Capillary rise in porous media

Journal of Colloid and Interface Science234,35–43 (2001)doi:10.1006/jcis.2000.7241, available online at http://www.idealibrary.com on

Capillary Rise in Porous Media

Marcelo Lago and Mariela Araujo

Reservoir Department, PDVSA Intevep S.A., Caracas, Venezuela

Received January 14, 2000; accepted September 28, 2000

Capillary rise experiments were performed in columns filled withglass beads and Berea sandstones, using visual methods to registerthe advance of the water front. For the glass bead filled columns,early time data are well fitted by the Washburn equation. However,in the experiments, the advancing front exceeded the predicted equi-librium height. For large times, an algebraic behavior of the velocityof the front is observed (T. Delker et al., Phys. Rev. Lett. 76, 2902(1996)). A model for studying the capillary pressure evolution in aregular assembly of spheres is proposed and developed. It is basedon a quasi-static advance of the meniscus with a piston-like motionand allows us to estimate the hydraulic equilibrium height, withvalues very close to those obtained by fitting early time data to aWashburn equation. The change of regime is explained as a tran-sition in the mechanism of advance of the meniscus. On the otherhand, only the Washburn regime was observed for the sandstones.The front velocity was fitted to an algebraical form with an expo-nent close to 0.5, a value expected from the asymptotic limit of theWashburn equation. C© 2001 Academic Press

Key Words: porous medium; capillary pressure; Washburn equa-tion; capillary rise.

aTmsg

nre

r-sit

spielea

thetedhe-

l weom-ion.tedii in

de-tionrted

ion 2ilib-ultsles,s, are

uidry[5,

re-t theis-terg the

stheoir

the

INTRODUCTION

Capillary rise in porous media has been studied for myears, since the pioneering work of Washburn in 1921 [1].interest in these phenomena derives from its relevance toareas such as packing of particulates [2–12], consolidatedples [13–17], and fibrous materials [18–20]. It is interestinnote that some features of the fluid rise cannot be describedthe Washburn equation for large enough times, as showHackett in his early work [4]. This has motivated recent intein the study of capillary rise in porous media.

Recently, Delkeret al. [5] showed that the fitting of expeimental capillary rise data of water in glass bead columngood only for short times. An algebraical expression descrthe behavior of the front’s speed for larger times, suggesthat the height does not have an equilibrium value as expefrom the Washburn formalism. This peculiar behavior was aciated with a pinning effect, the advancing front being stopbefore reaching the critical height of hydraulic equilibrium, wa velocity in the new regime described by a critical exponcharacteristic of the pinning–depinning transition. Neverthethis explanation is not consistent with the fact that the adv

35

nyheanyam-to

withbyst

isbesingctedso-ed

thnt,ss,nc-

ing front rises beyond the equilibrium height predicted fromWashburn equation fitted for early times. The work presenhere is a contribution to the understanding of capillary rise pnomena in porous media. By using a simple capillary modeare able to calculate the expected equilibrium height, which cpares favorably with that predicted by the Washburn equatThe deviation from this equation for large times is associawith a change in the mechanism of advance of the meniscthe porous sample.

In Section 1, the Washburn equation in porous media isrived and the conditions for its validity are stated. The equais also extended for application to a porous sample suppoby a plug and a velocity-dependent capillary pressure. Sectpresents a simple model used to predict the hydraulic equrium height, followed by Section 3 where the details and resfrom experiments performed in two types of porous sampglass columns filled with glass beads and Berea sandstonegiven. Finally, we end with a discussion and conclusions.

1. THEORETICAL FORMALISM

The Washburn equation (1) was derived originally for a liqrising in a cylindrical capillary tube by the effects of capillaforces. The derivation for capillary rise in a porous sample15] such as the one represented in Fig. 1 goes as follows.

Assuming that Darcy’s law is valid in the two monophasicgions, a macroscopic capillary pressure could be defined ainterfacePc. Fluids are considered incompressible and immcible, and it is assumed that the contact of fluid 1 with the wareservoir does not produce a noticeable pressure drop. Usinnotation shown in Fig. 1, the velocity of the frontv is given by

v = dh

dt= k

φ

Pc− (ρ1− ρ2)gh

µ1(h+ zR)+ µ2(L − H − zR), [1]

wherek is the permeability andφ the porosity of the porousample,ρi andµi denote the densities and viscosities offluids, andzR is the portion of the sample below the reservlevel. Capillary rise occurs whenPc > 0 andρ1− ρ2 > 0.

