ROCK FABRIC AND TEXTURE FROM DIGITAL CORE ANALYSIS

SPWLA 46th Annual Logging Symposium, June 26-29, 2005

ROCK FABRIC AND TEXTURE FROM DIGITAL CORE ANALYSIS

M. Saadatfar1, M. L. Turner1, C. H. Arns1, H. Averdunk1, T.J. Senden1, A. P. Sheppard1, R. M.Sok1,2, W. V. Pinczewski2, J. Kelly3 and M.A. Knackstedt1,2,∗

1Department of Applied Mathematics, Research School of Physical Sciences and Engineering,Australian National University, Canberra, Australia

2School of Petroleum Engineering, University of New South Wales, Sydney, Australia3Woodside Energy, Perth, Western Australia

∗Corresponding Author: [email protected]

Copyright 2005, held jointly by the Society of Petrophysicists and Well Log An-alysts (SPWLA) and the submitting authors.

This paper was prepared for presentation at the SPWLA 46th Annual LoggingSymposium held in New Orleans, Louisiana, United States, June 26-29, 2005.

ABSTRACT

Descriptions of rock fabric and texture are of great valueto geologists and petrophysicists as they can be used infacies analysis and in the interpretation of the environ-ment of deposition. Grain shape and size informationis used to correlate to petrophysical properties. Theyare also of importance to the production technologist forcompletion design and sand strength/failure prediction.Textural descriptions are traditionally obtained via petro-graphic and petrological analysis including thin sectionanalysis, particle sieving techniques and laser diffractionstudies. All methods have limitations in quantitativelydescribing the full 3D rock fabric and assumptions andinterpretations in the processing of data can skew or dis-tort predictions of textural data.

We have previously demonstrated the ability to image,visualize and characterise sedimentary rock in three di-mensions (3D) at the pore/grain scale via X-ray com-puted microtomography (Arns et al., 2004; Arns et al.,2005). We now demonstrate the ability to directly mea-sure rock fabric and texture from 3D digital images ofcore fragments. The mathematical procedure to extractindividual particles from a full core image is describedand its accuracy demonstrated. A single core fragmentimage can yield more than 8,000 individual grains. Wedescribe methods for mathematically characterizing theindividual grains including grain size (max/min and mode,skewness, sorting, kurtosis) and shape (sphericity, round-ness). These are measured in parallel with textural infor-mation (sorting, grain contacts, matrix/grain support).

A comparison of grain size analysis from digital imagedata to laser diffraction studies on sister core materialis shown; good agreement between estimates of grainsize is obtained. Comprehensive grain shape data ob-tained over thousands of grains shows significant vari-

ability within samples and systematic shape changes withgrain size. Measures of grain contacts and grain overlaparea show that many grains are loose within the pack andlarger grains can have coordination numbers greater than20. Anisotropy in grain orientation is also directly mea-sured.

The rock fabric and texture derived from digital 3D im-ages is more comprehensive, systematic and quantitativethan current analysis techniques. This analysis, coupledwith studies of 3D pore structure and the ability to di-rectly measure petrophysical properties from 3D images,will enable one to embark on a systematic study of theeffect of grain size, shape and cementation on transportand elastic properties of core material.

METHODOLOGY

In this section we describe the samples considered in thisstudy; a simple polymeric foam structure, two soil sam-ples, a poorly consolidated reservoir core and a poorlysorted reservoir sand. We briefly describe the methodsused to image the 3D structure of the porous samplesand to identify the distinct phases. We discuss the com-putational methods used to extract individual grains fromthree dimensional images and discuss the mathematicaltools used to analyse the size and shape of the extractedgrains and the fabric of the samples.

Tomographic Imaging

A high-resolution and large-field X-ray µCT facility hasbeen used (Sakellariou et al., 2003; Sakellariou et al.,2004a; Sakellariou et al., 2004b) to image the samples.The CT has a cone beam geometry. Details of the equip-ment and experimental methodology used to image themicrostructure of sedimentary rock have been given pre-viously (Sakellariou et al., 2004a; Knackstedt et al., 2004;Arns et al., 2005). The resolution chosen is dependent onthe grain size of the material. For grains of 100−300µmwe find that 4-10 µm resolution is sufficient.

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Figure 1: A core sample of 7.5mm x 7.5mm cross sectionmounted on the tomography rotation stage before imag-ing.

Samples imaged

Five samples are considered in this study. Details of theimage size and image resolution for each of the samplesis given in Table 1.

1. A microcellular polymeric foam was chosen as amodel system for study. The foam is characterisedby reasonably homogeneous spherical pores em-bedded within a solid matrix. This system of highlyspherical and rounded pores is used to validate thefabric and texture analysis algorithms and to illus-trate reference values for a sphere pack. Slicesthrough the sample and a 3D rendered image show-ing the spherical structure of the foams are givenin Fig 2(a) and Fig. 3(a).

2. Two samples (Soil A and B) of unconsolidated re-golith material from the saturated zone (depth=50.5m)are considered. The samples come from a fairlyextensive facies that appears homogenous on a cmscale. XRD analysis shows that the mineral phaseis made up of 88% quartz, 9% montmorilloniteand 3% kaolin. A PQ diamond coring techniquewas used to remove the material from the groundin a reasonably undisturbed state. Imaged slicesof the samples are shown in Fig 2(b) and (c) withan aluminium tube of diameter 1.5 cm surroundingthe core (left image). We note that the cut of soilA seems to contain larger grains with some layersexhibiting finer grains (Fig 2(b) right). A 3D ren-dered image of a small subset of Soil A is shownin Fig 3(b).

3. A sample of unconsolidated core from a prospec-tive oil reservoir was considered. Slices of thesample are shown in Fig 2(d). A 3D rendered im-age of a small subset of this sample is shown inFig 3(c).

