Journal of Colloid and Interface Science 368 (2012) 558–577

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Journal of Colloid and Interface Science

www.elsevier .com/locate / jc is

Contact line extraction and length measurements in model sedimentsand sedimentary rocks

Elena Rodriguez ⇑, Maša Prodanovic, Steven L. BryantDepartment of Petroleum and Geosystems Engineering, University of Texas at Austin, 1 University Station C0300, Austin, TX 78712, USA

a r t i c l e i n f o

Article history:Received 28 May 2011Accepted 28 October 2011Available online 10 November 2011

Keywords:Contact line lengthInterfacial areaColloid retentionDrainageImbibitionLevel set methodX-ray imagesModel sediments

0021-9797/$ - see front matter Published by Elsevierdoi:10.1016/j.jcis.2011.10.059

⇑ Corresponding author. Permanent address: 1200Houston, TX 77056, USA.

E-mail addresses: erodriguez@alumni.utexas.netutexas.edu (M. Prodanovic), steven_bryant@mail.utex

a b s t r a c t

The mechanisms that govern the transport of colloids in the unsaturated zone of soils are still poorlyunderstood, because of the complexity of processes that occur at pore scale. These mechanisms are of spe-cific interest in quantifying water quality with respect to pathogen transport (e.g. Escherichia coli, Cryptos-poridium) between the source (e.g. farms) and human users. Besides straining in pore throats andconstrictions of smaller or equivalent size, the colloids can be retained at the interfaces between air, water,and grains. Theories competing to explain this mechanism claim that retention can be caused by adhesionat the air–water-interface (AWI) between sediment grains or by straining at the air–water–solid (AWS)contact line. Currently, there are no established methods for the estimation of pathogen retention inunsaturated media because of the intricate influence of AWI and AWS on transport and retention. Whatis known is that the geometric configuration and connectivity of the aqueous phase is an important factorin unsaturated transport. In this work we develop a computational method based on level set functions toidentify and quantify the AWS contact line (in general the non-wetting–wetting–solid contact line) in anyporous material. This is the first comprehensive report on contact line measurement for fluid configura-tions from both level-set method based fluid displacement simulation and imaged experiments. Themethod is applicable to any type of porous system, as long as the detailed pore scale geometry is available.We calculated the contact line length in model sediments (packs of spheres) as well as in real porousmedia, whose geometry is taken from high-resolution images of glass bead packs and sedimentary rocks.We observed a strong dependence of contact line length on the geometry of the sediment grains and thearrangement of the air and water phases. These measurements can help determine the relative contribu-tion of the AWS line to pathogen retention.

Published by Elsevier Inc.

1. Introduction

Particles of colloidal size (effective diameters between 0.01 and10 lm) are naturally present in the subsurface. Some examples in-clude humic materials, silicate clays and mineral precipitates, andviruses and bacteria. These colloid particles themselves can be con-taminant (bacteria and viruses) or they can act as carriers of con-taminants such as pesticides or heavy metals [1]. The presenceand transport of colloids in the subsurface strongly affect groundwater quality and consequently the transport and retention mech-anisms of these particles are of specific interest in quantifyingwater quality with respect to pathogen transport (e.g. Escherichiacoli, Cryptosporidium) between the source (e.g. farms) and humanusers. In a completely different, but not less important field, inor-ganic colloids, such as clay particles, can cause problems in oil

Inc.

Post Oak Blvd., Apt. 1613,

(E. Rodriguez), masha@ices.as.edu (S.L. Bryant).

reservoirs, affecting the reservoir properties of sandstones. Theseparticles can move within the reservoir due to drag forces duringoil and gas production. This phenomenon is known as fines migra-tion and consists of the release of the fine particles from the porousmedia, their movement with the flow of permeate, and eventuallytheir capture within the porous medium or their path out of themedium. Migration and capture in oil reservoirs can cause a reduc-tion in permeability and therefore a decline in oil production.

In the unsaturated zone of the sediments, particles can be re-tained at the interfaces between air, water, and grain. This reten-tion is in addition to straining at pore throats and constrictionsof smaller or equivalent size.

Because of the complexity of processes at the pore scale, themechanisms that govern retention in the unsaturated zone are stillpoorly understood. Theories competing to explain this mechanismclaim that retention can be caused by adhesion at the air–water-interface (AWI) between sediment grains [2,3] or by straining atthe air–water–solid (AWS) contact line [4,5]. At this time, the dis-cussion about the relative contributions of AWI and AWS to reten-tion which started several years ago is still unsettled. Another

E. Rodriguez et al. / Journal of Colloid and Interface Science 368 (2012) 558–577 559

factor that makes the particles susceptible to retention at theseinterfaces is their affinity for the aqueous phase and thereforethe water saturation. Also important are factors like pH or ionicstrength of the particle solution. For example, Torkzaban et al.[6] investigated the factors that control the attachment to theAWI in column experiments using solutions of bacteriophages atdifferent pH and ionic strengths. The experiments showed thatretention at AWI and SWI (solid–water-interface) increased asthe pH decreased and that electrostatic interactions were moreimportant than hydrophobicity regarding attachment to the AWI.On the other hand, Gao et al. [7] credit capillary and friction forcesrather than electrostatic (DLVO) forces to be responsible for theretention of colloids at the AWmS.

More recently, Bridge et al. [8] studied the movement of colloidsduring drainage in quartz sands and observed colloid mobilizationby AWI while the AWS contact plays the part of fixing the colloidsto the SWI.

Currently, there are no established methods for the estimationof particle retention in unsaturated media because of the intricateinfluence of AWI and AWS on transport and retention. The geomet-ric configuration and connectivity of the aqueous phase and airphase are clearly important factors in unsaturated transport. Butreliable quantitative estimates of the specific interfacial area arestill difficult to obtain: measurements with interfacial tracers typ-ically include contribution from wetting films in drained pores[9,10], and direct observation (e.g. from epoxy-filled sections [11]or from high-resolution X-ray images [12,13]) is tedious to acquire.No estimates of lengths of AWS contact have been reported to date.

In this paper we quantify of the length of the air–water–solid (ingeneral non-wetting–wetting–solid) contact line in simple geome-tries and granular materials. When colloidal retention data is avail-able (expressed for example as number of colloidal particles pervolume of column effluent) the length of the contact line and theinterfacial area between wetting and non-wetting phases can beused to make predictions about where the colloidal size particlesare trapped when compared with experimental visualizations, andthus improve our understanding of the underlying mechanisms.

We used a novel computational, level set method based, pro-gressive quasi-static algorithm (LSMPQS) to reveal the configura-tion of the air–water–solid (AWS) contact lines. LSMPQS tracksthe pore scale motion of interfaces assuming capillary forces aredominant [14]. It has been implemented to compute the locationof an interface between two immiscible fluids confined by arbitrarysolid surfaces. Thus the method implicitly determines the locationof contact lines as the intersection of any pair of interfaces (e.g.,the intersection of the wetting–non-wetting interface with the wet-ting–solid interface) as a function of applied capillary pressure. Thevolume fraction occupied by water, the total interfacial areas, andthe contact line length are recorded for each configuration.

While energy, mass and momentum conservation equations forthe ‘‘common lines’’ have been developed [15,16] and the impor-

Fig. 1. (a) Displacement of wetting phase (water) by non-wetting (air) in a simple pore thand solid (w) phases.

tance of contact lines in the modeling of multiphase flow in porousmedia has been explored, [17,18] to our knowledge, this is the firstcomprehensive work on quantifying contact line lengths in disor-dered porous media. McClure et al. [19] report the ability to mea-sure contact lines but their focus is primarily on interface areas andcurvatures.

The contact lines have been identified and computed in modelsediments (random packs of spheres). We validate these computa-tions using fluid configurations extracted from high resolutionimages of glass beads packs and consolidated rock formations(Fontainebleau sandstone and sucrosic dolomite).

