characterization of the pore structure of reservoir rocks ... · cients of reservoir rocks bridge...

Characterization of the pore structure of reservoir rocks with the aidof serial sectioning analysis, mercury porosimetry and network

simulation

Christos D. Tsakiroglou a,*, Alkiviades C. Payatakes a,b

a Institute of Chemical Engineering and High Temperature Chemical Processes ± Foundation for Research and Technology, Hellas, P.O. Box 1414,

GR-26500 Patras, Greeceb Department of Chemical Engineering, University of Patras, GR-26500 Patras, Greece

Abstract

Processes of ¯uid transport through underground reservoirs are closely related with microscopic properties of the pore structure.

In the present work, a relatively simple method is developed for the determination of the topological and geometrical parameters of

the pore space of sedimentary rocks, in terms of chamber-and-throat networks. Several parameters, such as the chamber-diameter

distribution and the mean speci®c genus of the pore network are obtained from the serial sectioning analysis of double porecasts.

This information is used in the computer-aided construction of a chamber-and-throat network which is to be used for further

analysis. Mercury porosimetry curves are ®tted to either 2-parameter or 5-parameter non-linear analytic functions which are

identi®ed by the median pressures, mean slopes and breakthrough pressures. A simulator of mercury intrusion/retraction, incor-

porating the results of serial tomography, in conjuction with the experimental mercury porosimetry curves of the porous solid are

used iteratively to estimate the throat-diameter distribution, spatial correlation coe�cients of pore sizes and parameters charac-

terizing the pore-wall roughness. Estimation of the parameter values is performed by ®tting the simulated mercury porosimetry

curves to the experimental ones in terms of the macroscopic parameters of the analytic functions. The validity of the pore space

characterization is evaluated through the correct prediction of the absolute permeability. The method is demonstrated with its

application to an outcrop Grey-Vosgues sandstone. Ó 2000 Elsevier Science Ltd. All rights reserved.

Keywords: Pore structure; Mercury porosimetry; Pore network; Capillary pressure; Permeability; Network connectivity; Pore size;

Fractal dimension; Spatial correlations; Serial tomography

1. Introduction

The microstructural properties of sedimentary porousrocks (e.g. sandstones, limestones, dolomites, etc) in-¯uence a variety of transport processes either of indus-trial interest, such as the oil and gas recovery from rockreservoirs, or of environmental signi®cance, such as thecontamination of underground aquifers with seawater,industrial chemicals, agricultural chemicals and land®llleachates, the in situ remediation of contaminatedgroundwater, etc.

The geometrical and topological properties of thepore structure specify to a great extent the various

pore-scale mechanisms of multi¯uid transport, as forinstance, the meniscus motion, the ¯uid disconnectionand redistribution, the hydrodynamic dispersion ofmiscible liquids, the interfacial mass-transfer, thetransport-deposition of colloids, the di�usion and re-action of gas mixtures, etc [15,30,39,45,53]. Meso-scopic transport coe�cients of underground reservoirs(e.g. capillary pressure curves, absolute and two- orthree-phase relative permeabilities, hydrodynamic dis-persion coe�cients, electrical formation factor andresistivity index) are complex functions of parametersof the pore structure [7,18,22,32,54], dimensionlessparameters of ¯ow rates coupled with physicochemicalproperties of ¯uids [2,12,29,57], wettability [6,28,55]and saturation history [15,45]. The transport coe�-cients of reservoir rocks bridge the gap betweenthe pore scale dynamics of multiphase processes andthe spatial/temporal evolution of phenomena at the

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Advances in Water Resources 23 (2000) 773±789

* Corresponding author. Tel.: +30-61-965212; fax: +30-61-965223.

E-mail addresses: [email protected] (C.D. Tsakiroglou),

[email protected] (A.C. Payatakes).

0309-1708/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved.

PII: S 0 3 0 9 - 1 7 0 8 ( 0 0 ) 0 0 0 0 2 - 6

macroscopic scale of an oil reservoir or an aquifer[8,17,42].

Optical techniques are widely used for the charac-terization of the pore structure of relatively macropor-ous materials such as granular packs, sandstones, soils,etc. Image analysis techniques allow us to determinegeometrical properties of the three-dimensional porespace by processing data of two-dimensional randomsections [15] with the aid of the principles of quantitativestereology [56], and estimate topological properties ofthe interconnected pore structure by analyzing data ofserial sections [13]. Algorithms of reconstruction of thethree-dimensional pore network from data of two-di-mensional pore features of serial sections (serial to-mography) have been developed for the simultaneousdetermination of both topological and geometricalproperties of porous media [3,26,33,34].

Mercury porosimetry is the fastest method of deter-mining the capillary pressure curves, in which is em-bedded information about a wide range of pore sizes(equivalent capillary cylindrical diameter from �180 lmdown to �0.005 lm). For this reason, mercury porosi-metry is the most important method of analysis of thepore structure for a wide variety of porous materials[15]. The conventional interpretation of capillarypressure data is based on the dubious assumption thatthe pore space can be represented as a bundle of parallel,non-intersecting cylindrical tubes of equal length but ofdi�erent diameter [14]. Based ``on ideas of percolationtheory [15,45], several researchers have developed sim-ulators of mercury intrusion/retraction in pore networkmodels, using more sophisticated representations of thepore structure [1,9,18,21,27,36,41,43,49,53]. Such simu-lators have extensively been used to study the e�ects ofgeometrical and topological parameters of pore net-works (pore size distribution, coordination number, etc)in the form of capillary pressure curves. Special atten-tion has been paid to the e�ects of the spatial pore sizecorrelations on the capillary properties of porous media[10,19,37,44,50]. Nevertheless, little progress has beenmade on the use of the pore network simulators for thesolution of the reverse problem of characterization ofthe pore structure in terms of network models[18,37,38,41,52].

In the present work, a systematic procedure for thecharacterization of the pore structure of sedimentaryrocks such as sandstones, by deconvolving the mercuryintrusion/retraction curves is developed. The procedureof combining the data of serial tomography with theanalysis of mercury porosimetry data using a pore net-work simulator of mercury intrusion/retraction is dem-onstrated in the case of a sample of Grey-Vosguessandstone. The predictability of the absolute per-meability of the porous sample is used for the evaluationof the validity of the estimated microscopic parametervalues.