4. A single 8 mm diameter plug of a texturally com-plex sample from a prospective gas reservoir wasexamined. The sample exhibited extremely poorsorting and some clay content. Orthogonal slicesthrough the original tomogram are shown in Fig.2(e).

[a]

[b]

[c]

[d]

[e]

Figure 2: Orthogonal slices of the five samples imaged.(a) Foam, (b) Soil A, (c) Soil B, (d) unconsolidated sandand (e) poorly sorted sand. The phase surrounding thegranular core in (c) is an injected epoxy resin.

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Table 1: Details of the sample type, tomogram resolu-tion (µm), image size (voxels), grain phase fraction φgr,porosity φ and clay/silt fraction φs/c for the samples con-sidered in this study.

Sample Res. Size φgr φ φs/c

Foam 6.72 10003 72% 28% 0%Soil 1 17.0 10003 58% 8% 34%Soil 2 9.4 20003 62% 9% 29%Uncon. 6.72 10003 70.8% 29.2% 0%Poor Sort 8.6 10003 78.4% 7.2% 14.4%

[a]

[b]

[c]

Figure 3: Rendered 3D images of subsets of samples im-aged (left) and a typical grain (right). (a) Foam, (b) SoilA, (c) unconsolidated sand.

Phase Identification

The tomographic image consists of a cubic array of re-constructed linear x-ray attenuation coefficient values,each corresponding to a finite volume cube (voxel) of thesample. An immediate goal is to differentiate the atten-uation map into distinct pore and grain phases for eachof the samples imaged. Ideally one would wish to have amulti-modal distribution giving unambiguous phase sep-aration of the pore and various mineral phase peaks. In

particular one would like to obtain a clear bimodal dis-tribution separating the pore phase from mineral phasepeaks. This simple phase extraction is possible on thepolymeric foam sample. From the original 10003 imagewe extract a 456×456×912 subset (the region of the im-age which contains only foam) for analysis. The intensityhistogram (Fig. 4(a)) shows two distinct peaks associatedwith the two phases. The peak centered around 27500is associated with the polymeric material. The lowerpeak around 17000 is associated with the pore phase.For an intensity histogram with two distinct phase peaksit is sufficient to do a simple threshold segmentation at22500 followed by an isolated cluster removal to removenoise artifacts (isolated phase fragments of ≤ 20 vox-els). Comparison of the grey-scale and binarised imageof a slice of foam A is shown in Fig. 4(b) and (c). Theresultant solid phase fraction at the measured attenuationcutoff is (ρ/ρs)image = 28%. This is in good agreementwith the lab measured density of 29%.

[a] 0 10000 20000 30000 40000 50000xray intensity (arbitrary units)

0

0.001

0.002

0.003

norm

alis

ed p

roba

bilit

y

[b]

[c]

Figure 4: (a) Intensity histogram for the full image vol-ume of Sample A. (b) Grey-scale x-ray density map ofa slice of Sample A, and (c) the same slice after phaseseparation into pore and solid phases.

Unfortunately in rock and soil samples, the presence ofpores at scales below the image resolution leads to aspread in the low density signal making it difficult to un-ambiguously differentiate the pore from the microporousand solid mineral phases. One also may wish to under-take three-phase identification– phase separation of theresolvable pore phase, intermediate (silt/clay) phase andthe grain phase. To quantitatively analyse tomograms wehave developed a well-defined and consistent method tolabel each voxel (Sheppard et al., 2004). The first stagecomprises a nonlinear anisotropic diffusion (AD) filter(Perona and Malik, 1990) which removes noise whilepreserving significant features, i.e. the boundary regionsbetween the phases. The second stage applies an un-sharp mask (UM) sharpening filter (Pratt, 2nd Ed 1991)which has proven itself in practice to be highly effec-tive at sharpening edges without overly exaggerating thenoise. Finally, the phase separation is performed using acombination of watershed (Vincent and Soille, 1991) andactive contour methods (Caselles et al., 1997).

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2000 4000 6000 8000 10000 12000 140000

0.01

0.02

0.03

0.04

0.05

Figure 5: Normalized intensity histogram for the full im-age volume of the Soil B sample.

We describe the multiphase identification process for SoilB. From the original 20003 of the soil sample shown inFig. 2(c) we extract a 1080 × 1050 × 1900 subset (theregion of the image which contains only the sample) forphase separation. The intensity histogram for the regionis given in Fig. 5 after filtering. Three phase segmenta-tion was performed on the image. Absolute thresholdswere chosen for the void phase I < 6500, grain phaseI > 9000 and intermediate (clay/silt) phase 7000 <I < 8500. Phase separation in the ambiguous inten-sity regions between these phases was then made us-ing the active contours and watershed methods to delin-eate the three phases. A subset of the central slice alongwith the resultant pore, clay and grain phases are shownin Fig. 6. The resultant phase fractions obtained wereφpore = 9.4%, φgrain = 61.7%, φclay/silt = 28.7%.In the other soil sample and the poorly sorted sand weundertake this three phase segmentation process and ob-serve a significant clay fraction (Table 1). The unconsol-idated sand is clean and exhibits only trace amounts ofan intermediate phase.

GRAIN EXTRACTION

In sedimentary rocks the solid phase is made up of dis-tinct grains and cements. However, due to the process ofsedimentation and diagenesis the solid phase will appearcontinuous within the image as all grains are in contactwith at least three other grains. One must develop anautomated method to distinguish between particles in animage. In this section we describe two methods we useto extract distinct grains from the mineral phase of 3D to-mographic images. Both methods are based on introduc-ing grain boundaries at minima of a distance function forthe grain phase. For poorly cemented sands the methodis robust. For more strongly cemented materials (partic-ularly quartz sands with quartz cement) the partitioningis more difficult and will need to be rigorously tested.

