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Chertkov, V.Y. 1995b. Evaluation for soil of crack net connectedness Scott, G.J.T., R. Webster, and S. Nortcliff. 1986. An analysis of crackand critical stress-intensity factor. Int. Agrophysics. 9(3):189–195. pattern in clay soil: Its density and orientation. J. Soil Sci.

Chertkov, V.Y. and I. Ravina. 1998. Modeling the crack network of 37:653–668.swelling clay soils. Soil Sci. Soc. Am. J. 62:1162–1171. Scott, G.J.T., R. Webster, and S. Nortcliff. 1988. The topology of pore

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Ringrose-Voase, A.J. and C. Nys. 1990. One-dimensional image analy-

Three-Dimensional Quantification of Macropore Networks in Undisturbed Soil Cores

Johan Perret, S. O. Prasher,* A. Kantzas, and C. Langford

ABSTRACT exploitable by plant roots (Scott et al., 1988a). Similarly,the more continuous the macropores are, the moreThe role of macropores in soil and water processes has motivatedfreely gases can interchange with the atmosphere. Con-many researchers to describe their sizes and shapes. Several ap-

proaches have been developed to characterize macroporosity, such tinuous macropores also have a direct effect on wateras the use of tension infiltrometers, breakthrough curve techniques, infiltration and solute transport in soil.image-analysis of sections of soils, and CAT scanning. Until now, According to Sutton (1991), the size of pore openingsefforts to describe macropores in quantitative terms have been concen- is more important for plant growth than is the overalltrated on their two-dimensional (2-D) geometry. The objective of

soil porosity. Although existing pores constrain the pen-this study is to nondestructively quantify the three-dimensional (3-etration of roots, they provide favorable conditions forD) properties of soil macropores in four large undisturbed soil col-root growth. Several studies have shown that the pres-umns. The geometry and topology of macropore networks were deter-

mined using CAT scanning and 3-D reconstruction techniques. Our ence of continuous macropores in soil significantly bene-results suggest that the numerical density of macropores varies be- fits root growth (Bennie, 1991). One of the most impor-tween 13 421 to 23 562 networks/m3 of sandy loam soil. The majority tant factors influencing soil fertility, besides water andof the macropore networks had a length of 40 mm, a volume of 60 nutrient content, is soil aeration (Hillel, 1980; Glinskimm3, and a wall area of 175 mm2. It was found that the greater the and Stepniewski, 1985). Large soil pores are the pathslength of networks, the greater the hydraulic radius. The inclination

available for gas exchange between soil and atmosphereof the networks ranged from vertical to an angle of ≈558 from vertical.(Sutton, 1991). In natural soils, water movement followsResults for tortuosity indicated that most macropore networks had apaths of least resistance (i.e., preferential flow paths).3-D tortuous length 15% greater than the distance between their

extremities. More than 60% of the networks were made up of four Intuitively, large and continuous pores facilitate waterbranches. For Column 1, it was found that 82% of the networks had transport. It is now well known that the size and connec-zero connectivity. This implies that more than 4/5 of the macropore tivity of soil pores play a major role in the flow character-networks were composed of only one independent path between any istics of water and the transport of solutes through soiltwo points within the pore space.

(Ma and Selim, 1997). Jury and Fluhler (1992, p. 192)stated that “fluid transport through well defined struc-tural voids is not predictable unless the distributions of

Soil structure consists of a 3-D network of pores. the voids, aperture sizes and shapes, depths of penetra-Large pores play an important role in allowing tion, and interconnectivity are known.”

roots, gas, and water to penetrate into the soil. The The importance of macropores in many soil–plant–higher the macropore density, the more the soil can be roots processes has motivated many researchers to de-

scribe their sizes and shapes. Several approaches haveJohan Perret and S.O. Prasher, Dep. of Agricultural and Biosystems been developed to characterize macroporosity. AmongEngineering, McGill Univ., 21 111 Lakeshore Rd., Ste-Anne-de-Belle- them are tension infiltrometers (Everts and Kanwar,vue, QC, Canada, H9X-3V9; A. Kantzas, Dep. of Chemical and Petro-

1993; Timlin et al., 1994; Logsdon et al., 1993), break-leum Engineering, Univ. of Calgary, 2500 University Dr. N.W., Cal-gary, AB, Canada, T2N-1N4; and C. Langford, Dep. of Chemistry, through curve techniques (Ahuja et al., 1995; Jabro etUniv. of Calgary, 2500 University Dr. N.W., Calgary, AB, Canada,T2N-1N4. Received 28 Sept. 1998. *Corresponding author (prasher@

Abbreviations: CAT, computer-assisted tomography; CT, computedagreng.lan.mcgill.ca).tomography; ECD, equivalent cylindrical diameter; HU, HounsfieldUnits; 2-D, two-dimensional; 3-D, three-dimensional.Published in Soil Sci. Soc. Am. J. 63:1530–1543 (1999).

