Mobility of Foam in HeterogeneousMedia: Flow Parallel and Perpendicular

to StratificationDicksen Tanzil, SPE, George J. Hirasaki, SPE, and Clarence A. Miller, SPE, Rice U.

SummaryExperiments in homogeneous and heterogeneous sand-packed col-umns showed that the foam mobility in the two cases could differby two orders of magnitude. The difference is because of thegeneration of foam lamellae by snap-off for flow across an abruptincrease in permeability. This mechanism is shown to be depen-dent on the degree of permeability contrast and the gas fractionalflow. It has important implications for the degree of gravity seg-regation of gas and liquid in field-scale recovery processes.

IntroductionFoam has been used to improve the sweep efficiency of enhancedoil recovery1–4 and aquifer remediation processes5–8 because of itseffect in reducing gas mobility. Gas, as a bulk fluid, has a very lowviscosity compared with oil and water. However, when gas is in adispersed phase, as in foam, its apparent viscosity is greatly in-creased (i.e., its mobility is greatly reduced). Yet foam still has alower average density than resident liquids in a reservoir or aqui-fer. This density contrast tends to promote gravity segregation.Stone9 and Jenkins10 developed models to predict the extent ofgravity segregation in water-alternating-gas (WAG) injection. Akey parameter in this model is the permeability anisotropy or thekv/kh ratio, where kv and kh are vertical and horizontal permeabili-ties, respectively. This anisotropy is based on single-phase flow.We infer here from 1D experiments and calculations that the ef-fective anisotropy for foam flow in stratified systems may be muchgreater than for single-phase flow. The implication is that gravitysegregation for foam flow in stratified systems may be much lessthan predicted by the Stone-Jenkins model.

BackgroundField Test. Anisotropy in foam mobility was seen in an aquiferremediation field test that we conducted in a heterogeneous allu-vial formation beneath Hill Air Force Base, Utah.6,8 Foam wasused to divert injected surfactant solution to zones of lower per-meability and to the base of the aquifer where the contaminant waslocated. The field test showed that foam could be generated in situand propagated even in a shallow, unconfined aquifer. Air wasinjected intermittently with aqueous surfactant solution at low in-jection pressure (2 to 8 psi above hydrostatic). The in-situ forma-tion of foam was evident from the reduced injection rate observedwhile maintaining constant injection pressure and from foam pro-duction at the multilevel monitoring well located 15 ft (4.6 m)away from the injector (Fig. 1). Simulation results8 demonstratedthat, in the absence of foam, gas would not have been able topropagate far enough to reach the monitoring well. Yet the pho-tograph in Fig. 1 clearly shows foam being produced from the twoupper monitoring intervals (B and C, see Fig. 2). While the lowestmonitoring interval A is producing emulsified contaminant in thephotograph, foam was intermittently produced there as well, eventhough it is below the injection interval. The observations indi-

cated that foam was propagated horizontally, despite relativelyhigh foam mobility.

In simulating the field test using a modified UTCHEM reser-voir simulator,8 an anisotropy in foam mobility must be used inorder to history-match field observations. With an effective foamviscosity of 1 cp, matching required use of a ratio of vertical tohorizontal gas mobilities (�v/�h)g of 0.1 wherever foam is present.This anisotropy factor is used in addition to explicit modeling ofpermeability layering. The additional resistance of foam to verticalflow is attributed to both capillary entry effect and foam generationby snap-off at permeability discontinuities. Fig. 2 shows the simu-lated foam propagation profiles at three different times. As can beseen, the simulation results indicate that a considerable amount ofgas was propagated horizontally from the injector to the extractor.

Capillary Effects in Multiphase Flow. The effects of capillarityin impeding nonwetting-phase flow across layers in series are wellknown.11 These capillary effects are important for any type ofmultiphase flow in heterogeneous or stratified porous media, es-pecially when flow rates are low, permeability varies over shortdistances (inches or centimeters), and permeability contrasts arelarge.12,13 Such conditions exist in many reservoirs and aquiferswith episodic or periodic fluctuations in the depositional process.In such layered media, capillarity causes the entrapment of thenonwetting phase in high-permeability regions upstream of low-permeability layers.14,15 This capillary entrapment can signifi-cantly affect oil recovery even on the reservoir scale.16 Van Duijnet al.17 showed that when an oil/water displacement front arrivesat a sudden permeability decrease, the front will stop, waiting forthe oil saturation to build up before it enters the low-permeabilityregion. The same entrapment effect creates resistance to gas flowperpendicular to stratification.

In the presence of surfactants, capillarity also contributes tofoam generation by snap-off. Yortsos and Chang12 studied thevariation of capillary pressure for nonwetting-phase flow acrosspermeability variation in series. They showed that, for flow froma low-permeability to high-permeability region, capillary pressuredecreases (wetting-phase saturation increases) in the low-permeability region as it approaches the permeability increase.Similar findings have also been made by Chaouche et al.14 inbead-pack experiments and pore-network simulations. Further-more, van Duijn et al.17 showed that, during drainage, capillarypressure might become discontinuous at a permeability jump, in-dicating that one of the flowing phases loses continuity upstreamof the permeability jump. Van Lingen18 pointed out that the sub-sequent pressure drop across the sudden permeability increase in-duces snap-off. Snap-off at the sudden permeability increase maybe important to foam flow in heterogeneous reservoirs.

Between the two mechanisms by which capillarity impedesfoam flow perpendicular to stratification, foam generation by snap-off is potentially more important than capillary entrapment. In thepresence of foam-stabilizing surfactant, foam generated by snap-off can reduce gas mobility by orders of magnitude. Therefore, thisstudy focuses on snap-off at a sudden permeability increase.

Foam Generation by Snap-Off. Researchers have identified threemechanisms for foam generation in porous media: leave-behind,lamella division, and snap-off.19–21 Foam is generated by leave-behind when two gas fingers invade adjacent liquid-filled pore

Copyright © 2002 Society of Petroleum Engineers

This paper (SPE 78601) was revised for publication from paper SPE 63228, first presentedat the 2000 SPE Annual Technical Conference and Exhibition, Dallas, 1–4 October. Originalmanuscript received for review 27 October 2000. Revised manuscript received 12 January2002. Manuscript peer approved 4 February 2002.