Figure 6: (Top left), Tomographic grey scale data: Topright, pore phase (black): Bottom left, clay phase (black)and bottom right grain phase (black).

Partitioning via Erosion and Voronoi tesselation

The first algorithm is based on an erosion/dilation pro-cess followed by a voronoi partitioning–the method is il-lustrated in Fig. 7 for both a 2D and 3D sample. In thisalgorithm the grain phase image in Fig. 7(a)-(b) is firsteroded which breaks apart the individual grains (Fig. 7(c)-(d)). During the erosion process we monitor the numberof distinct grains until a maximum is reached. On reach-ing this maximum we assume that the grain phase is nowbroken up into all original distinct grains. Each grain isidentified and the original grain shape restored by per-forming a dilation (opposite of erosion) back to the orig-inal image.

After defining the separate particles, we now partition thespace using the Voronoi tessellation so that each Voronoicell contains a rock grain (Fig. 7(e)-(f)). A Voronoi re-gion associated with a feature is a set of points closer tothat feature than any other features. The Voronoi cellsare first defined by the center of mass of each identifiedgrain (Voronoi seed) and then by growing the Voronoiseeds at the same rate via dilation. When the bound-ary of two growing seeds meet a planar boundary is de-fined between them. The process continues until all thegrain phase is filled. The Voronoi regions associated witheach grain (Voronoi cells) are convex polyhedra in 3D(Fig. 7(e)-(f)) and the boundary between two adjacentcells is a plane which defines the overlap between the twodistinct grains. The final grain configuration for the orig-inal image shown in Fig. 7(a)-(b) is shown in Fig. 7(g)-(h))

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[a] [b]

[c] [d]

[e] [f]

[g] [h]

Figure 7: (a-b) A segmented image of a grain pack in(a) 2D and (b) 3D. (c-d) Images (a-b) after erosion. (e-f) shows the original images after separating the con-stituent grains via the Voronoi tesselation and (e) showsthe boundaries separing the distinct grains in (a). (g-h)shows the original grain pack with colours labelling thedistinct grains.

Partitioning via Watershed Algorithm

The first algorithm performs well, but can lead to the lossof the smallest grains during the erosion process. A sec-ond algorithm overcomes this limitation; grain identifica-tion is done in a two stage process, which takes Euclideandistance data (distance to the nearest grain boundary) ofthe grain phase as input. The basic assumption is that theboundaries between grains which are not isolated coin-cide with the watershed surfaces of the Euclidean dis-tance function (Vincent and Soille, 1991). The entiregrain space can be thought of as the union of spherescentred on every grain voxel. Each sphere radius is givenby the Euclidean distance value of the voxel at its cen-tre. We first identify all the voxels that are not coveredby any larger Euclidean spheres. Each one of these vox-els, which are at maxima of the distance function in theirlocal neighbourhood, forms a seed that will grow into a

single grain in the next stage of the algorithm.

The watershed transformation (Fujimoto, 1999) is per-formed by region growing, in which each region starts asa single seed voxel. Voxels that lie on the boundaries ofthe regions are processed in reverse Euclidean distanceorder, i.e. voxels with high distance values are processedfirst. When a voxel is processed, it is assigned to the re-gion on whose boundary it lies, or, if it lies on more thanone region boundary, the region whose boundary it firstbecame part of. At the end of the algorithm, the grainspace will be partitioned into grains whose boundarieslie on the watershed surfaces of the Euclidean distancefunction. The algorithm is parallelised using an imple-mentation of the ”time warp” discrete event simulationprotocol (Soille, 2nd Ed 2003). The resultant grain sep-aration is nearly identical to the Voronoi based partition-ing, but includes more information on smaller grains.

GRAIN SIZE, SHAPE AND TEXTURE

The granular image is now separated into distinct grainswhich are stored in a simple database with informationon the coordinates of each grain center of mass, the coor-dinates of all voxels of each grain, the number of neigh-bouring grains and all neighboring grain labels along withgrain overlap information. From the individual grain dataa number of grain size, shape, texture and fabric proper-ties can be derived.

Grain Size distribution

From the set of individual grains identified one can im-mediately define a comprehensive sediment size distri-bution and distribution measures including median andmean grain size, sorting, skewness and kurtosis. Thegrain volume is measured by counting the voxels in eachdistinct grain. Size according to the Wentworth gradescale is reported. In this grading scale one considers alogarithimic grading (Krumbein, 1934) based on:

φg = − log2d , (1)

where d is the particle size in mm. Here we define sizevia the volume of the particle and report the nominal di-ameter d of the sphere having the same volume as theparticle. Grain size distribution descriptors for each sam-ple are defined according to Folk and Ward (Folk and

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Ward, 1957):

φ̄g =φg16

+ φg50+ φg84

3,

Mdφg= φg50

,

σ =φg84

− φg16

4+

φg95− φg5

6.6,

Sk =φg84

+ φg16−2φg50

2(φg84− φg16

)+

φg95+φg5

−2φg50

2(φg95− φg5

),

K =φg95

− φg5

2.44(φg75− φg25

),

where φ̄g is the mean of the grain size distribution, Mdφg

its median, sorting is given by the standard deviation σ,Sk and K are Skewness and Kurtosis, and φgn

indicatesthe nth percentile of φg .

