On the Geometry and Topology of 3D Stochastic Porous Media

Journal of Colloid and Interface Science 229, 323–334 (2000) doi:10.1006/jcis.2000.7055, available online at http://www.idealibrary.com on

On the Geometry and Topology of 3D Stochastic Porous Media M. A. Ioannidis1 and I. Chatzis Porous Media Research Institute, Department of Chemical Engineering, University of Waterloo, Waterloo, Ontario, N2L 3G1 Canada Received April 26, 1999; accepted June 19, 2000

The nature of geometric and topological information contained in statistical correlation functions was investigated systematically using simulated porous media, generated by the level-cut of Gaussian random fields. Pore space partitioning techniques based on multiorientation scanning were implemented to determine the pore and neck size distributions, coordination number distribution, and genus of a number of model porous media. These results were correlated with the statistical properties (porosity and correlation function) of the microstructure, revealing for the first time the extent of morphological diversity of a broad class of stochastically reconstructed porous media. It was found that the dominant factor explaining microstructural variability among the media studied is the dimensionless length of spatial correlation. Accordingly, the resolution at which the void space is discretized during simulation was shown to affect significantly the resulting pore and neck size distributions and specific genus. It was also found that the average coordination number of simulated porous media is independent of correlation length, but decreases slightly with decreasing porosity. ° C 2000

Academic Press

Key Words: porous media, microstructure; correlation; geometry; topology; stochastic reconstruction.

INTRODUCTION

Understanding the influence of the microstructure on macroscopic behavior is required for optimizing a vast array of engineering applications involving porous materials. Such applications include hydrocarbon recovery, cleanup of contaminated aquifers, drying, filtration, and catalysis (1). It is now well understood that transport and capillary phenomena in chaotic pore spaces cannot be explained, let alone predicted, in terms of a single parameter, such as the volume fraction of the void phase. Instead, detailed descriptions of the pore space must be sought that reflect information on local variations in the geometry of pore channels as well as information on the degree of pore space connectivity (2). Accurate 3D representations of the pore space of reservoir rocks are an essential element in the prediction of macroscopic petrophysical properties (e.g., permeability, formation factor, relative permeability) from the microstructure. Such considerations have prompted 3D reconstructions of the pore space 1

To whom correspondence should be addressed.

from data obtained experimentally by serial sectioning of pore casts (3, 4) or X-ray and magnetic resonance microtomography (5, 6). Additionally, 3D reconstructions of reservoir pore space have been obtained using stochastic simulation techniques, commonly based on the truncation of Gaussian random fields (7, 8). Stochastic simulation is capable of producing 3D replicas of the microstructure with specified porosity and two-point correlation function. Such information is readily obtained from 2D high-resolution binary micrographs of reservoir rock samples (9). The low cost and high speed of data generation as well as the ability to overcome present resolution constraints of computed microtomography (ca. 10 µm) are all considerations favoring stochastic simulation as an alternative to experimental acquisition of 3D volume data. By design, statistically reconstructed porous media and their real counterparts possess identical first-order (porosity) and second-order (two-point correlation function) statistics. Stochastic simulation, at least at its present state, is not constrained to match higher order statistics (e.g., three- and fourpoint correlation functions) that have been linked to rigorous bounds on effective properties (10). It is thus relevant to inquire about the extent of statistical equivalence between real and stochastically reconstructed porous media. This question has been addressed empirically by Yao et al. (11), who computed the three- and four-point correlation functions of the pore space of a Vosges sandstone sample. The higher order correlation functions from real and stochastically reconstructed 3D volume data were shown to be in good agreement, lending support to the hypothesis of statistical equivalence. Notwithstanding, it has been recently reported that by varying the simulation technique, model microstructures with statistically indistinguishable porosity, two- and three-point correlation functions can be created that have very different pore morphologies and transport properties (12). This finding casts serious doubt on the appropriateness of even higher order statistics as surrogate measures of the geometry and topology of porous media. Does statistical equivalence between real and simulated porous media imply geometric and topological equivalence? This is a separate question that must be resolved by comparing not only the statistical properties of the two pore spaces but also a number of key geometric and topological attributes. Although stochastically simulated porous media are known to possess a fractal solid–void interface, the surface fractal dimension