203June 2002 SPE Journal

bodies and leave behind a liquid lens. Lamella division may occurafter some lamellae have been generated, provided that pressuregradient is sufficient to mobilize the lamellae. When a flowinglamella encounters a branch in its flow path, it may split into two,one in each of the path branches. While leave-behind producescontinuous-gas foam, or “weak” foam, lamella division results indiscontinuous-gas foam, or “strong” foam. The terms “weak”and “strong” here refer to small and large reductions in foammobility, respectively.

Snap-off also generates strong discontinuous-gas foam. Roof22

was the first to investigate the mechanism for snap-off, which mayoccur when a nonwetting phase (in this case, gas) enters a poreconstriction initially filled with wetting liquid. For snap-off tooccur, pressure difference across the interface at the pore throatmust be greater than that at the leading interface. In other words,the capillary pressure Pc at the leading interface must be below acritical value Pc

sn for snap-off to take place (Fig. 3). In a straightcapillary tube, Rayleigh instability will cause the formation of a

liquid bridge at exactly one-half of the capillary entry pressure, orPc

sn=1

2Pc

e. Falls et al.20 showed experimentally that the critical cap-illary pressure for snap-off in bead packs is also about half of thecapillary entry pressure, Pc

sn≈ 1

2Pc

e.The low capillary pressure requirement creates a paradox. On

the one hand, the capillary pressure must be at least as high as Pce

for the gas to enter the pore constrictions. On the other hand, thecapillary pressure must drop below Pc

sn≈ 1

2Pc

e for snap-off to occur.This restricts the occurrence of snap-off to “germination sites”where the downstream pore bodies are at least twice as large astheir corresponding pore throats.23 Based on a hydrodynamicanalysis, Kovscek and Radke24 maintained that repeated snap-offis possible at such sites when the role of liquid flow along porecorners and crevices is considered. They attributed developmentof strong foam in homogeneous media to this mechanism. How-ever, Rossen25 recently argued against this mechanism on thebasis that snap-off at a site would cease as gas bubbles filled thepore body downstream from the constriction and caused local cap-illary pressure to increase. Consequently, no consensus currentlyexists on the mechanism for generation of strong foam in homo-geneous porous media, a topic we will discuss further in a subse-quent paper.

Snap-Off at Stepwise Permeability Increase. The occurrence ofsnap-off around the sudden transition from low to high permeabil-ity has been observed experimentally in heterogeneous glass-beadpacks.19,20 Ransohoff and Radke19 found that snap-off at the per-meability increase occurred regardless of flow rate. Falls et al.20

saw lamellae being generated, mobilized, and convected down-stream of the permeability increase. In addition, they observedfoam lamellae steadily created near the permeability increase onlywhen the injected fractional flow, fg, was below about 80%. Athigher fg, lamellae were generated intermittently. Falls et al.20

attributed the occurrence of snap-off to the accumulation of liquidimmediately upstream of the sudden permeability increase thatreduced capillary pressure below Pc

sn.Recently, Rossen26 employed a pore-network model to study

snap-off at a sudden permeability increase. Given Pcsn≈ 1

2Pc

e, snap-off requires a permeability contrast between the high and lowpermeability media of at least 4. For wet flow (e.g., because ofhigh gas mobility), a permeability contrast of just above 4 will besufficient to cause a complete blockage of flow caused by snap-off

Fig. 1—Monitoring Well MW-2 during the surfactant/foam aqui-fer remediation field test at Hill AFB. Foam is being produced atthe upper sampling intervals B and C, while emulsified contami-nant is being produced at interval A. See Fig. 2 for the locationof the sampling intervals.

Fig. 2—Simulated foam propagation profiles at three different times. IN-2 is the injector well, EX-2 is the extractor, HC-1 and 2 arehydraulic control wells, and MW-1 and 2 are multilevel monitoring wells. Air was injected intermittently with surfactant over theupper 3 ft of IN-2. Axes are in feet.

204 June 2002 SPE Journal

at the permeability boundary. The drier the flow, the higher theminimum permeability contrast required to cause complete block-age. Snap-off at the sudden permeability increase can be a repeti-tive process, as was observed by Falls et al.20 Rossen26 pointed outthat, as snap-off blocks gas flow at the permeability boundary, thepressure gradient will rise, leading to the rupture or mobilizationof the lamellae that were generated. As lamellae break or moveaway from the boundary, pressure gradient and capillary pressuredecrease, allowing repetitive cycles of snap-off and breakageor mobilization.

TheoryIn this study, we modify the steady-state approach of van Lin-gen.18 Let us apply it to the two-phase flow of gas and water in aporous medium with a step increase in permeability. Let the porousmedium be homogeneous except for the step change in permeabil-ity. The approach assumes Darcy’s law [i.e., (1) gas and waterflow with a constant viscosity ratio, (2) relative permeability ratiois finite, and (3) both phase pressures are continuous]. We assumephase continuity in order to find the conditions when the phasesbecome discontinuous (i.e., when the capillary pressure drops be-low the capillary pressure for snap-off).

Our choice of capillary pressure curves for the low- and high-permeability media are illustrated in Fig. 4. In place of the primarydrainage curves used by van Lingen,18 the capillary pressurecurves used in this study allow Pc to drop to Pc

sn at 1 − Sgr asrequired for snap-off. Such a decrease in local capillary pressureoccurs, for instance, in the low-permeability zone just upstream ofthe stepwise permeability increase when gas initially advances intothe high-permeability zone. The value of Sgr is a function of thehistory of the porous media. Therefore, it is quite possible for thelow- and high-permeability zones to have different values of Sgr.For simplicity, however, we assume the same Sgr to apply any-where in the porous medium.

The capillary pressure scales with permeability k and porosity� following the Leverett model,

Pc = ���

kJ�Sw�, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1)

where ��the interfacial tension and the Leverett J-functionJ(Sw)�the dimensionless capillary pressure as a function ofwater saturation.

The steady-state saturation gradient within each homogenoussection is given by18

�Sw

�x=

ut

kP�c��1 − fg��w

krw(Sw)−

fg�g

krg(Sw)�, . . . . . . . . . . . . . . . . . . . . . . . ( 2)

where x�distance, ut�total superficial velocity, ƒg�gas frac-tional flow, Pc�=the derivative of capillary pressure with respect tosaturation, ��viscosity, kr�relative permeability, and the sub-scripts w and g refer to water and gas, respectively. Except in thevicinity of a permeability change, capillary pressure and saturation

are constant, �Sw/�x�0. Setting Eq. 2 to zero, one can calculate Sw

far from the permeability change,

1 − fg

fg=

�g

�w

krw(Sw)

krg(Sw), . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 3)

As shown in Fig. 5, Sw is a monotonically decreasing function offg and �g/�w.

