PHYSICAL REVIEW E JUNE 1999VOLUME 59, NUMBER 6

Trapping and mobilization of residual fluid during capillary desaturation in porous media

Lucian Anton1,2 and R. Hilfer2,3

1Institute for Atomic Physics, INFLPR, Laboratory 22, P.O. Box MG-7, 76900 Bukarest, Romania2ICA-1, Universitat Stuttgart, Pfaffenwaldring 27, 70569 Stuttgart, Germany

3Institut fur Physik, Universita¨t Mainz, 55099 Mainz, Germany~Received 21 April 1998; revised manuscript received 12 November 1998!

We discuss the problem of trapping and mobilization of nonwetting fluids during immiscible two-phasedisplacement processes in porous media. Capillary desaturation curves give residual saturations as a function ofcapillary number. Interpreting capillary numbers as the ratio of viscous to capillary forces the breakpoint inexperimental curves contradicts the theoretically predicted force balance. We show that replotting the dataagainst a novel macroscopic capillary number resolves the problem for discontinuous mode displacement.@S1063-651X~99!00304-9#

PACS number~s!: 47.55.Mh, 81.05.Rm, 61.43.Gt, 83.10.Lk

usate

teilityateainbles

hethgevete

-

th

ertu

haeondsi

pnowte

di-olesys-.ble

ips

d of

orever abutare

c-,e-or-

-

I. INTRODUCTION

A displacement of one fluid by another within a poromedium poses challenging problems of microscale to mroscale transitions which have received considerable attion from physicists in recent years@1–20#. For reviews thereader is referred to@12,21#. Apart from the omnipresenquenched correlated disorder and structural heterogenfluid-fluid and fluid-solid interactions generate metastabiand hysteresis phenomena which have resisted quantitprediction and understanding. A central problem of grpractical importance is the prediction of residual nonwettfluid saturation after flooding the pore space with immisciwetting fluid@22–24#. Obvious interest in this problem arisefrom enhanced oil recovery@25,26# or in situ remediation ofsoil contaminants@27,28#.

Microscopically the laws of hydrodynamics governing tpore scale processes are well known. The complexity ofmicroscopic fluid movements and the lack of knowledabout the microstructure and wetting properties, howerenders a detailed microscopic treatment impossible. Insone has to resort to a more macroscopic treatment.

Desaturation experiments@29–36# show that the conventional macroscopic description is incomplete@23,24#. Weshall begin our discussion by reminding the readers ofincompleteness. We then exhibit another problem@37#. Thisarises from the fact that microscopic capillary numbseemingly cannot represent the force balance in a desation experiment.

Given these problems our main objective is to show tthe recent analysis of@37# gives the correct force balancbetween macroscopic viscous and capillary forces for ctinuous mode desaturation experiments. For so-calledcontinuous mode displacement it leads to bounds on theof residual blobs.

II. PROBLEMS OF TWO-PHASE FLOW EQUATIONS

Let us begin our discussion with the standard macroscoequations of motion for two-phase immiscible displacemeMacroscopic equations of motion describe multiphase flon length scales large compared to a typical pore diame

PRE 591063-651X/99/59~6!/6819~5!/$15.00

c-n-

ty,

ivet

g

e

r,ad

is

sra-

t

-is-ze

ict.

r.

Hence they are applied to laboratory samples with linearmensions on the order of centimeters as well as to whreservoirs measuring kilometers and more. Consider atem having lengthsLx ,Ly ,Lz in the three spatial directionsThe probe space is assumed to be filled with two immiscifluids denoted generically as water~index w! and oil ~indexo!. The equations read@38–40,37#

f]Sw

]t5“•H Kkw

r

mw

[“Pw2rwgOe] J ~1!

2f]Sw

]t5“•H Kko

r

mo

[“(Pw1Pc)2rogOe] J ~2!

and they are supplemented with the constitutive relationsh

kwr ~x,t !5kw

r„Sn~x,t !…, ~3!

kor~x,t !5ko

r„Sn~x,t !…, ~4!

Pc~x,t !5Pc„Sn~x,t !…, ~5!

where

Sn~x,t !5Sw~x,t !2Swi

12Swi2Sor

. ~6!