PERRET ET AL.: 3-D QUANTIFICATION OF MACROPORE NETWORKS IN SOIL CORES 1531

Table 1. Different classifications of pores based on their Equiva-al., 1994; Li and Ghodrati, 1994; Ma and Selim, 1994),lent Cylindrical Diameter.and image-analysis of sections of soils (Koppi and

Terminology ECD ReferenceMcBratney, 1991; Moran and McBratney, 1992; Singhet al., 1991; Vermeul et al., 1993). Warner et al. (1989), Micropore ,30 mm Jongerius (1957)

Mesopore 30 mm–100 mmGrevers et al. (1989), Anderson et al. (1990), HansonMacropore .100 mmet al. (1991), Tollner et al. (1995), Asare et al. (1995),Micropore ,30 mm Marshall (1959)

and Heijs et al. (1996) have also recognized the great Macropore .30 mmpotential offered by computer-assisted tomography Micropore ,75 mm Johnson et al. (1960)

Very fine pore 75 mm–1000 mm(CAT) scanning for characterizing soil macroporosity.Fine pore 1000 mm–2000 mmNevertheless, efforts to describe macropores in quan- Medium pore 2000 mm–5000 mmCoarse pore .5000 mmtitative terms have not yet resulted in a comprehensiveCryptovoid ,0.1 mm Brewer (1964)theoretical framework that allows a complete represen-Ultramicrovoid ,5 mmtation of their geometry. This is partly due to the fact Microvoid 5 mm–30 mm

that macropores are very difficult to observe and charac- Mesovoid 30 mm–75 mmMacrovoid .75 mmterize, bearing in mind that the macropore networksVery fine pore ,2 mm Russell (1973)are complex 3-D structures. Up to now, most of theFine pore 2 mm–20 mm

work done on the quantification of soil macropores con- Medium pore 20 mm–200 mmCoarse pore .200 mmcentrated on their 2-D geometry. A few researchers,Micropore ,0.3 mm McIntyre (1974)such as Hanson et al. (1991), Heijs et al. (1996), andMinipore 0.3 mm–30 mmPerret et al. (1997), have used reconstructive imagery, Macropore 30 mm–300 mmSuper pore .300 mmallowing 3-D visualization of the macropore space, butBonding pore .0.005 mm Greenland (1977)have not quantified soil macroporosity in three dimen-Residual pore ,0.5 mmsions. Transport phenomena in porous media strongly Storage pore 0.5 mm–50 mm

depend on the 3-D pore structure (Chatzis and Dullien, Transmission pore 50 mm–500 mmFissure .500 mm1977). Very little work has been done to characterizeMacropore .1000 mm Bouma et al. (1977)the macroporosity of intact soil cores in terms of itsMacropore .60 mm Bullock and3-D parameters. Tollner et al. (1995) pointed out the

Thomansson (1979)need for additional research to investigate reliable ap-

Macrofissure 200 mm–2000 mm Reeves et al. (1980)proaches for computing tortuosity and connectivity of Enlarged macrofissure 2000 mm–10 000 mmsoil macropores. Imaging in three dimensions and quan- Macropore .3000 mm Beven (1981)tification of 3-D parameters of soil macroporosity are Micropore ,10 mm Luxmoore (1981)

Mesopore 10 mm–1000 mmcritical in order to accurately correlate soil pore struc-Macropore .1000 mm

ture with preferential flow phenomena and much addi-Macropore .1000 mm Luxmoore et al. (1990)

tional work is needed in this area (Hanson et al., 1991).The present investigation is a study of the 3-D proper-

ties of soil macropores. The main characteristics of a ropore, its definition may become hazy and ambiguous. Byporous medium that affect fluid flow are porosity, nu- strict definition, macropore implies a large pore. However,merical density, pore shape (size, length, volume, hy- large is a relative term and this lack of clarity has led to severaldraulic radius, tortuosity), and pore interconnectivity or conflicting definitions.

It would be useful to have a general agreement on poregenus per unit volume (Constantinides and Payatakes,terminology, just as one exists for the definitions of sand, silt,1989). Therefore, the objective of this study is to quanti-and clay. However, up to now there has been little or notatively determine the above parameters of soil mac-consensus for the definition and terminology used for classify-ropores in four undisturbed soil columns through a 3-Ding pores in general. A large number of classification schemesreconstruction from 2-D matrices generated by an x-ray based on the equivalent cylindrical diameter (ECD) and sev-

CAT scanner. The results of this work show promise eral contradicting definitions can be found in soil literaturefor future studies in the area of soil macropore quantifi- (Table 1).cation. These definitions fail to remove ambiguity from the de-

limitation between macropores and micropores. However, inTerminology this research study, the definition proposed by Luxmoore et

al. (1990, p. 144) was followed. They stated that “The termVarious standard geometrical parameters are meaningfulmacropore includes all pores in a profile that are (generally)to quantify the structure of 3-D macropore networks, such asdrained at field capacity, with the latter being 1 mm or moretheir relative position, length, volume, specific wall area, andin equivalent diameter.”orientation. Such parameters are explicit and do not need to

Up to now, definitions of macropore make no specific refer-be defined. However, several terms that are used to describeence to its size or shape in a 3-D context. The lack of informa-soil structure and their 3-D attributes are fuzzier and need totion describing their shapes has led to the generality used inbe clarified. Some of these terms are defined below:defining macropores (Kwiecien, 1987). Part of the goal of thisstudy is to characterize the shapes and other 3-D parametersMacropore of macropore networks with the aid of computer programs,in order to describe macropore geometry and, eventually, toAt first, the definition of a macropore may seem simple.