Fig. 6 illustrates what happens around the permeability in-crease. Let us begin in the upstream low-permeability region faraway from the permeability increase, where the water saturation isSw

−��Sw(ƒg,�g/�w) and the capillary pressure is Pc(Sw−�,klow).

Arrows in Fig. 6 indicate the path followed by the fluids asthey flow from klow to khigh. As the fluids approach the high-permeability region, capillary pressure decreases to maintain con-tinuity in phase pressures. Capillary pressures immediately up-stream and downstream of the jump are equal to Pc(S

+w,khigh) where

S+w�Sw

−��Sw(ƒg,�g/�w). There is no gradient in the downstreamhigh-permeability region. Spatial profiles of saturation and capil-lary pressure corresponding to Fig. 6 are plotted in Fig. 7. Theprofiles were obtained by solving the first-order differential equa-tion Eq. 2 using the finite difference method with S+

w�Sw(ƒg,�g/�w) as the boundary condition. The region where snap-off occursin the low-permeability zone is marked by the thick lines in Figs.6 and 7. Specifically, snap-off occurs in the region wherePc(Sw

+ ,khigh)<Pc<Pcsn. For the limiting case where downstream per-

meability is infinite, we have the capillary end effect. The modelshows that capillary end effect will always result in snap-off as the

Fig. 3—Snap-off occurs as the nonwetting phase passesthrough a pore constriction when the pressure differenceacross the interface at the throat (PcT) is greater than the pres-sure difference at the leading interface (PcF).

0 1gr

S

lowk

highk

Primary drainage

Water Saturation, Sw

Capillary

Pressure,Pc

cP

1Ð

low(k )sn

cP (k )e

low

Fig. 4—Capillary pressure curves for the low- and high-permeability media.

0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

µg /

µw = 0.02

1

10

10000

fg

Sw

Fig. 5—Water saturation as a function of gas fractional flow fordifferent values of µg/µw for a typical sandpack.

205June 2002 SPE Journal

capillary pressure in the medium drops to zero as it approaches theinfinite permeability at the outlet of a column (Fig. 6).

Required Permeability Contrast. For snap-off to occur at all, thecapillary pressure at the permeability jump must be less than thecritical capillary pressure for snap-off,

Pc�S+

w,khigh� � P snc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 4)

Let Pcsn�a·Pc

e and Nk ≡ khigh/klow be the permeability contrast. Bysubstituting Eq. 1 into Eq. 4, rearranging, and assuming constantporosity �, the requirement for snap-off at the step permeabilityincrease becomes

Nk � �1

a

J�S +w �

Je �2

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 5a)

where Je�the dimensionless capillary entry pressure. That is,there exists a minimum permeability contrast for snap-off, whichdepends on the exact value of the parameter a≡ Pc

sn/Pec ≈

1

2, the shape

of the Leverett J-function, and S+w�Sw(ƒg,�g/�w).

The dependence on a is intuitive: the higher are a and thus Pcsn,

the broader is the region for snap-off (see Figs. 6 and 7). Thedependence on Leverett J-function reflects the effect of the pore-size distribution in the porous media. This effect on snap-off hasbeen well studied by Rossen26 using the pore-network model. Hefound that a broad distribution requires higher permeability con-trast for snap-off at a sudden permeability increase, except underwet flow conditions.

Wet flow occurs in ordinary gas-water flow where �g/�w is lowapproximately 0.02. Except at very high fg, water saturation Sw

is high (see Fig. 5) so that J(S+w)≈ Je (regardless of pore-size dis-

tribution). Therefore, we can approximate the requirement forsnap-off as

Nk �≈1

a2for

�g

�w�� 1 or fg �� 1. . . . . . . . . . . . . . . . . . . . ( 5b)

For a≈ 1

2, a permeability contrast of at least about 4 is required

for snap-off to occur at a sudden permeability increase.

Effects of fg and µg/µw. Increase in fg and �g/�w leads to a de-crease in water saturation, as shown in Fig. 5. Consequently,S+

w=Sw(fg ,�g /�w) decreases and J(S+w)/Je increases. According to

Eq. 5a, the result is an increase in the required permeability con-trast for snap-off. This larger permeability contrast requirement fordrier flow is in agreement with Rossen.26

Let us now consider the effects of fg and �g/�w in a heteroge-neous porous medium with a given permeability contrast Nk. Theseeffects are illustrated in Fig. 8. Higher fg or �g/�w results in lowerS+

w ; therefore, there is a higher capillary pressure in the down-stream high-permeability region. At Point C in Fig. 8, Pc(S

+w) is

equal to Pcsn and no snap-off occurs at the sudden permeability

increase beyond this point. Therefore, there exist a maximum fgand a maximum �g/�w for foam generation by snap-off at suddenpermeability increase.

Repeated Snap-Off. Foam bubbles created by snap-off at a sharppermeability increase between homogeneous regions accumulatenear the permeability boundary until there is sufficient pressuregradient to mobilize the lamellae. The mobilization of the bubbles,in turn, allows further snap-off to occur.

Consider a pore throat in which initial snap-off has just oc-curred. The newly formed lamella gives rise to local discontinuityin the gas phase, with the capillary pressure upstream of thelamella discretely greater than the capillary pressure downstream.As gas continues to be injected, the newly formed lamella is dis-placed by the gas, and we again have local-phase continuity andthus local capillary-pressure continuity across the permeabilityboundary. Further snap-off will occur at the pore throat, providedthat the local water saturation in the high-permeability side is

Fig. 7—Water saturation and capillary pressure profiles atsteady state as gas and water flow from klow to khigh (left toright). Snap-off occurs at the sudden permeability increase ascapillary pressure drops below Pc

sn. Arrows indicate values atx=−�.