Grain Shape derived from Spherical Harmonics

In this section we discuss how to mathematically char-acterize the shape of any arbitrary particle and we applythis mathematical description to the particles extractedfrom the 3D images. Unfortunately the discrete or vox-elated representation of particles leads to significant er-rors in any calculation related to the surface of the parti-cle due to the roughness of a voxelated surface. For ex-ample the digital surface area of a sphere is 50% largerthan the actual value. Work on a range of oblate andprolate spheroids (Garboczi, 2002) shows the error canvary from 28% to 43%. The procedure we use to moreaccurately generate the 2D surface of each 3D particleis based on spherical harmonic shape descriptors of thegrain surface. This method has been previously used toapproximate molecular orbital surfaces (Duncan and Ol-son, 1993), to characterize the shape of an asteroid (Zu-ber et al., 2000) and to estimate the shape of cement par-ticles (Garboczi, 2002) from digital images.To approximate the rough surface of a voxelated object toa continuous and smooth surface via spherical harmonicswe first determine the position of all the surface pointsin terms of polar coordinates. This is done by findingthe center of mass (Cm) of the object and the line seg-ments (R) starting from the Cm and ending at the in-terface between the particle (solid phase) and the back-ground (pore phase), at all given spherical polar coordi-nate angles (θ and φ). The angles are chosen based on120-point Gaussian quadrature (Press et al., 1992). Thismethod of finding the surface points works only for par-ticles in which the line segment connecting the center ofmass to the surface crosses the interface only once; thegrain is convex.The function R(θ, φ), the spherical harmonics form acomplete set:

r(θ, φ) =

∞∑

n=0

n∑

m=−n

anmY mn (θ, φ); −n≤m≤ n (2)

where Y mn (θ, φ) is a spherical harmonic function of or-

der (n,m) and (anm) are the Fourier coefficients. Theaccuracy of spherical harmonic analysis with respect tothe number of coefficients n used in the spherical har-monic expansion, has been previously studied in (Gar-boczi, 2002). Depending on the resolution of the digitalimage, different n are needed to estimate the shape prop-erties of the object within a reasonable agreement withthe exact (analytical) results. We have found, in agree-ment with (Garboczi, 2002), that for grains of more than1000 voxels, the error in the estimation of the granularshape is low for n ' 20. This was tested on a rangeof analytic shapes. For example, error analysis on ellip-soids of revolution of 20000 voxels led to a decrease inthe error in the surface area from 28% (voxel-based) to2% (spherical harmonics, n = 20). From this descriptionof the grain we can more accurately quantify sphericityand roundness.

Sphericity

Sphericity measures the degree to which a particle ap-proaches a spherical shape. Quantitatively, sphericitywas defined by Wadell (Wadell, 1932) as the ratio ofthe particle’s volume to the volume of a circumscribingsphere which may be taken as the volume of a spherewith a diameter equal to the longest axis of the particle;

ΨW = 3

√

Vp

Vcs. (3)

A more accurate definition of sphericity involves mea-suring the three linear dimensions of the grains. Frommeasurement of volume, surface area and curvature ofeach grain we derive an equivalent regular triaxial ellip-soid with Long, Intermediate and Small axes dL and dI

and dS respectively. Sphericity is then rewritten as;

Ψ = 3

√

dSdI

d2

L

. (4)

This definition is known as intercept sphericity. Sneedand Folk (Sneed and Folk, 1958) introduced maximumprojection sphericity as a modification to Wadell’s def-inition. Their objection to Wadell’s sphericity was thatalthough it correctly measures the degree to which a par-ticle approaches a sphere, it does not correctly expressthe dynamic behavior of the particle in a fluid. They be-lieved that the sphericity of a particle should express itsbehavior in a fluid where particles tend to orient them-selves with their maximum projection area normal to theflow. They defined sphericity as;

ΨP =3

√

d2

S

dLdI. (5)

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We report both the intercept and the maximum projectionsphericity on the smooth surface of grains defined viaspherical harmonics.

Roundness

Roundness is a shape characteristic related to the dis-tance traveled by a particle prior to its deposition (Pow-ers, 1953). Experimental studies indicate that roundnessis related to the degree of abrasions and wear suffered bythe particle. A qualitative definition of roundness refersto the sharpness of the corners and edges of the grain.However the main problem in determining the roundnessis recognizing the corners. There have been many propo-sitions for an accurate and efficient method to measureroundness (Wentworth, 1919; Wadell, 1935; Krumbein,1941). They are based on a visual comparison of indi-vidual grains which leads to a number of problems; firstthe number of grains studied is limited, secondly they arebased on 2D projection images and thirdly, it is objective– different operators may estimate different roundness.We measure the roundness of the particles by derivingthe Maximum Covering Sphere map (Coles et al., 1998;Hilpert and Miller, 2001; Arns et al., 2005) for eachgrain: define locally for every point within the grain, thediameter of the largest sphere which fully lies within thegrain and covers that point (Fig. 8). To measure the cur-vature over the whole grain we find the maximal cover-ing sphere along the surface of the grain and summingover all surface patches derive the average curvature Cav

for the grain. The roundness of the grain is then definedby Cav × R where R is the maximally inscribed spherewithin the grain.

[a] [b]

Figure 8: Illustration of the measurement of the round-ness of the grains. In [a] a 2D slice of a the poorly consol-idated reservoir core is shown with the grains in black. In[b] the corresponding field of maximal covering radii isshown for the granular phase with the gray scale propor-tional to covering disk radius (brighter indicates largercovering radii).

It is assumed that roundness of rock particles is a prop-erty independent of sphericity (Wadell, 1932) and it iscommon to analyse grain shape data using a Spheric-ity/Roundness plot introduced by Powers (Powers, 1953)

Figure 9: Sphericity/Roundness relationship asgiven in (Powers, 1953). Image obtained fromhttp://www.carboceramics.com/tools/tr physical.html.

(see Fig. 9) and hereafter referred to as a Powers plot.

Shape Class

Other classifications of the shape of grains are used insedimentology. One common classification is based onthe measure of the three primary axes of each grain andgives an indication of the relative length of these axes;from oblate (disk-shaped), equant (sphere or cube-like),to bladed, and prolate (rod-shape). This shape relation isplotted in a diagram commonly called a Zingg diagram(Zingg, 1935). The long, intermediate and small axes ofeach grain is measured as described previously.

