Society of Petroleum Engineers

SPE 26402

Simulation of Foam Transport in Porous MediaA.R. Kovscek, T.W. Patzek, and C.J. Radke, U. of California

SPE Members

Copyright 1993, Society of Petroleum Engineers, Inc.

rThis paper was prepared for presentation at the 68th Annual Technical Conference and Exhibition of the Society of Petroleum Engineers held in Houston, Texas, 3-6 October 1993.

This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper,as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflectany position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to pUblication review by Editorial Committees of the Societyof Petroleum Engineers. Permission to copy is restricted to an abstract of not more than 300 words. Illustrations may not be copied. The abstract should contain conspicuous acknowledgmentof where and by whom the paper is presented. Write Librarian, SPE, P.O. Box 833836, Richardson, TX 75083-3836, U.S.A. Telex, 163245 SPEUT.

Abstract

FOllin is an excellent fluid for achieving mobilitycontrol of gas in porous media. Practical applicationof fOllins for EOR processes, however requires apredictive model of foam displacement. Further,quantitative information on fOllin-flow behavior atreservoir flow rates llild pressures is required as inputto llily field-scale modeling.

An experimental and mechanistic-modeling studyis reported for the transient flow of fOllin through 1.3,..lIn2 (1.3 D) Boise sandstone at backpressures inexcess of 5 MPa (700 psi) over a quality range from0.80 to 0.99. Total superficial velocities range fromas little as 0.42 to 2.20 m/day (1.4 ftlday to 7 ft/day).Sequential pressure taps and gllimna-ray densitometrymeasure flow resistance and in-situ liquid saturations,respectively. We garner experimental pressure andsaturation protiles in both the transient and steadystates.

Adoption of a mean-size foam-bubbleconservation equation along with the traditionalreservoir simulation equations allows mechanisticfoam simulation. Since fOllin mobility dependsheavily upon its texture, the bubble populationbalance is both useful and necessary as the role offOllin texture must be incorporated into any modelwhich seeks accurate prediction of flow properties.Our model employs capillary-pressure-dependentkinetic expressions for lamellae generation andcoalescence and also a tenn for trapping of lllinellae.

References and illustrations at end of paper

309

Additionally, the effects of surfactant chemicaltrmlspOlt moe included.

We find quantitative agreement betweenexperimental and theoretical saturation and pressureprotiles in both the trmlsient and steady states.

Inh'oduction

Fornu is useful for controlling mobility of gasesin porous media. FOllin is relatively cost effectivebecause it is mainly gas with stabilization of thegaslliquid interface provided by a relatively lowconcentration of surfactant (of order I wt%) withinthe aqueous phase. Since the gaseous portion of fornuis dispersed, gas-phase flow mobility is greatlyreduced and hence gravity override and viscousfingering through high-pelmeability streaks may bereduced. However, practical implementation of fOllinsfor mobility control in enhanced oil recovery (EOR)processes has been hindered because a generalunderstanding and a predictive model of fornu flowdoes not cUlTentIy exist.

Most previous studies were Eddisionian andfocused upon tile steady state. Although transientflow (i.e., displacement) is tile most relevant to EOR,a reliable expeIimental data set tIlat includes trmlsientpressure and in-situ saturation protiles (along tilelength of a core) does not exist for fOllin flow. Themost notable attempts at modeling fornu flow havefocused citllcr on predicting transient flow I or onpredicting steady state results2,3, but not botIl.Additionally, tile trmlsient expeIiments of Friedmann

2 SIMULATION OF FOAM TRANSPORT IN POROUS MEDIA SPE 26402

et all were for gas frontal advance rates betweenroughly 10 and 1000 m/day.

In recognition of the above issues, we undertooka simultaneous experimental and simulation study oftransient foam displacement. Here we demonstrate theusefulness and generality of the population-balanceapproach4 for simulating transient and steady statefoam flow in porous media.

The propagation of foam fronts within Boisesandstone at low displacement rates are trackedexperimentally under a variety of injection modes andinitial conditions. Specifically, three different types offoam displacement are considered: (1) simultaneousinjection of gas and surfactant solution at constantmass injection rates into a core completely saturatedwith surfactant solution, (2) simultaneous injectionof gas and surfactant solution into a brine-filled coreagain at constant mass injection rates, and (3) gasinjection into a surfactant saturated core at a fixedinjection and exit pressures. Total superficialvelocities in the transient mode are generally 1 m/day(3 ft/day) or less. Under steady-state conditions, theliquid flow rate is varied while holding the gas flowrate constant (and vice versa) and measuring theresulting pressure-drop behavior. We concentrate onoil-free systems to avoid confusing fmun propagationwith foam/oil interaction.

Pore-Level Schematic of Foam Flow

Gillis and Radke5 proposed Fig. 1 as a summaryof the pore-level distribution of foam. In this highlyschematic picture, cross-hatched circles refer to water­wet sand grains. Wetting surfactant solution isdenoted as the dotted phase. Foam bubbles are eitherunshaded or darkly shaded, depending upon whetherthey are stationary or flowing. For illustrativepurposes only, the largest pore channels lie at the topof the figure while the smallest lie at the bottom.

In compliance with strong capillary forces,wetting liquid occupies the smallest pore space andclings to the surface of sand grains as wetting films.The aqueous wetting phase maintains continuitythroughout the pore structure shown in Fig. 1 so thatthe aqueous-phase relative permeability function isunchanged in the presence of foam6- 11 . Minimalamounts of liquid transport as lmnellae. Unshadedflowing fomn transports as u'ains of bubbles throughthe largest mId least resistive flow channels. Becausethe smallest pore channels are occupied solely bywetting liquid and the largest pore channels cmTyflowing foam, bubble trapping occurs in theintennediate-sized pores.

Fomn reduces gas mobility in two ways. First,stationary or trapped fomn blocks a Im'ge number ofchannels that otherwise carry gas. Gas tracerstudies1,5 measure the fraction of gas trapped withina fomn at steady state in smldstones to lie between 85

310

and 99%. Second, bubble trains within the flowingfraction encounter drag because of the presence of porewalls and constrictions12, and because the gas/liquidinterfacial area of a flowing foam bubble is constantlybeing rearranged by viscous and capillary forces13.

These trains are in a constant state ofrearrmlgement. Fomn texture arises through a balancebetween varied and complicated foam generation anddestruction mechanisms. Regardless of whether foambubbles are generated in situ or externally, the aremolded and shaped by the porous medium3,14.Bubbles and lmnella transport some distance, aredestroyed, and then reformed. Further, trains haltwhen the local pressure gradient is insufficient tokeep them mobilized, and other trains then begin toflow. No single bubble or train is conserved over anylarge distmlce (Le., the length of several pore bodies).Bubble trains exist only on a time averaged sense.More thorough reviews of foam generation,coalescence, and transport on the pore level are givenin refs. 14 mId 15.