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does not vary strongly with either porosity or correlation length and is, therefore, of limited use as a measure of microstructural variability (13, 14). At present, little is known about the relationship between statistical properties commonly used as input to stochastic simulation (e.g., porosity and two-point correlation function) and the geometry and topology of model porous microstructures. From a practical standpoint, such information is useful in assessing the ability of the stochastic method to produce faithful models for a wide variety of reservoir rocks. Furthermore, elucidating the geometric and topological information content of statistical correlation functions is an important step toward developing improved methodologies for simulating the pore structure of diverse porous media. The most rigorous approach to quantitative characterization of the internal geometry of a porous medium is to partition the pore space into discrete and well-defined collection of individual pores. According to Dullien (2), pores can be rigorously defined as portions of the void space confined by solid surfaces and planes erected where the hydraulic radius of the pore space exhibits local minima. The latter regions are referred to as pore necks or throats. Partitioning of the pore space into its constituent pores requires that all local constrictions (i.e., necks or throats) be identified. This task can be accomplished by scanning the 3D microstructure from one or more orientations (15), by morphological thinning (6), or by a gradient search on the morphological skeleton (16). Upon partitioning of the pore space, geometrical information is contained in the pore and neck size distributions. Topological information is rendered in terms of the distribution of pore (or node) coordination number (Z ), defined as the number of necks (or links) associated with each pore (or node). Alternatively, topological information is contained in genus (G) or first Betti number (e.g., 17). The geometric and topological characteristics of a Berea sandstone sample, reconstructed from actual serial section data and also by means of 3D stochastic simulation, have been determined using multiorientation scanning techniques (18, 19). Close agreement was observed between the pore volume, neck area, and coordination number distributions of the real and simulated media. This work provided the only direct evidence in support of geometric and topological equivalence between real and stochastically simulated porous media. Generalizations on the extent of this equivalence, however, cannot be based on results from a single sample. A possible, albeit not very practical, way to proceed would be to directly compare statistical, geometric, and topological properties for a large number of real porous media and their simulated counterparts. Alternatively, one could conduct a systematic investigation of the structure of simulated porous media, aimed at establishing general qualitative and quantitative trends among the geometric, topological, and statistical properties of model microstructures. The aim of the present study is to explore the relationships between the low-order statistical properties (e.g., porosity and two-point correlation function) and the most prominent geometric and topological attributes of stochastically simulated microstructures (pore volume, neck area, and coordination number

distributions, genus). A number of isotropic model pore spaces, possessing different statistical properties, are generated using a stochastic simulation scheme based on the single-level cut of Gaussian random fields (8). The pore volume, neck area, and pore coordination number distributions of each of these media are determined using multiorientation scanning (3, 15). Novel results are obtained that are discussed in terms of the physical length scale (i.e., correlation length) describing the spatial distribution of porosity and in terms of the resolution with which the pore space is reconstructed from knowledge of low-order statistical properties. PORE SPACE RECONSTRUCTION AND CHARACTERIZATION METHODS

Statistical Description and Generation of Model Pore Spaces The goal of stochastic reconstruction is the generation of pore spaces with specified porosity (φ) and correlation function (R Z (u)). These statistical quantities are formally defined in terms of a binary phase function Z (r), conveniently assigned the value of unity if a point r in space is occupied by void or the value of zero otherwise (7): φ = Z (r) R Z (u) =

[1]

[Z (r) − φ][Z (r + u) − φ] . [φ − φ 2 ]

[2]

For isotropic porous media, the correlation depends only on the modulus u of the lag vector u separating two points in space. Methods for measuring porosity and correlation on 2D binary images of the pore space are well established. It should be borne in mind that the spatial resolution of both experimental 2D images and 3D computer reconstructions is always finite. That is, one always deals with a discrete map of the phase function in two or three dimensions, with Z (r) defined on unit elements of the pore space with finite area and volume (i.e., pixels or voxels). Statistical analysis of several hundred images from a large number of reservoir rock samples supports a stretchedexponential model for the correlation function (9) R Z (u) = exp[−(u/λ)ω ].

