Predicting the Capillary Imbibition of Porous Rocks from Microstructure

Transport in Porous Media 49: 59–76, 2002.© 2002 Kluwer Academic Publishers. Printed in the Netherlands.

59

Predicting the Capillary Imbibition of Porous Rocksfrom Microstructure

DAVID BENAVENTE1,2,�, PETER LOCK3, Ma ÁNGELES GARCÍA DELCURA2,4 and SALVADOR ORDÓÑEZ1,2

1Departamento de Ciencias de la Tierra y el Medio Ambiente, Universidad de Alicante, Ap. 99,03080 Alicante, Spain2Laboratorio de Petrología Aplicada, Unidad Asociada CSIC-UA, Spain3T. H. Huxley School of Environment, Earth Sciences and Engineering, Imperial College of Science,Technology and Medicine, London SW7 2BP, U.K.4Instituto de Geología Económica, CSIC-UCM, Madrid, Spain

(Received: 24 November 2000; in final form: 5 December 2001)

Abstract. The kinetics of capillary imbibition into porous rocks is studied experimentally andtheoretically. The Washburn law is modified by introducing various corrections relating to the mi-crostructure of the rocks, such as tortuosity, pore shape (obtained experimentally), and applying theeffective medium approximation (EMA) in order to calculate the effective radius that defines thehydraulic conductance and the topology of the capillary imbibition. The application of the EMAshows that capillary imbibition is mainly produced in 1-D, and the pore structure is constituted bydifferent pore throats in series, linked by chamber pores. The capillary process has been discussedas a function of their petrography and pore structure. Our study of the Washburn equation and theaddition of correction factors for the pore structure allows a very accurate prediction of the weightrate.

Key words: capillary imbibition, sorptivity, effective medium approximation, pore shape, porousrocks.

1. Introduction

Capillary imbibition is a transport phenomenon in porous rocks, the understandingof which is of extreme importance in many fields such as groundwater engineer-ing, petroleum engineering, soil physics, civil engineers, engineering geology andbuilding materials.

The imbibition kinetics of water in porous media was analysed theoreticallyby Washburn (1921). In his simple model, the porous medium is represented by acollection of parallel tubes, each having the same radius. When the predictions ofthis model are compared to experimental data (Dullien et al., 1977; Hammeckeret al., 1993), the apparent capillary radius is usually found to be several ordersof magnitude smaller than the typical pore diameter in the sample. Hence, thesimple cylindrical model (Washburn, 1921) is not really appropriate for quantifying

�Author for correspondence: e-mail: [email protected]

60 DAVID BENAVENTE ET AL.

the capillary imbibition kinetics into the porous network of a sedimentary rock.There are many theoretical geometrical models, based mainly on Washburn’s law,which describe the imbibition phenomenon, taking into account different geomet-rical shaped pores (Dullien et al., 1977; Hammecker et al., 1993; Hammecker andJeannette, 1994; Leventis et al., 2000). These models were constructed so as tohave a simple mathematical formulation, but they have a poor agreement with theactual pore structure, and the results are sometimes not in very good agreementwith experimental data. Due to the fact that these models were built by suppos-ing that capillary rise is produced either in vertical tubes or in an arrangement ofvertical geometrical elements, one-dimensional flow was assumed in these models.

The aim of the present paper is to modify the Washburn law by introducingvarious corrections relating to the microstructure of the rocks, and applying the ef-fective medium approximation in order to calculate the effective radius that definesthe hydraulic conductance and the topology of the capillary imbibition.