Assuming that the capillary pressure is constant duringrising process, the hydraulic equilibrium heighthe is calculatedfrom the condition

Pc = (ρ1− ρ2)ghe. [2]

0021-9797/01 $35.00Copyright C© 2001 by Academic Press

All rights of reproduction in any form reserved.

lt

t

ffe

r

o

m

ht

, is itrialds.a. Ifes,r a

[22]

es

rss,heretedthe

rom

hen

o

36 LAGO AND

FIG. 1. Schematic diagram of a capillary rise experiment in a porous sam

Another relevant parameter related to the microscopic veity of fluid 1 advancing through the porous structure only byeffect of gravitational forces is given by

vg = k

φ

(ρ1− ρ2)g

µ1. [3]

From Eqs. [1] to [3] the front velocity can be written as

v = vg(he− h)

h+ zR+ µ2

µ1(L − zR− h)

. [4]

For capillary rise, for example when water displaces air,term with the viscosity ratio can be neglected, thus Eq. [4] gsimplified to

v = vghe− h

h+ zR. [5]

This relation gives a straight line in a plot of the velocity asfunction of the inverse height as measured from the base oporous sample. This result has been used by Hackett in Reto get the values ofvg andhe+ zR from experimental data. ThWashburn equation is obtained after the integration of Eq.with the initial conditionh = 0 att = 0,

t = he+ zR

vgln

(he

he− h

)− h

vg. [6]

For small values ofh andzR¿ he the asymptotic behavioof Eq. [6] gives

t = h2

2hevg. [7]

A similar behavior is obtained when the imbibition processcurs in a cylindrical capillary [1] or in a porous sample, bo

ARAUJO

ple.

oc-he

heets

athe

. (4)

[5]

c-th

empirical behavior during capillary rise [5, 20, 21] derived fro

v = v1

(t

t1

)−b

, [8]

with b < 1. Assuming that at timet1 the front is at positionh1

with a velocityv1 , and integrating, the expression for the heigbecomes

h = h1+ v1t11− b

((t

t1

)1−b

− 1

). [9]

On the other hand, when a glass bead pack is preparedquite often necessary to use a piece of low-permeability mateas a plug in its lower portion to avoid the loosening of the beaThis plugging piece has a permeabilitykp and cross section areSp, which are generally different from those of the sampleDarcy’s equation is used in the region where this plug residit can be shown that the previous Eqs. [4] to [6], obtained fohomogeneous system, hold in terms of effective quantities

z∗R = zR+ L∗p − Lp, [10]

L∗ = L + L∗p, [11]

whereLp is the plug’s length. The effective plug length satisfi

L∗p = LpSk

Spkp. [12]

The initial estimation ofL∗p, a parameter which accounts fothe influence of the plugging piece on the imbibition procecan be done independently. If a displacement process, wfluid 1 flows by the effect of gravity, is performed as represenin Fig. 2, we get from Darcy’s law inside the bead pack andplugging piece

q1

φSL1= vg

L + L∗T, [13]

with L1 the total length of the region filled with fluid 1 andq1 themeasured volumetric flux. The rhs of (13) can be estimated fexperience, and it is equal to the inverse of the timeτ neededfor the fluid to move across the bead pack and the plug. WL = 0, i.e., for the case of the plug only, parameterτ has a valueof τp which can be easily found. Using Eq. [13] for these twcases it is easily shown that

L∗p =τp

τ − τpL , [14]

vg =L∗pτp= L + L∗p

τ. [15]

ofuce

As another extension to the traditional Washburn equation,we may consider the velocity dependence of the macroscopic

IN

l

y

aene

r2

ing,

r

[2],i in-, it isme-cii.on-]. Itith ae is

di-

om-, itn-

ingn be

CAPILLARY RISE

FIG. 2. Schematic diagram of an independent experimental setup with gbead columns for the estimation of Washburn’s parameters.

capillary pressure as obtained experimentally for sintered gbeads (21a),

Pc(v) = Pc(0)

[1− 300

(µφv

γ

)0.5], [15a]

whereγ is the interfacial tension, andµ an average viscositfor the two fluids and the static capillary pressurePc(0) whichsatisfies Eq. [2].

