stochastic reconstruction of chalk samples containing...

Transport in Porous Media 57: 1–15, 2004.© 2004 Kluwer Academic Publishers. Printed in the Netherlands.

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Stochastic Reconstruction of Chalk SamplesContaining Vuggy Porosity Using a ConditionalSimulated Annealing Technique

M. S. TALUKDAR1,∗, O. TORSAETER1 and J. J. HOWARD2

1Department of Petroleum Engineering and Applied Geophysics, Norwegian University of Scienceand Technology, Norway2Geoscience Building, Phillips Petroleum Company, OK, USA

(Received: 4 December 2001; in final form: 27 January 2004)

Abstract. The issue of stochastic reconstruction of chalk samples containing appreciable amountof vuggy porosity in the form of foraminifer shells using a conditional simulated annealing (SA)technique is addressed. A new indicator function suitable for the algorithm is defined to representchalk microstructures. Most of the chalk samples consist of vuggy porosity associated with largecavities found in foraminifer shell fragments or other dissolution features. These features have signi-ficant influence on fluid and electric transport properties depending on how they are interconnectedand how much contribution they have to the overall porosity. Stochastic reconstructions using thewidely used conditioning and truncation of Gaussian random fields method or the conventional SAtechnique are not adequate to reproduce them accurately. In conditional SA, the special features ofinterest are masked and are excluded during perturbation steps of the algorithm. We demonstrate theapplicability of this technique in reconstructing foraminifer shells of different shapes and statisticsin chalk samples using different combinations of the morphological descriptors.

Key words: reconstruction algorithm, conditional simulated annealing, chalk, vuggy porosity, fo-raminifer shells, geostatistics, two-point correlation function, lineal path function, chord distribution.

1. Introduction

Attempts have been made in the recent years to compute macroscopic properties(e.g. fluid permeability, electrical conductivity, capillary pressures, etc.) either di-rectly from fluid or electric flow simulation through a description of the pore spacegeometry and topology or converting the porous media into an equivalent regular-network model from known geometric and topological information. Reliable porespace geometric and topological information are accessible by experiments, forexample, X-ray computed microtomography (Coles et al., 1994, 1996; Spanneet al., 1994; Hazlett, 1995) and scanning laser confocal microscopy (Fredrich et al.,1995) or by explicit modeling of the grain depositional and diagenetic processes

∗Present address: Norsk Hydro ASA, P. O. Box 7190, 5020 Bergen, Norway. Tel.: +47-55-99-2667; Fax: +47-55-99-6639; e-mail: [email protected]

2 M. S. TALUKDAR ET AL.

(Bakke and Øren, 1997). The aforementioned experimental techniques are notsuited for routine application. Most importantly, their resolutions are not suffi-cient to image the sub-micron size pores that are abundant in chalk. On the otherhand, chalk microstructures are too complex to reproduce by explicit modelingof the grain depositional and diagenetic processes. Other viable alternative is toreconstruct rock samples by stochastic techniques. The conditioning and trunca-tion of Gaussian random fields (GRF) is a widely used stochastic reconstructiontechnique (Quiblier, 1984; Adler et al., 1990; Ioannidis et al., 1997; Levitz, 1998;Biswal et al., 1999; Bekri et al., 2000; Kainourgiakis et al., 2000; Liang et al.,2000a). The approach is mathematically elegant and computationally efficient,but unfortunately limited to imposing the pore fraction (porosity) and pore–poreautocorrelation function of the reference (real) medium as the only reconstructionconstraints. Levitz (1998), who attempted an off-lattice GRF method, observedthat these constraints are insufficient to reproduce the granular appearance of theclay-coated silica particles. Kainourgiakis et al. (2000) used an on-lattice GRFmethod to reconstruct the microstructure of a random pack of spherical particlesgenerated by a sequential deposition algorithm. The appearance of the solid matrixin the reconstructed random pack was amorphous and showed none of the dis-tinct morphology of the solid–pore interfaces that characterize sphere aggregates.Biswal et al. (1999) also failed to reproduce the well-sorted quartz grains of aFontainebleau sandstone sample. In a recent study, Bekri et al. (2000) attemptedto reconstruct chalk samples consisting of significant amount of vuggy porosityin the form of hollow foraminifer shells using an on-lattice GRF method. Thereconstruction technique failed to reproduce not only these features but also theautocorrelation function itself. The initial slope of the correlation function, whichis related to specific surface area of the porous medium, was poorly simulated.

