three-dimensional quantification of macropore networks in undisturbed soil cores

Three-Dimensional Quantification of Macropore Networks in Undisturbed Soil CoresJohan Ferret, S. O. Prasher,* A. Kantzas, and C. Langford

ABSTRACTThe role of macropores in soil and water processes has motivated

many researchers to describe their sizes and shapes. Several ap-proaches have been developed to characterize macroporosity, suchas the use of tension infiltrometers, breakthrough curve techniques,image-analysis of sections of soils, and CAT scanning. Until now,efforts to describe macropores in quantitative terms have been concen-trated on their two-dimensional (2-D) geometry. The objective ofthis study is to nondestructively quantify the three-dimensional (3-D) properties of soil macropores in four large undisturbed soil col-umns. The geometry and topology of macropore networks were deter-mined using CAT scanning and 3-D reconstruction techniques. Ourresults suggest that the numerical density of macropores varies be-tween 13 421 to 23 562 networks/in3 of sandy loam soil. The majorityof the macropore networks had a length of 40 mm, a volume of 60mm3, and a wall area of 175 mm2. It was found that the greater thelength of networks, the greater the hydraulic radius. The inclinationof the networks ranged from vertical to an angle of = 55° from vertical.Results for tortuosity indicated that most macropore networks had a3-D tortuous length 15% greater than the distance between theirextremities. More than 60% of the networks were made up of fourbranches. For Column 1, it was found that 82% of the networks hadzero connectivity. This implies that more than 4/5 of the macroporenetworks were composed of only one independent path between anytwo points within the pore space.

SOIL STRUCTURE consists of a 3-D network of pores.Large pores play an important role in allowing

roots, gas, and water to penetrate into the soil. Thehigher the macropore density, the more the soil can be

Johan Ferret and S.O. Prasher, Dep. of Agricultural and BiosystemsEngineering, McGill Univ., 21 111 Lakeshore Rd., Ste-Anne-de-Belle-vue, QC, Canada, H9X-3V9; A. Kantzas, Dep. of Chemical and Petro-leum Engineering, Univ. of Calgary, 2500 University Dr. N.W., Cal-gary, AB, Canada, T2N-1N4; and C. Langford, Dep. of Chemistry,Univ. of Calgary, 2500 University Dr. N.W., Calgary, AB, Canada,T2N-1N4. Received 28 Sept. 1998. * Corresponding author ([email protected]).

Published in Soil Sci. Soc. Am. J. 63:1530-1543 (1999).

exploitable by plant roots (Scott et al., 1988a). Similarly,the more continuous the macropores are, the morefreely gases can interchange with the atmosphere. Con-tinuous macropores also have a direct effect on waterinfiltration and solute transport in soil.

According to Sutton (1991), the size of pore openingsis more important for plant growth than is the overallsoil porosity. Although existing pores constrain the pen-etration of roots, they provide favorable conditions forroot growth. Several studies have shown that the pres-ence of continuous macropores in soil significantly bene-fits root growth (Bennie, 1991). One of the most impor-tant factors influencing soil fertility, besides water andnutrient content, is soil aeration (Hillel, 1980; Glinskiand Stepniewski, 1985). Large soil pores are the pathsavailable for gas exchange between soil and atmosphere(Sutton, 1991). In natural soils, water movement followspaths of least resistance (i.e., preferential flow paths).Intuitively, large and continuous pores facilitate watertransport. It is now well known that the size and connec-tivity of soil pores play a major role in the flow character-istics of water and the transport of solutes through soil(Ma and Selim, 1997). Jury and Fliihler (1992, p. 192)stated that "fluid transport through well defined struc-tural voids is not predictable unless the distributions ofthe voids, aperture sizes and shapes, depths of penetra-tion, and interconnectivity are known."

The importance of macropores in many soil-plant-roots processes has motivated many researchers to de-scribe their sizes and shapes. Several approaches havebeen developed to characterize macroporosity. Amongthem are tension infiltrometers (Everts and Kanwar,1993; Timlin et al., 1994; Logsdon et al., 1993), break-through curve techniques (Ahuja et al., 1995; Jabro et

Abbreviations: CAT, computer-assisted tomography; CT, computedtomography; ECD, equivalent cylindrical diameter; HU, HounsfieldUnits; 2-D, two-dimensional; 3-D, three-dimensional.

