spe 84296 robust determination of the pore space ... determination... · are part solid and part...

SPE 84296

Robust Determination of the Pore Space Morphology in Sedimentary RocksDmitry B. Silin, SPE, Lawrence Berkeley National Laboratory; Guodong Jin, SPE, UC Berkeley; andTad W. Patzek, SPE, UC Berkeley / Lawrence Berkeley National Laboratory

Copyright 2003, Society of Petroleum Engineers, Inc.

This paper was prepared for presentation at the SPE Annual Technical Conferenceand Exhibition held in Denver, Colorado, U.S.A., 58 October 2003.

This paper was selected for presentation by an SPE Program Committee follow-ing review of information contained in an abstract submitted by the author(s).Contents of the paper, as presented, have not been reviewed by the Society ofPetroleum Engineers and are subject to correction by the author(s). The material,as presented, does not necessarily reflect any position of the Society of PetroleumEngineers, its officers, or members. Papers presented at SPE meetings are subjectto publication review by Editorial Committees of the Society of Petroleum Engi-neers. Electronic reproduction, distribution, or storage of any part of this paperfor commercial purposes without the written consent of the Society of PetroleumEngineers is prohibited. Permission to reproduce in print is restricted to an ab-stract of not more than 300 words; illustrations may not be copied. The abstractmust contain conspicuous acknowledgment of where and by whom the paper waspresented. Write Librarian, SPE, P.O. Box 833836, Richardson, TX 75083-3836,U.S.A., fax 01-972-952-9435.

AbstractWe present a new robust approach to study the mor-phology (shapes and connectivity) of the pore space of asedimentary rock. Our approach is based on the long-established, fundamental concepts of mathematical mor-phology. In particular, we propose an efficient and stablealgorithm which distinguishes between the “pore bodies”and “pore throats,” and establishes their respective vol-umes and connectivity. Our algorithm is extensively testedon the 3D digital images of computer-generated and nat-ural sandstones. The algorithm tests on a pack of equalspheres, for which exact results can be verified visually,confirm its stability. Computer-generated pore space im-ages are used to investigate the impact of image resolutionon the algorithm output.

Presently, the proposed algorithm produces a stick-and-ball diagram of the rock pore space. One of distinctivefeatures of our approach is that no image thinning is ap-plied. Instead, the information about the skeleton is storedthrough the maximal balls associated with each voxel.These maximal balls retain information about the entirepore space. Comparison with the results obtained by athinning procedure preserving some topological propertiesof the pore space shows that our method produces morerealistic estimates of the number and shapes of pore bodies

and pore throats, and the pore coordination numbers.Based on the information about the maximal ball distri-

bution, we simulate mercury injection and compute a di-mensionless drainage capillary pressure curve. We demon-strate that the calculated capillary pressure curve is a ro-bust descriptor of the pore space geometry and, in particu-lar, can be used to determine the quality of computer-basedrock reconstruction.

Introduction

Fluid transport inside a permeable rock is determined bythe void space geometry and connectivity, and the solidsurface/fluid chemistry. The ever-changing distribution offluids in the pores of a gas- and oil-bearing rock must beunderstood to develop a successful hydrocarbon recoveryprocess. The process-dependent redistribution of reser-voir fluids during production and injection determines howmuch of the initial hydrocarbons will be recovered and howmuch will be left trapped. Although the length-scale of anoil field is measured in kilometers, the ultimate successof an oil&gas recovery scheme is the net result of count-less displacement events at a scale measured in micronsor nano-kilometers. Recent advances in micro-imaging ofnatural rocks, combined with advances in pore-networkflow modeling, allow researchers and engineers to gain abetter understanding of pore-level displacement mecha-nisms. In particular, credible predictions are now possibleof the impact of the rock wettability and fluid propertieson the relative permeabilities and capillary pressures, aswell as on the trapped oil and gas saturations.

One of the difficulties in studying the microscopicstructure of a porous medium is the absence of an elegantmean-field theory, such as the multiple-continua model.The microscopic rock models require storage and process-ing of huge amounts of data to characterize a tiny pieceof rock. Existing methods of extracting pore networksfor flow modeling from the microscopic rock images arecomputationally intensive and incorporate numerous ad-

2 D. B. SILIN, G. JIN, AND T. W. PATZEK SPE 84296

justable parameters. For example, a 5-micron-resolutionimage of a rock cube 2.5-millimeters on the side consists of125 million voxels. In comparison, a reservoir simulationwith an equal number of gridblocks would be a challengeeven for a high performance parallel computer.

A microscopic image of rock is a three-dimensionalarray of cubic atoms or voxels. Each voxel is assigneda nonzero value if it is attributed to the pore space andzero otherwise. A group of neighboring voxels can make aloosely-defined “pore throat” or a “pore body.” In model-ing, the pore throats control fluid flow, whereas the porebodies provide fluid storage. Even at this level, rock de-scription is approximate. First, the number and size of thevoxels in an image are limited by the resolution and view-ing angle of imaging device. Second, the image itself isoften an interpretation of the reflection, absorption, atten-uation, and diffraction patterns of electromagnetic waves.Each such interpretation is a solution of a series of inverseproblems. The inversion errors are then combined with er-rors inevitably produced by assignments of the voxels thatare part solid and part void space. The representative-ness of a digital image, and the determination of minimumresolution necessary to image a given rock adequately, areissues yet to be appropriately addressed in the literature.

In this context, attempts to develop efficient pro-cedures of computer reconstruction of sedimentary rocksseem to be promising.1,2 If such a procedure adequatelyreproduced a rock, it would be possible to create the corre-sponding digital image with an arbitrarily high resolution.One has to bear in mind, though, that verification of sucha procedure requires comparison with the digital images ofnatural rocks. Thus, indefinite refinement of resolution ofthe computer-generated rock images may be unrealistic.

The literature on processing images of natural rockscan be split into two periods. The first period is character-ized by the development of basic concepts of mathematicalmorphology. A systematic presentation of early develop-ment and results of the theory of mathematical morphol-ogy was given in the monographs by G. Matheron in 19753

and J. Serra in 1982.4 The concepts of skeleton and me-dial axis were inspired by early works of Th. Motzkin5

and J. Blum,6 and played a pivotal role in all subsequentinvestigations.

More recently, the revolutionary progress in imagingtechniques7–10 and computing power induced a new waveof morphological studies. Pore-network modeling made itpossible to gain the fundamental understanding of multi-phase fluid flow in porous media. In conjunction with im-age analysis, it became possible to build models of porousmedia capable of predicting the fluid transport propertiesof rock, so important in oil industry.11–18 Surveys of pore-network modeling are given in Refs.19–22

Extraction of pore networks from the microscopic 3Dimages of rocks still poses challenging problems, even ifwe neglect errors introduced by image processing and in-terpretation, and concentrate purely on geometric issues.Many computer “skeletonization” algorithms are based onthinning methods.8,9, 23–29 Thinning relies on removing the

“redundant” elements of an image, while preserving certaintopological properties of the entire pore space. Below, wediscuss some unwanted side-effects of this approach. Thedifficulties of porting the basic topological concepts,30 suchas connectivity, Euler-Poincare characteristic, etc., to dis-crete digital images are well known.31 Intuitive extensionof topological analysis of digital images may be insufficientfor rigorous analysis. Tests of thinning algorithms on sim-ple computer-generated images show that a refinement ofthe resolution can lead to less accurate results. A differ-ent method of skeletonization, based on a complete catalogof shape primitives for 2D and 3D objects, was developedin Refs.24,26,32,33 Although the latter method has beenextended to higher dimensions, its stability with respectto the starting point or image rotation has not been in-vestigated. Characterization of the pore space geometrywithout application of thinning algorithms was proposedin Refs.,34–36 where some elements of the algorithms pro-posed here were developed. In particular, the characteri-zation of skeleton as the set of centers of the maximal ballswas employed. In Refs.,37–40 the capillary pressure curveswere computed without pore network extraction. We alsoelaborate on this approach.

