Site and Bond Fractions on Bethe Trees

Powder

Site

Technology.

and

Bond

39 (1984)

21 - 28

Fractions

on

21

Bethe

Trees

G. MASON* Petroleum

(Received

Recovery June

Resecrch

27.1983;

Center.

in revised

New

form

Mexico October

Insiitute

of Nining

and Technology.

Socrrro.

X.11 rl_S__-1.)

1. 19S3)

SUMMARY Percolation theory. which was originally developed by mathematical physicists, can be applied to adsorption-desorption. to imbibition-desaturation and to mercury porosimetry. It is not always clear what past authors have done and how their models relate to each other_ By classifying systems as ‘site’ or ‘bond’ fractions being accessible, it is possible to categorise past work into four groups. Using a Bethe tree as the network of bonds and sites, the solutions to the four main problems are given by a simple analysis_ Two of these solutions are used in a modei of imbibition-desafurafion.

in the porous material will be modelled throughout by a Bethe tree. Levine et al. [4] have reviewed the application of percolation theory to the imbibition and desaturation of fluids in porous media. They state that the analysis of Iczkowski [S] does not correspond to the analysis of Fisher and Essam [6]. Without dwelling too much on the details, it will nevertheless be shown here that Iczkowski’s [S] analysis does correspond to that of Fisher and Essam (61. It turns out that in the applica.tion of percolation theory, there are four possible combinations of variables and soludons for all four are derived in a simple way here. The nomenclature of Levine et al_ [4] will be followed where appropriate.

INTRODUCTION

DEFINITIONS

Percolation theory has been increasingly applied to the behavior of liquid phases in porous media. The mathematical analyses of percolation theory published in the past do not directly relate to porous media, and a degree of translation is required to relate a model of porous material into the requirements of percolation theory- Indeed, the translation is crucial. This paper will emphasise the importance of using a suitable model for a particular process. It is highlighted by the comparison of two models of desaturation, one by Larson and Morrow [I] and Doe [2], the other by Chizmadzhev [3] plus others_ There is also a description of the mechanism that pertains during imbibition and a demonstration of the effect of this mechanism. The connectedness of the pores

In percolation theory, two classic problems are normally considered: the site problem and the bond problem- During the process to be described by the theory, sites (or bonds) on the network are randomly changed from one condition to_ another, from ‘unavailable’ to ‘available’, say black to white- As more sites are changed from black to white, the probai biiity of a white site having white neighbours increases, and ‘clusters’ of white sites appear. If the probability of a site being white exceeds some critical value (the critical percolation probability), then infinite clusters of white sites appear- The application of percolation theory is usually to calculate an ‘accessible fraction’ (the fraction conne&d to infinite clusters) in terms of the probal%lity of sites (or bonds) being available. Other nomenclature includes ‘dammed’ and ‘undammed’, ‘permitted’ and ‘non-permitted’_ The accessible fraction can apply to either sites or bonds- The definition of accessibility

*Present address: Department of Chemical Engineering, Loughborough University of Technology, Loughborough, Leicestershire (U.K.)

@ Elsevier Sequoiaprinted

in The

Netherlands

22

which is used to specify the accessible fraction can be in terms of either the sites or the bonds. Examples of the various possibilities are giv& in Fig. 1_ In the site problem, the accessible fraction of sites is often given in terms of the probability of the sites being available. In the bond problem, the accessible fraction of bonds is often given in terms of the bonds being available. The solutions to these two problems for Bethe trees (see below) are given by Fisher and Essam [S] _ The pore space of a porous material can be modelled by a network of pores, each pore consisting of a cavity which is linked to its neighbours by constrictions. Frequently, most of the pore volume is associated with the cavities, so the pore volume corresponds to the sites of percolation theory. The pore cavities are connected together by the constrictions in the pore space, and these correspond to the bonds in percolation theory. For

analysis purposes, a network is required which connects the sites together using the bonds. A frequently used [7,8,9] network model is a Bethe tree which is a perpetually branching structure with no closed pathways.

