comparison of network generation techniques for unconsolidated

Comparison of Network Generation Techniques for Unconsolidated Porous Media

Riyadh Al-Raoush, Karsten Thompson, and Clinton S. Willson*

ABSTRACT First, we discuss network models and techniques fordetermining the pore-scale properties and characteris-While network models of porous materials have traditionally beentics; second, we describe the regular and random porousconstructed using regular or disordered lattices, recent developments

allow the direct modeling of more realistic structures such as sphere media systems used to validate and evaluate the tech-packings, microtomographic images, or computer-simulated materi- niques we are proposing; third, we present the improve-als. One of the obstacles in these newer approaches is the generation ments made to two network generation techniquesof network structures that are physically representative of the real based on their known limitations; fourth, we validatesystems. In this paper, we present and compare two different algo- these algorithms using regular packings; and finally, werithms to extract pore network parameters from three-dimensional evaluate and compare the network representations ofimages of unconsolidated porous media systems. The first approach,

random packings generated by these techniques.which utilizes a pixelized image of the pore space, is an extension tounconsolidated systems of a medial-axis based approach (MA). Thesecond approach uses a modified Delaunay tessellation (MDT) of the BACKGROUNDgrain locations. The two algorithms are validated using theoretical

Early models developed to account for pore-scalepackings with known properties and then the networks generatedproperties idealized the pore space as collections offrom random packing are compared. For the regular packings, bothcapillary tubes and provided simple analytical solutionsmethods are able to provide the correct pore network structure, includ-

ing the number, size, and location of inscribed pore bodies, the num- to predict continuum-scale properties such as perme-ber, size, and location of inscribed pore throats, and the connectivity. ability. However, these models failed to incorporate theDespite the good agreement for the regular packings, there were interconnectivity of the pore space. Thus the idea ofdifferences in both the spatial mapping and statistical distributions in representing the pore space as a two- or three-dimen-network properties for the random packings. The discrepancies are sional network emerged from the pioneering work byattributed to the pixelization at low resolution, non-uniqueness of the Fatt (1956a,b,c). Due to the complexity of the pore-inscribed pore-body locations, and differences in merging processes

space morphology, the pore bodies and throats are usu-used in the algorithms, and serve to highlight the difficulty in creatingally represented by simplified shapes. Pore bodies havea unique network from a complex, continuum pore space.been represented by spheres or cubes, while porethroats have been represented by cylinders or otherducts with non-circular cross-sections. Although net-Water and solute transport and the distributionwork models can be two- or three- dimensional, two-of phases in subsurface systems are directly re-dimensional networks have significant limitations inlated to the geometry and the topology of the pore spacehow they represent the interconnectivity of real three-and can have a strong influence on continuum scaledimensional media (Chatzis and Dullien, 1977). Com-properties. While many processes related to site assess-mon three-dimensional networks are random (e.g.,ment and remediation are usually studied using a contin-Lowry and Miller, 1995) or cubic lattices with coordina-uum approach, pore-scale modeling is considered a pow-tion number of six (or smaller if bonds are removed).erful tool to better understand the physical processesIn actual porous media systems, however, the connectiv-involved and to determine macroscale constitutive rela-ity is distributed, and can have values larger than sixtionships that can be difficult to obtain experimentally.(Kwiecien et al., 1990). One solution to this problem isThe most critical part of constructing a network modelto use physically representative networks, which do notis the identification of the pore structure. This can beemploy an underlying lattice but instead, map the truedone using either using computer-generated porous me-pore structure of a medium onto a network (Bryant etdia systems or through the use of non-destructive im-al., 1993a). Consequently, they retain more completelyaging techniques (e.g., synchrotron microtomography).the true pore morphology and any inherent spatial cor-The significant challenge lies in the generation of net-relations.work structures that are physically representative of the

Network models have been used to study a wide rangereal materials of interest.of single and multiphase flow processes including rela-In this paper we present the development of andtive permeability (Blunt and King, 1990, Rajaram etcompare two improved techniques for generating physi-al., 1997, Fischer and Celia, 1999); the effect of porecally realistic network models of granular porous media.structure on relative permeability and capillary pressurehysteresis in two phase systems (Jerauld and Salter,R. Al-Raoush and C.S. Willson, Dep. of Civil and Environmental1990); prediction of permeability (Bryant et al., 1993a);Engineering, Louisiana State Univ., Baton Rouge, LA 70808; K.

Thompson, Dep. of Chemical Engineering, Louisiana State Univ., investigation of the functional relationship between cap-Baton Rouge, LA 70808. Received 6 June 2002. *Corresponding au- illary pressure, saturation, and interfacial areas (Reevesthor ([email protected]). and Celia, 1996); prediction of permeabilities and resi-Published in Soil Sci. Soc. Am. J. 67:1687–1700 (2003). Soil Science Society of America Abbreviations: MA, medial-axis based approach; MDT, modified De-

launay tessellation.677 S. Segoe Rd., Madison, WI 53711 USA

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1688 SOIL SCI. SOC. AM. J., VOL. 67, NOVEMBER–DECEMBER 2003

dence time distributions for mechanical dispersion in With either approach (microtomography images orpacked beds (Thompson and Fogler, 1997); drainage computer-generated porous media), a significant chal-and imbibition (Lowry and Miller, 1995, Hilpert and lenge lies in the generation of network structures thatMiller, 2001); and phase distributions, interfacial areas, are physically representative of the real materials ofand mass transfer (Dillard and Blunt, 2000). interest. The objective of this paper is to present and