Expression [15a] is similar to the one obtained for capilltubes, except for the numerical factor 300, which in that casreplaced by 4 [23]. For capillary tubes, this velocity dependeis associated with a variation in the advancing contact anglglass bead packing, the change inPc is two orders of magni-tude larger than in capillary tubes, suggesting that the disocharacteristics of natural porous media plays a major role [Substituting Eqs. [15a] and [3] in [1] and solving forv gives

v

vg= dy

dx=√

a23 + 4(a1y+ a2)(1− y)− a3

2(a1y+ a2)

2

, [15b]

with

y = h+ zR

he+ zR, x = tvg

he+ zR, a1 = 1− µ2

µ1, a2

µ2

µ1

L

he+ zR,

and a3 = he

he+ zR300

(µφvg

γ

)0.5

.

Note that Eq. [15b] reduces to [4] ifa3 = 0.

integrated numerically. The velocity dependence of the mac

POROUS MEDIA 37

lass

ass

ryis

ce. In

der3].

scopic capillary pressure slows down the rise at the beginnbut does not change the final equilibrium heighthe. Numericaldata can be fitted by Eq. [6] using a smaller value of parametevg.

2. SIMPLE CAPILLARY MODEL

The macroscopic capillary pressure, which appears in Eq.is related to the capillary pressure that exists in the menisciside the pores of the sample. In a random packing of spheresdifficult to estimate such a parameter due to the complex geotry of the pore system and the different positions of the menisIn the analysis that follows, a regular packing of spheres is csidered. Their centers make a rhombohedral structure [24is assumed that the meniscus advances quasi-statically wpiston-like motion inside the pores and that the contact lincontained in a horizontal plane.

For a point of contact of a meniscus with a sphere as incated in Fig. 3, the differential forced f that acts vertically on asegmentdp of the contact line satisfies

d f

dp= γ cosα = γ sin(φ + α). [16]

For any array of spheres the meniscii shapes can be very cplex. However, in a cavity such as the one shown in Fig. 4is expected that relation (16) holds for any point of the cotact line as does in a toroidal ring [25]. The total force acton the meniscus limited by the region inside the square cacalculated from the contact line perimeter, giving

f = 2πRγ sinφ sin(φ + θ ). [17]

isro-

FIG. 3. Geometrical representation of the contact point of a meniscus witha sphere of diameterR.

A

o

e

eca

t

nt

an

ath

oint

√

−

38 LAGO AND

FIG. 4. Upper view of a horizontal cut of the regular pack of spheresradiusR at the contact line level, according to the model.

Equation [17] is similar to the relation reported for an isolatsphere [26]. On the other hand, the projected area (A) of themeniscus inside that region is

A =(

4

π sin2 φ− 1

)πR2 sin2 φ. [18]

The capillary pressurePc, which in this case is representativof the whole interface, is found as the ratio of the total vertiforce and the projected area of the meniscus,

Pc = 24

π sin2 φ− 1

sin(φ + θ )

sinφ

γ

R. [19]

Figure 5 shows how the capillary pressure changes withheight of the contact point of the meniscushc, given by

hc = R(1− cosφ). [20]

Mason and Morrow calculated similar curves for various cofigurations [26]. However, for the bead pack used here, inot possible to reach the valueshc = 0 andhc = 2R, since themeniscus position is limited by the spheres located abovebelow it, as indicated in Fig. 6. Using only the points of cotact with the spheres it can be shown thatφ is limited by thevaluesφmin = 0.25π andφmax= 0.75π . If the meniscus cur-vature is taken into account, its center could reach the topbottom spheres before the contact line, limiting even morerange of allowedφ values in dependence withθ . These new

RAUJO

f

d

l

he

-is

nd-

nde

FIG. 5. Capillary pressure as a function of the height of the contact pfor various contact anglesθ .

relations could be written [22]:

√2− sinφ − r

Rsin(φ + θ ) = 0; [21]

2− 1− cosφ − r

R(1+ cos(φ + θ )) = 0, with φ = φmin;

[22]√

2+1− cosφ − r

R(1+ cos(φ + θ )) = 0, with φ = φmax;

[23]

2−√

2= (cosθ − 1+√

2)r

R, with φ = φmin; [24]

2−√

2= (cosθ + 1−√

2)r

R, with φ = φmax. [25]

ng FIG. 6. Lateral view of a section of the regular packing of spheres.