ANALYSIS

For the desaturation of porous materials (replacement of water by air), the volume of liquid in the pores relates to the sites of the network. Because a pore (or site) can only empty when a meniscus passes through the constriction joining it to its neighbour, the behaviour of the system depends upon the constriction behaviourThe constrictions between the pores correspond to bonds. Thus, the pore accessible fraction is defined by the bond network. We thus have a mixture of the two classic percolation problems: the fraction

ACCESSfEXE

XCESSIBLE

II

(I) Accessible

bonds

by bond ovahbillty

[=]

defined C-l

(iill Accessible bonds [=I defined by site ovoilobility [O]

h)

Accesslbte sites [o] defmed by bond ovoifobility I-;

(iv) Accessible by site

sites

[ml

availability

defined [0]

Fig. 1. Examples of bond and site accessible fractions as defined by bond and site available fractions.

23 of the pores which empties corresponds to the fraction of sites on the infinite bond clusters. We will ignore the fact that as well as air entering the pore, the Iiquid has simultaneously to leave the pore, and this requires a connected pathway to the bulk liquid phase. In some processes, such as the desorption of capillary condensed gases, this is not important because the capillary condensed liquid can evaporate away through the vapour space_ For the desaturation of sphere packs, the residual pendular rings effectively connect most of the liquid together except at very low saturations [lo]. In mercury porosimetry, the sample is usually evacuated_ before intrusion of the mercury is started. For desaturation then, the volume of liquid removed corresponds to the sites of percolation theory, but the requirement of connection to infinite ciusters is determined by the bonds. Chizmadzhev 131, Iczkowski [5], Mason [lo], Tabuchi [ll] and Stinchcombe [12] all give solutions to this problem for a network consisting of a Bethe tree, although they do not necessarily specify it as such- In particular, the analysis of Stinchcombe [12] is worth following. i) Bond problem Let xb be the fraction of available bonds and P,(xb) the probability that a site belongs to an infinite cluster of bonds. The subscripts s and b refer to sites and bonds respectively. We define a probability Rb that any arbitrarily chosen one of the z (the co-ordination number of the Bethe tree: the number of bonds meeting at a site) branches emerging from a given site is not part of an infinite cluster (i-e_ is blocked)_ So I-

Ps(Xb) = R,’

(I)

If the chosen branch is finite, then either the bond with which it begins is not available (probability (1 -xi,)) or, if it is available, all of the other (z - 1) branches emerging from the neighbouring site are all finite (blocked): Rb = (1 -xb)

+ XbRb

Z-l (2)

So we can obtain (3) The probability of a site belonging infinite cluster of bonds is given by

to our

P+(Xb) = 1 -

Rb=

(4)

Solving eqn. (2) for R, (i-e. solving the (Z - I) powered equation) and substituting for R, in eqn. (4) gives the accessible fraction of sites. Ps(xb), as a function of the available fraction of bonds, _a~,,_This is the required solution. Let us also find the probability of a bond being part of an infinite bond cluster_ The solution to this has been given by Fisher and Essam [6], but we will derive it more simply here_ We note that a bond is not part of an infinite bond cluster if the sites at both ends are not part of infinite clusters_ The probability that (z - 1) bonds from a site are not connected to infinite clusters is R,=- * and from eqn. (2)

RI,

z-l_%*xb-l -

Xb

So the probability R 2cz-1’ that all of the bonds from the two &es at both ends of the bond are not part of infinite bond trees is

&

2(-_-,>

=

tRb +xb Xb2

l)’

Thus, the probability that a bond is part of an infinite cluster, pb(_xb), is t.he product of _rb (the probability of an individual bond being available) and the probability that not all of the 2(z - 1) outgoing bonds are not part of infinite clusters (I - R,2”-1)): Pb(xb)

=x,(1

-

R,“=-“j

= _T

(7)

Pb(xb) corresponds to Stinchcombe’s f I2 ] R(p) and eqn. (7) is very similar to Stinchcombe’s 1121 equation 29. So much for the bond problem. ii) Site problem Let us now consider the site problem. If -r, is the probability of a site being available, let R, be the probability that a bond leaving a site does not. connect to an infinite tree_ Because a bond is blocked if either (i) the site at the end is not available or (ii) the site is available and the other (z - 1) branches to it are blocked, we obtain R, = (1 -xs)

+x&‘--1

which is very similar to eqn. (2)_ The acces-

(8)

24

sible fraction of sites, P,(x,), is the probability that a site is available, xS, multiplied by the probability that not all of the bonds leaving that site are blocked.

incoming bonds connected to the same site are either not available (1 -xb) or if available (x,) are capped (w), then

P&x,)

This is the same as eqn- (1) if

= x,(1

-

R=)