The most critical part of constructing a network model compare two different algorithms to extract pore net-is the identification of the pore-structure. Two general work parameters from three-dimensional images of po-approaches are commonly used. The first attempts to rous media systems. The first approach is based on adirectly map a specific porous medium onto a network medial-axis analysis of a pixelized image of the porestructure. The second attempts to create an equivalent space. The second method is based on a modified Delau-network using distributions of basic morphologic pa- nay tessellation of the grain locations. In this work, therameters. The fundamental difference between the two two algorithms are validated using theoretical packingsmethods is that direct mappings provide a one-to-one with known properties (e.g., simple cubic and rhom-spatial correspondence between the porous medium bohedral packings). Then, networks generated fromstructure and the network structure, whereas the second random packings are compared. Because various ap-type of network is equivalent only in a statistical sense. proaches to network generation lead to a variety of

Typical parameters used for characterization include structures, it is hoped that this comparison of indepen-pore-body and pore-throat size distributions, throat- dent techniques will improve existing algorithms for net-length distribution, coordination number, spatial corre- work reconstruction, will validate the methods, and willlation within the system, which includes size correlation give new insight into the fundamental structure of com-between adjacent pore bodies (body-body correlation), mon porous materials.size correlation between adjacent pore throats (throat-throat correlation), and size correlation between neigh- Porous Media Systemsboring pore throats and pore bodies (body-throat corre-

For this work, we have chosen to use computer-simu-lation), and pore-body/pore-throat aspect ratio. Therelated sphere packings for several reasons: (i) to studyare different approaches used to identify the pore struc-the MDT; (ii) to compare the MDT (which allows forture including the use of inverse methods to fit modelcertain analytical calculations) to the MA-based ap-parameters to an experimentally measured pressure sat-proach (which relies on the digitized image) and (iii) touration relation and the use of mercury porosimetry to

obtain the pore-size distribution of the system. allow the creation of digitized images of the system atDirect mapping techniques require some method of arbitrarily high resolutions.

obtaining the pore structure before reconstruction of The two techniques are validated using cubic andthe network. Imaging techniques include the analysis rhombohedral packings. The first packing used has aof serial thin sections (e.g., Cousin et al., 1996, Vogel simple cubic structure having uniform, equally spacedand Roth, 2001) and non-destructive techniques such pore bodies and throats. The second packing has aas X-ray microtomography (e.g., Rintoul et al., 1996, rhombohedral structure, which contains two distinctLindquist et al., 2000). Non-destructive direct imaging pore-body sizes that are repeated throughout the regu-approaches (i.e., tomography) are attractive because lar structure (Kruyer, 1958). The third packing has thethey provide a detailed and unique description of the same rhombohedral structure, but pixelized at a higherpore-space geometry. Equally important is the fact that resolution.the connectivity and spatial variation in inscribed pore

1. The cubic packing consists of 512 spheres, digitizedbody and throat sizes are retained. Recent advances inat 25 pixels per sphere diameter. The size of themicrotomography have allowed researchers to obtainthree-dimensional image is 200 by 200 by 200 pix-three-dimensional images of porous media systems onels. The calculations are based on an interior vol-the order of 10 �m. The acquisition of these extremelyume of 175 by 175 by 175 pixels, where a sphererich data sets has motivated the development of algo-radius is cut out from each side to remove therithms designed to extract the pore network parameters.boundary effects. For this system, there is one poreA complementary tool to microtomography is com-per sphere, all pores have inscribed sphere radiiputer reconstruction of various types of porous mediaof 0.7320R and inscribed throat radius of 0.4142R,(Bakke and Oren, 1997, Nandakumar et al., 1999, Liangwhere R is the sphere radius (Kruyer, 1958). Theet al., 2000). While microtomography is important fromcoordination number is exactly six.an analytical perspective (especially for types of porous

2. The first rhombohedral packing consists of 512media that are not well characterized), the use of com-spheres, digitized at 25 pixels per sphere diameter.puter-simulated media allows one to avoid this experi-The size of the three-dimensional image is 200 bymentally intensive step for packed beds, well-sorted173 by 163 pixels. The calculations are based on asands, or certain sandstones. Additionally, the use ofvolume of 175 by 148 by 138 where a sphere radiuscomputer-generated porous media is preferable to di-is cut out from each side to remove the boundaryrectly constructing networks from statistical informationeffects. Analytical calculations by Kruyer (1958)because one can work with more fundamental materialindicate two pore sizes for the rhombohedral pack-properties (e.g., grain-size distributions rather then in-

ferred properties of the pore space). ing: a smaller one with radius 0.2247R and a larger

AL-RAOUSH ET AL.: COMPARISON OF NETWORK GENERATION TECHNIQUES 1689

one with radius 0.4142R; all pore throats have in-scribed radii equal to 0.1547R.

3. A second rhombohedral packing was created at ahigher pixel resolution for use with the MA algo-rithms. (Resolution does not affect the MDT algo-rithm since it works with the sphere locations di-rectly rather than with digitized packings.) Itconsists of 512 spheres, digitized at 45 pixels persphere diameter. The size of the three-dimensional

Fig. 1. The medial axis concept in two dimensions; circles representimage is 364 by 315 by 297 pixels. The calculationsthe solid phase and the dashed lines represent the medial axis.are based on a volume of 319 by 270 by 252 where

a sphere radius is cut out from each side to removetance transform of the inverted image. The local maxi-the boundary effects. For a granular media systemmum of the resulting image will represent points on thewith a d50 of 450 �m, the resolution of this systemobject’s skeleton. Each skeleton is equally distant fromwould be on the order of 10 �m. This is approxi-at least two different points on the object border. Themately the limit when applying synchrotron micro-border points closest to an interior point are called thetomography to typical porous media systems. Theprojection points of the interior point and can be ob-radius of the spheres in this packing is 22.5 pixels.tained from a distance transform (Goldak et al., 1991;