2. Effective Medium Approximation (EMA)

In order to relate imbibition rates to pore structure, we have used the effectivemedium approximation, EMA (Kirkpatrick, 1973). EMA has been successfullyapplied to pore space in order to predict the electric and hydraulic conductance andalso permeability (Doyen, 1988; Schlueter, 1995). Their models were used to inferan average conductance parameter for heterogeneous disordered networks fromthe statistics of local conducting elements. According to Kirkpartrick (1973), it ispossible to build a homogeneous network with the same topology but in which allconductances gi have a single value geff. In other words, it is necessary to consideran equivalent homogenous rock, which, by construction, has the same pore skel-eton and the hydraulic conductance as the real rock sample. This homogenised rockis obtained by smoothing the irregularities of pore structure. For straight cylindricalthroats of length y and circular cross-section of radius r, the hydraulic conductanceare given by

g = π

8

r4

y. (1)

Thus, in the homogeneous pore space, all the flow channels have the same hy-draulic conductance, given by geff ∞ reff

4/y. The pore length, y, is assumed ap-proximately constant (Doyen, 1988). As a criterion to fix geff, it is required thatthe incremental pressure changes induced when the individual conductances gi arereplaced by geff in this medium should average to zero. That is, a throat channelwith radius ri in the heterogeneous pore structure is replaced by a throat with radiusreff. In order to calculate geff, and therefore reff, it is necessary to determine geff

self-consistently from the experimental pore size distribution. This self-consistencycondition gives reff as the solution of the following implicit equation (Kirkpatrick,1973):

CAPILLARY IMBIBITION OF POROUS ROCKS FROM MICROSTRUCTURE 61

N∑i=1

geff(reff)− gi(ri)

gi(ri)+ [(z/2)− 1

]geff(reff)

= 0, (2)

where z is the spatial average coordination number. Using Newton’s method forfinding the zero of a function, Equation (2) can be solved for the effective poreconductance, thereby obtaining the characteristic throat radius reff. This operationgives rise to a microscopically homogeneous pore system in which all the throatchannels are uniform cylinders with radius reff (Doyen, 1988). Therefore, reff is aneffective value controlling the physical property involved (Schlueter, 1995).

One important topological parameter for the flow properties is the spatial aver-age coordination number z. It is defined as the average number of branches meetingat one node in the skeleton. Thus, Doyen (1988) assumed that z= 6, due to thefact the pore space connectivity is similar to that of a close random packing ofgrains. Upper and lower bounds on the effective conductivity are found from thetwo limiting cases z= 2 and z=∞. For z= 2, Equation (2) can be solved for

geff(reff) = N

(N∑i=1

1

gi(ri)

)−1

, (3)

whereas for z=∞, Equation (2) can be solved for

geff(reff) = 1

N

N∑i=1

gi(ri). (4)

When the coordination number is 2, the tubes are arranged in series, and theeffective conductance reaches its lowest possible value. On the other hand, whenthe coordination number is ∞, the tubes are arranged in parallel, and the effectiveconductance reaches its maximum value. The limiting values of both the effectiveconductance and the coordination number correspond to 1-D arrangements of thetubes. For a coordination number between z= 2 and z=∞, the fluid flow has a 2-Dor 3-D character and Equation (2) must be solved numerically to find the effectiveconductance (Schlueter, 1995).

The correlation of this model with the porous medium is based on supposingthat the pore chamber is placed in the node, that is, between the pore throats.Thus, the flow properties are controlled by the throat connecting the pores, andthe pore chamber does not regulate the flow properties, but it does influence theporosity (Doyen, 1988; Etris et al., 1988). This pore structure is in agreement withgeometric models, such as those of Dullien et al. (1977), Hammecker et al. (1993),and Hammecker and Jeannette (1994).

In samples with porosity higher than 10%, z increases due to the highest con-nectivity of the pore throats. This can be explained due to there is more connectiv-ity between pore throats. In this case the fluid flow has a 3-D character (Doyen,1988).


Having calculated geff from Equation (2), reff can be obtained from Equation(1). Therefore, the characteristic hydraulic radius reff obtained from EMA not onlydepends on the pore size distribution, but also on the topology of the pore network.

3. Materials

In this study, 12 porous rocks which are commonly used as building materials havebeen chosen for their different petrophysical and petrologic characteristics. Theserocks are divided into three groups according to the Folk classification (1962):biocalcirudite (BR), biocalcarenite (BC) and sandstone (SS).