However, as we come to consider the complexity of a mac- help clarify the definition of a soil macropore.

1532 SOIL SCI. SOC. AM. J., VOL. 63, NOVEMBER–DECEMBER 1999

draulic radius (Rh) of a pore can be simply computed (Eq.[2]) as

Rh 5Volume of pore

Wall area of pore[2]

Necks can therefore be easily located by identifying localminima in the hydraulic radii of macropores. The reason forusing the hydraulic radius as a measure of pore throat is thatit is a useful measure of size in the case of irregularly shapedpores (Dullien, 1992). The hydraulic radius can provide agood indication of the pore neck position and of the poreexpansion–contraction.

Topology of Macropore Networks

A complete description of pore structures requires geomet-rical as well as topological information (Macdonald et al.,1986). Topology deals with properties of an object in a spacethat remain unaltered when that space is deformed. The topol-ogy of macropore networks concerns essentially the numberper unit volume and the degree of connectivity of macroporeFig. 1. Tortuosity of a soil macropore.networks, regardless of their shape. The number of networks,defined later as numerical density, is a measure of the complex-Macropore Network and Branches ity of pore structure (Scott et al., 1988a). Topological parame-ters characterizing the morphology of a porous medium areIn a porous medium, a network is a set of macropores thatthe numerical density, coordination number, connectivity, andare interconnected such that there is a passage from any partgenus (Dullien, 1992). Each of these terms is defined below.of the set to every other part (Scott et al., 1988a). Thus, the

concept of a macropore network implies a 3-D structure. Abranch is a portion of the macropore network connecting to Numerical Density of Networksthe rest of the network.

The number of networks per unit volume, regardless oftheir size or shape, is the numerical density. Scott et al. (1988a)

Tortuosity pointed out that it is very difficult to determine this quantity.Up to now, this information was roughly estimated by cuttingTortuosity (t) is one of the most meaningful 3-D parametersparallel plane sections through soil. Essentially, numericalof pore structure. Carman (1937) introduced the concept ofdensity was only accessible in two dimensions. One of thetortuosity as the square of the ratio of the effective averagemajor drawbacks of this approach is that there is not a one-path in the porous medium (Le) to the shortest distance mea-to-one correspondence between the number of networks esti-sured along the direction of the pore (L). Several researchersmated in the 2-D sections and those in three dimensions (Scotthave reviewed this definition (Hillel, 1982; Marshall andet al., 1988a).Holmes, 1988; Jury et al., 1991; Sahimi, 1995) and have rede-

fined tortuosity (Eq. [1]):Coordination Number

t 5LLe

[1] One of the simplest concepts for characterizing pore topol-ogy is the coordination number (Z). It is defined as the number

Tortuosity is a dimensionless factor always greater than one, of pore throats that meet at a given point along a pore (Sahimi,which expresses the degree of complexity of the sinuous pore 1995). In other terms, the coordination number determinespath (Fig. 1). Tortuosity can easily be related to the conductiv- the number of branches meeting at one node. Until now, theity of a porous medium since it provides an indication of only approach to determining the coordination number hasincreased resistance to flow due to the pore system’s greater been to reconstruct a branch-node chart of the pore structurepath length (Dullien, 1979). The term continuity is sometimes (Fig. 2) from a series of parallel sections of the porous mediumused to describe pore tortuosity. Richter (1987) has defined (Dullien, 1992).pore continuity as the reciprocal of tortuosity.

Connectivity and Genus of Macropore NetworksHydraulic Radius in Three Dimensions The concept of connectivity (Ccon) can also be used to char-

acterize the topology of a complex system such as soil mac-Another relevant parameter is the hydraulic radius of themacropore network. Macropores are not regularly shaped. roporosity. Connectivity is a measure of the number of inde-

pendent paths between two points within the pore spaceThe neck (also known as the pore throat) of a pore is animportant feature of pore geometry, which directly controls (Macdonald et al., 1986). In other words, connectivity is the

number of nonredundant loops enclosed by a specific geomet-percolation rates. A neck is defined as the local minimum inpore space size (Kwiecien et al., 1990). The pore throat is rical shape. Each macropore network has a connectivity, which

is a positive integer equal to the number of different closedlocated where the minimum mean radius of curvature of agas–liquid interface is observed. This corresponds to the loca- circuits between two points in the network. If there is only

one open circuit, the connectivity is equal to 0 (zero); thetion of maximum capillary pressure. Maximum capillary pres-sure within a macropore is very difficult to measure. However, connectivity is 1 if the circuit is closed. The term connectivity

density is sometimes used to define the connectivity per unitthe hydraulic radius is a good approximation of the meanradius of curvature (Kwiecien, 1987; Dullien, 1992). The hy- volume (Scott et al., 1988b). Figure 3 shows four different

PERRET ET AL.: 3-D QUANTIFICATION OF MACROPORE NETWORKS IN SOIL CORES 1533

Fig. 2. Schematic representation of (a) pore space, and (b) coordination number on a branch-node chart (after Dullien, 1992).

shapes to illustrate the notion of connectivity (Fig. 3a, b, c, were placed under a set of 10 300-W light bulbs for a periodof 6 wk. The cores were rotated periodically to accelerate evap-and d display a connectivity of 1, 1, 2, and 3, respectively).