2. Theoretical model of the pore structure

2.1. Pore space connectivity

The topological properties of porous media quantifythe network skeleton and are independent of any mi-croscopic details concerning pore shapes and sizes[13,31]. For a porous sample, there are two classes ofstructural aspects that are reported by the topologicalproperties. These are: (1) the number of disconnectedparts of pore network (sub-networks) in the unit volumeof the sample, and (2) the total connectivity of the net-work in the unit volume of the sample. The concept ofconnectivity of a pore structure, which is a measure ofthe degree to which the pores are multiply connected,can be most directly understood for a network of points(nodes) and lines (bonds) in space. For such a networkcomposed of n nodes and b bonds, the quantitativemeasure of the connectivity is called the 1st Betti num-ber [13,31], p1, and is de®ned by

p1 � bÿ n� p0; �1a�where p0 is the 0th Betti number [13,31], which is equalto the number of disconnected parts of a network. Ingeneral, for the examination of the topology of a com-plex multiply connected closed surface in space, it isuseful to introduce the concept of its deformation re-tract [13,31], which is formed by shrinking the surfacecontinuously until it approaches a bond-and-node net-work (Fig. 1). In this manner, the 0th and 1st Betti

Fig. 1. A multiply-connected 2D porous structure

(Gmax � 9;Gmin � 1).

774 C.D. Tsakiroglou, A.C. Payatakes / Advances in Water Resources 23 (2000) 773±789

numbers of the deformation retract of a set of closedsurfaces correspond to the number and connectivity ofthe set of surfaces. The serial sectioning analysis of porestructures is based upon forming the deformation retractof the interface separating the solid matrix from the porespace and determining the Betti numbers of this network[33±35,59]. A direct measure of the connectivity of amultiply connected closed surface is the genus. The ge-nus G of a multiply connected closed surface is de®nedas the number of (non-self-intersecting) cuts that may bemade upon the surface without separating it into twodisconnected parts [13]. In practice, the genus of a closedsurface (namely, the interface between the pore spaceand the solid matrix) is numerically equal to the 1st Bettinumber of its deformation retract [31]. It is possible toput bounds on the estimate of genus for a given sample.These two bounds, Gmax and Gmin are determined bytaking into account and ignoring, respectively, in thecalculations, the bonds crossing the sample boundary(Fig. 1). Commonly, the average of Gmax and Gmin isused as an estimate of the mean genus, hGi.

The 1st Betti number, p1, (or genus, G) is a scale-dependent parameter, Eq. (1a), and for su�ciently largenetwork sizes (or sample volumes), it becomes a linearfunction of the network size (or the sample volume)[13,31,33±35]. Instead of hGi, the speci®c genus, hGvi,which is equal to the mean genus per unit volume is usedas a measure of the connectivity of pore structures. Aftera su�cient number of serial sections have been analyzed,the speci®c genus is estimated from the constant slope ofa plot of the mean genus vs sample volume [33].

The mean coordination number, hcni, of a bond-and-node pore network is a more familiar measure of thepore space connectivity, is de®ned as the mean numberof bonds adjoining to a node and is given by

hcni � 2� 2�p1 ÿ p0�n

: �1b�

Eq. (1b) is applied only to very large networks and re-quires reliable estimates of p1 and p0. A great number oflarge serial sections must be analyzed before a correctestimate of hcni is obtained from Eq. (1b) [31]. Usually,a small volume of the sample is analyzed and only thespeci®c genus, GV, is determined with satisfactory ac-curacy [35]. It must be pointed out that the speci®c ge-nus is a dimensional parameter, is a measure of theconnectivity density of the pore space and consequentlydepends not only on network connectivity but also onother geometrical characteristics of the pore structuresuch as the mean distance between adjacent pores [33].Therefore, the value of hGvi by itself gives no informa-tion about the connectivity of a pore structure whereasno comparison between di�erent structures can be madein terms of the values of this parameter. Nevertheless, ifthe value of hGvi is combined with other geometricalparameters of the pore space (as we will see below) then

both the network skeleton is automatically quanti®edand useful network properties, such as the distributionof coordination number, are obtained.

2.2. Pore network model

The pore space is modeled as a three-dimensionalnetwork of ``spherical'' chambers interconnectedthrough long ``cylindrical'' throats (primary network,Fig. 2) with roughness features superposed on the freesurface of chambers and throats (fractal network). Thechamber-diameter distribution (CSD), the throat-diam-eter distribution (TSD), and the primary porosity, ep,are used as input parameters for the construction of theprimary network, whereas the pore-wall roughness ismodeled as a fractal surface (as it is described below) ofa given fractal porosity ef . Initially, the skeleton of thenetwork is taken to be a regular cubic lattice, and soeach chamber is connected to six neighbors through sixthroats (coordination number, cn � 6). Then, throatsare removed according to the TSD and by retaining theaccessibility of each chamber to the boundary throats.This procedure is continued until the speci®c genus hGvi(and hence the connectivity) of the pore networkmatches that of the prototype porous medium.

The assignment of sizes to the nodes and bonds ofnetwork is made according to the CSD and TSD, re-spectively. This procedure can be done completely atrandom so that no correlation exists among the sizes ofchambers and those of neighboring throats (uncorre-lated network). On the other hand, the assignment ofsizes can be done in such a way that a c±t or a c±c and c±t correlated network is obtained [50]. Two input pa-rameters are used in the procedure of constructing cor-related networks [50]: (1) the ``voting'' exponent, ns, thattakes the values ÿ1 (negative c±t correlation), 0 (no c±tcorrelation) and �1 (positive c±t correlation) and (2) thefraction of ``seed'' chambers, as, that ranges from 0 to 1,

Fig. 2. Chamber-and-throat network model.

C.D. Tsakiroglou, A.C. Payatakes / Advances in Water Resources 23 (2000) 773±789 775

and the smaller its value the stronger the c±c correla-tions. After the construction of a correlated pore net-work has been completed, the statistical correlationcoe�cients qc±t and qc±c of the chamber sizes with themean sizes of adjoining throats and the mean sizes ofneighboring chambers are respectively calculated. Thesecoe�cients are used as measures of the spatial correla-tions between the sizes of throats and those of theiradjacent chambers and between the sizes of neighboringchambers, respectively.