3.1. BIOCALCIRUDITE (BR)

BR-1 is an unsorted biocalcirudite with fragments of bryozoans ranging in sizefrom 2 to 8 mm. This stone contains fragments of red algae, molluscs and echino-derms. Some grains of quartz and feldspars (0.03–0.1 mm) are also present. Thisbiocalcirudite has abundant interparticle porosity, and intraparticle porosity relatedto bryozoans (Figure 1(a)). The cement in this stone is scarce. The main cementconsists of microcrystalline drusy calcite cement. In some points of the stone, thiscement shows as a meniscus distribution. Over the plates of echinoderms showssyntaxial calcite spar overgrowths.

BR-2 is an unsorted biocalcirudite with grains ranging in size from 0.2 to 10 mm.This biocalcarenite has abundant interparticle porosity and some of intraparticaleporosity is related to foraminifera and bryozoans. BR-2 contains detritical grainsof quartz and feldspars (0.1–0.4 mm). It has three authigenic silica forms: chalce-dony, idiomorphic quartz and quartz overgrowths. Authigenic glauconite is alsoabundant. The calcite cement is scarce. This cement consists of equant spar equicry-stalline mosaic, with minor syntaxial calcite spar overgrowths around echinodermfragments (rim cement).

3.2. BIOCALCARENITE (BC)

BC-1 is a well-sorted biocalcarenite with foraminifers smaller than 0.3 mm andquartz grains from 0.04 to 0.08 mm. Feldspar, chert and mica grains are also present.Intraparticle porosity is abundant, but it is poorly connected. The interparticleporosity is filled by microcrystalline sparite cement (Figure 1(b)). Glauconite andsilica cements (criptocrystalline quartz and calcedonite) are also present.

Bateig stone is a biocalcarenite with four commercial varieties: Blue (BC-2),Fantasy (BC-3), Layer (BC-4), White (BC-5). Bateig stone shows interparticle andintraparticle as well as channel porosity by the classification system of Choquetteand Pray (1970), (Ordóñez et al., 1994, 1997; Benavente et al., 1999).

3.3. SANDSTONE

SS-1 is a coarse biocalcarenite (bioclasts from 1.5 to 2 mm) with ∼=10% of otherclasts. This rock shows some layers of sandy biocalcarenite (30–35% of other


Figure 1. (a) BSE – SEM photomicrograph of the BR-1 sample. The rock is composed mainlyof calcite bioclast (light grey), and some quartz and feldspars grains (dark grey). The porosityis mainly interparticle porosity and some intraparticle. (b) BSE – SEM photomicrograph ofthe BC-1 sample. The rock is composed mainly of foraminifers and other calcite fossils (lightgrey), and some quartz and feldspar grains (dark grey). The porosity is interparticle and intra-particle. (c) BSE – SEM photomicrograph of the SS-1 sample. The rock is composed mainlyof quartz (light grey) and clay minerals (dark grey). The porosity is interparticle. (d) Detail ofthe clay minerals in the SS-1 sample. The pore space is impregnated with epoxy resin and air(black).

clasts). The other clasts (0.5–0.75 mm) are quartz, feldspars and rock fragments:limestones, sandstones, chert, quartzites and slates. The main bioclasts are bryo-zoans and red algae. Fragments of molluscs, foraminifers and plates and spinesof echinoderms with syntaxial rim cement are also present. Both interparticle andintraparticle porosity are abundant. Cement is microcrystalline drusy calcite. Glau-conite and iron oxides are minor component of this rock.

SS-2 is a sandy biocalcarenite with fragments of bryozoans, red algae, echino-derms, foraminifers and molluscs from 0.3 to 2 mm. SS-2 shows banded structurewith layers of different amount of interparticle porosity. Sandy-quartz,feldspars, sandstone and chert are 10–15% of stone and their size mode is 0.2–0.5 mm. The interparticle and intraparticle porosity are abundant. Cement in thisstone is scarce. The main cement is equant spar drusy mosaic (Choquette and Pray,1990).

SS-3 is a well-sorted sandstone that consists of monocrystalline quartz grainswith rare feldspars, metamorphic rocks, chert and muscovite grains. Primary in-terparticle porosity has been partially filled by silica crystallization, which formsfrequent overgrowths on quartz grains (Figure 1(c)). Clay minerals are present in


the interparticle porosity where the illite is the main clay mineral (Figure 1(d)).The siliceous cement is abundant, and contributes to a high level of coherence inthe stone (Benavente et al., 1999).