The genus of a pore system is defined as the largest number oration.A 1-mm-i.d. polyethylene tube was inserted into one of theof nonintersecting cuts that can be made through a shape

without disconnecting any part from the rest (Dullien, 1992). soil columns to verify the ability of the CAT scanner to portraythe size and the location of a known macropore.In Fig. 3a and b, only one cut can be made through the pore

structure without creating two independent networks (the cutsare represented as ellipses intersecting the pore). Thus, the X-Ray CAT Scanninggenus of both structures shown in Fig. 3a and 3b is equal to

A modified medical ADVENT HD200 whole-body CAT1. Since two and three cuts can be made in the pores shownscanner (Universal Systems, Solon, OH) was used for thisin Fig. 3c and 3d, their genus is equal to 2 and 3, respectively.study at the TIPM laboratory in Calgary, AB. Computed to-A general theorem of topology states that the genus is numeri-mography (CT) or computer-assisted tomography (CAT) iscally equivalent to connectivity. Macdonald et al. (1986)a method of diagnostic imaging used for nondestructive im-strongly suggested that an accurate determination of the genusaging of cross-sectional slices of the human body or an object.would help to elaborate new and better flow models.This scanner incorporates a fourth-generation scan geometryAlthough these topological concepts have recently beenwith scan times as short as 2s/scan and a high pixel resolutiongiven increased attention in the field of petroleum recovery,(up to 195 3 195 mm). In most CAT scanners, the actualthese concepts have not yet been used to describe the spatialcollection of patient data occurs in the gantry where the patientcharacteristics of a soil macropore network. For this study,lies horizontally. However, the ADVENT HD200 was modi-we propose evaluating the coordination number and the con-fied to allow vertical scanning. For that purpose, the CATnectivity–genus of soil macropores in order to describe theirscanner gantry was rotated 908 and positioned on a metalcomplex geometry.frame designed to hold the whole gantry horizontally. Figure4 shows the rotated gantry of the CAT scanner.MATERIALS AND METHODS

During CAT scanning, each column was placed verticallySoil Cores in the scanner unit so that the x-ray beam intersected the soil

column perpendicularly to its longitudinal axis. A bubble levelIn July 1995, four undisturbed soil columns, 800 mm inindicator was used to ensure that the soil column was vertical.length and 77 mm in diam., were taken from a field site atThe longitudinal axis of the core was positioned at the centerthe Macdonald Campus of McGill University in Ste-Anne-of the gantry of the scanner. During the computer tomographicde-Bellevue, QC. The columns were extracted from an unculti-process, the x-ray tube rotated around the soil column. A pre-vated field border that had been covered for many years bycollimator modulated the thickness of the x-ray beam. Fora combination of quack grass [Elytrigia repens (L) Desv. exthis study, the collimator was set to a thickness of 2 mm. TheNevski], white clover (Trifolium repens L.), and wild oattransmission and detection of this thin, rotating x-ray beam(Avena fatua L.). Periodic mowing during the summer wasthrough the soil column resulted in a large number of attenua-the only cultural practice used. The land slope was ,1%.tion measurements taken at discrete angles. For this purpose,Column size was selected based on the need for a sample thatan array of 720 detectors was located within the gantry. Oncewas large enough to represent macropore distribution, yetcollected, the data were mathematically reconstructed to gen-small enough to be handled easily when full of soil. The hy-

draulic bucket of a backhoe was used to drive polyvinyl chlo-ride (PVC) pipes in small increments of about 80 mm. Theobjective was to obtain soil cores that were disturbed as little aspossible to obtain samples that were representative of naturalconditions. The soil belonged to the Chicot series. The Chicotseries is a type of soil encountered in the Montreal area follow-ing the Canadian soil taxonomy. These soils are developedfrom sandy materials over a calcareous till and, as a result,they are generally well drained (Lajoie and Baril, 1954). Thesoil was predominantly a sandy loam with an A horizon thick-ness of ≈0.4 m.

The soil columns were scanned under dry as well as satu- Fig. 3. Illustration of the concept of connectivity and genus (fromMacdonald et al., 1986).rated conditions. To reach dry conditions, the soil columns

1534 SOIL SCI. SOC. AM. J., VOL. 63, NOVEMBER–DECEMBER 1999

Fig. 4. View of the rotated gantry of the Advent HD200.

erate a 512 by 512 matrix. To produce a mean x-ray energybeam in the Compton energy range, the CAT-scan systemparameters were adjusted to 120 peak kV and 50 mA.

The position of the core was set mechanically for each scanwith a digital indexing ruler having a precision of 60.001 mm.A total of 360 sections or scans was obtained for each column,leaving no space between two consecutive scans.