In recent years, a much more flexible method known as the simulated annealing(SA) technique is immerging as an attractive stochastic reconstruction technique(Yeong and Torquato, 1998a, b; Liang et al., 2000b; Manwart et al., 2000; Talukdaret al., 2002a, b). Using this method, Yeong and Torquato (1998a, b) imposedthe pore-phase two-point correlation and lineal path functions as constraints inthe reconstruction of a Fontainebleau sandstone sample. Manwart et al. (2000)reconstructed Berea and Fontainebleau sandstone samples from information onthe pore-phase two-point correlation function, pore-phase lineal path function, andpore size distribution function. Liang et al. (2000b) imposed the neighborhood rankdistribution together with two-point correlation function. In a recent study using theSA technique, Talukdar et al. (2002a) showed that the solid-phase chord distribu-tion function contains additional information that is critical for the reconstructionof the morphology of particulate media exhibiting short-range order. However, inanother study involving reconstruction of chalk samples with no vuggy porosity,Talukdar et al. (2002b) showed that imposing chord distribution functions resultsonly in minor improvement over what is achieved by using the pore-phase two-point correlation function as the only constraint. During this work, the authors

STOCHASTIC RECONSTRUCTION OF CHALK SAMPLES 3

attempted to reconstruct 2D chalk samples containing significant amount of vuggyporosity (some of the images used by Bekri et al., 2000). Conventional SA withas many as 10 morphological constraints (pore-phase autocorrelation functions intwo orthogonal directions, pore- and solid-phase chord distribution functions intwo orthogonal and diagonal directions) were not adequate to accurately replicatethe foraminifer shells (Talukdar, 2002).

The conditional simulated annealing (CSA) is a technique in which the targetedfeatures can be marked and kept unchanged without affecting the global spatialcorrelation statistics. Hazlet (1997) used this technique to duplicate a slice of thetarget image (computed microtomography dataset of a Berea sandstone) into areconstructed 3D sample. The motivation of the present work is to demonstratethe applicability of this technique to reconstruct 2D and 3D chalk samples contain-ing varying amount of vuggy porosities in the form of hollow foraminifer shells.The paper is organized as follows. In Section 2 we briefly review the petrology ofchalk and the effects of vuggy porosity on transport properties. The definitions ofthe morphology of chalk microstructure and the algorithms used to reconstruct itfrom limited morphological information are described in Section 3. In Section 4we describe the reconstruction results demonstrating the ability of the CSA toreconstruct vuggy porosity. We summarize in Section 5 with concluding remarkson the significance of our findings.

2. Petrology of Chalk and the Effects of Vuggy Porosity on TransportProperties

North Sea chalks found at the Greater Ekofisk Region are fine-grained limestonescomposed predominantly of skeletal debris from the pelagic unicellular algae, coc-colithophore (Sulak, 1991) (see Figure 1(a)). Upon death, the external skeletonor coccosphere, begins to disaggregate into wheel-shaped coccoliths as it settlestowards the ocean floor (see Figure 1(b)). During sedimentation and early stagesof burial these fragile coccoliths continue to disaggregate into small platelets thatrange in size from 1 to 2 µm (Feazel et al., 1985). Whole coccoliths are commonlyfound, but coccospheres are rarely preserved (Sulak, 1991) (see Figure 1(c)). Otherparticles are also found in North Sea chalks, the most important of which are silt-sized grains of silica and other carbonate fragments of biogenic origin. The mostrecognizable of these fragments are shell remnants from Foraminifers, a unicellularorganism that produces uni- or multi-chambered shells that range from 1 to 20 µmin size (Siemers et al., 1994) (see Figure 1(d)).

Two kinds of porosity are observed in chalk: interparticle porosity contributedby the pore space between the fine-grained coccolith particles, and vuggy porosityassociated with large cavities found in Foraminifer shell fragments or other dissol-ution features. Lucia (1983) suggested that the chalk permeability is controlledby the amount of interparticle pore space, presence of separate vugs (vugs areinterconnected through the interparticle pore network) contribute little, if any, to


Figure 1. (a) Coccolithophore, (b) a separate coccolith. Coccolithophore is the organism in-cluding the spherical cell and external coccoliths. Courtesy of Hans Schrader and Dag IngeBlindheim, University of Bergen, Norway, and Jeremy Young, British Natural History Mu-seum, London. (c) Scanning electron microscope image of a 30% porosity North Sea chalk(×5000) showing the coccoliths and coccolith platelets. (d) Backscatter scanning electronmicroscope image (BSE) of a 33% porosity North Sea chalk (1024 × 1024 pixels at 200×)showing separate and touching vugs in the form of foraminifer shells. Pores are shown inblack.