SS-4 is a well-sorted dolomitic calcarenite. This stone contains grains oflimestone, dolostone and quartz from 1 to 2 mm. Minor amounts of chert andfeldspar are also present. Interparticle porosity is abundant. Cement is scarceand composed mainly of dolomite. Traces of siliceous cement are alsopresent.

SS-5 is a sandy biocalcarenite with grains from 0.1 to 0.5 mm. It containsforaminifers and fragments of bryozoans with minor amounts of molluscs, redalgae and plates of echinorderms. The sand fraction is constructed quartz, feld-spars, chert and mica (20–25% of rock). Interparticle porosity is partially filledby microcrystalline and mesocrystalline calcite cement with intercrystalline poro-sity. This rock is partially recrystallized with glauconite forming a minor compo-nent.

In order to measure the throat pore size distribution of rocks, the mercury in-trusion porosimetry (MIP) technique was used (Figure 2). The porosity data wasobtained using the Autoscan-33 mercury porosimetry.

4. Capillary Imbibition Kinetic Experiment

Four samples were used in each test. The samples were each in the form of 2.5 × 2.5× 4 cm prisms, and distilled water was used. Clean and dry samples were intro-duced vertically in a container and left in an upright position on their minor base(2.5 × 2.5). The water was added until it covered 2 mm of the sample height. Thislevel was maintained constant during the test. The container was covered to avoidevaporation. The temperature was constant (20◦C) throughout the experiment. Thetime intervals used were short at the beginning (0, 3, 6, 12, 20, 30, 60 min), andlonger later on (2, 5, 8, 24 and 48 h). In each measurement the sample is blotted(with wet paper), quickly weighed and returned to the container as soon as pos-sible. Thus, the adsorbed water, W(t) [kg], as a function of time is calculated asfollows:

W(t) = WS(t)−W0, (5)

where WS(t) [kg] is the weight of the sample with water, and W0 [kg] is the weightof the sample at t = 0. The results were plotted as adsorbed water per area of thesample throughout imbibition occurs versus square root of time, that is, W(t)/S

versus√t (Figure 3). With this kind of representation, the capillary imbibition

kinetics show two parts: a first part that defines water adsorption, and a secondpart defining the saturation part. The slope of the curve during water adsorptionis the water absorption coefficient C [kg/(m2·h0.5)] (Table I). This parameter isessentially equivalent to the sorptivity parameter in soil physics (Zimmerman andBodvarsson, 1991).

CA

PILL

AR

YIM

BIB

ITIO

NO

FPO

RO

US

RO

CK

SFR

OM

MIC

RO

STR

UC

TU

RE

65

Figure 2. Accumulative mercury intrusion and pore size distribution curves of porous rocks.

66D

AV

IDB

EN

AV

EN

TE

ET

AL

.

Figure 2. (continued)


Figure 3. Experimental weight of adsorbed water per unit area, W(t)/S versus square root oftime,

√t , of all samples.

Table I. Experimental water absorption coefficient, Cexp

Sample Cexp [kg/(m2·h0.5)]

BR-1 3.81 ± 0.16

BR-2 6.11 ± 0.15

BC-1 0.91 ± 0.10

BC-2 1.00 ± 0.08

BC-3 0.97 ± 0.06

BC-4 3.17 ± 0.12

BC-5 1.12 ± 0.01

SS-1 1.96 ± 0.03

SS-2 2.71 ± 0.15

SS-3 1.18 ± 0.18

SS-4 0.73 ± 0.03

SS-5 2.61 ± 0.19

5. Derivation of Imbibition Rate in a Rock with Cylindrical Pores

The Hagen–Poiseuille equation, which is based on the assumption of steady-stateflow, may be written as