Fig. 5. Illustration of (a) the six nearest neighboring voxels rule andData from each scan were recorded on a magnetic tape,(b) 26 neighboring voxels rule; (c) superposition of consecutivetransferred to a SUN4 workstation running under the UNIX2-D matrices for the 3-D algorithm.operating system, and converted to a bulk density value. First,

the scans were stored in matrices composed of CT numbersFour computer programs were developed to reconstruct,that were expressed in a dimensionless quantity known as

visualize and quantify 3-D macropore structures in soil col-Hounsfield Units (HU). The CT number for water is roughlyumns. The first program, called Filterjo.pro, thresholds the0 and 2420 for air. The CT values are a function of the electronmacropores in the 360 2-D bulk density matrices before isolat-density (bulk density) and atomic number of the material. Iting macropore networks in three dimensions. Each pixel canhas been previously demonstrated that the CT numbers arerepresent only two states of the dry soil columns, pore space,linearly related to the bulk density of the soil (Anderson etor soil matrix. The first task accomplished by Filterjo.pro isal., 1988). The soil columns were mounted inside a core holderto partition 2-D matrices into regions of 1 for pores and 0 forassembly made of a hollow Plexiglas annulus, partitioned intosoil matrix. The pores contain either water or air (i.e., densityfour chambers filled with two liquids. Water and mineral oil,1). Thus, the pore can be isolated by applying a thresholdwere used as reference materials. By plotting CT numbers vs.on all the pixels in the bulk density matrices having a valuethe bulk density of reference materials and using a simpleless than 1. This transformation is called segmentation. Thelinear regression, a calibration curve that relates bulk densityprogram then regroups pixels belonging to the same poreto the CT number of the scanned material was derived. A(clustering) by following a set of rules described by Perret etFORTRAN 77 program was developed to compute the cali-al. (1997). A filtering subroutine is then executed. For thatbration line of each scan. Once the linear calibration equationspurpose, a criterion is used to determine if the pore belongswere established, a computer algorithm transformed CAT-to the macropore domain. The criterion is based on the sizescan arrays into matrices of bulk density. Part of the algorithmof the pore. If the pore has an ECD ,1 mm, it is removedalso allowed for computation of the soil section’s porosityfrom the matrix. After filtering, a median smoothing is applieddistribution on a voxel (i.e., volume element) basis. The re-on each matrix with a neighborhood of two pixels. This processmaining analysis was done using the PV-WAVE language onis similar to smoothing with an average filter, but it does nota 300 MHZ Pentium II with 128 Mb of RAM.blur edges larger than the neighborhood. Median smoothingwas used since it has been found to be effective in removing3-D Reconstruction of Macropores noise (Visual Numerics, 1994). Each pixel in the resultingmatrix is then multiplied by 21. Therefore, matrix elementsThe PV-WAVE language was chosen for computer pro-

gramming in this study. The PV-WAVE language is a compre- have a value of 0 (i.e., soil) or 21 (i.e., macropores). Theresulting 2-D matrices are then stored in a file ready for thehensive programming environment that integrates state-of-

the-art numerical and graphical analyses. This programming 3-D analysis.A second computer program, called Netjo.pro, was devel-language is widely used for analyzing and visualizing technical

data in many fields, such as medical imaging, remote sensing, oped to recognize and isolate 3-D macropore networks. Thefirst step in developing this program was to establish a set ofand engineering. PV-WAVE is an ideal tool for working with

large arrays such as our CAT-scan data, because of its array- rules, which were used to determine how the macropore spacesin each 2-D matrix were connecting to each other in 3-D.oriented operators and ability to display and process data

in the ASCII and binary I–O formats. Another reason that Several different clustering criteria were used. The first algo-rithm was based on the six nearest neighboring voxels rule inmotivated this choice was PV-WAVE’s ability to be used

under both UNIX and PC environments. three dimensions (Fig. 5a). With this algorithm, similar voxels

PERRET ET AL.: 3-D QUANTIFICATION OF MACROPORE NETWORKS IN SOIL CORES 1535

(with a value of 21) are clustered together if they are beside,above, or below one another. In other words, all clusteredvoxels are joined by at least one planar face. Although thisalgorithm was fast, it did not cluster all voxels belonging to anetwork in a single pass when the network’s 3-D structurewas very complex and showed a high degree of connectivity.Therefore, a second clustering algorithm was developed, basedon the 26 neighboring voxels rule (Fig. 5b). This algorithmclusters all voxels belonging to the macropore domain aroundthe voxel of interest if they share a face, an edge, or even acorner by visiting the top and bottom 2-D matrices. In otherwords, two voxels will be registered as part of the same groupof pore volume if they have a value of 21 and share a commoncorner. Thus, the clustering algorithm examines the 26 nearestneighboring voxels in three dimensions. To do so, each cross-section of the soil column is analyzed by superimposing itsadjacent sections (Fig. 5c). A 3-D filtering algorithm was incor-porated in Netjo.pro to eliminate all networks having a length#10 mm. These macropores were removed because we as-sumed they were not contributing to preferential flow. Theoutput of this program is a large 3-D matrix that containsmatrix elements of 0 (i.e., soil matrix domain) and 1, 2, 3, . . . ,n for the macropore domain, where each integer representsa network. In other words, each voxel belonging to a networkhas an integer value. For instance, if a soil column has 45independent macropore networks, all voxels of the last net-work will have a value of 45. This approach was successfullyimplemented in the recognition and reconstruction of mac-ropore networks.

The third computer program, called Rview.pro, producesa list of vertices and polygons that describe the 3-D surfaceof macropores. Each voxel is visited to find polygons formedby the macropores. The polygons are then combined and ren-dered to reconstruct an exact 3-D representation of the mac-ropore networks. The reconstructed image allows the visual-ization of 3-D macropore networks (Fig. 6).