2. Methods

2.1. Level set method

Measuring the contact line length requires detailed knowledgeof wetting and non-wetting interface positions in the granularmedium under investigation. We use both LSMPQS simulationand microtomography images as a source of such knowledge.LSMPQS algorithm [14] determines the geometry of capillaritycontrolled fluid configurations and thus readily provides porescale interfacial areas (wetting/non-wetting, wetting/solid, non-wetting/solid). The fluid interfaces are confined by solid surfaceswhich correspond to the grains in the porous medium. The contactlines exist at the intersection of these interfaces with grain sur-faces. Fig. 1a shows non-wetting phase (air) displacing wettingphase (water) between two solid grains. The point contacts atthe intersection between fluid–fluid interface and solid in the 2Dschematic will become contact lines in 3D.

The level set method is a numerical method for propagatinginterfaces [20]. The fluid locations are defined by an arbitrary func-tion u(x), where x is a position vector, whose value is zero at theinterface between the two fluids, less than zero for the non-wetting phase and larger than zero for the wetting phase. Thisinterface is allowed to move normal to itself with a velocity whichdefines the physics of the problem. Its motion is governed by thefollowing equation:

ut � Fjruj ¼ 0 ð1Þ

Therefore the physics of the problem are given by the definitionof F. In our porous medium application, changes in capillary forcesare driving the displacement of one fluid by another. Thus F is de-fined as:

F ¼ Pc � rj ð2Þ

where Pc is the capillary pressure, r is the interfacial tension be-tween fluids and j is twice the mean curvature of the interface.For our application F is defined as follows:

roat (s = solid). (b) Level set function for the non-wetting fluid (u), wetting fluid (uw)

560 E. Rodriguez et al. / Journal of Colloid and Interface Science 368 (2012) 558–577

Fðx; tÞ ¼ a0 � b0jðx; tÞ ð3Þ

where a0 is the capillary pressure like term, b0 is the interfacial ten-sion like term and j is twice the mean curvature of the interface.We look for steady state solutions of the level set Eq. (1) with theabove definition of F. Its non-trivial solution is the constant curva-ture interface:

j ¼ a0=b0ð¼ Pc=rÞ ð4Þ

Note that the resulting surface is a function of the ratio of cap-illary pressure and interfacial terms, rather than their individualvalues. The steady-state solution (i.e. the limit at large time) ofEq. (1) for different curvature determines the location of the inter-face. Another level set function w is defined for the solid. This func-tion is stationary, since the solid phase does not move, and it isequal to zero on the interface between solid and pore space andnegative in the pore space. The region where the level set functionsu and w are both equal to zero correspond to the interface betweensolid and non-wetting phases; cf. Fig. 1b. Similarly, the locus ofpoints where u is positive and w is zero corresponds to the inter-face between solid and wetting phases. The locus of points whereu is zero and w is negative corresponds to the interface betweennon-wetting and wetting phases. To identify triple contact pointsmore conveniently, we define an auxiliary level set function forthe wetting phase uw, that will be less than zero where u is posi-tive and w is negative (see Fig. 1b) and larger than zero elsewherebut at the interface. As a result the triple contact points will be thelocations where all three level set functions are equal to zero.

The solving of the problem involves the calculation of Heavi-side functions and its derivative, the Dirac delta function, for thethree level set functions and it is described in the next section.The algorithm was coded in C and added as a new functionalityto the LSMPQS software package developed by Chu and Prodanov-ic [21].

2.2. Contact line identification and length measurement

We devised a method that uses phase-segmented voxelizeddata regardless of whether the data comes from simulation(LSMPQS; and is therefore in itself smooth) or segmented X-raytomography image of fluid phases in the pore space. Thus whencomparing simulation and experimental results, we avoid any is-sues with comparing length measurements based on differentmethods for different sources of fluid phase information.

The procedure to find the point contacts involves the calcula-tion of Heaviside and Dirac delta functions. The Heaviside functionH, also called unit step function, is a discontinuous function whosevalue is equal to zero for negative arguments and equal to one forpositive arguments. The Heaviside function of the level set functionu(x) will be:

HðuðxÞÞ ¼0 uðxÞ < 01 uðxÞ > 0

�ð5Þ

To avoid the discontinuity at u(x = 0) in the numerical calcula-tions we used the ‘‘smoothed’’ version of Eq. (5), selecting a toler-ance e whose value is of the order of the width of the numericalcell, such that:

HðuðxÞÞ ¼0 uðxÞ < �e12þ

uðxÞ2e þ 1

2p sin puðxÞe

� ��e < uðxÞ < e

1 uðxÞ > e

8>><>>:

ð6Þ

The Dirac delta function is defined as the derivative of the Heav-iside function, therefore:

dðuðxÞÞ ¼ H0ðuðxÞÞ ¼0 uðxÞ < �e12e 1þ cos puðxÞ

e

� �� ��e 6 uðxÞ 6 e

0 uðxÞ > e

8>><>>:

ð7Þ

We are interested in finding the locations where our level setfunction is equal to zero, which now corresponds to the interval�e < u(x) < e. Therefore, examining Eqs. (5)–(7), the contact ‘‘re-gion’’ exists for the voxels where the three Dirac delta functions(solid, wetting and non-wetting) are positive.

While the contact line position and identification in the porousmedia is conceptually simple, due to discretization (in simulation)and finite resolution (in imaging), its extraction and precise lengthmeasurement are not trivial. We thus first identify rather thickcontact regions of voxels around triple contact points (typically 3voxels) instead of a thin string of voxels. Since the Heaviside func-tion is smoothed to avoid a discontinuity in the numerical calcula-tion (see previous section) the Dirac delta function identifies asbelonging to the contact line the voxels where the value of thelevel set function u(x) is between the positive and negative valuesof an specified tolerance. This set of voxels is larger than just thevoxels where the value of the level set function is equal to zero.

We ‘‘thin’’ these thick regions of voxels using medial axisapproach implemented in 3DMA-Rock package (W.B. Lindquist ofSUNY Stony Brook). The purpose of the medial axis is to obtain areduced representation of an object that is easier to analyze; whileit has been extensively used in various image analysis applications,its first use in porous media applications was by Lindquist et al.[22]. The medial axis of an object can be seen as its skeleton. Forthis reason, the medial axis of the thick region of voxels shouldbe a very good representation of the contact line, which is a one-dimensional object. The resulting contact line is a collection of dig-itized links and nodes, and subsequent length measurement isstraightforward.

To obtain the skeleton of the digitized object, the object voxelsare carefully eroded, layer by layer, while preserving the objecttopological and geometric properties. A voxel is removed only ifits removal does not induce a local change in topology, such asbreaking the object in two parts or creating a hole or cavity.

We normalize the computed length in order to be able to com-pare different systems. We divided the computed contact linelength (Lc) by a characteristic length of the system that we chooseto be the radius of the grains (R) to obtain a dimensionless contactline length, LcD. Then we divide this magnitude by the total volumeof the domain (Vb), also made dimensionless (VbD) by dividing itsvalue by the cube of the characteristic length (R3). Therefore:

Normalized specific contact line length ¼ LcD

VbDð8Þ

In this way, we obtain an intrinsic value for the contact linelength. This normalized specific contact line length is then inde-pendent of, for example, the number or size of spheres in a packing,as we will see later.

3. Results of contact line length

3.1. Analytical test for contact line length validation

We first simulated drainage in a simple system of two equal sizespheres contained in a box as shown in Fig. 2a, where we can easilydetermine the contact length analytically and then compare it withthe result from simulation. By observation of the last step of drain-age, the canonical shape of the contact line has been identified astwo circles bounding the pendular ring between the two spheres(Fig. 2b). Once the pendular ring was completely formed we calcu-lated the length of the contact line as the length of the medial axis

Fig. 2. (a) Two spheres of radius R in a box. (b) Last step of drainage, where a pendular ring between the two spheres has been formed (red = non-wetting phase in contactwith sphere surfaces, blue = wetting phase, green = contact line). (c) The two rings that make the medial axis of the contact line. (For interpretation of the references to colorin this figure legend, the reader is referred to the web version of this article.)