Expel'imenlal Appal'alus and Pl'ocedures

Persoff et al.16 originally designed the apparatusused for the experimental foam floods. More detail isalso located in refs. 15 and 17. The centerpiece of theapparatus is a vertically mounted, 60-cm long, 5.1­em diameter, 1.3-llm2 Boise sandstone core with aporosity of 0.25. The core is epoxy-mounted into a316 stainless steel sleeve designed to withstandpressures up to 20 MPa (3000 psi). AMity-mitedome-loaded backpressure regulator (Grove Valve andRegulator Company, Emeryville, CA) maintains corebackpressure.

Nitrogen gas and foamer solution are injected atthe top of the core to prevent buoyancy-driven gasflow. For injection at fixed mass flow rates, a Brooks5850C mass flow controller (Emerson Electric,Hatfield, PA) meters nitrogen flow yielding gas Darcyvelocities from 0.30 to 2.1 m/day (1 to 7 ft/day) at 5MPa (700 psi) backpressure. An ISCO 500D syringepump (Instrumentation Specialties Company,Lincoln, NE) provides liquid flow. Liquid velocitiesas low as 0.009 m/day (0.03 ft/day) are employed.

Liquid saturation profiles are measured byscanning gamma-ray densitometry utilizing a 47 mCiCs-137 source. After a simple calibration wheregmnma-ray bemn intensity (counts/s falling within a662 keV peak) is measured at preselected points alongthe core at 0% (Id) and 100% liquid saturation (Iw),the liquid content at mlY previously calibrated point inthe core is found from the Beer-Lmnbert law, Sw =[lnOd/I)]/[lnOd - Iw )], where I is the intensitymeasured at any unknown saturation. Mounting ofthe gmnma-ray source and detector on a trmlslatingcaiTiage allows the core to be scanned.

SPE 26402 A. R. KOVSCEK, T. W. PATZEK, AND C. J. RADKE 3

Pressure taps are located at the core inlet, outlet,and at 10 cm (4 in.) intervals along the core, and aresealed with Swagelok O-seal (Crawford FittingCompany, Solon, Ohio) fittings. Pressure ismeasured using a single Paroscientific 43 KTpiezoelectric quartz-crystal pressure transducer(Paroscientific, Redmond, WA). A Scannivalve 12L7multiplexing valve (Scannivalve, San Diego, CA)allows all pressure taps to be visited sequentially andrapidly. An HP-9000 (Hewlett Packard Co.,Mountain View ,CA) controls the apparatus andrecords all data.

The foamer is a saline solution containing 0.83wt% NaCI (1. T. Baker, reagent grade) with 0.83 wt%active C14-16 a-olefin sulfonate surfactant (BiotergAS-40, Stepan). Water is provided by a Bamstead FiStreem II glass still (Barnstead, Thermolyne Corp.,Dubuque, Iowa). The solution surface tension is 33mN/m measured by the Wilhelmy plate metllOd, andhas a viscosity of 1 mPa·s. Bottled bone-dry nitrogenis the gas source.

The core is first flushed with copious amounts(20-100 PV) of 0.83 wt% brine at 7 MPabackpressure. Periodically, the backpressure isreleased and then reapplied. This treatment removesvirtually all gas and surfactant from the core. Becausetrace amounts of isopropanol or methanol can have adeleterious effect on foam production, no alcohols areused as foam breakers or as cleaning solvents on mlYportion of the experimental appm'atus. For thoseexperiments where the core is presaturated withaqueous surfactmlt solution, at least 5 PV of fomnersolution is injected to satisfy rock adsorption ofsurfactant. Measurement of the surfactant elutioncurve for the core reveals little detectable surfactantadsorption at Ule surfactant concentration employed.After 1 PV of foamer solution is injected, theconcentration of surfactant in the inlet and effluentstremns is equal.

In experiments where liquid mld gas m'e injectedat constant mass flow rate, the gas/liquid mixture isnot foamed before injection. The initial injection ratesare not altered until a steady state pressure drop isachieved. After steady state is reached, the liquid andgas rates are varied independently to reach a series ofnew steady states. In experiments where gas alone isinjected at a fixed inlet pressure, the bone-dry nitrogenstream is first passed through a 0.001 m3 (1 liter)stainless steel bomb filled with 0.83 wt% brine tosaturate tlle nitrogen with water vapor. In allexperiments, the progress of foam propagation istracked by frequent pressure and saturation sweeps.Discussion of the experimental results is deferreduntil after review of ule fomn displacement modeldescliption.

311

Foam Displacement Model

Previously we outlined a population-balancefoam displacement model15,17 that is easy toimplement, fits simply into the framework of currentreservoir simulators, employs a minimum number ofparameters, and directly embodies pore-level events.This previous work, however, neglected capillarypressure, Pc, in both Ule flux of gas and liquid and didnot explicitly account for the role of capillarypressure in foam-Imnellae coalescence. These neweffects are included here.

The first step in formulating the model is toWlite in sk1lldard reservoir simulator form the requisitematerial balance equations for tlle gaseous andaqueous phases including the transport and rockadsorption of surfactant1S,17. Including capillarypressure gradients in the transport of gas and liquid issk'lndard (c.f., ref. 18).

The effective resistance of the gas phase is astrong function of fomn texture l -3,12,13. Therefore,mechanistic prediction of fornu flow in porous mediais impossible without a conservation equationaccounting for the evolution of foam texture4 .Following Patzek4 and also refs. 1 and 2 we write atransient population balance on Ule average flowingand trapped bubble size. Rates of accumulation,trapping, convection, generation, and coalescence offomn bubbles m'e incorporated into a species (i.e.,bubble) balance, just as they are for any molecularspecies in a reservoir simulator,

Mechanistic rate equations for the generation offomn by capillary snap-off and capillary suctioncoalescence m'e also constructed15,17. However, wewish now to include capillary-pressure informationdirectly in the expression for foam coalescence. Therate of coalescence is, therefore, written as

(1)

where vf is the local interstitial fornu velocity (vf =Uf/<jlS 0, nf is the local flowing fomn texture (I.e., thenumber of bubbles per nowing gas volume), mld k-lthe constant for fomn coalescence. Equation (1)teaches that a greater nux of lmnellae (vfllf) leads toincreased coalescence. k-l renects the number of foamtermination sites. More coalescence sites appear as Pcincreases, as shown by the work of Jimenez andRadkel9. This rate COnSk1l1t is additionally affected bysurfactant type and concentration. The Ulin lamellaepictured in Fig. 1 m'e stabilized by surfactant adsorbedat the gas-liquid interface. Thus, different surfactantstructures achieve varying degrees of stabilization.The ability of a lmnella to withstand large capillarypressures before rupturing catastrophically is

4 SIMULATION OF FOAM TRANSPORT IN POROUS MEDIA SPE 26402

where P*c,max is a limiting value for Pc* and Cos isa reference surfactant concentration for strong net

where the scaling factor kO-1 is taken as a constant.Equation (2) allows the coalescence rate to increasesmoothly from zero at Sw equal to 1 to very large

values as Pc approaches Pc*. As desired, in thevicinity of the limiting capillary pressure, Pc*, theslope of the curve given in Eq. (2) approachesinfinity.