In this work, we assume that an appropriate digitalimage of rock has been obtained, and we concentrate ex-clusively on the analysis of the geometry of this image.The image itself may have originated from computer to-mography, a computer model of rock deposition, or anyother source. We are no longer concerned with the issuesdiscussed in the two previous paragraphs. Instead, we dis-cuss how a digital image can be analyzed, and investigatehow the results of this analysis are affected by variationsof the input data. Our approach is based on the tools andconcepts of mathematical morphology, as well as on anefficient implementation of the object-oriented algorithmdesign. Thus, all computer code used in this research hasbeen written in C++.

In the next section, we briefly overview the salientconcepts of mathematical morphology. Then we describehow these concepts are implemented as objects in the al-gorithm. In the last section, we illustrate our approachwith computations performed for digital images of bothcomputer-generated and natural rocks.

Basic concepts of mathematical morphologyThe fundamental concepts of mathematical morphology,such as skeleton, medial axis, thinning, etc., are widelyused in the literature on image processing. These conceptsare usually illustrated in two dimensions where, for exam-ple, the medial axis of a channel-like structure is a curveconnecting the points “in the middle” of that channel. Thefact that in 3D the picture is tremendously more compli-cated is usually de-emphasized. However, it can be demon-strated that the skeletonization of a 3D channel, which isa graph of a set-valued mapping, can lead to extremely ir-regular structures.41,42 Normally, in image-processing lit-erature it is implicitly assumed that in natural rocks such“exotic” structures are hardly possible and can be ignored.

SPE 84296 ROBUST DETERMINATION OF THE PORE SPACE MORPHOLOGY. . . 3

Following this tradition, we also assume that the construc-tions below are all legitimate and well defined.

To avoid ambiguity, we explain the basic mathemat-ical morphology concepts used in this article. As a refer-ence, we use the book by Serra,4 which apparently is thefirst systematic text on the subject. First, let us define theskeleton of a set. Sometime, this concept is confused withthe medial axis. Strictly speaking, the medial axis is asubset of the skeleton, and they are not equivalent to eachother. Each point in R

3 is characterized by three Cartesiancoordinates (x, y, z). For brevity, with each point we asso-ciate a position-vector r = (x, y, z). The distance betweentwo points is then defined as

dist(r1, r2) =√

(x1 − x2)2 + (y1 − y2)2 + (z1 − z2)2

A ball of radius R centered at r0 is a set of all points rsuch that dist(r, r0) ≤ R. We will denote such a ball byBR(r0). Let M be a set in R

3. Usually, M will be the porespace of a rock. The complementary set will be denotedby M c. A ball BR(r0) is called a maximal ball if, first, itis a subset of M , and, second, it is not contained in anyother ball, which is a subset of M . A point r from M is apoint of the skeleton S(M) if it is the center of a maximalball. Clearly, for a given center r0 ∈ S(M) such a ballis unique. Any ball touching M c at two or more pointsis a maximal ball and, therefore, its center is an elementof the skeleton. The inverse, in general, is incorrect. Thedefinition of skeleton is dimension-independent, thereforea 2D version is straightforward.

For simple geometric objects in 2D, the above-formulated definition describes structures, which resemblewhat one intuitively would call a skeleton, see Fig. 1. Formore complicated 2D objects, the skeleton can look a bitbizarre. A rigorous application of the definition of skeletonresults then in the structure shown in Fig. 2 as the dashedlines. The vertical “bones” of the skeleton correspond tothe small perturbations on the upper boundary of the fig-ure, that vanish to the right. In 3D, the skeletons of simpleconvex bodies may have complex geometry. In Fig. 3, theskeleton of a rectangular parallelepiped makes a branching“film” cutting the pore space into parts disconnected fromeach other.

Fig. 1 - The skeleton of a simple 2D object.

Due to the above-mentioned side-effects of skeletongeneration, methods of thinning were developed to elimi-nate the redundant parts, like the branches leading to the

Fig. 2 - The skeleton of a 2D object with perturbed boundary.

Fig. 3 - The structure of the skeleton of a simple 3D object,such as a rectangular parallelepiped, may not resemble whatone intuitively calls a skeleton.

corners in Fig. 2. Thinning is usually based on the sequen-tial applications of elementary operations in a hit-or-miss4

transformation. Without going into details, this transfor-mation finds a domain near the boundary of a set, whichcan be eliminated. Iterative elimination of the redundantdomains found by this transformation thins the set andleads to a structure that can be called the skeleton. Un-fortunately, the definition of hit-or-miss transformation in-cludes a pair of auxiliary sets that must be defined in such away that the transformation leads to a desired result. Theproper choice of these sets is a very difficult operation.Examples4 of hit-or-miss transformations are in 2D, andapplicability of the results to an irregular 2D set seems tobe low. The difficulties related to developing a reasonablethinning algorithm based on a hit-or-miss transformationare only aggravated when one deals with an irregular 3Dstructure, such as a digitized image of a porous rock.

There is a group of algorithms for thinning 2D and3D images based on the idea of sequential elimination ofindividual voxels, where each voxel is removed only if thisoperation preserves certain “topological” properties of theset. Such properties usually include connectivity. Moresophisticated algorithms27 also check values of topologicalinvariants, e.g., Betti numbers. The elimination procedureis iterated until no further voxel can be removed. The re-maining set of voxels is then declared as the medial axis.Seemingly, such an algorithm is based on the fundamentaltopological properties, and should lead to robust sensibleresults. However, severe difficulties exist that strongly im-pact the efficiency of this algorithm. Originally, Betti num-


bers were defined for differentiable manifolds with bound-aries, see e.g., Ref.43 The continuum Betti numbers cannotbe automatically applied to a cubic cut-off of the digitizedrock pore space, and their “discretizations” are often am-biguous. Even if sensible discrete topological invariantscan be defined, their recalculation at each iteration of thealgorithm can be a daunting computational task. Finally,the verifiability and reproducibility of the results is prob-lematic: a different orientation of the image, and multiplechoices of the order in which the voxels are eliminated canalter the result significantly. For example, let us thin a 2Dset in Fig. 4, assuming that two pixels are connected ifthey have at least a common vertex and the pixel-removalprocedure must preserve connectivity. If we start with re-moving sequentially pixels 3, 1 and 5, then pixels 2 or4 cannot be removed without breaking the connectivityof the remaining set. Alternatively, sequential removal ofpixels 1, 5, 2 and 4 leads to a set consisting of the sin-gle pixel 3. The two thinned sets complement each other.In other words, applying the same principle and preserv-ing connectivity, we obtain two mutually exclusive results,depending on the starting pixel. This conclusion appar-ently holds true when the skeletonization is performed us-ing shape primitives.33 Using such primitives for measur-ing lengths of the skeleton elements will not lead to robustresults because extraction of the skeleton by thinning isnonunique.