2.3. Fractal roughness model

In the past, experimental and theoretical analysesrevealed that pore cusps and roughness features weremainly intruded by mercury over the high-pressure re-gion [10,18,52,58]. The fractal roughness model devel-oped in [52] is further treated here to represent theirregular morphology of throats and chambers. Theroughness features are right triangular prisms for thethroats and right circular cones for the chambers of theprimary network (Fig. 3). With regard to the roughnessmodel described in Ref. [52], some modi®cations aremade here: (a) the number of layers of roughness fea-tures extends to in®nity (fractal structure of in®nitespeci®c surface area), (b) the parameters required for thedetermination of the roughness of such a network are

the characteristic number of features per linear dimen-sion of the previous layer, nr, and the surface fractaldimension, Ds. The angle of sharpness is di�erent forthroats, wt, and chambers, wc, and is so adjusted that thefractal dimension of throats, Dst, matches that ofchambers, Dsc, namely

Ds � Dst � Dsc; �2a�where

Dst �ln n2

r sin wt

ÿ �ln nr sin wt� � ; �2b�

Dsc � 2 ln�nr�ln�2nr

��tan wc=p

p � : �2c�

2.4. Calculation of absolute permeability

The absolute permeability of chamber-and-throatnetworks is calculated by analogy to the solution ofelectric circuits [11,18]. The pressure, the hydraulicconductance and the volumetric ¯ow rate are equivalentto the potential, the electrical conductance and thecurrent ¯ow. Kircho�Õs rules formulate that the sum ofthe voltages is zero for every closed loop of conductorsand that the algebraic sum of the currents ¯owing intoeach node is also zero. From the application of theserules to the pore network, a system of coupled linearequations for the potential at each node is obtained.This system is solved using an iterative Gauss±Seidelmethod. Substituting the volumetric ¯ow rate at theinlet of network and the pressure drop along it inDarcyÕs law, the absolute permeability is obtained. Thefractal roughness features of throats are included in thecalculation of hydraulic conductances as cylindricaltubes of equivalent cross-section area, whereas theroughness features of chambers are considered as dead-end pores and are ignored.

3. Serial sectioning analysis of porecasts

The experimental procedure of serial tomography asapplied to a porous sample of a sedimentary rock con-sists of the following stages [59]:

(a) Construction of double porecasts. The poroussample is evacuated and is impregnated with heatedliquid WoodÕs metal at high-temperature (T � 85±95°C), by applying a su�ciently high-pressure(P � 100±150 atm) for a period of 2±4 h. Then, thesystem is cooled at room temperature so that WoodÕsmetal is solidi®ed. The host rock is dissolved with hy-dro¯uoric acid and the pore cast is impregnated with theepoxy-resin ERL-4206 [47] colored with Oracet Red(MERCK). The whole system is heated to 60±70°C and

Fig. 3. Fractal model of the pore-wall roughness: (a) self-similar

fractal surface of the throats; (b) throat roughness features; (c)

chamber roughness features.


is kept at this temperature for 8±12 h, until polym-erization and hardening of the resin is completed.

(b) Production of serial sections. A small piece of theporecast is sliced and is glued on a microscope glassslide. The production of polished successive sections iscarried out by grinding and polishing the surface of theporecast, with a Minimet Polisher/Grinder (BUEH-LER). The amount of material, that is removed at eachstep, is properly adjusted by regulating the speed, loadand operation time of the equipment.

(c) Image acquisition. Each new polished surface ofthe sample is placed under an optical microscope, itsimage is taken with a videocamera, and is recorded in avideotape for further analysis (Fig. 4(a) and (b)).

(d) Image digitization and data storage. The video-recorded images (dimensions: 1.65 mm�1.24 mm, Fig.4(a) and (b)) are sent to an image analysis card(IMAGING TECHNOLOGY) which is installed in thehost computer. Only the central region (dimensions:1.08 mm�1.04 mm, Fig. 4(c) and (d)) of the primaryimages is digitized. The digitization is performed auto-matically by a software package which uses as par-ameters the length of resolution of the picture (number

of pixels/area) and the breakthrough gray-level betweenWoodÕs metal and resin.

For the determination of the topological and geo-metrical properties of porous media, the algorithmGETPO has been developed [33] which reconstructs thethree-dimensional pore network from data of two-di-mensional pore features of serial sections. Primarily, thealgorithm is used for the determination of the genus. Inaddition to the genus, the algorithm GETPO recon-structs a chamber-and-throat network that has the sametopology with the actual pore network and approxi-mates its geometry. For our purposes, the algorithmGETPO is used mainly for the determination of themean speci®c genus, GV, and the chamber diameterdistribution, CSD.

The results of the application of the algorithmGETPO to data of 44 serial sections of a double pore-cast of an outcrop Grey-Vosgues sandstone (sampleGV-1, Fig. 4) are given in Table 1. Using the Marquardtparameter estimation method [5], the chamber diameterdistribution (CSD) and the throat diameter distribution(TSD) were ®tted with bimodal distribution functions ofthe form

Fig. 4. (a) and (b) Video-recorded pictures of two successive sections of a porecast of Grey-Vosgues sandstone. The WoodÕs metal (pore space)

appears white and the resin (solid matrix) appears black. (c) and (d) Digitized negative images of the foregoing successive sections. Pixel size: 2 lm ´ 2

lm. Resolution of digitization: 500 ´ 482 pixels/image. Mean distance between serial sections: 7.3 lm.


fc�D� � cc fc1�D� � �1ÿ cc�fc2�D�; �3a�and

ft�D� � ct ft1�D� � �1ÿ ct�ft2�D�; �3b�respectively, where cc (ct) and 1ÿ cc�1ÿ ct� are thecontribution fractions of the log-normal componentdistribution functions fc1 and fc2 (ft1 and ft2) to the totaldistribution function fc (ft) (Table 1).

In general, every method that determines all geo-metrical properties of the pore space solely fromthree-dimensional reconstructed structures has someshortcomings, especially with reference to the lessmacroporous materials [33]. Speci®cally, throats of sizesmaller than the lowest limit of resolution (which isdetermined by the microscope magni®cation) are notmeasured or are assigned to larger sizes because theyare either completely or partly `ìnvisible''. Also, somevery small throats may not be impregnated by WoodÕsmetal and therefore they are not seen at any magni®-cation. In addition, short throats intervening betweenserial sections are not seen and are identi®ed as parts ofchambers. Therefore, the TSD obtained from serialsectioning analysis is expected to be overestimated withrespect to the `àctual'' one and the combination ofserial tomography with mercury porosimetry is neces-sary for the determination of the `àctual'' throat sizedistribution.