Following validation of the algorithms using the regu- Brandt, 1992; Lohman, 1998). Figure 1 shows the con-lar packings, the two methods are compared using a cept of the medial axis in a simple, two–dimensionalcomputer-generated random sphere pack. For random cross-section.packings, which are generally of more interest than reg- Baldwin et al. (1996) developed a morphological thin-ular packings, the network structure created by these ning algorithm to obtain a representation of the skeletonalgorithms is not unique. Hence, it is of interest to com- (medial axis) of the pore space and partition the voidpare the networks that are generated by each, as well space into corresponding pores and throats. No attemptas differences in network geometry that are introduced was made to use the medial axis itself as tool to charac-by parameters in each algorithm. In this paper, we focus terize pores and throats; the thinned images were usedon results obtained using a random sphere pack con- instead. In their work, the pores are identified by sum-taining 1000 uniformly sized spheres, with a porosity of mation of the voxels bounded by the solid matrix and38%. The pack is contained in a cubic domain, and is the locations corresponding to local minima are definedperiodic in all directions. The use of periodicity prevents as pore necks. The thinning algorithm is based on aedge effects from influencing the tessellation. However, segmentation algorithm that leads to misidentificationbecause the medial axis algorithm does not employ peri- or less accurate partitioning of the void space. Lindquistodicity, the networks were cropped during comparison et al. (2000) developed algorithms to calculate effectiveof the two algorithms. The packing was created using pore body and pore throats radii and other geometrica collective rearrangement technique (Liu and Thomp- properties such as coordination number based on theson, 2000). For the medial axis calculations, this system medial axis of the pore space. Their algorithms primarilyis pixelized using a resolution of 25 pixels per sphere have been applied to consolidated systems, and it is notdiameter. Thus, the original size of the three-dimen- clear whether they are applicable to higher porositysional image is 250 by 250 by 250. After cropping one materials (e.g., unconsolidated media).sphere radius from each side, the image is 200 by 200by 200. Medial Axis Generation

The 3DMA software package (http://www.ams.sunysb.Network Generation Using Medial Axis edu/~lindquis/3dma/3dma.html; verified 13 June 2003)Based Approach is used to construct the medial axis on the “binary”images. The 3DMA medial axis algorithm utilizes theThe medial axis of an object is the skeleton of the

object running along its geometrical middle. For a po- technique developed by Lee et al. (1994) and modifiedby Lindquist and Lee (1996). A burning algorithm isrous medium, it provides simple and compact basic in-

formation about the topology and geometry of the void applied as follows: each voxel in the solid phase is la-beled with integer 0, voxels in the exterior region (be-space. In the case of an image with a range of intensity

values, the first step in defining the medial axis is to yond the boundary of the image) are labeled with inte-ger �1, and voxels in the void space are initiallyapply a segmentation algorithm to separate the phases

of the image based on the intensity values. A discrete unlabeled. The algorithm proceeds by simultaneouslyassigning unlabeled voxels in the void space with an“burn” algorithm (Kwok, 1988; Sirjani and Cross, 1991;

Lindquist and Lee, 1996) is applied to the segmented integer n�1 for each voxel neighboring a grain-phasevoxel. This procedure is iterated at a rate of one layerbinary image to obtain a discrete representation of the

continuum pore space. per iteration until all voxels in the pore space are la-beled. There are two methods to define the neighbor-The medial-axis skeleton can be obtained by inverting

the image so that the background becomes the new hood of a voxel. The first is face touching, where eachvoxel can be neighbored by a maximum of six voxels.foreground and vice versa and then computing the dis-


The second is boundary touching, where each voxel canbe neighbored by a maximum of 26 voxels. For complexthree-dimensional systems, the 26-connectivity ap-proach is preferred. Determination of the medial-axisvoxels depends on the burning direction; if the burnenters a voxel from two different directions that arenot perpendicular, the voxel is considered a medial-axisvoxel. If the burn enters two neighboring voxels fromdifferent directions, both voxels are considered as me-dial axis voxels. The result of this burning algorithm isa set of media1 axis voxels each with a unique burn

Fig. 2. Example showing the merging of two inscribed pore bodiesnumber, which is related to its distance from the closestthat occupy the same void space; if there is any overlap, the largestgrain voxels.pore is chosen.The medial axis in its initial form is made up of a

network of one-dimensional paths and nodes at paththe minimum burn number along the path connectingintersections. The spatial location and the burn numbertwo nodes. Because a path is a set of medial-axis voxelsof each voxel in the medial axis is recorded and is usedthat each have exactly two neighboring voxels (on theas the basis from which to calculate the locations andmedial axis) and there might be multiple occurrences ofsizes of inscribed pore body and pore throats as well asthe minimum burn number along the path, the dilationthe coordination numbers. It should be noted that thealgorithm described above is applied to all the “mini-medial axis is very sensitive to the noise and the bound-mum” voxels along the path. The smallest radius foundary of the solid phase.is considered as the inscribed radius. Figure 3b presentsa schematic of this process.Inscribed Pore Radius Calculations Figure 4 graphically illustrates the overall process of

To provide a unique measure of pore location and creating inscribed pore body and throat data from thesize, we have chosen to focus on calculating the inscribed medial axis. Figure 4a is a periodic random packing ofpore body radii (i.e., the radius of the largest inscribed uniform diameter spheres; Fig. 4b is the medial axissphere in a pore). A node is defined as an intersection generated from that system and Fig. 4c shows the in-of three or more paths on the medial axis. Because of scribed pores, with the radius of the sphere equal to thethe complex, three-dimensional nature of the medial inscribed pore radius.axis, a node might consist of more than one voxel havingthe same maximum burn number. In this case, one voxel Validation of the Medial Axis-Based Algorithmis chosen randomly as the center of the node. A dilation