3

1 2

4

5b)

3

3

1 2

4

5a)

2

4

Fig. 4 - The result of a thinning procedure strongly depends onthe selection of the stating point: the remaining set of pixels infigure a) is complementary to the result in figure b), althoughboth are derived from the same original configuration.

A few other concepts are often used in digital imageanalysis. One of them is the just-introduced connectivity.In a 3D image there are 3 possible definitions of neighborvoxels. By the first definition, two voxels are neighbors ifthey have a common face. By the second one, neighborvoxels must have at least one common edge. Finally, bythe third definition, two voxels are neighbors if they have

a common vertex. According to the maximum numberof neighbors of a voxel, the voxel connectivity is called 6-connectivity (first definition), 18-connectivity (second defi-nition), and 26-connectivity (third definition). Two voxelsare connected if they are end-points of a chain of neighborvoxels.

The definitions of pore bodies and pore throats aremore complicated. Intuitively, in a fluid-bearing rock, porebodies are the larger pore space openings where most ofthe fluids is stored. To a large extent, pore bodies de-termine the rock porosity. Pore throats are the narrowgateways that connect the pore bodies and determine therock permeability. Although these two concepts are intu-itively clear, no commonly-accepted rigorous definition isavailable. Some authors define pore bodies as junctions ofthree or more branches of the medial axis. As one can inferfrom the example in Fig. 3, even for a non-discretized 3Dset such a definition makes little sense. In 2D, a structuresimilar to that shown in Fig. 2, by this definition, wouldhave 4 distinct pore bodies with the coordination numbersequal to 3. The coordination number of a pore body is thenumber of pore throats connecting it to other pore bodies.Clearly, the coordination number depends on how the porebody and its pore throats are defined. Image discretizationimposes more ambiguity and amplifies unpredictability ofthe results.

Pore body and pore throat detection algorithmOne of the weak points of thinning algorithms is that oncea voxel is deleted from the set, all information related tothis voxel is removed as well. It is impossible to undo thesedeletions and their cumulative effect can distort the result.In addition, the dependence of the result on which voxel isto be removed first when there are multiple choices, makesprocessing images by parts impossible.

Here we propose a procedure in which the voxels arenot removed, but instead the information is stored in anaggregate format. In the interior part, far enough from theboundary, the result of image analysis depends neither onthe orientation of the image nor on the selection among themultiple choices of voxels to be removed. The proposed al-gorithm consists of several steps. Using an object-orientedapproach, this algorithm can be coded relatively easily.

At this stage, our algorithm does not produce a com-plete pore-network ready for single- or multi-phase flowsimulations.13–15,17 However, it robustly characterizes thepore connectivity via a stick-and-ball representation.

Building voxel objects. A voxel object is one of the ba-sic elements of our algorithm. It corresponds to a voxel inthe image, therefore it has three coordinates. Further, eachvoxel “knows” its maximal radius, i.e., the radius of thecorresponding maximal ball. In addition, each voxel hasas properties two lists of pointers to other voxels. Initially,both these lists are empty, and their roles are explained be-low. In all calculations, we define unit length as the linearsize of one voxel.

First, the algorithm calculates the maximal ball radius


for each voxel as follows. Starting from a zero-radius ball,i.e., the voxel itself, the radius of the ball is incrementedby one step until the ball hits a solid-phase voxel. A roughestimate of the complexity of such a search is equal to thetotal number of voxels in the pore space times the numberof voxels in a maximal ball that can be inscribed in thepore space. Clearly, the algorithm complexity increasesrapidly with refinement of the image resolution. Also notethat discrete versions of the balls of small radii have shapesthat only remotely resemble spheres, see Fig. 5.

a) b)

c)

Fig. 5 - A discretized ball or sphere mimics a spherical shapeonly for a sufficiently large radius: a) R = 1; b) R =

√3; c)

R =√

13.

After all maximal balls have been constructed, someof them are subsets of the others, as in Fig. 6. Notethat there are configurations, which are only possible fordiscrete sets. Both included balls and the correspondingvoxels carry information about the pore space that is al-ready stored in the including ball or balls. Therefore, thesecond step in the algorithm is the removal of the includedballs. To perform this operation, it is convenient to storeall voxels in a sorted list,44 where the sorting is by themaximal radius. In addition, to facilitate the search, anobject called reference table is used. A reference table in-cludes an array having the same dimensions as the image.An entry of this table with coordinates (i, j, k) is a pointerto the pore voxel V centered at (i, j, k) or a zero pointer ifthe voxel with coordinates (i, j, k) is in solid phase. To findall the maximal balls included in a given voxel centered atV0 = (i0, j0, k0), and with the maximal ball radius R0, onechecks all the voxels pointed by the reference table entriesat (i, j, k) such that

dist (V,V0)2 ≤ R2

0

Squared values are used in the inequality above to avoidoperations with floating-point numbers and the relatedround-off errors. As an included maximal ball is detected,the corresponding entry in the reference table is set to azero pointer. After this operation is finished, the list of

voxels is updated according to the remaining nonzero en-tries in the reference table.

It is interesting to note that in a linear space of realnumbers a ball of radius R1 centered at c1 includes a ballof radius R2 centered at c2 if and only if

R2 + dist(c1, c2) ≤ R1

This is not true for digital images: there are exampleswhere the inclusion holds true, but the inequality fails.This is another paradox of discretization.

A

Fig. 6 - Voxel A is a zero-radius maximal ball included in twoother maximal balls of radius 1. Such a paradox is possibleonly in discrete images.

According to our definition, the set of centers of thevoxels remaining in the list after removal of the includedmaximal balls is the skeleton of the pore space. Clearly,the order, in which these voxels are calculated, does notmatter. If a large image must be processed, it can be splitinto parts, each part can be analyzed separately, and theresults can be merged into the skeleton of the whole porespace. Therefore, a computer with a modest memory sizecan be used to process large image.

Examples in Figs. 1–3 suggest that only finding theskeleton is insufficient. To characterize the pore space, the“redundant ribs” corresponding to the corners and similarstructures should be removed. Here, we use the fact thatthe voxel objects already “know” the radii of their maximalballs. Note that the maximal radius decreases along a ribleading into a corner. Therefore, a hierarchy of the skeletonvoxels needs to be established. We distinguish betweenmaster and slave voxels. If the maximal balls of two voxelsoverlap, the one with the larger radius is called master andthe other one is called slave. Thus, the third step of thealgorithm is search for the slaves of each voxel. Use of thereference table facilitates this task. First, for each currentvoxel a domain where potentially overlapping voxels canbe located is determined from the radius of the maximalball of the current voxel. Then, using the pointers providedfor these locations by the reference table, the slave voxelsare found. After finishing this operation, each voxel’s twolists of pointers include, respectively, the pointers to themasters and slaves of this voxel. For small radii, for whichthe balls have bizarre shapes (see Fig. 5), the hierarchyis built slightly differently, depending on the connectivityconvention adopted in the analysis. Zero-radius voxels aredeclared slaves of all neighbor voxels with positive radii.