IN

e4

o

ioa

ern isd inen

theead

ita

thecan

esultthe

ghtases,try

usnd

9a.ithowerent,theted. Inentaters to

CAPILLARY RISE

FIG. 7. Limit values ofφ angles as a function of the contact angleθ .

r is the radius of curvature of the meniscus and is considnegative (r < 0) when the concavity is downward. Equation [2may be combined with Eq. [21] or [22] in order to find, forgivenθ , the values ofφmin, and analogously forφmax. Numericallimit values of theφ allowed range are shown in Fig. 7.Pc valuesfor the limit values ofφ can be obtained using Eq. [19] and aplotted as a function of the contact angle in Fig. 8.

For the most favorable case of imbibition, which shouldcur whenθ = 0, Pc has its maximum value athc = R, whichcorresponds toφ = π/2. Thus whenθ = 0, the minimum valueof the capillary pressure is found just before touching the upsphere, in the position given byφmax= 0.75π . Using Eq. [19],

Pminc =

(2

8π− 1

)γ

R∼= 1.293

γ

R. [26]

During the rising process through several layers of sphethe capillary pressure will remain in the range of values shoin Fig. 5, between the limits set by Fig. 7, with an exceptfor the case when the displacement front goes from one lto another, near the contact points between the spheres,

FIG. 8. Limit values of capillary pressurePc as a function of the contactangleθ .

POROUS MEDIA 39

red]

a

re

c-

per

res,wnnyerwith

values that are significantly larger. The front will stop in a laywhere, for the first time, the pressure due to the liquid columlarger than the minimum value of the capillary pressure foun(26). An expression for the equilibrium capillary height, whheÀ R, can be obtained from Eq. [2] in the form

Pminc ∼= (ρw − ρa)ghe. [27]

Substitution of Eq. [27] on [26] gives a relation betweenequilibrium height and the diameter of the spheres in the bpack. For perfect wettingθ = 0 we get

he∼= 2.587γ

(ρw − ρa)γd. [28]

It is worth noting that if Eq. [19] is carefully analyzed,will be found that, whenφ = φmax, the capillary pressure hasmaximum aroundθ ≈ 0.0854π ∼= 15.4◦, a commonly observedcontact angle for the glass–water–air system [27]. Followingsame arguments, it is found that the equilibrium height thatbe reached with this angle is

hmaxe ∼=

2.818γ

(ρw − ρa)γd, [29]

which is 9% larger than the one estimated in Eq. [28] forθ = 0.Therefore, we have reached the apparently contradictory rthat with the smallest contact angle it is not possible to getlargest equilibrium height. The reason the equilibrium heidoes not decrease monotonically as the contact angle increas in a cylindrical capillary, is related to the particular geomeof the pore space in the regular sphere packing.

3. MATERIALS AND METHODS

Experiments were performed in two different types of poromedia, initially dry: glass columns filled with glass beads aBerea sandstones.

Glass Bead Packings

The experimental setup is drawn schematically in Fig.A glass pyrex tube of 10 mm internal diameter is filled wglass beads. It is assembled vertically and connected by its lextreme to a water reservoir. At the beginning of the experimthe porous sample is dry and water is allowed to rise byopening of a control valve. The height of the front is estimavisually as a function of time, with the aid of a video cameraorder to avoid fluid loses by evaporation during the experim[5, 8], a plastic hose is connected to the upper part of the wreservoir tank. This hose contains water drops on its wallguarantee that the air inside it remains water saturated.

Glass beads spheres withρ = 2.46± 0.02 g/cm3, previously

D

illb

cn

t

ol-he

sidew atmpo or

ing,

. 9b.y,n amns,

sed..6%

Forace-the

d onater

astic

thekedeadyighttime.areenceites

outd byangenc-tedthepro-

entofpped

40 LAGO AN

FIG. 9. Schematic diagram of the experimental setup used in the caprise study with glass columns filled with glass beads (a) and sandstones (

thickness and average pore size less than 40µm was attachedto the bottom of the glass column. The beads were subjeto a rigorous cleaning process that consists of immersioconcentrated HNO3 at 70% for 4 days. Parameterτp, whichaccounts for the effect of the plug over permeability, is estimaafter the cleaning process and has a value around 44.0± 3 s.