0

AB

C

Water Saturation, Sw

Capillary

Pressure,

lowk

Pc

grS

cP

1Ð

low(k )sn

wS ( f

g, µ

g/µ

w)

increasing fg

increasing µg/µ

w

Fig. 8—Increase in gas fractional flow or gas/water viscosityratio causes an increase in Pc(Sw) at the high-permeability side.As snap-off generates foam, apparent gas viscosity increasesuntil Pc(Sw)=Pc

sn at Point C where snap-off ceases.

0

2

Water Saturation, Sw

Capillary

Pressure,

snap-off

+w

Sk

lowk

Ð ∞wS

}N

k=

snap-off

�=k

N

=k0

�

}

}=

�= ∞

=∞

high

Pc

khigh low

/k

grS

cP

1Ð

low(k )

sn

wS ( f

g, µ

g/µ

w)

Fig. 6—Snap-off occurs as gas and water flow along the pathshown by the arrows from klow to khigh. Imbibition capillary pres-sure curves are used to allow capillary pressure to drop belowthe critical capillary pressure for snap-off, Pc

sn.

206 June 2002 SPE Journal

sufficiently high that the local capillary pressure remains less thanPc

sn(klow).As more lamellae are generated, the downstream gas mobility

decreases, resulting in an apparent increase in �g/�w and thus indecreasing water saturation. Consequently, capillary pressure andsaturation downstream of the permeability boundary move fromPoint A to Point B in Fig. 8, where the capillary pressure is stillless than Pc

sn(klow), and snap-off continues. However, snap-offceases when the capillary pressure is equal to or greater thanPc

sn(klow). That is, snap-off ceases when the conditions in thehigh-permeability region reach Point C or any point to the left ofPoint C.

After snap-off ceases at a location, the downstream apparentgas mobility will increase because of continuing gas flow acrossthe nearby boundary. Consequently, apparent �g/�w will decreaseand saturation and capillary pressure will move toward Point A inFig. 8. This will result in the resumption of snap-off, and the cyclewill repeat itself. In effect, this becomes self-regulating foam gen-eration, in which the downstream saturation and capillary pressureare maintained near Point C.

Gradual Increase in Permeability. The above discussion pre-sented the theory of foam generation by snap-off at a stepwiseincrease in permeability. Yet a truly stepwise (i.e., discontinuous)change in permeability never exists in the physical world. In bothman-made and naturally occurring porous media, variations in theporous medium characteristics always occur over a finite distance.

Instead of a stepwise increase, let us now consider a gradualincrease in permeability as shown in Fig. 9, in which it increaseslinearly from klow to khigh to over a finite distance Ltrans. When thecapillary pressure drops below Pc

sn at any point along the 1Dporous medium, snap-off occurs. The capillary pressure is a func-tion of both permeability and water saturation through Eq. 1. Thewater saturation profile, in turn, is obtained from Eq. 2, which canbe written in the dimensionless form as18

�Sw

�xD=

Nc,trans

dJ�dSw��1 − fg�

krw�Sw�−

fgM

krg�Sw��. . . . . . . . . . . . . . . . . . . . . . . ( 6)

Taking the transition length Ltrans as the characteristic length, thedimensionless distance is defined as xD ≡ x/Ltrans. The dimension-less group

Nc,trans ≡ut �w Ltrans

� �khigh�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 7)

is a modified capillary number which includes the transition lengthLtrans and khigh as the characteristic permeability. The parameter Mis the viscosity ratio, M=�g /�w.

Similar to the earlier stepwise-permeability-increase problem,water saturation profile can be obtained by solving Eq. 2 (or Eq. 6),with S+

w obtained from Eq. 3 as the boundary condition. The resultwould be the “^-shape kink” in the water saturation profile around

the transition zone as shown in Yortsos and Chang12 for thegradual increase in permeability.

With the gradual increase in permeability, the occurrence ofsnap-off depends not only on the permeability ratio Nk and gasfractional flow fg, but also on the capillary number of the transitionregion, Nc,trans. Fig. 10 shows the maximum values of Nc,trans forsnap-off. Below the Nc,trans curves, the capillary pressure dropsbelow Pc

sn somewhere in the transition region. Above the curves,the capillary pressure always stays above Pc

sn, and no snap-offoccurs. Each curve approaches some finite value of Nk as Nc,trans

approaches zero. At Nc,trans�0, corresponding to the limiting caseof stepwise permeability increase, the occurrence of snap-off de-pends only on Nk and fg. Consistent with our earlier discussion, Nk

of at least about 4 is needed for low fg. The permeability contrastrequirement increases with increasing fg.

With the permeability increasing over a finite distance, Nc,trans

is nonzero. The larger the value of Nc,trans the greater the perme-ability contrast required for snap-off. From the definition ofNc,trans in Eq. 7, large Nc,trans corresponds to large transition lengthand/or high velocity. Therefore, less permeability contrast is re-quired for snap-off when the transition length or velocity is small.

ExperimentalAir was injected into sandpacked glass columns saturated withsurfactant solution to observe foam generation in situ. The surfac-tant used was sodium di(1,3-dimethylbutyl) sulfosuccinate (Aero-sol� MA-80I from American Cyanamid) at 4 wt% active concen-tration in water. The CMC is approximately 1 wt%. The surfactantwas selected for its phase behavior with chlorinated hydrocarbons(for aquifer remediation purposes)6,8,27 and not for its foam-ing characteristics.

Transparent glass columns were used to allow visual observa-tion. All columns were of equal dimensions: 1 ft (30.5 cm) inlength and 1 in. (2.54 cm) in internal diameter. The injection endhad two parallel inlets, which allowed simultaneous coinjection ofgas and liquid. Gas and liquid were injected using two syringepumps: ISCO model 314 and Harvard model 55–4140, respec-tively. Injection pressure was automatically recorded on a PC us-ing a Validyne pressure transducer system. The outlet of the col-umn was left open to the atmosphere.

ResultsTransient Foam Generation at Abrupt Permeability Increase.To study foam flow and generation in heterogeneous media, we

packed a glass column with sands of two different grain sizes. Thecolumn was divided into four sections, each 3 in. (7.6 cm) long(Fig. 11a). Labeled Column A in Table 1, the first and thirdsections were packed with coarse sand of about 70 darcy perme-ability, and the second and fourth sections with fine sand of about

klow

khigh

Permeability

Ltrans

klow

khigh

Distance

Ltrans

Fig. 9—Gradual (linear) increase in permeability of distance Ltrans.

