estimating particle size distributions from a network model of porous media

Ž .Powder Technology 104 1999 169–179www.elsevier.comrlocaterpowtec

Estimating particle size distributions from a network model of porousmedia

Toby Mathews a, G. Peter Matthews a,), Stephen Huggett b

a Department of EnÕironmental Science, 602 DaÕy Building, UniÕersity of Plymouth, Drake Circus, Plymouth, PL4 8AA, UKb School of Mathematics and Statistics, UniÕersity of Plymouth, Drake Circus, Plymouth, PL4 8AA, UK

Received 3 July 1998; received in revised form 9 February 1999; accepted 9 February 1999

Abstract

A new method is described for estimating the particle size distribution of consolidated porous materials using a network model of voidstructure based on mercury porosimetry. It is compared to the only other comparable method, a modification of that developed by Mayer

w Ž . xand Stowe R.P. Mayer, R.A. Stowe, J. Coll. Sci. 20 1965 893 for obtaining the radii of uniform spheres in regular packings. Themethods are tested on unconsolidated materials, namely four very different sands, a sample of glass beads of narrow size range and abinary mixture of the coarsest and finest of the sands. The new method compares favourably with the modified Mayer and Stowetechnique when estimating the particle sizes of the unmixed sands, and is an improvement for the glass beads and the sand mixture. Thenetwork model used to estimate particle size distributions also successfully models the experimental liquid permeabilities of the samplesto within one order of magnitude. q 1999 Elsevier Science S.A. All rights reserved.

Keywords: Particle size; Porosimetry; Network model; Mercury intrusion; Porous media

1. Introduction

During the preparation and development of porous ma-terials, or in the study of the migration of fluids throughthem, it is often of interest to be able to infer the particle

Ž .size distribution PSD of the hypothetical unconsolidatedsolid phase. Examples would be the inference of theparticle size distribution of the sand grains that formed asandstone prior to cementation, or the individual sizes ofthe carbon granules which, when consolidated, form afilter or a catalyst substrate.

Exact solutions have long been available which allowthe calculation of the particle size distribution of regularlypacked mono-sized spheres, derived from the pressurersaturation characteristics of a wetting fluid such as water

w xor benzene 2 , or of a non-wetting fluid such as mercuryw x1 , the latter calculation being currently available in the

w xcontrol software for mercury porosimeters 3 . However,the mathematics for the deduction of the size distributionof poly-disperse andror randomly packed particles is in-

) Corresponding author. Tel.: q44-1752-233021; Fax: q44-1752-233021; E-mail: [email protected]

tractable. In previous publications, we have described amethod of inferring an approximate, but nevertheless use-ful, void structure for random porous media, based on the

Žextent of mercury intrusion with applied pressure the.‘mercury intrusion curve’ . In this work, we test whether

this void structure model can be used to infer the particlesize distribution of the assumed-spherical particles of thesolid. The method is tested using porous samples made upfrom unconsolidated grains, from which its likely successon consolidated samples can be judged.

2. Exact solutions for regularly packed spheres

It is useful to consider the approximations, successesand deficiencies of the present method within the context

w xof the exact solutions mentioned earlier. Haines 2 de-scribed in detail the intrusion and extrusion of a wettingfluid around the points of contact of regularly packedspheres. His considerations were expressed mainly as thedevelopment of a capillary ‘pressure deficiency’ plottedagainst Tra, T being the surface tension of the wettingfluid and a the radius of the spheres. The results werecompared qualitatively to real randomly packed, approxi-mately mono-disperse samples such as glass beads, lead

0032-5910r99r$ - see front matter q 1999 Elsevier Science S.A. All rights reserved.Ž .PII: S0032-5910 99 00059-5

( )T. Mathews et al.rPowder Technology 104 1999 169–179170

shot, starch, sand and the confection ‘Hundreds and Thou-sands’.