Fig. 3. Wetting–non-wetting interfacial areas from experimental data and LSMPQSsimulation. Experimental measurements extracted from [24–27]. Here the simu-lation accounts for interfaces between bulk wetting and non-wetting phases, andbetween the surface of grains (presumed to hold wetting phase film) and non-wetting phase in drained pores.

E. Rodriguez et al. / Journal of Colloid and Interface Science 368 (2012) 558–577 561

obtained with 3DMA-Rock. We compared this length with the ana-lytical result, where pendular ring configuration is assumed to be atoroid. Two cases were tested. In one the spheres are in point con-tact and in the second, the spheres are separated by a gap of widthequal to 10% of the radius of the spheres. The curvature used forthe calculations does not necessarily reflect the curvature at whichmost pendular rings are present in sphere packs. Table 1 shows therelative error of the calculation for different resolutions (dx). Therelative error is between 2% and 10% in all cases. This test givesus confidence that the length calculated by means of medial axisis a fairly accurate measure of the simulated contact line.

Table 1 reports contact line length in units of grain radius forthis simple case. In packings of grains, we will report the contactline length per unit bulk volume. This intrinsic value can be usedto compare contact line lengths of systems of different sizes.

3.2. Contact line length validation

There are no previous contact line measurements (experimentalor numerical) in porous media with which we can compare our re-sults. Analytical solutions are not feasible for disordered packingsexcept at drainage endpoints when all wetting phase is held aspendular rings. For example, a dense disordered packing of equalspheres of radius R has about 6 point contacts per sphere. If thependular rings are formed at a curvature C = 1.48R�1, then fromTable 1, the specific contact line length will be:

Specific contact line length ¼ 8R1 contact

� 6 contacts1 sphere

� 12� ð1� /Þ

43 pR3

ð9Þ

where division by two eliminates double counting the contacts ateach sphere. The normalized value (divide length by R and volumeby R3) is thus 3.7. At C = 10R�1, the analytical value of the length ofthe ring is 4.65R and therefore we get a normalized specific contactline length of 2.1. This calculation neglects the contribution of me-nisci in pore throats. It is known that during drainage and imbibi-tion menisci make a substantial contribution to interfacial area

Table 1Contact line length from analytical test and simulation (via 3DMAa) in two simple cases.

Two spheres in point contact curvature = 1.48R�1, analytical length = 8R T

dx Length, R (3DMA) Relative error (%) L

0.02 8.36 4.31 50.04 8.20 2.50 50.05 8.69 8.63 50.08 7.56 5.25 5

a The criteria for the 3DMA trimming are as follows: (1) 26 connectivity between voxelremoved. (3) Branch leaf paths of less than 20 voxels have been removed to reduce noi

relative to pendular rings, and it is reasonable to expect similar con-tribution to contact line length. Thus we anticipate magnitudes ofnormalized specific contact line length to be between 1 and 10.

Interfacial area measurements are more easily available. Sincecontact lines are the intersection of fluid/fluid interfaces withfluid/solid interfaces, it is reasonable to expect contact line lengthto correlate with fluid–fluid interfacial area. Thus comparing thesimulated interfacial areas with experimental results can providesome confidence in the contact line lengths predicted from thesame fluid/fluid/solid configurations.

There are two contributions to interfacial area as commonlymeasured by interfacial tracers. One is the interface of the bulkconnected volumes of wetting and non-wetting phases, the otheris the film of wetting phase covering the solid surfaces in drained

The length is measured at a constant curvature for different voxel sizes (dx).

wo spheres with a gap of 0.1R curvature = 2.89R�1, analytical length = 5.63R

ength, R (3DMA) Relative error (%)

.92 5.15

.76 2.31

.12 9.06

.31 5.68

s is considered. (2) Isolated voxels, needle eye paths and surface remnants have beense effects. See 3DMA documentation [23] for medial axis modification case 4.10.

562 E. Rodriguez et al. / Journal of Colloid and Interface Science 368 (2012) 558–577

pores. Fig. 3 shows the sum of these contributions in the LSMPQSsimulation of drainage in a dense, disordered pack of equalspheres. The experimentally determined interfacial area is subjectto variability because of the time limitation for the diffusion of thetracer from the bulk water phase to the interface and hence to thefilms. This limitation results in smaller interfacial areas. As shownin Fig. 3, our results for the total wetting–non-wetting interfacialarea follow the trend of experimental results obtained from tracertechniques. Given the limited amount of data (capillary pressure,for instance, is not available), and disparate sources, we find thecomparison very good.

In the absence of experimental measurements of contact linelength, the fact that our simulated interfacial area follows the sametrend as the experiments provides some confidence that our esti-mation of contact line length should be a good approximation.

4. Contact line length measurements in model sediments

We simulated drainage and imbibition displacements in twopacks of different numbers of randomly arranged, densely packedspheres of the same radius, as the one shown in Fig. 4a. The spheresare packed into a periodic cubic domain using the cooperativere-arrangement algorithm developed by Thane [28]. The periodic-

Fig. 4. (a) Periodic cubic pack of randomly distributed spheres of the same size (radiusdifferent cubic packs of same size spheres. (c) Normalized specific contact line length dularge (623) different packs of same size spheres.

ity allows spheres to extend beyond the faces of the cube to avoidboundary artifacts. The curvature vs. saturation plot is shown inFig. 4b. In Fig. 4c we compare normalized specific contact linelength during drainage and imbibition. One pack has 91 sphereswhile the other has 623 spheres, making its volume six times lar-ger, yet the normalized specific length of the contact line does notdiffer considerably.

The main numerical parameter affecting the length of the con-tact line is the resolution or voxel size (dx) used in the simulations.Fig. 5 shows the normalized contact line length for the same packof 91 spheres for two different voxel sizes (dx = 0.04R anddx = 0.08R). The length for the smaller dx (better resolution) is con-siderably larger than for the larger dx, being almost twice as largeat the last steps of drainage.

This difference can be explained by looking at the representa-tion of contact line for two different resolutions in a single porethroat as shown in Fig. 6.

Fig. 6a shows the voxel representation of the contact line for aresolution dx = 0.04R after the pore has been drained and the wet-ting phase is held as pendular rings between the three grain con-tacts. Once we calculate the medial axis of the voxel output inFig. 6b, we can see two isolated rings of contact line in the centerof the picture corresponding to a single pendular ring. The same

R and voxel size dx = 0.08R). (b) Curvature vs. wetting phase saturation plot for tworing drainage and imbibition vs. wetting phase saturation for small (91 spheres) and

Fig. 5. Normalized specific contact line length vs. wetting phase saturation duringdrainage and imbibition for the same pack of spheres using two differentresolutions (voxel size, dx of 0.04R and 0.08R).

E. Rodriguez et al. / Journal of Colloid and Interface Science 368 (2012) 558–577 563

ring is at the left of the image in Fig. 6a. In Fig. 6c we compute ex-actly the same state (dimensionless curvature = 11) using a resolu-

Fig. 6. Voxel representation of the contact line (green) in a pore throat for two differenwetting/non-wetting interface = 11. Medial axis representation of the contact line in threferences to color in this figure legend, the reader is referred to the web version of thi

tion dx = 0.08R, and instead of two lines, we see a single thick lineof voxels for contact line where the pendular ring is. When we cal-culate the medial axis in Fig. 6d we obtain a single contact line.Moreover, we also see incomplete paths of contact line elsewhereon the sphere surfaces at this coarser resolution. Therefore, largervoxel sizes are not able to resolve the contact line associated withpendular rings and lead to an underestimate of the contact length.We computed contact line length accurately previously for a voxelsize of dx = 0.08R for the case of a pendular ring between twospheres of radius R (see Table 1). The main difference here is thecurvature of the wetting–non-wetting interface (C = 1.5 in Table1 vs. C = 11 in Fig. 6). At larger curvatures, the radius of the ringof contact line decreases, bringing the ring closer to the locationof minimum separation between the two spheres. For coarse reso-lution we have smaller number of voxels defining the space be-tween the grains and the contact line associated to each spherecan merge into a single line. This effect would be more noticeableas the width of the gap between spheres decreases.

