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Toward Improved Prediction of Reservoir Flow Performance simulating oil and water flows at the pore scale John J. Buckles, Randy D. Hazlett, Shiyi Chen, Kenneth G. Eggert, Daryl W. Grunau, and Wendy E. Soll 112 Los Alamos Science Number 22 1994

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Page 1: Toward Improved Prediction of Reservoir Flow Performance · Toward Improved Prediction of Reservoir Flow Performance simulating oil and water flows at the pore scale John J. Buckles,

Toward Improved Prediction of ReservoirFlow Performance

simulating oil and water flows at the pore scale

John J. Buckles, Randy D. Hazlett, Shiyi Chen, Kenneth G. Eggert, Daryl W. Grunau, and Wendy E. Soll

112

Los Alamos Science Number 22 1994

Page 2: Toward Improved Prediction of Reservoir Flow Performance · Toward Improved Prediction of Reservoir Flow Performance simulating oil and water flows at the pore scale John J. Buckles,

In today’s world oil market, economicproduction of oil and gas resourcesrequires carefully engineered recov-

ery projects of increasing technicalcomplexity and sophistication. Hydro-carbons do not reside in cavernouspools awaiting discovery. Rather theyare found

—sometimes at enormousdepths—within the confines of tinypores in rock. Although the pores maybe interconnected, the resulting path-ways still present a significant resis-tance to the flow of oil toward a welldrilled into the hydrocarbon-bearingstrata. In addition, since water residesin some of the pores, hydrocarbons andwater are recovered simultaneously atthe well-head. Thus, even when largeamounts of oil are known to be in a

reservoir, often only a relatively smallfraction of it can be recovered withconventional pumping technology. Themost common method of enhancing oilrecovery, which accounts for much ofthe oil production in the U.S., is the in-jection of water at strategic locations todisplace the oil toward the productionwells.

Modern recovery projects of the typedepicted in Figure 1 require very largecapital investments. A single off-shorewell drilled to a depth of 15,000 feetcan cost up to $100 million. To besuccessful, a project must have a siz-able hydrocarbon target, the promise ofextracting oil or gas at a sufficientlyhigh rate, a strategy for water separa-tion and disposal, and a scheme for

transportation to a refinery. As explo-ration efforts are driven to more remoteenvironments, the infrastructure costsfor hydrocarbon production escalate.Large potential returns are inevitably ac-companied by large monetary risk.

Obviously, economic analysis mustbe performed before deciding whether aparticular investment should be made.Computer simulations of reservoir flowperformance are one of the essential in-gredients in the analysis. On the basisof both field and laboratory data aboutthe distributions of different rock typesand the properties of each type, we atMobil attempt to predict the average, orbulk, flow behavior of the fluidsthrough the hydrocarbon-bearing rockof a reservoir. Many times our simula-

Toward Improved Prediction of Reservoir Flow Performance

Number 22 1994 Los Alamos Science 113

Figure 1. A Modern Oil-Recovery ProjectThis illustration of a modern oil-recovery project shows both production wells (black arrows indicate hydrocarbons) and injection

wells (blue arrows indicate water) drilled into oil-bearing strata deep beneath the Earth’s surface. The enlargement at left shows the

porous network in the oil-bearing rock. Water (blue) that has been injected under pressure into the reservoir is displacing oil

(black). The regular planar grid between the surface and the reservoir represents in two dimensions the rectangular-block geometry

used in bulk flow simulations of reservoir performance. Plotted on the grid are lines of constant altitude in the top layer of the

reservoir, which is not parallel to the surface but rather has hills and valleys. Bulk-flow simulations of reservoir-flow performance

are an important ingredient in the decision of whether to invest in major recovery projects. Improved prediction of bulk-flow para-

meters through understanding the physics at the pore scale is the goal of the lattice-Boltzmann simulations of multiphase flow

through porous media.

Water (blue) pushingoil (black) throughporous rock

Injection wellProduction well

Grid for reservoir-scalesimulations

Horizontal wellWater flooding

Page 3: Toward Improved Prediction of Reservoir Flow Performance · Toward Improved Prediction of Reservoir Flow Performance simulating oil and water flows at the pore scale John J. Buckles,

tion-based predictions over- or under-estimate reservoir performance becausethey do not properly model multiphaseflow. Such errors can be the Achillesheel of the investment analysis, leadingto very expensive errors in productionstrategy or even to a mistake in the de-cision of whether or not to invest.