The variables in these equations are the pressure fielwater denoted asPw and the water saturationSw . Thesaturation is defined as the ratio of water volume to pspace volume. Pressures and saturations are averages omacroscopic region much larger than the probe size,much smaller than the system size. Their argumentsthe macroscopic space and time variables (x,t). The satura-tions obey Swi,Sw,12Sor , where the two numbers0<Swi , Sor<1 ar two parameters representing the irreduible water saturation,Swi , and the residual oil saturationSor . The residual oil saturation gives the amount of oil rmaining in a porous medium after water injection. The nmalized saturationSn varies between 0 and 1 asSw variesbetweenSwi and Sor . The permeability of the porous medium is given by the absolute~single phase flow! permeabil-

6819 ©1999 The American Physical Society

tie

un-f-ti

io

o

e

e

beFoalst

peo

ionthe-

reth

n

nio

eoein

thiztrt

enllaari-oie-

ap-

ofromis

di-

ceanes

ryndoutoilIn

isdon

tu-ple,oilter-, a. Intedoilde

the,

ves

The

6820 PRE 59L. ANTON AND R. HILFER

ity tensorK . The porosityf is the volume fraction of porespace. The two fluids are characterized by their viscosimw ,mo and their densitiesrw ,ro . The termsrgOe representthe gravitational body force whereeT5(0,0,21) is a unitrow vector pointing along the negativez axis andg is theacceleration of gravity. The orthogonal matrix

O5S cosay 0 sinsy

0 1 0

2sinay 0 cosay

D ~7!

describes an inclination of the system. Here a rotation arothe y axis with tilt or dip angleay was assumed. The macroscopic capillary pressurePc is defined as the pressure diference between the oil and the water phase. The constiturelation for Pc assumes that the capillary pressure functPc depends only on the saturation@41#. In addition toPc ,the dimensionless relative permeabilities are assumed tfunctions of saturation only,kw

r 5kwr (Sw), ko

r5kor(Sw). They

represent the reduction in permeability for one phase duthe presence of the other phase@38,39#. The three constitu-tive relationskw

r (Sw), kor(Sw), andPc(Sw) are assumed to b

known from experiment. Equations~1! and ~2! are couplednonlinear partial differential equations which mustcomplemented with large-scale boundary conditions.laboratory experiments the boundary conditions are typicgiven by a surface source on one side of the sample, aface sink on the opposite face, and impermeable walls onother faces. For a geosystem the boundary conditions deupon the well configuration and the geological modelingthe reservoir environment.

The problem with the macroscopic equations of mot~1! and~2! arises from the experimental observation thatparametersSor andSwi are not constant and known, but dpend strongly on the flow conditions in the experiment@29–36,26#. Hence they may vary in space and time. More pcisely, the residual oil saturation depends strongly onmicroscopic capillary number Ca5mwv/sow , wherev is atypical flow velocity andsow is the surface tension betweethe two fluids. The capillary desaturation curveSor(Ca) isshown in Fig. 1 for unconsolidated glass beads and sastones@32,36#. Such curves contract clearly the assumptthat the functionskw

r (Sw), kor(Sw), andPc(Sw) depend only

on saturation. Instead they show thatkwr (Sw), ko

r (Sw), andPc(Sw) depend also on velocity@42,43# and pressure. Hencthey depend on the solution and cannot be considered tconstitutive relations characterizing the system. The depdence shows that the system of equations of motion iscomplete.

The breakpoint in a capillary desaturation curve markspoint where the viscous forces, which attempt to mobilthe oil, become stronger than the capillary forces, whichto keep the oil in place. For this reason the capillary desaration curves are usually plotted against Ca, which represthe microscopic force balance between viscous and capiforces. From Fig. 1 it is seen that the theoretical force bance corresponding to Ca51 on the abscissa and the expemental force balances represented by the various breakpat Ca!1 differ by several orders of magnitude. Plotting r

s

d

ven

be

to

rlyur-hendf

e

-e

d-n

ben--

eeyu-tsryl-

nts

sidual saturation against the correct ratio of viscous to cillary forces should result in a breakpoint at Ca'1 @37,39#.