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

al., 1994; Li and Ghodrati, 1994; Ma and Selim, 1994),and image-analysis of sections of soils (Koppi andMcBratney, 1991; Moran and McBratney, 1992; Singhet al., 1991; Vermeul et al., 1993). Warner et al. (1989),Grevers et al. (1989), Anderson et al. (1990), Hansonet al. (1991), Tollner et al. (1995), Asare et al. (1995),and Heijs et al. (1996) have also recognized the greatpotential offered by computer-assisted tomography(CAT) scanning for characterizing soil macroporosity.

Nevertheless, efforts to describe macropores in quan-titative terms have not yet resulted in a comprehensivetheoretical framework that allows a complete represen-tation of their geometry. This is partly due to the factthat macropores are very difficult to observe and charac-terize, bearing in mind that the macropore networksare complex 3-D structures. Up to now, most of thework done on the quantification of soil macropores con-centrated on their 2-D geometry. A few researchers,such as Hanson et al. (1991), Heijs et al. (1996), andFerret et al. (1997), have used reconstructive imagery,allowing 3-D visualization of the macropore space, buthave not quantified soil macroporosity in three dimen-sions. Transport phenomena in porous media stronglydepend on the 3-D pore structure (Chatzis and Dullien,1977). Very little work has been done to characterizethe macroporosity of intact soil cores in terms of its3-D parameters. Tollner et al. (1995) pointed out theneed for additional research to investigate reliable ap-proaches for computing tortuosity and connectivity ofsoil macropores. Imaging in three dimensions and quan-tification of 3-D parameters of soil macroporosity arecritical in order to accurately correlate soil pore struc-ture with preferential flow phenomena and much addi-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 aporous medium that affect fluid flow are porosity, nu-merical density, pore shape (size, length, volume, hy-draulic radius, tortuosity), and pore interconnectivity orgenus per unit volume (Constantinides and Payatakes,1989). Therefore, the objective of this study is to quanti-tatively determine the above parameters of soil mac-ropores in four undisturbed soil columns through a 3-Dreconstruction from 2-D matrices generated by an x-rayCAT scanner. The results of this work show promisefor future studies in the area of soil macropore quantifi-cation.

TerminologyVarious standard geometrical parameters are meaningful

to quantify the structure of 3-D macropore networks, such astheir relative position, length, volume, specific wall area, andorientation. Such parameters are explicit and do not need tobe defined. However, several terms that are used to describesoil structure and their 3-D attributes are fuzzier and need tobe clarified. Some of these terms are defined below:

MacroporeAt first, the definition of a macropore may seem simple.

However, as we come to consider the complexity of a mac-

Table 1. Different classifications of pores based on their Equiva-lent Cylindrical Diameter.

Terminology

MicroporeMesoporeMacroporeMicroporeMacroporeMicroporeVery fine poreFine poreMedium poreCoarse poreCryptovoidUltramicrovoidMicrovoidMesovoidMacrovoidVery fine poreFine poreMedium poreCoarse poreMicroporeMiniporeMacroporeSuper poreBonding poreResidual poreStorage poreTransmission poreFissureMacroporeMacropore

MacrofissureEnlarged macrofissureMacroporeMicroporeMesoporeMacroporeMacropore

ECD<30 [Am30 [Am-100 iim>100 (Am<30 [Am>30 u,m<75 (Am75 fun-1000 (Am1000 (Am-2000 [Am2000 (Am-5000 (Am>SOOO (Am<0.1 (Am<5 (Am5 u.m-30 (Am30 (Am-75 [Am>75 (Am<2 [Am2 iim-20 (Am20 u.m-200 (Am>200 [Am<0.3 (Am0.3 |Am-30 iim30 (Am-300 (Am>300 (Am>0.005 (Am<O.S (Am0.5 (iin-50 (Am50 (Am-500 (Am>500 (Am>1000 M-m>60 (Am

200 (Am-2000 (Am2000 (Am-10 000 (Am>3000 (Am<10 iim10 [Am-1000 (Am>1000 iim>1000 [Am

Reference

Jongerius (1957)

Marshall (1959)

Johnson et al. (1960)

Brewer (1964)

Russell (1973)

Mclntyre (1974)

Greenland (1977)

Bouma et al. (1977)Bullock and

Thomansson (1979)Reeves et al. (1980)

Beven (1981)Luxmoore (1981)

Luxmoore et al. (1990)

ropore, its definition may become hazy and ambiguous. Bystrict definition, macropore implies a large pore. However,large is a relative term and this lack of clarity has led to severalconflicting definitions.

It would be useful to have a general agreement on poreterminology, just as one exists for the definitions of sand, silt,and clay. However, up to now there has been little or noconsensus for the definition and terminology used for classify-ing pores in general. A large number of classification schemesbased on the equivalent cylindrical diameter (ECD) and sev-eral contradicting definitions can be found in soil literature(Table 1).