(9)

The fraction of accessible given by (see eqn. (7))

bonds,

R,+x,-1’

Pb(XS) = l-

>

-G

which concludes

the site problem.

iii) Comparison Levine et al. [4]

with Iczkowski

P,,(x,),

w = (1 -xb

+x,w)“-1

(11)

is

R,=wx,+l-x,

(10)

which, because of the way in which w and Rb have been defined and are used, is actually the case_ The fraction of sites which are not accessible ( W in Iczkowski’s nomenclature) is given by Iczkowski 153 as

[S] and

Iczkowski [S] essentially follows the reasoning of eqns. (1) - (4). His variables are different, and the bond probabilities are equated in slightly different places (Fig. I). He lets the probability of an incoming bond not being connected to an infinite tree (capped, as he calls it) be w_ Because a bond will be capped if all of the other (z - 1)

(z-1)

(12)

157= (WXb + 1 -xb)=

(13)

As (1 - W) corresponds to PS(xb), this is in agreement with eqn. (4). For z = 4, Levine et aZ_ [4] calculated w (from eqn_ (11)) as 3 2xb

w=l--

4 -$(I-

%r’2

(14)

(their equation 38)- This can be used together with eqn_ (13) to give Iv and hence (1 - W), the fraction of sites accessible. Levine et al. [4] identified (1 - W), which corresponds to PS(xb) with Pb(xb)/x,, and pointed out that the function giving P,,(x,,) from (1 - W) so obtained does not agree with the functions listed by Fisher and Essam [S]_ The reason for the lack of agreement is that the fraction of accessible bonds, Pb(xb), is not given by %(xIJ)

= xIPs(xb)

(15)

It is given by eqn. (7): P,,(xb) (if

R,, = (1-q-J

+

xb Rb

2-I

=

x,,(l- R,,Z
(7)

andas Rb = [ 1 -

P,(xb)]l’=

(4)

we obtain Pb(Xb) 'Xb{l

- [l -PS(xb)]=-1”=3

(16)

Using eqn, (16)

instead of eqn. (15) makes the derivation of Pb(xb) from eqn. (14) agree with the Fisher and Essam [6] function for 2 = 4_

(ii)w=

(I-x~ +

~~"1 Z-I

Fig. 2. Diagram showing the different ways in which Stinchcombe [12] (i) and Iczkowski [5] (ii) regard the same problem.

(17) This was established by numerical of the functions using test values.

evaluation

25

iv) Es-amples If the co-ordination number z analytic solutions can be obtained R, using eqns. (2) and (8). This explicit expressions to be obtained bond and site accessible fractions. forz

is small, for Rb and enables for ah the These are

= 3s

Rt, = (1 -xX,)/x, P&b)

= 1 -

P&b)

=Xb[l-

desaturation (the replacement of water by air in a water-wet system) which is different to that of Chizmadzhev 133 and others [&lo, ll]_ The Larson [7] model envisages a pore (site) which can fill at one capillary pressure and empty at another - in essence, the Everett domain theory [13], but including the effect of pore blocking. The connectivity

(18)

(1 -

X,)3/Xb3

h

(19)

(1 -.Xx,)4/x,4]

R, = (1 -_-r,)jx,

(20) (21)

P&r,)

=..r,.l- (1 -X,)3/&3]

(22)

P&c,)

= 1 -

(23)

(1-&,4/X,”

I

and for z = 4: Rb

=xb-112(1 _ %),,,_ ;

Ps(xb)

=

P&I,)

=

(24)

1 - Rb4

(25)

xb(l

(26)

-Rb?

P&s) pb(xs)

=x,(1

2,1i2-

-Rs4)

= (1 -Rs6)

$

(27) (23)

I

I

00

R,,x,-"*(+

02

03

9-Z

CY,xLZSLE

05

56 ‘Rx;ICN

97

09

09

10

x,-_x*

Fig_ 3. _4ccessible fractions of bonds and sites in terms of available fractions of bonds and sites for a Bethe tree with coxirdination number z = 35

(29)

These functions are given in graphical form in Fig. 3 and Fig. 4. For z = 4, these functions relate to the desaturation curves given by Iczkowski [5] and Mason [lo] as foIIows_ xb is the available fraction of bonds and corresponds to Iczkowski’s [5] p and Mason’s [lo] (1 -p)p&b) is the accessible fraction of sites and corresponds to the fraction of the pore space filled with air_ This is Iczkowski’s [5] (1 - W) and Mason’s [lo ] ‘fractional saturation’_