A more easily usable approach was developed by Mayerw xand Stowe 1 for fluid penetration in packed arrays of

uniform spheres. These packings were defined in terms ofŽ .a single packing angle, s see Fig. 1 , calculated from the

porosity of the sample. The packings have largest accessŽ .openings ‘pore-throat entries’ varying in shape from

Ž .‘square’ for the most porous highest values of s pack-Žings to ‘triangular’ for the closer packed smallest values

.of s structures. A relationship is given which relates theporosity, surface free energy of the mercury and experi-mental breakthrough pressure to a single particle diameter,d , characteristic of all particles in the packing. A singlec

example is given by Mayer and Stowe, in which thecalculated diameter for a packing of glass beads is shownto be within the measured size range for the beads.

w xCurrent mercury porosimeter software 3 extends thisprocedure, applying it to every point on a mercury intru-sion curve rather then a single breakthrough pressure.Effectively this represents the porous material as beingcomposed of spherical shells of regularly packed uniformspheres, with the smallest on the outside and largest in themiddle. A simplified cross-section of such a structure isshown in Fig. 2. The packing angle s is assumed to be thesame throughout the sample. At successively higher pres-sures mercury breakthrough occurs through the progres-sively smaller voids between packings of progressivelylarger spheres until the material becomes saturated.

The aim of this study was to improve upon the concep-tual shortcomings of this approach, termed here the Modi-

Ž .fied Mayer and Stowe MMS method. We do this by firstgenerating a void space structure using our modelingpackage ‘Pore-Cor’, described below. The geometrical in-verse of this void structure is an irregularly shaped solidphase structure. In this work, we determine the sizes of aregularly spaced array of spherical particles which liewithin this solid phase. There are a series of approxima-tions implicit in this calculation, including the simplified

Ž .Fig. 1. a Illustration of the packing angle, s , in an array of uniformŽ . Ž .spheres and examples of b the square and c triangular openings

formed by packing angles of 908 and 608 respectively.

Fig. 2. A cross-section through a solid as represented by the MMSmethod of estimating particle size distributions.

geometry of the void space structure, and the fact that thespheres do not fully represent the solid phase structure. Wehave sought to determine whether, despite these approxi-mations, the new method provides a useful guide to parti-cle size distributions, and one which is better than thecurrent MMS method. The judgment is made by compar-ing the modeling predictions with measurements made bythe suppliers using sieves, and our own measurementsusing a Malvern Instruments Mastersizer X laser diffrac-tometer.

3. Void structure model

The field of network modeling of meso- and macro-por-w xous media is well developed 4–7 . The void space model

Žwhich we have developed, named ‘Pore-Cor’ pore-level.properties correlator , can simulate a wide range of proper-

w xties of meso- and macro-porous media 8 . It conceptual-izes the porosity of a material as a network of voidsconnected by smaller channels. It has been successfullyemployed in the modeling of a range of materials, such assandstones, paper coatings and tablets, and a variety oftheir properties, for example porosity, permeability, colloid

w xflow and tortuosity 9–12 . The model described here hasthe particular characteristics that it has an explicit geome-try upon which all properties are calculated from firstprinciples, and that the fitting parameters used to convergeon experimental data are characteristics such as connectiv-ity which can be checked experimentally as being within arealistic range. The absence of arbitrary fitting coefficientsallows it to be used on any porous medium.

3.1. Geometry

Pore-Cor is a network model of porous media whichrepresents the void space of samples as an array of cubicpores connected by cylindrical throats, which are the

( )T. Mathews et al.rPowder Technology 104 1999 169–179 171

constricted connections between pores. The pores are cen-tered on a regularly spaced 10=10=10 lattice of nodes,positioned using Cartesian coordinates x, y, z, the dis-tance between which is termed the pore row spacing; thepore row spacing is used to adjust the porosity of thesimulated porous media. Each array of up to 1000 poresand their connecting throats is described as a unit cell, Fig.3, and for modeling purposes the unit cell connects andrepeats infinitely in all directions. Other parameters in-clude the connectiÕity, which is the average pore coordina-

Ž .tion number number of connected throats per pore andthroat and pore skews, the skew of the throat and poresize distributions respectively. The throat skew, s, is thenumber percentage of the throats of smallest size, e.g., thethroat skew on Fig. 9a linear fit, shown below, is 1.97%.The number percentage, N , of a particular throat diame-d

ter, d, varies linearly over the logarithmic size axis be-tween d and d such that:min max

Ž . Ž . Ž .Ž . Ž .2 1y s q s log d q s log d y 2y s log dmax minN s 1Ž .d Ž . Ž .log d ylog dmax min

For calculation purposes in this paper, we define aprimary position p, which is the position of a pore centredisplaced by half the pore-row spacing in the x, y and zdirections. Each primary position is at the centre of a cubedefined by the centres of eight adjacent pores.