After building the hierarchy, there are many voxelswhich are masters and slaves at the same time. For ex-ample, Fig. 7 shows a part of rectangular pore space (thethick solid “C”), and the corresponding part of the skeleton(the dashed lines). It is a simplified 2D illustration, wherevoxels A, B, and C are shown as circles whose radii arethe respective maximal radii. By definition, voxel B is amaster of voxel C and a slave of voxel A. In this case, voxelA characterizes the size of the opening, whereas voxel Bcharacterizes the narrowing into the corner, towards voxelC. Therefore, to characterize the pore, one retains only thevoxel with the largest maximal ball. This characterizationis done through a four-step enhanced hierarchy procedure,whose purpose is to retain only the largest master voxels.First, the master voxels, which themselves have no mastersare selected. These voxels correspond to the local maximaof the maximal ball radii. In Fig. 7 such a voxel is A. Sec-ond, for all other voxels the lists of masters are cleared. InFig. 7, after such an operation voxel B will “forget” thatA is its master, and voxel C will “forget” about its twomasters A and B. Third, using a depth-first type clustersearch algorithm,45 each “super” master seeks all affiliatedslaves. For example, in Fig. 7, voxel A will first find voxelB as an “immediate” slave, and then it will find voxel C asa slave of voxel B. This is an implementation of the prin-ciple “a slave of my slave is also my slave”. Fourth, the“intermediate” master voxels, like voxel B, are demoted toslaves and a new list of master voxels is created. Again, inFig. 7, after this operation only voxel A will be a mastervoxel, and all other voxels will be its slaves.

A

B

C

Fig. 7 - Voxel B is a slave of voxel A, and a master of voxelC at the same time. After a rearrangement of the hierarchy,voxel C will have only one master, voxel A.

Once the procedure described in the previous para-graph is carried out, one obtains the following picture. Alist of master voxels, corresponding to the local maxima ofthe radii of maximal balls, is created. In a neighborhood ofa voxel from this list, the maximal inscribed ball radii areequal to or less than the maximal ball radius at this voxel.If such a master voxel, say V ∗, is isolated, i.e., does notoverlap with any other master voxel, then in all directionsfrom the center of V ∗ the pore space narrows. Thus, anisolated master voxel can be associated with a pore body.The volume of the union of the maximal balls of all slaves

for whom V ∗ is the only master, may be called the porebody volume. In Fig. 7, all the shown balls will be partsof the pore body associated with voxel A.

Sometimes, two master voxels may overlap. In sucha case, the respective maximal radii must be equal, be-cause otherwise the voxel with the smaller maximal radiuswould have been declared a slave of the other one by theprocedure described above. In Fig. 7, if the pore spacewere continued to the right, voxel A would have a neigh-bor master voxel of the same radius. In such a case, it isnatural to merge the respective pore bodies into one porebody. Thus, by virtue of our construction, a pore bodyis associated with an isolated master voxel or with a clus-ter of master voxels in case of overlapping. The ball (orone of the balls) defining a pore body will be a ball in thestick-and-ball diagram we are about to construct.

Now, let us check the slave voxels. Some of them,having only a single master, are parts of the respectivepore bodies. However, there are slave voxels having twoor more master voxels. Clearly, if a slave voxel, let uscall it V∗, has exactly two masters, then the pore bodiescorresponding to these master voxels are connected via V∗.For example, in Fig. 8, both voxels B and C are slaves ofboth voxels A and D. Hence, A and D are connected via Band C. It is natural to call the union of all maximal ballsassociated with the voxels connecting two given mastervoxels a pore throat. In the stick-and-ball diagram, a porethroat connecting two pore bodies is depicted as a straightline segment connecting the centers of the balls associatedwith these pore bodies.

CB DA

Fig. 8 - Both voxels B and C have voxels A and D as theirmasters. B and C can be called the pore throat connectingthe pore bodies related to A and D.

Algorithm verification. To verify a physical model, nu-merical or analytical simulations should be compared withfield or laboratory measurements. To verify a numericalalgorithm, a test problem with a known analytical solu-tion can be considered, and the numerical result can becompared with the exact one. In image analysis, verifica-tion of an algorithm, like the one described above, is nota straightforward problem. On one hand, the test imagemust have sufficiently complicated geometry with internalopenings and connections between them. On the other,


the image should be small and simple so that the resultsof computations will be transparent and verifiable. Thesetwo requirements are almost always mutually exclusive.

To verify our algorithm, a computer-generated pack-ing of equal spheres was used. The spheres are packedin layers. Every layer where all spheres touch each otheris sandwiched between two layers where each sphere istangential to four spheres from one layer below and fourspheres from one layer above Fig. 9a. The whole packis shown in Fig. 9b, and its porosity, if the stencil 9a isapplied indefinitely, is about 26%.

a) b)

Fig. 9 - a) A stencil of the sphere pack; b) the whole spherepack.

In some cases, the the results can be reasonably goodif the sphere pack is aligned with the coordinate axes but,still, the algorithm may fail in other situations. Therefore,to test our algorithm, the whole pack of spheres was ro-tated, as in Fig. 10. Then, a part of the image cut by acube was analyzed. The pore space of this part is shown inFig. 11. The constructed stick-and-ball diagram is plottedin Fig. 12.

Fig. 10 - Analysis of a rotated sphere pack is a more chal-lenging problem.

Additional verification of the algorithm can be per-formed by analyzing the shapes of pore bodies and porethroats detected by the procedures described above. Since

Fig. 11 - The pore space of the sphere pack in Fig. 10.

Fig. 12 - Stick-and-ball diagram of the pore space in Fig. 11.

these pore bodies and throats are only conventional termsto characterize the geometry of the pore space, there is noclear boundary where a pore body ends and a pore throatbegins. Thus, no quantitative criterion of whether the porebody and pore throat shapes are detected correctly can beapplied, and the shape analysis can be performed only vi-sually. In Fig. 13a, two views of a pore throat are shown.Only the boundary voxels are displayed, so only “the walls”of the pore throat are seen. Nevertheless, the similaritywith the channel formed by the three tangential spheresin Fig. 13b is clear. Similarly, Fig. 14a shows two pro-jections of a pore body. Again, only the boundary voxelsare displayed. The detected pore body structure can becompared with the shape of the void space surrounded bythe six spheres shown in Fig. 14b. Such a combination ofspheres is periodically repeated in Figs. 9b and 10.

The pore throats and pore bodies which are at theboundary of the image are cut and their shapes may beirregular. This fact has to be taken into account when theentire image is analyzed by parts and the partial resultsare later merged together: the parts into which the wholeimage is divided must overlap. The size of this overlapdepends on the sizes of grains and pores, and on the imageresolution.

The sphere-pack in Fig. 10 is computer-generated, andits image can be digitized at an arbitrarily high resolution.


a)

b)

Fig. 13 - a) Three views of a pore throat. The pore throatshape resembles the opening in the three tangential spheres,b).

a)

b)

Fig. 14 - a) Two views of a pore body. It is clearly seen thatthe pore body is an opening confined by six spheres in shownb).

Images of the same pack obtained at different resolutionshave been analyzed. The computer time for most opera-tions of the algorithm grows quadratically with the imagerefinement, Fig. 15. The most expensive task is compu-tation of the maximal balls. We remark that the currentscaling of computation time with image size exceeds the-oretical estimates. Further optimization of the code willreduce the number of computations.