These definitions fail to remove ambiguity from the de-limitation between macropores and micropores. However, inthis research study, the definition proposed by Luxmoore etal. (1990, p. 144) was followed. They stated that "The termmacropore includes all pores in a profile that are (generally)drained at field capacity, with the latter being 1 mm or morein equivalent diameter."

Up to now, definitions of macropore make no specific refer-ence to its size or shape in a 3-D context. The lack of informa-tion describing their shapes has led to the generality used indefining macropores (Kwiecien, 1987). Part-of the goal of thisstudy is to characterize the shapes and other 3-D parametersof macropore networks with the aid of computer programs,in order to describe macropore geometry and, eventually, tohelp 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

_ Volume of poreWall area of pore [2]

Fig. 1. Tortuosity of a soil macropore.

Macropore Network and BranchesIn a porous medium, a network is a set of macropores that

are interconnected such that there is a passage from any partof the set to every other part (Scott et al., 1988a). Thus, theconcept of a macropore network implies a 3-D structure. Abranch is a portion of the macropore network connecting tothe rest of the network.

TortuosityTortuosity (T) is one of the most meaningful 3-D parameters

of pore structure. Carman (1937) introduced the concept oftortuosity as the square of the ratio of the effective averagepath in the porous medium (Le) to the shortest distance mea-sured along the direction of the pore (L). Several researchershave reviewed this definition (Hillel, 1982; Marshall andHolmes, 1988; Jury et al., 1991; Sahimi, 1995) and have rede-fined tortuosity (Eq. [1]):

Tortuosity is a dimensionless factor always greater than one,which expresses the degree of complexity of the sinuous porepath (Fig. 1). Tortuosity can easily be related to the conductiv-ity of a porous medium since it provides an indication ofincreased resistance to flow due to the pore system's greaterpath length (Dullien, 1979). The term continuity is sometimesused to describe pore tortuosity. Richter (1987) has definedpore continuity as the reciprocal of tortuosity.

Hydraulic Radius in Three DimensionsAnother relevant parameter is the hydraulic radius of the

macropore network. Macropores are not regularly shaped.The neck (also known as the pore throat) of a pore is animportant feature of pore geometry, which directly controlspercolation rates. A neck is defined as the local minimum inpore space size (Kwiecien et al., 1990). The pore throat islocated where the minimum mean radius of curvature of agas-liquid interface is observed. This corresponds to the loca-tion of maximum capillary pressure. Maximum capillary pres-sure within a macropore is very difficult to measure. However,the hydraulic radius is a good approximation of the meanradius of curvature (Kwiecien, 1987; Dullien, 1992). The hy-

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 NetworksA 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 macroporenetworks, regardless of their shape. The number of networks,defined later as numerical density, is a measure of the complex-ity of pore structure (Scott et al., 1988a). Topological parame-ters characterizing the morphology of a porous medium arethe numerical density, coordination number, connectivity, andgenus (Dullien, 1992). Each of these terms is defined below.

Numerical Density of NetworksThe number of networks per unit volume, regardless of

their size or shape, is the numerical density. Scott et al. (1988a)pointed out that it is very difficult to determine this quantity.Up to now, this information was roughly estimated by cuttingparallel plane sections through soil. Essentially, numericaldensity was only accessible in two dimensions. One of themajor drawbacks of this approach is that there is not a one-to-one correspondence between the number of networks esti-mated in the 2-D sections and those in three dimensions (Scottet al., 1988a).

Coordination NumberOne of the simplest concepts for characterizing pore topol-

ogy is the coordination number (Z). It is defined as the numberof pore throats that meet at a given point along a pore (Sahimi,1995). In other terms, the coordination number determinesthe number of branches meeting at one node. Until now, theonly approach to determining the coordination number hasbeen to reconstruct a branch-node chart of the pore structure(Fig. 2) from a series of parallel sections of the porous medium(Dullien, 1992).

Connectivity and Genus of Macropore NetworksThe concept of connectivity (Ccon) can also be used to char-

acterize the topology of a complex system such as soil mac-roporosity. Connectivity is a measure of the number of inde-pendent paths between two points within the pore space(Macdonald et al., 1986). In other words, connectivity is thenumber of nonredundant loops enclosed by a specific geomet-rical shape. Each macropore network has a connectivity, whichis a positive integer equal to the number of different closedcircuits between two points in the network. If there is onlyone open circuit, the connectivity is equal to 0 (zero); theconnectivity is 1 if the circuit is closed. The term connectivitydensity is sometimes used to define the connectivity per unitvolume (Scott et al., 1988b). Figure 3 shows four different


branch

(a) (b)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,and d display a connectivity of 1, 1, 2, and 3, respectively).