the permeability, however, the presence of touching vugs (vugs are interconnectedby direct contact with each other) may contribute to the permeability significantlyand the capillary pressure may be significantly different. Ioannidis et al. (1997)observed a quite variable resistivity and formation factor values depending on thefraction of vuggy porosity present in the sample. An increase in cementation ex-ponent ‘m’ is also expected because their contribution to porosity is counted as ifthey are matrix pores. It is evident from this discussion that the vugs have influenceon fluid and electric transport properties of chalk. The influence depends on howwell they are interconnected. If the vugs are not well connected they may work asa prime source for oil trapping in some chalk reservoirs, specially, in a water-wetsystem. Considering the above facts, it is worthwhile to investigate whether chalkreconstruction including reliable representation of vuggs and variable amount ofvuggy porosity is possible. This kind of reconstructed samples may be very usefulfor further studies on the effect of fluid and electric transport properties.


3. Stochastic Reconstruction from Limited Morphological Descriptors

3.1. MORPHOLOGICAL DESCRIPTORS

Traditionally a binary phase function Z(�r) (Adler et al., 1990) is used to define thestructure of a porous material,

Z(�r) ={

1 �r points to pore space,

0 otherwise.(1)

For reconstruction of vuggy porosity (in the form of hollow foraminifer shells) inchalk using a CSA, we modify the above phase function as follows and refer to asan indicator function:

I (�r) =

1 �r points to pore space not belonging to foraminifer shells,

0 �r points to solid space not belonging to foraminifer shells,

2 �r points to solid space belonging to foraminifer shells,

3 �r points to pore space belonging to foraminifer shells.

(2)

The reasoning for this modification will be evident in the following section. Fora digitized medium, the porosity, φ, can be calculated from the indicator function(Eq. (2)) simply by counting the number of pore-phase voxels (non-vuggy andvuggy, i.e. indicators 1 and 3, respectively) and then dividing it by the total numberof sample voxels.

For a statistically homogeneous medium, the two-point probability function ofthe pore-phase, S2(�u), can be defined from the function Z(�r),

S2(�u) = 〈[Z(�r)] · [Z(�r + �u)]〉, (3)

where, �u is a lag vector. A surrogate of S2(�u) is the so-called autocorrelationfunction, Rz(�u) which is just a normalization of S2(�u) with respect to porosity,

Rz(u) = S2(u)− φ2

(φ − φ2). (4)

For a statistically homogeneous and isotropic porous media, φ is a constant, andS2(�u) and Rz(�u) are only functions of the modulus of the lag vector, that is, S2(�u) =S2(u) and Rz(�u) = Rz(u). The function S2(u) and Rz(u) may be determined fromcross-sectional images of the pore space or from small-angle scattering experi-ments.

We calculated the two-point correlation function by successively translating asampling rod of u pixels in length a distance of one pixel at a time, spanning thewhole image, counting the number of occurrences that the two end-points of therod fall in pores and finally, dividing the number of occurrences by the total numberof trials. This function was calculated along the two orthogonal directions. Periodicboundary conditions were implemented in order to approximate an infinite medium


with a finite grid (Adler et al., 1990). In a recent study, Talukdar and Torsaeter(2002) have shown that for reconstruction of a small chalk sample the correlationfunctions are to be computed with the application of periodic boundary conditionsto avoid finite size effects. However, it is not critically important when the samplesize is reasonably large.

The slope of S2(u) or Rz(u) at the origin is related to the specific surface area(the interfacial area per unit volume), s, which for digitized media is given byYeong and Torquato (1998a):

s = −2DdS2(u)

du

∣∣∣∣u=0

= −2D(φ − φ2)dRzdu

∣∣∣∣u=0

, (5)

where, D is the dimensionality of the space. A characteristic length scale of thepore space is provided by the correlation length λ̄, defined as the integral of S2(u)

or Rz(u) (Ioannidis et al., 1996),

λ =∫ ∞

0(φ − φ2)S2(u) du =

∫ ∞

0Rz(u) du. (6)