GRAIN FABRIC ANALYSIS

In this section we describe the methodology for obtainingfabric data from the grain images including quantifyingthe grain packing characteristics (grain to grain contacts,grain overlap area) and orientation.

Grain Connectivity

One of the most important characteristics of packing isthe frequency with which grains touch each other. Fromthe grain partitioning algorithm one can immediately iden-tify all grain-to-grain contacts within the full volume.Due to potential biasing at the boundaries we do not con-sider any grain that is in contact with the edge of theimage volume.

Grain Overlap

The total overlap area between grains can also be de-rived directly from the grain partitioning throughout thefull volume. This leads to an estimate of the degree of

7 ZZ


cementation of the rock. The shape of the grain con-tact is not ascertained by the algorithm. Due to the na-ture of the partitioning algorithm, illustrated in Fig. 7,most contacts are long (are approximately planar). Con-cave/convex and sutured contacts cannot be discerned.The measure of the grain overlap area does allow one todifferentiate tangential contacts (small overlap areas) tolong contacts.

Grain orientation

From measurement of the three primary axes of the grainsone can obtain grain orientation information. We reportthe orientation of the primary axis dL of each grain todescribe the degree of preferential orientation within thegranular medium.

RESULTS

Here we describe the results for the fabric and textureanalysis on the five tomographic samples. The total num-ber of grains imaged in each sample is given in Table 2.The rock and soil samples all include over 4000 imagedgrains.

Table 2: Statistics derived from the images of the fivesamples. We give the number of grains captured per im-age (No.). We also report the mean volume of the grain,the median grain size, the dispersion (sorting or stan-dard deviation), the skewness and kurtosis of the grainsize distribution. All properties are defined in φg units(Eqn.(1)) and based on the definitions of Folk and Ward(Folk and Ward, 1957).

Foam Soil1 Soil2 Uncon PoorSort

No. 1521 5063 5795 8310 4103φ̄g 1.74 1.01 1.58 2.70 2.62

Mdφg1.68 1.10 1.50 2.66 2.69

σ .37 .35 .52 0.42 1.04Sk 0.07 -0.07 0.30 0.24 -.45K 1.09 .97 1.01 1.21 1.30Z 6.02 5.23 5.63 5.10 7.45

Foam sample

The foam, which is characterised by reasonably homo-geneous spherical pores embedded within a solid ma-trix and a reasonably narrow distribution of the bubblesizes, is used to validate the fabric analysis algorithmsand to give reference values for a homogeneous spherepack. The analysis is performed on the pore phase asthis phase mimics a homogeneous spherical grain pack.Data in Table 2 indicates that the system is well sorted,is slightly positively skewed and mesokurtic. In Fig. 10

0 1 2 3φ=-log2(diameter)

Wentworth grain size scale

S A N D

Coars Medium Fine

0.35% 77.54 22.11

0 1 2 3 4φ=-log2(diameter)

0

10

20

30

40

50

60

70

Freq

uenc

y

Figure 10: Grain size data for the ideal foam system.

we summarise the grain size data; the grains are primar-ily of medium grade with some fine grades (1 < φg < 3).

In Fig. 11 we summarise the grain shape data; we findthat, as expected, the sphericity and roundness of the’grains’ are both close to 1.0. The data for all grainssuperimposed on the Power plot shows that all grains liein the upper right region. The data on the Zingg dia-gram also lies strongly in the upper right quadrant furtherunderlining the strongly compact spherical shape of the’grains’.In Fig. 12 and Table 2 we summarise the grain fabricdata. The average connectivity of the grains is Z̄ =6.02 and we observe a homogeneous coordination num-ber distribution in Fig. 12(a) with a maximum coordina-tion number of Zmax = 11. The grain overlap area dis-tribution in Fig. 12(b) is homogeneous with the averageoverlap area of the order of .04 mm2. Despite the ho-mogeneity of the system a very strong anisotropy is ob-served in the grain orientation (Fig. 12(c)). This is con-sistent with the manufacturing process which is based onan extrusion process and leads to slight shearing of thesystem resulting in a small but consistent elongation ofthe foam structure in one direction.

Soil Sample A

Data in Table 2 is based solely on the statistics measuredon the resolved grains which lie within the sand gradescales (very coarse → very fine). The results indicate thatthe first soil system is well to very well sorted, is nearsymmetrical and mesokurtic. In Fig. 13 we summarise

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[a]0 0.02 0.04 0.06 0.08

Volume [mm3]

0

0.2

0.4

0.6

0.8

1

Sphe

rici

ty

Intercept SphericityMaximum Projection

[b] 0 0.02 0.04 0.06 0.08

Volume [mm3]

0

0.2

0.4

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0.8

1

Rou

ndne

ss

[c]

[d] 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1S/I

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

I/L

Bladed Prolate (Roller)

Oblate (Disk) Equant (Compact)

Figure 11: Grain shape data for the ideal foam system.(a) Sphericity and (b) roundness are plotted as a func-tion of grain volume. (c) Sphericity/roundedness and (d)Zingg diagram for the system.

the grain size data; the grains are primarily of coarse andmedium grade (0 < φg < 2).

In Fig. 14 we plot data from a laser particle size exper-iment on a sister core of the soil sample. The particlesize data shows a bimodal distribution with a significantfraction (29.1%) of the grains lying in the silt/clay rangeand most of the remaining grains within the coarse tomedium sand grades. The binarization of this core (re-call Table 1) led to an estimate of the intermediate phasevolume (clay/silt fraction) as 34% in reasonable agree-ment with the particle size data. We compare the his-togram from the digital image data to the data from theparticle size analysis for the sand fraction in Fig. 13(b).The agreement is good.