The genus of a pore system is defined as the largest numberof nonintersecting cuts that can be made through a shapewithout disconnecting any part from the rest (Dullien, 1992).In Fig. 3a and b, only one cut can be made through the porestructure without creating two independent networks (the cutsare represented as ellipses intersecting the pore). Thus, thegenus of both structures shown in Fig. 3a and 3b is equal to1. Since two and three cuts can be made in the pores shownin Fig. 3c and 3d, their genus is equal to 2 and 3, respectively.A general theorem of topology states that the genus is numeri-cally equivalent to connectivity. Macdonald et al. (1986)strongly suggested that an accurate determination of the genuswould help to elaborate new and better flow models.

Although these topological concepts have recently beengiven increased attention in the field of petroleum recovery,these concepts have not yet been used to describe the spatialcharacteristics of a soil macropore network. For this study,we propose evaluating the coordination number and the con-nectivity-genus of soil macropores in order to describe theircomplex geometry.

MATERIALS AND METHODSSoil Cores

In July 1995, four undisturbed soil columns, 800 mm inlength and 77 mm in diam., were taken from a field site atthe Macdonald Campus of McGill University in Ste-Anne-de-Bellevue, QC. The columns were extracted from an unculti-vated field border that had been covered for many years bya combination of quack grass [Elytrigia repens (L) Desv. exNevski], white clover (Trifolium repens L.), and wild oat(Avena fatua L.). Periodic mowing during the summer wasthe only cultural practice used. The land slope was <1%.Column size was selected based on the need for a sample thatwas large enough to represent macropore distribution, yetsmall 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). Thesoi] was predominantly a sandy loam with an A horizon thick-ness of ~QA m.

The soil columns were scanned under dry as well as satu-rated conditions. To reach dry conditions, the soil columns

were placed under a set of 10 300-W light bulbs for a periodof 6 wk. The cores were rotated periodically to accelerate evap-oration.

A 1-mm-i.d. polyethylene tube was inserted into one of thesoil columns to verify the ability of the CAT scanner to portraythe size and the location of a known macropore.

X-Ray CAT ScanningA modified medical ADVENT HD200 whole-body CAT

scanner (Universal Systems, Solon, OH) was used for thisstudy at the TIPM laboratory in Calgary, AB. Computed to-mography (CT) or computer-assisted tomography (CAT) isa method of diagnostic imaging used for nondestructive im-aging of cross-sectional slices of the human body or an object.This scanner incorporates a fourth-generation scan geometrywith scan times as short as 2s/scan and a high pixel resolution(up to 195 X 195 (jun). In most CAT scanners, the actualcollection of patient data occurs in the gantry where the patientlies horizontally. However, the ADVENT HD200 was modi-fied to allow vertical scanning. For that purpose, the CATscanner gantry was rotated 90° and positioned on a metalframe designed to hold the whole gantry horizontally. Figure4 shows the rotated gantry of the CAT scanner.

During CAT scanning, each column was placed verticallyin the scanner unit so that the x-ray beam intersected the soilcolumn perpendicularly to its longitudinal axis. A bubble levelindicator was used to ensure that the soil column was vertical.The longitudinal axis of the core was positioned at the centerof the gantry of the scanner. During the computer tomographicprocess, the x-ray tube rotated around the soil column. A pre-collimator modulated the thickness of the x-ray beam. Forthis study, the collimator was set to a thickness of 2 mm. Thetransmission and detection of this thin, rotating x-ray beamthrough the soil column resulted in a large number of attenua-tion measurements taken at discrete angles. For this purpose,an array of 720 detectors was located within the gantry. Oncecollected, the data were mathematically reconstructed to gen-

o(a) (b) (c) (d)

Fig. 3. Illustration of the concept of connectivity and genus (fromMacdonald et al., 1986).


(a) (b)

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 ±0.001 mm.A total of 360 sections or scans was obtained for each column,leaving no space between two consecutive scans.