The pore-phase lineal path function L(�u) is another useful characteristic of mi-crostructure. This quantity measures the probability that a line segment spanningfrom �r to �r + �u lies entirely within pore phase. A closely related function is the so-called chord distribution function, Ci(u), where the subscript i refers to either poreor solid (Coker and Torquato, 1995; Roberts and Torquato, 1999). A chord is thelength u between intersections of a line with the two-phase interface. The chorddistribution function can be directly interpreted in terms of microstructural fea-tures, as it contains phase connectedness and correlation information along a linealpath (Roberts and Torquato, 1999). That is, if Ci(u) �= 0 for large values of u, thereexist connected regions of phase i at scale u. Importantly, the value of u at whichCi(u) is maximum provides an estimate of the length scale associated with phasei. In this study, L(�u) was calculated along the two orthogonal directions, whereas,Ci(u) was calculated along the two orthogonal and two diagonal directions, where‘x-diagonal’ is the direction from top left to bottom right and ‘y-diagonal’ is frombottom left to top right of the image.

3.2. RECONSTRUCTION OF CHALK SAMPLES CONTAINING FORAMINIFER

SHELLS BY CSA

SA, a global optimization technique, has a rich history in geostatistical applica-tions (see Ouenes et al., 1994, for a review). Recently, it has been endowed as apowerful tool for stochastic reconstruction of porous media (Yeong and Torquato,1998a, b; Liang et al., 2000b; Manwart et al., 2000; Talukdar and Torsaeter, 2002;Talukdar et al., 2002a, b). The capability of the SA technique can be greatly en-hanced through a different variant of the algorithm known as the CSA. In this


technique, the special features of interest are masked and are excluded duringperturbation steps of the algorithm. Hazlet (1997) explored this technique andduplicated a slice of the computed microtomography dataset (target image) into thereconstructed 3D samples. Here, we briefly describe the technique in the contextof reconstructing hollow foraminifer shells:

The idea behind this technique is to gradually transform an initial random struc-ture (‘high-energy’) containing required number of hollow foraminifer shells intoa ‘minimum-energy’ configuration where the global correlation statistics are satis-fied. This is achieved by reducing an ‘energy function’ (defined later) to a groundlevel in the SA process. Construction of the initial structure (including foraminifershells) is achieved following three steps: (a) generate a medium of size L×M

pixels (L×M ×N voxels in 3D) containing only 0’s (solid space not belonging toforaminifer shells), (b) place required number of foraminifer shells of size, shapeand porosity similar to that observed in the BSE images at random or specified lo-cations. This step replaces some of the 0’s with 2’s and 3’s at the locations wherethe foraminifer shells are placed (described further below), and (c) adjust porosityof the sample (make equal to the experimental porosity) by replacing requirednumber of 0’s not belonging to foraminifer shells (equal to a fraction φnv = φ−φv,where, φv and φnv are the vuggy and non-vuggy porosities, respectively) with 1’s.

Except Bekri et al. (2000)’s image (see Section 4.1), only circular (2D) andspherical (3D) foraminifer shells were constructed. The reasons for selecting spher-ical shells are two folded; (a) foraminifer shells in the real chalk sample are quitespherical [see Figure 1(d)], and (b) ease of implementation. The foraminifer shellsin Bekri et al. (2000)’s case were placed manually to make them visually similar tothose in the real images. The algorithm for construction and placement of sphericalshells consists of the following steps: (a) select a point (random or specified) in thestarting image (consisting of 0’s only), (b) select a radius of the foraminifer shellto be placed at this location (the radii of shells are obtained from image analysis),(c) count total number of pore voxels for this shell (i.e. sphere volume in voxelsmultiplied by the observed fraction of pores in this shell size). This enables somesolid voxels to exist inside the sphere if the shell porosity in less than 1.0, (d)place required number of pore voxels in this sphere (randomly or in some order),(e) take precautions so that the sphere cannot expand beyond sample boundary, (f)place a solid cover of required thickness around the sphere (can be obtained fromimage analysis). This cover can be fully closed or open at selected positions, (g)label solid voxels (inside shell and cover) with 2 and pore voxels with 3, (h) repeatthis procedure at a different location (random or specified) to generate shells ofdifferent sizes. Note that the touching vuggs can be constructed by simply puttingoverlapping spheres of required sizes (there are images where no touching vuggscan be observed).