4. Analytic description of mercury porosimetry curves

4.1. Basic equations

The comparison of experimental with simulatedmercury porosimetry results can highly be simpli®ed byusing parametric analytic functions to describe themercury intrusion and retraction curves. Experimentalanalyses [14,15,40] and pore network simulations[18,21,36,43,49] have shown that most mercury porosi-

metry curves exhibit sigmoidal shape. In addition, eachcurve is subject to the limitation that mercury saturationgoes from 0 to 1, for intrusion and from its residualvalue S0 to 1, for retraction, as the capillary pressurechanges from very low to very high values. All afore-mentioned characteristics of mercury porosimetrycurves are embedded into the following 2-parameter(2-P) analytic functions.

(a) Mercury intrusion curves

S � �1� aI�eÿbI=P

1� aIeÿbI=P; bI > 0; �4a�

S � �1� aI�1� aIeÿbI=P

; bI < 0; �4b�

(b) Mercury retraction curves

S � S0 � �1� aR ÿ S0�eÿbR=P

1� aReÿbR=P; bR > 0; �5a�

S � �1� aR ÿ S0� � S0eÿbR=P

aR � eÿbR=P; bR < 0: �5b�

The functional form of the curves changes with thesign of parameters bI, bR, Eqs. (4a), (4b), (5a), (5b) inorder to keep its consistency with the aforementionedlimitations of mercury porosimetry curves. The medianpressures PI, PR specify the pressure range of intrusionand retraction curves whereas the mean slopes sI, sR

specify the width of the corresponding curves. Theseparameters are de®ned at saturation values S � 1=2 formercury intrusion and S � �1� S0�=2 for mercury re-traction, respectively, and are given below.

(a) Characteristic parameters of mercury intrusioncurves

PI � bI

ln aI � 2� � ; bI > 0; �6a�

PI � bI

ln�aI=�2aI � 1�� ; bI < 0; �6b�

Table 1

Topological and geometrical properties of the reconstructed pore space (sample: GV-1)

No Property Parameter values

1 Speci®c genus GV � 2:9� 10ÿ6 lmÿ3

2 Chamber diameter distribution (mean value, standard deviation) Bimodal distribution function

fc�D� : hDci � 29:1 lm; rc � 28:5 lm

Component lognormal distribution functions

fc1�D� : hDc1i � 22:0 lm; rc1 � 10:0 lm

fc2�D� : hDc2i � 43:5 lm; rc2 � 44:2 lm

Contribution fraction: cc � 0:33

3 Throat diameter distribution (mean value, standard deviation) Bimodal distribution function

ft�D� : hDti � 18:4 lm; rt � 11:3 lm

Component lognormal distribution functions

ft1�D� : hDt1i � 15:3 lm; rt1 � 4:6 lm

ft2�D� : hDt2i � 23:3 lm; rt2 � 16:1 lm

Contribution fraction: ct � 0:62


sI � �aI � 2� ln2�aI � 2�4bI�aI � 1� ; bI > 0; �7a�

sI � ÿ�2aI � 1� ln2�aI=�2aI � 1��4bI�aI � 1� ; bI < 0: �7b�

(b) Characteristic parameters of mercury retractioncurves

PR � bR

ln�aR � 2� ; bR > 0; �8a�

PR � ÿ bR

ln�aR � 2� ; bR < 0; �8b�

sR � �1ÿ S0��aR � 2� ln2�aR � 2�4bR�aR � 1� ; bR > 0; �9a�

sR � ÿ�1ÿ S0��aR � 2� ln2�aR � 2�4bR�aR � 1� ; bR < 0: �9b�

The parameter values (aI,bI) and (aR,bR) of intrusionand retraction curves are estimated separately for eachsample using a properly adjusted unconstrained Mar-quardt method of non-linear least squares [5]. The ex-perimental mercury porosimetry curves of a carbonateand a sandstone are compared to the correspondingones given by the analytic ®tting functions in Fig. 5(a)and (b). Although the 2-P functions ®t satisfactorily tocurves of narrow pressure range (carbonate C-4, Fig.5(a)) they are unable to ®t curves of a wide pressurerange (sandstone BS-1, Fig. 5(b)). In order to improvethe ®tting to mercury porosimetry curves, 5-parameter(5-P) functions resulting from the linear combination oftwo 2-P functions are used.

(a) Intrusion curve

S � cIS1�P ; aI1; bI1� � �1ÿ cI�S2�P ; aI2; bI2�; �10a�

Sj�P ; aIj; bIj� � �1� aIj�ebIj=P

1� aIjeÿbIj=P;

bIj > 0; j � 1; 2: �10b�(b) Retraction curve

S � cRS1�P ; S0; aR1; bR1�� 1ÿ cR�S2�P ; S0; aR2; bR2�; �11a�

Sj�P ; S0; aRj; bRj� � S0 � �1� aRj�eÿbRj=P

1� aRjeÿbRj=P;

bIj > 0; j � 1; 2: �11b�The median pressure and mean slope of each com-

ponent curve are given by Eqs. (6a), (6b), (7a), (7b), (8a),(8b), (9a), (9b). The ®tting of the curves of sandstoneBS-1 is improved substantially with the 5-P functions(Fig. 5(b)). The description of mercury porosimetrycurves in terms of the parameters of the speci®c analytic

functions makes easier and more e�cient any compari-son between experimental curves of di�erent porousrocks, or between experimental curves and simulatedresults.

In mercury porosimetry, the breakthough pressure[15] is de®ned as the minimum pressure at which mer-cury forms a connected pathway along the poroussample for a ®rst (during intrusion) or last (during re-traction) time. In terms of percolation theory, thebreakthrough pressure is related with the percolationthreshold of the pore network [45] and depends ongeometrical and topological properties of the porestructure [10,15,19,44,45,54]. It has been found [23,40]that the measured breakthrough pressure correspondsgraphically to the in¯ection point on a mercury intru-sion plot (this is where the mercury intrusion curve be-comes convex downward).

The breakthrough pressures of intrusion, PIB, andretraction, PRB, are de®ned at the in¯ection points ofthe mercury intrusion and retraction curves, respec-tively. These pressures are determined by setting thesecond derivative of each analytic function of Eqs. (4a),(4b), and (5a), (5b) equal to zero. Finally, there areobtained

bI ÿ 2PIB ÿ �2PIB � bI�aIeÿbI=PIB � 0; �12a�

bR ÿ 2PRB ÿ �2PRB � bR�aReÿbR=PRB � 0: �12b�When the mercury intrusion/retraction curves are

®tted with 5-P functions, then PIB and PRB are regardedas the in¯ection points of the low-pressure 2-P compo-nent functions. Such an approach is based on the real-istic assumption that the low-pressure componentfunction re¯ects the ®lling or the emptying of the pri-mary pore network while the high-pressure componentfunction re¯ects the ®lling or the emptying of the porewall roughness. Note that the parameters PIB and PRB

are not strictly equal but are expected to be comparableto the real capillary pressures at which mercury forms anetwork spanning cluster for a ®rst and a last time, re-spectively. These parameters are additional quantitativemeasures of comparison between experimental andsimulated mercury porosimetry curves.