The collaboration reported here be-tween Mobil and Los Alamos NationalLaboratory is designed to improvereservoir-flow predictions by modelingfluid flow at the scale of individualpores, tens of microns across, and usingthose results to predict bulk-flow para-meters. Ultimately our results will beincorporated into reservoir-scale com-puter simulations. Relative permeabili-ties are sensitive bulk parameters usedin the simulations because they canchange by factors of 2 or 3 dependingon details of the fluid-fluid and rock-fluid interactions at the pore scale.Moreover such differences can translateinto significant differences in the oiland water flow rates seen at the well-head. An improved ability to predictrelative permeabilities from easy-to-gather data is one of the major goals ofour collaborative effort.

Parameters for porous-media flow.Permeabilities and relative permeabili-ties are defined by Darcy’s law for flowthrough porous media. This empiricallaw says that the rate of flow through aporous rock is proportional to the pres-sure drop per unit length along the di-rection of flow. The constant of pro-portionality is called the permeability k.Specifically, the flow rate of a fluid, Q,along a direction x is proportional tothe pressure gradient

Dp/Dx along thatdirection:

Q 5 ,

where A is the cross-sectional area of

∆p}∆x

kA}m

Toward Improved Prediction of Reservoir Flow Performance

114 Los Alamos Science Number 22 1994

1.0

0.8

0.6

0.4

0.2

00 1.0

Rel

ativ

e pe

rmea

bilit

y, k

r

Oil relative permeability, k roil

Water relative permeability, k r

water

Water wet

Oil wet

flow1.adb 7/26/94

0.2 0.4 0.6 0.8Water saturation, Swater

Figure 2. Typical Relative Permeability Curves for Oil and WaterRelative permeability versus water saturation (the fraction of the pore space occupied

by water) is plotted for two rock types, one with a greater affinity for oil (oil-wet) and

one with a greater affinity for water (water-wet). The curves were deduced from experi-

ment. They show that oil can be recovered more easily from water-wet rock than from

oil-wet rock. Note that if the rock is oil-wet (dashed lines), the relative permeability of

oil falls precipitously as the fraction of the pore space occupied by oil goes below 80

percent (or equivalently, as the water saturation increases above 20 percent). For

water-wet rock, the relative permeability of oil falls more gradually and not until the oil

saturation goes below 70 percent.

Page 4: Toward Improved Prediction of Reservoir Flow Performance · Toward Improved Prediction of Reservoir Flow Performance simulating oil and water flows at the pore scale John J. Buckles,

the porous rock perpendicular to the di-rection of flow and m is the viscosity ofthe fluid. The permeability k describesthe ease with which fluids flow throughthe porous network. Under a givenpressure gradient, fluid will flow moreslowly through a rock of low perme-ability than through one of high perme-ability. For the most part, the poregeometry of a rock (size and degree ofconnectedness of the pores) determinesits permeability.

To describe the bulk flow in oilreservoirs, where pressure gradientscause oil, water, and gas to compete forspace to flow, the concept of permeabil-ity is extended to include the sensitiveparameters mentioned above, the rela-tive permeabilities of each fluid present.Consider the case in which both oil andwater occupy the pore space of a rock.Then the application of a pressure gra-dient will cause both oil and water toflow simultaneously. Darcy’s law ismodified to describe the flow rate ofeach fluid:

Qtotal 5 Qoil + Qwater

5 (kroil + kr

water ) ,

where Qoil is the flow rate of oil, Qwateris the flow rate of water, kr

oil is the rela-tive permeability of oil, and kr

water isthe relative permeability of water. Rel-ative permeabilities depend not only onthe pore geometry, which is fixed, butalso on variable quantities including thesaturation of each fluid (fraction of thepore space occupied by each fluid), andthe spatial distribution of each fluidwithin the pores. The sum of the tworelative permeabilities is typically lessthan one because fluid-fluid interactionsincrease the resistance to flow.