The annealing process is simulated by selecting a pore pixel and a solid pixel atrandom and then by interchanging their indicator function values. This interchangeslightly modifies the correlation statistics of the sample and therefore changes


the ‘energy’ of the system. The main feature of the CSA which distinguishes itfrom conventional SA is that during iteration steps, a pore pixel and a solid pixelnot belonging to the foraminifer shells (i.e. indicator function values 1 and 0, re-spectively) are only chosen. This selection process preserves sample porosity andforaminifer shells. The ‘energy’ is measured in terms of deviations from a set of tar-get, experimentally determined, functions conveying morphological information,for example, S2(u), Rz(u), L(u) and Ci(u). It must be emphasized here that whilethe special features of interest are masked during annealing perturbations, they arecompletely open during computation of the statistical functions (indicator functionvalue 2 and 3 are replaced with 0 and 1, respectively). This ensures preservation ofthe global correlation statistics.

The ‘energy function’ is generally defined in terms of an arbitrary number (n)of user-defined reference functions as:

E =∑n

umaxn∑u=0

[fn(u)− f̃n(u)]2, (7)

where, fn and f̃n are the simulated and reference functions, respectively. Any com-bination of the morphological descriptors may be used in the objective function.Each reference function f̃n is matched to a maximum lag umax

n . A pixel interchangeis accepted with a probability pa given by the Metropolis rule (Metropolis et al.,1953),

pa ={

1 if �E � 0,

e−�E/T if �E > 0,(8)

where, �E is the change in energy in two consecutive interchanges and T is acontrol parameter representing the ‘temperature’ of the system. After a certainnumber of interchanges, T is reduced according to a schedule known as annealingschedule. The process is continued until the evolving system’s fn matches the targetf̃n within a tolerance limit. The readers are referred to our recent papers (Talukdarand Torsaeter, 2002; Talukdar et al., 2002a, b) for the annealing schedule used inthis study.

4. Results and Discussion

4.1. 2D RECONSTRUCTIONS

To demonstrate the applicability of the technique, two 2D samples have beenreconstructed using the morphological descriptors and foraminifer shell statis-tics obtained from two different North Sea chalk samples. The first sample is a33% porosity (helium porosimetry) chalk of the Greater Ekofisk Region (codename ‘e32’). A total of five BSE images were used to obtain the morphologicaldescriptors and foraminifer shell statistics. The images were captured at a pixelresolution of 0.5 × 0.5 µm2 (image size 1024 × 1024 pixels) in order to get good


statistics for the foraminifer shells. One of the five images is shown in Figure 1(d).Note that the image contains some special features (big solid object shown inwhite) in addition to foraminifer shells. Statistics of the foraminifer shells and solidobjects obtained from several microscopic images are shown in Table I.

A sample of size 512 × 512 pixels has been reconstructed constrained by pore-phase S2(u) and L(u) computed in two orthogonal directions. The reconstructedsample contains five foraminifer shells of different sizes and one solid object. The

Table I. Statistics of foraminifer shells in chalk samples

Foraminifer shell Solid object

Sl. No. Approximate internal Approximate external Sl. No. Approximate

dia (pixels) dia (pixels) dia (pixels)

1 71 92 1 115

2 92 112 2 56

3 102 117 3 102

4 36 41 4 66

5 87 107 The solid cover

6 97 117 thickness of the

7 120 130 foraminifer shells are

8 61 92 calculated from internal

9 31 46 and external diameters.

10 46 56

11 102 117

Figure 2. 2D reconstruction of sample e32; (a) reconstructed sample (512 × 512 pixels). Porespace is shown in black. One of original images is shown in Figure 1(d). (b) Comparison ofpore-phase S2(u) and L(u) between the real image (average over five images) and the recon-structed image (evaluated globally, i.e. both vuggy and non-vuggy porosities were included).Average of orthogonal x- and y-directional values are plotted.


starting energy of the system was 0.1516 and the simulation was termi-nated as soon as the system energy fell below 10−5. The reconstructed sampleis shown in Figure 2(a), and S2(u) and L(u) functions of the simulated mediumare compared with the target ones (average over 5 images) in Figure 2(b). Thesefunctions were calculated globally, that is, both vuggy and non-vuggy porositieswere considered. The vuggy porosity φv of the sample is 2.4%. The reconstructedimage looks very similar to the real image except that the hollow foraminifer shellsare separated [one of original images is shown in Figure 1(d)]. As stated previ-ously, it is straightforward to generate touching vugs by simply placing overlappingshells instead of placing at random locations. As can be observed, S2(u) and L(u)have matched quite well. The specific surface area s (cf. Eq. (5)) of the simulatedmedium is 1.009 µm−1 which is remarkably close to that of the target images(s= 0.996 µm−1). The correlation length λ (cf. Eq. (6)) is also matched withinan error less than 0.2% (λtarget = 0.676 and λsimulated = 0.675) which indicates anaccurate match of S2(u) at all lags.