Because Pc* varies with surfactant concentration,another function is necessary for simulations wherethe porous medium is not presaturated withsurfactwIt. Recent work by Aronson et al.23 measuredpressure drop in 2.3-JlID2 beadbacks for N2 fowns at agas fractional flow of about 90% and also Pc* forsingle foam films at a variety of surfactant (sodiumdodecyl sulfate) and brine (NaCl) concentrations. Thiswork showed that at elevated brine concelllrations(roughly 1 wt%) even small concentrations ofsurfactant (0.03 wt%) produced substantial beadpackpressure drops and large rupture pressures for singlefoam films. The following function for Pc* at highbrine concentrations is suggested by their work

determined by the molecular structure andconcentration of the surfactant.

The capillary-pressure dependence, hencesaturation dependence, of k-l (Pc) is quite drwnatic.The experiments of Khatib et al.20 show that forstrongly foaming solutions k-l (Pc) is small for lowcapillary pressures but rises steeply as Sw decreasesand Pc increases. Since moving lamellae are rapidlystretched under larger Pc they become very thin,fragile, and therefore highly vulnerable to breakage.At low aqueous phase saturations sufficient time doesnot exist for surfactant solution to flow into a rapidlystretched lamella thereby thickening and stabilizingit19 . In fact, the study of single foam filmsdemonstrates that a characteristic or limiting capillarypressure (Pc *) exists for film breakage21 ,22depending strongly upon surfactant fonnulation andconcentration. When Pc* is met or exceeded fownfilms spontaneously rupture. Thus, a Pc near thatcorresponding to Pc * leads to a rate of foamcoalescence approaching infinity20. Cognizant ofthese facts we write

Capillary PressureWe include capillary pressure via the Leverett J­

function. The following form of the J-functionapproximates the capillary pressure relation for ourBoise sandstone

where <l> is the rock porosity, K is the absolutepermeability, and 0' is the surface tension of thefowner solution.

J(Sw) = Pc (1-)112 =( ().067 )0.2 (4)0' K Sw - 0.15

fown generation. We chose values of 0.3 atm (30kPa) and 0.083 wt% respectively for these twopw·wneters. This function allows Pc* to increaserapidly and smoothly from 0 as the surfactantconcentration increases and finally to plateau. Hence,Pc* is small when Cs is small and consequently therate of coalescence is hU'ge WId fown cannot fonn.

Assumptions and parameter fillingIn the simulatin equations, the aqueous phase is

assumed incompressible and nonvolatile, while thegas (i.e., N2) is assumed insoluble and ideal.Gravitational effects w'e neglected. Further, it isassumed that when the core is presaturated withsurfactant, the surfactant is present in equalconcentration throughout the aqueous phase WId thatrock adsorption has been satisfied. In this instance,the surfact<Ult mass balance is automatically obeyed.If the core is not presaturated with surfactant, we setrock adsorption to zero because the surfactant elutioncurves24 for this clean sandstone displayed nosignificant adsorption loss. Also it is assumed thatonce fown traps it cannot be displaced. This allowsfor simulation of so-called continuous-gas foruns2.

The requisite conservation equations andconstitutive relations are incorporated into a standardfinite-difference simultaneous solution (SS) simulatorwith explicit upstream weighting of the phasemobilities and solved (c.f., ref. 18). The fourprimitive unknowns are pressure, gas-phasesaturation, surfactant concentration, and bubbledensity. Further numerical details are availableelsewhere24.

Numerical values of parameters for thepopulation balance portion of the model aredetermined by steady-state measurements.Specifically, steady-state flow trends, saturation, WIdpressure drop profiles must be matched. Fortunately,this drastically narrows our range of parameterchoices. The matching procedure requires only onesteady-state pressure profile along with theaccompanying steady state trends of pressure dropversus gas velocity at fixed liquid rate and pressure

(2)

(3)P~ = P~,max tanh (~i)

312

SPE 26402 A. R. KOVSCEK, T. W. PATZEK, AND C. J. RADKE 5

dmp versus liquid velocity at fixed gas rate. These areeasily obtained within one experimental run. Forfurther details of the matching procedure see refs.15,17,24. Once the foam displacement parameters aredetermined, tllere is no need to make adjustments toaccommodate different types of transient injection orinitial conditions. Parameter values used' here aretaken from ref. 15 except for kO-1. Since a newfunction for lie fOllin coalescence rate constant isemployed, kO_I is adjusted to 0.017 cm- I .

The solid lines in Figs. 2 and 3 show lie resultsof tlle steady-state parllineter matching proceduredescribed above. Figure 2 reports the steady statesystem pressure drop versus liquid velocity atconstant gas velocity while Fig. 3 displays tlle steadystate pressure drop versus gas velocity (at exitpressure) relationship with liquid velocity heldconstant at two different levels.

In the steady state we find an excellent fitbetween experiment and lieoretical prediction. In Fig.2, the model pressure gradient increases linearly llildoverlies the experimental data (symbols) almostexactIy. In Fig. 3 pressure drop at two different givenliquid velocities is shown to be independent of gasvelocity. At the liquid velocity of 0.028 m/day inFig. 3 (open circles) the fOllin simulator overpredictstIle experimental pressure drop slightly, but matchesalmost exactly at a liquid velocity of 0.077 m/day.The overestimation of system pressure gradient isunderstood by complli'ing tIle constant liquid velocityused in Fig. 3 (0.028 m/day) to the results in Fig. 2.The experimental point at 0.028 m/day on Fig. 2 didnot fall on the model predicted line. The data takenduring that portion of tIle current experiment appearto have slightIy depressed pressure drops. In general,simulation mimics experiment well in the steady­state mode.

The steady-state pressure-drop trends are a resultof tlle adjustment of fOllin texture as flow rateschange. When gas velocity is varied under constmltliquid flow rate conditions, foam texture coarsens,viscosity decreases, and constant pressure drop ismaintained. When liquid velocity is increased whilegas rates are held constant, fOllin texture increaseslinearly wili Vw and hence viscosity is adjusted soliat Newtonian behavior is found.