Fluid distributions at the pore scaleare governed, in turn, by fluid-fluid in-

teractions and rock-fluid interactions.The latter determine an empirical quan-tity called wettability, the “preference”of a rock to be in contact with one fluidover another. In the jargon of the field,a rock is called water-wet and water iscalled the wetting fluid if, in the pres-ence of water and oil, the rock surfaceinteracts more strongly with water thanwith oil so that water wets (spreads outon) the rock surface and excludes directcontact with oil. A rock is called oil-wet and oil is called the wetting fluid ifthe rock “prefers” oil over water. Atone time, experts thought all oil reser-voirs were water-wet. Now most ex-perts agree that reservoirs are predomi-nantly of intermediate or mixed wetta-bility. Intermediate wettability refers tothe absence of any preferential interac-tions. Mixed wettability, on the otherhand, implies that some regions are oil-wet and others water-wet on scales allthe way down to individual pores.

Although it is well known that wetta-bility influences the size and shape ofpathways available for fluid flow and istherefore a controlling parameter for rel-ative permeabilities, an explicit quantita-tive relationship between wettability andrelative permeability has not yet beenderived. Our current pore-scale studies,which include explicit modeling offluid-fluid and fluid-rock interactions,should help to uncover that and otherfundamental relationships among the

empirical parameters used to describemultiphase flow through porous media.

Current data for reservoir-scalesimulations. Obtaining relative perme-ability data is both difficult and expen-sive. Nevertheless, those data are cru-cial to large-scale simulations ofreservoir flow performance. In the sim-ulations the reservoir is treated as a setof tens of thousands of block-shapedregions, 100 to 1000 feet on a side,with each block assigned a permeabilityk and relative permeability curves foroil, water, and gas. These curves typi-cally show relative permeabilities ver-sus saturation of one of the fluids (seeFigure 2). The saturation of a fluid isthe fraction of the pore space occupiedby that fluid.

Relative permeabilities are derivedfrom bench-top experiments on rocksamples that are specially cut duringthe well-drilling process by a drill bitwith a hollow core. The rock “cores”are typically 1 to 2 inches in diameterand 3 to 4 inches in length. The coresare carefully selected to represent largersections of reservoir rock. Retrieval ofsuch cored reservoir rock is expensiveand has historically been limited to atmost about a half-dozen samples perreservoir. The laboratory experimentsattempt to simulate flow under subsur-face conditions of the fluids found inreservoirs. Hydrocarbons and salty

Dp}Dx

kA}m

Toward Improved Prediction of Reservoir Flow Performance

Number 22 1994 Los Alamos Science 115

Table 1. Permeabilities Deduced from Simulation and Experiment

Micromodel-1

Micromodel-2

Beadpack

Berea

Sandstone

Sandstone

23

37

1100

1

7.4

7.4

21.5

32

1000

1.2

8.6

21.5

~80 µm

~80 µm

50 µm

10 µm

20 µm

20 µm

Pore depth = 50 µm

Pore depth = 74 µm

Flow in long direction

Flow in short direction

Porous mediumk (experiment)

[darcy]k (simulation)

[darcy]Imaging

resolution Comments

latticetab1.adb 7/26/94

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water are pumped through the samplesat the pressures of thousands of poundsper square inch and temperatures be-tween 100 to 250 degrees Fahrenheit.Relative permeabilities, as defined byDarcy’s law, are computed from themeasured flow rates and applied pres-sure gradients. It is only through suchexperiments that the effects of wettabil-ity on multiphase flow are incorporatedinto the simulations. Unfortunately thelaboratory experiments may not alwaysreflect the conditions present in the sub-surface. For example, the wettability ofthe samples may have been altered bythe invasion of drilling lubricants dur-ing the sample-retrieval process. Infact, both the quality and quantity ofdata on permeability and relative per-meability limit the accuracy of reser-voir-scale simulations.

Lattice-Boltzmann simulations ofpore-scale flow. The lattice-Boltzmannmethod was developed at the Laborato-ry. This method enables us to modelexplicitly, in the complex geometry of aporous network, the influence of fluid-fluid and fluid-rock interactions on bulkflow parameters. Details of the lattice-Boltzmann model for the flow of im-miscible fluids are described in thecompanion article, “Lattice-BoltzmannFluid Dynamics.” The algorithm wasdesigned to run most efficiently onlarge massively parallel computers suchas the Laboratory’s CM-5 ConnectionMachine. Here we present some com-parisons between two- and three-dimen-sional lattice-Boltzmann simulations ofsingle-fluid and two-fluid flow and cor-responding laboratory experiments.