Figure 3. 2D reconstruction of sample p13 (Bekri et al., 2000); (a) one of the five target im-ages (400 × 400 pixels), (b) reconstructed sample (400 × 400 pixels). Pore space is shown inblack. (c) Comparison of pore-phase Rz(u) between the real image (average over five images)and the reconstructed image (evaluated globally). Average of orthogonal x- and y-directionalvalues are plotted. (d) Comparison of solid-phase Cs(u) between the real image (average overfive images) and the reconstructed image (evaluated globally). Average of both orthogonal anddiagonal x- and y-directional values are plotted.


To further verify the effectiveness of the technique, we considered another chalksample from the Ekofisk field. This was sample p13 from the study of Bekri et al.(2000), a BSE image of which is shown in Figure 3(a). For this sample, image anal-ysis yields s= 0.615 µm−1 and φ= 0.39. Vuggy porosity (ca. 9%) in the form ofhollow foraminifer shells is evident. The readers should recall that the GRF methodwas not adequate to reproduce the microstructure of this sample, resulting in com-puted values of permeability that were very different from the experimental one(Bekri et al., 2000). Also, reproduction of the hollow shells was not possible withany combination of the correlation functions using the conventional SA method(Talukdar, 2002). We apply CSA to reconstruct foraminifer shells of arbitrary shapeand porosity (containing some solid fragments in the shells). The constraints usedin this reconstruction were the pore-phase Rz(u) (measured along the two ortho-gonal directions) and the solid-phase Cs(u) (measured along the two orthogonaland two diagonal directions). The result of a 2D reconstruction (400 × 400 pixels)is shown in Figure 3(b), whereas a comparison between the target and simulatedcorrelation functions is shown in Figure 3(c and d). It can be readily observed thatthe reconstructed sample looks visibly quite similar to the real image. The specificsurface area s (cf. Eq. (5)) of the simulated medium is 0.616 µm−1 which is remark-ably close to that obtained from image analysis (s= 0.615 µm−1). The correlationlength λ (cf. Eq. (6)) of the simulated medium is 1.581 which is also very close tothat of the target images (λtarget = 1.579). The following rough estimates of forma-tion factor and permeability may be readily obtained: F = 0.39−2.2 = 7.94 andk= 8.48 mD, the latter using k= 64φ2/226Fs2 proposed by Liang et al. (2000a)with s= 0.615 µm−1 and φ= 0.3 (i.e. assuming that resistance to fluid flow isdominated by interparticle porosity). A slightly higher value of cementation ex-ponent (m= 2.2) is consistent with previous theoretical computations of the effectof vuggy porosity on electrical conduction (Ioannidis et al., 1997). The estimatesof both permeability and formation factor are in reasonable agreement with theexperimental values (Fexp = 7.75, kexp = 6.12 mD) reported by Bekri et al. (2000).

4.2. 3D RECONSTRUCTIONS

For further demonstration of the applicability of the technique, we reconstructedthree 3D samples with different amount of vuggy porosities. Samples of size 1503

voxels each were reconstructed using pore-phase S2(u) and L(u) of sample e32computed along the three orthogonal directions (target z was taken to be the averageof x- and y-directional values). The samples have a vuggy porosity of 0.1, 2.4 and5.7%, respectively where the second sample corresponds to the real chalk medium.Five foraminifer shells (completely closed) and one solid object were constructedin each of the samples but the sizes of the shells were different resulting in differentvuggy porosities. The agreement between the target and simulated S2(u), λ andL(u) was similar to that in the first 2D case and hence are not repeated here.The microstructures of the first (0.1% vuggy porosity) and the third (5.7% vuggy


Figure 4. Reconstruction of a 3D sample with 0.1% vuggy porosity; (a) microstructure. Thepore space is shown opaque with ends in black. The solid is transparent. (b) A random slicethrough the reconstructed sample. Pores are shown in black.

