Pore-scale measurements of solute breakthrough using ...hopmans.lawr.ucdavis.edu/papers-ppt-zip/volkerwrr2000.pdf · physical processes of water movement and contaminant trans-

Pore-scale measurements of solute breakthroughusing microfocus X-ray computed tomography

V. Clausnitzer and J. W. HopmansDepartment of Land, Air and Water Resources, University of California, Davis

Abstract. X-ray computed tomography (CT) offers distinct advantages to studyfundamental physical processes of water movement and contaminant transport in porousmedia. Tomography provides nondestructive and noninvasive cross-sectional or three-dimensional representations of porous media and has the potential to measure phasedistribution and species concentration at the pore scale. Sources of error are discussed forthe application of industrial microfocus CT to quantitative studies of flow and transport.Specifically, effective resolution and measurement uncertainties due to photon randomnessare considered for a miscible displacement experiment. A calibration method for themeasurement of solute concentration is proposed that accounts for the effect of beamhardening, which is characteristic for polychromatic industrial X-ray sources. The resultsof an X-ray microfocus CT experiment are presented, emphasizing the need to correct forbeam hardening and describing the inherent spatial variability of solute breakthroughthrough a glass-bead porous medium with an effective spatial resolution of approximately85 mm.

1. Introduction

X-ray computed tomography (CT) provides nondestructivecross-sectional or three-dimensional object representationsfrom the attenuation of electromagnetic radiation. Attenua-tion depends on the density and the atomic constituents of thematerial that is scanned. Since it has the potential to noninva-sively measure phase distribution and species concentration,X-ray CT offers significant advantages to study fundamentalphysical processes of water movement and contaminant trans-port in porous media.

Following the introduction of CT in the medical sciences[Hounsfield, 1973], the technique was adopted by Earth scienceresearch beginning in the early 1980s. Linear relationshipswere established between attenuation and soil bulk density[Petrovic et al., 1982; Anderson et al., 1990] and between atten-uation and volumetric water content [Anderson et al., 1988;Brown et al., 1987; Crestana et al., 1985; Hainsworth and Ayl-more, 1983; Hopmans et al., 1992]. Tomography has also beenused to measure two- or three-dimensional heterogeneity ofsoil bulk density and water content [e.g., Hopmans et al., 1994]and to monitor the transport and breakthrough of solutes inporous media [Steude et al., 1990; Vinegar and Wellington,1987]. These investigations were performed using medicalscanners, which provide a spatial resolution of about 0.1–1.0mm.

The application of industrial scanners, in contrast to medicaldevices, is not subject to biological dose restrictions and allowsspatial resolutions of 1 mm or less [Haddad et al., 1994] ifcombined with a synchrotron X-ray source providing paralleland monochromatized radiation. Industrial scanners have alsoproduced effective results when equipped with X-ray sourcesthat emit polychromatic radiation, such as produced by con-ventional tube sources. The introduction of microfocus X-ray

sources and cone-beam reconstruction methods [Jasti et al.,1993; Martz et al., 1993] has provided the capability to obtainhigh-resolution (;10 mm) three-dimensional data sets from asingle object rotation thereby drastically reducing the totalacquisition time needed [e.g., Klobes et al., 1997].

On the basis of the same principles of X-ray CT the radia-tion of g rays consists of high-energy photons emitted fromtransitions of atomic nuclei. Using radioactively decayingsources, monochromatic g-ray beams can be obtained at theexpense of longer acquisition times because of small photonfluxes. Brown et al. [1993] present a quantitative analysis ofsuch a system for porous-media applications.

As an alternative to radiation-based CT, magnetic field gra-dients are applied in nuclear magnetic resonance (NMR) im-aging to measure nuclear spin density within an object. UsingNMR, spatial information about porous medium propertiescan be obtained. NMR imaging can provide a spatial resolu-tion of 10–100 mm and superior time resolution. As an exam-ple, the technique has been successfully applied recently tostudy dispersion in porous media [e.g., Greiner et al., 1997]. Themajor limitation, however, is its inability to study samples con-taining ferromagnetic or paramagnetic materials [Gladden andAlexander, 1996] thereby excluding many soils.

Given the successful history of application of CT in generaland the increasing availability of industrial microfocus X-rayscanners in particular, this paper is intended as a contributionto define the realm of applicability of microfocus X-ray CTregarding quantitative studies of flow and transport in porousmedia. Specifically, effective resolution and measurement un-certainty due to photon randomness is considered in miscibledisplacement experiments.

2. Theory

2.1. Physical Background

X rays, like visible light, are a form of electromagnetic (EM)radiation with a wavelength of 0.3 pm–3.0 nm. The correspond-

Copyright 2000 by the American Geophysical Union.

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WATER RESOURCES RESEARCH, VOL. 36, NO. 8, PAGES 2067–2079, AUGUST 2000

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ing high energies of X rays enable their penetration of objectsimpermeable to light, allowing physical characterization fromattenuation measurements. For a detailed treatment of theprinciples of EM radiation the reader is referred to Knoll[1979].