This analysis suggests that the reason why the length is almosttwice as large at the last steps of drainage when using larger voxelsize (Fig. 5) is because most of the contact line at these points isassociated with pendular rings.

An image of the contact line in the last step of drainage for apack of 91 spheres for two different resolutions is shown inFig. 7. The rings of contact line are easier to distinguish for the finerresolution case on the left.

t resolutions (a) dx = 0.04R and (c) dx = 0.08R when dimensionless curvature of thee same pore throat for (b) dx = 0.04R and (d) dx = 0.08R. (For interpretation of thes article.)

Fig. 7. Contact line configuration in the last step of drainage in a pack of 91 spheres for two different resolutions (a) dx = 0.04R and (b) dx = 0.08R.

Fig. 8. Normalized specific contact line length vs. dimensionless curvature duringdrainage and imbibition for a pack of 91 spheres of the same radius R using aresolution dx = 0.04R.

564 E. Rodriguez et al. / Journal of Colloid and Interface Science 368 (2012) 558–577

4.1. Hysteresis and (dis)similarity of contact line lengths and areas

In Fig. 5 we observed that the contact line length in these spherepacks increased as saturation decreased until reaching a maximumvalue at a wetting phase saturation near 20%. Beyond that satura-tion the contact line length started to decrease until the drainageendpoint. The behavior of the contact line length for imbibitionis similar; however in Figs. 4 and 5, the contact line length exhibitssome hysteresis from intermediate to small water saturations(Sw < 0.5) in the case of coarse resolution (dx = 0.08R), the lengthduring imbibition being larger than the length during drainage ata given water saturation. The better resolution simulation suggestsno major hysteresis; still the contact line length curves for drain-age and imbibition do not completely match.

During imbibition, the decrease in capillary pressure is causingthe pendular rings to expand and therefore the length associated tothem increases. Recall that the imbibition simulation assumes thatall pendular rings are connected via wetting films to the bulk wet-ting phase and thus can expand in response to a decrease in capil-lary pressure. If imbibition starts from a drainage endpoint wheremost of the wetting phase is present in form of pendular rings (aswe saw is the case in these sphere pack cases), the shrinkage of thependular rings that occurred during the last steps of drainage is re-versed in the first steps of imbibition. The fluid/fluid interfacesthus are following reversibly the last steps of drainage and no sig-nificant hysteresis in contact line length is expected. A pendularring will keep expanding with the decrease in curvature until asnapoff event occurs that forms a meniscus. Because imbibitionand drainage events of the same pore do not occur at the same cur-vature we see hysteresis in the curvature–saturation plot. To see ifthis translates into hysteresis for the contact line length, in thenext sections we will analyze contact line behavior in simplegeometries.

We are going to further investigate dependence of contact linelength by looking at its behavior with curvature (or, equivalently,capillary pressure), rather than saturation. Plots of contact linelengths against saturation should not be construed as claims thatlengths only depend on saturation. Contact lines, areas, curvaturesand saturations are interrelated in a complex manner and thus notwo-dimensional plot between any pair cannot be expected to givea functional relationship. We exemplify this using the finer resolu-tion (dx = 0.04R) simulations already presented in Fig. 5. Fig. 8shows the corresponding lines of contact line length vs. dimension-less curvature for the same pack of 91 spheres of Fig. 4a.

Fig. 8 shows that contact line length does show hysteresis withrespect to curvature being the length larger for imbibition for cur-

vatures between 2 and 8. At large curvatures (8 < C < 12) corre-sponding to small wetting phase saturations we do not observemajor hysteresis since as the fluid displacement is following thereversible path of the last steps of drainage: the pendular ringsare expanding but not coalescing. The curvature C � 8 indicatesthe point where irreversible events (during previous drainage ofpores) may have happened, or are taking place now during imbibi-tion (snap-off in throats, meniscus merger in pores).

The curves of interfacial area vs. wetting phase saturation inFig. 9 show that interfacial area also exhibit hysteresis during imbi-bition, being larger than the area during drainage at a given wet-ting phase saturation. It is further instructive to plot interfacialarea vs. contact line length. We calculate normalized (dimension-less) specific area as follows:

Normalized specific area ¼Aw—nw

Grain surfaceBulk volumeðGrain radiusÞ3

ð10Þ

where grain surface is equal to 4pR2.The results are shown in Fig. 9b where we see regions where the

increase in contact line length is associated to an increase in inter-facial area as well as regions where the increase in contact linelength is associated to a decrease in interfacial area. Observation

Fig. 9. (a) Wetting–non-wetting interfacial area vs. water saturation for a computer generated pack of 91 spheres of radius R using a resolution dx = 0.04R. (b) Normalizedspecific wetting–non-wetting interfacial area vs. normalized specific contact line length for a computer generated pack of 91 spheres of radius R using a resolution dx = 0.04R.

E. Rodriguez et al. / Journal of Colloid and Interface Science 368 (2012) 558–577 565

of the two plots in Fig. 9 shows that there is no simple functionalrelationship between any pair of properties.

The hysteresis observations interfacial areas are consistent withthose by Culligan et al. [12] and Chen et al. [29], but we add an-other dimension (contact line area) to the discussion. We next ex-plore both areas and contact lines in a simple pore where, asopposed to larger systems, the visualization of areas and lengthsis tractable.

4.1.1. Analysis of interfacial area and contact line length in a singlepore

We can understand why the wetting–non-wetting interfacialarea shows hysteresis (with respect to saturation) during simula-tions of drainage and imbibition in computer generated packs ofspheres by analyzing the behavior of the wetting–non-wettinginterface during drainage and imbibition in a simple pore.

In Fig. 10a we see all drainage steps in a simple 2D pore. Betweenthe first position of the interface (step 1) and the rest of the drainagesteps there is an irreversible jump (Haines jump, [30]). The jumpcauses the initial single meniscus to split into two different ones(one in each adjacent throat). After that jump, the next 19 drainagesteps are small and reversible. If drainage continues from step 20,then the throat in the upper right corner drains in another irrevers-ible interface jump. If on the other hand the drainage simulation ishalted at step 20 and imbibition is simulated from that step,Fig. 10b, interfaces will reversibly trace the path followed duringdrainage for several steps. When the interface returns to positioncorresponding to the second drainage step, however, the next imbi-bition step does not reverse the Haines jump to the pore throat onthe left side of the domain. Instead the interfaces continue advanc-ing into the pore body until two menisci merge at the tip of the righthand side solid disk. This causes an instability (Melrose type of theimbibition event [31]) and the interface will irreversibly jump to anew location (in this case, outside the pore/geometry shown – theentire pore imbibes). If we plot interfacial area for these processes(as shown in Fig. 10d and e), we see the higher areas during imbibi-tion beyond the Haines jump point. For imbibitions from step 21the difference in areas is even more prominent because of twoHaines jumps during drainage.

We repeated the same exercise in a 3D pore body included be-tween 4 same radius spheres having 4 pore throats of differentsizes, shown in Fig. 11. We did four drainage simulations with thispore, starting drainage through each of the four pore throats. The

resolution for the simulation was dx = 0.04R. Fig. 12 shows theinterfacial area vs. water saturation for drainage and imbibitionthrough the pore throat indicated with an arrow in Fig. 11. Thedrainage finished after 17 steps in this case and two Haines jumpsoccurred, between steps 9 and 10 and between steps 14 and 15.Imbibition was started from step 17. Interfacial areas for drainageand imbibition coincide until the point corresponding to theHaines jump at step 15 (the interfacial areas for imbibition followthe line of steps 17, 16 and 15 of drainage), as happened for the 2Dcase.