In Fig. 15 we summarise the grain shape data. The Spheric-ity data lies between values of 0.4 and 0.7 with a slightbias to larger sphericity for larger grains. The roundnessdata lies in the rounded to well rounded range with largervalues observed for smaller grains. The data for all grainssuperimposed on the Powers plot shows a spread of grain

[a] 0 2 4 6 8 10 12Coordination number

100

200

300

400

Freq

uenc

y

[b] 0 0.02 0.04 0.06 0.08 0.1

Overlapping area [mm2]

0

100

200

300

400

500

600

700

Freq

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y

[c]

Figure 12: Grain fabric data for the homogeneous foam;(a) coordination number distribution and (b) overlap areadistribution. In (c) the orientation of the longest axis isshown along a hemisphere– there is a strong preferencefor a single direction. The direction of the elongated axisis shown for each grain and the length of the lines is pro-portional to the equivalent radius of the grain.

[a]0 1 2 3

φ=-log2(diameter)


S A N D

Coars Medium Fine

48.14% 51.69% 0%

[b]0 0.5 1 1.5 2

φ=-log2(diameter)

0

1

2

3

4

5

6

Freq

uenc

y pe

rcen

t

Laser sizing

Figure 13: Grain size data for the first soil sample. In(b) we also show the grain size distribution for the sandfraction obtained by laser particle sizing (Fig. 14 on asister plug). The agreement is excellent.

9 ZZ


Figure 14: Grain size data for soil sample obtained bylaser particle sizing.

shape across the sample. The data on the Zingg diagramalso varies over a wide range indicating the breadth ofgrain shapes measured on this sample.

In Fig. 16 and Table 2 we summarise the grain fabricdata. The average connectivity of the grains is Z̄ = 5.23and we observe a broader coordination number distribu-tion in Fig. 16(a). No grains of Z < 3 are noted as ex-pected since a stable packing for any grain would ini-tially have a minimum of 3 neighbours. The peak in thedistribution lies at a coordination number of Z = 3 andmore than 50% of the grains have a Z ≤ 4. Z = 3corresponds to a loose grain and Z = 4 is the minimumcoordination number in 3D for a grain to be stable. Thelarge number of grains of low coordination indicates thatmany grains may be loose within the pack and may in-dicate that the pack is weakly grain supported or matrixsupported. We also note that a small number of grainsexhibit very high coordination. In Fig. 16(b) we plot thecoordination number versus grain volume. We note thatthe grains of high coordination correspond to the grainsof largest volume as expected. In Fig. 17 we show twosnapshots from a rendered 3D movie of a highly coordi-nated (Z = 22) grain showing that many smaller grainsare attached to this large grain. The grain overlap areadistribution in Fig. 16(c) is positively skewed with themean overlap area of the order of .4 mm2. No anisotropyin the grain orientation is observed in Fig. 16(d)– the di-rection of the longest axis seems equally weighted alongall orientations.

Soil Sample B

In contrast to Soil A, data in Table 2 indicates that thesecond soil system is only moderately sorted, is stronglypositively skewed and mesokurtic. This highlights theheterogeneity in this soil despite the proximity of the two

[a]0 0.07 0.14 0.21 0.28

Volume [mm3]

0

0.2

0.4

0.6

0.8

1

Sphe

rici

ty

Intercept sphericityMaximum projection

[b] 0 0.07 0.14 0.21 0.28

Volume [mm3]

0

0.2

0.4

0.6

0.8

1

Rou

ndne

ss

[c]

[d]0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

S/I

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

I/L

Bladed Prolate (Roller)

Oblate (Disk) Equant (Compact)

Figure 15: Grain shape data for the first soil sample. Ar-rangement as in Fig. 11.

samples. In Fig. 18 we summarise the grain size data; thegrains are primarily of medium grade with some coarseand fine fractions (0 < φg < 3). A small amount of veryfine/silt fraction (φg = 4− 6) is also observed. We againcan compare this analysis to the data from laser particlesizing on a sister core by overlaying the histogram fromthe digital data onto the data from the particle size anal-ysis for the sand fraction in Fig. 18(a). The digital datais now biased to lower grain sizes compared to both theSoil A sample and the laser data. This again highlightsthe heterogeneity in grain sizes within this soil.

In Fig. 19 we summarise the grain shape data. Similarto the data for Soil A (recall Fig. 15), sphericity datalies between values of 0.4 and 0.7 with a very slightbias to larger sphericity for larger grains. The round-ness data lies in the rounded to well rounded range withlarger values observed for smaller grains. The data for allgrains superimposed on the Power plot shows a similarspread of grain shape across the sample. The data on theZingg diagram also varies over a wide range indicating

10


[a] 0 5 10 15 20Coordination number

0

50

100

150

200

Freq

uenc

y

[b]0 0.15 0.3 0.45

Volume [mm3]

0

5

10

15

20

25

30

Coo

rdin

atio

n nu

mbe

r

y = 5.03 + 0.00015 x

[c]0 0.4 0.8 1.2

Overalapping area [mm2]

0

200

400

600

800

Freq

uenc

y

[d]

Figure 16: Grain fabric data for soil A sample.

Figure 17: Two images of a highly coordinated grain (or-ange) and its neighbouring grains from a rendered 3Dmovie.

0 1 2 3φ=-log2(diameter)


S A N D

Coars Medium Fine

18.01% 73.20% 8.81%


2

4

6

8

Freq

uenc

y

Laser sizing

Figure 18: Grain size data for the second soil sample.

[a]0 0.05 0.1 0.15

Volume [mm3]

0

0.2

0.4

0.6

0.8

1

Sphe

reic

ity


[b] 0 0.05 0.1 0.15 0.2

Volume [mm3]

0

0.2

0.4

0.6

0.8

1

Rou

ndne

ss

[c]

[d]0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

S/I

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

I/L

Bladed

Oblate (Disk)

Prolate (Roller)

Equant (Compact)

Figure 19: Grain shape data for the second soil sample.Arrangement as in Fig. 11.

the breadth of grain shapes measured on this sample.In Fig. 20 and Table 2 we summarise the grain fabricdata. The average connectivity of the grains is Z̄ = 5.63and we observe a slightly broader coordination numberdistribution in Fig. 20(a) than in Fig. 16(a). The peak inthe distribution lies at a coordination number of Z = 3and Z = 4 again indicating that a large number of grainsare loose or potentially loose within the pack. The grainoverlap distribution in Fig. 20(b) is positively skewedwith the mean overlap area of the order of .4 mm2. Noanisotropy in grain orientation is observed (Fig. 20(c)).