Data from each scan were recorded on a magnetic tape,transferred to a SUN4 workstation running under the UNIXoperating system, and converted to a bulk density value. First,the scans were stored in matrices composed of CT numbersthat were expressed in a dimensionless quantity known asHounsfield Units (HU). The CT number for water is roughly0 and - 420 for air. The CT values are a function of the electrondensity (bulk density) and atomic number of the material. Ithas been previously demonstrated that the CT numbers arelinearly related to the bulk density of the soil (Anderson etal., 1988). The soil columns were mounted inside a core holderassembly made of a hollow Plexiglas annulus, partitioned intofour chambers filled with two liquids. Water and mineral oilwere used as reference materials. By plotting CT numbers vs.the bulk density of reference materials and using a simplelinear regression, a calibration curve that relates bulk densityto the CT number of the scanned material was derived. AFORTRAN 77 program was developed to compute the cali-bration line of each scan. Once the linear calibration equationswere established, a computer algorithm transformed CAT-scan arrays into matrices of bulk density. Part of the algorithmalso allowed for computation of the soil section's porositydistribution on a voxel (i.e., volume element) basis. The re-maining analysis was done using the PV-WAVE language ona 300 MHZ Pentium II with 128 Mb of RAM.

3-D Reconstruction of MacroporesThe PV-WAVE language was chosen for computer pro-

gramming in this study. The PV-WAVE language is a compre-hensive programming environment that integrates state-of-the-art numerical and graphical analyses. This programminglanguage is widely used for analyzing and visualizing technicaldata in many fields, such as medical imaging, remote sensing,and engineering. PV-WAVE is an ideal tool for working withlarge arrays such as our CAT-scan data, because of its array-oriented operators and ability to display and process datain the ASCII and binary I-O formats. Another reason thatmotivated this choice was PV-WAVE's ability to be usedunder both UNIX and PC environments.

Top Scan

r Middle Scan

Bottom Scan

(c) 1 Macropore 1 Soil Matrix

Fig. 5. Illustration of (a) the six nearest neighboring voxels rule and(b) 26 neighboring voxels rule; (c) superposition of consecutive2-D matrices for the 3-D algorithm.

Four computer programs were developed to reconstruct,visualize and quantify 3-D macropore structures in soil col-umns. The first program, called Filterjo.pro, thresholds themacropores in the 360 2-D bulk density matrices before isolat-ing macropore networks in three dimensions. Each pixel canrepresent only two states of the dry soil columns, pore space,or soil matrix. The first task accomplished by Filterjo.pro isto partition 2-D matrices into regions of 1 for pores and 0 forsoil matrix. The pores contain either water or air (i.e., density<1). Thus, the pore can be isolated by applying a thresholdon all the pixels in the bulk density matrices having a valueless than 1. This transformation is called segmentation. Theprogram then regroups pixels belonging to the same pore(clustering) by following a set of rules described by Ferret etal. (1997). A filtering subroutine is then executed. For thatpurpose, a criterion is used to determine if the pore belongsto the macropore domain. The criterion is based on the sizeof the pore. If the pore has an BCD <1 mm, it is removedfrom the matrix. After filtering, a median smoothing is appliedon each matrix with a neighborhood of two pixels. This processis similar to smoothing with an average filter, but it does notblur edges larger than the neighborhood. Median smoothingwas used since it has been found to be effective in removingnoise (Visual Numerics, 1994). Each pixel in the resultingmatrix is then multiplied by -1. Therefore, matrix elementshave a value of 0 (i.e., soil) or —1 (i.e., macropores). Theresulting 2-D matrices are then stored in a file ready for the3-D analysis.

A second computer program, called Netjo.pro, was devel-oped to recognize and isolate 3-D macropore networks. Thefirst step in developing this program was to establish a set ofrules, which were used to determine how the macropore spacesin each 2-D matrix were connecting to each other in 3-D.Several different clustering criteria were used. The first algo-rithm was based on the six nearest neighboring voxels rule inthree dimensions (Fig. 5a). With this algorithm, similar voxels


(with a value of -1) 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 — 1 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,the algorithm visits each voxel for every section until it findsa voxel belonging to Network 1. Then, using the six nearestneighboring voxels rule, Branjo.pro isolates and clusters eachvoxel of the first branch of the network. As the branch isbeing clustered, the program computes the perimeter, surfacearea, and centroid of the branch in each section. When theprogram finds a connection to a new branch, it stores thelocation of the connection so that it can investigate propertiesof this new branch at a later time. Once the program hasreached the end of the branch, it computes the branch's tortu-osity, orientation, length, volume, wall area, hydraulic radius,and the number and location of other connecting branches.At that stage, the program moves on to the location of a storedconnection (if present) and evaluates the properties of thenew branch. Once all the branches of a particular networkhave been analyzed, results are written to an ASCII file forfurther analysis on a worksheet. Then, the program repeats thesame process for the second macropore network (Network 2)and so on to Network n. A detailed flow diagram of Branjo.prois given in Fig. 7.

RESULTS AND DISCUSSIONNumerical Density, Relative Position, and

Length of Macropore NetworksFigure 8 shows the number, vertical position, and

length of macropore networks found in soil columns.