It is quite difficult to specify the rigorous requirementsfor a “sufficient” resolution of a rock image. However, nu-merical experiments with images of the various computer-generated sphere-packs show that for an adequate descrip-tion of the pore space, the resolution should be at least oneorder of magnitude finer than the representative sphere ra-

dius. Therefore, for example, the resolution of 5 micronsmay miss fine features of a rock with the grainsize below50 microns. For imaging chalks or diatomites a super-highresolution imaging technique, such as the one proposed inRef.,10 is needed.

104

105

106

107

108

100

101

102

103

104

Image size, voxelsC

PU

tim

e, s

ec

HierarchyMasters & SlavesPore bodies & LinksVoxel setupMaximal balls

Fig. 15 - A log-log plot of the computational time in secondsfor most image-processing operations versus the image size invoxels.

Next, we present an example of application of theabove-described algorithm to an image of Fontainebleausandstone kindly provided to us by Schlumberger. Theporosity of the sample is about 17%. From the whole im-age of 512 × 512 × 512 voxels, a portion of the size of 200voxels in each dimension was selected. The image of thepore space of the analyzed sample is shown in Fig. 16.First, the maximal balls of all voxels are found and dis-played in Fig. 17. Note that these balls are shown asspherical objects of respective radii, not as discrete balls,examples of which are displayed in Fig. 5. For this reason,the flat surfaces cut by the boundary of the image, Fig. 16have round shapes in Fig. 17. The next step of the algo-rithm is finding the master voxels. The result is presentedin Fig. 18. The displayed spheres have the radii of themaximal balls corresponding to the master voxels. Someof these balls may overlap. When the distance between twomaster voxels is small enough, the size of the throat con-necting their associated maximal balls is comparable withthe radius of the balls∗, and these voxels are gathered ingroups. Also, our classification of “masters–slaves” doesnot account for isolated voxels. There is a probability thatsuch voxels are isolated due to the insufficient resolutionof the image or the pore space may have disconnecetd cav-ities. In most cases, the isolated voxels are singletons, i.e.,their maximal radii are zero. Fig. 19 shows the result ofgrouping and “clouds” of isolated singletons.

Finally, Fig. 20 shows the grouped master voxels, the∗We remind the reader that all overlapping maximal balls corre-

sponding to the master voxels must have equal radii.


Fig. 16 - The pore space of a 200 × 200 × 200 image ofFontainebleau sandstone. The porosity is about 17%.

Fig. 17 - Maximal balls of the image presented in Fig. 16.

singletons, and the links between each pair of the mastervoxels. We remind here that each master voxel stores de-tailed information about the respective pore body throughthe list of all slave voxels. Similarly, each link is presentedaccording to the algorithm as a list of slave voxels havingtwo masters. Such a list also provides detailed informationabout the actual geometry of the pore throat, cf. Figs. 13and 14.

The lists of voxels for each pore body and each porethroat provide information about their volumes and shapesand can be used for pore-network modeling of the transportproperties of the rock.

The dimensionless capillary pressureWhen the pore space is shared between two immiscible flu-ids in equilibrium, the wetting fluid occupies the cornersof large pores and small pores, while the non-wetting fluidoccupies the central parts of the pores it invaded. An inter-

Fig. 18 - Master voxels of the image presented in Fig. 16.

Fig. 19 - Grouped master voxels and “freelance” singletonsof the image presented in Fig. 16.

face between these two fluids is a surface whose curvature isdetermined by the capillary pressure. Although in realitythe fluid interfaces are not spherical, they can be approx-imated by spheres. The information about the maximalballs, obtained with the algorithm discussed above, can beused to reconstruct a capillary pressure curve. A proce-dure similar to the one described here was considered inRef.40

Let us assume a capillary pressure level and calculate themean radius of curvature† corresponding to an equilibriuminterface at this pressure. The corresponding sphere radiuswill be called the threshold radius. It is natural to assumethat all the maximal balls whose radii exceed the currentthreshold radius are filled with the non-wetting fluid. Theremaining part of the pore space is filled with the wet-ting fluid. Calculation of the volumes occupied by eachfluid yields the fluid saturations corresponding to a given

†For a sphere, the mean radius of curvature = 1/2 of the radiusof the sphere.46


Fig. 20 - The reconstruction of the pore structure and theconnectivity of the pore space in Fig. 16.

capillary pressure. Of course, such a calculation is ap-proximate for at least two reasons. First, keeping in mindthe bizarre shape of “spheres” of small radii, cf. Fig. 13,one can expect a systematic error that is especially largefor high capillary pressures. More accurate calculations athigh capillary pressures require higher image resolutions.Second, we do not ask how the non-wetting fluid reachedits equilibrium locations. In other words, there can beenclaves of non-wetting fluid, which are entirely discon-nected (trapped). This issue can be addressed by mod-eling an invasion-percolation process using cluster searchalgorithms.45

We begin by applying the approach described in theprevious paragraph to the images of a tilted pack of equalspheres shown in Figs. 10–11. Since these images can beobtained at an arbitrarily high resolution, it is possible toinvestigate how changing the resolution affects the result.In Fig. 21, we present the dimensionless capillary pressurecurves obtained at different resolutions. The abscissa isthe wetting fluid saturation and the ordinate is our dimen-sionless capillary pressure, defined as the inverse curvatureradius in voxel units. To calculate the physical capillarypressure, one has to rescale the computed values accordingto the actual resolution of the image and twice the coeffi-cient of surface tension in the Young-Laplace equation.

Five capillary pressure curves have been obtained atdifferent resolutions and are presented in Fig. 21. To makethese curves comparable, an additional normalization hasbeen applied, so that the curvatures are in the same physi-cal length units for all curves. In the middle part of the sat-uration interval, the results are remarkably similar. How-ever, at low saturations, at which the wetting fluid residesin the narrow pores and in the corners of the large pores,the results obtained at different resolutions differ signifi-cantly. At saturations close to unity, the non-wetting phasebecomes disconnected, and can be trapped. Therefore, inreality, the capillary pressure curves would not smoothlycontinue up to S = 1 as it is shown in Fig. 21, but in-

stead abruptly go to zero at a certain endpoint saturationS = S∗ < 1.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

50

100

150

200

250

Wetting fluid saturation

Norm

ali

zed

dim

ensi

on

less

cap

illa

ryp

ress

ure

Fig. 21 - These dimensionless capillary pressure curves vswetting fluid saturation were obtained from the images of thepore space of an equal-sphere pack (Fig. 16) at different res-olutions.

After testing and verification, the procedure describedabove was applied to analyze images of Fontainebleausandstone. The images of four samples with averageporosities between 12% and 21% were analyzed. Each im-age has dimensions of 512 × 512 × 512 voxels. For anal-ysis, each image was split into parts. It is interesting tonote that the variations of porosity between different partswithin each sample were large, up to 100%. For each part,the capillary pressure curve was calculated, and the resultswere compared. This comparison was performed to inves-tigate stability, and, therefore, credibility of the capillarypressure curves. Significant differences among the curvesobtained from the different parts of the image would meanthat our method has no predictive capability, at least forthe images of the sizes and resolutions considered.

