “Flow of Fluids throughPorous Medi&” Chem, Eng. Pros., 43 (1949) 601-612.DeIshad M., M. Delshad, D. Bhuyan, G.A. Pope and L.W.Lake: “Effects of Capillary Number on the ResidualSaturation of Three Phase Micellar Solution,” paper SPE14911 Proceedings of the SPE/DOE 5th Symposium onEnhanced Oil Recovery, TUISLOklahom~ Apr. 20-23, 1986.Lake, L.W.: “Enhanced Oil Recovery, ” Prentice-Hall,Englewood Cliff, New Jersey, (1989),Jerauld, G.R.: “General Three-phase Relative PermeabilityModel for Prudhoe Bay; SPERE, (Nov. 1997) 255-263.Morrow, N.R. and B. Songkran: “Surface Phenomena inEnhanced Oil Recovery,” D. O. Shah (cd.), Plenum Press,New York City, (1982) 387-411,Jin, Minquan.: “A study of Non-Aqueous Phase LiquidCharacterization and Surfactant Remediation,” Ph.D.Dissertation, U. of Texas, Austin, 1995.Delshad M.: “Trapping of Micellar Fluids in BereaSandstone: Ph.D. dissertation, The U. of Texas, Austin,1990.Stegemeier, G.L.; “Mechanisms of Entrapment andMobilizationof Oil in Porous Media” improved OiI Recoveryby Surfacttit and Polymer Flooding, Shah, D. O. and R S.Schechter (eds.), Academic Press, New York City (1977) 55-91.Chatzis, I. and N.R Morrow: “Correlation of CapillaryNumber Relationships for Sandstones, ” paper SPE 10114presented at the 56th Annual Conference of the SPE, SanAntonio, Texas, Oct. 5-7, 1981.Chang, Yih-Bor., G.A. Pope and K. Sepehrnoori: “A higher-order finite-difference compositional simulator,” Journal ofPetroleum Science and Engineering, 5 (1990) 35-50.

Wu, Wei-Jr, P. Wang, M. Delshad, C. Wang, G.A. Pope andM. Sharma “Modelinz Non-Equilibrium Mass TransferEffects for a Gas Con~ensate F;eld; paper SPE 39746presented at the Asia Pacific Conference held in KualaLumpur, Malaysia, March 23-24, 1998.Wang, P., G.A, Pope and K. Sepehrnoori: “Development ofEquations of State for Gas Condensates for CompositionalPetroleum Reservoir Simulation,” Submitted to Industrial &Engineering Chemistry Research, Nov. 1997.Narayanaswmy, G: “Well Deliverability af Gas CondensateReservoirs,” M.S. Thesis, U. of Texas, Austin, 1998.Blom, S. M. P and Hagoort, J.: “The Combined Effect of Near-Critical Relative Permeability and Non-Darcy Flow on Wel IImpairment by Condensate Drop-Out,” paper SPE 39976presented at the SPE Gas Technology Symposium, Calgary.Albert% Canad& March 15-18, 1998.Narayanaswasny, G., M. M. Sharma and G.A. Pope: “Effect ofHeterogeneity on the Non-Darcy Flow Coefficient” paperSPE 39979 presented at the SPE Gas TechnologySymposium, Calga~, Alberta, Canada, March 15-18, 1998.

649

... .. .

0

-

: I

b

:/.

./. ..”-..

,0 . ...”.... \/.”..”4 ‘l;- ~a.

1-?’

—.

c

;iiii

-

. .. .

‘\.‘i

●\ “.

\ i,,

\ .,<

:c

652

L—-.— —

\

.1

II

I

.

0

I1.1.II1

I

654