A multiplicity of such jumps during transition between equilib-rium states integrated over multiple pores in a porous mediumcauses hysteresis of capillary-pressure–saturation curves in gen-eral porous media [11]. Morrow [11] and Gray and Hassanizadeh[15] did a careful theoretical analysis suggesting that the energystored in interfacial areas, and possibly contact line lengths, canbe used to find a more unique relationship between capillary pres-sure, saturation (i.e. volumes), areas and contact line length. Somepreliminary experimental work [12,29] indeed points that mightbe the case. We have added contact line possibility of explicit mea-surements in this paper. Our focus in this paper is verification ofthe methods, and we will focus on exploring these relationshipsby comparing to carefully crafted experiments in our future work.

To study the behavior of the contact line length, we simulateddrainage and imbibition in a simple 3D pore between threespheres in point contact, like the one shown earlier in Fig. 6.The capillary pressure curve and the interracial area vs. saturationcurves are shown in Fig. 13a and b respectively. The interfacialarea–saturation curve in Fig. 13b has been calculated accordingto Eq. (10) in Section 4.1.

An irreversible Haines jump occurs at a curvature of 11, wherethe wetting phase saturation Sw is reduced from 0.53 to 0.09. FromSw = 0.09 the increase in curvature reduces the wetting phase sat-uration slowly in a reversible manner. Imbibition is started fromthe drainage endpoint and the initial decrease in curvature makesthe wetting fluid to follow the reversible path of drainage untilreaching the wetting phase saturation of 0.09. At this point, thenext imbibition step does not reverse the Haines jump and theinterface. The interface keeps advancing through the pore throatuntil reaching a wetting phase saturation equal to 0.57, after whichpoint the pore is completely imbibed. The curves of normalizedspecific contact line length vs. water saturation and curvature areshown in Fig. 14a and b respectively.

Fig. 10. Drainage and imbibition steps in a 2D pore (in alternating red and green color). Locations of interfaces are shown for (a) all the drainage steps; (b) imbibition fromstep 20, with the starting location shown in blue; and (c) imbibition from step 21. The corresponding trends of interfacial area vs. water saturation are shown for (d)imbibition from step 20. (e) Imbibition starting from step 21. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of thisarticle.)

Fig. 11. Geometry of a pore body between four spheres used to simulate drainageand imbibition. Simulations have been conducted starting drainage from all fourpossible pore throats.

Fig. 12. Interfacial area vs. water saturation for drainage and imbibition in a porebody between four identical spheres.

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Fig. 13. (a) Curvature vs. wetting phase saturation for drainage and imbibition in a pore between three spheres of radius R in point contact (dx = 0.04R). (b) Normalizedspecific wetting–non-wetting interfacial area vs. curvature for drainage and imbibition in a pore between three spheres of radius R in point contact (dx = 0.04R).The dottedline indicates a Haines jump.

Fig. 14. (a) Normalized specific contact line length vs. wetting phase saturation for drainage and imbibition in a pore between three spheres of radius R in point contact(dx = 0.04R). (b) Normalized specific contact line length vs. curvature for drainage and imbibition in a pore between three spheres of radius R in point contact (dx = 0.04R). Thedotted line indicates a Haines jump.

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The contact line length–saturation curve in Fig. 14 exhibitssome hysteresis after wetting phase saturation equal to 0.09,where the Haines jump occurred during drainage. However thehysteresis is more noticeable in the contact line length–curvatureand interfacial area–saturation plots.

In the contact line length–curvature plot (Fig. 14b), contact linelengths for drainage and imbibition start to differ at a curvature of11, where the Haines jump occurs in drainage. At the next curva-ture during imbibition, the pore does not imbibe, and it still sup-ports the advance of the non-wetting phase, generating contactline. The first coalescence event during imbibition does not occuruntil a curvature equal to 8.25, and the pore eventually imbibesat a curvature of 4.65.

In the contact line length–saturation plot (Fig. 14a), at smallersaturations at the beginning of imbibition the fluid is followingthe reversible path of the last steps of drainage. During imbibition,

the pore is not suddenly imbibed at a wetting phase saturation of0.09, rather the interface continues advancing until reaching asaturation of 0.57 where the pore imbibes. Therefore from satura-tion 0.09–0.57 in imbibition the advancing meniscus is generatinginterfacial area that was ‘‘skipped’’ during drainage because of theHaines jump. This causes the hysteresis in interfacial area shown inFig. 13b.

We can look at the interface and contact line configurations at awetting phase saturation of 0.57. For drainage, this saturation isthe point where the non-wetting fluid is about to drain the pore.For imbibition, at that point the non-wetting fluid has been stea-dily moving through the pore body since the beginning. The config-uration of the interfacial phase is shown in Fig. 15 where we seehow different the status of drainage and imbibition is at the samewater saturation. The normalized specific interfacial area (Aw–nw/VB) is equal to 0.017 for drainage and 0.047 for imbibition.

Fig. 15. Wetting non-wetting interfacial area configuration at the same wetting phase saturation Sw during (a) drainage and (b) imbibition in a pore throat between threespheres in point contact. (a) Drainage, Sw = 0.57, C = 9, Aw–nw/VB = 0.017. (b) Imbibition Sw = 0.57, C = 4.7, Aw–nw/VB = 0.047.

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The corresponding contact line configuration is shown inFig. 16. Even though the configuration is completely different fordrainage and imbibition the value of (LcD/VbD) does not differ much,being equal to 3.3 for drainage and 3.9 for imbibition for Sw = 0.57.Therefore a larger interfacial area does not necessarily imply a pro-portionately larger contact line length.

This exercise suggests that the increase or decrease in interfa-cial area during drainage and imbibition processes in a pore bodybetween four same-size spheres does not necessarily lead to a var-iation in contact line length of the same proportion. At the samewetting phase saturation there is a 60% increase in area from drain-age to imbibition (Fig. 15) while there is a 15% increase on contactline length (Fig. 16). Still, contact line length shows significanthysteresis as a function of curvature or as a function of wetting–non-wetting interfacial area in single pores and sphere packs.Please refer to [32] for a further investigation on contact line lengthhysteresis.

The contributions of pendular rings and meniscus to contactline length may give some insight into the hysteresis (or lack ofthereof) in sphere packs. In the next section, we make an analyticalestimate of contact line length in a computer generated pack ofspheres that involves counting the number of pendular rings andmeniscus at every step of drainage.

Fig. 16. Wetting non-wetting contact line configuration at the same wetting phase satspheres in point contact. (a) Drainage, Sw = 0.57, C = 9, LcD/VbD = 3.3. (b) Imbibition Sw = 0

4.1.2. Analytical estimate of contact line length in a computergenerated pack of spheres

We estimated an analytical solution for the contact line lengthin every step of drainage in a computer generated pack of spheres.We started by assuming that at a given saturation the contact lineis associated to pendular rings and menisci. Depending on thestage of drainage there will be more pendular rings or more menis-ci present. The drainage simulation for this estimate is indepen-dent of the LSMPQS approach, being based on a pore networkmodel [33]. From the network model we counted the number ofcomplete pendular rings (i.e. rings surrounded by non-wettingphase) and assumed that all of them are between spheres in pointcontact. This is a reasonable approximation since the average ofpoint contacts per sphere in packs of randomly distributed spheresof the same size has been estimated to be close to 6 (5.8 in [34], 5.9in [35], 5.6 in [36]). Then the contact line was calculated analyti-cally for the given curvature and assuming a toroidal shape forthe pendular ring. We also counted the number of menisci at everydrainage step in the computer generated pack of spheres and esti-mated the amount of contact line associated with a meniscus.

Fig. 17 shows the contact line length associated to pendularrings and menisci, together with the total contact line (sum of me-nisci and pendular rings length).

uration Sw during (a) drainage and (b) imbibition in a pore throat between three.57, C = 4.7, LcD/VbD = 3.9.

Fig. 17. Normalized specific contact line length vs. wetting phase saturation from anetwork model simulation of drainage in a computer generated pack of spheres ofradius R showing the contribution of pendular rings and menisci at every step.