ALTZRNATIVE

MODELS

i) Accessible fraction of sites on bond and site trees It is worth pointing out that Larson and Morrow [ 11, who used a Bethe tree network, and also Doe 123, have formulated a model of

Fig. i. Accessible fractions of bonds and sites in terms of available fractions of bonds and sites for a E&he tree with co-ordination number z = 1_

26

requirement for emptying (also filling) is that such a pore can only empty (also fill) if it is connected to an infinite tree of air- (and also water-) filled sites. In percolation theory terminology, this model equates the empty fraction to the accessible fraction of sites on the infinite trees defined by available sites. It will be remembered that the model used by Chizmadzhev [3] and others 15, 10, 111 regarded the empty fraction as the accessible fraction of sites as defined by available bonds. This difference is crucial. If we consider some particular liquid-filled pore, then if we use the model of Chizmadzhev [3] and ot.hers [5,10, 113, the probability of the pore emptying increases as more of the constrictions (bonds) become connected to the air. Indeed, if we consider a pore with four bonds each with a probability of passing a meniscus x,,, then if all four bonds are connected to the accessible fraction, the probability of such a pore remaining filled is (1 -x~)~. This ignores, as before, the requirement of the connection of the liquid by a pathway to the bulk liquid. If we use the Larson [l] model, the probability of the site being empty is x,, and is unchanged as constrictions after the first become connected to the air. a) Imbibition and desatumtion The Larson [l] and also the Doe [ 2] model is much better as a model of imbihition (water displacing air), because for irnbibition the pore filling curvature is likely to be associated with the single central hole of the pore. The process modelled is the imbibition into an initially dry porous material by a wetting fluid. The permitted sites are those that would fill at a particular capillary pressure, and the accessible fraction of such sites gives the fractional liquid saturation. If we have a relation between xb and xs, it is possible to show the fraction filled during imbibition, P&E,), and the fraction filled during desaturation, [ 1 - P&,)], on the same graph. Mason 1143 has proposed that for the pore space of a packing of spheres, the meniscus that passes through the largest bond constriction has the same curvature as the meniscus which is associated with the filling of the central site. The hypothesis implies that all hysteresis in imbibition-desaturation, adsorption-desorption and mercury porosimetry has its origins in percolation effects_ The hypoth-

esis can be expressed

mathematically

as

Xbes = 1 -x7+114 where x~_~ is the equivalent value of _x, but in terms of xb_ So, using eqns. (24) and (25), we can obtain a ‘saturation’ given by [ 1 - PJx,,)] for desaturation and, using eqns. (27), (28), and (30), we can obtain the saturation during imbibition, given by P&-c& as a function of xb_ These are shown on Fig. 5. In both cases, the residual saturations of air and water are zero, because, as has been mentioned before, this model ignores the fact that for a pore to fill or empty, a connection to infinite trees of both water- and air-filled bonds and sites is necessary. Note that imbibition commences with a sharp threshold, a fact observed by Haines 1151 for a system where sufficient suction has been applied to destroy the funicular state. I-PsWbJ

Fig. 5. The desaturation fraction filled [l -P&Q)] uersusxb and the imbibition fraction filled [Ps(x,)] UersKsxb.s for Z = 4. The fraction emptied during desaturation is the fraction of sites connected to

infinite bond trees. The fraction

filled during imbibi-

tion is the fraction of sites connected to infinite site trees.

Indeed, the results on Fig. 5 can be taken further using the meniscus radii calculated from tetrahedral pores in a random sphere packing [lS]_ Now we no longer need eqn. (30) but can directly relate xb and x, to normalised meniscus curvatures. These can be directly compared with Haines’ [15] experimental results (Fig. 6). The desaturation comparison has already been given IlO] and

2’7 1

*

.

0 -

O

0.1

02

0.3

0.4

FRACTION

.

3

i

.