Unconsolidated Reservoir Core

Data in Table 2 indicates that the unconsolidated reser-voir sand is well sorted, exhibits slight positive skewnessand is platykurtic. In Fig. 21 we summarise the grainsize data; the resolved grains are primarily of a fine tovery fine grade (2 < φg < 4). No experimental laserparticle sizing data was available for this core.

In Fig. 22 we summarise the grain shape data. The Spheric-

11 ZZ


[a] 0 5 10 15 20 25Coordination number

100

200

300

400

500

600

Freq

uenc

y

[b] 0 0.5 1 1.5 2 2.5Overalapping area [mm

2]

0

500

1000

1500

Freq

uenc

y

[c]

Figure 20: Grain fabric data for soil B sample.

ity data lies between values of 0.7 and 0.9. The round-ness data lies primarily between 0.4 and 0.6 (roundedgrains) and shows a bias to larger values observed forsmaller grains. The data for all grains superimposed onthe Power plot shows a similar grain shape within thesample. The data on the Zingg diagram lies primarily inthe compact quadrant.



S A N D

Very FineMedium Fine

3.33%

S I L T

0.51%19.96%77.19%


0

50

100

150

200

250

300

Freq

uenc

y

Figure 21: Grain size data for the unconsolidated reser-voir core sample.

[a]0 0.003 0.006 0.009

Volume [mm3]

0

0.2

0.4

0.6

0.8

1

Sphe

reic

ity


[b]0 0.003 0.006 0.009

Volume [mm3]

0

0.2

0.4

0.6

0.8

1

Rou

ndne

ss

[c]

[d]0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

S/L

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

I/L

Bladed

Oblate (Disk)

Prolate (Roller)

Equant (Compact)

Figure 22: Grain shape data for the unconsolidated reser-voir core sample. Arrangement as in Fig. 11.

In Fig. 23 and Table 2 we summarise the grain fabricdata. The average connectivity of the grains is Z̄ = 5.10and we observe a broad coordination number distribu-tion in Fig. 23(a). This low Z̄ = 5.10 may indicate thelooseness of the packing despite the high grain volumefraction. Also the peak in the distribution lies at a coor-dination number of Z = 3, 4 indicating that a large num-ber of grains are loose within the pack. The grain overlapdistribution in Fig. 23(b) is narrow and positively skewedwith the mean overlap area of the order of .04 mm2. Noanisotropy in grain orientation is observed (Fig. 23(c)).

Poorly Sorted Core

Data in Table 2 indicates that the unconsolidated reser-voir sand is very poorly sorted σ = 1.04, is stronglynegatively skewed and very platykurtic. In Fig. 24 wesummarise the grain size data; the resolved grains varyfrom very coarse to a very fine grade (−1 < φg < 4).Again no laser particle sizing data was available for di-rect comparison.

12


[a] 0 5 10 15 20 25 30 35Coordination number

100

200

300

400

500

600

Freq

uenc

y

[b] 0 0.05 0.1 0.15 0.2


0

200

400

600

800

Freq

uenc

y

[c]

Figure 23: Grain fabric data for the unconsolidated reser-voir core sample.

In Fig. 25 we summarise the grain shape data. The grainsare not spherical with sphericity values of 0.4 to 0.7.The roundness data lies primarily between 0.5 and 0.9(rounded to well rounded grains) and as seen previously,shows a bias to larger values for smaller grains. The datafor all grains superimposed on the Power plot shows a

-1 0 1 2 3 4φ=-log2(diameter)


S A N D

Very coars sand Medium Fine

0.961% 23.20% 35.97%

Coars sand Silt

0.719% 39.32%

-1 0 1 2 3 4

φ=-log2(diameter)

0

10

20

30

40

50

Freq

uenc

y

Figure 24: Grain size data for the poorly sorted reservoircore sample.

[a] 0 0.01 0.02 0.03 0.04

Volume [mm3]

0

0.2

0.4

0.6

0.8

1

Sphe

reic

ity


[b] 0 0.01 0.02 0.03 0.04

Volume [mm3]

0

0.2

0.4

0.6

0.8

1

Rou

ndne

ss

[c]

[d]0 0.2 0.4 0.6 0.8 1

S/I

0

0.2

0.4

0.6

0.8

1

I/L

Bladed

Oblate (Disk)

Prolate (Roller)

Equant (Compact)

Figure 25: Grain shape data for the poorly sorted reser-voir core sample. Arrangement as in Fig. 11.

broad distribution of grain shape within the sample. Thedata on the Zingg diagram lies primarily in the prolatequadrant.

In Fig. 26 and Table 2 we summarise the grain fabricdata. The average connectivity of the grains is Z̄ = 7.24and we observe a very broad coordination number distri-bution in Fig. 26(a). Again, the peak in the distributionlies at a coordination number of Z = 3 but one also ob-served some grains (largest) with coordination numbersof Z ≥ 60. The grain overlap distribution in Fig. 26(b)is broad, strongly skewed with the mean overlap areaper grain of the order of .3 mm2. We observe someanisotropy in the grain orientation in Fig. 26(c). It is notclear if the axis corresponding to the preferential align-ment of the grains is along the bedding plane of the core.