4.2. Application to simulated curves

Details about the simulator of mercury intrusion/re-traction in chamber-and-throat networks are reported inRefs. [49±52]. 2-P functions were ®tted to existing sim-ulated curves of smooth-wall pore networks [49±51] andthe results are shown in Table 2(a)±(c). By using a cubicprimary pore network with log normal CSD(hDci � 40 lm; rc � 15 lm) and TSD �hDti � 10 lm;rt � 5 lm�, simulated mercury intrusion/retractioncurves were produced as functions of the fractalroughness properties Ds and ef /et. 5-P functions were


®tted to these curves and the results are shown in Table2(d). Based on the results of Table 2(a)±(d), a parametercovariance matrix was constructed (Table 3), where thevariation of each structural parameter of the pore net-work was correlated with the variation of the parame-ters of the corresponding ®tting functions. Thecovariance matrix of Table 3 can be used as a guidewithin a scheme of iterative parameter estimation, as wewill see below. The 5-P functions are more suitable thanthe 2-P functions for the ®tting of almost all mercuryporosimetry curves without excluding the necessity ofusing 8-P or higher parameter order functions to ®texceptionally complex curves. However, each of thesefunctions is nothing but the linear combination of 2-Pfunctions. Hence, the use of 2-P functions is aimed at

clarifying the physical meaning of the characteristicparameters of the ®tting functions and understandingthe correlations of these parameters with the actualproperties of the pore structure (Tables 2(a)±(d) and 3).Such correlations are di�cult to be deconvoluted di-rectly from 5-P functions. In the following, 5-P func-tions are used to ®t experimental and theoreticalmercury intrusion/retraction curves.

5. Method of characterization of pore structures

The microstructural characterization of a sample of agiven rock, in terms of a chamber-and-throat network,is performed with an algorithm that combines data of

Fig. 5. Fitting of mercury porosimetry curves to 2- and 5-parameter functions: (a) ®tting to the curves of a carbonate; (b) ®tting to the curves of a

sandstone.


serial tomography with experimental mercury poros-imetry curves and simulations of mercury intrusion/re-traction in pore networks. The iterative procedure usedto estimate the pore structure parameters is illustrated inFig. 6. The following steps are included in this scheme ofanalysis.

(a) The porosity of the sample is measured with im-bibition method [15].

(b) The absolute permeability of the sample ismeasured with gas permeametry [15].

(c) The mean speci®c genus hGvi and the chamber-diameter distribution CSD are obtained from serialsectioning analysis of a double porecast of the poroussample.

(d) The mercury porosimetry curves of the sample aremeasured and then are ®tted with 5-P analytic functions.

Table 2

Parameters of the ®tting functions of simulated mercury intrusion/retraction curves

hDti (lm) rt (lm) PI (MPa) sI (MPaÿ1) PR (MPa) sR (MPaÿ1) PIB (MPa) PRB (MPa)

(a) E�ects of the throat diameter distribution

13 5 0.1074 35.885 0.03331 13.587 0.1057 0.0308

10 5 0.1340 32.998 0.04023 4.53 0.1323 0.0364

10 3 0.1385 43.686 0.04273 4.502 0.1376 0.040121

10 8 0.1254 17.032 0.02386 7.964 0.1192 0.0186

16 8 0.0823 26.077 0.02545 26.239 0.07824 0.0226

hcni PI (MPa) sI (MPaÿ1) PR (MPa) sR (MPaÿ1) PIB (MPa) PRB (MPa)

(b) E�ects of the mean coordination number

6 0.1390 49.768 0.04318 13.371 0.1383 0.04112

5 0.1449 42.740 0.04423 10.701 0.1440 0.04213

4 0.1495 35.015 0.0435 8.206 0.1482 0.04119

3 0.1597 18.778 0.0426 5.683 0.1555 0.04000

hDti (lm) rt (lm) Corellation

type

PI

(MPa)

sI

(MPaÿ1)

PR

(MPa)

sR

(MPaÿ1)

PIB

(MPa)

PRB

(MPa)

(c) E�ects of the spatial pore size correlations

10 5 Uncor 0.1340 32.998 0.04023 4.53 0.1323 0.0364

10 5 c±t 0.1184 19.937 0.03211 4.59 0.1135 0.0265

10 5 c±c & c±t 0.1221 11.176 0.02235 7.36 0.1087 0.0171

13 5 Uncorel. 0.1074 35.885 0.03331 13.587 0.1057 0.0308

13 5 c±t 0.0976 23.733 0.02903 14.128 0.0933 0.0259

13 5 c±c & c±t 0.1031 14.672 0.02281 15.489 0.0936 0.0190

Ds nr ef /et cI PI1

(MPa)

sI1

(MPaÿ1)

PI2

(MPa)

sI2

(MPaÿ1)

(d) E�ects of the fractal roughness properties

2.25 6 0.418 0.775 0.1340 31.871 0.2733 1.098

2.25 14 0.243 0.785 0.1346 31.348 0.2198 1.558

2.25 28 0.149 0.655 0.1340 41.614 0.1803 2.759

2.30 8 0.381 0.790 0.1336 30.703 0.4862 0.524

2.30 16 0.247 0.655 0.1350 37.588 0.1850 2.655

2.30 34 0.149 0.695 0.1334 38.967 0.2394 1.459

2.35 8 0.412 0.775 0.13371 29.354 0.6813 0.367

2.35 18 0.256 0.655 0.1350 33.400 0.2283 1.795

2.35 40 0.153 0.725 0.1331 36.518 0.3503 0.803

Table 3

Covariance matrix

Variation of the parameters of the pore structure Variation of the parameters of ®tting functions

hDti PI ¯ sI PR � ct sR rt PI ¯ sI ¯ PR ¯ sR hcni PI ¯ sI PR � ct sR qc±t PI ¯ sI ¯ PR ¯ sR � ct

qc±c PI� ct sI ¯ PR ¯ sR Ds PI1� ct sI1 ¯ PI2 sI2 ¯ef /et PI1� ct sI1 ¯ PI2 sI2 ¯


(e) The throat diameter distribution TSD and theprimary porosity are estimated from the experimentalmercury intrusion curve. A ®rst estimate of the TSD isobtained by di�erentiating the mercury intrusion curveaccording to the conventional method of analysis that isbased on the tube bundle model of the pore structure[14].