Q = dv

dt= πr4

H

8η

�P

y, (6)


where v is the volumetric uptake, Q is the volumetric flowrate, rH is the hydraulicradius, η is the viscosity of the fluid (1.003 × 10−3 Ns/m2 for water at 20◦C), �Pis the pressure drop, and y is the length of the tube. The hydraulic radius, rH , isdefined as

rH = A

P, (7)

where A is the cross-sectional area and P is the perimeter of the pore.The Washburn equation is derived by applying Equation (6) to the laminar flow

of a liquid meniscus in a capillary of radius r. In the special case of capillary risein vertical direction, y represent the height of rise. The volumetric uptake in thecapillary tube is

dv = πr2dy. (8)

The capillary pressure �P can be written as

�P = 2γ cos θ

r, (9)

where γ is the interfacial tension (0.0728 N/m for water at 20◦C) and θ is the con-tact angle (in general to be considered zero Schultze (1925a, b). Inserting Equations(8) and (9) in (6), we can obtain the Washburn equation (1921):

dy

dt= rγ cos θ

4ηy⇒ y(t) =

√rγ cos θ

2ηt, (10)

when y0 = 0 for t = 0. In these unsaturated conditions, the cumulative volumetricuptake of one pore can be calculated as

v(t) = πr2y(t) = πr2

√rγ cos θ

2ηt. (11)

In order to calculate the mass of absorbed water in one pore, w(t), is necessaryto multiply the volume by the density, ρ (998 kg/m3 for water at 20◦C) as follows:

w(t) = ρv(t) = πρr2

√rγ cos θ

2ηt. (12)

Taking into account that r, γ , θ and η are constants, the height and weight kin-etics in a cylindrical tube or a pore can thus be formulated as simple square rootfunctions (Hammecker et al., 1993):

y(t) = B√t , w(t) = A

√t , (13)

where

B =√rγ cos θ

2ηand A = πρr2

√rγ cos θ

2η. (14)


From the experimental point of view, the height kinetic coefficient B in a porousrock is relatively easily measured, taking into account that the rock is comprised ofparallel capillary tubes. To estimate the weight kinetic coefficient A, it is necessaryto take into account the weight of the entire pore space. If we suppose the rock hasn pores, the incremental weight of the rock, W(t), due to capillary flow is

W(t) = n ·w(t) = nπρr2

√rγ cos θ

2ηt. (15)

By definition, the porosity is

φ = Vp

Vb= nπr2L

SL= nπr2

S, (16)

where Vp is the pore volume, Vb is the bulk volume, L is the sample length and Sis the macroscopic area of the sample. Substituting nπr2 =φS from the Equation(15) in the incremented weight of the rock, W(t), one obtains

W(t) = φSρ

√rγ cos θ

2ηt. (17)

The expression of the incremented weight per area can be written as follows:

W(t)

S= φρ

√rγ cos θ

2ηt = C

√t , (18)

where

C = φρ

√rγ cos θ

2η. (19)

6. Correction for Pore Shape: Roundness

At this point, we introduce a pore shape factor to quantify the deviation fromcircularity of the cross-section of the pore throat that controls the flow properties.This pore shape factor is called ‘roundness’. In order to calculate the roundness,analysis of the backscattered electron images (BSEI) has been performed.

However, with BSEI, information is obtained about the pore chambers, but notof the pore throats. Due to the fact that both pore chambers and pore throats areinterparticle porosity, and due to the textural characteristics of these porous rocks,we have supposed that the shape of the pore chamber is the same as that of the porethroat.

The pore shape factor may be defined as the deviation from circularity of thecross-section of the hydraulics radius, that is,

δ = rH

(rH )C. (20)


This pore shape factor is called ‘roundness’, δ. This coefficient gives a valuebetween 1 and 0. If δ= 1, the cross section of the pore is a perfect circle. The ratiodecreases from 1 as the cross section departs from a circular form.

By the definition of the hydraulic radius, Equation (7), the hydraulic radius ofthe pore with circular cross-section can be written as

(A

P

)C

= A√4πA

=√A

4π, (21)

where A is the cross-sectional area and P is the perimeter. The pore shape factorcan be defined as ratio between hydraulic radius with non-circular cross-sectionand with circular cross-section, and it can be written as follows:

A/P

(A/P )C= 2

√πA

P 2= δ. (22)

Therefore, we can introduce the roundness, δ, in the water absorption coeffi-cient, C, as follows:

C = φρ

√rδγ cos θ

2η. (23)

In order to calculate the pore shape, the analysis of the backscattered elec-tron images (BSEI) has been conducted. Image analysis was performed using theUTHESCA ImageTool�. The BSEI was taken with a JEOL JSM-840 from polishedthin sections.