Then the side length, C, of a pore at primary position,p, is given by:

C sc max d ,d ,d ,d ,d ,d 2Ž .Ž .p p , x p , y p , z p ,yx p ,yy p ,yz

where d is the throat diameter at position p in the xp, x

direction and c is the pore skew, subject to the conditionthat, C Fd .p max

Fig. 3. An example of a unit cell generated by Pore-Cor. Shown is themodeled void structure of Redhill 30 sand. The large, vertical scale bar isequivalent to 2190 mm, the small horizontal bars show 438 mm.

3.2. Calculation of permeability

A long standing problem in the study of porous mediahas been the question of how to calculate the permeabilityof a solid from a knowledge of the geometry of the voidspace within it. The absolute permeability k of a poroussolid is traditionally defined in terms of Darcy’s law. Withreference to a cell of the solid of unit volume, this may bewritten:

dV kA dPcell cellsy 3Ž .ž /d t hlcell cell

Ž .where h is the viscosity of the fluid, dVrd t is thecell

volumetric flow rate across the cell, dP rl is thecell cell

pressure gradient across the length l of the cell, andcell

A is the cross-sectional area. Many attempts have beencell

made to calculate k from primary parameters such as thediameters, lengths and positions of the pores and throats.Other workers have described equations based on charac-teristic parameters such as porosity, the total externally-accessible surface area per unit volume of the solid, the

Žcharacteristic throat diameter d often loosely referred toc.as the characteristic pore diameter , the tortuosity t, and

w xthe formation factor F 13 . The most successful to datew xhas been that of Thompson et al. 14 :

1 d2c

ks 4Ž .226 F

The equation predicts permeabilities correct to a factorof 7, for a range of sandstone and limestone samplescovering several orders of magnitude of experimental per-meability.

An incompressible fluid flowing through a tube takesup a parabolic velocity profile, with maximum flow ratedown the centre of the tube. If the flow at the walls isassumed to be zero, integration over the velocity profileyields the Poiseuille equation:

dV p r 4dPtube tube

sy 5Ž .ž /d t 8m ltube tube

Ž .dVrd t is the volume flow rate, r the radius of thetube tube

tube and d P rl is the pressure gradient along thetube tube

tube. Poiseuillian flow has been shown to occur for oilw xdisplacement in capillaries down to 4 mm in diameter 15 .

If we now assume that Poiseuillian flow occurs acrossthe whole cell in the yz direction, i.e., from the top to thebottom face of the unit cell. Then

dV p dPcell4sy V r 6Ž .Ž .tubes ; z cellž /d t 8m lcell ,yz cell

V is an averaging operator over the whole unit celloperating on the fourth power of the individual radii rtube; z

of all tubes lying parallel to the z axis. It is calculated bymeans of the ‘Dinic’ network analysis algorithm. V is

Ž .defined such that Eq. 4 is satisfied, and generates a termwhich is related to the effective Poiseuillian capacity of the


Fig. 4. Calculation of the hard sphere diameter, D , the minimumh

diagonal distance between pairs of pores.

cell for flow in the yz direction. Since at this stage of thecalculation, all the tube lengths l are identical andtube; z

l s l rb , where b is the number of tubes in the ztube; z cellŽ .direction in the unit cell in this case 10 , we can include

these lengths in the averaging function, so that

dV p r 4tubes ; z

sy V dPž / ž /d t 8m b lcell ;yz tubes ; z cell

p r 4dPtubes ; z

sy V 7Ž .ž /8m l btubes ; z cell

By considering tubes in the "x and "y directions as well,Ž .and comparing with the Darcy equation, Eq. 3 , it follows

that

p r 4 ltube cellks V 8Ž .ž /8b l Atube cellcell

Once this equation is corrected for the square cross-sectionof the a liquid permeability may be calculated. A fullerdiscussion of the permeability calculation is given else-

w xwhere in the literature 13 .

4. Theory

In addition to more sophisticated calculations describedbelow, two bounding calculations were made for the sizesof solid spheres associated with each primary position. Ifeach sphere were infinitely compressible, it could be de-formed into a shape that fully occupied the solid phasespace associated with each primary position. If this de-formed sphere were then allowed to return to a sphericalshape with the same volume, then the diameter of thissphere would be given by:

3 8 121 13 3 23= S y C q p r LÝ Ýk kž /8 4ks1 ks1)D s2 9Ž .c 4p

where D is the diameter of the compressible sphere, S isc

the Pore-Cor pore row spacing, C is side length of thecubic pores and r is the radius of cylindrical throats. Dc

then provides the upper bound to the sphere diameter atthe chosen primary position. The corresponding lower

Ž .Fig. 5. Illustration of the iterative process for calculating the diameter, D , of a sphere that makes contact with four pores. a Shows the initial array of4Ž . Ž . Ž . Ž .pores and b–e show the calculation of the individual points of contact. It should be noted that although c and d look similar, in d the sphere makes

Ž .contact with cube ‘3’. f Shows the sphere making contact with the edge of the imaginary cube bounding the volume available for the spherical particle.