Figure 5 shows portions of three-dimensional imagesalgorithm is performed to find the radius of the poreof the first two test systems and the location of the poresas follows: (i) start from the chosen voxel on the nodein these systems, calculated from the medial-axis-basedand create a sphere with a radius of one voxel; (ii) insertmethod. (The lighter spheres are the packing; darkerthis sphere in the original segmented data and check ifspheres are the inscribed spheres denoting pore loca-there is any overlap between the inscribed sphere andtions). Table 1 shows the values calculated for the threethe surrounding grains. If there is an overlap, the proce-test systems along with the analytical and MDT-baseddure stops and this radius is considered as the radius ofvalues (which will be discussed later). For all three sys-the pore. If there is no overlap, the algorithm proceedstems, the MA algorithm predicted the exact numberby increasing the radius by one voxel (i.e., dilation ofof pores and for a majority of the pores, the correctthe sphere radius), and repeats the previous step until

an overlap between the grain-phase and sphere occurs.In some cases, two or more nodes (and hence pore

bodies) occupy what can be considered one void space.Physically, this might not be appropriate because theidea of the pore network is to provide a unique andsimple representation of the void space. Therefore, amerging criterion is adopted in which two inscribed porebodies are merged if they overlap. When merging isimplemented, the largest inscribed pore body is chosento be the inscribed pore body of that void space (Fig. 2).It is noted, however, that this approach may underrepre-sent the inscribed pore body sizes for the system.

Inscribed Throat Radius Calculations Fig. 3. (a) The inscribed pore radius-finding algorithm where the burnnumber is local maxima, and (b) the inscribed throat radius whereThe same dilation algorithm for calculating the pore the burn number is a local minima. Shaded circles represent the

radii is used to calculate the throat radii. The center of solid phase, dashed lines represent the medial axis, and the circlesrepresent the inscribed pore body and throats.the inscribed-throat sphere is located at the voxel with


Fig. 4. (a) Small random packing of uniform spheres, (b) medial axis generated on this packing, and (c) location of inscribed pore bodies onthe medial axis.

coordination number. For the cubic packing, the MA significant problems in dealing with a digitized imageof a continuum pore space.algorithm slightly underpredicted both inscribed pore-

body and pore-throat sizes. The values obtained caneasily be explained since the system has a finite resolu- Delaunay-Tessellation-Based Approachestion and the medial axis is not necessarily in the exact

The Delaunay Tessellationcenter of the pore body and/or throat. Examination ofthe inscribed pore bodies and throats calculated for the The Voronoi diagram and its dual, the Delaunay tes-two rhombohedral packings show that the MA algo- sellation, are techniques for spatial analysis that are usedrithm does underpredict some of the throat/body sizes, for numerous applications (Okabe, 2000), including thebut that the prediction gets better as the resolution characterization of porous media. Voronoi polyhedraincreases. It is to be noted that in the rhombohedral have long been used to analyze local structure in randompacking with high resolution (Table 1) there are rela- packings (e.g., Finney, 1970, 1976). Mellor (1989) arguedtively few inscribed pore throats (21 throats) with a that the simplical cells formed by the Delaunay tessella-radius that is the absolute minimum (i.e., one pixel). tion are more appropriate than Voronoi cells for charac-These throats should belong to the group of throats that terizing random packings, and this approach has beenhave a radius somewhere between one or two pixels; used widely since. In his original work, Mellor used thethe error ultimately stems from the pixelization of the Delaunay tessellation to measure geometric properties

of the Finney packing (Finney, 1968). He then extendedoriginal spheres. This observation illustrates one of the

Fig. 5. (a) 512–sphere cube packing, (b) location of the inscribed pore bodies in the cubic packing, (c) 512-sphere rhombohedral packing, and(d) locations of the inscribed pore bodies in the rhombohedral packing.


Table 1. Networks of regular packings as obtained from theoretical calculations (TC), medial axis (MA), and modified DelaunayTessellation (MDT) based approaches; note that all distances are non-dimensionalized by the sphere diameter (in voxels).

Cubic packing† Rhombohedral‡ Rhombohedral§

Method TC MDT MA TC MDT MA TC MDT MA

Pore propertiesNumber of pores 343 343 871 871 871 871Number of matches 343 871 871Max. inscribed radius 0.366 0.365 0.360 0.207 0.207 0.160 0.207 0.209 0.222Min. inscribed radius 0.366 0.365 0.360 0.112 0.112 0.040 0.112 0.113 0.067Ave. inscribed radius 0.365 0.360 0.144 0.105 0.144 0.127Distance between matched pores from

each method (center to center)(min/max/ave) 0.04 : 0.04 : 0.04 0.0 : 0.133 : 0.068 0.0 : 0.111 : 0.049

Throat propertiesMax. coordination number 6 6 8 8 8 8Min. coordination number 6 6 4 3 4 4Ave. coordination number 6 6 5.4 5.22 5.3 5.4 5.3Max. inscribed radius/Ds (pixel/pixel) 0.207 0.208 0.160 0.077 0.077 0.120 0.077 0.078 0.111Min. inscribed radius/Ds (pixel/pixel) 0.207 0.208 0.160 0.077 0.077 0.040 0.077 0.078 0.022Ave. inscribed radius/Ds (pixel/pixel) 0.208 0.160 0.077 0.063 0.078 0.084

† Diameter of spheres (Ds) � 25 pixels, 3D size � 175 � 175 � 175 pixels.‡ Diameter of spheres (Ds) � 25 pixels, 3D size � 175 � 148 � 138 pixels.§ Diameter of spheres (Ds) � 45 pixels, 3D size � 319 � 270 � 252 pixels.

this approach to create a topologic model of the inter- Fogler (1998) used a similar approach to model reactiveflows in which the packing structure changes over time.connected pore structure, which was used to model

capillary-dominated immiscible displacement (Mellor,1989; Mason and Mellor, 1995). Limitations of the Delaunay Tessellation