2.1.1. Emission. Industrial and medical X-ray sourcesgenerate X rays by bombarding a specific target material (e.g.,tungsten) with electrons in an electron-ray tube. X-ray photonsare emitted from the target by two main mechanisms. First,deflection of incident electrons after collision with atomic elec-tron clouds in the target results in conversion of kinetic energyinto electromagnetic radiation; that is, radiation is emitted asthe electrons are decelerated. The resulting continuousbremsstrahlung X-ray energy spectrum has an upper bound atthe X-ray photon energy corresponding to complete conver-sion of the kinetic energy of an electron. For example, themaximum X-ray energy is 125 keV for a tube with a sourcepotential of 125 kV (Figure 1). Second, target atoms becomeexcited as orbital electrons are transferred to a higher energylevel by the incident radiation. As an excited atom returns to itsground state, characteristic X-ray photons are emitted with atotal energy equal to the difference between the excited andground energy levels. The characteristic photon-energy valuesdepend on the atomic constituents of the target material. Inthe resulting X-ray source spectrum the contributions by exci-tation appear as distinct peaks superimposed on the continu-ous bremsstrahlung spectrum. As an example the theoreticalenergy spectrum for a tungsten target using a source potentialof 125 kV is shown in Figure 1 by the bold curve.

2.1.2. X-ray interactions. Along the path through the ob-ject the photons of the X-ray beam interact with the atoms ofthe object material. Within the energy range of typical indus-trial sources (,1 MeV), photons are removed from the beamby two mechanisms. First, photons can be scattered out of thebeam path following the collision with object electrons. Sec-ond, they can disappear because of photoelectric absorption byan object atom whereby the atom ejects a photoelectron froma shell whose binding energy is less than the absorbed photon’senergy, with the excess energy converted to kinetic energy ofthe photoelectron. As a result of both removal mechanisms,the X-ray beam is attenuated by losing a certain fraction of its

intensity I [photons/s] per increment ds [L] of straight pathlength:

dI/ds 5 2mI , (1)

where the linear attenuation coefficient m [L21] is the prob-ability per unit path length that a given photon is removedfrom the beam. For a stationary heterogeneous porous me-dium, m is spatially distributed, whereas in transient flow andtransport experiments, m can also be a function of time owingto miscible or immiscible fluid displacement. Integration of (1)along the beam path yields

I/I0 5 exp S2EL

m dsD , (2)

where I0 and I are the X-ray beam intensities entering andexiting the object, respectively, and L is the length of the paththrough the object.

Owing to the physical nature of the attenuation process, m isrelated to the electron density encountered by the beam. Con-sequently, m depends on the mass fractions of the object ma-terial’s atomic constituents and total mass density. As a result,m is linearly related to concentration c of a specific solute inliquid solution. In addition, attenuation is a function of theX-ray radiation frequency or photon energy level E . The con-tributions to attenuation by scattering and photoelectric ab-sorption for many elements have been tabulated as a functionof photon energy [e.g., Hubbell, 1969], which allows computa-tion of theoretical linear attenuation coefficient values forcomposite materials. Figure 2 presents such theoretical rela-tionships of m as a function of E for the materials used in theexperimental study to be discussed in section 3.2. Although mgenerally decreases with increasing E , distinct peak valuesappear at the characteristic energy levels for photoelectric ab-sorption. For example, the specific increase at 33.3 keV in thetwo m(E) curves for NaI solutions of 50 and 100 mg/mL iscaused by photoelectric absorption, involving the atomic Kshell of iodine. Since such relationships between m and photonenergy for different materials are so distinct, theoretical m(E)

Figure 2. Linear attenuation coefficient as a function ofbeam energy for different materials. NaI values indicate con-centration in water solution.

Figure 1. Theoretical energy spectra for tungsten, computedfor a detector of 10 mm diameter at a distance of 10 cm fromthe source at 125 kV potential and an electron current of 0.1 mA.

CLAUSNITZER AND HOPMANS: MEASUREMENT OF SOLUTE BREAKTHROUGH2068

plots are used to select optimum source potential and the typeof filtering of the polychromatic X-ray sources needed.

2.1.3. Detection. Beam intensity values are determinedby integrating a measure of either the photon incidence rate orthe energy deposition rate over a defined time interval Dt . Incontrast to traditional devices that generate and count electri-cal impulses for each incident photon, scintillating detectorsemit visible light when exposed to radiation. Planar X-ray-sensitive scintillators, which are made of glass or inorganiccrystals, provide an instantaneous, two-dimensional radio-graphic projection of the object. The increase in dimensionreduces the required scanning time for a given object positiondramatically as beam energy is measured simultaneously for allbeams, without the need for vertical and/or horizontal objecttranslations that are required by line or point detectors. Theradiographic image can be recorded by any camera for a se-lected exposure interval Dt . The image is usually mirrored atan angle sufficient to place the camera away from the X-raybeam path.

2.2. CT Principles

In contrast to radiology, which projects cumulative X-rayattenuation of an object on a two-dimensional radiograph,computed tomography (CT) provides cross-sectional or three-dimensional representations of the scanned object. The tech-nological and mathematical principles of CT are covered indetail by McCullough [1975], Brooks and DiChiro [1975, 1976],McCullough and Payne [1977], Panton [1981], and Kak andSlaney [1988].