(f) A theoretical chamber-and-throat network of ®-nite dimensions is constructed (Fig. 7). The input valuesof certain parameters (0) are slightly di�erent from theiroutput values (2) because the construction of the porenetwork is made under some limitations due to the re-quirements concerning the realistic representation of thepore space:

1. No intersection of adjacent chambers is allowed.2. Throats adjoining to a chamber are not allowed to in-

tersect outside the chamber.3. Each chamber remains accessible to the network

boundaries through at least one uninterrupted con-tinuous pathway of throats and chambers.

4. Throats are removed randomly and according to theestimated TSD until the speci®c genus of the theo-

retical pore network matches that of the poroussample.

Obviously, the di�erences between input (0) and output(2) parameter values are expected to increase as theoverlapping between the TSD and the CSD increases orthe degree of the spatial pore size correlations (ar-rangement of small or large sizes in adjacent positions)increases. The distribution of the coordination numberis calculated after the construction of the irregular net-work has been completed. The fractal properties of thepore network (Fig. 7) are so selected that the total po-rosity of the simulated network matches that of theporous sample.

(g) Mercury intrusion and retraction in the theoreticalchamber-and-throat network are simulated [49] and theabsolute permeability of the pore network is calculated.

(h) The simulated mercury intrusion and retractioncurves are ®tted with 5-P analytic functions.

(i) The values of the characteristic parameters of theanalytic functions of simulated mercury intrusion/re-traction curves are compared with the correspond-ing ones of the experimental curves. In addition, the

Fig. 6. Method of characterization of a pore structure in terms of chamber-and-throat network models.


experimental value of absolute permeability is comparedwith its theoretical value. If signi®cant di�erences arefound, the values of the pore network parameters alter.The procedure is repeated until the experimental mer-cury porosimetry curves and the absolute permeabilityare predicted satisfactorily. The following e�ects of porenetwork parameters on mercury porosimetry curves(Table 2(a)±(d)) are taken into account to estimate thenew parameter values.1. As the mean value of the TSD increases, the steepest

part of the intrusion curve (which corresponds to the®lling and emptying of the greatest fraction of thepore network and takes place over the low-pressureregion) moves to lower pressures whereas the widthof the intrusion and retraction curves decreases(Tables 2(a) and 3).

2. As the TSD widens, both mercury intrusion and re-traction curves move toward lower pressures, thewidth of intrusion curve increases and the width ofretraction curve decreases (Tables 2(a) and 3).

3. As the mean coordination number decreases, theintrusion curve moves to higher pressure range, itswidth increases, the loop of hysteresis between theintrusion and retraction curves widens and the re-sidual mercury saturation increases (Tables 2(b)and 3).

4. As the degree of the spatial pore size correlations in-creases, the intrusion curve widens whereas the mer-

cury retraction curve moves to lower pressure rangeand its width decreases (Tables 2(c) and 3).

5. When the fractal dimension increases, the intrusioncurve widens toward high-pressures whereas the re-traction curve widens and moves to higher pressuresas well (Tables 2(d) and 3).

6. The speci®c genus of the simulated network is verysensitive to the length of periodicity of the lattice.The primary porosity et in¯uences the length ofperiodicity of the pore network and hence it has apronounced e�ect on the mean value of coordinationnumber that is reached when the speci®c genus of thenetwork matches its experimental value.

7. The absolute permeability of the pore network [11] isa�ected mainly by the TSD and the pore networkconnectivity (namely, the coordination number distri-bution).

6. Results and discussion

The above methodology was applied to the estima-tion of the parameter values of a sample (GV-1) of anoutcrop Grey-Vosgues sandstone.

(a) The porosity of the sample was measured andfound to be et � 0:18.

(b) The gas permeability of the sample was measuredand found to be k � 110 md.

Fig. 7. Computer-aided construction of theoretical chamber-and-throat networks.


(c) The speci®c genus hGvi and the chamber diameterdistributuion, CSD were determined by using serialsectioning analysis of a double porecast (Table 1).

(d) The experimental mercury porosimetry curveswere ®tted with 5-P analytic functions and the optimumparameter values were determined (Fig. 8). The mercuryintrusion curve was decomposed to two componentcurves extending over the low- and high-pressure range,respectively (Fig. 8).

(e) By assuming that the coe�cient cI (Fig. 8) isroughly comparable to the ratio of the primary to thetotal porosity ( ep=et � 0:55), the low-pressure compo-nent of intrusion curve was di�erentiated accordingto the conventional method of analysis and a log nor-mal TSD with parameter values hDti � 12:2 lm;rt � 2:45 lm was obtained. Theoretical simulation hasshown that the actual TSD of an uncorrelated networkis usually wider than that obtained with di�erentiationbecause of ``shadowing'' network e�ects [49]. A lognormal TSD (0) with parameter values hDti � 12:5 lm;rt � 4:0 lm and a primary porosity ep � 0:11 wasselected.

(f) When the construction of the primary chamber-and-throat network was completed the output log nor-mal TSD (2) as well as any other network parameterwere calculated (NET1, Tables 4 and 5). The choice ofthe fractal properties (Ds� 2.45, nr� 12) was made sothat the size range of roughness features be su�cientlywide because the slope of the high-pressure component(sI2) of intrusion curve is quite low (GV-1, Table 6).

(g) Mercury intrusion in and retraction from thenetwork were simulated and the theoretical absolutepermeability was determined (NET1, Table 4).

(h) The optimum values of the characteristic par-ameters of the analytic ®tting functions of the simulatedintrusion/retraction curves were determined (NET1,Table 6).