The bead packing process is initiated by pouring, withoutterruption, glass beads into a glass column filled with wa

ARAUJO

ary).

tedin

ted

in-er.

compaction of the beads, with a final porosity of 37.4± 0.4%.At this time, water is permitted to flow through the packed cumn, to determineτ from Eq. [13] and therefore to estimate tparameters of the Washburn equation,z∗R andvg. The drying ofthe bead pack is performed by placing the glass column inan oven for about 20 min in a sequence of steps with air floroom temperature. The air flow is applied with a vacuum puattached to its lower extreme. This process is repeated twthree times, beginning with an oven temperature of 80◦C to avoidrapid evaporation, which may produce damage to the packand then arising the temperature to 120◦C.

Rock Samples

The experimental setup for this case is represented in FigA piece of rock sample with a cylindrical shape, initially dris placed vertically and partially immersed (about 0.5 cm) ivessel containing water. Similar to the case of the glass colua video camera was used to register the front advance.

Two Berea sandstones of similar characteristics were uThe samples had a diameter of 3.8 cm, 38-cm length, 21porosity, and permeability of 1.0× 10−8 cm2. They were cov-ered with a shrinkable thermoelastic transparent material.cleaning, distilled water was used through a series of displments until the effluent was clear (after 1 or 2 liter). Fordrying process an oven at a temperature of 100◦C was used. Thecriterion used to decide whether the rock was dry was basethe fact that when the sample cooled down, there was not wcondensation on the internal side of the shrinkable thermoelmaterial.

4. RESULTS AND DISCUSSIONS

Figure 10 shows the typical behavior of the height andvelocity of the front during a capillary rise process in the pacglass columns. This representation in log–log scale was alrused by Delker and collaborators [5]. It is seen that the he(measured at 2-mm intervals) has a soft dependence withHowever, small jumps with amplitude smaller than 1 mmobserved in the experiments. This is evidence for the existof sites of minimum capillary pressure that become pinning sfor the interface [5, 8].

During most of the rising process, the front advanced withleaving a significant amount of air trapped, as was detectevisual observation or captured by the video camera. The chof color in the area of the front was abrupt, reflecting a step fution in water saturation. Only in a few experiments that laslong times was air trapping noticeable. As the front rose,amount of trapped air increased until the water saturationfile around the front became blurred, making the measuremof the actual front position difficult. A systematic analysisthis effect was not made since most experiments were stobefore this occurred.

The first part of the imbibition process was satisfactorily fittedto a Washburn type of equation, such as Eq. [6], from which

IN

in

ew

m

.

a

otio

i

o

ion.gerus’s

thedom-e-re-

d thetionlid.

hringh is

lity

d by

burn

CAPILLARY RISE

FIG. 10. Time evolution of the height (a) and velocity (b) of the frontbead packs.

parametershe andvg were estimated. The last part of the ascwas fitted the algebraical form given in Eq. [9]. Figure 10 shosome of the fits performed on the data. The excellent agreewith the measured data is quite evident. The transition regbetween the two regimes was in the time range of 1× 103 to2× 103 s when the front’s velocity was about 10−2 mm/s.

Figure 11 displays the behavior of thehe fitted parameterIt is clear thathe decreases as the diameter of the spherecreases, similar to what is observed with the value estimfrom Eqs. [28] and [29] for contact angles of 0 and 15.4◦, respec-tively. The agreement between the calculated and the estim(with the Washburn equation) equilibrium heights is quite godespite the simplicity of the model. In a natural bead packspace between the spheres changes in shape and size, makadvance of the meniscus a more complex process. On thehand, the porosity of the bead pack used in the experimentsaround 35%, whereas for the regular packing such as theconsidered in the model it is 26%; this difference should grise to an overestimation of the equilibrium height.