[3]

Such a model was first introduced by Sinha et al. (20) to describe the correlation structure of rough surfaces. The correlation function introduces a characteristic length scale of the pore space, expressed by the correlation parameter λ in [3] or, in a modelindependent manner, by Z∞ ˜ = R Z (u) du. λ

[4]

0

Given the discrete nature of both pore space photomicrographs

MICROSTRUCTURE OF STOCHASTIC POROUS MEDIA

and 3D simulated microstructures, the distance u between two points in space is originally measured in terms of the number of elementary cubes separating two points in space. Thus, λ or ˜ are originally determined as the number of elementary cubes λ contained in one correlation length. Of interest is the physical ˜ which is readily attached length scale of spatial correlation ( L), to the correlation function, provided that the size (α) of the ˜ elementary cubes is known (e.g., L˜ = λα). The present work focuses on multiple realizations of a number of exponentially correlated, isotropic porous microstructures, generated using a hybrid Fourier transform-linear filtering method (8). This method combines the speed of discrete Fourier transform with the low residentmemory requirements of the linear filtering method of Adler et al. (7) and is well suited for creating large pore space models (NC = 256 voxels per side were used here). Ability to generate such large models is important if a broad range of correlations is to be examined because the con˜ must hold to ensure the statistical homogeneity of dition NC À λ Z (r) (7). Following stochastic reconstruction, all simulated media were operated on with a smoothing filter designed to reduce the roughness of the rock–pore interface. Since no constraint is imposed upon the connectivity of a pore space during its stochastic reconstruction, a certain amount of nonpercolating solid and void phases was found. This is generally a problem if the purpose of stochastic simulation is to exactly replicate a fully connected pore space with a priori prescribed porosity and correlation function. Here, however, the purpose of stochastic simulation is simply to create a number of statistically different microstructures. Accordingly, all geometric and topological parameters, characterizing the connected (percolating) fraction of void space, were related to porosity and autocorrelation measured after smoothing and filtering out all isolated clusters of solid and void phases. Identification of nonpercolating clusters was achieved by clustering the void and solid phases according to the 26- and 6-neighbor rules, respectively, using the cluster multiple labeling algorithm of Hoshen and Kopelman (21). A posteriori statistical description of the model microstructures consisted of determining the connected porosity (φo ) and the correlation function R Z (u). The best-fit parameters (λ, ω) describing the correlation ˜ function according to [3] and the integral correlation scale λ were also computed. The latter quantity was determined by carrying out the integration in [4] up to u = NC . Representative cross sections through some of the model microstructures are depicted in Fig. 1. The corresponding correlation functions are shown in Fig. 2. Typical 3D representations of the void space of simulated microstructures are shown in Fig. 3. Geometric and Topological Characterization by Multiorientation Scanning Partitioning of the pore space into its constituent pores was accomplished using a suite of multiorientation scanning algorithms first developed by Kwiecien et al. (3) and refined by Zhao et al. (15). Only a brief description of these algorithms is given here— the reader is referred to the original publications for additional

325

FIG. 1. Cross sections through 3D statistically simulated microstructures ˜ = 7.01. (b) Model E: (void phase shown in black). (a) Model D: φo = 0.188, λ ˜ = 12.6. (c) Model I: φo = 0.078, λ ˜ = 7.18. φo = 0.187, λ

details. Using a discrete 3D representation of the microstructure in computer memory, the software initially detects local constrictions (necks) along pore space pathways by scanning the microstructure from nine different orientations (three orthogonal and six diagonal orientations). Scanning from each orientation produces a list of potential necks by identifying local minima in the cross-sectional area of overlapping two-dimensional features. The aforementioned analysis unavoidably results in identification of overlapping necks, i.e., necks found during scanning from different orientations, which share common voxels. This problem is resolved by assuming that the smallest of overlapping necks provides the most accurate estimate of a local constriction in the pore space. By combining solid and neck voxels into one phase and clustering the remaining void voxels using the 26-neighbor rule, individual pores emerge as separate clusters.