Fig. 6. A pore configuration that would result in D < D .4 h

bound, which we term the hard sphere diameter D , ish

equal to the minimum diagonal distance between oppositepores surrounding the primary position. An example isshown in Fig. 4, in which the central sphere is touching theopposite pores 1 and 8. D is given by the equation:h

'D s 3 SyC yC 10Ž . Ž .h I II

C and C are the side lengths of the two diagonallyI II

opposed pores giving the smallest value of D .h

Within the range defined by these two bounding calcu-lations lies the actual sphere diameter associated with eachprimary position, i.e., the diameter of the largest spherewhich can be fitted between the eight pores and up totwelve throats without distortion. In practice the mathemat-ics required to find this value is analytically impossible,and would require a laborious, Monte Carlo-like calcula-tion for its determination. An approximation to the solutioncan be found by ignoring the throats, which are usuallysmaller than, and never larger than, the pores, and thencalculating the diameters of spheres which touch varyingnumbers of adjacent pores. The hard sphere calculation

represents a two-contact calculation, and we now describethree- and four-contact calculations carried out by vectormathematics. To a good approximation, the actual spherediameter is equal to the largest of the two-, three- andfour-contact diameters.

The method of calculating the three- and four-contactdiameters, D and D , was as follows. The starting point3 4

was a spherical ‘seed’ particle centred on the mid-pointbetween the eight nodes of the pores, with diameter D <3

S. The diameter of the seed particle was then increased to asize at which it made contact with the inner-corner of thelargest of the eight pores surrounding it. If p is a vectordefining the position, in three-dimensions, of the centre ofthe seed particle, and c is the vector defining the position1

Žof the first contact point with the corner of the largest.pore , then the position vector u of a point lying on the

line through p and c can be defined,

us 1ql pylc . 11Ž . Ž .1

It is then possible to move u by incrementing l fromzero until the distance from u to c is equal to the1

distance to one of the remaining seven inner-corners of thepores. Thus a second contact point, c , is found, giving2

three non-collinear points, p, c and c , and a sphere1 2

diameter D equal to the distance from u to c or c . It is2 1 2

now possible to move the centre of the sphere away fromthe final position of u in a plane determined by p, c and1

c , perpendicular to the line between c and c . Its2 1 2

position vector now, z, is given by,

mzs 1qm uy c qc 12Ž . Ž . Ž .1 22

m is incremented until another contact point, c , is found,3

equidistant from the sphere centre to c and c . Mean-1 2

while the diameter has incremented to the value of D .3

Table 1Ž .PSDs as supplied sieved and as measured by laser diffraction using a Malvern Instruments Mastersizer X

Particle Redhill 30 Chelford 60 Redhill 65 Redhill HH Redhill HH– Ballotini beadssize, mm Redhill 30

As sup- Master- As sup- Master- As sup- Master- As sup- Master- As sup- Master- As sup- Master-plied, % sizer, % plied, % sizer, % plied, % sizer, % plied, % sizer, % plied, % sizer, % plied, % sizer, %

‘Pan’ 0.00 0.01 0.1 0.00 0.10 0.00 51.30 43.17 25.65 22.16 nra 76.2863 0.00 0.02 0.3 0.00 0.40 0.00 29.10 28.21 14.55 33.30 nra 33.7290 0.10 0.10 2.7 0.34 3.00 0.50 13.40 18.34 6.75 12.55 nra 0.00

125 0.40 0.54 23.7 7.76 17.10 9.10 5.00 8.60 2.70 15.93 nra 0.00180 2.10 2.31 38.5 25.62 37.70 25.76 0.70 1.35 1.40 15.84 nra 0.00250 19.30 10.75 24.2 42.20 33.80 36.74 0.30 0.04 9.80 0.22 nra 0.00355 48.50 30.38 8.3 20.77 7.50 22.14 0.20 0.21 23.85 0.00 nra 0.00500 27.50 34.95 1.9 3.31 0.30 5.69 0.00 0.03 13.75 0.00 nra 0.00710 3.00 15.12 0.2 0.01 0.10 0.07 0.00 0.00 1.50 0.00 nra 0.00