It turned out that the dimensionless capillary pres-sure calculation is stable with respect to the porosity vari-ation within each image. In addition, the capillary pressurecurves obtained for the parts of different images are veryclose. Cubic parts of sizes of 90, 180, and 270 voxels ineach dimension have been considered. In the first case,each image was presented as the union of 6 × 6 × 6 = 216partial images of 90×90×90 voxels. The computed curvesare shown in Fig. 22. The capillary pressure curves cal-culated for each part of a single image show some scatter,cf. Fig. 22a–d. The most interesting observation is thatthe scatter within each part is comparable with the scatteramong all 864 curves computed from all the image partsand shown in Fig. 22e.

The red part of each curve corresponds to the satura-tions at which the non-wetting phase is disconnected. Inthis analysis, we used the 26-connectivity. It appears thatthere is no correlation between the percolation threshold


a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

c)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

d)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

e)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Fig. 22 - These dimensionless capillary pressure curvesvs. wetting fluid saturation were computed for parts of aFontainebleau sandstone image. The parts have different av-erage porosities: a) φ = 12%; b) φ = 13%; c) φ = 17%; d)φ = 21%. Plots a–d include each 216 curves, and e includesall 864 curves.

saturation and the porosity. In Fig. 23, the scatteredcircles have abscissas equal to the threshold wetting phasesaturations, at which the nonwetting phase percolates, andthe ordinates are the porosities of the respective parts. Thevertical and horizontal red lines correspond to the medialvalues of the end-point saturations and the porosities. Theblack dashed line is the best linear fit. Note that the slopeof this line is close to zero. Also, the scatter of the pointsin Fig. 23 is in a sharp contrast with the almost coalescingcurves in Fig. 22. Thus, at least at the scale and reso-lution of the analyzed images, the threshold saturation ispractically independent of porosity.

If larger image parts are analyzed by the method de-scribed above, the dimensionless capillary pressure curvespractically collapse together, whereas the scatter of theporosity-endpoint saturation computations does not no-ticeably change, see Fig. 24.

The dimensionless capillary pressure curves computedfor different images may be different. Within each image,the curves computed for its different parts practically co-incide with each other. Thus, the two groups of curves inFig. 25 are clearly distinct. The lower group of curves cor-responds to the image of average porosity 17%, whereas theupper group of curves is a combination of curves computedfor both 12%- and 13%-porosity images. The curves for the

a)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.08

0.09

0.1

0.11

0.12

0.13

0.14

0.15

0.16

0.17

0.18

b)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

c)

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

d)

0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.14

0.16

0.18

0.2

0.22

0.24

0.26

Fig. 23 - The rock porosity vs. 1 - breakthrough saturationof nonwetting phase for parts of the Fontainebleau sandstoneimage. The parts have different average porosities: a) φ =12%; b) φ = 13%; c) φ = 17%; d) φ = 21%. Plots a–d includeeach 216 curves, and e includes all 864 curves.

a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

b)

0.74 0.76 0.78 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.940.11

0.115

0.12

0.125

0.13

0.135

0.14

c)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

d)

0.86 0.88 0.9 0.92 0.94 0.96 0.98 10.196

0.198

0.2

0.202

0.204

0.206

0.208

0.21

0.212

0.214

0.216

Fig. 24 - Dimensionless capillary pressures vs. wettingphase saturation (a and c), and the corresponding porosityvs. (1 - breakthrough nonwetting phase saturation) distribu-tions (b and d) for the 270× 270× 270-voxel fragments of theFontainebleau sandstone image. The average porosities are:a–b φ = 12%; c–d φ = 21%.

21%-porosity images are not displayed because they fall in-between these two groups. Such a non-monotonic behav-ior of the capillary pressure curves can be a consequenceof the image segmentation, of under-representativeness ofthe image size, or of the geometry of the rock pore space.

Thus far, the capillary pressure curves have been com-puted without modeling percolation. Now, let us considerprimary drainage. The non-wetting fluid must somehowget into the rock and remain connected to the inlet asit displaces the wetting fluid. To model drainage, we as-sume that originally the pore space is occupied by the wet-ting fluid, which is pushed by the inflow of the nonwettingfluid. We assume that the flow is macroscopically one-


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Fig. 25 - These two groups of dimensionless capillary pressurecurves vs wetting phase saturation are clearly distinct: theupper group corresponds to the porosities of 12% and 13%,whereas the lower group corresponds to the 17%-porosity.

dimensional, i.e., the fluids flow in and out only throughtwo opposite faces of the cubic rock sample. To verifythe results, three directions of flow were modeled for eachimage. Since no information about the orientation of therock in its natural environment was available, we call thesedirections X, Y and Z. The idea is as follows. We as-sume that the pressure of the wetting fluid inside the rockis constant, whereas the pressure of the non-wetting fluidnear the inlet face increases. As the pressure increases, thenon-wetting fluid enters the largest pores and then contin-ues its propagation.47 As its invasion continues, the non-wetting fluid remains a single cluster connected to the inletface. At a certain capillary pressure breakthrough occurs,and a sample-spanning cluster of pores containing the non-wetting fluid is formed. In simulations, a modification ofa breadth-first cluster search algorithm45 was used.

In Fig. 26, we show the results for rock drainage sim-ulated with invasion percolation. The entire image waspartitioned into 3× 3 × 3 = 27 parts of 180 voxels in eachdirection. The results for different images are similar andare not presented. It turns out that the flow directiondoes not affect the results, i.e., the rock is isotropic. Themean breakthrough saturation is almost the same for dif-ferent parts and in different directions. The red part of thedimensionless capillary pressure curve corresponds to thenon-wetting phase saturations below percolation thresh-old.

Thus far, we assumed the 26-connectivity among thevoxels. In Fig. 27, we present the results of analogoussimulations for the 18-connectivity. As one would expectintuitively, the scatter of the breakthrough saturations ishigher than that for the 26-connectivity, but the differenceis not dramatic.

Analyzing image parts of different sizes, one noticesthat the average wetting fluid saturation at which the non-wetting phase percolates increases with the image size,Fig. 28. In larger images, the non-wetting phase spansthe sample at lower capillary pressures. Clearly, the firstspanning path of the non-wetting fluid increases in lengthwith the size of the image. Therefore, the larger imageshave better-connected, wide pore paths, which are cut inthe smaller images.

a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.16

0.17

0.18

0.19

0.2

X

b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

c)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.16

0.17

0.18

0.19

0.2

Y

d)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

e)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.16

0.17

0.18

0.19

0.2

f)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Fig. 26 - The rock porosity vs wetting fluid saturation atbreakthrough of nonwetting phase (a, c and e), and the di-mensionless drainage capillary pressures vs wetting fluid satu-ration (b, d and f) for 27 parts of the Fontainebleau sandstoneimage with the average porosity 17%. a–b drainage in the X-direction X; c–d drainage in the Y -direction; e–f drainage inthe Z-direction.

To relate the dimensionless capillary pressure curvescalculated in this paper to a capillary pressure curve from,e.g., a mercury injection experiment, additional scalingmay be needed. Besides the possible distortion of the cal-culated curve at low wetting fluid saturations, error is alsointroduced by assuming spherical interfaces, and approxi-mating the spheres with the discrete structures shown inFig. 5. At this time, we cannot devise such a scaling be-cause we lack mercury injection data for a core whose im-ages were also acquired. However, the stability of the di-mensionless capillary pressure curves with respect to theselection of the analyzed image suggests that mercury in-jection may not be required.