Briefly, the common principle of CT scanners is to measurethe spatial distribution of attenuation from I values obtainedfrom the scanning of the object using many different beamdirections. The term “beam” is used here to include all possiblestraight lines from the X-ray origin at the source to points onthe face of an individual detector cell. Following (2) and as-suming I0 is fixed and known, each I measurement can beconverted into a value for the integrated attenuation along thecorresponding beam path. To calculate an approximate solu-tion of the spatial attenuation distribution, the scanned portionis considered to consist of a set of nonoverlapping volumeelements or voxels. Each voxel is assumed to be homogeneousin composition and represented by a single m value. After asufficient number of independent integrated-attenuation mea-surements have been obtained, the discrete approximation ofthe m distribution within the three-dimensional object is recon-structed. Various mathematical approaches to solve the recon-struction problem for a given scanner geometry are availableand have been compared in the literature [e.g., Kak and Slaney,1988; Brooks and Di Chiro, 1975].

Main differences between available industrial scanners in-clude the number of beams/detectors simultaneously active,beam geometry, and whether the object is moved relative toradiation source and detector assemblage or vice versa. A fan(two-dimensional) or cone (three-dimensional) geometry in-herently magnifies the object by casting an enlarged radio-graphic projection of the object onto the detector line or plane,as shown in Figure 3 for the cone-beam scanners used in thisstudy. Consequently, the spatial resolution obtainable fromthese beam types is larger, while the object size must be re-duced correspondingly. Note that the spatial resolution refersto the minimum distance over which phases can be discrimi-nated and is not necessarily equal to the voxel size. Moreover,voxel size is not necessarily equal to the detector size but is a

user-defined parameter in the reconstruction. As resolutionincreases, voxels can be made smaller.

2.3. X-Ray CT Equipment

The reported experiments were performed at ScientificMeasurement Systems (SMS), Inc., Austin, Texas. Two differ-ent cone-beam systems were used, both equipped with indus-trial microfocus X-ray tubes of 10-mm source spot size andplanar detector arrays. Each scan took approximately 30 min tocomplete.

2.3.1. System A. Source current and potential settingswere 0.1 mA and 125 kV, respectively. Individual 58.6 3 58.6mm2 detector cells were combined in groups of four, resultingin an effective detector-cell size of 117.2 3 117.2 mm2. A totalof 360 radiographic projections per scan were taken with a6.37-fold magnification using 18 increments and 1-s exposuretime. The three-dimensional reconstruction by cone-beambackprojection was performed with the RECON code [Martz etal., 1990] resulting in cubic voxels of 23 mm side length.

2.3.2. System B. Source current and potential were set to0.08 mA and 124.7 kV, respectively. Individual detector cellsmeasured 20 3 20 mm2. A total of 720 exposures per scan weretaken in 0.58 increments at 1-s exposure time each. The mag-nification factor was 1.375. Reconstruction was performed bySMS using proprietary software. As the algorithm used as-sumes parallel horizontal beam planes, reconstruction was lim-ited to 30 circular cross sections. Final voxel size was 17.1 317.1 3 14.5 mm3.

2.4. Error Sources

2.4.1. Instrument. Local defects in the scintillator or inthe charge-coupled device chips of the digital camera maycause erroneous and unlikely low or high beam intensitiesresulting in ring artifacts after reconstruction. We used a sim-ple algorithm to identify those extreme readings and to replacetheir respective value by the average reading of all adjacentdetector units at the expense of a small loss in spatial resolu-tion for the affected voxels.

Errors can also be introduced by temporal variations of theposition of the X-ray source or object, for example, by tem-perature changes caused by the stepper motors that rotate theobject between scans. This error is particularly significant in

Figure 3. Cone-beam scanner geometry.

2069CLAUSNITZER AND HOPMANS: MEASUREMENT OF SOLUTE BREAKTHROUGH

transient experiments, where several scans of the same objectare taken. To eliminate such errors, radiographs for the sameobject angle are compared for all scans to exclude or to correctfor any shift in the projection of the object.

Also, the correction for variations in the X-ray beam sourceintensity is critical in the quantitative analysis of scan se-quences. Biased changes (drift) in I0 may occur because oftarget heating, even under conditions of constant source po-tential and current. Drift can also be caused by an unstablefocus of the electron beam on the target, resulting in fluctua-tions in size and/or position of the “beam spot.” Drift is bestcorrected before the reconstruction algorithm is applied to theradiographic projections. A change in I0 by a factor a for agiven photon-energy level will result in a difference of ln a inln (I), independent of the attenuation encountered by therespective beam. Assuming that drift affects all photon-energylevels similarly, temporal changes in ln a for each detector linecan be detected by measurements of I0 as a function of timefor beams that pass only through air. Changes in the ln a valuecan then be used to correct for ln (I) values for the otherbeams in the detector line.