Pore geometry is a major factor de-termining flow in porous media, so wehad to replicate realistic geometries inboth the simulations and the experi-ments. To simulate two-dimensionalflow in the laboratory, we took advan-tage of computerized etching techniques

to create glass micromodels of thin sec-tions of rock. The digitized patternused to etch each glass micromodel wasalso used to create the two-dimensionalpore geometry for the correspondinglattice-Boltzmann computer simulation.The three-dimensional experiments in-volve either beadpacks (beads of vari-ous sizes packed together to approxi-mate the geometry of porous rock) oractual core samples from reservoirs.X-ray tomography performed at the Na-tional Synchrotron Light Source atBrookhaven National Laboratory isused to capture the three-dimensionalpore structure of the rock samples inthe digitized form needed for the simu-lations. The digital rendering of thex-ray image, which is used in the simu-lations, can have a resolution of onemicron (much less than average porediameters, which are typically tens ofmicrons).

Table 1 lists the results of the single-fluid simulations, which yield valuesfor the permeability k for differenttypes of porous media. The simulationsare in good agreement with experimentexcept for the case of flow across theshortest dimension of the sandstonesample. The discrepancy may be dueto differences in the degree of homo-geneity between the simulated regionand the experimental sample. One ofour present research goals is to deter-mine the size and resolution of the sim-ulated domain required to accuratelypredict permeabilities and other macro-scopic flow properties from the detailsof pore structure, material texture, androck-fluid interactions.

In the lattice-Boltzmann model forthe flow of immiscible fluids, the vis-cosity of each fluid may be varied inde-pendently. Also, the surface tension atfluid-fluid interfaces and the contactangle of each fluid with the rock sur-face may be varied. (See Figure 6 in“Lattice-Boltzmann Fluid Dynamics”

Toward Improved Prediction of Reservoir Flow Performance

116 Los Alamos Science Number 22 1994

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Toward Improved Prediction of Reservoir Flow Performance

Number 22 1994 Los Alamos Science 117

Figure 3. Water Displacing Oil at Different Capillary Numbers These two-dimensional lattice-Boltzmann simulations show flow through a two-dimensional glass micromodel for three values of the

capillary number, the ratio of viscous to interfacial (or capillary) forces. Here capillary number is defined as Nca

5 mU/fs, where m is

viscosity, U is displacement velocity, and f and s are porosity and interfacial tension, respectively. The pores are originally filled

with the blue wetting fluid (oil). The red non-wetting fluid (water) is injected at left. Different displacement velocities were used to

simulate the effect of different capillary numbers on the efficiency with which the pores are swept. Note that as the velocity de-

creases, or the capillary number decreases, the red non-wetting fluid (water) becomes less effective at displacing the blue wetting

fluid (oil). At low capillary number, when interfacial forces dominate, the simulation produces an irregular displacement front.

These results are realistic compared with published laboratory results.

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for the definition of contact angle.)The contact angle of a fluid is 08 if thefluid completely wets the rock and in-creases to 1808 as the fluid changesfrom strongly wetting to strongly non-wetting. In lattice-Boltzmann simula-tions we vary the strength of rock-fluidinteractions to control the contact angleand thus the wettability conditions.

One quantity used to characterize theflow is capillary number—the ratio ofthe viscous forces (the shear forceswithin each fluid) to the forces at thefluid-fluid interfaces. When the capil-lary number is low, interfacial forcesare much stronger than viscous forcesand tend to control the flow. For exam-ple, the interfacial forces will tend todraw the wetting fluid into the smallestpores and thereby block the invasion ofthe nonwetting fluid. We are using

lattice-Boltzmann simulations to deter-mine the quantitative effects of wetta-bility and capillary number on relativepermeabilities and displacement effi-ciency (the efficiency of displacing onefluid by injection of a second fluid).