Figure 5. Reconstruction of a 3D sample with 5.7% vuggy porosity; (a) microstructure. Thepore space is shown opaque with ends in black. The solid is transparent. (b) A random slicethrough the reconstructed sample. Pores are shown in black.

porosity) reconstructed samples are shown in Figures 4 and 5, respectively. Themicrostructures are visibly different due to the fact that the total porosity is samefor all samples (φ= 33%) but the interparticle porosity is different due to differentamount of vuggy porosities.

Let us now examine very roughly what would possibly happen to the fluid trans-port properties of these three samples. Use of equation k = 64φ2/226Fs2 (where,


Table II. Expected permeability of the 3D reconstructed samples

Sample φnon−vuggy s(µm−1) k (mD)

1 0.329 1.088 2.3

2 0.306 1.075 2.0

3 0.273 1.066 1.6

F = 0.33−2.2 = 11.5 and s is calculated from Eq. (5)) gives a rough estimate of kas shown in Table II.

Clearly, this reconstruction technique paves the way for further analysis of theeffects of the amount of vuggy porosity on fluid and electric transport properties ofchalk using numerical tools, such as, network modeling.

5. Conclusions

The potential of the CSA technique in reconstructing chalk samples containingappreciable amount of vuggy porosity in the form of foraminifer shells is demon-strated. A new indicator function suitable for the algorithm is defined to representthe structure of the chalk samples. The results demonstrate that reliable replicationof vuggy porosity in the form of hollow foraminifer shells of different shapes andstatistics is possible without loss of global correlation statistics. The techniquepaves the way to analyze the effects of the amount of vuggy porosity on fluidand electric transport properties of chalk using numerical tools, such as, networkmodeling.

Acknowledgement

Saifullah Talukdar gratefully acknowledges the financial support from PhillipsPetroleum Company, Norway for his PhD studies.

References

Adler, P. M., Jacquin, C. G. and Quiblier, J. A.: 1990, Flow in simulated porous media, Int. J.Multiphase Flow 16(4), 691–712.

Bakke, S. and Øren, P. E.: 1997, 3D pore-scale modeling of sandstones and flow simulations in thepore networks, SPEJ 2, 136–149.

Bekri, S., Xu, K., Yousefian, F., Adler, P. M., Thovert, J.-F., Muller, J., Iden, K., Psyllos, A., Stubos,A. K. and Ioannidis, M. A.: 2000, Pore geometry and transport properties in North Sea chalk,J. Petrol. Sci. Eng. 25, 107.

Biswal, B., Manwart, C., Hilfer, R., Bakke, S. and Øren, P. E.: 1999, Quantitative anal-ysis of experimental and synthetic microstructures for sedimentary rock, Physica A 273,452–475.


Coker, D. A. and Torquato, S.: 1995, Extraction of morphological quantities from a digitized medium,J. Appl. Phys. 77, 6087–6099.

Coles, M. E., Spanne, P., Muegge, E. L. and Jones, K. W.: 1994, Computed microtomography ofreservoir core samples, in: Proceedings of 1994 International Symposium of SCA, Stavanger,Norway.

Coles, M. E., Hazlett, R. D., Muegge, E. L., Jones, K. W., Andrews, B., Dowd, B., Siddons, P., Peskin,A., Spanne, P. and Soll, W. E.: 1996, Developments in synchrotron X-ray microtomography withapplications to flow in porous media, Paper SPE 36531 presented at the 1996 Annual TechnicalConference and Exhibition, Denver, Colorado, USA, 6–9 October.

Feazel C. T., Keany, J. and Peterson, R. M.: 1985, Cretaceous and Tertiary chalk of the EkofiskField area, central North Sea. in: Roehl and Choquette (eds), Carbonate Petroleum Reservoirs,Springer-Verlag, New York, Inc., pp. 497–507.

Fredrich, J., Menendez, B. and Wong, T.-F.: 1995, Imaging the pore structure of geomaterials,Science 268, 276–279.

Hazlett, R. D.: 1995, Simulation of capillary-dominated displacements in microtomographic imagesof reservoir rocks, Transp. Porous Media 20, 21–35.

Hazlett, R. D.: 1997, Statistical characterization and stochastic modeling of pore networks in relationto fluid flow, Math. Geol. 29, 801–822.

Ioannidis, M. A., Kwiecien, M. J. and Chatzis, I.: 1996, Statistical analysis of the porous microstruc-ture as a method for estimating reservoir permeability, J. Petrol. Sci. Eng. 16, 251–261.

Ioannidis, M. A., Kwiecien, M. J. and Chatzis, I.: 1997, Electrical conductivity and percolationaspects of statistically homogeneous porous media, Transp. Porous Media 29, 61–83.