Formally, in a similar, but more cumbersome manner,one can model fluid displacement in imbibition. However,imbibition to a large extent is a spontaneous process andaccurate accounting for time is needed. The current modelis insufficient for such simulations.

Statistical methods for numerical reconstruction ofnatural rocks have been discussed in a large number ofworks, see e.g., Refs.48–50 An important issue for such anactivity is a reliable criterion for comparison of the sim-ulated and natural rock. Usually, a two-point correlationfunction is used as such a criterion. In fact, a dimensionlesscapillary pressure curve also can be used to compare theimages of natural and computer-generated rocks. Indeed,


a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.16

0.17

0.18

0.19

0.2

b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

c)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.16

0.17

0.18

0.19

0.2

d)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

e)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.16

0.17

0.18

0.19

0.2

f)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Fig. 27 - Same procedure is in Fig. 26 but assuming 18-connectivity. The variability of the breakthrough saturationsis not very dramatic.

80 100 120 140 160 180 200 220 240 260 2800.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Linear size of the image

Wet

ting

fluid

satu

rati

on

atth

e

non-w

etti

ng

fluid

bre

akth

rough

Fig. 28 - The dependence of the average breakthrough satu-ration on the image size.

the results presented in this section demonstrate that thecalculated capillary pressure curve does not depend on theporosity or the size of the sample, rather it depends on theintrinsic geometry of the pore space.

Conclusions

In this paper, we have presented a new robust approach tostudy the pore space morphology. Our approach is basedon the fundamental concepts of mathematical morphology,

summarized, e.g., in Ref.4 We have proposed an efficientand stable algorithm which distinguishes between the porebodies and pore throats, and establishes their respectivevolumes and connectivity. Our algorithm has been testedon the images of computer-generated and natural sand-stones. The algorithm tests on a pack of equal spheres,for which exact results can be verified visually, confirmedits stability. The impact of the resolution on the resultshas been investigated using computer-generated sandstoneimages of different resolutions.

The proposed algorithm produces a stick-and-ball dia-gram of the rock pore space. One of distinctive features ofour approach is that no image thinning is applied. Instead,the information about the skeleton is stored as the radiiof the maximal balls associated with the voxels. In fact,these maximal balls retain information about the entirepore space. A comparison of our results with the resultsof a thinning procedure, which preserves some topologicalproperties of the pore space,27 shows that our method pro-duces more realistic estimates of the number of pore bodiesand pore throats, and of the pore body coordination num-bers.

Using information about the maximal ball distribu-tion, we have computed a dimensionless drainage capil-lary pressure curve which simulates mercury injection. Weconclude that the calculated capillary pressure curve is arobust descriptor of the pore space geometry, and it canbe used to determine the quality of computer reconstruc-tion of natural rocks. In addition, an appropriate scal-ing of this curve should predict the capillary pressure ofthe rock based on its 3D image. This scaling has not yetbeen developed due to lack of appropriate experimentaldata. We have observed that the nonwetting phase break-through saturation in drainage practically does not dependon the sandstone porosity. Moreover, at least within theconsidered length scales, this breakthrough occurs at lowercapillary pressures in larger sandstone samples.

AcknowledgmentsThis research was supported by the Assistant Secretary forFossil Energy, Office of Natural Gas and Petroleum Tech-nology, through the National Petroleum Technology office,Natural Gas and Oil Technology Partnership under USDepartment of Energy contract no. DE-AC03-76SF00098to Lawrence Berkeley National Laboratory. Partial sup-port was also provided by gifts from ChevronTexaco andConocoPhillips to UC Oil, Berkeley. We are thankfulto Schlumberger for providing the synchrotron images ofFontainebleau sandstone.

References1. P. E. Øren and S. Bakke. Process based reconstruction of

sandstones and prediction of transport properties. Trans-port in Porous Media, 46(2-3):311–343, 2002.

2. G. Jin, T.W. Patzek, and D. B. Silin. Physics-based recon-struction of sedimentary rocks. SPE 83587. In SPE West-ern Regional /AAPG Pacific Section Joint Meeting, LongBeach, California, U.S.A., 2003. SPE.


3. G. Matheron. Random Sets and Integral Geometry. Wiley,New York, 1975.

4. J. Serra. Image Analysis and Mathematical Morphology.Academic Press, New York, 1982.

5. Th. Motzkin. Sur quelque proprietes caracteristiques desensemples bornes non convexes. Atti Acad. Naz. Lincei,21:773–779, 1935.

6. H. Blum. An associative machine for dealing with the vi-sual field and some of its biological implications. In E. E.Bernard and M. R. Kare, editors, Biological Prototypes andSynthetic Systems, volume 1, pages 244–260, NY, 1962.Plenum Press. Proc. 2nd Annual Bionics Symposium, heldat Cornell University, 1961.

7. Lars Oyno, B. G. Tjetland, K. H. Esbensen, Rune Solberg,Aase Schele, and Tore Larsen. Prediction of petrophysi-cal parameters based on digital vidoe core images. SPEReservoir Evaluation and Engineering, 1:82–87, February1998.

8. W. B. Lindquist and A. Venkatarangan. Investigating 3Dgeometry of porous media from high resolution images.Phys. Chem. Earth (A), 25(7):593–599, 1999.

9. Arun Venkatarangan, W. Brent Lindquist, John Dunsmuir,and Teng-Fong Wong. Pore and throat size distributionsmeasured from synchrotron x-ray tomographic images offontainebleau sandstones. Jopurnal of Geophysical Reser-ach – Solid Earth, 105(9):21509–21527, 2000.

10. Liviu Tomutsa and Velimir Radmilovic. Focussed ionbeam assisted three-dimensional rock imaging at submicronscale. Technical Report LBNL-52648, Lawrence BerkeleyNational Laboratory, May 2003. Submitted to 2003 Inter-national Symposium of the Society of Core Analysts.

11. M. J. Blunt and P. King. Relative permeabilities fromtwo- and three-dimensional pore-scale metwork modeling.Transport in Porous Media, 6:407–433, 1991.

12. M. J. Blunt and H. Sher. Pore network modeling of wetting.Physical Review E, 52(December):6387, 1995.

13. S. Bakke and P. E. Øren. 3-d pore-scale modelling of hetero-geneous sandstone reservoir rocks and quantitative analysisof the architecture, geometry and spacial continuity of thepore network. spe 35479. In European 3-D Reservoir Mod-elling Conference, pages 35–45, Stavanger, Norway, 1996.SPE.

14. P. E. Øren, S. Bakke, and O. J. Arntzen. Extendingpredictive capabilities to network models. SPE Journal,(December):324–336, 1998.

15. T. W. Patzek. Verification of a complete pore network sim-ulator of drainage and imbibition. SPE Journal, 6(2):144–156, 2001.

16. A. Al-Futaisi and T. W. Patzek. Impact of wettability ontwo-phase flow characteristics of sedimentary rock: Quasi-static model. Water Resources Research, 39(2):1042–1055,2003.

17. Dmitry Silin and Tad Patzek. NetSimCPP: Object-OrientedPore Network Simulator. Lawrence Berkeley National Lab-oratory, 2001.