Fig. 6. Three-dimensional reconstruction of macropore networks inColumn 1.

Each vertical bar represents a macropore network. Aninteger number has been assigned to each network(from left to right) for reference. For example, Network18 of Column 1 starts at a depth of =250 mm and endsat a depth of 680 mm. The total number of networksper soil column is summarized in Table 2. Column 3has the greatest number of networks (i.e., 79 networks).The numerical density was calculated from the numberof networks per soil column (Table 2).

It can be observed that the numerical density variessignificantly from one soil column to another, althoughthey were taken from the same site with a distanceof only =0.5 m between them. The macroporosity wasevaluated for each soil column (Table 2) and was inaccordance with the observations made by Edwards etal. (1990), who reported macroporosities ranging from0.4 to 3.8%.

One would expect that the number of macroporenetworks per unit volume would have a consequentialeffect on macroporosity. More precisely, it would makesense that soil columns with a large numerical densitywould have a large macroporosity and vice-versa. How-


START ProgramsBRANJO.PRO J

Declare variables /

Visit each pixel of 2DMatrix

Nogo to next2D section

:3D Clusteringregroup voxels

belonging to branch

[Calculate and StoreM-Perimeter

- Surface Area•Position ofCentroidJ

Yes

Yes3D Analysis of the Branch

Compute and store:-Tortuosity-Orientation

-Volume-Wall Area

-Hydraulic Radius- Number and Location of

Connecting Branches

Gotoconnectingbranch at

Rem(n,x,y,z)

Yes-

/ Write Results )( to output file /

Fig. 7. Flow diagram of the PV-WAVE program Branjo.pro.

ever, our results do not confirm this supposition or indi-cate a direct relationship between numerical density andmacroporosity. For instance, Column 1, which has thesmallest numerical density (i.e., 13 421 macropore net-works/m3), has the greatest macroporosity. Column 3(with a numerical density of 23 562 networks/m3) exhib-its a much smaller macroporosity (i.e., 2.59%). Theseresults can be explained by the difference in the averagenetwork volume. The average network volume in Col-umn 1 is more than 2.5 times that of Column 3. Thisexplains the relatively high macroporosity of Column1. Therefore, one cannot use numerical density as anindication of macroporosity.

The vertical length and position of macropore net-

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.

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 ofthe network to convey fluids. The greater the hydraulicradius, the greater its transport capacity. The hydraulicradius of every network found in the soil columns ispresented in Fig. 12.

It can be observed that there is no apparent relation-ship between hydraulic radius and depth. However,longer networks have greater hydraulic radii thanshorter networks. For instance, the artificial macroporeof Column 4 running from top to bottom of the soil hasthe greatest hydraulic radius of the column. Since longpores have a greater ability to convey water, these re-sults are expected.

The frequency distributions of hydraulic radii were


Networks of Column

c0

100

200 -

E 300 -

£Q. 400 -

o500 -

600

700

) 10

II MlI1,

1I

1

20

1

30 40

Irr'I1 Ml

Ih

Jl

1 Networks of Column 2

5

1

1

0 00 -

100 -

200 -

| 300

£a 400Q

500

600 -

700

''li

10 20 30

„

I

jlhM

1K

40 5(

'' I-,!

'I-nr

Networks of Column 3

0 10 20 30 40 50 60 70 80

Networks of Column 4

0 10 20 30 40 50 60

100 -

200

| 300

JCa 400 -0)a

500

600

700

ll'jll V-"1\1'

T\\IV\\

100 -

200

| 300 -

£0. 400»Q

500

600

700

\\"'I'v

"""„

-4 I1

_|L_

l|,,1

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 SoilProperty column 1 column 2 column 3

Soilcolumn 4

Number of networks 45 47 79 54Numerical density

(networks/in3) 13421 14018 23562 16106Macroporosity 3.8% 2.18% 2.59% 2.79%Volume (mm3) 67.4 Q, = 24t 51.2 16 57.7 25 52.5 21Geometric mean Q2 = 48 42 41 53

Q3 = 129 142 114 93Q, = 9674 6132 3079 9948

Tortuosity (mm/mm) 1.25 Q, = 1.10 1.18 1.09 1.28 1.14 1.26 1.12Geometric mean Q2 = 1.17 1.14 1.19 1.18

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

Average hydraulicradius (mm) 0.14 ± O.OSt 0.13 ± 0.07 0.12 ± 0.05 0.13 ± 0.04

t Qi, Qi, Q3, and Q4 represent the 1", 2nd, 3rd, and 4th quartiles.I Standard deviation.