The pore roundness was calculated at a magnification of 100×. The field ofview with smaller magnifications is wider than with higher magnifications, but theirresolution is lower. On the other hand, higher magnifications have better resolution,but the field of view is smaller, making it sometimes difficult to measure the poreshape. Using 100× magnification, it is possible to balance both problems. The poreroundness for the different rocks at 100× is shown in Table II.

There are other methods that may be used in order to obtain the pore shape.One of them is the fractal theory. This has widely been used in the modellingand characterisation of the pore structure (Pape, 2000; Tsakiroglou and Payatakes,2000). The parameter that quantifies roughness is the fractal dimension which

Table II. The pore roundness factors for the different rocks at 100×

Sample BR-1 BR-2 BC-1 BC-2 BC-3 BC-4

δ 0.601 0.518 0.540 0.497 0.522 0.488

Sample BC-5 SS-1 SS-2 SS-3 SS-4 SS-5

δ 0.497 0.547 0.529 0.565 0.526 0.546


can be obtained, for example, by image analysis, mercury porosimetry, absorptiongases and/or X-ray-scattering (Bale and Schmidt, 1984; Wong et al., 1986; Tsakiro-glou and Payatakes, 1993; Schlueter, 1997). Other approximations are based on thedecreasing the original pore hydraulic radius (Blunt et al., 1992). We think that aconsideration of pore shape not only provides an alternative approach to estimatingthe water adsorption coefficient in comparison with the previous methods, but it iseasily obtained, and gives an interesting and realistic view of the pore structure.

7. Correction for Pore Orientation: Tortuosity

Due to the fact pores are virtually never straight tubes, it is necessary to correct forthe pore orientation by introducing the tortuosity, τ (Berryman and Blair, 1987).The tortuosity can be derived for randomly orientated cylindrical tubes in threedimensions (Schlueter, 1995).

Over short sections of a tube, the pressure gradient is �p/& where & is directedalong the tube axis. The applied pressure gradient is �p/y with respect to thespecimen axis (Berryman and Blair, 1987). If the tube axis is at an angle ' withrespect to the specimen axis, then the Hagen–Poiseuille equation, (6), is given by

Q = dv

dt= πr4

H

8η

�P

&= πr4

H cos'

8η

�P

y. (24)

The spatial average of y component is over all channel orientations and it isgiven by 〈cos2 '〉. The average square cosine of the angle is related by the tor-tuosity by τ =〈cos2 '〉−1 and is equal to 3 when the channels are straight andrandomly orientated (Doyen, 1988; Dullien, 1992). In this study, the tortuosity isassumed equal to 3.

Moreover, the relationship between macroscopic and microscopic properties ofa porous medium as well as the fact that macroscopically measured variables in aporous medium can be carried out with the aid of appropriate experimental and/ortheoretical scale-up procedures (Dullien, 1992). Also, due to fact that the waterabsorption coefficient (or sorptivity) is related to the square root of the permea-bility, that is, C ∝ √

k (Zimmerman and Bodvarsson, 1991), and the permeabilityis inversely related with to the tortuosity, τ (Bear, 1988), it is reasonable to expectthat the water absorption coefficient, C, is inversely proportional to square root ofthe tortuosity. Thus from both reasoning, Equation (23) can be rewritten as follows:

C = φρ

√rδγ cos θ

2τη. (25)

Therefore, the amount of adsorbed water W(t) per unit area, S, (kg/m2) varieslinearly with the square root of time, and its proportionality constant C has units


of [kg/(m2·h0.5)]. This coefficient depends on the characteristics of the material (φ,r, δ and τ ) and of the fluid (ρ, η, θ and γ ).