Comparison of Theory and DisplacementExperiments

The following four test cases illustrate theefficacy of our population-balance method inreproducing a variety of transient foam-flow behavior.First, we consider two eXllinples of simultaneousinjection of gas and surfactant solution at differentconstant mass injection rates into a core completelysaturated Willl surfactllilt solution. Next, we exploresimultmleous injection of gas and surfactllilt solution,

313

again at constmlt mass injection rates, into a core liatis initially free of surfactllilt. Lastly, we inject gas ata fixed injection pressure into a surfactant-saturatedcore. In this case no liquid is injected.

In the transient mode, we wish to determine tllelength of time required for lie system to come tosteady state and to verify the existence and track liemovement of fOllin displacement fronts within lieporous medium.

Sjmultaneous Injectjon into a Surfactant-Saturated~

In the first eXllinple, injection rates are quite low.Gas is injected at a rate of 0.43 m/day relative to lieexit pressure of 4.8 MPa and foamer solution isinjected at 0.046 m/day into a surfactant saturatedcore. This yields a quality or gas fractional flow of90% at tlle core exit. Figures 4 and 5 display liemodel results in addition to the experimentalsaturation and pressure profiles. Figure 6 displays liefomn texture profiles generated by lie simulator. Thepopulation-balance parllineters employed are exactlyidentical to tllOse above that reproduced steady statefOllin behavior. Theoretical results are presented assolid lines. Unfortunately, no experimental meliodcurrently exists to directly measure foam texture insitu. Dashed lines simply connect lie individual datapoints. Elapsed time is given nondimensionally inpore volumes (PV) which is lie ratio of lie totalvolumetric flow rate (at exit pressure) multiplied byelapsed time lliIll divided by tlle void volume of liecore.

Steep saturation fronts are measured llild predictedat all time levels (Fig. 4) whereby aqueous-phasesaturation upstrellin of the front is roughly 30%,about 5 units above connate, and downstream it is100%. Model fronts lli"C somewhat steeper and sharperthan those measured experimentally, but lietheoretical saturation profiles track experimentalresults very well. From lie saturation profiles it isapparent that foam moves lirough the core in apiston-like fashion. After the front passes a particularlocation, saturation changes very little. Even lioughnitrogen and surfactant solution are injectedseparately, rapid foam generation and liquiddesaturation still occurs very near lie core inlet. Gasbreakthrough at tlle core outlet occurs at roughly 1PV, and little or no change experimentally ortlleoretically occurs in the saturation profile afterbreakthrough.

The model further predicts tlmt Sw is higher attIle core inlet. Aqueous phase saturation is around40% at x/L equal to zero, but drops rapidly toapproximately 30% by x/L equal to 0.15. Since nofomn is injected, fOllin bubble density is essentiallyzero at tIle inlet, effective flowing-foam viscosity isequal to the gas viscosity, and, consequently, Sw ishigher tllml in the remainder of the core. Including lie

6 SIMULATION OF FOAM TRANSPORT IN POROUS MEDIA SPE 26402

dispersive action of capillary pressure in the materialbalance fluxes obviates steep gradients in aqueousphase saturation. Without capillary pressure effects inthe material balance fluxes, Sw is 76% at the inletand drops to 30% by x/L equal to 0.2015 ,17.Minssieux25 detected such a region of high Sw nearthe inlet of a sandpack. A region of net foamgeneration exists near the inlet by implication. Foamtexture increases rapidly, but the region where rates ofgeneration and coalescence are out of balance is finite.Unfortunately due to equipment limitations, fewexperimental data are available in this region.

The region of net foam generation is alsowiUlessed in the transient pressure profiles of Fig. 5.Both the experimental and model results (solid lines)show that pressure gradients near the inlet are shallowindi~ating that flow resistance is small. Steepgradients are found downstream of the inlet region.These steep gradients confirm the existence of astrong foam piston-like front moving through thecore. In general, large pressure gradients are witnessedwhere aqueous-phase saturation is low and vice versa.Hence, we infer experimentally that foam texturemust be coarse near the inlet and the fraction of foamflowing there large.

These inferences are born out in Fig. 6 whichrcports model-predicted foam texture as a function ofdimensionless distance and time. At all time levels,foam bubbles are coarsely textured near the inlet, butbeyond the first fifth of the core, foam texturebecomes nearly constant at each time level. Fioure 6also confirms that foam moves through the c~lumnin a piston-like fashion consistent with theexperimental data in Figs. 4 and 5. Furtherconsideration of these three figures shows that thesaturation, pressure, and foam texture fronts trackexactly both experimentally and theoretically. Highpressure gradients and fine foam textures are seenwhere liquid saturation is low and vice versa.

We notice one more interesting feature of Fig. 6.At times of 0.65 and 0.80 PV the bubble densitydownstream of the inlet region exceeds the foamtexture at steady state. This effect arises because thecompressibility of N2 is included. A foam bubblecreated upstream finds itself out of equilibrium withthe local pressure (that is, smaller or more dense thanthe local pressure demands) when it transportsdownstream. Hence, the steady-state texture isovershot somewhat as finely textured flowin o foamfills the initially liquid-filled regions near th~ foamfront. Coalescence forces coarsen the bubbles overtime to the equilibrium density. At steady state, thefoam t~xture decreases away from the inlet region.Essentially, the bubbles expand and hence theirnumber density decreases as they flow downstreaminto lower pressure areas. No overshoot in bubbletexture is found in the calculations when the gasphase is made incompressible24.

314

In the second example (Figs. 7 to 9), foamdisplacement rates are roughly 3 times larger. Gas isinjected at 1.2 m/day relative to the exit pressure of5.0 MPa and foamer solution is injected at 0.11m/day again into a surfactant-saturated core whereinitially Sw is 100%. The gas fractional flow of 92%is slightly larger than in the first example. It isimportant to reiterate that model parameters areidentical to those used to generate Figs. 2 to 6. Onlythe injection rates are changed.

Examination of Figs. 7 and 8 shows that, again,the experimental (symbols and dotted lines) andthcoretical (solid lines) transient saturation andpressure profiles agree quite well. Sharp piston-likedisplacement is evidcnced. Because higher rates areused incurring a larger pressure drop across the coreand because the gas is compressible, the foam frontprogresses down the core more slowly than it did inthe first example. In Fig. 4 gas breakthrough issomewhat after 0.80 PV while in Fig. 7 it is closerto I PV. Again steady-state liquid saturation is higherat the core inlet and the pressure gradient shallowerthan it is farther downstream in the core, because ofthe region of ~et foam generation near the beginningof the core. Figure 9 confinlls that a piston-like frontof foalU develops that tracks exactly with thesaturation and pressure profiles. The effects ofcompressibility on bubble texture are even moredramatic in Fig. 9 than they are in Fig. 6.