The effectiveness of this simple approach is demonstrated inFigure 4. The rapidly oscillating values represent measuredmicrofocus source intensity as a function of time during asequence of scans lasting approximately 30 min each. Usingsystem A, intensity values were obtained at the center horizon-tal detector line by averaging I readings from 100 beamsthrough air. All values were divided by a fixed reference aver-age obtained in advance. The drift in I0 due to heating isevident for nearly all scans of the sequence depicted in Figure4, most strongly for the initial scans (the large gap between thefirst and second one reflects the time needed for initial objectpositioning). The source voltage was turned off during reposi-tioning of the object between subsequent scans, which mostlikely caused the jumps in I0 at the beginning of each scan.Continuous application of the source voltage, rather than turn-ing the voltage off and on between scans, can eliminate thisshort-term drift. The long-term trend in I0, however, is dom-inating and causes most of the variation of the reconstructed mvalues (solid line and diamonds in Figure 4) that were notcorrected for drift. Each attenuation value in Figure 4 repre-

sents the mean of 3 3 106 voxels of the 0.8-mm-thick Plexiglaswall. For comparison, mean m values for the same voxel set arealso shown after the described drift correction method wasapplied to the radiographs, indicating that correction of thedrift-induced error largely eliminated temporal variations.

2.4.2. Systematic errors. Systematic errors lead to appar-ent attenuation variations of the X-ray beam that are notcaused by differences in the electron density of the scannedmedium.

2.4.2.1. Partial-volume effect: The reconstructed linearattenuation coefficient for voxels containing a phase interfacewill, in general, be underestimated. The theoretical average mvalue based on the volume fractions for the phases presentwithin the voxel is a maximum value that represents the effec-tive linear attenuation coefficient only if beam and interfaceare normal to each other. For all other alignments the effectivevalue of m is smaller and has a minimum value if the beamdirection is parallel to the interface (e.g., compare with elec-trical analog of serial verses parallel resistors). For radiation-based CT the partial-volume effect is trivial if the voxel size ismuch smaller than m21 [Hsieh et al., 1998]. Since this criterionis overly satisfied for the microfocus systems and porous mediaused in this study, errors caused by the partial-volume effectare insignificant.

2.4.2.2. Aliasing and blurring: So-called aliasing artifacts(streaks forming symmetric patterns) arise during image re-construction because of undersampling of each projected ra-diograph (finite number of detector cells) and the finite num-ber of projections. Theoretically, the spatial samplingfrequency for each radiograph should be at least twice themaximum frequency determined by the spatial pattern withinthe image (the “Nyquist rate”) to avoid aliasing [e.g., Kak andSlaney, 1988, p. 180]. Thus, even for optimum scanning systemswith infinitely small X-ray emission spot and detector cells,some aliasing will occur for objects with sharp interfaces. How-ever, several sources of blurring or unsharpness combine toinevitably smooth each projection thereby filtering out thehighest radiograph frequencies and reducing the potential foraliasing. First, a certain amount of blurring is introduced by thefinite size of the detector cells (i.e., final beam width). Indeed,this effect alone may be sufficient to prevent aliasing fromhorizontal projection sampling if rotation increments aresmaller than one half a detector-cell width [Kak and Slaney,1988, p. 188]. Second, in scanners with a spot X-ray source, thatis, with a fan or cone geometry, the spot has a finite size (often;10 mm). Denoting the ratio of source-detector to source-object distance as magnification K , it follows from simple geo-metric arguments that the projection of an object point ontothe detector plane will be blurred over a distance equal to thespot size multiplied with (K 2 1). Third, additional blurring iscaused by the varying penetration depth of the X-ray photonswithin the thickness (;2 mm) of the scintillating detectionlayer, since a perfect camera focus can be obtained only for asingle penetration depth.

The combined effect of unsharpness contributions is bestmeasured by obtaining a CT image of a well-defined interfacewithin the object. For example, Figure 5 shows reconstructed mvalues for the vertical glass-liquid interface (represented byvertical dashed line) at the inside of a water-filled capillarytube that was scanned using system A. To (nearly) exclude theinfluence of noise, attenuation values in Figure 5 representaverages over vertical columns of 60 voxels each. Horizontalbars indicate 61 standard error (SE). After reconstruction the

Figure 4. Time dependency of relative source intensity (un-labeled curve, left axis) and corresponding mean reconstructedvoxel attenuation values for Plexiglas with and without driftcorrection (right axis).


well-defined interface is blurred over at least three voxellengths as indicated by the rectangle outlined by shading. Thusattenuation values for voxels adjacent to phase interfaces willbe in error, if interpreted as a concentration measurement.Hence blurred voxel values may be inadvertently interpreted asa different phase. In heterogeneous systems with large inter-facial areas the fraction of blurred voxels may be potentiallylarger than each of the individual phases [Clausnitzer and Hop-mans, 1999]. If one half of the blurred phase-interface length isconsidered as a measure of the effective spatial resolution, aresolution value of 35–45 mm is obtained, which is approxi-mately equal to the resolution value of 40 mm measured inde-pendently from radiographs of test-pattern objects.

2.4.3. Polychromatic beam. Even though specific attenu-ation is a function of beam energy, the reconstruction compu-tations implicitly assume a single effective beam energy. Be-cause X-ray sources are typically polychromatic, “beam-hardening” artifacts result; that is, voxels near thecircumference of a homogeneous cross section of a scanningobject appear to be of higher attenuation. In reality, photons ofstrongly attenuated energy levels (those with relatively lowenergy) are eliminated from the beam at a faster rate thanphotons of weakly attenuated energy levels (those with rela-tively high energy). Consequently, beams that pass near thecenter of the scanned object, and whose path through theobject is longer, become relatively more penetrating as theirenergy spectrum becomes increasingly skewed toward higherphoton-energy values. This change in the energy spectrum isdemonstrated in Figure 1, where the complete beam spectrumas emitted by a tungsten source is compared to beam spectraresulting from attenuation by materials that were used in ourflow study (Plexiglas, water, and glass). It is evident that thebeam spectrum changes noticeably even after relatively smallpenetration depths and that the lower-energy part of the spec-trum is increasingly eliminated as penetration depth increases.