Column 1Column 2Column 3Column 4

Network Length (mm)Fig. 9. Frequency distributions of the length of macropore networks

in the four soil columns.

also assessed for every column and are shown in Fig.13. The distributions of hydraulic radii are almost sym-metrical with a mode of =0.13. Here again, networks inall soil columns show a similar trend.

Inclination and TortuosityFigure 14 shows the frequency distributions of the

inclination of macropore networks. The inclination ofnetworks ranges from vertical to an angle of about 55°from vertical for some networks. The overall tendency

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 ofmacropore networks found in the four columns. Thedistributions are similar and skewed to the right, witha mode of = 1.15. Most of the networks have a tortuosityin the range 1 to 1.4. Thus, the majority of the macroporenetworks have a 3-D tortuous length 15% greater thanthe distance between their extremities. Some macroporenetworks have a tortuosity as high as 2.4.

Number of Branches, Branch-Node Chart,and Connectivity

Figure 16 shows the distributions of the number ofbranches per network for all four soil columns. The


100 1000 10000

Network Volume (mm )Fig. 10. Frequency distributions of volume of macropore networks in the four soil columns.


100 1000 10000 100000

Wall Area (mm )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

Column 1Hydraulic Radius (mm)0 0.1 0.2 0.3

Column 2Hydraulic Radius (mm)

0 0.1 0.2 0.3 0.4

0 - —————

100 - ———— -

200 -j ————

1 300 —————

£ i l lfc 40° ————— ~<a | iO

500 -j ———— -1i

|700 •] ————

1

\

1

' ', "I1

11

T~t—

|l1

-j+U

i

§

'• : :'l100 -. ——— '--

200 : — i— ' ——'•

E ' ,1!£ 300 ——— l! ——

—— « 400 ————— -« I0 500—— V|

600 :-t— 700 || :

i , ;i : !

1 1 \ j

! iIj j

'; i i; '. I

i j1 !

i i

_ _ _ _ —————— J ————————— ̂ ___

Column 3Hydraulic Radius (mm)0 0.1 0.2 0.3 0.4

Column 4Hydraulic Radius (mm)0 0.1 0.2 0.3

U —————— -r

100 —— h ——

I200 —————

I 3°°

~ 400 ———— -)0) IQ II

500 ——— p-

I600 ——— h1-

700 —— ——

i

"

^1

1

i— r ~

I 1, • " I100 ————————. 1

_L ———————— 200 ——— —— J- -? , |u ,

——— § 30° — H|;i1 ! .e- —————— a «o ——— j-^

« I.

-j ! 500 ^ ——— 1 ——

! : :"'——— , ————— : 600 - ——————— -

i '

J ————————— 700 —————— : —

| 'i

f~vIj

:

j

j

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 numbermode of the distributions suggests that most macropore of branches.networks are made up of approximately four branches. Results presented above were obtained by analyzingThe distributions are skewed to the left, indicating the each network in the soil columns. As mentioned earlier,

Column 1

Column 2

Column 3

Column 4

0)

o>

Hydraulic Radius (mm)Fig. 13. Frequency distributions of hydraulic radius of macropore networks in the four soil columns.


Column 1Column 2Column 3

Column 4

Angle from Vertical (degree)Fig. 14. Frequency distributions of macropore network inclination.

Tortuosity (mm/mm)Fig. 15. Frequency distributions of tortuosity of macropore networks.

• Column 1-Column 2-Column 3• Column 4

=*=20 30 40 50 60 70

- Number of Branches / NetworkFig. 16. Frequency distributions of number of branches per macropore network.

8>,oc0)3

ofLJL

25 -

20

15

10 -

5

A —»— Column 1L\Q — »- Column 2

j^VVb -*- Column 3

UlA\ ~^~ Column 4

/F TSfl*/ / rM/ M AAA a

2.4

80

a macropore network is a set of branches that are inter-connected. The parameters that have been evaluatedfor each network (i.e., number of networks, relativeposition, length, wall area, volume, hydraulic radius,orientation, and tortuosity) can be assessed in a similarfashion for each branch of every macropore network.However, on average, the number of branches per soilcolumn was calculated to be 288. To include analysis ofeach branch of all four soil columns in this paper would

be too exhaustive. Therefore, we decided to limit theinvestigation to branches of five large networks of Col-umn 1 only.