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FIG. 2. Average correlation functions of the statistically simulated microstructures shown in Fig. 1.

FIG. 3. 3D representations of the void phase in statistically simulated porous media (only one quarter of the computational domain is shown). (a) Model I: ˜ = 7.18. (b) Model D: φo = 0.188, λ ˜ = 7.01. φo = 0.078, λ

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In the course of pore space partitioning, a fraction of the pore space is identified as bound by necks, solid, and the sample boundary. This pore space corresponds to incomplete or “edge” pores that are not counted in the statistics. Since no feature touching the boundary is ever identified as a neck, all necks found are internal and therefore, are included in the statistics. Furthermore, a number of pores are always found that have only one neck associated with them (Z = 1). These so-called “dead-end” pores possess individual volumes that are very small compared to the average pore volume. Their existence may be attributed to random irregularities in the solid–void interface. To simplify the analysis, we eliminated all dead-end pores by assigning their volume to the pore they are connected to and deleting the associated necks from the neck data set. Finally, a large number of pores with Z = 2, connected in series, are also detected in the model pore networks. These are not solely a consequence of the convergent–divergent nature of the pore space, but also a consequence of applying a strict criterion for pore space partitioning, i.e., any local narrowing of the crosssectional area is a potential neck. Pores with Z = 2 in series are linked together, so that only one pore with Z = 2 remains that is bounded by the smallest necks along the pore space channel. A 3D illustration of pores and associated necks is provided in Fig. 4. Upon partitioning of the pore space, quantitative geometrical information is rendered in the form of number frequency distributions of pore volume (VP ) and neck area ( AN ). For purposes of comparison only, equivalent pore and neck dimensions (in number of voxels) are simply computed as D = (VP )1/3 and d = (AN )1/2 , respectively. The mean (µ) and variance (σ 2 ) of

the pore and neck size distributions compactly represent information about the distribution of equivalent sizes. Pore connectivity was measured in terms of the distribution of pore coordination number (Z ), i.e., the number of necks associated with each pore. In relation to coordination number statistics, the connectivity or topological complexity of a pore network may be expressed alternatively in terms of the genus or first Betti number, G = b − n + 1,

[5]

where b and n are the number of branches and nodes in the network. This parameter is a measure of the number of independent paths between two points in the pore space. The value of G may also be interpreted as the maximum number of independent cuts that can be made in the branches of the network without disconnecting it into two separate networks. Clearly, only pores with Z = 3 or higher are valid nodes in a topological context; i.e., G is not related to the number of pores with Z = 2. This is because both b and n decrease (increase) by 1 for each node with Z = 2 removed (added). For networks containing a sufficiently large number of nodes, it is easy to show that the genus per node (G o ) is related to coordination number statistics by (22) G o = hZ 3 i/2 − 1,

[6]

where hZ 3 i is the average coordination number of pores with Z ≥ 3. The equation above was used here to compute the genus G from the coordination number statistics.

FIG. 4. 3D illustration of individual pores extracted from a statistically simulated microstructure (necks are shown in black).

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TABLE 1 Statistical, Geometric, and Topological Properties of Stochastically Simulated Porous Media #

φo

λ

ω

˜ λ

n

hZ 3 i

G

µPSD

σPSD /µPSD

µNSD

σNSD /µNSD

A B C D E F G H I

0.251 0.245 0.240 0.188 0.187 0.178 0.135 0.128 0.078

8.91 6.05 5.07 7.25 13.0 5.19 5.41 6.53 7.15

1.10 1.13 1.15 1.09 1.04 1.14 1.09 1.08 1.01

8.56 5.79 4.86 7.01 12.6 5.01 5.30 6.38 7.18

192 ± 17 436 ± 40 741 ± 18 353 ± 31 106 ± 18 688 ± 86 567 ± 54 352 ± 57 231 ± 39

4.1 ± 0.11 4.1 ± 0.06 4.1 ± 0.09 4.1 ± 0.14 4.2 ± 0.36 4.2 ± 0.15 3.8 ± 0.05 3.8 ± 0.05 3.8 ± 0.09