0.0

0.5

1.0

1.5

2.0

2.5

0 5 10 15 20 25 30 35 40 45

Nk = khigh /klow

Nc,trans

No snap-off

Snap-off

fg = 0.5

0.6

0.7

0.8

0.9

x 10Ð6

Fig. 10—Maximum transition-region capillary number for snap-offat different values of permeability ratio and gas fractional flow.Snap-off occurs below the curves; no snap-off above the curves.

207June 2002 SPE Journal

16 darcy. The boundaries between sections of different grain sizesin Column A were sharp, with transition regions of less than 1⁄32 in.(1.5 mm). The permeability ratio Nk between the coarse and finesands was calculated from the difference in capillary entry pres-sure. It was approximately 4.4, close to the estimate of approxi-mately 5 obtained from the grain size ratio using the Blake-Kozenyequation. The overall harmonic mean permeability of the columnwas 26 darcy.

In all experiments presented in this paper, gas was injected at0.6 ft/day (0.2 m/day) superficial velocity (calculated at standardtemperature and pressure, STP) with the column placed horizon-tally. The generation of a strong foam front in the vicinity of thestep permeability increase 6 in. (15 cm) from the inlet was ob-served at an injected gas fractional flow, fg, of 67%. Upstream ofthe permeability increase, foam was fairly weak, and gravity over-ride was observed (Fig. 11a). Visual observation with a 20× mag-nifying glass clearly showed weak continuous-gas foam in theupstream coarse-sand region. The magnification was not powerfulenough to observe foam texture in the fine sand. However, thepresence of weak foam is clear from the gravity override and lowpressure drop.

Strong foam was formed once gas began to enter the down-stream high-permeability section. Gas accumulated immediatelydownstream and formed a piston-like foam front [0.4 PV (porevolume) profile of Fig. 11a]. Visual inspection with the magnify-ing glass showed discontinuous foam bubbles near the permeabil-ity jump with each bubble approximately the same size as the porebodies. The strong piston-like front was propagated downstream(see 0.65 and 1.3 PV profiles) until it reached the end of thecolumn. Gas saturation upstream of the permeability transition alsoincreased as the front propagated, displacing liquid in regions pre-viously unswept by gas. Nevertheless, as the experiment pro-gressed, gas upstream of the step permeability increase alwaysremained continuous, while downstream gas was discontinuous.

The generation of strong foam at the permeability increase wasalso evident from the pressure history shown in Fig. 12 (thick solidline). Initial invasion of the column by gas resulted in only amodest rise in the pressure drop. As gas began to flow through thepermeability increase (Arrow 2 in Fig. 12), the pressure drop wasonly 0.31 psi, which is approximately equal to the capillary entrypressure of the low-permeability medium. The pressure drop beganto increase after the strong piston-like foam front was formeddownstream of the permeability increase. First, the pressure dropremained quite modest as gas continued to displace the liquid inthe previously unswept regions upstream of the permeability in-crease. Then, it began to increase rapidly after the upstream regionwas fully swept by gas and the piston-like front had covered theentire cross-section downstream. Indeed, the pressure drop hadrisen to approximately 2.6 psi when the piston-like front reachedthe last fine-sand section (Arrow 3 in Fig. 11) and to more than 10

psi when the foam front reached the column outlet (Arrow 4).About 1.6 PV of gas had been injected at STP when gas break-through occurred. Using the ideal gas law, the actual volume of gasinjected was calculated to be approximately 1 PV at gas break-through. As the pressure drop started to rise rapidly after around0.6 PV of gas injected, the actual injected superficial gas velocitydropped to approximately 0.3 ft/day, or about half the nominalvelocity at STP.

The apparent gas viscosity �g,app in the column can be calcu-lated through Darcy’s law:

�g,app =1

�rg=

k

ug

p

x, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 8)

where �rg�the relative gas mobility, ug�the superficial gas ve-locity at injection conditions, and p/x�the pressure drop acrossdistance x. At breakthrough, the overall �g,app across the columnwas about 5,000 cp. Assuming negligible pressure drop upstreamof the permeability increase, we can estimate �g,app for the strongfoam that was created at the permeability increase. We estimated�g,app to be approximately 15,000 cp in the coarse-sand regiondownstream of the permeability increase from the pressure differ-ence between Arrows 3 and 2 in Fig. 12. Pressure drop, hence�g,app, appeared to keep increasing even after breakthrough, indi-cating that new lamellae kept being generated and foam texturecontinued to become finer. However, the experiment was contin-ued for only a short time after breakthrough because of safetyconsiderations, because the injection pressure exceeded an im-posed safety limit of about 10 psi.

Comparison With Homogeneous Column. A similar experimentwith ug=0.6 ft/day and fg�67% was performed in a homogeneouscolumn packed with sand similar to that in the low-permeability

0

2

4

6

8

10

12

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

PV of Gas Injected at STP

Pressure

Drop,psi

Heterogeneous

Homogeneous, low perm

f g = 67%

u g = 5 ft/day

1

4

3

2

fg= 67%

ug= 0.6 ft/day

Fig. 12—Pressure drop during transient foam experiments withfg=67% and ug=0.6 ft/day in the heterogeneous (thick solid) andhomogeneous (thin solid) columns. Arrows indicate the posi-tions of gas front in the heterogeneous column at given times:(1) gas enters the fine sand section, (2) gas enters the coarsesand section and forms a strong foam front, (3) the strong foamfront enters the last fine sand section, and (4) gas breaksthrough.

TABLE 1ÑCOLUMN PROPERTIES AND RESULTING

APPARENT VISCOSITIES IN FOAM EXPERIMENTS

Fine Sand

(mesh size)

Coarse Sand

(mesh size) Nk

g,app

at fg = 67%

Column A 70/100 24/48 4.4 5,000 cp

Column B 70/100 48/75 3 < 50 cp

Column C 70/100 16/30 13* 4,000 cp

*Gradual permeability change

µ

0.4 PV 0.65 PV 1.3 PV

3� 6� 9� 12�

70 darcy 16 darcy 70 darcy 16 darcy

0.3 PV

3� 6� 9� 12�

9 darcy(b) Homogeneous column

(a) Heterogeneous column

Fig. 11—Location and shape of the gas displacement front dur-ing transient foam experiments in (a) heterogeneous columnwith alternating high and low permeabilities, and (b) homoge-neous column with low permeability. fg=67% and ug=0.6 ft/day.