The distributions of branch lengths of Networks 6,18,19, 32, and 44 in Column 1 are presented in Fig. 17a.Only networks that have a length greater than averagewere selected. As indicated in Fig. 8, Networks 6, 18,19, 32, and 44 of Column 1 have a length greater thanthe average macropore network. Fig. 17a suggests that


(a

80 -

~

x 6° -oc<o§• 40 -£

11

II20 | I

0 i

) (

^ "\ ,

— •— Network 6

-•—Network 18—*— Network 19-x- Network 32—o— Network 44

A A3 100 200 300 400

Length (mm)

100

(b)1.4 1.6

Tortuosity (mm/mm)1.8

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]:

7 =^av 7 fZ-J\ Ji [3]

where Z-, is the number of branches connected to a nodeof type i, and /j is the relative frequency of such nodes.A branch-node chart was constructed for Network 6 ofColumn 1 to illustrate the concept of average coordina-tion number (Fig. 18). A branch-node chart is a repre-

Column 1 - Network 6

Z = 3.09

Genus = 2Connectivity = 2

lillliill I o— Non-connecting Node

siiWHp •— Connecting NodemiiSim

Fig. 18. Branch-node chart for Network 6 of Column 1.

sentation of the 3-D arrangement of the pore networksin a 2-D plane. Network 6 was selected because of itshigh number of branches. The branch-node chart ofNetwork 6 gives an idea of its ability to transmit a fluid.More precisely, it indicates branches that may act as


432 -1 -

0 -10 -20 -30 -40 -5060 -70 -80 -90

u|u' p-u"uur|

"

u u~

10 15 20 25 30 35 40 45

Macropore Network

Fig. 19. Connectivity of macropore networks and their number ofbranches (Column 1).

preferential flow paths, as well as a dead-ended set ofbranches that will not be part of the main channels.

Forty-two connecting nodes can be found on Network6. Three branches connect on 38 nodes. The remainingfour nodes connect four branches. Therefore, the coor-dination number of Network 6 can be calculated as inEq. [4]:

38 4Zavc = 3 X ^ + 4X — = 3.09-avg 42 42 [4]

The average number of branches meeting at a node is3.09. Like tortuosity, this gives an indication of increasedresistance to flow due to the degree of branchedness ofthe macropore network.

The genus or connectivity of Network 6 was alsoevaluated. This was achieved by counting the numberof nonredundant loops enclosed in Network 6. Twononredundant loops can be isolated in Fig. 18. Thus,the connectivity of Network 6 is equal to 2. Similarly,the connectivity was calculated for every network ofColumn 1. Results are shown in Fig. 19. The numberof branches per network should increase the probabilityof finding nonredundant loops in the 3-D structure ofthe networks. Therefore, the connectivity and the num-ber of branches were plotted on the same graph to verifythis relationship for each macropore network. However,no direct relationship can be observed in Fig. 19. Mac-ropore networks with one branch do not contain loopsand therefore have a connectivity equal to 0. As men-tioned earlier, the term connectivity density is sometimesused to define the connectivity per unit volume. In thecase of Column 1, the connectivity density is equal to4772 loops/m3.

SUMMARY AND CONCLUSIONSX-ray CAT scanning has been a useful approach to

nondestructively quantify threshold macroporosity ofundisturbed soil columns. The main characteristics ofthe geometry and topology of macropore networks weredetermined using 3-D reconstruction techniques. Forthat purpose, several programs were written in the PV-WAVE programming language.

Our results suggested that the numerical density var-

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-age, macropore networks had a hydraulic radius of0.13 mm.

Our results on network inclination suggest that itranges from vertical to an angle of about 55° from thevertical. The overall tendency of network inclinationdistributions suggests that the smaller the inclination,the greater the number of macropores.

Results for tortuosity indicated that the majority ofthe networks had a tortuosity between 1 and 1.4. Themode of the tortuosity distributions suggested that mostmacropore networks had a 3-D tortuous length 15%greater than the distance between its extremities. It wasfound that some macropore networks had a tortuosityas high as 2.4.

More than 60% of the networks were made up offour branches. Our results for Column 1 suggested that82% of the networks had a connectivity of 0 (zero). Theconnectivity density was equal to 4772 nonredundantloops/m3.

The 3-D arrangement of networks of soil macroporesplays a determining role in the rate of water and solutemovement through soil. These results can be used todetermine and quantify the effect of 3-D geometry ofmacropore network on solute transport through soilcolumns.

ACKNOWLEDGMENTSThe authors wish to thank Daniel Marentette for his help

and suggestions in the technical part of this work. The authorsalso gratefully acknowledge the financial support provided bythe Natural Sciences and Engineering Research Council ofCanada (NSERC) and the Environmental Science and Tech-nology Alliance Canada (ESTAC).

three-dimensional quantification of macropore networks in undisturbed soil cores

Documents