The reason that the simplical cells in the DelaunayThe direct use of the Delaunay tessellation for map-tessellation are a good basis for characterizing packings

ping pore structure has two significant problems. First,is their geometry. Specifically, if the centers of thethe tessellation restricts the network to a fixed coordina-spheres in the packing are used to perform the tessella-tion number (equal to exactly four in three dimensions).tion, then the largest voids tend to be found in theIn reality, experimental analyses show that disorderedinterior of tetrahedrons, while the tightest constrictionsporous media contain a distribution of coordinationin the packing are necessarily projected onto the facesnumbers with a mean value somewhat larger than z �of the tetrahedrons. Thus, each tetrahedron represents4. For packed beds of spheres, Yanuka et al. (1986)a pore with four throats emanating outward across theused serial sectioning to obtain an average coordinationfour faces (see Fig. 2.17–2.19 from Mellor [1989] ornumber of z � 4.3. Jerauld and Salter (1990) suggest thatFig. 5 from Bryant et al. [1993a]). These considerationsthis value should be amended to z � 5.3 by discardingprovide the rationale for characterizing pore and pore-individual-pore coordination numbers equal to either 1throat geometry from the tessellation. An added benefitor 2. The impact of this limitation (i.e., a fixed coordina-is that the tessellation uniquely defines the connectivitytion number of z � 4) depends on the process of interest.of the system because each tetrahedron shares its fourFor the modeling of permeability or other single-phasefaces with four neighboring tetrahedrons in the space-processes, its impact may be relatively small. However,filling array. Hence, the tessellation can be used to mapit is well known that the coordination number stronglyout the interconnected network structure of the packing.affects percolation properties in a network (Jerauld etFollowing Mellor’s work, the Delaunay tessellational., 1984a, b), which would suggest that altering a net-has been used in a number of applications for pore-work’s coordination number would affect the modelingspace characterization. Vrettos et al. (1989) used it asof multiphase flow (Larson et al., 1981; Bryant anda basis for modeling various transport processes in aBlunt, 1992), the flow of yield-stress fluids (Rossen andrandom packing. However, the use of a parking tech-Gauglitz, 1990), etc.nique to generate their computer-simulated packing re-

The second problem associated with the tessellation,sulted in a porosity of nearly 0.6, which is physicallywhich is of equal or greater concern, is the incorrectunrealistic. Bryant et al. (1993 a, b) again used the tessel-identification of pore locations and sizes. Specifically,lation of the Finny packing as a model porous material,the Delaunay tessellation tends to subdivide single voidsbut also included compaction and grain growth so as tointo multiple pores (where these larger voids are definedgenerate structures more representative of consolidatedeither intuitively or by other geometric arguments). Thismaterials. They performed dynamic modeling, showingproblem is most easily conceptualized using a regularquantitative agreement between predicted and mea-packing as illustrated using Fig. 6. Figure 6a is a sche-sured permeabilities in sandstones, and they demon-matic of a two-dimensional packing on a nearly squarestrated the effect that pore-scale spatial correlations inlattice. Clearly, each of the open spaces centered be-the packing have on transport properties. Thompsontween four particles should be identified as a discreteand Fogler (1997) applied the Delaunay tessellation topore. Similarly, the constrictions between pairs of parti-computer-simulating sphere packings to create network

models of heterogeneous structures, and Fredd and cles should represent pore throats, thus giving a fixed


Fig. 6. Schematic of proper pore identification by merging simplical cells of the Delaunay tessellation.

coordination number equal to four for this particular drons. The consequences are analogous to what wasstructure. This intuitive network is shown in Fig. 6b and described in two dimensions: too many pores, a coordi-6c. In contrast, parts d through f of Fig. 6 illustrate a nation number of 4 instead of 6, and the creation ofhypothetical network structure derived from a Delau- physically unreasonable pore throats. While similar ar-nay tessellation. In two dimensions, simplical cells in a guments can be made for other regular packings (e.g.,Delaunay tessellation are triangles, which forces the the rhombohedral structures discussed in this paper)packing structure to be discretized as shown in Fig. 6d. definitive arguments regarding random packings areUsing the analogy between the tessellation and the pore harder to make. However, using visualization, geometricstructure described above, pores are placed into each analysis, and coordination-number data, there is cer-of the Delaunay triangles, while the common faces (i.e., tainly evidence that similar problems exist when Delau-sides of the triangles in two dimensions) define the pore nay tessellations are used to discretize random packings.throats. Though the sides of all triangles in the tessella-tion do indeed represent local minima in particle-parti-

Modified Delaunay Tessellationscle spacing, it can be argued that the diagonal sides areinappropriate choices for pore throats. This point is Despite the problems mentioned above, the Delaunayfurther emphasized by comparing the two network tessellation is a very powerful tool for pore-space char-structures (Fig. 6c and 6f), which clearly share similar acterization, and is therefore chosen as a basis for theoverall topology but differ with respect to important MDT technique developed as part of this research. Indetails: the Delaunay network contains twice as many particular, the following important attributes are re-pores, it has a fixed coordination number of 3 instead tained: uniqueness of the spatial map; computationalof 4, and it contains pore throats where there should efficiency (Thompson, 2002); ease of defining the net-be none. Finally, if the sum of the volumes of all network

work interconnectedness.pores is made to equal the total void volume (which itWe propose a conceptually simple solution to theshould for a physically representative network), then

problems illustrated above, which is to allow for theindividual pores created by the Delaunay tessellationmerging of multiple tetrahedrons into larger polyhe-must be split into smaller volumes. This limitation isdrons in cases where a single void space has been unnec-important because the network statistics will not beessarily subivided. The major challenge to implementingrepresentative of the true void sizes in the packing.this approach is the proper identification of sets of tetra-In three dimensions, an analogous problem occurs inhedrons that should be merged.simple cubic packings: pores should clearly be defined

The algorithm described here relies on the examina-as the relatively large voids that exist in the center oftion of inscribed spheres (contained within the voideach group of eight spheres in the lattice, as shownspace) in an effort to identify multiple tetrahedrons thatearlier in Fig. 6b. However, a tessellation will divide

each cubic pore into either five or six Delaunay tetrahe- comprise a single pore. The logic for this approach is