The last program, called Branjo.pro, was developed to iso-late and characterize each connecting branch in a macroporenetwork. The first task accomplished by the program is to readthe 3-D matrix generated by Netjo.pro. Then, the programthresholds all the voxels having a value of 1 (i.e., first mac-ropore network). Starting at the top section of the soil column,

Fig. 6. Three-dimensional reconstruction of macropore networks inthe algorithm visits each voxel for every section until it findsColumn 1.a voxel belonging to Network 1. Then, using the six nearest

neighboring voxels rule, Branjo.pro isolates and clusters eachvoxel of the first branch of the network. As the branch is Each vertical bar represents a macropore network. Anbeing clustered, the program computes the perimeter, surface integer number has been assigned to each networkarea, and centroid of the branch in each section. When the (from left to right) for reference. For example, Networkprogram finds a connection to a new branch, it stores the

18 of Column 1 starts at a depth of ≈250 mm and endslocation of the connection so that it can investigate propertiesat a depth of 680 mm. The total number of networksof this new branch at a later time. Once the program hasper soil column is summarized in Table 2. Column 3reached the end of the branch, it computes the branch’s tortu-has the greatest number of networks (i.e., 79 networks).osity, orientation, length, volume, wall area, hydraulic radius,

and the number and location of other connecting branches. The numerical density was calculated from the numberAt that stage, the program moves on to the location of a stored of networks per soil column (Table 2).connection (if present) and evaluates the properties of the It can be observed that the numerical density variesnew branch. Once all the branches of a particular network significantly from one soil column to another, althoughhave been analyzed, results are written to an ASCII file for they were taken from the same site with a distancefurther analysis on a worksheet. Then, the program repeats the

of only ≈0.5 m between them. The macroporosity wassame process for the second macropore network (Network 2)evaluated for each soil column (Table 2) and was inand so on to Network n. A detailed flow diagram of Branjo.proaccordance with the observations made by Edwards etis given in Fig. 7.al. (1990), who reported macroporosities ranging from0.4 to 3.8%.RESULTS AND DISCUSSION

One would expect that the number of macroporeNumerical Density, Relative Position, and networks per unit volume would have a consequentialLength of Macropore Networks effect on macroporosity. More precisely, it would makesense that soil columns with a large numerical densityFigure 8 shows the number, vertical position, and

length of macropore networks found in soil columns. would have a large macroporosity and vice-versa. How-

1536 SOIL SCI. SOC. AM. J., VOL. 63, NOVEMBER–DECEMBER 1999

works can be evaluated, as shown in Fig. 8. This providesa good indication of the long networks that might havea significant impact on vertical water and chemical dis-placement. The artificial macropore (i.e., polyethylenetubing) running through the soil in Column 4 has beenreadily detected and identified as Network 1.

Figure 9 shows the frequency distributions of thelength of the macropore networks in the four soil col-umns. The distributions peak at ≈40 mm for all soilcolumns. This indicates that the majority of the mac-ropore networks have a length of 40 mm. As expected,the distributions are skewed to the left, showing thepresence of a few long networks, especially for Columns1 and 4.

Volume, Wall Area, and Hydraulic RadiusFrequency distributions were evaluated to represent

the tendency of volume networks in the four soil col-umns. Since the distributions were substantially skewed,results are displayed on a semi-log graph (Fig. 10). Thegeometric mean was used to measure the central ten-dency of the network volume frequency distributions,since it is a useful summary statistic for highly skeweddata. The results for each column are presented in Ta-ble 2.

The mode is equal to 60 mm3 and is about the samefor the four columns. Knowing that the majority of themacropore networks have a length of 40 mm, this im-plies that most networks have an ECD of ≈1.4 mm. Thenetwork volume distributions suggest that about 2.5%of the networks have a volume .7500 mm3. In otherterms, 2.5% of the networks have a volume equivalentto a capillary of 2.4 m by 2 mm in diam. or a sphericalcavity of 24 mm in diam.

The distributions of the wall area of macropore net-works have been also evaluated (Fig. 11). The resultssuggest the same bimodal pattern for all soil columns.The mode of the distributions is ≈175 mm2. A secondarypeak, however, can be observed for networks having awall area of 1200 mm2. The prevailing macropore net-work length of 40 mm implies that two sizes of networksmay be found in this soil. The first category, whichaccounts for about 18 to 28% of the networks, repre-sents networks with an equivalent diam. of 1.4 mm, andthe second category delineates networks with an ECDof ≈9.5 mm.

Fig. 7. Flow diagram of the PV-WAVE program Branjo.pro. As mentioned earlier, the hydraulic radius is a usefulmeasure of size in the case of irregularly shaped pores.Just like tortuosity, it is a good indication of the ability ofever, our results do not confirm this supposition or indi-

cate a direct relationship between numerical density and the network to convey fluids. The greater the hydraulicradius, the greater its transport capacity. The hydraulicmacroporosity. For instance, Column 1, which has the

smallest numerical density (i.e., 13 421 macropore net- radius of every network found in the soil columns ispresented in Fig. 12.works/m3), has the greatest macroporosity. Column 3

(with a numerical density of 23 562 networks/m3) exhib- It can be observed that there is no apparent relation-ship between hydraulic radius and depth. However,its a much smaller macroporosity (i.e., 2.59%). These

results can be explained by the difference in the average longer networks have greater hydraulic radii thanshorter networks. For instance, the artificial macroporenetwork volume. The average network volume in Col-

umn 1 is more than 2.5 times that of Column 3. This of Column 4 running from top to bottom of the soil hasthe greatest hydraulic radius of the column. Since longexplains the relatively high macroporosity of Column

1. Therefore, one cannot use numerical density as an pores have a greater ability to convey water, these re-sults are expected.indication of macroporosity.