Table 2Porosity, water saturation and normalized specific contact line length for differentpacks of glass beads (I = 100% hydrophilic, E = 50% hydrophilic 50% hydrophobic,O = 100% hydrophobic) obtained from CT imagesa by means of LSMPQS and 3DMA-Rock analysis as described in the text.

Sample Porosity (%) Sw LcD/VbD

I2 37.5 0.0775 1.04I3 38.0 0.0805 1.34E2 37.0 0.0553 0.60E3 36.2 0.0566 0.64O1 38.0 0.036 0.27O2 37.9 0.0347 0.28

a Courtesy of Dr. Willson of Louisiana State University.

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The contribution of the pendular rings to the total contact linelength is larger than the contribution of the menisci at wettingphase saturations smaller than 0.38. At saturations smaller than0.2 the contact line length associated to pendular rings starts todecrease because the effect of the increase in curvature (whichreduces contact line length) overcomes the number of new ringsformed. Nevertheless their contribution to contact line length isstill larger than the contribution of the menisci. This suggests thatthe contact line length is dominated by pendular rings at low watersaturations while the contribution of menisci and pendular ringsbalances at higher wetting phase saturations.

During imbibition, coalescence of two or three pendular rings ascapillary pressure decreases creates one meniscus. On the otherhand, a Melrose event (two menisci that merge) destroys menis-cus, because the merger yields one remaining meniscus and leavesno pendular ring. Our simulations of imbibition in granular mate-rials suggest that coalescence of pendular rings is rare during imbi-bition (also showed by Gladkikh and Bryant [37]), because menisciin pore throats tend to merge at larger curvatures (and thus causepores to imbibe) than the curvatures at which rings coalesce.Therefore at the beginning of imbibition (small Sw) and until mostof the pendular rings have merged as pores imbibe, the contact linewill be dominated by the contribution of the pendular rings. Theearly period corresponds to the growth of the pendular rings,reversing the shrinking that happened during drainage. Accordingto Fig. 5 for a sphere pack and Fig. 14 for a single pore, this is trueuntil Sw , 0.15, where we see that the contact line length andinterfacial areas for drainage and imbibition are almost identical(first 3 points at low Sw in Figs. 5 and 14). At Sw > 0.15 we havemeniscus advancing through the pores generating interfacial area,as shown in Fig. 13b for the single pore case, until they start tomerge (Melrose event) and the interfacial area starts to decrease.

The interfacial area associated to a meniscus is larger than thearea associated to a pendular ring, as we showed in Fig. 15, andfor that reason interfacial area at intermediate saturations reachesa maximum. However we also saw that a larger interfacial area doesnot necessarily means larger contact line (Fig. 13b vs. Fig. 14a;Fig. 16) therefore the number of new menisci created duringimbibition that greatly contribute to the increase in interfacial areabefore they coalesce, do not contribute in the same magnitude to aincrease in contact line length as the interface moves through thethroat. Also, at the same saturation we have smaller curvature forimbibition. Smaller curvatures increase the contact line length of

pendular rings, but not as much the length associated to a menis-cus. This is shown by the line for imbibition in Fig. 14b, before coa-lescence. Thus to notice an increase in the contact line length duringimbibition, new pendular rings would have to be created with thedecrease in curvature, which clearly is not the case. Presumably itis the coalescence of pendular rings into menisci that creates theapproximate balance (lack of hysteresis) during imbibition insphere packs (cf. Fig. 5).

In the next section we analyze contact line length from high-resolution images of real porous media and compare the resultswith the contact line length from model porous media.

5. Results of contact line length from high resolution X-rayimages of porous media

5.1. Contact lines in high resolution X-ray images of glass beads ofdifferent hydrophobicity

We analyzed high resolution X-ray CT images of drainage end-points in porous media made of glass beads of diameters rangingbetween 0.3 and 0.42 mm (courtesy of Dr. Willson from LouisianaState University). The images are of size 3003 with a voxel sizeequal to 10.92 lm. Assuming an average sphere diameter of0.36 mm, the resolution of the CT image in terms of sphere radiusR is 0.06R. Three different types of packs were analyzed, one con-taining 100% hydrophilic beads, one containing 50% hydrophilicand 50% hydrophobic beads and a last one containing 25% hydro-philic and 75% hydrophobic beads. From these images we can ex-tract the contact line between air water and solid phases andcompare with the results in simulations of drainage in computergenerated packs of spheres.

Table 2 shows the results of the contact line length for the dif-ferent bead packs extracted from their corresponding images,together with the porosity and the water saturation of the pack.These last two characteristics of the bead packs were calculatedby voxel count.

We observed that at these water saturations the majority of thecontact line exists as isolated paths (Fig. 18), which we believe be-long to pendular rings. It is not possible to see complete pendularrings because of the error due to digitization (‘‘smoothing’’ of dig-itized inputs).

In Fig. 19 we compared the contact line length extracted fromthese microtomography images with the contact line length calcu-lated from LSMPQS simulations in sphere packs. Since in the spherepacks we assume a strongly water wet surface, the direct compar-ison is made with the results from the packs of 100% hydrophilicbeads. The contact line length extracted from the images with aresolution of dx = 0.06R is closer to the results of LSM simulationfor a poor resolution (dx = 0.08R). As we could see in Fig. 18a thedouble rings of contact line around pendular rings are not apparentbecause of digitization effect. Digitization cause information to be

rings

Fig. 18. Contact line configuration in (a) a pack of 100% hydrophilic beads (sample I2 in Table 2) and (b) a pack of 50% hydrophilic beads (sample E3 in Table 2).

Fig. 19. Normalized specific contact line length vs. water saturation from drainageand imbibition simulation in a sphere pack compared with the contact line lengthextracted from microtomography images of glass bead packs (I2 and I3 in Table 2).

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lost since all the information that we have at a voxel in the digi-tized (segmented) image is whether it belongs to a phase or an-other, making the interface to show a ‘‘staircase effect.’’ Eventhough LSM simulation has a ‘‘reinitialize’’ routine available tosmooth the surfaces, this still translated in loss of information.For this reason, even though the resolution of the CT image(dx = 0.06R) is better than the resolution of our coarse simulation(dx = 0.08R) the results appear to be closer. Thus it is likely that ahigher resolution image would have yielded a larger contact linelength, which would agree well with the sphere pack simulation.

5.2. Contact lines in high resolution X-ray images of sedimentary rocks

We have also tested the performance of the algorithm for con-tact line length calculation in images of sedimentary rocks. Highresolution images of Fontainebleau sandstone and sucrosic dolo-mite samples were provided by Dr. Knackstedt of AustralianNational University. The dry images (solid and air) were comple-mented with wet images (solid, air and water), corresponding tothe last step of imbibition for the sandstone sample and the laststep of drainage for the dolomite sample. Fig. 20 shows somesample slices from these images.

The data of these images are contained in cubes of 5003 voxels,each of size 3.5 lm. We simulate drainage and imbibition in these

porous media geometries using LSMPQS. Of primary interest hereis whether the relative magnitudes of the contact lines in thesesedimentary rocks are comparable to the trends observed for themodel sediments. Because of computational time limits we simu-lated drainage and imbibition in 2503 subsamples of a 5003

sample.Fig. 21a and b shows the images for two 2503 subsamples of

sandstone and dolomite respectively.We calculated the contact line length using the same approach

as described above for the computer generated packs of spheres.Fig. 22 shows simulated contact line vs. water saturation for 2503

subsamples of sucrosic dolomite and Fontainebleau sandstone.The normalized specific contact line length shows hysteresis, beinglarger for imbibition than for drainage especially at low watersaturations. This behavior was not observed in the sphere packs(cf. Figs. 4a and 5).