18. I. Hidajat, A. Rastogi, M. Singh, and K. Nohanty. Trans-port properties of porous media reconstructed from thin-sections. SPE Journal, pages 40–48, March 2002.

19. V. M. Entov. The micromechanics of flow in porous me-dia. Soviet Academy Izvestia. Mechanics of Gas and Fluids,(6):90–102, 1992.

20. B. Berkowitz. Percolation theory and its application togroundwater hydrology. Water Resources Res., 29(4):775–794, 1993.

21. B. Berkowitz and R. P. Ewing. Percolation theory andnetwork modeling applications in soil physics. Surveys inGeophysics, 19:23–72, 1998.

22. M. J. Blunt. Flow in porous media - pore-network modelsand multiphase flow. Current Opinion in Colloid & Inter-face science, 6(3):197–207, 2001.

23. Ta-Chin Lee, Rangasami L. Kashyap, and Chong-NamChu. Building skeleton models via 3-d medial surface/axisthinning algorithm. Graphical Models and Image Process-ing, 56(6):462–478, November 1994.

24. P.P. Jonker and A.M.Vossepoel. On skeletonization algo-rithms for 2, 3 ... N dimensional images. In D. Dori andA. Bruckstein, editors, Shape, Structure and Pattern Recog-nition, pages 71–80. World Scientific,, Singapore, 1995.Proc. SSPR’94, Nahariya, Israel, Oct.4-6, 1994.

25. Kalman Palagyi and Attila Kuba. A 3D 6-subiteration thin-ning algorithm for extracting medial lines. Pattern Recog-nition Letters, 19:613–627, 1998.

26. P.P. Jonker. Morphological operations on 3D and 4D im-ages: From shape primitive detection to skeletonization. InG. Sanniti di Baja G. Borgefors, I Nystom, editor, DiscreteGeometry for Computer Imagery, number 1953 in Lec-ture Notes in Computer Science, pages 371–391. Springer-Verlag, 2000. Proc. 9th Int. Conf. DGCI (Uppsala, S,Dec.13-15).

27. L. Portuaud, P. Porion, E. Lespessailles, C. L. Benhamou,and P. Levitz. A new method for three-dimensional skele-ton graph analysis of porous media: application to tra-becular bone microarchitecture. Journal of Microscopy,199(2):149–161, 2000.

28. W. B. Lindquist. Network flow model studies and 3D porestructure. In Z. Chen and R. E. Ewing, editors, Fluid Flowand Transport in Porous Media: Mathematical and Numer-ical Treatment, volume 295 of Contemporary Mathematics.American Mathematical Society, 2002.

29. Robert M. Sok, Mark A. Knackstedt, Adrian P. Sheppard,W. V. Pinczewski, W. B. Lindquist, A. Venkatarangan, andLincoln Paterson. Direct and stochastic generation of net-work models from tomographic images; effect of topologyon residual saturation. Transport in Porous Media, 46:345–372, 2002.

30. P. M. Adler. Porous Media, Geometry and Transports.Butterworth-Heinemann Series in Chemical Engineering.Butterworth-Heinemann, Boston, 1992.

31. Hans-Jorg Vogel. Digital unbieased estimation of the eu-ler poincare characteristic in different dimensions. AcraStereol., 16(2):97–104, 1997.

32. P.P. Jonker and A.M.Vossepoel. Mathematical morphol-ogy in 3D images: Comparing 2D & 3D skeletonizationalgorithms. In K.Wojciechowski, editor, BENEFIT Sum-mer School on Morphological Image and Signal Process-ing, volume 2, pages 83–108. Silesian Technical University,ACECS, Gliwice, Poland,, 1995. Summer school held inZakopane, Poland, Sept. 27-30.


33. P.P. Jonker. Skeletons in n dimensions using shape primi-tives. Pattern Recognition letters, 23(4):677–686, 2002.

34. J. F. Delerue, E. Perrier, A. Timmerman, M. Rieu, andK. U. Leuven. New computer tools to quantify 3D porousstructures in relation with hydraulic properties. In J. Feyenand K. Wiyo, editors, Modelling of transport processes insoils, pages 153–163. Wageningen Pers, The Netherlands,1999.

35. J. F. Delerue, E. Perrier, Z. Y. Yu, and B. Velde. Newalgorithms in 3D iamge analysis and their application tothe measurement of a spatialaized pore size distribution insoils. Physics and Chemistry of the Earth, Part A: SolidEarth and Geodesy, 24(7):639–644, 1999.

36. Jean-Francois Delerue and Edith Perrier. DXSoil, a libraryfor 3D image analysis in soil science. Computer & Geo-sciences, 28:1041–1050, 2002.

37. Z. R. Liang, C. P. Fernandes, F. S. Magnani, and P. C.Philippi. A reconstruction technique for three-dimensionalporous media using image analysis and fourier transform.Journal of Petroleum Science and Engineering, 21:273–283,1998.

38. C. P. Fernandes, F. S. Magnani, P. C. Philippi, and J. E.Daıan. Multiscale geometrical reconstruction of porousstructures. Physical Review E, 54(2):1734–1741, 1996.

39. Z. R. Liang, P. C. Philippi, C. P. Fernandes, and F. S.Magnani. Prediction of permeability from the skeleton ofthree-dimentional pore structure. SPE Reservoir Evalua-tion and Engineering, 2(2):161–168, 1999.

40. F. S. Magnani, P. C. Philippi, Z. R. Liang, and C. P.Fernandes. Modelling two-phase equilibrium in three-dimensional porous microstructures. International Journalof Multiphase Flow, 26(1):99–123, 2000.

41. D.B. Silin. On upper semicontinuous multivalued map-pings. Sov. Math., Dokl., 35(3):587–590, 1987.

42. D. B. Silin. Some properties of upper semicontinuous mul-tivalued mappings. Proc. Steklov Inst. Math., 185:249–262,1990.

43. J. R. Munkers. Elements of Algebraic Topology. Addison–Wisley, Menlo Park, CA, 1984.

44. Alexander Stepanov and Meng Lee. The Standard TemplateLibrary. Hewlett Paccard Laboratories, 1995.

45. D. B. Silin and T. W. Patzek. Object-oriented clus-ter search for an arbitrary pore network. Technical Re-port LBNL-51599, Lawrence Berkeley National Laboratory,Berkeley, CA, January 2003.

46. C. E. Weatherburn. Differential Geometry of Three Di-mensions, volume 2. At the University Press, Cambridge,1930.

47. D. Wilkinson and J. F. Williamsen. Invasion percolation:A new form of percolation theory. J. Phys. A. Math Gen.,16:3365–3376, 1983.

48. J. G. Berryman and S. C. Blair. Use of digital image analy-sis to estimate fluid permeability of porous materials: Ap-plication of two-point correlations functions. Journal ofApplied Physics, 60:1930–1938, September 1986.

49. Z. Liang, M. A. Ioannidis, and I. Chatzis. Geomet-ric and topological analysis of three-dimensional porous

media: Pore space partitionanig based on morphologicalskeletonization. Journal of Colloid and Interface Science,221:13024, 2000.

50. M. A. Ioannidis and I. Chatzis. On the geometry and topol-ogy of 3D stochastic porous media. Journal of Colloid andInterface Science, 229:323–334, 2000.

spe 84296 robust determination of the pore space ... determination... · are part solid and part...

Documents