Fig. 7. Schematic of three potential overlap conditions encounteredduring pore merging.

illustrated in Fig. 7. Part a of the figure is a schematic Fig. 9. Schematic of single inscribed sphere in a multi-tetrahedronpore. The single sphere is identified regardless of which triangleof a nearly square pore in two dimensions, divided intois used to seed the local optimization.two triangles. The inscribed circles are overlapping to

a large extent due to the local geometry of the particles.vertex circles for the triangle on the right. In the three-(Note that the inscribed circles are allowed to extenddimensional packings, the analogous situation generatesbeyond the boundaries of the triangle itself.) Figureslarge spheres that do not lie entirely within the pore7b and 7c are similar, but the geometry is increasinglyspace and therefore do not effectively characterize theelongated, thereby decreasing the amount that the in-void structure. Furthermore, due to their large size, theyscribed circles overlap. Once the location of the (largest)can envelop any number of neighboring spheres in theinscribed triangles (or tetrahedrons in three dimensions)packing, and the subsequent merging of cells generatesare known, they can be used to determine whether sim-unrealistically large pores that have no near neighbors.plical cells should be merged, based on the amount that

Consequently, the algorithm consists of a local opti-they overlap. In the first schematic, intuitive argumentsmization to find the largest inscribed sphere that can fitwould suggest that the two cells should be merged towithin the void space immediately adjacent to a tetra-form a single quadrilateral pore. In the last case, thehedron. The optimization is performed using a three-pores should clearly remain separate. Arguments fordimensional simplex search, modified according to thethe middle case are more nebulous, emphasizing thealgorithm of Nelder and Mead (1965): seed points toneed to define a merging criterion based on the degreebegin the searches are the centers of mass of the tetrahe-of overlap. The appropriate choice for this criterion isdrons, and the optimization routine locates the inscribeddiscussed later.sphere. Using this approach, pores are defined using aA related point should be made regarding how themore physically meaningful criterion (equivalent to theinscribed spheres are defined. In the two-dimensionaldefinition used by Nolan and Kavanagh [1994]) becauseschematics (Fig. 7), the inscribed circles are uniquelyit ensures that the inscribed spheres lie entirely withindefined as those that contact the set of three vertexthe void phase of the packing. An added benefit in thecircles. In three dimensions, tests showed that this ap-context of the Delaunay tessellation is that convergenceproach works well for packings of uniform-diameterof the local optimization disassociates the resulting in-spheres, and is ideal from a performance standpointscribed sphere from any particular tetrahedron in thebecause the location and radius of the inscribed spherestessellation. This latter point is illustrated in Fig. 9,can be found analytically by solution of a set of fourwhich is a schematic of a two-dimensional pore definednonlinear algebraic equations. However, as spheres inby six particles and tessellated into four triangles. Forthe packing become distributed in size, this approachthis particular geometry, the optimization could be initi-breaks down because of the existence of relatively flatated from any one of the four triangles, but each searchtetrahedrons, which leads to the formation of very largewould converge on the single inscribed circle shown.inscribed spheres. This problem is shown schematicallyAn added benefit of this approach is that the final loca-(in two dimensions) in Fig. 8. The unshaded circle istions of merged pores necessarily correspond to localthe unique inscribed circle that touches the three graymaxima (in inscribed sphere size), which is not true ifpores are tied to tetrahedrons.

Unfortunately, because many real pore structurescontain multiple local maxima in the same void, thecritical overlap for merging must still be defined some-what arbitrarily. A reasonable criterion seems to bewhether the center of an inscribed sphere lies within aneighboring inscribed sphere. This choice is discussedbriefly below, and probably warrants additional re-search. Once this criterion is imposed, multiple tetrahe-drons become merged together, and the resulting poresare polyhedrons with essentially arbitrary shapes. ThisFig. 8. Schematic of unrealistically large inscribed radii. This condi-modified Delaunay tessellation provides for a bettertion can occur when only the simplical-cell spheres are used for

the inscribed sphere calculations. one-to-one correspondence between discrete network


Fig. 10. Schematic of network generation algorithm using the modified Delaunay tessellation algorithm.

pores and the true voids within the packing, and it also Second, a tessellation was performed on the rhombo-hedral packing, which generated six tetrahedrons perallows for a variable coordination number. At the samesphere. Subsequent application of the merging algo-time, the important attributes of the Delaunay tessella-rithm identified two different pore structures, each oftion that were described above are retained.which repeats through the regular packing, as shown inFigure 10 illustrates the overall process for the peri-Fig. 12 using visualization. The two pore sizes were inodic 1000-sphere random packing used in this study.exact agreement with the theoretical values given in Ta-The figure shows the packing, the Delaunay tessellation,ble 1 (within the numerical accuracy specified duringand the network structure labeled ‘MDT1’ in laterthe run). Further analysis showed that 2/3 of the poresresults.in the packing are four-sphere clusters (which have thesmaller value of the inscribed radius), while 1/3 of theValidation using Regular Packingspores are six-sphere clusters having the larger radius.