The proximity between the measured and fitted values sh

POROUS MEDIA 41

ntsent

ion

in-ted

atedd,heng thetherwasoneve

wn

anism for the advance of the meniscus is a piston-like motTherefore, the change in the behavior of the height for lartimes suggests a change in the mechanism of the meniscmotion from piston-like to a more slow process, such asones studied by Lenormand [28]. These processes becomeinant at very low velocities and account for the fluid displacment. The change in the mechanism of advance should beflected in a change in the macroscopic capillary pressure, anWashburn equation, which was obtained under the assumpthat the capillary pressure term was constant, is not longer va

On the other hand, thevg fitted parameter had a value whicwas always smaller than the one estimated independently duthe packing preparation, as is shown in Fig. 11a. In this grapalso plotted the dependence ofvg obtained from substituting inEq. [3] the Carman–Ergun correlation (C–E) for the permeabi[29], with the form

k = 1

180

φ3

(1− φ)2d2. [30]

FIG. 11. (a) Equilibrium heighthe as a function of sphere diameterd.Results obtained from a Washburn fitting are compared with those predicteEqs. [28] and [29]. (b) Parametervg as a function of sphere diameterd. Resultsobtained during the packing procedure (Experimental) and from a Wash

o-ch-fitting are compared with those predicted using the Carman–Ergun correlation(Eq. [30]). It is also shown this last curve corrected by a factor of 0.67.

“-s

n

g.e

it

.7

vva

eaeeginin

i-aosn

is

ing

ime

aneThe

42 LAGO AND

The correlation of Eq. [30] gives values ofvg very close tothose measured during the packing preparation and labeledperimental.” Hence, the fitted value ofvg in the Washburn equation should be an underestimate of its real value. These remight be an effect of not including the velocity dependencePc such (as given by Eq. [15a], for example) as in the traditioWashburn equation.

Using Eq. [16b], a representative curve of the front heievolution was generated and was fitted with the Washburn Eqwith the same procedure. The apparent parameters obtainthis case were 0.94∗ he and 0.67∗ vg. In Fig. 11a we also plottedthe Carman–Ergun curve, corrected by a factor of 0.67, whis closer to the fitted values ofvg. It should be mentioned thathe fitted value ofvg is very sensitive to variations ina3 andz∗R.

For the last part of the rising process, the values of exponb, obtained by fitting to a potential form, are in the range of 0and 0.95 and are not enough to demonstrate the existencedependence with the diameter of the spheres. A similar behafor the front height evolution as shown in Fig. 10a was obserfrom the data reported by Hackett [4] for imbibition of oil insand pack. The fitted data are presented in Fig. 12.

For the rock samples the results of the capillary rise expments are shown in Figs. 13a and 13b for the front heightits velocity, respectively. In this case, the clear visible changthe rising curve observed in the packed columns was not sA fitting of all the data points to the Washburn Eq. [6] sugests that this equation describes these points correspondthe first part of the process, where it is difficult to estimate,dependently, the value of parametershe andvg, but with theirproduct remaining aroundhevg ≈ 3.2± 0.2 mm2/s. Using thepermeability and porosity values measured independently,found thatvg = 5.1× 10−2 mm/s, which gives a “contradictory” value of he = 6.3 cm. It can be seen from Fig. 13a ththe front overpasses this calculated equilibrium height withany visible change in the front height evolution. This suggethat whether the equilibrium height of the Washburn equatio

FIG. 12. Experimental results reported in Ref. [7] during the capillary rof oil in a sand pack with an average grain size of 168µm. The fitting to the

ARAUJO

Ex-

ultsofal

ht[6]d in

ch

ent4of aior

ed

ri-ndinen.-g to-

t is

tuttsis

e

FIG. 13. Time evolution of the height (a) and velocity (b) of the advancfront in Berea sandstones.

higher than the estimated one or not, there is not a visible regchange during the capillary rise in the used rocks.

The algebraical Eq. [9] was also fitted to the data withexponentb = 0.57± 0.01, a value close to 0.5; this is thnumber expected from the asymptotic Washburn equation.

esFIG. 14. Time evolution of the square of the height for Berea samples.

IN

tt

sw

ec

c

aqnodr

o

n

J.,

CAPILLARY RISE

correspondent fitting to the asymptotic Eq. [7] was done onlyheights smaller than 10 cm, leading tohevg ≈ 3.1± 0.1 mm2/s.

In the case of rock samples, it is quite common to presenresults by using plots of the square of the height as a funcof time, as done in Fig. 14. In this plot it can be seen thatevolution of the square of the height has some curvature,gesting that the asymptotic equation could not reproducegood precision the entire imbibition process.