We now proceed to show that a partial understandingthe macroscopic force balance can already be obtained fthe traditional equations of motion. The main result of thanalysis is a preliminary experimental validation of themensional considerations introduced in@37#.

III. DISCONTINUOUS VERSUS CONTINUOUS MODEDISPLACEMENT

Before embarking on a discussion of the force balanduring capillary desaturation, it is crucial to emphasizeimportant difference in the way capillary desaturation curvare measured.

~i! In the first method of measuringSor(Ca), the oil in afully oil saturated sample is displaced with water at a velow Ca. After the oil flow at the sample outlet stops aseveral pore volumes of water have been injected withproducing more oil, the flow rate is increased. Again theflow is monitored until no more oil appears at the outlet.this way the flow rate is increased iteratively untilSor hasfallen to zero. After the first injection the oil configurationdiscontinuous~or disconnected! and it remains disconnectethroughout the rest of the experiment. This mobilizatimode will be called discontinuous.

~ii ! In the second method of measuring capillary desaration curves one again starts from a fully saturated samand performs a waterflood at a given value of Ca. Afterflow ceases at the outlet, the residual saturation is demined. Then the sample is again saturated fully with oilnew value of Ca is chosen, and the injection is repeatedthis experiment the injection starts always with a connecoil phase contrary to the previous experiment, where thephase is discontinuous at higher Ca. This mobilization mowill be called continuous.

The capillary number required to reach a givenSor isknown to be much lower in the continuous mode than indiscontinuous mode@36,35#. This is also seen in Fig. 1

FIG. 1. Experimentally measured capillary desaturation cur~capillary number correlations! for bead packs@36# ~solid lines withcircles and squares! and sandstone@32# ~star symbols! as a functionof microscopic capillary number Ca5mwv/sow .The dashed line isthe desaturation curve for continuous mode displacement.dash-dotted line marks the plateau value for sandstone.

otha

ouldhcuqu

s-b

re

o

q

in

pic

vis-ers

the

es.of

o-

PRE 59 6821TRAPPING AND MOBILIZATION OF RESIDUAL FLUID . . .

which shows continuous mode data and discontinuous mdata for the unconsolidated glass beads. From the factthe equations of motion are only valid in the subintervSwi,Sw,12Sor it follows that they cannot be applied tdiscontinuous mobilization mode experiments. They shohowever, be valid for the continuous mode to the extent tthe equations themselves are correct. Therefore we disnext the macroscopic force balance predicted by these etions from the continuous mobilization mode.

IV. MACROSCOPIC FORCE BALANCE

It was shown in@37# that the balance of macroscopic vicous and capillary forces is not represented appropriatelythe microscopic capillary number Ca, and that a new macscopic capillary numberCa should be used instead. Here wgeneralize those results to anisotropic and inclined pormedia and apply them to replot theSor data obtained for thecontinuous mode displacement. To this end we rewrite E~1! and ~2! in dimensionless form. Introducing the matrix

L5S Lx 0 0

0 Ly 0

0 0 Lz

D ~8!

and definingl and L through

l 5~detL !~1/3!, ~9!

L5 l L , ~10!

one defines dimensionless quantities

x5Lx5 l L x, ~11!

“5L21“5~ l L !21

“. ~12!

Definek and K through

K5S kxx kxy kxz

kxy kyy kyz

kxz kyz kzz

D , ~13!

k5~detK !~1/3!, ~14!

K5kK . ~15!

The time scale is normalized as

t5LxLyLzf t

Q~16!

using the volumetric flow rateQ. Time is measured in unitsof injected pore volumes. The pressure is normalized usthe macroscopic equilibrium capillary pressure as@44,37#

P5PbP, ~17!

where

Pb5Pc@~Swi2Sor11!/2# ~18!

deatl

,atssa-

yo-

us

s.

g

is the pressure at an intermediate saturation.Pc(Sw) is theequilibrium capillary pressure function.

With these definitions the dimensionless macroscocapillary numbers for oil and water, defined as

Caw5mwQ

Pbkl, ~19!

Cao5Caw

mo

mw

, ~20!

give an expression of the balance between macroscopiccous and capillary forces. The macroscopic gravity numb

Grw5mwQ

rwgkl2, ~21!