1000 0.10 4.39 0.1 0.00 0.00 0.00 0.00 0.00 0.05 0.00 nra 0.001410 0.00 1.43 0.0 0.00 0.00 0.00 0.00 0.00 0.00 0.00 nra 0.002000 0.00 0.00 0.0 0.00 0.00 0.00 0.00 0.00 0.00 0.00 nra 0.00


To find the fourth contact point, c , the sphere is4

moved along the line,

wszqn n. 13Ž .n in the above equation is normal to the plane through c ,1

c and c , and given as,2 3

c yc = c ycŽ . Ž .3 2 2 1ns . 14Ž .

< < < <c yc c yc3 2 2 1

As before the sphere’s position is altered by minutelyincrementing n until a fourth, equidistant, contact point,

c , is found. Fig. 5a–e give a graphical representation of4

five successive stages in the four contact point method, orŽD calculation. ‘D ’ is used to denote the procedure,4 4

although in certain cases, described below, less than four.points of contact may be made .

In certain instances the configuration of pores aroundthe primary position is such that the sphere can makecontact with the edge of the cubic volume defined by thecentres of the eight pores surrounding it. If the particlewere allowed to continue growing it would therefore‘bulge’ into an adjacent volume, Fig. 5f. Due to the extra

Ž . Ž . Ž . Ž . Ž .Fig. 7. Experimental and simulated mercury intrusion curves for a Redhill 30 sand, b Redhill 65 sand, c Redhill HH sand, d Chelford 60 sand, eŽ . Ž . Ž .Ballotini beads and f the 50–50 mixture of Redhill HH and Redhill 30 sands. Thick line Experimental intrusion curve; medium-thick line Pore-Cor,

Ž .Linear fit; thin line Pore-Cor, Logarithmic fit.


level of complexity this would add to the calculation theD calculation is aborted whenever a sphere makes contact4

with this imaginary boundary and the particle size, D ,4

taken to be the diameter at this point.In most cases, D )D )D fD . In some unusual4 3 2 h

cases, for example when all the largest pores are to oneside of the primary position, Fig. 6, D -D . Overall, the4 h

final sphere diameter D was taken to be the maximum ofD and D .4 h

5. Experimental

Four different Redhill sands, obtained from the quarryin Redhill, Surrey were modeled using this new technique.They were Redhill 30, Chelford 60, Redhill 65 and RedhillHH. They contained between 96.7 and 99.3% SiO , with2

other main constituents Fe O , Al O and K O. A binary2 3 2 3 2

mixture of Redhill 30 and Redhill HH sands, the coarsestand finest grades respectively, and a sample of soda glassBallotini beads of narrow size range were also studied.Experimentally derived particle size ranges are summa-rized in Table 1. ‘Pan’ refers to particles -63 mm whichdrop through to the final pan during sieving.

Mercury porosimetry curves were obtained for the sam-ples, using a Micromeritics Autopore III instrument. Theintrusion curves, shown with the simulated curves in Fig.7, were then input into Pore-Cor. The simulated intrusioncurves were fitted to the experimental curves either byŽ .i minimizing the sum of the squared differences be-tween the logarithms of the effective pore-entry diame-ters corresponding to each point on the intrusion curve

Ž .and the equivalent points on the simulation, log fit , orŽ .ii by minimizing the linear difference between the

Ž .values, linear fit , as explained in a previous publica-w xtion 8 .

It can be seen that on the logarithmic abscissa usually usedto plot mercury intrusion curves, the logarithmic fits lookthe more satisfactory. The linear fits can be used as avisual measure of the sensitivity of the final calculatedPSD to both the type and closeness of the fit. The parame-ters corresponding to the fitted curves are given in Table 2.

Table 2Pore-Cor parameters and results for simulated sand samples

Sample Type of fit Throat skew Pore skew Connectivity

Redhill 30 Log 1.67 1.25 2.9Linear 1.97 2.2 3.7

Redhill 65 Log 1.17 1.52 2.9Linear 1.87 2.4 3.7

Redhill HH Log 0.92 1.39 2.8Linear 1.12 1.29 3.8

Redhill HH– Log 0.85 1.1 2.8Redhill 30 Linear 0.85 – 4.0Chelford 60 Log 0.87 1.25 2.9

Linear 0.87 1.25 2.9Ballotini Log 0.77 1.2 2.9Beads Linear 0.97 1.2 3.8

Fig. 8. The Pore-Cor simulated unit cell for Redhill HH sand. The large,vertical scale bar is equivalent to 601.19 mm, the small horizontal barsshow 120.24 mm.