In Figure 1 the unattenuated (i.e., source) spectrum wascomputed using the program TUBDET, developed at the Non-destructive Evaluation Division at Lawrence Livermore Na-tional Laboratory as an extension of the original work by Taoet al. [1985]. Attenuated spectra were subsequently derivedfrom the elemental composition and density of each phaseusing available attenuation properties. The scattering and pho-toelectric-absorption values for each energy level and element

were obtained from tables published by the National Institute ofStandards and Technology (NIST) [1995], based on the work byChantler [1995]. Values for energies up to 100 GeV are givenby Hubbell [1969]. Hence, if the atomic composition of theobject is known, the systematic error associated with polychro-matic radiation can be estimated by simulation based on theknown atomic attenuation properties using the following analysis.

The generalization of the definition of the linear attenuationcoefficient, m [ I21 dI/ds , for polychromatic radiation de-pends on whether beam intensity is measured as photon orenergy flux. When using photon flux, the total beam intensity Iis the specific photon intensity i(E) [photons/eV] integratedover the photon energy range R , produced by the respectivesource. From

m 5

d ER

i~E! dE

F ER

i~E! dEG ds

(3)

it follows that

m 5

ER

di~E!

ds dE

ER

i~E! dE

(4)

because E is independent of s . Since at each specific energylevel (E) di(E)/ds is identical to m(E)i(E), the effectivelinear attenuation coefficient for polychromatic radiation canbe expressed as

m 5

ER

m~E!i~E! dE

ER

i~E! dE

; (5)

that is, it represents an i(E) weighted average of all m(E) overthe photon energy range. Note that i(E) decreases with s(Figure 1). If the detector measures energy flux (as, e.g., scin-tillators do), i in (5) should be replaced by Ei [McCullough,1975], where E denotes the photon energy. As an example ofthe latter case, Figure 6 shows the effective linear attenuationcoefficient m encountered by an unfiltered beam from thetungsten source of Figure 1 as it penetrates a theoretical ho-mogeneous glass-water mixture (50% each by volume), to-gether with the energy value for a hypothetical monochromaticbeam with equivalent attenuation (effective monochromaticbeam energy) at the given penetration depth. For monochro-matic radiation the linear attenuation coefficient would beconstant with penetration depth, depending only on the singu-lar beam energy. However, for polychromatic radiation theeffective attenuation decreases with penetration depth, be-cause of the selective removal of photons of the more stronglyattenuated energy levels.

2.4.4. Photon randomness. All radiographic and radia-tion-tomographic measurements are subject to noise becauseof the random nature of both photon emission at the source

Figure 5. Glass-water interface represented by mean voxelattenuation values.


and photon interaction with the electrons of the object mate-rial. Let the subscript j refer to the photon beam arriving atdetector j . Provided the X-ray source is drift-free, the averagephoton-emission rate for the jth beam l j is fixed for a chosencombination of source potential and current. It can then beassumed that (I0) j, the number of photons emitted into the jthbeam over a period Dt , follows a Poisson distribution with amean and a variance of l jDt .

Along its path each photon contributing to (I0) j will eitherbe removed from that beam by interaction with electrons (scat-tering and/or photoelectric absorption), or it will pass throughthe object and contribute to the intensity value measured bythe jth detector cell (Ij). However, scattered photons whosescattering angle is sufficiently small may, instead, arrive atsome other detector cell. As this applies to all beams, a givendetector reading Ij will generally be an overestimate. Since theintensity of scattered photons is nearly independent of rota-tional object position, this error can be relatively easily cor-rected [Kak and Slaney, 1988, p. 125] by subtracting the knownor estimated scatter contribution from each detector reading.Scatter-corrected I values are assumed in the following anal-ysis.

According to (2) the probability, Ij/(I0) j, of photons passingthe object without removal from the beam path is given by

Ij

~I0! j5 exp S 2E

Lj,k

m dsD , (6)

where Lj ,k denotes the length of the jth beam path at the kthobject position (rotation step). Consequently, the expectedvalue and variance of the number of detected photons, E(Ij ,k)and Var (Ij ,k), respectively, are equal to

E~Ij,k! 5 Var ~Ij,k! 5 exp S2ELj,k

m dsD l jDt , (7)

for which the measured value Ij ,k is an estimate. Using thisestimate, the coefficient of variation (CV) for Ij ,k is (Ij ,k)20.5,implying decreasing uncertainty by emission and attenuationrandomness with increasing photon count. Hence, if the un-certainty of the values of Ij ,k can be reduced, the uncertainty of

reconstructed values for the linear attenuation coefficient willdecrease as well. Kak and Slaney [1988, p. 197] provided asolution for the spatial distribution of uncertainty in recon-structed m values for a single-slice filtered-backprojection al-gorithm. In general, for homogeneous objects the maximumuncertainty occurs at the center of the object because thebeams with the longest path through the object experience thelargest absolute attenuation.