Gro5Grwmo

mw

rw

ro

, ~22!

express the viscous to gravity force balance. Finallygravillary numbers

Glw5rwgl

Pb, ~23!

Glo5Glwro

rw

, ~24!

express the ratio between gravitational and capillary forcThe well known bond number, measuring the magnitudebuoyancy forces, is given as

Bo5Glw2Glo ~25!

in terms of the gravillary numbers.With these definitions the dimensionless equations of m

tion may be rewritten as

]Sw

] t5“•$Akw

r @Caw21

“ Pw2Grw21g#%, ~26!

2]Sw

] t5“•$Ako

r@Cao21

“~ Pw1 Pc!2Gro21g#%, ~27!

where the dimensionless matrix

A5L21K L215S l 2kxx

Lx2k

l 2kxy

LxLyk

l 2kxz

LxLzk

l 2kxy

LxLyk

l 2kyy

Ly2k

l 2kyz

LyLzk

l 2kxz

LxLzk

l 2kyz

LyLzk

l 2kzz

Lz2k

D ~28!

contains generalized ‘‘aspect ratios.’’ The vector

gT5~ LOe!T5S 2Lx

lsinay,0,2

Lz

lcosayD ~29!

using the

6822 PRE 59L. ANTON AND R. HILFER

TABLE I. The sample and flow parameters used in the calculations. The permeabilities for the glass bead packs were obtainedrelation between their radius and permeabilities published in@25# on p. 47. The capillary pressure was estimated using the Leverett-j-functioncorrelation~30!. For sandstone thej function from Ref.@51# was used and for bead packs the one from Ref.@50# was used. Thej functionwas evaluated close to (Swi2Sor11)/2 according to Eq.~18!.

Permeability Porosity Surface tension Cap. pressureSample k (10212 m2) f sow ~N/m! Pb ~Pa!

sandstone 799~Ref. @32#! 0.14 ~Ref. @32#! 0.28 ~Ref. @32#! 3.3731022 ~Ref @32#! 1.73104 ~Ref. @51#!

70 mm bead pack~Ref. @36#! 10 ~Ref. @25#! 0.36 ~Refs.@25,53#! 2.831023 ~Ref. @36#! 2.43102 ~Ref. @50#!

115 mm bead pack~Ref. @36#! 30 ~Ref. @25#! 0.36 ~Refs.@25,53#! 1.1831022 ~Ref. @36#! 5.83102 ~Ref. @50#!

f t

ion

af

e-te

isn

aa

roteetatwi

-

tattio

esb

-re

m

urntaThhiu

inle

ointve

the

the

ical

a

a-w-the

pli-f theeualoseeedith

pic

represents the effect of dip angle and geometric shape osystem on the gravitational driving force.

V. APPLICATION TO EXPERIMENT

We are now in a position to plot the capillary desaturatcurves against the macroscopic capillary numberCa, whichrepresents the balance between macroscopic viscouscapillary forces. To do so we have searched the literaturecapillary desaturation measurements and we found R@22,25,29–31,45–48,35#. Unfortunately none of the publications contains all the necessary flow and medium parameto calculateCa. Measuring all the flow parameters for a dplacement process is costly and time consuming, and hethey are rarely available~see also@49#!. While permeability,porosity, and the fluid parameters such as viscositiessurface tensions are usually available capillary pressure drelative permeabilities and residual saturations are nottinely measured. Hence we extract the required paramefrom different publications assuming that they have bemeasured correctly and are reproducible anywhere and atimes. In spite of all the uncertainties, it is well known ththe capillary pressure curves of unconsolidated sandsvarious grain sizes can be collapsed using the Leverett-j cor-relation @50,41,51#. The Leverett-j correlation states essentially that

Pc~Sw!5sowAf/k j~Sw!. ~30!

Often the formula contains in addition an average conangle at a three-phase contact. We do not include the weangle as it is generally unknown, and including it would nchange our results significantly. Similar to the capillary prsure data, theSor data for unconsolidated sands seem towell established@34,36# in spite of larger fluctuations of theresults. Because mostPc and Sor data are available for unconsolidated sand and standard sandstones such as BeFontainebleau, we limit our analysis to these two cases.