Two of the Pore-Cor unit cells corresponding to theseparameters are shown in Figs. 3 and 8. The unit cells werethen used for the calculation of pore and throat sizedistributions, as exemplified in Fig. 9, and absolute liquidpermeabilities.

An artifact which arises from the inefficient packing ofthe pores and throats within the Pore-Cor unit cell, coupledwith the smallness of the unit cell, is that the modeling isaffected by the maximum and minimum experimental pres-sures on the intrusion curves, even if these pressures varyoutside the range over which intrusion takes place. Tocontrol this effect, the experimental curves were eachtruncated at equivalent diameters of -d r5 and )5 d ,c c

where d is the pore size at the point of inflection of thec

intrusion curve.

6. Results

The results of the particle size calculations take theform of a list of 1000 sphere diameters, best viewed asfrequency histograms. The results, along with experimentalmeasures of PSDs for the samples, are shown in Fig. 10.For clarity, three histograms are shown for each sample,covering the same size intervals. The middle histogramshows the experimental measurements of particle size dis-tributions, measured by laser diffraction for all samples,and by sieving for all the sand samples. The top histogramfor each sample shows the bounding calculations for thetwo types of fit to the mercury intrusion curve. The bottomhistograms show the D and MMS calculations.4

Ideally, the quality of the results should be judged inthe light of statistical analysis. However, in this case, such


Ž . Ž . Ž . Ž .Fig. 9. The pore and throat size distributions of the Pore-Cor simulations of for a Redhill 30 sand, b Redhill 65 sand, c Redhill HH sand, d ChelfordŽ . Ž .60 sand, e Ballotini beads and f the 50–50 mixture of Redhill HH and Redhill 30 sands. The top graphs represents the log fit, the bottom shows the

linear fit, except for Chelford 60 where both fits gave the same results.

an analysis is difficult. The major problem is that theŽfrequency of the ‘Pan’ fraction i.e., the fraction of parti-

.cles smaller than 63 mm is often too large to be ignored orcombined with another size category. Since it refers to asize range with only an upper limit, and is part of adistribution on a logarithmic scale, it precludes calculation

of the distribution mean, and hence also the standarddeviation. Two other statistical analyses, namely x 2 andF-test, avoid this problem. However, x 2 analyses in thiscase are overly sensitive to the spreads of the distributionon the size axis, and F-tests can only be carried out on thestandard deviation of the percentage readings, with no size

Ž . Ž . Ž . Ž . Ž . Ž .Fig. 10. Particle size distributions for a Redhill 30 sand, b Redhill 65 sand, c Redhill HH sand, d Chelford 60 sand, e Ballotini beads and f the50–50 mixture of Redhill HH and Redhill 30 sands. For each sample there are three histograms, described in the legend. Top—o Compressible Spheres,

Ž .Log Fit k Incompressible Spheres, Log Fit; Middle—B As Supplied Sieved I Laser Diffraction; Bottom—9 D Calculation $ MMS Calculation.4


Table 3Experimental and Pore-Cor modeled permeabilities

Sample Type of fit Experimental Modeledpermeability, permeability,darcies darcies

Redhill 30 Log 7.99 4.38Linear 3.22

Redhill 65 Log 3.78 1.64Linear 1.44

Redhill HH Log 1.07 0.30Linear 0.82

Redhill HH– Log 0.16 0.33Redhill 30 Linear 1.95Chelford 60 Log 2.07 3.24

Linear 3.24Ballotini Log 1.46 0.18Beads Linear 0.52

information included, and therefore shed little light onthese results. One is therefore left with a visual, qualitativejudgement as the criterion for success in this case.

It can be seen from Fig. 10 that, as expected, thecompressible and hard sphere methods provide the upperand lower bounds for the D method. In the cases of the4

four unmixed sand samples and the glass beads, the PSDsproduced by the D calculation reproduce the shapes of the4

experimental distributions reasonably accurately, thoughskewed slightly towards larger particle sizes.