(i) The theoretical results were compared with theexperimental data and new estimates of the parametervalues were made based on data of Table 3. The TSDwas kept nearly constant while the mean coordinationnumber was decreased by adjusting the value of theprimary porosity et. The other parameters were changedaccordingly (NET2, Tables 4 and 5). In this manner, theagreement of theoretical with experimental results wasimproved substantially with respect to both mercuryporosimetry curves (NET2, Table 6) and absolute per-meability (NET2, Table 4). Consecutive iterations re-vealed that simulations on pore networks with unimodalTSDs were unable to predict simultaneously the capil-lary behavior (Fig. 9(a)) and permeability of the poroussample (Tables 4 and 6). For this reason, simulations onpore networks with bimodal TSDs, Eq. (3b), were at-tempted (POR1±POR2, Tables 4 and 5). Although withthe use of bimodal TSDs the predictability of the ex-perimental data was improved (Fig. 9(b), Tables 4 and6), some important characteristics of the capillary be-haviour of the porous sample GV-1, such as the rela-tively low cI value (which means that the contributionfraction of the low-pressure component curve on theoverall intrusion curve is almost the same with thecorresponding one of the high-pressure componentcurve) were not well-predicted by using uncorrelatednetworks and bimodal TSDs (Table 6). With the use ofstrongly c±c and c±t correlated chamber-and-throatnetworks and bimodal TSDs (CP1±CP3, Tables 4 and 5)the agreement of simulated results with the experimentaldata became satisfactory (Fig. 10, Tables 4 and 6).

The breakthrough pressures and permeability of thenetwork CP3 are very close to the corresponding ones ofthe sample GV-1 (Fig. 10(b)). The parameters of theanalytic ®tting functions of mercury intrusion curve ofthe CP3 network are also very close to the correspond-ing ones of GV-1 sample but there is a discrepancy with

Fig. 8. Mercury porosimetry curves of Grey-Vosgues sandstone GV-1 (et � 0:18; k � 110 md).


respect to the mean slope of the low-pressure componentfunction of the mercury retraction curve, sR1 (Table 6).The relatively high slope sR1 of the CP1±CP3 networks(Table 6) is due to the fact that the boundary e�ects onthe last part of mercury retraction curve increase sig-ni®cantly in strongly correlated pore networks [50].Simulations in pore networks of dimensions much largerthan those used here (20 ´ 20 ´ 20) are needed so that®nite-size e�ects are eliminated and the mean slope ofthe low-pressure component function of the retractioncurve is reduced.

In Fig. 11, the TSDs of the strongly correlated sim-ulated networks (TSD1±TSD3) are compared with theTSDs obtained from the serial sectioning analysis(TSD4, Table 1) and the conventional method of anal-ysis of porosimetry data but by using the tube bundle

model (TSD5). The serial sectioning analysis (TSD4,Fig. 11) is unable to account for throats of small sizes(smallest pore feature traced �2 lm) and hence theprediction of the ¯ow properties of the reconstructedthree-dimensional pore space becomes questionable [24].

On the other hand, new techniques have appeared,where the three-dimensional pore structure is recon-structed from only one section with the aid of an auto-correlation function. These techniques have nolimitation on the pore size range and seem very prom-ising for the accurate determination of certain meso-scopic transport properties of homogeneous porousmedia [20,60]. However, these three-dimensional recon-structed structures must be combined with time-con-suming computational tools, such as cellular automata[45] to simulate multiphase transport processes.

Table 6

Parameters of the analytic ®tting functions of mercury intrusion/retraction curves

Pore net cI PI1

(MPa)

sI1

(MPaÿ1)

PI2

(MPa)

sI2

(MPaÿ1)

cR PR1

(MPa)

sR1

(MPaÿ1)

PR2

(MPa)

sR2

(MPaÿ1)

GV)1 0.545 0.1233 24.86 1.04 0.243 0.590 0.0570 2.650 2.11 0.037

NET1 0.515 0.1560 9.619 0.191 1.873 0.925 0.0841 1.717 68.94 0.0018

NET2 0.795 0.1347 12.76 2.50 0.100 0.880 0.0826 2.65 17.56 0.008

NET3 0.805 0.1256 14.69 1.19 0.209 0.780 0.0590 5.44 1.680 0.0734

POR1 0.750 0.1273 18.792 1.450 0.1743 0.700 0.0585 7.626 3.236 0.0405

POR2 0.770 0.1262 17.746 0.997 0.264 0.690 0.0522 9.350 2.250 0.0529

CP1 0.625 0.1191 13.338 0.9426 0.2654 0.590 0.0422 13.216 1.825 0.0706

CP2 0.585 0.1210 20.008 0.9537 0.2727 0.630 0.0434 11.305 1.986 0.0557

CP3 0.63 0.1187 20.76 1.017 0.246 0.675 0.0382 13.655 2.01 0.0635

Table 4

Parameters of the theoretical chamber-and-throat networks used in simulations

NET hDci(lm)

rc

(lm)

hDti(lm)

rt

(lm)

hcni GV ´ 106

(lmÿ3)

DS ef /et qc±c qc±t k

(md)

NET1 31.48 23.32 12.67 4.65 5.85 2.867 2.4 0.368 )0.151 0.446 176

NET2 30.91 22.36 12.66 4.70 4.95 2.889 2.45 0.371 )0.180 0.438 135

NET3 29.07 22.69 12.55 1.92 4.10 2.890 2.35 0.372 )0.183 0.386 69.7

POR1 28.45 20.94 10.93 5.07 4.35 2.835 2.50 0.380 )0.16 0.441 99.8

POR2 28.50 20.87 10.44 4.12 4.35 2.850 2.45 0.371 )0.18 0.424 80.6

CP1 29.87 28.00 10.28 6.45 4.90 2.860 2.50 0.434 0.380 0.596 106.7

CP2 28.96 25.25 9.27 9.46 4.40 2.860 2.45 0.399 0.384 0.598 71.8

CP3 28.72 23.98 9.43 9.15 4.65 2.855 2.45 0.396 0.381 0.579 96.0

Table 5

Parameters of the component distribution functions of the CSD and TSD

NET cc hDc1i (lm) rc1 (lm) hDc2i (lm) rc2 (lm) ct hDt1i (lm) rt1 (lm) hDt2i (lm) rt2 (lm)