The experimentalSor data analyzed here are taken froRef. @32# for sandstones and from Ref.@36# for unconsoli-dated glass beads. In Fig. 1 we show the capillary desattion curve for oil-water displacement in a typical sandsto~sample No. 799 from@32#! using star symbols. These dawere obtained in the continuous mode of displacement.values of the surface tension and fluid viscosities for texperiment are given in Table I. We also show continuo

he

ndorfs.

rs-ce

ndta,u-rsnall

th

ctngt-e

a or

a-e

ess

modeSor data for unconsolidated glass beads from Fig. 636. To calculateCa we have used the values given in TabI.

Plotting the data againstCa, we obtain Fig. 2. It is seenthat the continuous mode displacements give a breakpfor Ca'1 while the discontinuous mode displacements hatheir breakpoint at a higher value. This is consistent withidea that the traditional equations of motion~1! and ~2!should be applicable to the continuous mode but not todiscontinuous case.

The result obtained here is consistent with the theoretpredictions from@37#. To fully validate our use ofCa as acorrelating group for plotting continuous modeSor data,however, it would be desirable to varyCa by varying thesystem sizel. If the Sor curves obtained for different mediand length scalesl also show their breakpoint atCa'1, thiswould give further evidence for the applicability of the trditional equations of motion. We consider it possible, hoever, that deviations appear indicating a breakdown ofequations also for continuous mode displacement@23,24#.

As stated above, the equations of motion are not apcable to discontinuous mode displacements because oconstraintSwi,Sw,12Sor . Nevertheless we can use thgroup Ca to estimate an upper bound for the size of residblobs. As the flow rate is increased, the residual blobs, whsize is so large that the viscous drag forces on them excthe capillary retention forces, will break up and coalesce w

FIG. 2. Same as Fig. 1 but plotted against the macroscocapillary numberCa5mwQ/(Pbkl) from Eq. ~19!. Note that thebreakpoint for continuous mode displacement occurs aroundCa'1.

ths

iz

t.

sge-

PRE 59 6823TRAPPING AND MOBILIZATION OF RESIDUAL FLUID . . .

other blobs which may again break up and coalesce furdownstream@34,33#. The condition that the viscous forcedominate the capillary forcesCa>1 predicts that after a floodwith Ca the porous medium contains only blobs of linear ssmaller than

l blob<mwQ

Pbk. ~31!

.

tt.

d

s

al

he

er

e

This result is of importance for microscopic models@52# ofbreakup and coalescence during immiscible displacemen

ACKNOWLEDGMENTS

One of us~R.H.! thanks Dr. P. E. O” ren for many usefuldiscussions. We are grateful to the Deutsche Forschungmeinschaft for financial support.

oir

ia

e

-

il

il

-

@1# M. Blunt and P. King, Phys. Rev. A42, 4780~1990!.@2# U. Oxaal, Phys. Rev. A44, 5038~1991!.@3# U. Oxaal, F. Boger, J. Feder, T. Jo”ssang, P. Meakin, and A

Aharony, Phys. Rev. A44, 6564~1991!.@4# R. Hilfer, Phys. Rev. B44, 60 ~1991!.@5# R. Hilfer, Phys. Rev. B45, 7115~1992!.@6# R. Hilfer, Phys. Scr.T44, 51 ~1992!.@7# M. Blunt, M. King, and H. Scher, Phys. Rev. A46, 7680

~1992!.@8# V. Frette, J. Feder, T. Jo”ssang, and P. Meakin, Phys. Rev. Le

68, 3164~1992!.@9# P. Meakin, J. Feder, V. Frette, and T. Jo”ssang, Phys. Rev. A

46, 3357~1992!.@10# M. Kataja, K. Hiltunen, and J. Timonen, J. Phys. D25, 1053

~1992!.@11# B. No”st, B. Hansen, and E. Haslund, Phys. Scr.T44, 67

~1992!.@12# M. Sahimi, Rev. Mod. Phys.65, 1393~1993!.@13# M. Blunt and H. Scher, Phys. Rev. E52, 6387~1995!.@14# V. Horvat and H. Stanley, Phys. Rev. E52, 5166~1995!.@15# S. Schwarzer, Phys. Rev. E52, 6461~1995!.@16# L. Furuberg, K. Maloy, and J. Feder, Phys. Rev. E53, 966