202 ± 25 459 ± 34 785 ± 45 365 ± 44 119 ± 29 740 ± 68 514 ± 40 321 ± 49 206 ± 40

9.45 8.17 7.30 9.49 11.1 8.49 8.78 9.44 9.79

0.60 0.55 0.51 0.56 1.21 0.54 0.53 0.53 0.54

8.98 7.10 6.07 7.71 9.60 6.12 6.19 7.07 7.01

0.72 0.63 0.56 0.62 0.82 0.55 0.54 0.58 0.57

RESULTS AND DISCUSSION

The main statistical, geometric, and topological attributes of all simulated media are summarized in Table 1. All values listed in this table are averages over five different realizations of each microstructure. The simulated media covered a wide range of porosity values (0.078 < φo < 0.251). The range of correlation ˜ < 12.6. The lengths studied, however, was limited to 4.86 < λ reason is that the number of internal pores in a cube of fixed ˜ (see size (NC = 256 here) decreases rapidly with increasing λ Table 1). A proper analysis of synthetic microstructures exhibiting longer correlations would require the generation of sys-

tems much larger than possible with the computational resources available for this study. Effect of Porosity and Correlation Length on the Pore and Neck Size Distributions The characteristic pore and neck size distributions of the simulated media of Fig. 1 are shown in Figs. 5 and 6, respectively. With no exceptions, the two-parameter Weibull distribution function (23) f (D) = γβ(D − 1)β−1 exp[−γ (D − 1)β ]

FIG. 5. Pore size distributions of the media shown in Fig. 1. Each curve is the average over five realizations.

[7]

MICROSTRUCTURE OF STOCHASTIC POROUS MEDIA

329

FIG. 6. Neck size distributions of the media shown in Fig. 1. Each curve is the average over five realizations.

was found to provide a good fit of the measured data. In all cases, the pore and neck size distributions exhibited significant overlap, as anticipated from visual examination of the microstructures shown in Fig. 3. Evidently, the average pore-to-throat size aspect ratio (µPSD /µNSD ) of the simulated pore spaces is relatively low (see Table 1) and increases slightly with decreasing ˜ From these observations, it appears that the stochastic φo and λ. simulation method employed produces models that might best represent the pore space in well-sorted, clean reservoir rocks. It is worth nothing that the permeability of a wide variety of rock samples has been empirically correlated to the length scale L˜ (9) and to 2D estimates of the average pore size (24). Such observations imply a relationship between the average pore size and the average size of necks that control fluid flow. Such a relationship is observed in statistically simulated porous media that may, therefore, have a much wider applicability, at least insofar as prediction of permeability is concerned. Although the mean and variance of the pore and neck size distributions vary within relatively narrow ranges (see Table 1), a signficant trend is evident from the data. Specifically, the pore and neck size distributions are independent of porosity (compare models D and I), but become significantly broader as the correlation length increases (compare models D and E). A dual ˜ interpretation of this observation is afforded by recalling that λ measures the length of pore space correlation as a multiple of elementary cubes of size equal to the resolution limit of the reconstruction (α). Prior to assigning a size α to each elementary

cube, two model microstructures with the same porosity, but dif˜ may be viewed either as realizations of the ferent values of λ, ˜ at two different resolutions (difsame porous medium (same L) ˜ ferent α), or as realizations of two different media (different L) at the same resolution (same α). If, for example, models D and E (see Table 1) are viewed as different porous media reconstructed using cubes of the same size (α), then a broader distribution of pore and neck sizes is present in the porous medium possessing a greater correlation length (see Figs. 5 and 6). This is easily understood by considering that L˜ is a characteristic length scale of the model pore space and, as such, it sets an upper limit on pore size. The size of the smallest pore or throat that may be found in a simulated porous medium, however, always corresponds to the size of a single voxel. If, on the other hand, models D and E are viewed as realizations of the same porous medium (i.e., same ˜ using cubes of different size α (i.e., αD /αE = λ ˜ E /λ ˜ E ), value of L) then the differences shown in Figs. 5 and 6 must be attributed to differences in spatial resolution. To illustrate this point, the pore size distributions of models D and E were plotted in terms of the absolute pore size (α D) in Fig. 7. This figure reveals that the pore and throat size distributions of the coarser realization (model D) are shifted toward larger sizes. The above observation has important implications for the choice of voxel size in stochastic simulation. Some workers (7, 25) have tacitly assumed that the pore structure can be simulated without loss of information by sparsely sampling the experimental correlation function. This operation results in stochastic