Figure 3 shows the results of a two-dimensional simulation for three valuesof the capillary number. The pores areinitially filled with the blue wettingfluid (oil) and are being invaded by thered nonwetting fluid (water). The sim-ulation reproduces the correct qualita-tive relation between capillary numberand oil recovery efficiency under oil-wet conditions—namely, the recoverybecomes less efficient as the capillarynumber decreases. In part (a), wherethe capillary number is highest, viscousforces dominate and we see more uni-form or piston-like displacement of the

oil and higher displacement efficiency.In part (c), viscous forces are a hundredtimes smaller, so capillary forces domi-nate and the displacement front hasmany fingers. Also, the displacementefficiency is much lower than in (a).The blue (oil) and red (water) distribu-tions in Figure 3 are similar to the oiland water distributions observed by nu-clear-magnetic-resonance micro-imag-ing of flow through a mixture of glassand polystyrene beads.

Relative permeabilities of the porenetwork in Figure 3 under various con-ditions can be deduced from the resultsof lattice-Boltzmann simulations. Inparticular, Figure 4 shows the relativepermeabilities of the wetting (blue/oil)and nonwetting (red/water) fluids versusthe saturation of the wetting fluid fortwo of the three values of the capillary

Toward Improved Prediction of Reservoir Flow Performance

118 Los Alamos Science Number 22 1994

1.0

0.8

0.6

0.4

0.2

0.0

1.0

0.8

0.6

0.4

0.2

0.0

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Rel

ativ

e pe

rmea

bilit

y, k

r

Saturation of wetting fluid, Swet

Capillary number = 10–2 Capillary number = 10–3

Nonwetting fluid Wetting

fluidNonwetting

fluidWetting fluid

Rel

ativ

e pe

rmea

bilit

y, k

r

Saturation of wetting fluid, Swet

flow4.adb 7/26/94

Figure 4. Relative Permeability Curves for Oil and Water Predicted by Lattice-Boltzmann Simulations These two plots show relative permeabilities kr for the wetting fluid (oil) and the nonwetting fluid (water) as a function of the

wetting-fluid saturation for two values of the capillary number. Each data point was derived from lattice-Boltzmann simulations of

flow through the network shown in Figure 3. The curves are similar to those obtained in the laboratory experiments. The relative

permeability of the wetting fluid increases, that is, the fluid flows more easily, as its saturation (the fraction of pore space it occu-

pies) increases. Similarly, the relative permeability of the nonwetting fluid decreases as its saturation (equal to 1 2 Swet) decreases.

Note how different values of the capillary number change the shape and position of the curves.

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number used in our simulations. Again,the curves are typical of those seen inthe laboratory. The input to our simu-lations includes pore geometry, fluidviscosities, and interfacial tensions.The capability to predict relative per-meability, an empirical quantity, fromsuch a simulation is new and is stillbeing validated against experimentalflow data.

Although we are still validating thelattice-Boltzmann simulation technique,we have already begun a study of theeffects of wettability on two-fluid flowin two- and three-dimensional simula-tions. Figure 5 shows the residual oilsaturation (the percentage of the oil thatremains after being flushed with water)versus wettability for oil-wet, water-

wet, and neutral-wet conditions as ob-tained from some early two-dimensionalsimulations using lattice-gas rather thanlattice-Boltzmann methods. As expect-ed, the simulations show that the mostefficient displacement of oil by water isobtained at neutral wettability, whereasit is most difficult to extract oil if therock is oil-wet. A major input to suchsimulations are the rock-fluid interac-tions that define the wettability condi-tions. At Mobil we are developingtechniques to map wettability on thebasis of mineral distribution and fluidproperties, so that we can incorporaterealistic complex wettability distribu-tions into our simulations. Figure 6shows a typical three-dimensional lattice-Boltzmann simulation in which

Toward Improved Prediction of Reservoir Flow Performance

Number 22 1994 Los Alamos Science 119

10

8

6

4

2

01 2 3 4 5

Wettability index

Res

idua

l oil

satu

ratio

n (p

erce

nt)

Oil-wet

Water-wet

Neutral-wet

flow5.adb 726/94

Figure 5. Residual Oil Saturation and Wettability These data were obtained from lattice-gas simulations run on the system shown in Fig-

ure 2. The residual oil saturation is the percentage of pore space occupied by oil after

the oil-filled pores have been flushed with water. The wettability conditions are varied

by changing the collision rules at the fluid-rock boundaries. The rules implement slip

or non-slip boundary conditions for all particles near a rock surface. Intermediate wet-

tabilities are simulated by random fluctuations between the two types of boundary con-

ditions. The wettability index is a dimensionless quantity that measures the relative

affinity of a rock sample to draw in oil or water spontaneously.