208 June 2002 SPE Journal

section in the heterogeneous experiment. The permeability of thecolumn was 9 darcy. Gravity override dominated, and gas break-through occurred after less than 0.3 PV of gas was injected (Fig.11b). Pressure history is shown as the thin solid line in Fig. 12.Apparent foam viscosity �g,app was about 20 cp at breakthroughand later became steady at �g,app of about 50 cp. Therefore, ap-parent foam viscosity in the homogeneous and heterogeneous col-umns differed by some 2 to 3 orders of magnitude.

Other experiments showed that strong foam with apparent vis-cosities of hundreds of cp could be obtained with the same sur-factant formulation in homogeneous sandpacks by using muchhigher gas injection rates, even if only momentarily.28,29 The co-herent foam bank that developed when the injection rate was in-creased could be maintained and propagated even after the ratewas subsequently decreased.

Another experiment in the homogeneous column was per-formed for ug=0.6 ft/day and fg�100%, or gas-only injection. Theresulting pressure drop and observations were virtually identical tothose of fg�67%. �g,app was steady at about 50 cp before the foameventually dried out. That is, at constant ug , there was essentiallyno effect of fg observed in the homogeneous medium, and pressuredrop was independent of the two values of injected liquid velocity.

Effects of Foam Quality in Heterogeneous Media. Althoughfoam quality (i.e., gas-fractional flow fg) was shown to have noeffect on pressure drop and foam mobility in the homogeneouscolumn,29 it does influence the formation of strong foam in het-erogeneous media. As mentioned earlier in our theoretical discus-sion, there exists a maximum fg at which foam generation ceases.In addition, Falls et al.20 pointed out that foam is steadily createdby snap-off at the permeability increase only when fg is below acertain value.

The strong foam front in the heterogeneous column experimentat fg�67% moved in a “stop-and-go” manner. Although the over-all movement was piston-like, its detailed movement was muchmore complicated. Some highly mobile continuous-gas fingers pe-riodically appeared, shooting ahead of the strong discontinuous-gas front. Gravity override sometimes occurred ahead of the pis-ton-like front, in which gas saturation was visibly lower. Theseobservations showed that some lamellae had broken, allowingnewly continuous gas to move ahead of the discrete gas bubbles.However, the continuous-gas fingers invariably stopped only a fewgrain diameters ahead of the low-mobility foam front. Afterward,the piston-like front would move, albeit very slowly, and catch upwith the few gas fingers.

For the strong foam to not only form but also be maintained andpropagated, snap-off must occur often enough, in the repetitivecycle of foam formation followed by breakage or mobilization, toprovide sufficient lamellae and prevent the formation of long con-tinuous gas paths. At fg=67%, lamella generation appears to be

sufficiently frequent. In Fig. 13, we show the effects of differentvalues of fg with other experimental conditions being equal. Whenfg was 87% or less, the strong foam front was not only generatedin the vicinity of the permeability increase, but also propagated tothe end of the column. That is, given a sufficient amount of liquidcoinjected with gas, foam lamellae kept being replenished by foamgeneration at the permeability increase, and the strong foam frontwas maintained and propagated at least for 6 in. (15 cm) to the endof the column. Pressure drop increased because of the propagationof the strong low-mobility foam. In both fg�67% and fg�87%experiments, the apparent viscosity �g,app at breakthrough wasabout 5,000 cp.

At higher injected gas-fractional flow, weaker foam was gen-erated. At fg�95% (see Fig. 13), the pressure drop increased moreslowly, indicating fewer lamellae generated and weaker foam. Inaddition, the strong foam front propagated only for a distance ofabout 3 in. (7.5 cm) from the location of permeability increase. Asfoam began to enter the last fine-grain section of the column, muchweaker foam appeared ahead of the piston-like front, exhibitingvisibly lower gas saturation and gravity override. Most or all of thegas seemed to flow through unobstructed continuous-gas paths,and no apparent movement of the original piston-like front wasobserved thereafter.

Foam was even weaker at fg�100% when no liquid was coin-jected. Although a piston-like front developed immediately down-stream of the step permeability increase, weaker foam appearedahead of the strong foam front when it had propagated for onlyabout 0.4 in. (1 cm). Pressure drop in this experiment remainedonly slightly above the capillary entry pressure of the fine sand.

The results showed pressure drop becoming greater as the liq-uid fractional flow was increased (i.e., fg decreased). It confirmedthe model presented in our theoretical discussion as well as theobservation of Falls et al.20 According to the model, increasing fgresults in increasing Pc at the step permeability increase, to a pointthat it does not meet the capillary pressure criterion for snap-off.Foam mobility obtained at fg�87% was approximately equal tothat at fg�67%. One possible explanation is that the rate of lamellaformation was about the same for any fg below 87%. However, itis also possible that the reduction in mobility was becoming lim-ited by coalescence30 as gas saturation and capillary pressure in-creased. If the limiting capillary pressure associated with coales-cence is less than the capillary pressure for snap-off, than foamwill break before it reaches Pc

sn .

Effect of Permeability Contrast and Gradual Permeability In-crease. As mentioned earlier in our theoretical discussion, perme-ability contrast Nk of at least about 4 is required for snap-off givena≡ Pc

sn/Pce ≈ 1

2. In addition, greater Nk causes Pc

sn(klow) to rise rela-tive to the high-permeability capillary pressure curve. That is, itallows Point C in Fig. 8 to move further to the left and snap-off tooccur for a wider range of Pc(S

+w). To study the dependence of

foam mobility on Nk , we prepared two additional columns withvarying permeability contrasts. All the columns had four sectionsdiffering in permeability. In addition to Column A, discussed pre-viously, that had a permeability ratio Nk of 4.4, Column B withNk=3 and Column C with Nk=13 were prepared (Table 1). Thefine-grain sections in all the columns were packed with the samegrain size. Only the coarse sand was varied to provide differentvalues of Nk . A very sharp transition between the different sectionswas obtained for the Nk=3 column as it was for the Nk=4.4 col-umn. For the Nk=13 column, however, the fine particles penetratedas far as 0.6 in. (1.5 cm) into the coarse-sand sections, creatingwhat is effectively a gradual change in permeability rather than astepwise change.