Fig. 23 shows the configuration of the contact lines in a step ofdrainage in one of the Fontainebleau sandstone subsamples. Pairsof circles of contact line around pendular rings are not as easy toidentify in these samples as for the model sediments because ofthe more complex geometry of the pore space. The resolution orvoxel size for the simulation is given by the resolution of the imageand cannot be changed. Taking an average grain size (diameter) of250 lm for Fontainebleau sandstone (measured from the image)the resolution for the Fontainebleau sandstone images is dx =0.03R, therefore the contact line is comparable with the simulationsat resolution dx = 0.04R. Comparing with the results for contact linelength in the periodic pack of spheres (recall Fig. 5) we observe thatthe normalized specific contact line lengths for drainage are similarin the sandstone, dolomite and sphere pack simulations.

Fig. 24 shows the curves of contact line length during drainagecurves for these three materials and Fig. 25 shows the correspond-ing curves for imbibition. A main difference between the three por-ous media is the saturation at which the maximum length occurs.This saturation is smaller in the sphere pack than in the sedimen-tary rocks, during both drainage and imbibition displacements.

The magnitude of the maximum length is larger in the spherepacks than in the rocks for drainage. For the sphere packs, sincethere is no hysteresis in contact line length, the maximum lengthduring imbibition is reached at similar saturation, and it has simi-lar value, than during drainage.

However for the rocks we observe significant hysteresis, thecontact line length being much larger during imbibition than dur-ing drainage. For sandstone and dolomite we do not have clearlydefined pendular rings because of the complex geometry, and thuswe do not see the reversible first steps at the beginning of imbibi-tion observed in the sphere packs (Fig. 5). In this case drainage

Fig. 20. (a) 500 � 500 slice of dry Fontainebleau sandstone, (b) 500 � 500 slice of dry sucrosic dolomite, (c) wet Fontainebleau sandstone at the last step of imbibition, (d) wetsucrosic dolomite at the last step of drainage. White: grains, black: air, gray: water (courtesy of Dr. Knackstedt of Australian National University).

Fig. 21. (a) 2503 subsample of Fontainebleau sandstone and (b) 2503 cube subsample of sucrosic dolomite.

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concludes after several pores are drained (Haines jumps) and nopendular rings are created (notice that the drainage endpointoccurs at larger wetting phase saturations for the rocks inFig. 24). Therefore we start imbibition after the Haines jumpsand as shown in Fig. 14b there is a large difference in contact linelength for the same curvature.

For clarification, we also computed the interfacial area betweenwetting and non-wetting phases for drainage and imbibition inthese samples and plot interfacial area vs. wetting phase saturation(Fig. 26) and contact line length vs. interfacial area (Fig. 27) for the

same sandstone and dolomite samples as in Fig. 22. Notice that inFig. 27 we are plotting the specific normalized area (AD/VbD)whereas we were plotting interfacial area normalized by solid areain Fig. 26 (the difference is only in the scaling parameter).

As was the case for computer generated packs of spheres, theinterfacial area during imbibition is larger than the area duringdrainage for both sandstone and dolomite in Fig. 26. Also, duringimbibition, an increase in interfacial area is not associated withan increase in contact line length, as shown in Fig. 27. The differ-ence of this last result with respect to the sphere packs (cf.

Fig. 22. Normalized specific contact line length vs. water saturation for LSMPQSsimulations of drainage and imbibition in (a) 2503 sample of Fontainebleausandstone and (b) 2503 sample of sucrosic dolomite.

Fig. 23. Contact line (green) and non-wetting phase (red) during drainage in a 2503

subsample of Fontainebleau sandstone (Sw = 0.55, curvature = 3.66). (For interpre-tation of the references to color in this figure legend, the reader is referred to theweb version of this article.)

Fig. 24. Comparison normalized specific contact line length vs. water saturation forsimulations of drainage in a sphere pack (Fig. 5, for dx = 0.04R), a 2503 sample ofFontainebleau sandstone (Fig. 22a), and a 2503 sample of sucrosic dolomite(Fig. 22b).

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Fig. 9b)) is noticeable, as we observe here how both contact linelength and interfacial area keep increasing at the start of imbibi-tion, while only area had a large increase during imbibition inthe sphere packs.

In Fig. 26 we observe a similar interfacial areas trend to that ofsphere packs (recall Fig. 9): the interfacial area is larger for imbibi-tion than for drainage and reaches a maximum value around 0.15.However, the main difference is that the interfacial area at theimbibition endpoint is larger for sandstone and dolomite than forthe sphere packs (compare a normalized interfacial area value of0.03 at (1 � Snwr) = 0.96 for a sphere pack in Fig. 9 with a value ofabout 0.10 for both rocks at (1 � Snwr) = 0.8 in Fig. 26).

The reason for this discrepancy is the larger imbibition endpointsaturation and therefore larger number of blobs of trapped non-wetting phase in the sandstone and dolomite packs than in thesphere packs, which create more interfacial area than the connectedbulk phases, which can be seen in Fig. 28 where the non-wettingphase configuration it is shown at a wetting phase saturation ofSw = 0.91 for a sphere pack (from imbibition in Fig. 9a) andSw = 0.78 for Fontainebleau sandstone (last step of imbibition inFig. 26a). The normalized interfacial wetting–non-wetting areasfor these cases are 0.05 and 0.10 respectively. Notice that some ofthe trapped blobs in the sandstone span multiple pores. These

trapped blobs also contribute to more contact line than the mainbulk phase, therefore explaining the hysteresis at the end ofimbibition.

Further investigation of the hysteresis phenomena in sedimen-tary rocks is shown in [32].

5.2.1. Analysis of wet images of Fontainebleau sandstone and sucrosicdolomite

Segmented files of wet images corresponding to the last step ofdrainage in the dolomite and the last step of spontaneous imbibi-tion in the same Fontainebleau samples were also provided. Thecontact line from the wet images was extracted and its lengthwas calculated and compared with the length that resulted from

Fig. 25. Comparison normalized specific contact line length vs. water saturation forsimulations of imbibition in a sphere pack (Fig. 5, for dx = 0.04R), a 2503 sample ofFontainebleau sandstone (Fig. 22a), and a 2503 sample of sucrosic dolomite(Fig. 22b).

Fig. 26. Interfacial area between wetting and non-wetting phases vs. saturation forsimulations of drainage and imbibition in 2503 samples of (a) Fontainebleausandstone and (b) sucrosic dolomite.

Fig. 27. Normalized specific wetting–non-wetting interfacial area vs normalizedspecific contact line length for simulations of drainage and imbibition in 2503

samples of (a) Fontainebleau sandstone and (b) sucrosic dolomite.

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the LSMPQS simulation of drainage (or imbibition) in the samesubset of the dry sample. Tables 3 and 4 show the results.

The contact line extracted from the image is always larger thanthe one from LSMPQS simulation, but this difference is consider-able in the case of the drainage endpoint in dolomite, where thecontact line length from the image is around five times larger.The wet image for dolomite in Fig. 20d corresponds to the last stepof drainage (at a wetting phase saturation of 0.22). In Fig. 29 wecompare the configuration of the wetting phase from the image(a) with the configuration for our LSMPQS simulations of drainagein this rock (b), in a 1003 subsample.

After observation of the wetting phase configurations we con-clude that the main cause of the difference between contact linelength computed directly from the wet image and contact linelength computed from the displacement simulation is the presenceof wetting phase as thin films covering the grains. LSMPQS doesnot capture these films since they incorporate physics from a scalenot captured by our current resolution. Fig. 30a and b shows theconfiguration of the contact line and non-wetting phase in a subsetof the wet sample of Fontainebleau and the configuration from sec-ondary imbibition simulation in the same subsample at similar

Fig. 28. Top view of the trapped non-wetting phase configuration at imbibition endpoint in (a) computer generated pack of spheres (dx = 0.08R) Sw = 0.91. (b) Fontainebleausandstone, Sw = 0.78.

Table 3Normalized specific contact line length calculated from the wet images (full sample).

Sample Porosity (%) Sw LcD/VbD

Dolomite 21.43 0.22 8.78Fontainebleau 19.33 0.64 2.10

Table 4Normalized specific contact line length calculated from subsamples of the wet imagesand normalized contact line length from drainage and imbibition simulations.