The vertical length and position of macropore net- The frequency distributions of hydraulic radii were

PERRET ET AL.: 3-D QUANTIFICATION OF MACROPORE NETWORKS IN SOIL CORES 1537

Fig. 8. Number, relative position, and vertical length of macropore networks in the four soil columns.

Table 2. Selected properties of macropore networks for each of the soil columns.

Soil Soil Soil SoilProperty column 1 column 2 column 3 column 4

Number of networks 45 47 79 54Numerical density

(networks/m3) 13 421 14 018 23 562 16 106Macroporosity 3.8% 2.18% 2.59% 2.79%Volume (mm3) 67.4 Q1 5 24† 51.2 16 57.7 25 52.5 21Geometric mean Q2 5 48 42 41 53

Q3 5 129 142 114 93Q4 5 9674 6132 3079 9948

Tortuosity (mm/mm) 1.25 Q1 5 1.10 1.18 1.09 1.28 1.14 1.26 1.12Geometric mean Q2 5 1.17 1.14 1.19 1.18

Q3 5 1.31 1.24 1.35 1.34Q4 5 2.34 2.05 2.37 2.33

Average hydraulicradius (mm) 0.14 6 0.05‡ 0.13 6 0.07 0.12 6 0.05 0.13 6 0.04

† Q1, Q2, Q3, and Q4 represent the 1st, 2nd, 3rd, and 4th quartiles.‡ Standard deviation.

1538 SOIL SCI. SOC. AM. J., VOL. 63, NOVEMBER–DECEMBER 1999

suggested by the data is that the greater the inclination,the fewer the number of macropore networks. However,the inclination fluctuates erratically and there is no evi-dence of a clear trend.

As mentioned earlier, tortuosity is a dimensionlessfactor always greater than 1, which expresses the degreeof complexity of the pore path. A macropore networkwith a tortuosity of 1 implies that the length of theeffective or tortuous path of network is equal to theshortest distance measured along its direction. In otherwords, it indicates that the network follows a straightpath. As tortuosity increases and moves away from 1, thepath of the macropore network becomes more tortuous.

Figure 15 shows the distributions of the tortuosity ofFig. 9. Frequency distributions of the length of macropore networksmacropore networks found in the four columns. Thein the four soil columns.distributions are similar and skewed to the right, witha mode of ≈1.15. Most of the networks have a tortuosityalso assessed for every column and are shown in Fig.in the range 1 to 1.4. Thus, the majority of the macropore13. The distributions of hydraulic radii are almost sym-networks have a 3-D tortuous length 15% greater thanmetrical with a mode of ≈0.13. Here again, networks inthe distance between their extremities. Some macroporeall soil columns show a similar trend.networks have a tortuosity as high as 2.4.

Inclination and TortuosityNumber of Branches, Branch-Node Chart,Figure 14 shows the frequency distributions of the and Connectivityinclination of macropore networks. The inclination of

networks ranges from vertical to an angle of about 558 Figure 16 shows the distributions of the number ofbranches per network for all four soil columns. Thefrom vertical for some networks. The overall tendency

Fig. 10. Frequency distributions of volume of macropore networks in the four soil columns.

Fig. 11. Frequency distributions of wall area of macropore networks in the four soil columns.

PERRET ET AL.: 3-D QUANTIFICATION OF MACROPORE NETWORKS IN SOIL CORES 1539

Fig. 12. Hydraulic radius of macropore networks in the four soil columns.

distributions for each column follow the same trend. The presence of very few networks with a large numberof branches.mode of the distributions suggests that most macropore

networks are made up of approximately four branches. Results presented above were obtained by analyzingeach network in the soil columns. As mentioned earlier,The distributions are skewed to the left, indicating the

Fig. 13. Frequency distributions of hydraulic radius of macropore networks in the four soil columns.

1540 SOIL SCI. SOC. AM. J., VOL. 63, NOVEMBER–DECEMBER 1999

Fig. 14. Frequency distributions of macropore network inclination.

Fig. 15. Frequency distributions of tortuosity of macropore networks.

Fig. 16. Frequency distributions of number of branches per macropore network.

a macropore network is a set of branches that are inter- be too exhaustive. Therefore, we decided to limit theinvestigation to branches of five large networks of Col-connected. The parameters that have been evaluated

for each network (i.e., number of networks, relative umn 1 only.The distributions of branch lengths of Networks 6,position, length, wall area, volume, hydraulic radius,

orientation, and tortuosity) can be assessed in a similar 18, 19, 32, and 44 in Column 1 are presented in Fig. 17a.Only networks that have a length greater than averagefashion for each branch of every macropore network.