As mentioned above, the Delaunay tessellation incor- All of the throats in the rhombohedral packing are equalrectly discretizes most regular packings because of the in size, with inscribed radii equal to 0.156R, again ingeometric limitations of the simplical cells. Therefore, agreement with Kruyer (1958). Coordination numbersone important validation of the modified Delaunay tes- for the two types of pores are 4 and 8 respectively,sellation is its ability to produce correct morphologic differing significantly from the coordination numbersdata for regular packings. As an aside, we note that reported by Yanuka et al. (1986). We believe the valuesvery small random perturbations (approximately 10�6 � presented here are correct, and that simpler empiricalDsphere in magnitude) were imposed on the spheres in or statistical analyses may not be appropriate for allthe regular packings because the Delaunay tessellation types of pore structures, especially those with extremeis unique only for random structures (Bowyer, 1981). porosity values such as the rhombohedral packing

First, a tessellation was performed on the cubic pack- (which has the lowest possible porosity of all monodis-ing, which generated five tetrahedrons per sphere. Sub- perse unconsolidated packings).sequent application of the merging algorithm correctly

The Critical Merging Length in the Modifiedidentified that these groups of five should be merged,Delaunay Tessellation Algorithmand that all parameters in the resulting network struc-

ture were exactly correct: the number of pores (one For regular packings, the network structure createdpore per sphere), the coordination number (six), and by the modified Delaunay tessellation is independentthe pore and throat sizes. Computer-aided visualization of the merging criterion chosen (for reasons illustratedwas also performed, as illustrated in Fig. 11. in Fig. 9). However, for random packings, the critical

merging length has a significant effect on the network

Fig. 11. Visualization of the cubic packing and sample of a pore and Fig. 12. Visualization of the rhombohedral packing and sample of thetwo distinct pores and their connectivity as obtained using theits connectivity as obtained using the modified Delaunay tessella-

tion method. modified Delaunay tessellation method.


Table 2. Statistics for Delaunay and modified Delaunay networks.

DT MDT1 MDT2

Pore propertiesNumber 6209 3941 2611Void volume � 102 (max/min/ave) 21.9 : 0.0954 : 5.17 82.5 : 0.861 : 8.14 115 : 2.10 : 12.3Inscribed radius (max/min/ave) 0.432 : 0.0987 : 0.214 0.432 : 0.108 : 0.195 0.432 : 0.108 : 0.191

Throat propertiesCoordination # (max/min/ave) 4 : 4 : 4 14 : 2 : 4.69 16 : 2 : 5.54Effective radius (max/min/ave) 0.403 : 0.0785 : 0.166 0.455 : 0.0785 : 0.154 0.555 : 0.0785 : 0.148Inscribed radius (max/min/ave) 0.427 : 0.0697 : 0.152 0.406 : 0.0577 : 0.132 0.413 : 0.00290 : 0.120Throats with aspect ratio �0.9% 22.4 13.4 6.01

structure. While the merging length can be varied over if the respective inscribed spheres contacted one another.(To help illustrate the differences between MDT1 anda large range, we compare only three networks for the

sake of brevity, which are denoted as follows: (i) DT, MDT2, refer to Fig. 7: The cells in Fig. 7a would bemerged in both of the modified networks. The cells ina network created directly from the Delaunay tessella-

tion; MDT1, a modified Delaunay network; the merging Fig. 7b would remain separate using the MDT1 criterion,but would be merged using the MDT2 criterion.).criterion was set so that two tetrahedrons were merged

if the center of one inscribed sphere was inside the Table 2 contains morphologic data for the three net-works, and illustrates the significant differences betweenother; MDT2, a modified Delaunay network, the merg-

ing criterion was set so that two tetrahedrons were merged the three networks. Particularly noteworthy are the fol-lowing points. The number of pores decreases substan-tially as the pores become merged. While the minimumvoid volume increases dramatically, the inscribed sphereradii are essentially unchanged (because inscribedspheres are allowed to bulge out of their tetrahedronsfor any of the networks). The average coordinationnumber shows a moderate increase, while the maximumcoordination number changes from a value of four (forthe Delaunay tessellation) to 14 and 16 respectively forthe two merged networks. The last row in the tablegives the fraction of throats with a high aspect ratio,defined as Rt,ins/Rp,ins. High-aspect-ratio throats are thosethat exhibit only a slight constriction (e.g., the diagonalsin Fig. 6a), and thus reductions in their number is gener-ally desirable.

Comparison between Medial Axis and ModifiedDelaunay Tessellation-Based Approaches

Comparisons of the network generation techniquesare made in this section. For all of these comparisonsthe MDT2 algorithm was used. When comparing MAversus MDT2 networks, we search for cases where thesame pore is identified by both algorithms. These arereferred to as ‘matched pores’ in the tables below. Thecriterion for matching is an overlap of the inscribedspheres found by the two respective methods.

Comparison for Regular Packings

As mentioned previously, the comparison of the twoapproaches is based on two types of packings: regularand random packings. A comparison of the regularpackings is provided in Table 1.

For the cubic packing, there is a good agreementbetween the two methods. The slight differences in thelocations and radii of the pores from both methods aredue to the fact that the MDT-based parameters areobtained directly from the actual sphere locations andradii, while the MA-based parameters are obtainedfrom a pixelized representation of the system. There isFig. 13. Histogram of distance between centers of matched pores nor-also a good agreement in the case of the rhombohedralmalized by the sphere diameter. (top) Twenty-five pixel diameter

sphere resolution and (bottom) 45-pixel diameter sphere resolution. packing for both high and low resolutions. Figure 13


Fig. 14. Uniform random packing system with the dark spheres representing the locations of inscribed pore bodies from the medial axis (MA)and modified Delaunay tessellation (MDT) method. Light color represents grains; dark color represents inscribed pore bodies.

shows the distributions of distance between matched continuum pore space. Also note however that the aver-age throat radius becomes closer to the true value atpores for the two resolutions. Clearly, as the resolution

increases, the distance between the centers of the the higher resolution, indicating that higher resolutionsdo indeed improve the description in an average sense.matched pores decreases. In the medial-axis-based