Sample Porosity (%) Sw LcD/VbD (LSM) LcD/VbD (from image)

DM1 (sub 110) 22.13 0.21 1.70 8.90DM2 (sub101) 21.02 0.21 1.95 8.88DM3 (sub 111) 22.90 0.20 1.30 7.67FB1 (sub 001) 18.73 0.63 1.50 2.11FB2 (sub 011) 19.54 0.61 1.65 2.28FB3 (sub 010) 19.78 0.66 1.60 1.92

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water saturation respectively. While the configuration of the non-wetting fluid is reasonably similar in both cases, we can see more‘‘density’’ of contact lines in the result from the image. This ac-counts for the larger values of normalized specific contact linelength in the images.

Fig. 29. Wetting phase configuration in a 1003 subsample of sucros

The water films are much more evident in the images of thedolomite than for sandstone, as shown in Fig. 31. The more angulargeometry of the dolomite makes the wetting fluid to remain in theroughness of the grain as thin films to a greater extent than in thesandstone. Also, more water remains in form of films and pendularrings at low water saturations (20% in this dolomite sample) thanat larger saturations (60% for the sandstone sample). Because thefilms are thin, about 1 or 2 voxels thick between the solid andthe main non-wetting fluid phase, most of the film voxels are iden-tified by the image processing algorithm as contact lines. Conse-quently the reported value of contact line length is much largerin the images than in the simulations in the same void space asthe image.

Fig. 32 shows the contact line in a close up view of sucrosicdolomite. Patches of contact line with a grid-like structure are evi-dent. These presumably correspond to water films. The actual con-tact line will be only the perimeter of these patches. While theLSMPQS method is able to identify pendular rings, we are currentlyworking in correctly identify water films. Notice that these waterfilms did not exist for the case of the sphere pack because of thesmoothness of the spheres.

Fig. 33 shows the contact line length computed from processingthe wet images that correspond to the void space of the same 2503

subsamples of sandstone and dolomite where we run the simula-tions of drainage and imbibition (shown in Fig. 22).

ic dolomite (a) from image and (b) from simulation (Sw = 0.22).

Fig. 30. (a) Contact line (green) and non-wetting phase (red) in a 2503 subsample (sub 001; cf. Table 4) of Fontainebleau sandstone extracted from the wet image (Sw = 0.63).(b) Contact line in the same 2503 cube subsample from the simulation step at similar water saturation (Sw = 0.60). (For interpretation of the references to color in this figurelegend, the reader is referred to the web version of this article.)

Fig. 31. (a) Contact line (green) and non-wetting phase (red) in a 2503 subsample (sub 110, cf. Table 4) of sucrosic dolomite extracted from the wet image (Sw = 0.22). (b)Contact line length in the same 2503 subsample from the LSMPQS simulation step at similar water saturation (Sw = 0.21). (For interpretation of the references to color in thisfigure legend, the reader is referred to the web version of this article.)

Wetting phase films

Fig. 32. Contact line (green) extracted from image shown over grain surface in a1003 subset of sucrosic dolomite. (For interpretation of the references to color inthis figure legend, the reader is referred to the web version of this article.)

E. Rodriguez et al. / Journal of Colloid and Interface Science 368 (2012) 558–577 575

We attempted to estimate the amount of contact line associatedwith water films in the wet images corresponding to the drainageendpoint in sucrosic dolomite. We ‘‘coarsened’’ the segmented wet

dolomite images by replacing boxes of a given number of voxels byonly one voxel whose value reflects the majority on the box. Thislocal averaging tends to eliminate wetting films (the voxels tendto be reassigned as solid or as non-wetting phase) but preserve vol-umes of the wetting phase associated with the bulk phase. Thecoarsened image provides a better basis for comparison with ourLSMPQS calculations since the latter only accounts for the contactline associated to the bulk phase and trapped phases. The differ-ence between the contact line length associated to the main bulkphase and the total contact line from the image yields the contactline associated to wetting films. The percentage of contact lineassociated to wetting films calculated in this way ranged between80% and 90%. This amount of films is enough to explain the differ-ence between the contact line length from simulation and fromimages for dolomite in Fig. 33b. The percentage of contact lineassociated to the wetting film is expected to be smaller for the Fon-tainebleau imbibition endpoint, since less amount of wetting filmspresent at larger wetting phase saturations. To validate this state-ment we would need wet images of the imbibition endpoint in thissandstone sample to estimate a percentage contact line associatedto wetting films in a similar manner than what we did for the dolo-mite sample.

Fig. 33. Same simulations as Fig. 22 (solid curves) for the normalized specificcontact line length vs. water saturation showing the contact line length estimatedfrom the wet images as a green star in both plots. (a) 2503 sample of Fontainebleausandstone. (b) 2503 sample of sucrosic dolomite.

576 E. Rodriguez et al. / Journal of Colloid and Interface Science 368 (2012) 558–577

6. Conclusions

We have been able to identify and quantify the length of thecontact line between wetting, non-wetting and solid phases duringdrainage and imbibition processes in different models for porousmedia using a level set based method (LSMPQS). In addition tocomputer generated models of sediments we also tested the con-tact line calculation in model geometries extracted from high res-olution images of real sediments and sedimentary rocks.

The samples analyzed through high resolution images were col-umns of glass beads of different hydrophobicity, Fontainebleausandstone and sucrosic dolomite samples. Besides simulatingdrainage and imbibition in these media to estimate contact lines,we also extracted the contact lines from high resolution imagesof wet porous media, at their drainage or imbibition endpoints.These are the first estimates of the contact line length in realisticporous systems to date.

The contact line length was normalized in order to provide anintrinsic quantity and facilitate comparison of systems of differentsize and geometry. The normalized specific contact line length isremarkably similar for drainage displacements in sphere packsand rocks. However, while no hysteresis was observed for imbibi-tion in sphere packs, the contact line lengths for imbibition dis-

placements in rocks were significantly larger than the lengths fordrainage.

The main factor affecting the computation accuracy of the con-tact line length is the image resolution or voxel size for the simu-lation. In unconsolidated systems (sphere packs), most of thecontact line at small wetting phase saturations exists in pendularrings between spheres in point contact or having small gaps be-tween them, and a fine resolution is necessary to distinguish thetwin circles of contact lines associated with the pendular rings.Our analysis suggests that the contact line associated with pendu-lar rings is the one that dominates drainage and imbibition pro-cesses in sphere packs.

For the dolomite and sandstone cases, the high resolution of thesample images allowed the identification of water films, which sat-isfy our algorithm’s definition of contact lines. In this case, thecomparison of the level set method simulation and result fromthe images give a corrected contact line estimate as well as an esti-mate of the amount of water films.

Our algorithm applied to LSMPQS simulations of drainage andimbibition passes several validation tests: it predicts lengths smal-ler than an upper bound obtained by independent calculation (forsphere packs), it is consistent with observations in glass beadpacks, and it is consistent with observations in real rocks. There-fore we conclude that our calculated lengths are reasonablyaccurate.

Knowledge of the contact line lengths is useful for investigatingtrapping of colloids during transport in the unsaturated zone ofsediments (shallow aquifers) or in hydrocarbon reservoirs (oil/water or gas/water). In an upcoming work we use contact linelength calculations in sphere packs to estimate trapping of colloidsin unsaturated columns of glass beads. The energy associated tocontact lines is also a factor to consider in the theories of thermo-dynamics of multiphase flow.

Acknowledgments

Funding for the work reported in this paper was provided by theNational Institute of Food and Agriculture, US Department of Agri-culture, under Agreement No. 2007-35102-18162. Any opinions,findings, conclusions, or recommendations expressed in this publi-cation are those of the author and do not necessarily reflect theview of the US Department of Agriculture. We are grateful to Prof.Clinton Wilson of Lousiana State University and Prof. Mark Knack-stedt of Australian National University for providing us with theexperimental data to analyze.

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