Simultaneous Injection into a Brine-Filled CoreIn Figs. 10 to 13 we inject gas and surfactant

solution at fixed mass injection rates into a corecompletely saturated with brine containing nosurfactant. The gas injection rate is 0.43 m/day whilethe foamer solution injection rate was 0.077 m/day togive a gas fractional flow of 85%. The systembackpressure was 5.0 MPa. Because surfactant is notinitially present throughout the core, a slowerpressure response than the above two cases isanticipated.

In the transient theoretical saturation profilesshown in Fig. 10 we see that at short times (Le.,0.10 PV) two saturation fronts exist. The first frontis located at roughly xlL equal to 0.35 and is thedistance that unfoamed gas travels into the core. Theexperimental and theoretical locations of this firstfront agree well. Little liquid is displaced by this frontbecause gas mobility is high in the absence of foam,and, consequently, gas breakthrough is quite rapidwhen the porous medium is not saturated withsurfactant solution. The second front is atapproximately xlL equal to 0.06 and is quite steep andsharp. This second saturation front corresponds to thedistance surfacL:'U1t has propagated. Foam fonns quiterapidly when surfactant is present. This secondsaturation front is too close to the core inlet at a timeof 0.10 PV to be detected experimentally.

SPE 26402 A. R. KOVSCEK, T. W. PATZEK, AND C. J. RADKE 7

After gas breakthrough, the foam piston frontcontinues down the core pushing out most of theliquid tIlat the first displacement front left behind.Quite good agreement between simulation andexperiment is witnessed even at tIle later times of 1.0and 1.6 PV. Foam-front propagation is slow becausefoam transports only as quickly as surfactant. Foamcoalescence is infinite whenever surfactantconcentration is zero.

These points are weIl illustrated on Figs. 11 and12. Figure 11 presents the transient foam texturehistory whereas Figure 12 contains the surfactantpropagation history. Comparison of these two figuresshows that foam texture is quite fine when surfactantconcentration is high but falls off dramatically wheresurfactant concentration is low. In the absence of anysurfactant, foam texture is zero. In otIler words, acontinuous channel of unfomned gas exists.

Figure 13 presents tIle transient pressure profilesfor this case. Because tIle theoretical saturationprofiles track well with experiment, we expect thepressure profiles to track well also. Exmnination ofFig. 13 shows that tllis is indeed true. Not only dotlleoretical and experimental foam-front locations inFig. 13 match well, but also do the predicted andexperimentally detennined pressure gradients. Wheresaturation is low and surfactant concentration high,pressure gradients are quite steep and vice versa. As inthe earlier ca.<;es, tIle pressure gradients near the coreinlet are shallow reflecting tIle region of net foamgeneration near tIle core inlet shown in Fig. 11. Thesystem pressure drop reaches steady state in about 3.5PV.

Fixed-Pressure Gas InjectionIn the last mode of forun generation, shown in

Figs. 14 to 16, gas alone is injected into a surfactant­saturated core such tImt forun is generated at a fixedpressure drop. Initially tIle experimental pressure dropestablished over tIle core was 380 kPa (55 psi),however as gas discharged from tIle cylinder, theregulator allowed the pressure to faIl to about 300kPa (44 psi). The experimental decline in theinjection pressure was well documented, thus it was asimple matter to include a declining gas injectionpressure into the numerical simulation of thisexperiment. Figure 14 displays the experimental mldsimulated pressure profiles. Exrunination of thesystem pressure drop at x/L equal to zero shows tIlattIle declining injection pressure was indeed accuratelymodeled. Because of the decline in injection pressure,choice of a gas flow rate for nondimensionalizingtime is not clear. Roughly an hour after gasbreaktIlCough, tIle effluent gas rate on tIle 0.1 MPa (1aun) side of tIle backpressure regulator stabilized at

5.1 cm3/s mld remained constant. This rate is chosento nondimensionalize time in both tIle experiment and

315

simulation. Again the simulation parruneters areidentical to tIlOse used for tile tIlCee earlier cases.

Figure 15 compares the experimental andsimulated saturation profile~. Several aspects of thisgraph are worthy of note. First, t1le saturation profilesmatch moderately well. At times longer t1lan 0.43 PVthe predicted front lags somewhat behind theexperimental front indicating tIlat t1le simulator ispredicting too efficient of a forun displacement. Thus,the experimental saturation front is moving t1lroughthe core more rapidly than is the simulated one.Concomitantly, aqueous-phase saturations upstreamof tile saturation front do not match as well as t1leydid for tile fixed-rate injection schemes. At 0.43 PVtIle tIleoretical saturation upstrerun of t1le front isslightly above 31 %, whereas the actual averagesaturation is closer to 45%. Afler forun breakt1lrough,the experimental saturations continue to declineslowly as do tile simulated ones. Given enough time,the experimental saturations, as well as tIle simulatedones, must decline to connate saturation, because noliquid is injected. In fact, tile simulated profile at 3.8PV indicates correctly that the core slowly drys outfrom the front towards the back. The actualexperiment was not run long enough that connateliquid saturation and complete collapse of t1le foamwere reached.

Pressure profiles prior to fomn breaktIlrough areshown in Fig. 14 at times of 0.18 and 0.43 PV.Again some discrepancies between t1le tIleoretical andactual pressure profiles are seen. In general t1lough thematch between tile two is acceptable. As suggested bythe saturation profiles, t1le simulated pressure profileat 0.43 PV lags behind the actual profile. Carefulstudy of Fig. 14 shows tIlat the agreement at 3.8 PVis quite good.

When forun is generated with a fixed pressuredrop across a core, it is customary to also report t1leeftluent gas rate at a vru·iety of time levels to quantifygas pnxIuction26,27. For instance, at time levels of1.1, 2.0, and 3.8 PV the experimental effluentsuperficial gas velocities ru·e 0.24, 0.24, and 0.25cm/s respectively while tIle model yields rates of0.19, 0.21, and 0.21 cm/s. The difference betweenexperiment and simulation here is consistent Willl t1leresults discussed in Figs. 14 and 15. The simulatorpredicts slightly too large of a reduction in gasmobility. This additionaly mmmer of comparison alsoshows that tIle theoretical model predicts foambehavior adequately.