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Toward Improved Prediction of Reservoir Flow Performance

120 Los Alamos Science Number 22 1994

Figure 6. Water Displacing Oil in Three DimensionsThis figure shows snapshots taken at different timesteps dur-

ing a three-dimensional lattice-Boltzmann simulation of the

forced displacement of oil (yellow) by water (blue) within a

porous rock. The pore geometry in this simulation replicates

that in a portion of a rock sample obtained from a Mobil off-

shore oil reservoir over 10,000 feet below sea level. The geom-

etry was determined from a computed tomography (CT) scan

that provided a three-dimensional digital image of the pore

structure at a resolution of 20 microns. The rock is shown as

transparent in order to make visible the oil and water occupy-

ing the pores. The second and third snapshots show that

water continually displaces oil as the simulation progresses.

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water is displacing oil in a water-wetsystem. During this simulated displace-ment, we recover bulk flow properties,such as mass fluxes, local velocities,and pressure profiles, from which wecan compute relative permeabilities.Such computer experiments provide farmore insight into and control over thedisplacement process than wet-lab ex-periments.

The state-of-the-art technology re-sources required for such a project arebeyond what private companies—evenlarge ones—can afford. The Laborato-ry and Mobil Corporation have com-bined their expertise and their resourcesto apply Laboratory-developed lattice-Boltzmann technology to tough re-search problems in the oil industry.We expect the results of this work tohave major implications as U.S. reser-voirs mature and require more complexand risky recovery schemes.

Further Reading

I. Field Roebuck, Jr. 1979. Applied PetroleumReservoir Technology. IED Exploration, Inc.

H. C. Slider. 1983. Worldwide Practical Petro-leum Reservoir Engineering Methods. PennWellPublishing Company.

L. F. Koederitz, A. H. Harvey, and M. Honar-pour. 1989. Introduction to Petroleum ReservoirAnalysis. Contributions in Petroleum Geologyand Engineering, volume 6. Gulf PublishingCompany.

F. A. L. Dullien. 1992. Porous Media: FluidTransport and Pore Structure, second edition.Academic Press.

Toward Improved Prediction of Reservoir Flow Performance

Number 22 1994 Los Alamos Science 121

John J. Buckles earned his B.S. in chemical en-gineering from Purdue University in 1976 and hisPh.D. in chemical engineering from the Universi-ty of Illinois in 1983. For the past eleven yearshe has been employed at Mobil Research andDevelopment Corporation. Buckles’s researchinterests include enhanced oil recovery and pore-scale flow modeling.

Daryl W. Grunau was a graduate research assis-tant at the Laboratory in 1990 and again from1991 to 1993. His work included the develop-ment of theory and computer code for research incomputational fluid dynamics using the lattice-gas and lattice-Boltzmann methods. He is cur-rently an applications engineer at Thinking Ma-chines Corporation. Grunau received his B.A. inmathematics and computer science from TaborCollege in Kansas in 1987 and his M.S. andPh.D. in applied mathematics from ColoradoState University in 1989 and 1993.

Randy D. Hazlett earned his B.S. and Ph.D. inchemical engineering from the University ofTexas, Austin, in 1980 and 1986. Since 1986 hehas been employed at Mobil Research and Devel-opment Corporation. Hazlett’s research interestsinclude flow in porous media, interfacial phe-nomena, improved oil recovery, and imaging andimage processing.

Standing (left to right): John J. Buckles, Shiyi Chen, and Daryl W. Grunau; seated: Randy D. Hazlettand Mary E. Coles

Wendy E. Soll has been a staff member of theGeoanalysis Group in the Earth and Environmen-tal Sciences Division since 1993. She was apostdoctoral fellow at the Laboratory from 1992to 1993. Her research includes the study of mul-tiphase flow and transport in porous media utiliz-ing lattice-Boltzmann pore-scale modeling. Shereceived her B.S.E. in mechanical engineeringfrom Princeton University in 1984. At the Mass-achusetts Institute of Technology she earned herM.S. in mechanical engineering in 1987 and herSc.D. in civil engineering in 1991. Soll is amember of the American Geophysical Union, theSociety of Petroleum Engineers, and the Ameri-can Society of Mechanical Engineers.

The biographies of co-authors Shiyi Chen andKenneth G. Eggert appear on page 109.