09

0.9

HAINES= THEORY

0.5 LlDUlD

0.6

0.7

1.0

FILLED

Fig_ 6_ Comparison of desaturation-irnbibition curses gi=en by Gaines [15]. uncorrected for the effect of the r dictions (z = 4) combined with pore size distributions height of the packing, with the Bethe tree percolation p-e of a random sphere packing [16]_ The theory and experimeni can be brought more into agreement if z = 3-S.

agrees quite well. The imb‘S;*‘ i I,ron comp2rison does not agree as well. Neither of the two theoretical curves predicts the residual saturations, and this is because the theory is only concerned with the entry of air during desaturation and the entry of water during imbibition, and neglects the connection of the opposite phase. ii) Accessible fraction of bonds on bond and site trees Chatzis and Dullien [ 171, who used a 2D regular structure for their network, chose to identify the fraction that is empty during desaturation (replacement of oil by water) with the accessible fraction of bonds, accessibility being defined by connection to an infinite bond tree_ This contrasts with the approach of Chizmadzhev 133 and others 15, 10, 113, which identified the accessible fraction with the sites, accessibility being defined by infinite bond trees.

iii) Mechanism of blob entrapment The Bethe lattice accessible fraction model has also been used with respect to residual saturation and the entrapment of blobs (7, S]_ When it is used as such, the accessible fraction refers to the ability of the liquid to Zeaue a pore during imbibiiion. In all of the previous discussions and analyses, this mechanism has not been involved_

CONCLZiSlON

The literature ->f the application of percolation theory to the uesaturation and imbibition of liquid from and into a porous material is full of confusion for the casual reader. Infinite clusters can be defined either in terms of available bonds or in terms of available sites. The accessible fraction (the fraction of infinite clusters) can also be in terms of bonds or sites. Figure 7 gives a

28

ACKNOWLEDGEMENT

SITES

This work was completed while the author was on sabbatical leave at the Petroleum Recovery Research Center, New Mexico Institute of Mining and Technology, SOCOZO, New Mexico, U8.A. Partial funding was protided by the U.S. Department of Energy, Contract No. DE-ASl9430BC10310 and the New Mexico Energy Research and Development Institute, Project. No. 2-69-3309_

+, rx,) lMaIalTION iI nns lwsk

I

SITES

nil

I

woson

DESORPTION

DESATIJRATION i, LOxso” 8 Mrncl. DESORPT~ON il Doe

REFERENCES 1

R_ G_ Larson and N_ R_ Morrow. noZ_, 30 (1981)

Powder

Tech-

123.

P. H. Doe, Ph_D_ Thesis, Univ. of Bristol, 1975. Yu. A_ Chizmadzhev, Soviet Electrochemistm, 2 (1966) 1. 4 S. Levine. P_ Reed, G_ Shutts and G_ NeaIe, Powder Technor_. I7 (1977) 163_ 7 5 R_ P_ Iczkowski, 2nd. Eng. Chem. Fundam.. (1968) 572. 6 M. E. Fisher and J. W_ I&am. J_ Math. Phys_. 2 (1961) 609_ 7 R. G. Larson. L. E. Striven and H. T_ Davis, 2

3

Fig. 7. Schematic representation of the various site and bond possibilities and their use as models of particular processes.

Nature.

8

schematic representation of these possibilities and the manner in which they have been used by various authors. Furthermore, the analysis can be applied to the phase entering the system or to the phase Ieaving the system (in the real world we need both). This gives eight different possibilities all giving related functions However, provided that one can relate the important mechanisms and properties of red systems to the idealised param eters of percolation theory, it is possible to arrive at simplified analyses for relatively complex processes such as the example of imbibition and desaturation given here-

Chem.

9

268

(i977)

R. G. Larson, Eng.

Sci,

R. G. Larson, Chem.

Eng.

409_

L. E. Striven 36 (1981)

and H. T. Davis, 57.

H. T. Davis and L. E. Striven,

Sci_, 36 (1981) 75. J. ColZoid Interface Sci..

G. Mason, 41 (1972) 208. 11 T_ Tabuchi, Soil Science, 102 (1966) 161. J_ Phys C, Solid State Phys-. 12 R. B. Stinchcombe, 6 (1973) Ll. 13 D. H. Everett, in E. A. Flood (Ed_), The Solid-Gas Interface. Marcel Dekker, New York, 1965, p_ 1055. 1-i G. Mason, J_ CoZZoid Interface Sri., 88 (1982) 36_ 15 W_ B. Haines. J Agric. Sci. 20 (1930) 97. ScL. 35 (1971) 16 G. Mason, L CoZZoid Interface 279_ I. Chatzis and F. A. L. Dullien, J. Canad. Pet. 17 10

Tech_,

16 (1977)

97_