Further exrunination of Fig. 14 reveals anot1lerinteresting feature: the pressure gradients, bot1l fromexperiment and simulation, are steepest immediatelyupstream of the foam front. Farther upstream t1legradients lessen. The simulated bubble profiles ofFig. 16 explain tIlis behavior. Because only gas isinjected mld foam generation requires some liquid tobe present in order for snap-off and lamellae creation

8 SIMULATION OF FOAM TRANSPORT IN POROUS MEDIA SPE 26402

to occur, foam texture coarsens rapidly far upstreamof the foam front. Due to the reduced availability ofliquid, foam generation cannot keep pace withcoalescence, which is quite high because saturation islow and correspondingly the capillary pressure islarge. At the displacement front, foam textures arefine (see the profiles at 0.18 and 0.43 PV) becausecoalescence has not had time to catch up withgeneration yet. As the foam piston front movesthrough the core, foam texture at the front becomescoarser. This is a result of the gas advancing andcausing a decline in the pressure drop through thegaseous phase, even though the injection pressure isremaining (fairly) constant. As a consequence, gasvelocity and foam generation also decline. The bubbleprofile at 3.8 PV shows that given enough time, theflowing foam coalesces and the texture declinestoward zero. Even if all flowing foam coalesces, asubstantial portion of the porous medium containstrapped foam that impedes gas flow.

Conclusions

We have demonstrated that a foam displacementmodel based on the bubble population-balanceapproach well predicts experimental foamdisplacement under a variety of injection conditions inone-dimension. In general, we find good quantitativeagreement between experiment and theory in both thetransient and steady states. The numerical values ofparameters required for the m.odel are found by fittingsteady-state trends and thus are not difficult to obtain.Hence, all simulated results shown here are producedfrom a single set of parameters. Because ourpopulation-balance formulation is mechanistic, it isgeneral. Thus, extension to large field scales shouldbe possible without parameter adjusunent.

Direct incorporation of the role of foam textureinto the simulator is the key to its success. Foamtexture governs foam flow in porous media. A changein the flow velocity of either wetting liquid or gasmust be accommodated by a change in texture and intum a change in flow resistance. In the transient andsteady-state modes, fine foam textures are predicted tolead to large pressure gradients and low liquidsaturations, whereas coarse textures lead to lessergradients and higher liquid saturations.

Specifically, we draw the following conclusionsfor foam displacement and flow in 1.3 ,.un2 (1.3 D)Boise sandstone at 5 MPa backpressure and for totalsuperficial velocities between 0040 and 2.1 m/day.

When gas and liquid are injected simultaneouslyinto an initially aqueous surfactant-solution saturatedcore the resistance to gas flow builds rapidly in time.Steady state is generally achieved in about 2 PV, andthe steady-state aqueous-phase saturation is roughly30%. The population-balance approach accuratelypredicts the location of saturation and pressure fronts.

316

When the porous medium is completely filledwith brine but devoid of surfactant, the pressureresponse is slow. Two displacement fronts emerge.Unfoamed gas moves rapidly through those portionsof the core where surfactant is. absent. Wheresurfactant is present, foam forms and the seconddisplacement front builds. The second foam fronttracks surfactant propagation through the core.Pressure gradients are large and saturations low wheresurfactant and foam are present and vice versa. Againthe population-balance approach mimics theexperimental data

When gas alone is injected into a core saturatedwith surfactant solution at a fixed pressure drop, astrong foam displacement front forms rapidly. Theflow mobility of gas is reduced by the presence offoam during the displacement and for several PV aftergas breakthrough. Although the simulator predicts aslightly larger reduction in gas mobility than is foundexperimentally, the agreement between thepopulation-balance approach and experiment is quiteacceptable.

Finally, we find both experimentally andtheoretically that a region of net foam generationexists very close to the inlet face of a linear core.Unfoamed surfactant solution and nitrogen areconverted rapidly into a finely textured foam in thisregion.

Nomenclature

Cs surfactant concentration, wt%k rate constant, units depend on rate expressionI intensity of gamma-ray beam (counts/sec)I Leverett I-functionK permeability, m2L length of core, mnf number density of flowing foam (# of

bubbles/volume of flowing foam)p pressure, PaPV total pore volumes injectedPc capillary pressure, Pnw-Pw, Par foam generation or coalescence rate (#of

bubbles/(time)(volume of gas»Si pha..<;e saturationUi Darcy velocity of phase i, mlsVi interstitial phase velocity of phase i, m/sx spatial variable, m

Greek J,etters~ porosity(J surface tension, N/m

SlIhscrj[lts-I denotes coalescence rate constantc coalescenced <hyf flowing foam

SPE 26402 A. R KOVSCEK, T. W. PA1ZEK, AND C. J. RADKE 9

g generationmax maximumnw nonwetting phasew wetting phase

Superscriptso denotes reference value* value corresponds to the limiting capillary

pressure

Acknowledgement

This work was supported by the U. S.Department of Energy under contract No. DC03­76SFOOO98 to the Lawrence Berkeley Laboratory ofthe University of California. P. Persoff providedinvaluable assistance in setting up the experiments.

References

1. Friedmann, F, Chen, W. H., and Gauglitz, P.A: "Experimental and Simulation Study ofHigh-Temperature Foam Displacement in PorousMedia," SPERE (February 1991) 37-45.

2. Falls,A H., Hirasaki, G. J., Patzek, T. W.,Gauglitz, P. A, Miller, D. D., andRatulowski, T.: "Development of a MechanisticFoam Simulator: The Population Balance andGeneration by Snap-Off," SPERE (August 1988)884-892.

3. Ettinger, R. Aand Radke, C. J: "Influence ofFoam Texture on Steady Foam Flow in BereaSandstone," SPERE (February 1992) 83-90.

4. Patzek, T. W.: "Description of Foam Flow inPorous Media by the Population BalanceMethod," in Surfactant Based Mobility ControlProgress in Miscible-Flood Enhanced OilRecovery," Smith, D. H. Ed., ACS SymposiumSeries No. 373, 326-341 (1988).

5. Gillis, J. V.and Radke, C. J.: "A Dual GasTracer Technique for Determining Trapped GasSaturation During Steady Foam Flow in PorousMedia," SPE 20519, presented at 65th AnnualTechnical Conference, New Orleans, LA,September, 1990.

6. Bernard, G. G., Holm, L. W., and Jacobs, W. L.:"Effect of Foam on Trapped Gas Saturation andon Permeability of Porous Media to Water,"SPEJ (December 1965) 295-300.

7. Holm, L. W.: "The Mechanisms of Gas andLiquid Flow Through Porous Media in thePresence of Foam," SPEJ (December 1968)

317

359-369.

8. Huh, D. G.; Handy, L. L.: "Comparison ofSteady- and Unsteady-State Flow of Gas andFoaming Solution in Porous Media;" SPERE(February, 1989) 77-84.

9. De Vries, AS.and Wit, K: "Rheology ofGas/Water Foam in the Quality Range Relevantto Steam Foam," SPERE (May, 1990) 185-92.

10. Friedmann, F.; Jensen, J. A: "Some ParametersInfluencing the Formation and Propagation ofFoams in Porous Media," SPE 15087,presented at the SPE California RegionalMeeting': Oakland, CA, April, 1986.