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FIG. 7. Effect of spatial resolution on pore size distributions. Models D and E are taken to represent low- and high-resolution realizations, respectively, of the same porous medium.

FIG. 8. Pore coordination number distributions of the simulated media shown in Fig. 1. Each histogram is the average over five realizations.

MICROSTRUCTURE OF STOCHASTIC POROUS MEDIA

FIG. 9. (a) Correlation between pore size and and pore coordination number and (b) joint probability density function (Model D).

331

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realizations at a resolution coarser than the one of the original photomicrograph (i.e., a value of α greater than the one corresponding to a single pixel in the original binary image). ˜ is used to preserve the This means that a smaller value of λ ˜ resulting in a distortion of physical length scale of correlation L, the number frequency distribution of pore and throat sizes. Qualitatively, such a loss of geometrical information, even when statistical information (i.e., porosity, correlation function) remains intact, is expected from sampling theory. The present study provides, however, a quantitative illustration of this fact in terms of well-defined pore structure parameters. Effect of Statistical Properties on Coordination Number and Genus Exponentially decaying coordination number distributions were invariably observed in all simulated media, as shown in Fig. 8. It was also generally observed that large pores might be located in nodes of all coordination numbers, whereas small pores appear to be correlated with nodes of low connectivity. This is shown in Fig. 9a where the size of each pore in a simulated medium is plotted against the pore’s coordination number and in Fig. 9b as a joint probability histogram. Correlations of pore size to coordination number, similar to the ones reported here, are reasonable to expect in real porous rocks, but their account seems to have escaped the attention of previous

FIG. 10.

investigations. This finding is significant for the development of network simulators patterned after the geometry and topology of stochastically simulated pore spaces (26). Despite the rather broad range of porosity values covered in this study, the average coordination number of simulated media decreases only slightly, from a value of 4.2 for media with φo > 0.178 to a value of 3.8 for media with φo < 0.135, ir˜ In excellent agreement, we have respective of the value of λ. recently reported hZ 3 i = 4.12 for multiple realizations of a stochastically simulated porous medium with φo = 0.185, using a completely different method of pore space partitioning (16). Few measurements of the average coordination number of real porous media are available in the literature for comparison with the theoretical results. Analysis of serial section data from a Berea sandstone sample (φo = 0.178) by Kwiecien (18) has yielded hZ 3 i = 4.6, whereas Lindquist and Venkatarangan (27) have found hZ 3 i = 3.4 for a Fontainebleau sandstone sample (φo = 0.121) using medial-axis analysis of synchrotron X-ray microtomographic data. Neither of these values is at odds with the present findings. A complementary view into the topological characteristics of stochastically simulated porous media is offered by analysis of the genus (G). This parameter was calculated as the product nGo using the data of Table 1, with G o computed from Eq. [6]. As shown in Fig. 10, the data support a relationship of

˜ on the genus of statistically simulated porous media. Solid line corresponds to G ∝ λ ˜ −2 . Effect of correlation parameter λ

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MICROSTRUCTURE OF STOCHASTIC POROUS MEDIA

FIG. 11. Effects of porosity and spatial resolution on the connectivity of simulated media, measured in terms of the genus per correlation length.