Kainourgiakis, M. E., Kikkinides, E. S., Steriotis, Th. A., Stubos, A. K., Tzevelekos, K. P. andKanellopoulos, N. K.: 2000, Structural and transport properties of alumina porous membranesfrom process-based and statistical reconstruction techniques, J. Colloid Interface Sci. 231,158.

Levitz, P.: 1998, Off-lattice reconstruction of porous media: critical evaluation, geometrical confine-ment and molecular transport, Adv. Colloid Interface Sci. 76/77, 71–106.

Liang, Z., Ioannidis, M. A. and Chatzis, I.: 2000a, Permeability and electrical conductivity of porousmedia from 3D stochastic replicas of the microstructure, Chem. Eng. Sci. 55, 5247.

Liang, Z., Ioannidis, M. A. and Chatzis, I.: 2000b, Reconstruction of 3D porous media using simu-lated annealing, in: Proceeding of the XIII International Conference on Computational Methodsin Water Resources, Alberta, Canada, pp. 25–29.

Lucia, F. J.: 1983, Petrophysical parameters estimated from visual descriptions of carbonate rocks: afield classification of carbonate pore space, JPT 35(3), 629.

Manwart, C., Torquato, S. and Hilfer, R.: 2000, Stochastic reconstruction of sandstones, Phys. Rev.E 62, 893.

Metropolis, N., Rosenbluth, A., Rosenbluth, M., Teller, A. and Teller, E.: 1953, Equation of statecalculations by fast computing machines, J. Chem. Phys. 21, 1087–1092.

Ouenes, A., Bhagavan, S., Bunge, P. H. and Travis, B. J.: 1994, Application of simulated annealingand other global optimization methods to reservoir description: myths and realities, Paper SPE28415 presented at the 69th Annual Technical Conference and Exhibition, New Orleans, LA,USA, 25–26 September.

Quiblier, J. A.: 1984, A new three-dimensional modeling technique for studying porous media,J. Colloid Interface Sci. 98, 84–102.

Roberts, A. P. and Torquato, S.: 1999, Chord-distribution functions of three-dimensional randommedia: approximate first-passage times of Gaussian process, Phys. Rev. E 59, 4953–4963.

Siemers, W. T., Caldwell, C. D., Farrel, H. E., Young C. R., Yang-Logan, J. and Howard, J. J.: 1994,Chalk lithofacies, fractures, petrophysics and paleontology: an interactive study of chalk reser-voirs, Paper presented at the EAPG/AAPG Special Conference on Chalk, Copenhagen, Denmark,7–9 September.


Spanne, P., Thovert, J., Jacquin, C., Lindquist, W., Jones, K. and Adler, P.: 1994, Synchrotroncomputed microtomography of porous media: topology and transports, Phys. Rev. Lett. 73,2001–2004.

Sulak, R. M.: 1991, Ekofisk field: the first 20 years, Paper SPE 20773 presented at the 65th AnnualTechnical Conference and Exhibition of the Society of Petroleum Engineers held in New Orleans,LA, USA, 23–26 September.

Talukdar, M. S.: 2002, Ekofisk chalk: core measurement, stochastic reconstruction, network mod-eling and simulation, PhD Dissertation, Department of Petroleum Engineering and AppliedGeophysics, Norwegian University of Science and Technology, Trondheim, Norway, May 2002(ISBN 84-471-5416-1, ISSN 0809-103X).

Talukdar, M. S. and Torsaeter, O.: 2002, Reconstruction of chalk pore networks from 2D backscatterelectron micrographs using a simulated annealing technique, J. Petrol. Sci. Eng. 33, 265–282.

Talukdar, M. S., Torsaeter, O. and Ioannidis, M. A.: 2002a, Stochastic reconstruction of particulatemedia from two-dimensional images, J. Colloid Interface Sci 248, 419–428.

Talukdar, M. S., Torsaeter, O., Ioannidis, M. A. and Howard, J. J.: 2002b, Stochastic reconstructionof chalk from 2D images, Transport Porous Media 48, 101–123.

Yeong, C. L. Y. and Torquato, S.: 1998a, Reconstructing random media, Phys. Rev. E 57, 495–506.Yeong, C. L. Y. and Torquato, S.: 1998b, Reconstructing random media. II. Three-dimensional media

from two-dimensional cuts, Phys. Rev. E 58, 224–233.

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