11. Sanchez, J. M.; Schechter, R S.; Monsalve, A.:"The Effect of Trace Quantities of Surfactant onNitrogenlWater Relative Permeabilities;" SPE15446, presented at the 61st SPE AnnunalTechnical Conference: New Orleans, LA,October, 1986.

12. Falls, A. H., Muster, J. J., and Ratulowski, J.:"The Apparent Viscosity of Foams inHomogeneous Beadpacks," SPERE (May 1989)155-164.

13. Hirasaki, G. J. and Lawson: "Mechanisms ofFoam Flow in Porous Media: ApparentViscosity in Smooth Capillaries," SPEJ(April 1985) 176-190.

14. Chllinbers, K. T. and Radke, C. J.: "CapillaryPhenomena in FOllin Flow Through PorousMedia," in Interfacial Phenomena in PetroleumRecovery, M01TOW, N. R Ed., Marcel DekkerInc., New York (1991) Ch. 6 191-255.

15. Kovscek, A R. and Radke, C. J.: "Fundamentalsof Foam Transport in Porous Media," in Emuns..in the Petroleum Industry. Schramm, L. L. Ed.,ACS Advances in Chemistry Series, to appear1994.

16. Persoff, P, Radke, C. J., Pruess, K., Benson, S.M., and Witherspoon, P. A:"A LaboratoryInvestigation of Foam Flow in Sandstone atElevated Pressure;" SPERE (August 1991)365-371.

17. Kovscek, A. R; Radke, C. J.: "AComprehensive Description of Transient FoamFlow in Porous Media;" No. FS-9, presented atDOEINIPER Field Application of Foams for OilProduction Symposium: Bakersfield, CA,February 1993.

10 SIMULATION OF FOAM TRANSPORT IN POROUS MEDIA SPE 26402

18. Aziz, K.; Settari, A.: Petroleum ReservoirSimulation, Applied Science Publishers Lm:London, 1979;pI25-199.

19. Jimenez, A. I. and Radke, C. J.: "DynamicStability of Foam Lamellae Flowing Through aPeriodically Constricted Pore," in Oil-FieldChemistIy: Ellhanced Recovery and ProductionStimulation, Borchardt, J. K.and Yen,T. F. Eds.,ACS Symposium Series No. 396,460 - 479 (1989).

20. Khatib, Z. I., Hirasaki, G. 1., and Falls, A. H.:"Effects of Capillary Pressure on Coalescence andPhase Mobilities in Foams Flowing ThroughPorous Media," SPERE (August 1988) 919-926.

21. Khristov, K, Krugljakov, P., and Exerowa, D.:"Influence of tbe Pressure in the Plateau-GibbsBorders on the Drainage and the Foam," Colloidand Polymer Sci. (1979) ill 506-511.

22. Bergeron, V and Radke, C. J.: "EquilibriumMeasurements of Oscillatory DisjoiningPressure," Langmuir (December, 1993)3020-3026.

23. Aronson, A. S.; Bergeron, V.; Fagan, M. E.;Radke, C.J.: "The Influence of DisjoiningPressure on Foam Stability and Flow in PorousMedia," Colloids and Surfaces, to appear.

24. Kovscek, A. R.: PhD thesis, University ofCalifornia, Berkeley, in preparation (1993).

25. Minnsieux, L.: "Oil Displacement by Foamsin Relation to Their Physical Properties inPorous Media," JPT (January, 1974) 100-108.

26. Hanssen, J. E.: "Foam as a Gas-Blocking Agentin Petroleum Reservoirs. I: EmpericalObservations and Parametric Study," Journal ofPetroleum Science and Engineering, to appear.

27. Hanssen, J. E.: "Foam as a Gas-Blocking Agentin Petroleum Reservoirs. II: Mechanisms of GasBlockage," Journal of Petroleum Science andEngineering, to appear.

318

0.120.1

DUg.. =0.43 m/day, p(1) =4,8 MPa

<> Ug.. =1.2 m/day, p(1) =4,8 MPa

6 Ug.. = 0,43 m/day, p(1) = 5,0 MPa

0.02 0.04 0.06 0.08liquid velocity (m/day)

Figure 2: Experimental (symbols) and model (solid line) steady­state pressure gradient versus liquid velocity at fixed gas injectionrate.

<>

o

dimensionless distance, x/L

Figure 4: Experimental (symbols connected by dashed lines) andmodel (solid lin~s) transient aqueous-phase saturation profiles forsimultaneous injection of gas and foamer solution at fixed massrates. Porous medium is presaturated with surfactant solution.

~1

(I:l

I... I : I •

=-= 0.8 I./ ..'; 0.11 PV .: 0,46 PV 1/ 0,80 PV".. 1 p

~ J •

:I....0'""'" 0.6..

;j(3'"

.i.fIl 0.4:I=.. -":I~ 2,0 PV" 0.2 Us = 0,43 mfday L =0,60 m

uO' = 0,046 mfday backpresSlD'e = 4,8 MPat

0

0 0.2 0.4 0.6 0.8 1

8

7

-.e 6:::!:'-" 5...=..1 4..01)

t 3:IfIlfIl

t 2Cl.

1

0

oo

III III

o

III

o uO' =0.028 m/day, p(1) =4,8 MPa

III uO' = 0,077 m/day, p(1) = 5.0 MPa

o

IIIw

o

III

Figure 1: Pore-level schematic for a flowing foam. Flowingbubbles are unshaded and trapped gas is darkly shaded,

o 0.5 1 1.5 2gas velocity (m/day)

Figure 3: Experimental (symbols) and model (solid line) steady­state pressure gradient versus gas velocity at fixed liquid injectionrate.

Co)......co

8

7

i6"'a

=-~ 5..c":; 4e01)

t 3:I'"rl

2..Cl.

1

0

1

1

0.80 PV

0.65 PV

0.8

u. = \.2 mldayu

w=o.lI m1dayL=O.60 m

backpressure =5.0 MPa

2.0PV

0.46 PV

0.2 0.4 0.6 0.8

dimensionless distance, x/LFigure 6: Model transient flowing-foam textures for a porousmedium presaturated with surfactant. Gas and foamer solution aresimultaneously injected at fixed mass rates.

dimensionless distance, x1LFigure 8: Experimental (symbols connected by dashed lines) andmodel (solid lines) transient pressure profiles. Porous medium ispresaturated with surfactant solution. Gas and fowner solution aresimultaneously injected at fixed mass rates.

o

5000

4000,......=-~..:o:

e. .......c 3000..-a,......." .........:Ie.

'"'" . 2000f:,....e.~

~g;

1000

r~"0

0 0.2 0.4 0.6

1

1

...