For given scanner and object geometry the photon count canbe increased by raising the source current and thus the photonemission rate l or by extending the acquisition time Dt . Toavoid overheating the target anode when increasing the sourcecurrent, a larger source spot size may be required with a con-sequent loss in spatial resolution. Conversely, lower photoncounts can also reduce the spatial resolution if the associatedhigher noise requires local averaging of adjacent detector read-ings to achieve an acceptable CV. Thus, in many cases a trial-and-error procedure is required to determine a compromisebetween noise level and spatial and temporal resolution for agiven object. For example, the settings for source current,acquisition time, effective detector size, and number of pro-jections per scan reported for CT systems A and B were ob-tained from a series of test scans and represent an optimalcompromise for the respective system and considered objectcomposition and size.

3. Quantitative Analysis of CT Measurements3.1. Calibration

To verify the linearity of attenuation versus solute concen-tration, m(c), for the reported polychromatic radiation of theused CT systems, six capillary tubes of 0.82-mm inner diameterwere filled with sodium iodide (NaI) solution with concentra-tions of 0, 20, 40, 60, 80, and 100 mg/mL. The tubes were sealedand placed vertically inside a Plexiglas tube of 4.76-mm innerand 7.94-mm outer diameter. To minimize beam-hardeningeffects on the final calibration, the radial distances from thevertical axis of the Plexiglas tube to the center of each capillarytube were approximately identical (1.5 mm). The capillarytubes were scanned using system A, yielding 60 horizontalslices from a single complete rotation. Average attenuationvalues were obtained for each NaI concentration using allvoxels from the 60 slices combined (23,397) with a standarderror of 5 3 1024 cm21.

A linear relationship between solute concentration c and m

is expected if radiation is monochromatic and a negligiblechange in liquid volume due to the iodine dissolution is as-sumed. The slope is determined by the photon energy E andthe atomic composition of the solute. The intercept (m at zeroconcentration) is defined by the attenuation of the pure liquidphase at energy level E . Using the tabulated attenuation prop-erties of hydrogen, oxygen, sodium, and iodine [NIST, 1995],theoretical calibration lines, m(c), for NaI solution were com-puted for energy levels at 1-keV increments in the tungsten125-keV source spectrum. Four such curves at 50, 60, 70, and80 keV are presented in Figure 7 and compared with themeasured calibration curve (diamonds).

Also, based on the spectrum of a tungsten source beamhaving penetrated the plexiglas tube with capillaries near thevertical axis, the theoretical average effective beam energy Eeff

of 64.6 keV was computed from

Figure 6. Effective attenuation and equivalent monochro-matic energy for a beam penetrating a hypothetical homoge-neous glass-water mixture (50% each by volume).


Eeff 5

E0

125 keV

E@Ei~E!# dE

E0

125 keV

@Ei~E!# dE

, (8)

where [Ei(E)] represents the beam intensity as measured by ascintillating detector, using handbook values of attenuationvalues for glass, water, iodine, and Plexiglass. The computedEeff value of 64.6 keV corresponds well with the location of theobserved calibration line, suggesting that Eeff is a useful mea-sure when comparing attenuation of polychromatic with mono-chromatic beams.

Given that the relationship m(c) is linear for each specific E ,it follows from (5) that the effective calibration curve at a fixedradial distance r , m(c), for a polychromatic source must alsobe linear. The near-perfect fit of a straight line to the measureddata in Figure 7 confirms this theoretical result (R2 50.9992). It is important to note that while the effective m is,indeed, a linear function of c at a given radial distance r fromthe vertical center axis, the slope will vary with r because theenergy spectrum changes with r because of beam hardening.This will become important later in the analysis of iodidebreakthrough. Hereafter the symbol m always denotes the ef-fective linear attenuation coefficient.

3.2. Miscible Displacement Experiment

A 50-mm-long vertical flow cell was made from a Plexiglastube with 4.76-mm inner and 7.94-mm outer diameter and wasrandomly filled with precision-grade glass beads of 0.5-mmdiameter (Catophyte Inc., Jackson, Mississippi). Rubber septawere placed at both ends of the tube, leaving no free spacebetween the bead pack and the septa. Liquid inflow occurredthrough two needles penetrating the septum at the top of thetube. Both needle openings of 0.6-mm diameter were posi-tioned at the horizontal center of the septum. A constant flowrate of 100 mm3/h was maintained by a double syringe pump to“instantaneously” switch injection between water and 100-mg/mL NaI solution, with each liquid entering through one ofthe inflow needles.