5. SUMMARY

The results presented here for glass columns filled with bindicate a transition from a piston-like advance of the menisto a slower mechanism in the capillary rise phenomena. Thedraulic equilibrium height can be estimated by using a simgeometrical model with values consistent with those prediby the Washburn equation. Also, an excellent correlationtween the estimated equilibrium height and the average spdiameter was found. Regarding the Washburn parametervg, it ispossible that the velocity dependence of the macroscopic clary pressure, not considered in the traditional Washburn etion, gives the fitted parameter which is an underestimatioits real value. The behavior observed for the water front for ltimes was fitted to an algebraical form; however, in order totermine the specific mechanism of meniscus advance aftetransition, more experiments are needed.

For the case of sandstones, an algebraical form with anponent close to 0.5 describes the behavior observed in therange studied. This value is expected from the asymptotic fof the Washburn equation.

ACKNOWLEDGMENTS

The authors thank A. Rodr´ıguez, R. Paredes, V. Alvarado, Y. Araujo, a

POROUS MEDIA 43

for

theiontheug-ith

adsushy-pletedbe-here

pil-ua-of

nge-the

ex-timerm

d

REFERENCES

1. Washburn, E.,Phys. Rev.17(3), 273 (1921).2. Ayala, R., Casassa E., and Parfitt, G.,Powder Technol.51,3 (1987).3. Siebold, A., Walliser, A., Nardin, M., Oppliger, M., and Schultz,

J. Colloid Interface Sci.186,60 (1997).4. Hackett, F.,Trans. Faraday Soc.17,260 (1921).5. Delker, T., Pengra, D., and Wong, P.,Phys. Rev. Lett.76(16), 2902

(1996).6. Lu, T., Biggar, J., and Nielsen, D.,Water Resource Res.30(12), 3275

(1994).7. Diggins, D., Fokkink, L., and Ralston J.,Colloids Surf.44,299 (1990).8. Van Brakel, J., and Heertjes, P. M.,Powder Technol.16,75 (1997).9. Bruil, H., and Van Aartsen, J.,Colloid Polym. Sci.252,32 (1974).

10. Kilau, H., and Pahlman, J.,Colloids Surf.26,217 (1987).11. Yang, Y., Zografi, G., and Miller, E.,J. Colloid Interface Sci.122, 35

(1988).12. Grundke, K., Boerner, M., and Jacobasch, H.,Colloids Surf.58,47 (1991).13. Grattoni, C., Chiotis, E., and Dawe, R.,J. Chem. Tech. Biotechnol.64,17

(1995).14. Einset, E.,J. Am. Ceram. Soc.79(2), 333 (1996).15. Handy, L.,Pet. Trans. AIME219,75 (1960).16. Schembre, J., Akin, S., Castanier, L., and Kovscek, A.,SPE-46211

(1998).17. Babadagli, T., and Ersjaghi, I.,SPE-24044 (1992).18. Pezron, I., Bourgain, G., and Qu´ere, D.,J. Colloid Interface Sci.173,319

(1995).19. Marmur, A., and Cohen, R.,J. Colloid Interface Sci.189,299 (1997).20. Noble, J., and Arnold, A.,Chem. Eng. Commun.100,95 (1991).21. Philip, J. R.,Soil Sci.84,257 (1957).22. Lago, M., Msc. thesis, Universidad Sim´on Bolıvar (1998).23. Weitz, D., Stokes, J., Ball, R., and Kushnick, A.,Phys. Rev. Lett.59,2967

(1987).24. Graton, L., and Fraser, H.,J. Geol.43(8), 785 (1935).25. Lu, T. X., Nielsen, D. R., and Biggar, J. W.,Water Resource Res.31(1), 11

(1995).26. Mason, G., and Morrow, N.,J. Colloid Interface Sci.168,130 (1994).27. Araujo, Y., Toledo, P., Leon, V., and Gonzalez, A.,J. Colloid Interface Sci.

176(2), 485 (1995).28. Lenormand, R., and Zarcone, C.,SPEPaper No.13264, 17 pp. (1984).

he29. Macdonald, I., El-Sayed, M., Mow, K., and Dullien, F.,Ind. Eng. Chem.Fundam.18,199 (1979).