Pressure drops for similar experiments in the three columns areshown in Fig. 14. Injected superficial gas velocity ug was 0.6ft/day and injected gas fractional flow fg was 67% for all experi-ments. The breakthrough times for each experiment were shown asarrows. For Column B with Nk=3, pressure drop remained near thecapillary-entry pressure for the fine sand of 0.31 psi, compared tothe Nk=4.4 column where pressure drop was over 10 psi at break-through. Although there was a significant delay in break-

0

2

4

6

8

10

12

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

PV of Gas Injected at STP

Pressure

Drop,psi

f g = 67%

f g = 100%

f g = 95%

f g = 87%

Fig. 13—Effects of gas fractional flow fg on the pressure dropacross the heterogeneous column with ug=0.6 ft/day.

209June 2002 SPE Journal

through time in Column B (0.8 PV as opposed to 0.3 PV in thehomogeneous column), it was the result of capillary entrapmentowing to the heterogeneity. Visual inspection of Column B withNk=3 showed no formation of a strong piston-like gas front in thevicinity of the permeability increase. In fact, gravity override thatstarted upstream of the permeability increase simply continuedinto the coarse-sand section downstream. This result confirms thata permeability ratio of about 4 is required for the creation of strongfoam at a step increase in permeability.

Strong foam was observed in Column C with Nk=13. However,it was weaker than in Column A with Nk=4.4. At breakthrough, thepressure drop was only about 3 psi. Apparent viscosity across thecolumn at breakthrough was approximately 1,500 cp. The weakerfoam can be attributed to the mixing of fine- and coarse-sandparticles around the permeability increase, resulting in a gradualpermeability increase. Yet, despite the absence of an abrupt per-meability increase, strong foam was still obtained, albeit with lessmobility reduction initially. However, pressure drop continued toincrease after breakthrough, indicating further reduction in foammobility. After 1.8 PV of gas injected at STP, pressure dropreached 7.5 psi and �g ,app across the column was 4,000 cp, closeto that for the Nk=4.4 column at breakthrough. Therefore, theresult confirmed that strong foam can be generated by snap-offeven when the permeability increase is gradual, although higherpermeability contrast is required.

DiscussionWe have shown the effect of a sudden permeability increase onfoam formation in 1D flow. Given sufficient permeability contrastand sufficient fractional flow of water, gas flowing through anincrease in permeability will undergo snap-off owing to the cap-illary pressure reduction. When surfactant is present, stable foambubbles are produced.

Capillary end effects represent the limiting case of an infinitestep increase in permeability. Recently, Apaydin and Kovscek31

studied the capillary end effects on foam flow in a homogeneoussand packed column. They observed a steep pressure gradient thatbuilt up starting at the end of the column and working its wayupstream, indicating stronger foam developing at the end of thecolumn. Such behavior is consistent with our discussion of snap-off in the low-permeability zone at a sudden permeability increase.The increase in foam strength may have been caused by snap-offnear the end of the column as the capillary pressure dropped tozero at the outlet.

These studies on foam formation in the heterogeneous columnsand capillary end effects suggest that the capillary pressure forsnap-off Pc

sn needs to be accounted for in the “population-balance”modeling of foam.20,32 The dependence of snap-off on gas andliquid rates, as assumed in the population balance model ofKovscek et al.,32 does not predict foam generation at an abrupt

permeability increase or at the end of a column. Only by consid-ering the drop in capillary pressure can we correctly model snap-off in heterogeneous porous media and the capillary end effects.

Implications for 2D Flow. In our 1D column experiments with�g�0.6 ft/day, foam mobility of (50 cp)–1 was obtained in thehomogeneous column. In the heterogeneous column, with Nk=4.4and fg of 87% or less, overall mobility of about (5,000 cp)–1 wasobtained because of snap-off at the permeability increase. Let usnow consider the 2D stratified formation shown schematically inFig. 15, which consists of horizontal strata of alternating high andlow permeability. When flowing horizontally, fluids in each layersee a uniform permeability. Only when flowing vertically do thefluids cross the boundaries from one stratum to another. Therefore,we expect snap-off to occur for vertical flow perpendicular to thestrata, but not for horizontal flow parallel to the strata. Let thepermeability contrast Nk between high and low permeability stratabe 4.4 as in Column A. Assuming foam mobilities similar to thoseobtained in the experiments, we may expect horizontal foammobility equal to that in the homogeneous column, �h ,foam�(50 cp)–1, and vertical foam mobility similar to that in the hetero-geneous column, �v ,foam≈ (5,000 cp)–1. That is, the vertical andhorizontal mobilities of foam differ by more than 2 orders ofmagnitude, and the mobility anisotropy ratio for foam is (�v /�h)foam≈ 0.01.

According to the Stone-Jenkins model, one may estimate theextent of gravity segregation from the viscous-to-gravity forceratio (VGR). Applying Darcy’s law for linear geometry, it can bewritten as

Lg

L= VGR = �kh

kv

h

L�� �p

g�, . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 9)

where Lg�the distance before segregation, h and L�the forma-tion’s thickness and length, respectively, and �the density dif-ference between the underriding and overriding fluids. The Stone-Jenkins model accounts for neither the effect of snap-off in foamflow (it was formulated for ordinary gas/liquid flow) nor the effectof capillarity. For the stratified formation in Fig. 15 with Nk=4.4,we can calculate the vertical and horizontal permeabilities from theharmonic and arithmetic averages of the high and low permeabili-ties, respectively, giving kv /kh�0.6. Applying this value of kv /kh

in Eq. 9, however, will overestimate segregation for foam flow. Ifthe mobilities of foam in heterogeneous and homogeneous col-umns are representative of those perpendicular and parallel tostratification in layered reservoirs, kv /kh should be replaced by�v /�h , which is in the order of 0.01 in this case as indicated above.

Such use of �v /�h deduced from column experiments may bemore appropriate in estimating the segregation length from Eq. 9compared to the conventional methods of estimating kv /kh . Obvi-ously, at this point, this remains an assertion that needs to beconfirmed experimentally in 2D systems. If proven, however, itwould strongly affect the estimate of the effectiveness of foamprocesses in porous media. Foam applications on the field scaleoften showed apparent viscosity of only about 1 cp.4,6,8,33,34 Simu-lations using the single-phase value of kv /kh will predict segrega-tion at a short distance from the injection well. However, consid-ering snap-off at the heterogeneity boundary and the resulting very

Low perm

Low perm

High perm

High perm

Low perm

High perm

.