A single needle penetrating the bottom septum was used asliquid outlet. The bead pack was initially saturated with water.A 30-min period of water injection was followed by a 90-minsolution pulse and 240 min of water injection. The estimatedporosity value of approximately 0.41 cm3/cm3, computed frominjected water and tube volumes, corresponds to an averagepore water velocity of approximately 13.7 mm/h. Using systemB, scans were taken continuously for the duration of the ex-periment and lasted approximately 20 min each, resulting in atotal of 16 data sets. Each scan covered the same vertical rangeof 0.44 mm with its center located 20 mm below the inflow end.The pore volume corresponding to the 20 mm depth was ap-proximately 146 mm3, while the total volume of the solutionpulse was 150 mm3. Drift correction was applied to the recon-structed data sets based on the average m value of 3 3 106

voxels located within the wall of the Plexiglas tube. For prac-tical reasons, the size of each reconstructed data set was re-duced in the horizontal plane by retaining only the smallestsquare that completely contained the circular cross section ofthe bead pack for a total of 277 3 277 3 30 voxels of 233 mm3

each.As indicated in section 3.1, one cannot simply use the cali-

bration curve in Figure 7 to determine the concentration dis-tribution within the bead pack from liquid-phase m valuesbecause the effective X-ray energy changes as the beam passesthrough the medium. Hence a calibration method is neededthat incorporates horizontal radial distance r from the objectaxis. Since the first scan of the glass-bead pack used pure wateras the liquid phase, it provides the intercept (c 5 0) for them(c) calibration line at any given radial position r . However,to completely define the respective linear calibration line at r ,a second fixed point is required. Since the local iodide concen-tration at any r during the breakthrough experiment is un-known, the second calibration point was obtained from theobserved attenuation for the glass phase, mglass(r). Figure 8shows values of mglass as a function of r , with each valuerepresenting the mean of 9 3 9 3 9 (729) voxel values centeredwithin a specific glass bead. Horizontal bars represent 61 SEbased on the observed noise in the mglass values. The shape ofthe observed relationship mglass(r) was well described by aquadratic function, which was fitted through the data points.The nonlinear behavior is the result of beam hardening, caus-ing an apparent increase in attenuation from the center (r 50) to the perimeter of the Plexiglass tube. This increase in

Figure 7. Measured NaI calibration line (diamonds) andtheoretical NaI calibration lines at energy levels of 50, 60, 70,and 80 keV.

Figure 8. Fitted curve for the attenuation of glass as a func-tion of radial distance from the center axis.


mglass(r) corresponds to a decrease in effective photon energyEeff(r), that is, E of the equivalent monochromatic radiation(Figure 2). For each Eeff(r), there is a NaI concentrationcequiv(r) whose attenuation coefficient matches the corre-sponding observed value of mglass(r). According to Figure 2

the NaI concentration value at which the attenuation coeffi-cient matches that of glass decreases as Eeff decreases withinthe applicable range of 60–70 keV. The concentration cequiv(r)can be computed from the NIST [1995] tables using the chem-ical composition of the glass beads and iodide solutions. A plotof computed cequiv values as a function of r is shown in Figure9 together with a straight-line fit. Standard error values incequiv are based on the respective SE for mglass by applyingstandard error propagation. The values mglass(r) and cequiv(r)together provide the second point of the linear m(c) calibra-tion curve at radial position r .

The final calibration equation to compute a solute concen-tration at a pore location x within the three-dimensional voxelset is then given by

c~x! 5 F m~x! 2 mH2O~x!

mglass~r! 2 mH2O~x!G cequiv~r! , (9)

where mglass(r) and cequiv(r) are the respective fitted func-tional expressions and mH2O

(x) is the m value of the voxel atlocation x in the first data set obtained (i.e., before applicationof the NaI pulse). Note that the units of concentration aremass of solute per volume of liquid phase and that c representsresident concentration. To reduce the standard error due tophoton noise in the individual reconstructed voxel m values,the mean m value for a group of 5 3 5 3 5 adjacent liquid-phase voxels was considered a measurement at the centerlocation x of the voxel group. Given the voxel side length of17.1 mm, the effective spatial resolution of the local solutebreakthrough was thus approximately 86 mm, while the actualspatial resolution of the scanner had been measured at 28 mmfor negligible noise. A still larger group size would result ineven smaller SE values but would limit the number of appli-cable pore locations to a few large pore bodies because of thenecessary “safety distance” from the glass-liquid interface toprevent blurring effects.

An expression for the variance of the resulting local concen-tration values, s2[c(x)], at a given time t is obtained by sub-stituting the respective derivative expressions from (9) into theerror propagation equation while assuming that m, mH2O

, mglass,and cequiv are independent:

Figure 9. Fitted curve for the NaI concentration resulting inan attenuation identical to that of glass at the effective energycorresponding to the respective radial distance from the centeraxis.

Figure 10. Local breakthrough curves at 10 random posi-tions (see Figure 14 for locations): (a) 1, 2, 4, and 5, (b) 6–9,and (c) 3 and 10.