However, on average, the number of branches per soil were selected. As indicated in Fig. 8, Networks 6, 18,19, 32, and 44 of Column 1 have a length greater thancolumn was calculated to be 288. To include analysis of

each branch of all four soil columns in this paper would the average macropore network. Fig. 17a suggests that

PERRET ET AL.: 3-D QUANTIFICATION OF MACROPORE NETWORKS IN SOIL CORES 1541

Fig. 17. Frequency distributions of (a) length and (b) tortuosity ofbranches for selected networks of Column 1.

more than 60% of the branches of Networks 6, 18, and19 have a length of 10 mm. Since Network 32 has onlyone 150-mm long branch, and Network 44 has only one79-mm long branch, a single peak reaching 100% intheir distributions was observed.

The distributions of the tortuosity of these networkbranches are shown in Fig. 17b. The distributions donot suggest a trend, since tortuosity of the branchesseems to vary significantly from one network to another.Again two peaks reaching 100% can be observed forNetworks 32 and 44 for the same reason discussed pre-viously.

One of the simplest concepts for characterizing poretopology is the coordination number (Z), which is de-fined as the average number of branches meeting at aconnecting node. In mathematical terms, the averagecoordination number of a network can be written as inEq. [3]:

Zavg 5 on

i51

Zi fi [3] Fig. 18. Branch-node chart for Network 6 of Column 1.

where Zi is the number of branches connected to a node sentation of the 3-D arrangement of the pore networksof type i, and ƒi is the relative frequency of such nodes. in a 2-D plane. Network 6 was selected because of itsA branch-node chart was constructed for Network 6 of high number of branches. The branch-node chart ofColumn 1 to illustrate the concept of average coordina- Network 6 gives an idea of its ability to transmit a fluid.

More precisely, it indicates branches that may act astion number (Fig. 18). A branch-node chart is a repre-

1542 SOIL SCI. SOC. AM. J., VOL. 63, NOVEMBER–DECEMBER 1999

ies between 13 421 to 23 562 macropore networks/m3 ofsoil. No direct relationship could be observed betweennumerical density and macroporosity. The position andthe length of macropore networks were evaluated. Theartificial macropore installed in one of the soil columnswas readily detected. It was found that the majority ofthe macropore networks had a modal length of 40 mm, avolume of 60 mm3, and a wall area of 175 mm2. However,some macropore networks, although representing onlya small percentage, could reach a length of 750 mm, avolume of 10 000 mm3, and a wall area of 50 000 mm2.

The hydraulic radius in three dimensions was alsoassessed as an indication of the ability of the networksto convey water. It was found that the greater the lengthof networks, the greater the hydraulic radius. On aver-

Fig. 19. Connectivity of macropore networks and their number of age, macropore networks had a hydraulic radius ofbranches (Column 1).0.13 mm.

Our results on network inclination suggest that itpreferential flow paths, as well as a dead-ended set ofranges from vertical to an angle of about 558 from thebranches that will not be part of the main channels.vertical. The overall tendency of network inclinationForty-two connecting nodes can be found on Networkdistributions suggests that the smaller the inclination,6. Three branches connect on 38 nodes. The remainingthe greater the number of macropores.four nodes connect four branches. Therefore, the coor-

Results for tortuosity indicated that the majority ofdination number of Network 6 can be calculated as inthe networks had a tortuosity between 1 and 1.4. TheEq. [4]:mode of the tortuosity distributions suggested that mostmacropore networks had a 3-D tortuous length 15%Zavg 5 3 3

3842

1 4 3442

5 3.09 [4]greater than the distance between its extremities. It wasfound that some macropore networks had a tortuosityThe average number of branches meeting at a node isas high as 2.4.3.09. Like tortuosity, this gives an indication of increased

More than 60% of the networks were made up ofresistance to flow due to the degree of branchedness offour branches. Our results for Column 1 suggested thatthe macropore network.82% of the networks had a connectivity of 0 (zero). TheThe genus or connectivity of Network 6 was alsoconnectivity density was equal to 4772 nonredundantevaluated. This was achieved by counting the numberloops/m3.of nonredundant loops enclosed in Network 6. Two

The 3-D arrangement of networks of soil macroporesnonredundant loops can be isolated in Fig. 18. Thus,plays a determining role in the rate of water and solutethe connectivity of Network 6 is equal to 2. Similarly,movement through soil. These results can be used tothe connectivity was calculated for every network ofdetermine and quantify the effect of 3-D geometry ofColumn 1. Results are shown in Fig. 19. The numbermacropore network on solute transport through soilof branches per network should increase the probabilitycolumns.of finding nonredundant loops in the 3-D structure of

the networks. Therefore, the connectivity and the num-ACKNOWLEDGMENTSber of branches were plotted on the same graph to verify

this relationship for each macropore network. However, The authors wish to thank Daniel Marentette for his helpand suggestions in the technical part of this work. The authorsno direct relationship can be observed in Fig. 19. Mac-also gratefully acknowledge the financial support provided byropore networks with one branch do not contain loopsthe Natural Sciences and Engineering Research Council ofand therefore have a connectivity equal to 0. As men-Canada (NSERC) and the Environmental Science and Tech-tioned earlier, the term connectivity density is sometimesnology Alliance Canada (ESTAC).used to define the connectivity per unit volume. In the

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