method, the radius of a pore is very sensitive to thecalculated location of that pore because the radius is Comparison for Random Packingobtained by a dilation algorithm starting from the center Figure 14 shows three-dimensional cutouts of theof the pore (i.e., the node location) to the closest sur- packing and the inscribed pore bodies obtained fromrounding solid space. As mentioned earlier, a node in both network generation methods. Table 3 shows thethe medial axis can be made up of several voxels in which pore network parameters obtained from both methods.the same local maximum was identified. Therefore, the We will now discuss several particular examples to high-issue of which voxel to choose as the pore center can light our findings. In Fig. 15 and 16, which are two-lead to errors in calculating the radius of the inscribed dimensional slices through representative pores, thepore body at that location. A point worth noting is that gray spheres are the spheres, the dashed lines are thethe minimum throat size predicted by the MA algorithm inscribed pore bodies obtained from the MDT2 algo-decreases with increasing resolution. This effect is be- rithm, and the solid lines are the inscribed pore bodiescause of a relatively small number of inscribed pore obtained from the MA algorithm. (Note that examiningthroats (21) with a radius of one pixel at the higher a two-dimensional slice [i] makes the grains appear toresolution. In reality, these throats should belong to the have different radii; [ii] may not indicate the true in-group of throats that have a radius of two pixels. The scribed pore body radii; and [iii] may not show poreerror ultimately stems from the pixelization of the origi- body/grain contacts.) In the cases shown in Fig. 15, thenal spheres. This observation illustrates one of the signif- inscribed pore bodies are considered to be a good matchicant problems in dealing with a digitized image of a because of their overlap. Two different situations are

illustrated in Fig. 15. On the left is a case where bothTable 3. Comparison between Medial Axis (MA) and Modifiedmethods define two pores within the void space shownDelaunay Tessellation (MDT) based approaches on random

packing (sphere diameter � 25 pixels). (pores from each method did not get merged becausethere is no overlap). On the right, both methods defineMethod MA MDT2only a single pore in the void space. On the other hand,

Number of pore bodies 1980 1838 Fig. 16 illustrates a situation where there is no matchNumber of matches 1039 1124% matches 66.7 73.61% void space 17.22 29.21% aspect ratio �0.9 28.1 6.0% aspect ratio �0.8 29.8 18.8% aspect ratio �0.7 39.39 34.8Max. distance apart 0.668Min. distance apart 0.0Ave. distance apart 0.21Max. coordination number 7 16Min. coordination number 2 2Ave. coordination number 3.7 5.54Max. pore body radius (pixels) 12 11.43 Fig. 15. Two examples showing a match between inscribed pore bod-Min. pore body radius (pixels) 1.0 2.86 ies identified from MA (solid line) and MDT (dashed line)Ave. pore body radius (pixels) 4.14 5.15 methods.


Fig. 16. An example where there is no match between inscribed porebodies from MA (solid line) and MDT (dashed line) methods.

between the inscribed pore bodies defined by the re- Fig. 17. Distribution of the distance between the centers of matchedspective methods. While each method does define a inscribed pore bodies from the two methods for random sphere

packing.pore in the same general void space, the fact that thereis no overlap between pores from both methods classi-fies this case as non-match. This situation explains two

the algorithms, within the constraints of available com-features appearing in Table 3. First, approximately 30%puter memory and/or computation time.of the pores are not matched. Second, a discrepancy

Analysis of the medial axes created from the regularexists in the fraction of the void space occupied bypackings showed that, even in these ideal cases, recon-inscribed pore bodies when comparing the two methods.struction of the skeleton is sensitive to the resolutionFigure 17 shows the distribution of distances betweenof the image. In addition, the non-uniqueness of thethe centers of matched inscribed pore bodies from thenode to pore-body correlation can lead to an overesti-two methods. The largest distances are associated withmate of the number of pores. To provide a more physicalmatches between large pores. The number of inscribedrepresentation of the inscribed pore body locations,pore bodies from both methods that are separated bynode-merging algorithms were developed. Results alsoa distance larger than 0.2 sphere diameters is 156, orshowed that the pixelization effect (e.g., the finite sizeapproximately 15%.of the pixels) plays an important role the reconstructionFigure 18 shows a comparison of the inscribed poreof the medial axis and extraction of pore network pa-body and throat distributions from the two methods.rameters.As expected, the MA method generates inscribed pore

As shown in the characterization of the regular pack-bodies with smaller radii than the MDT-based method.ings, both methods are able to provide the correct poreBecause of this shift in the pore-size distribution, wenetwork structure, including the number and locationwould expect resulting differences in certain simulationsof the inscribed pore bodies, the number and locationand particularly in imbibition simulations that makeof the inscribed pore throats, and the connectivity. Beinguse of this parameter to estimate radii of curvature foranalytically based, the MDT method was able to extractfluid interfaces.the correct pore body and throat sizes. It was found that,because the MA-based method is resolution-dependentCONCLUSIONS and precise voxel locations of nodes are non-unique,the exact inscribed pore body and throat sizes wereIn this paper, we have presented and compared two

different methods for calculating the locations and sizes underestimated. As the resolution of the image in-creases, the agreement between the two methods im-of inscribed pores and pore throats in three-dimensional

porous media systems. Both methods involve significant proves. It should be noted, that while inscribed porebody-size distributions generated from the two methodsmodifications to existing techniques. The first tech-

nique, based on a medial axis, relies on discretization show good agreement and trend, no attempt was madeto compare the effectiveness of the two methods toof the three-dimensional image and can be applied to

any three-dimensional porous media system. The De- properly capture the spatial correlation of the system.Despite the good agreement of the regular packings,launay tessellation-based method is traditionally used

with sphere packings (in which particle locations are there were differences in the both the spatial mappingand the statistical distributions in network propertiesknown), but it could easily be used in conjunction with

tomographic images from which the centroids of parti- for random packings. The discrepancies between thetwo methods can be attributed to several factors: (i)cles had been determined. While the packings analyzed

in this work are limited in size, they are representative pixelization at finite resolution; (ii) the arbitrary processof choosing the center of a inscribed pore body (i.e.,of systems scanned using high-resolution microtomogra-

phy; no inherent restrictions on system size exist within node) from a group of voxels all having the same burn


Fig. 18. Inscribed (top) pore body and (bottom) throat-size distributions (right) MDT and (left) MA for the random sphere packing.

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comparison of network generation techniques for unconsolidated

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