8. Analysis of the Results

Effective radii were calculated from Equation (2) using Newton’s method. The in-put data for r in Equation (25) were the pore size distribution (obtained from singlemercury intrusion porosimetry) and the spatially-averaged coordination number, z.The results of this calculation are shown in Table III.

When the coordination number is 2, the effective radius reaches the lowest valuein comparison with mean radius, 〈r〉. When the coordination number is ∞, theeffective radius reaches the highest value. In the case of z= 6, the value of reff isclose to mean radius.

In order to obtain C from Equation (25), the viscosity of water was taken to be1.003 × 103 Ns/m2, the interfacial tension as 0.0728 N/m and the density equal to998 kg/m3. Following Schultze (1925a, b) a contact angle of 0◦ was assumed forsimplicity, and the tortuosity taken to be 3 (Doyen, 1988; Dullien, 1992). Porosityand reff for different coordination numbers were used from Table III. The roundnessparameter was taken from Table II and the results for C (Table IV) were comparedwith the experimental coefficients, Cexp in Table I.

The water absorption coefficient Cexp that best fits the experimental data isfound using a spatial average coordination number equal to 2. The calculation ofC from z= 6 and z=∞ give poor fits. They provide kinetic coefficient values thatare 100 or even 200 times higher than the experimental data.

Table III. Effective radius, reff, for different spatially-averaged coordination number, z, and theporosity, φ, obtained from MIP

Sample reff (z= 2)/µm reff (z= 6)/µm reff (z=∞)/µm 〈r〉/µm φ

BR-1 0.1654 19.7154 32.7811 18.0765 0.1186

BR-2 0.0155 6.5321 15.4918 6.5098 0.2755

BC-1 0.0068 0.1312 0.2703 0.1320 0.1554

BC-2 0.0082 0.0626 10.9560 0.5054 0.1270

BC-3 0.0094 0.2922 0.6649 0.2997 0.2040

BC-4 0.0096 0.8285 7.9655 1.2887 0.1600

BC-5 0.0075 0.0542 0.0944 0.0531 0.1535

SS-1 0.0171 7.6834 20.0127 7.8514 0.1730

SS-2 0.0136 10.3627 28.3107 11.7564 0.1778

SS-3 0.0246 4.6521 25.6316 7.9211 0.0640

SS-4 0.0072 0.3183 11.7207 1.6219 0.2979

SS-5 0.0369 0.5295 0.9811 0.5428 0.2175


Table IV. Calculate water absorption coefficient, C, for differentspatially-averaged coordination numbers, z

Sample Cz=2 Cz=6 Cz=∞

BR-1 7.79 85.01 109.62

BR-2 5.14 105.51 162.49

BC-1 1.96 8.62 12.37

BC-2 1.69 4.67 61.72

BC-3 2.97 16.59 25.02

BC-4 2.28 21.19 65.69

BC-5 1.95 5.25 6.92

SS-1 3.48 73.85 119.19

SS-2 3.14 86.71 143.32

SS-3 1.57 21.61 50.72

SS-4 3.82 25.39 154.10

SS-5 6.43 24.35 33.15

9. Discussion

The reff (z= 2) calculated from the EMA (Table III) give values lower than themean pore size of the rocks. This is due to the fact that reff is not a real radius ofthe rock pores, but rather it is a ‘statistical’ effective radius. The effective radiushas been defined from the throat size distribution and the spatial average coordi-nation number, using the effective medium approximation. Therefore, the effectivehydraulic radius not only contains information on the effective pore space but alsoon the topology of the fluid flow. As a consequence of this, the effective radiuscannot be directly compared with the actual pore size.

The calculated water absorption coefficients for z equal to or close to 2 aremuch closer to the experimental data than are the z= 6 and z= ∞ values. From thisresult, we can obtain important information about the pore structure that defines thecapillary imbibition. This implies that capillary imbibition takes place essentiallyin one-dimension and the pore structure is mainly constituted by different throatsin series, linked by chamber pores. That fact that values of z close to 2 providebetter estimates of C means the fluid flow is mainly produced in one-dimension.A small contribution of non-one-dimensional flow will also be produced. Thus, forexample, in the BC-4 rock, if z= 2.2, Cz=2.2 = 3.16 kg/(m2h0.5). We consider thisto be important since previous models assumed one-dimensional flow. Therefore,the application of the EMA to capillary imbibition demonstrates the one-dimensionflow.