~1996!.@17# B. Virgin, E. Haslund, and R. Hilfer, Physica A232, 1 ~1996!.@18# H. Kytomaa, M. Kataja, and J. Timonen, J. Appl. Phys.81,

7148 ~1997!.@19# B. Berkowitz and H. Scher, Phys. Rev. Lett.79, 4038~1997!.@20# J. Andrade, M. Almeida, J. M. Filho, S. Havlin, B. Suki, an

H. Stanley, Phys. Rev. Lett.79, 3901~1997!.@21# R. Hilfer, Adv. Chem. Phys.XCII , 299 ~1996!.@22# G. Willhite, Waterflooding, Vol. 3 of SPE Textbook Serie

~Society of Petroleum Engineers, Richardson, TX, 1986!.@23# R. Hilfer, Phys. Rev. E58, 2090~1998!.@24# H. Besserer and R. Hilfer~unpublished!.@25# L. Lake, Enhanced Oil Recovery~Prentice Hall, Englewood

Cliffs, 1989!.@26# P. O” ren, J. Billiotte, and W. Pinczewski, SPE Formation Ev

March , 70 ~1992!.@27# J. Salles, J. Thovert, and P. Adler, J. Contam. Hydrol.13, 3

~1993!.@28# R. Helmig, Multiphase Flow and Transport Processes in t

.

Subsurface~Springer, Berlin, 1997!.@29# H. Dombrowski and L. Brownell, Ind. Eng. Chem.46, 1207

~1954!.@30# J. Taber, SPE J.9, 3 ~1969!.@31# J. Taber, J. Kirby, and F. Schroeder, AIChE Symp. Ser.69, 53

~1973!.@32# A. Abrams, SPE J.15, 437 ~1975!.@33# I. Chatzis, M. Kuntamukkula, and N. Morrow, SPE Reserv

Eng.August, 902 ~1988!.@34# N. Wardlaw and M. McKellar, Can. J. Chem. Eng.63, 525

~1985!.@35# R. Larson, H. Davis, and L. Scriven, Chem. Eng. Sci.36, 75

~1981!.@36# N. Morrow, I. Chatzis, and J. Taber, SPE Reservoir Eng.Au-

gust, 927 ~1988!.@37# R. Hilfer and P. O” ren, Transp. Porous Media22, 53 ~1996!.@38# A. Scheidegger,The Physics of Flow Through Porous Med

~University of Toronto Press, Toronto, 1974!.@39# F. Dullien, Porous Media-Fluid Transport and Pore Structur

~Academic, San Diego, 1992!.@40# M. Sahimi, Flow and Transport in Porous Media and Frac

tured Rock~VCH Verlagsgesellschaft mbH, Weinheim, 1995!.@41# J. Bear,Dynamics of Fluids in Porous Media~Elsevier, New

York, 1972!.@42# P. deGennes, Europhys. Lett.5, 689 ~1988!.@43# F. Kalaydijan, SPE Proc.67, 491 ~1992!.@44# R. Hilfer and P. O” ren, Tech. Report F&U-LoU-94001, Stato

A/S, Trondheim~1993!.@45# E. Donaldson, R. Thomas, and P. Lorenz, SPE J.9, 13 ~1969!.@46# R. Ehrlich, H. Hasiba, and P. Raimondi, J. Petrol.26, 1335

~1974!.@47# E. L. duPrey, SPE J.13, 39 ~1973!.@48# S. Gupta and S. Trushenski, SPE J.19, 116 ~1979!.@49# R. Hilfer and P. O” ren, Tech. Report F&U-LoU-96015, Stato

A/S, Trondheim~1993!.@50# M. Leverett, W. Lewis, and M. True, Trans. AIME146, 175

~1942!.@51# J. Dumore and R. Schols, SPE J.26, 437 ~1974!.@52# L. Anton and R. Hilfer~unpublished first version of the manu

script!.@53# P. Wong, J. Koplik, and J. Tomanic, Phys. Rev. B30, 6606

~1984!.