.

.

.

Fig. 15—Schematic of a 2D stratified formation with alternatinghigh and low permeabilities.

0

2

4

6

8

10

12

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

PV of Gas Injected at STP

Pressure

Drop,psi

f g = 67%

N k = 3

N k = 13

(gradual)

N k = 4.4

u g = 5 ft/day

fg= 67%

ug= 0.6 ft/day

Fig. 14—Pressure drop across columns with different perme-ability contrast at fg=67% and ug=0.6 ft/day. Arrows indicate gasbreakthrough times.

210 June 2002 SPE Journal

low �v /�h for foam, we may expect delayed segregation and thefoam to be more effective than previously predicted.

Foam experiments conducted by Holt and Vassenden35 in a 2Dhomogeneous bead-pack provided interesting insights. Despite thepack’s homogeneity, Holt and Vassenden35 observed segregationlengths Lg extending at least 60% farther than predicted by theStone-Jenkins model using 1D coreflood mobility data. In fact,considering that the foam obtained in their 2D model was actuallyweaker than in the coreflood, they found that the segregationlength Lg was enhanced by at least a factor of three. They attrib-uted these results to an apparent reduction in vertical gas mobilityand suggested that the anisotropy was a result of lamellae beingoriented parallel to the direction of flow. It is also possible that theanisotropy was caused by slight heterogeneity that might haveresulted during packing. This slight heterogeneity may be negli-gible for single-phase flow, but it may have been responsible forthe apparent reduction in vertical gas mobility. Nevertheless, inheterogeneous porous media, we may expect the degree ofmobility anisotropy and the delay in gravity segregation to be evenmore significant.

In naturally occurring porous media, the increase in permeabil-ity is gradual rather than stepwise. As demonstrated earlier, thelonger the transition length, the larger the transition-region capil-lary number Nc ,trans and the greater the permeability ratio requiredfor snap-off. Nevertheless, lower velocity results in smaller Nc ,trans

and smaller permeability contrast requirement. Therefore, in radialflow from the well, snap-off at a permeability increase wouldbecome more important far away from the well. Generally, viscousforces would become lower far away from the well as buoyancystarts to dominate. However, in stratified media, foam generationthrough snap-off would lessen the effect of buoyancy and increasethe viscous forces in the far-from-well region.

ConclusionsSnap-off of the flowing gas at a step-permeability increase is animportant mechanism for foam generation. Foam mobility in ho-mogeneous and heterogeneous sandpacks may differ by two ordersof magnitude. The permeability contrast must be large enough, andthe gas-fractional flow must be low enough, for this foam genera-tion mechanism to be active. This mechanism has a self-regulatingfeature that maintains the capillary pressure, and, hence, satura-tion, around the step increase in permeability near the limitingcapillary pressure for snap-off. It also has important implicationsfor enhanced oil recovery and aquifer remediation. Reservoirs andaquifers are naturally stratified. If the mobility of foam flow per-pendicular and parallel to stratification in a reservoir or aquifer issimilar to flow in heterogeneous and homogeneous sand columns,the degree of gravity segregation may be much less than thatpredicted from the kv /kh based on single-phase flow.

Nomenclaturea � parameter defined as Pc

sn /Pce

fg � fractional flow of gash � formation thickness, L, ft

Je � the dimensionless capillary entry pressureJ(Sw) � Leverett J-function

k � permeability, L2, darcykr � relative permeability

kv /kh � permeability anisotropy ratioL � formation length, L, ft

Nk � permeability contrast, khigh /klow

Pc � capillary pressure, m/Lt2, psiPc

sn � limiting capillary pressure for snap-off, m/Lt2, psiSw � water saturation

STP � standard temperature and pressure (80°F, 14.7 psi)u � superficial (Darcy) velocity, L/t, ft/day

VGR � viscous-to-gravity force ratiox � distance, L, ft

� � pressure gradient, m/Lt2, psi� � porosity

� � viscosity, m/Lt, cp � density, m/L3, lb/ft3

� � interfacial tension, m/t2,dyne/cm�v /�h � mobility anisotropy ratio

Subscripts

app � apparentg � gash � horizontalt � totalv � verticalw � water

AcknowledgmentsThis work is supported by the Processes in Porous Media Consor-tium of the Energy and Environmental Systems Inst. (EESI) atRice U. Contributing donor companies to the consortium are Arco,Baker-Atlas, Chevron, ExxonMobil, Halliburton, Marathon, NorskHydro, Phillips, Saga, Schlumberger, and Shell. The field test atHill AFB was funded by the Advanced Applied Technology Dem-onstration Facility (AATDF) of the U.S. Dept. of Defense.

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SI Metric Conversion Factorscp × 1.0* E–03 � Pa · s

darcy × 9.869 233 E–01 � �m2

ft × 3.048* E–01 � min. × 2.54* E–00 � cmpsi × 6.894 757 E–00 � kPa

*Conversion factor is exact.

Dicksen Tanzil is currently research associate at BRIDGES toSustainability (Houston). e-mail: tanzil@alumni.rice.edu. Heholds a BS degree from Purdue U. and a PhD degree from RiceU., both in chemical engineering. George J. Hirasaki had a26-year career with Shell Development and Shell Oil Cos. be-fore joining the Chemical Engineering faculty at Rice U. in 1993.e-mail: gjh@rice.edu. His research interests are in NMR well log-ging, reservoir wettability, asphaltene deposition, emulsion co-alescence, and surfactant/foam aquifer remediation. He is amember of the National Academy of Engineering. Hirasakiholds a BS degree in chemical engineering from Lamar U. anda PhD degree in chemical engineering from Rice U. He wasnamed an Improved Oil Recovery Pioneer at the 1998 SPE/DOR IOR Symposium and was the 1989 recipient of the LesterC. Uren Award. Clarence A. Miller is Louis Calder Professor ofChemical Engineering at Rice U. and a former chairman of thedepartment. e-mail camill@rice.edu. His research interest is sur-factants, including their applications in improved oil recovery,aquifer remediation, and foam flow. He holds BS and PhD de-grees from Rice U. and the U. of Minnesota, respectively.

212 June 2002 SPE Journal