The glass beads comprise uniform spheres over a nar-row size distribution, Fig. 10e. It would be expected thatthe MMS method would produce an accurate answer forthis system, but in fact it produces a distribution up tounrealistically large sizes. It may be inferred that the MMSmethod is over-sensitive to the intrusion at low pressures.In this case an un-modified Mayer and Stowe method,

based solely on d , produces a result of just under 28 mm.c

This figure falls comfortably within the tallest peak on theexperimentally derived distribution and illustrates the limi-tations of extending the method over a range of intrusionpressures.

The mixed sand sample has, as expected, a bi-modalsize distribution which is clearly indicated by both experi-mental methods in the centre histogram of Fig. 10f. Thisbi-modality is also manifest in the two-step mercury intru-sion curve, Fig. 7f. Unfortunately, the Pore-Cor unit cellnetwork is not extensive enough to model bi-modal distri-butions, and the allowable simulated PSDs are currentlyrestricted to unimodal distributions. The consequence ofthis can be seen in Fig. 7f—the simulated curves have asingle point of inflection and are a poor fit to the shape ofthe two-step experimental intrusion curve. The fit at lowintrusion is also affected by the essential 5 d cut-offc

mentioned above. Nevertheless, the fitted curves do strad-dle the range of the experimental curve, and give a verymuch better approximation to the particle size distributionthan does the MMS method, Fig. 10f. There is even a hintof bi-modality in the linear fit result, but this is probablyan artifact.

Table 3 shows the permeabilities of the samples asmeasured experimentally and as modeled by Pore-Cor. Thepermeabilities of all samples were measured experimen-tally by constant head permeametry apart from those of thetwo finest samples, the Redhill HH sand and the Ballotinibeads, which were calculated using a falling head perme-ameter.

All modeled permeabilities are within one order ofmagnitude of those measured experimentally, a promis-ingly accurate result considering the difficulties inherent in

w xpredicting permeability 14 . As might be expected, the

2Ž .Fig. 11. Graph of experimental permeability versus logarithmic fit modeled permeability. Regression line ys0.5663 x, R s0.67 - - - ysx.


logarithmically fitted simulations generally produce themore accurate estimates of permeability. They are plottedagainst experimental permeability in Fig. 11. It can be seenthat Pore-Cor produces a particularly successful estimationof the permeability trend, with R2 s0.67 for the linearregression, and rather underestimates the trend, an exactprediction of which would follow the dashed line.

7. Summary

It is evident from the results that the distributionsproduced by the new four contact point method, or D4

calculation, fall between the bounds of the compressibleand hard sphere methods, and compare favourably withthose produced by the MMS method. Both the D and4

MMS calculations produced good approximations of thePSDs of the unmixed sand samples, but for the glass beadsand mixture of sands the D method provided a significant4

improvement.The D method presented here is based on a wide range4

of approximations, some of which may be addressed infuture work. Thus it would not be sensible to use themethod to study unconsolidated material, since the directexperimental methods described here are much more reli-able. Nevertheless, the theoretical approximations are lessgross than those within the MMS method, and it has beendemonstrated, by the trials with unconsolidated material,that the new method will be a useful tool in the study ofconsolidated material where no direct experimental meth-ods are available for particle size measurement.

The modeling method also predicts permeabilities to auseful degree of accuracy.

8. List of symbols

a Radius of sphereA Cross-sectional area of cellC Pore side lengthd Characteristic throat diameterc

d Diameter of throat connected to pore atp

position p.d , d Minimum and maximum throat diametersmin max

D Sphere diameter produced by new ‘D ’4 4

methodD Compressible sphere diameterc

D Hard sphere diameterh

F Formation factork Absolute permeabilityl LengthN Number percent of throats of diameter dd

P Primary positions Throat skewS Pore row spacing

t TimeT Surface tension of wetting fluidV Volume of fluid

Greek lettersb Number of tubes in the z directionl Incremental parameterh Viscosity of fluids Packing anglem Incremental parametern Incremental parameterV Averaging operatorc Pore skew

Subscriptscell Celltube Tubex, y, z Pertaining to the x, y or z axis direction,

may be positive or negative

Vector notationc Three-dimensional position vector of a cubic

pore cornern Vector normal to a planep Three-dimensional position vector of the

centre of a spherical ‘seed’ particleu, z, w Three-dimensional position vectors

Acknowledgements

This work was partly funded by the EPSRC.

References

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estimating particle size distributions from a network model of porous media

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