CONCLUSIONS

1. We demonstrate the ability to directly measure rockfabric and texture from 3D digital images of rockfragments. We describe two methods to distin-guish individual grains from a complex image. We

13 ZZ


[a] 10 20 30 40 50 60Coordination number

0

200

400

600

800

1000Fr

eque

ncy

[b] 0 0.5 1 1.5 2 2.5


0

500

1000

1500

2000

Freq

uenc

y

[c]

Figure 26: Grain fabric data for poorly sorted sample.

then describe methods for the characterisation ofindividual grains including grain size data (max/min,mean, median, mode, skewness, sorting and kurto-sis), grain shape data (sphericity, roundness) andtextural information (sorting, grain contacts, ma-trix/grain supported). Current measures of grainsize and shape are often based on visual compar-isons for 2D projections and are subject to opera-tor bias. The rock fabric and textural data obtainedfrom 3D images is more comprehensive, system-atic and quantitative than current analysis techniques.

2. We characterise five individual samples includinga model foam and 4 sedimentary samples. Thesamples encompass a broad spectrum of grain fab-ric and texture. The resultant analysis on up to8000 grains per image gives a wide distribution ofgrain size and sorting values. Grain size data basedon the Wentworth scale are analysed from very finesands/silts through to very coarse sands. Compar-ison of grain size data from the digital method todata from laser particle sizing on a sister core sam-ple is consistent.

3. Grain shape data varies considerably across sam-ples. The model homogeneous system is very stronglyrounded and spherical. The sedimentary systemsvary from subrounded to well rounded and have

sphericity values from 0.3 − 0.9. Systematic vari-ations in grain shape are noted with grain size. Aplot of the roundness and sphericity for all grainswithin a sample allows one to gauge mean particleshape as well as the spreads in the measures.

4. Grain fabric data varies with sorting and consoli-dation. The three poorly consolidated sedimentarysystems all have mean connectivity of 5.1 < Z <5.6. In contrast the more cemented and poorlysorted sand exhibits a higher connectivity Z =7.45. All samples exhibit a large number of grainswith coordination number of Z = 3 which corre-sponds to a loose grain. Grain overlap areas aremeasured and may be linked to the degree of ce-mentation of a rock.

ACKNOWLEDGEMENTS

The authors acknowledge the Australian Government fortheir support through the ARC grant scheme. We fur-ther thank the CRC for FCS, CRC LEME, BHP-BillitonPetroleum and Woodside Energy who have provided corematerial for this study and financial support for the facil-ity. We thank the A.N.U. Supercomputing Facility andthe Australian Partnership for Advanced Computing forvery generous allocations of computer time. We thankWayne Alger and Chris Cubitt from Woodside Energyfor comments on the submission and MS thanks Dr. E.Garboczi of NIST for discussions and assistance with thespherical harmonics calculations.

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ABOUT THE AUTHORS

M. Saadatfar: was awarded a BSc from the Institute ofAdvanced Studies in Zanjan Iran and is an ANU PhDcandidate.

M. L. Turner: was awarded a BSc (Hons) in Geologyfrom ANU and is now an ANU PhD candidate.

C. H. Arns: Christoph Arns was awarded a Diploma inPhysics (1996) from the University of Technology Aachenand a PhD in Petroleum Engineering from the Univer-sity of New South Wales in 2002. He is a ResearchFellow at the Department of Applied Mathematics at the

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Australian National University. His research interests in-clude the morphological analysis of porous complex me-dia from 3D images and numerical calculation of trans-port and linear elastic properties with a current focuson NMR responses and dispersive flow. Member: AM-PERE, ANZMAG, ARMA, DGG.

H. Averdunk: Holger Averdunk has studied MolecularBiology and Computer Science. He has worked in Bioin-formatics and the IT Industry. Currently he is working on3D filters and Segmentation methods.

T.J. Senden: Timothy John Senden received his train-ing and PhD in physical chemistry at the Australian Na-tional University in 1994. His principal techniques areatomic force microscopy and surface force measurement,but more recently micro-X-ray tomography. His researchcentres around the application of interfacial science toproblems in porous media, granular materials, polymeradsorption and single molecule interactions.

A.P. Sheppard: Adrian Sheppard received his B.Sc. fromthe University of Adelaide in 1992 and his PhD in 1996from the Australian National University and is currentlya Research Fellow in the Department of Applied Math-ematics at the Australian National University. His re-search interests are network modelling of multiphase fluidflow in porous material, topological analysis of complexstructures, and tomographic image processing.

R. M. Sok: Rob Sok studied chemistry and received hisPhD (1994) at the University of Groningen in the Nether-lands and is currently a Research Fellow in the Depart-ment of Applied Mathematics at the Australian NationalUniversity. His main areas of interest are computationalchemistry and structural analysis of porous materials.

W. V. Pinczewski: W.V Pinczewski holds BE (Chem.Eng)and PhD degrees from the University of New South Wales(UNSW). He is Head of the School of Petroleum En-gineering at UNSW. His research interests include im-proved oil recovery, multi-phase flow and transport prop-erties in porous media and network modelling.

J. Kelly: J.C. Kelly holds BSc Hons (Geology) and PhDdegrees from La Trobe University and is currently a Se-nior Petrophysicist with Woodside Energy. His main ar-eas of interest are formation evaluation and reservoir char-acterisation.

M.A. Knackstedt: Mark Knackstedt was awarded a BScin 1985 from Columbia University and a PhD in Chem-ical Engineering from Rice University in 1990. He is anAssociate Professor at the Department of Applied Math-ematics at the Australian National University and a Vis-

iting Professor at the School of Petroleum Engineeringat the University of NSW. His work has focussed on thecharacterisation and realistic modelling of disordered ma-terials. His primary interests lie in modelling transport,elastic and multi-phase flow properties and developmentof 3D tomographic image analysis for complex materi-als.

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ROCK FABRIC AND TEXTURE FROM DIGITAL CORE ANALYSIS

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