NET1 0.73 23.36 8.57 53.43 34.0 1.0 12.67 4.65 ± ±

NET2 0.745 23.23 8.59 53.33 32.7 1.0 12.66 4.70 ± ±

NET3 0.455 22.53 7.28 34.54 28.89 1.0 12.55 1.92 ± ±

POR1 0.92 23.92 11.7 80.6 30.9 0.58 8.67 5.5 14.05 1.61

POR2 0.91 23.71 11.3 76.9 31.2 0.33 15.01 1.11 8.2 3.07

CP1 0.635 22.83 9.87 42.11 41.74 0.60 13.73 2.47 5.09 7.08

CP2 0.845 22.47 11.3 64.3 44.03 0.545 13.89 2.24 3.74 11.60

CP3 0.89 22.96 11.70 75.32 40.98 0.59 13.85 1.8 3.07 11.44


The conventional method of analysis [14] gives a``pore-size distribution'' (TSD5, Fig. 11) that is com-parable to the TSDs obtained with the present method(TSD1, TSD2, TSD3 ± Fig. 11). However, no furtherinformation is provided about any other properties ofthe porous structure. For instance, the absolute per-meability of a bundle of parallel capillary tubes is givenby the following approximate relationship [15]

k � et

96r2

t

�� hDti2

��13�

which yields a value k � 205 md if the parameter valuesof TSD5 are used. Therefore, the conventional ``pore-size distribution'' is insu�cient to predict correctly themesoscopic single-phase transport properties of porousmaterials and hence the topological, correlational andfractal properties of the porous microstructure must be

taken into account, as well. Apparently, the estimatedTSDs of the correlated networks CP2 and CP3 (Fig. 11)as well as their component distribution functions (Table5) are very close each other. This means that the TSDstarts to converge to a unique bimodal distributionfunction after utmost three iterations. Also, both thecorrelation coe�cients qc±c, qc±t and the fractal proper-ties Ds, ef /et of the CP2 and CP3 seem to converge toconstant values (Table 4). The observed di�erence be-tween the permeabilities of CP2 and CP3 is due to theirdi�erent mean coordination number (Table 4). For anyfurther improvement of the prediction of the capillaryand ¯ow properties of the investigated porous sample,the TSD as well as the parameters qc±c, qc±t, Ds, ef /et ofthe simulated networks should be kept almost unalteredand only the mean coordination number be adjusted tomatch the experimental data.

Fig. 9. Comparison between 5-parameter analytic ®tting functions of the experimental (GV-1) and simulated mercury intrusion/retraction curves.

Uncorrelated chamber-and-throat networks with: (a) unimodal and (b) bimodal TSDs.


From petrographic analyses of Grey-Vosgues, it iswell-known [16] that a small amount of clay is presentupon the grain surface but does not form any secondarypore networks inside the solid matrix. Therefore, thepresence of clays may a�ect the fractal dimension, Ds,but not the ratio ef /et.

The determination of the surface fractal dimensionfrom the high-pressure characteristics of capillary pres-sure curves is not a new idea [48]. However, except forthe fractal dimension, the present method estimates thefractal roughness porosity so that the values of bothparameters are consistent with the values of the otherpore network parameters. Furthermore, there are manyother methods for the determination of the fractalproperties of porous materials at small length scales as

for instance, the fracture surface technique [25], the ni-trogen adsorption [46], the small angle X-ray scattering[4], the neutron scattering [58], etc. The reliability of thepresent method with respect to the calculation of thefractal dimension values could be evaluated in com-parison with corresponding values obtained from eitherthe analysis of two-dimensional images of random sec-tions at several magni®cations or nitrogen adsorptionisotherms, but this is beyond the scope of the presentwork.

With reference to the one-phase transport coe�-cients, the present methodology is self-predictive be-cause the consistency of the experimental with thetheoretical absolute permeability is included in the iter-ative procedure of pore structure parameter estimation(Fig. 6). Although the theoretical tools needed for thecalculation of multiphase transport properties of reser-voir rocks from microscopic properties of the porestructure have already been developed [12,54], there iscurrently a lack of experimental data about the relativepermeability and resistivity index curves of the investi-gated core of Grey-Vosgues. When such data are ob-tained, then the existing theoretical simulators will beable to be used for the evaluation of the reliability of thepresent method of pore structure characterization withrespect to the predictability of multiphase transportcoe�cients. Nevertheless, such an investigation can be-come the subject of a forthcoming publication.

7. Conclusions

A methodology was developed for the characteriza-tion of the porous structure of reservoir rocks in termsof chamber-and-throat networks. The chamber (equiv-alent spherical) diameter distribution and speci®c genus

Fig. 11. Comparison of the TSDs obtained from the present method with the TSDs obtained from the serial sectioning analysis and the di�eren-

tiation of the mercury intrusion curve.

Fig. 10. Comparison between 5-parameter analytic ®tting functions of

the experimental (GV-1) and simulated mercury intrusion/retraction

curves. Strongly c±c and c±t correlated chamber-and-throat networks

with bimodal TSDs.


of the porous structure are obtained with application ofa three-dimensional reconstruction algorithm [33] toserial section data of double porecasts. The throat(equivalent cylindrical) diameter distribution function,the spatial correlation coe�cients of the pore sizes andthe parameters of the fractal porosity of the pore-wallroughness are estimated by comparing the experimentalwith simulated mercury intrusion/retraction curves andthe experimental with the theoretical value of the ab-solute permeability. The mercury intrusion/retractioncurves are ®tted with 5-P analytic functions and anycomparison of experimental data with simulated resultsis made in terms of certain characteristic parameters ofthese functions. Application of the method to thecharacterization of the structure of an outcrop Grey-Vosgues sandstone (sample GV-1) indicated that thesimultaneous prediction of capillary (mercury intrusion/retraction curves) and ¯ow (absolute permeablity)properties is accomplished under the following pre-sumptions: (a) the throat diameter distribution is bi-modal, (b) strong c±c and c±t spatial correlationsbetween pore sizes exist, (c) the fractal dimension is ca.2.45 and the fractal roughness porosity is ca. 0.4 and (d)the mean coordination number of the pore networkranges from 4.4 to 4.7.

The most signi®cant novel contributions of thepresent method on the characterization of the structureof reservoir rocks are summarized below:· The comparison of experimental with simulated mer-

cury porosimetry curves is simpli®ed by using thecharacteristic parameters of analytic ®tting functions.

· The computer-aided construction of irregular net-works is based on the speci®c genus rather than onthe mean coordination number.

· The fractal dimension and fractal porosity are deter-mined simultaneously from mercury porosimetrycurves in a manner that keeps their values consistentwith the values of the other parameters of the porenetwork.

· The iterative procedure of parameter estimation issimpli®ed by using a covariance matrix into which in-formation concerning the e�ects of pore networkparameters on simulated mercury porosimetry curvesis embedded.

· The complete microstructural characterization of po-rous rocks in terms of chamber-and-throat networksprovides all necessary parameter values of the porespace to simulate multiphase transport processes insuch pore networks.

Acknowledgements

The present work was supported by the Institute ofChemical Engineering and High Temperature ChemicalProcesses. We thank Dr. D.G. Avraam for his valuable

assistance in capturing and processing the images ofserial sections. We also thank Mr. E. Skouras for hisassistance in recon®guring and running the algorithmGETPO in Unix.

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