˜ −2 . The decrease of the genus with increasing λ ˜ the form G ∝ λ is clearly attributed to the rapid decrease of the number of nodes n (see Table 1). Fewer pores are found in model media with large ˜ because in these media the values of the correlation parameter λ pore space is correlated over distances that are increasingly comparable to the size of the computational domain (NC ). Further elaboration of these results is possible by considering the genus per unit volume (G V ) (28, 29). Under conditions of constant resolution α and sample size NC , G V is proportional to G. In this case, the results of Fig. 10 imply that G V decreases quadratically ˜ Clearly, with the physical length scale of spatial correlation L. G V depends on the underlying physical scale of the pore space and is not an appropriate measure for making topological comparisons among porous media with different correlation scales. In their work with sinters of copper spheres, Pathak et al. (22) considered the nominal particle diameter as the underlying physical scale of the pore space and defined the volume per particle as a natural unit of volume. Accordingly, these authors selected the genus per particle as the relevant parameter for correlating connectivity data from sinters made with particles of different sizes. For stochastically simulated porous media the underlying physical scale is the dimensional correlation length L˜ and the relevant connectivity parameter is the genus per correlation length, ˜ 3 /Nc3 . The latter quantity is plotted against G C ≡ G V L˜ 3 = G λ porosity in Fig. 11, revealing that media with similar values of ˜ but widely different values of porosity, have approximately λ,

the same value of G C . Conversely, media with similar values ˜ have significantly of porosity, but widely different values of λ, different values of G C . These findings suggest that G C depends weakly on porosity, but is rather sensitive to the resolution of the stochastic reconstruction. It follows from the available data ˜ implying that a decrease in the resolution of the that G c ∝ λ, spatial discretization (i.e., using an elementary cube of larger size α) results in a linear decrease of the genus per correlation length. This finding provides a qualitative explanation for the significantly lower value of G V found by Macdonald et al. (29) for a set of 3D volume data from a Berea sandstone sample that was later re-examined by Kwiecien (18). In these studies, L˜ was exactly the same, since both studies dealt with the same sample. Macdonald et al. (29) found G V ≈ 4.88 × 10−7 µm−3 using a pixel size of 14.9 × 13.1 µm, whereas Kwiecien (18) found G V ≈ 8.2 × 10−7 µm−3 using a pixel size of 5.2 × 6.1 µm. CONCLUSIONS

A thorough geometric and topological characterization of a class of stochastically simulated porous microstructures was conducted. Partitioning of the void phase continuum was achieved using a method of multiorientation scanning for the detection of local constrictions (pore necks) along pore space channels. The study succeeded in revealing a number of hitherto unknown relationships between low-order statistical properties

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(porosity and correlation function) and several important geometric and topological attributes of statistically reconstructed porous media (pore and neck size distributions, coordination number distribution, genus). The resolution of the reconstruction process, quantified in terms of the number of elementry cubes corresponding to one correlation length (i.e., effective decay length), was found to have important effects on the breadth of pore and neck size distributions and on the magnitude of specific genus. In addition to clarifying the effects of limited resolution on the geometry and topology of model porous media, these findings provide insight into the capabilities of the most common class of stochastic simulation methods (single levelcut of Gaussian random fields) to reproduce the microstructural variability of natural porous media. In this respect, the present study establishes a general framework for evaluating stochastic simulation techniques, including some which have appeared in the recent literature (12, 30). APPENDIX: NOMENCLATURE

AN b d D G L n NC R Z (u) u VP Z (r) Z

Neck area Number of branches in pore network Equivalent neck size Equivalent pore size Genus or first Betti number Dimensional correlation length scale Number of nodes in pore network Size of computational domain (voxels) Autocorrelation function Dimensionless lag distance Pore volume Binary phase function Coordination number

Greek Letters α λ ˜ λ µ σ φo ω

Pixel or voxel size Dimensionless correlation parameter, Eq. [3] Dimensionless integral correlation length, Eq. [4] Mean of equivalent size distribution Standard deviation of equivalent size distribution Connected porosity Stretch exponent, Eq. [3]

Subscripts NSD PSD

Neck size distribution Pore size distribution ACKNOWLEDGMENTS

The authors gratefully acknowledge the financial support provided by the Natural Sciences and Engineering Research Council of Canada (NSERC). Stimulating discussions with Dr. M. Kwiecien are also greatly appreciated.

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