In samples with high porosity, z increases due to the highest connectivity of thepore throats (Doyen, 1988). This fact can be observed in the case of BR-2 rock.Thus, for example, if z= 2.5, Cz=2.5 = 6.16 kg/(m2h0.5). This can be explained by


the fact that there is more connectivity between pore throats. In this case the fluidflow has a 3-D character (Doyen, 1988).

The accuracy of the results not only depends on the approximations made in dif-ferent parts of this work, but also on the petrography of rocks and the experimentalprocedure. The petrography of these rocks, which depends on the texture, the mi-neralogy and the location of the different minerals in the rock (Hemmecker andJeannette, 1994) may change the water absorption value predicted by our model.Thus, for example, the contact angle can be modified by the rock mineralogy dueto the chemical composition and superficial heterogeneity of the different grains.In the specific case of clays which expand with water, their presence may notonly strongly modify the contact angle but may also close the pores and thereforemodify the pore structure. The proposed model requires information of the poresstructure by mean reff and the porosity, and this was obtained before the water flowsthrough porous rock. Therefore, this fact may modify the predicted values. Forexample, the clays in sample SS-3 (Figure 1(d)) change the water absorption duringthe capillary imbibition. This fact may explain the low value of the water absorptioncoefficient of rock SS-3. This fact is also produced in the BC-2 and SS-4 samples.The model cannot predict the capillary imbibition of a rock which contains a highamount of swelling clays. Furthermore, the model can only be applied when themodified pore structure is known. Referring to the experimental water absorptioncoefficient, two important sources of error can be identified: evaporation during thecapillary imbibition, and fitting the slope of the graph W(t)/S versus

√t .

We consider that the deduction of our simple model is a novel extension of theWashburn equation and a significant improvement from what has been used in thepast. Also the proposed equation is accurate in comparison with the experimentaldata and is of very easy application.

10. Summary

Our study of the Washburn equation leads to Equation (25) to predict the rate ofwater absorption. The addition of correction factors due to the pore shape, bymeans of roundness, δ, and the tortuosity, τ , and the application of the EMA inthe calculation of reff and its application in the Equation (25) gives accurate valuesof C.

The most important difference between our model and geometric previous mod-els (Dullien et al., 1977; Hammecker et al., 1993; Hammecker and Jeannette,1994; Leventis et al., 2000) is that our model is built from measures of pore struc-ture, whereas previous models were built from geometric formulations. We cannotcompare our model with previous geometric models, because these models predictthe capillary height and not the amount of water per area. From the experimentaldata, we think that our model, which is based on a more realistic view of the porestructure, allows a very accurate prediction of the weight rate.


The application of the EMA provides us with information about reff and z. Thestudy of the co-ordination number z is an important result of this work. It showsthat capillary imbibition is mainly produced in 1-D, and that the pore structure isconstituted by different pores in series, in previous models one-dimensional flow isassumed. In samples with high porosity, an increase of z has been observed. Thisis due to the fact that there is more connectivity between pore throats. In this casethe fluid flow has a 3-D character (Doyen, 1988).

The direct application of the proposed model is that a single mercury intru-sion porosimetry measurement (MIP) for these rocks, along with the BSEI image,allows the prediction of their capillary imbibition kinetics, and provides usefulinformation about the fluid transport phenomena.

Acknowledgements

This work was supported by the Research Project MAT 2000-0744 (MCYT, Spain).The visit of David Benavente to Imperial College was supported by a pre-doctoralresearch fellowship Generalitat Valenciana. The authors thank R. Fort for providingthe mercury porosimetry data, M. Palomo for preparing the thin sections, V. Lopezfor assistance with the back scattering electron images, and R. Zimmerman andother members of the Porous Media Research Group at IC for their hospitality andassistance.

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Predicting the Capillary Imbibition of Porous Rocks from Microstructure

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