0.8

0.8

O.72·PV

100u. z 0.43 mlday

...,.... l ...0.......'

Uwa 0.046 mIday ·s Uw = 0.046 mlday

L a 0.60 mldayS backpresaure = 4.8 MP........ 80backpressure = 4.8 MPa =.. L=0.6Om

=-c;::.. 60.....="U=Cu

"40

:is~:I~

llIl 20=icc

0

0.6

0.6

ci,.0.40 PV

0.4

0.4

L=O.60mbackpreISUI'C =S.O MPa

0.2

........ ilt'- ~-~'I' ! .... i:•.f"· •., • Ii.~:i.! :.a.r:: _ .\1.:: _ 1.5 PV

dimensionless distance, x1LFigure 5: Experimental (symbols connected by dashed lines) andmodel (solid lines) transient pressure profiles for simultaneousinjection of gas and foamer solution at fixed mass rates. Porousmedium is presaturated with surfactant solution.

dimensionless distance, x1LFigure 7: Experimental (symbols connected by dashed lines) andmodel (solid lines) transient aqueous-phase saturation profiles.Porous medium is presaturated with surfactant solution. Gas andfoamer solution are simultaneously injected at fixed mass rates.

oo

1

0.2

0.6

0.8

f'-l-

~;::f:I'll'"~1..g":Ii

2000

kpv1600

--..~lloo

~olIIIc .......1200..-a ~-- •.....,,'".........

a~... ."-- 800a~

c.400

0

0 0.2

c.>No

1

....~

0.8

..&-- •

0.6

ci

1:1-"-0--' 0 •••

0.4

E0.10 PV ""._-_.~

• O"'-Ej' 1.0 PV :I!!.~t"''''-',-:0:;:,,-:-__I .J3" •••O····· r.~::::::::-"'d

1.6 PV

I /

0.2

(!/

.~. J i• ..E]" _. -.t:~!.-...~; ..•-..6 •• -: • ...:_---0••• • -@-... --. A 'li .A· .. · 3.8 PV

Us = 0.43 m1day L = 0.60 mu" = 0.077 m1day backprcssurc = 5.0 MPa

r

dimensionless distance, xILFigure 10: Experimental (symbols connected by dashed lines) andmodel (solid lines) transient aqueous-phase saturation profiles.Gas and foamer solution are simultaneously injected at fixed massrates. Porous medium is initially free of surfactant.

1•fI.l

§0.8;:

fD..•

0.6'"..~

'S.'" 0.4

I'dD=..D~• 0.2

0

0

dimensionless distance, xlLFigure 9: Model transient flowing-foam textures for a porousmedium presaturated with surfactant. Gas and foamer solution aresimultaneously injected at fixed mass rates.

120tI:'"

's! 100r:l~

Ii80 I- ////~ I I I ,- 3.0 PV

~..•....\

I\

r:l 0.72 PV.. 60

\

Col0.94 PVr:l 0.40PV=

\

Col0.61 PV

~,t:J 40 0.33 PV

i

\

,t:J

011Us ~ 1.2 m1day

r:l 20 u" ~ 0.11 m1dayi= L =0.60

C backpressure = 5.0 MPa

0

0 0.2 0.4 0.6 0.8 1

c.>I\)......

1.6 PV

\Us =0.43 m1day

u" = 0.077 m1dayL=O.60 m

backprcssurc = 5.0 MPa

1.0 PV

r ~ ~ ~ UPV

o 0.2 0.4 0.6 0.8 1dlmslonless distance, xIL

Figure 12: Model transient surfactant concentration. Gas andfoamer solution are simultaneously injecled at fixed mass rates.

1

--~ 0.8i--r:l=;: 0.6•....r:l..Col

lil 0.4Col..[;'e'C 0.2il

0

dimensionless distance, xlLFigure 11: Model transient flowing-foam textures for a porousmedium initially free of surfactant. Gas and foamer solution aresimultaneously injected at fixed mass rates.

140

--..E 120!I:~

Ii 100Cl

'':;:

f 80

\1.6 PV

1:

\

..ColI:

= 60Col 1.0 PV..

\

~,t:J 40D,t:J Us = 0.43 m1dayOIlI: 20

u" = 0.077 m1day'i L= 0.60 mCl backpressurc = 5.0 MPa

C

0

0 0.2 0.4 0.6 0.8 1

dimensionless distance, x/LFigure 13: Experimental (symbols connected by dashed lines) andmodel (solid lines) transient pressure profiles. Gas and foamersolution are simultaneously injected at fixed mass rates. Porousmedium is initially free of surfactant.

10.8

gas only injectionL=0.60m

backprcssurc = 4.8 MPa

0.6

..• -.. ~.-.:..... ~

'......-.-.-:::....::t::::...._.~_

0.4

···.3.8 PV" ............. '.0.43 PV

•......0.18 PV

0.2dimensionless distance, x1L

Figure 14: Experimental (symbols connected by dashed lines) andmodel (solid lines) transient pressure profiles. Gas is injected atfixed pressure at core inlet. Porous medium is initially saturatedwith surfactant.

400

350

300..-.g"

g".lIIi 2500 ......."'.-....tg; 200; .III".-. 150..~g,,~

'S:100

50

0

010.8

Us ~ 1.2 mlday

Uw

= 0.11 mlday

L=O.60 mbackpressure ~ 5.0 MPa

5000

t30P~4000

..- '"

~~g. .....0 3000.."'.-...." .......:=g"IIIIII •

2000t.-.g,,~

g"

1000 l ~..0

0 0.2 0.4 0.6

wI\)I\)

10.8

3.8 PV .......J

0.43 PV

0.'0.4

0.18 PV

gas only injectionL =0.60 m

backprcssurc = 4.8 MPa

0.2dimensionless distance, x1L

Figure 16: Model transient flowing-foam textures. Gas is injectedat fixed pressure at core inlet. Porous medium is initially saturatedwith surfactant.

35....-.e 30!=-==

25.2..".... 20="u=0u 15":is~

oS 10DI)

=i 50C

0

01

3.8 PV

.. -...•

0.8

... ..-_ ...•

onC·~"l _0.43 PV'

0.2

gas only injectionL=O.60m

backpressurc = 4.8 MPa

...~. .... ...~ .1·······.. I..·•" ".:::..::::::::J... -0•••)-••0 ....•.•....•. _....'

o 0.4 0.6dimensionless distance, xIL

Figure 15: Experimental (symbols connected by dashed lines) andmodel (solid lines) transient aqueous-phase saturation profiles.Gas is injected at fixed pressure at core inlet. Porous medium isinitially saturated with surfactant.

1

o

0.2

0.4k

0.6

0.8

•(I)

C-o1..:=....~III

i...::lo"i