Figure 11. Concentration values estimated from equation(9) for pure water and 100 mg/mL NaI. True values are indi-cated by horizontal lines.


s2@c~x!# 5 S 1mglass~r! 2 mH2O~x!D

2

cequiv2 ~r!s2@m~x!#

1 S m~x! 2 mH2O~x!

@mglass~r! 2 mH2O~x!#2

21

mglass~r! 2 mH2O~x!D2

cequiv2 ~r! s2@mH2O~x!#

1 S mH2O~x! 2 m~x!

@mglass~r! 2 mH2O~x!#2D 2

cequiv2 ~r!s2@mglass~r!#

1 S m~x! 2 mH2O~x!

mglass~r! 2 mH2O~x!D2

s2@cequiv~r!# . (10)

The local value for the variance of the liquid-phase attenua-tion, s2[m(x)], was approximated by the estimated variance ofthe mean m value for the considered group of 125 liquid-phasevoxels. The value of s2[mH2O

(x)] is identical to the s2[m(x)]value for the first data set of the sequence (corresponding tothe water-saturated sample). The values of mglass(r) andcequiv(r) in (10) are those predicted from the fitted expressionsgiven in Figures 8 and 9, respectively. Variance values,

Figure 12. Sequence of three-dimensional iodide distribution images at selected times of 37, 58, 105, 145,186, and 245 min within the scanned volume of 4.76 3 4.76 3 0.44 mm3. Voxel concentration values aremapped using a gray scale with black and white corresponding to concentrations of 0.0 and 50 mg/mL,respectively.


s2[mglass(r)] and s2[cequiv(r)], were computed from the co-variance matrix of the respective optimized parameter set[Clausnitzer and Hopmans, 1995]:

s2@mglass~r!# 5 s2@A0# 1 s2@A1#r2 1 s2@A2#r4

1 2 ~Cov @A0, A1#r 1 Cov @A0, A2#r2

1 Cov @A1, A2#r3! , (11a)

s2@cequiv~r!# 5 s2@B0# 1 s2@B1#r2 1 2 Cov @B0, B1#r , (11b)

where Ai and Bi denote the fitted coefficients of ith order inthe expressions given in Figures 8 and 9, respectively. Finaluncertainty, =s2[c], varied between 2.9 mg/mL and 3.7 mg/mL, decreasing with increasing concentration values. The vari-ation in iodide breakthrough of both peak concentration andpeak travel time is presented in Figures 10a, 10b, and 10c,which show c(t) curves for 10 different pore locations x withinthe scanned volumetric slice of 0.4-mm thickness. Break-through curves were obtained from 16 scans, each separated byabout 20 min. The observed local breakthrough curves can beroughly classified into three groups. Breakthrough within thefirst group (Figure 10a) exhibits higher peak concentrationvalues with shorter peak travel times than the second group(Figure 10b), whereas the breakthrough of the third group(Figure 10c) shows distinct multiple peaks, demonstrating pos-sible unstable flow behavior. Observed variations are likelycaused by density-driven preferential flow during the down-ward displacement of water by the NaI solution whose densitywas initially 10% higher than that of water [Liu and Dane,1996a, b]. This interpretation was further supported by visualobservation when staining dye was added to the NaI solution.Preferential flow in channels with diameters less than the ef-fective spatial resolution of 86 mm can also explain the rela-tively small peak concentrations of the breakthrough curves.We note that differences in the 10 presented local break-through curves are noticeably reduced after 180 min of break-through, indicating a much more uniform flow during the sub-sequent displacement of the solution by the water, that is, bythe liquid of lower density.

The accuracy of the predicted concentration values was sup-ported by two independent tests. First, the calibration given by(9) was applied to liquid-phase voxels inside two capillariesattached to the outside of the flow cell, with one containingpure water and the other containing 100-mg/mL NaI solution.The two capillaries combined bracket the experimental con-centration range of the breakthrough experiment. Figure 11shows that the predicted values after calibration are approxi-mately identical to the true values, with deviations not largerthan the estimated uncertainty of 3–4 mg/mL.

Second, the measured mass of solute moving through theobserved cross sectional slice, when integrated over time, mustequal the total mass of applied of solute (15 mg). Voxel con-centration values (milligrams of iodide per cubic centimeter ofsolution) for the 16 data sets representing the breakthroughwere computed from

c~x , t! 5 Fm~x , t! 2 m~x , t0!

~mglass 2 mH2O!~r! G cequiv~r! , (12)

where m(x, t0) denotes the voxel attenuation value for thewater-saturated sample before the pulse was applied. The dif-ference between the values of mglass and mH2O

depends onradial distance from the center because of beam hardening. Ifonly local breakthrough is considered, the value of mH2O

isexplicitly known at each considered position x and can be useddirectly, as in (9). To compute the complete mass balance,however, a relationship is needed that predicts the difference(mglass 2 mH2O

) for any value of r . Values of the difference(mglass 2 mH2O

) as a function of r were computed using theknown mglass(r) values shown in Figure 8 and theoretical mH2O

values corresponding to the effective energy at the respectiveradial distance. The relationship (mglass 2 mH2O

)(r) was ex-pressed as a quadratic function of r whose coefficients wereobtained by fitting the computed difference values (R2 50.9875). The parametric expression for cequiv(r) is the sameas in Figure 9. As it should, (12) predicts c values of zero(subject to photon noise) for voxels located within the glassphase, assuming perfect drift correction. A sequence of thespatial distribution of iodide concentration values at selectedtimes of 37, 58, 105, 145, 186, and 245 min is presented inFigure 12. Two rather distinct regions can be recognized in thethree-dimensional representation, with relatively higher con-centration values predominantly found on the left and lowervalues found on the right. This observation supports the like-lihood of preferential flow within the bead pack.

The total mass M of solute passing a horizontal cross sectionA of the flow cell is given by

M 5 ETE

Aliquid

c~x , t!n z~x! dA dt , (13)

where T denotes the duration of the experiment, A liquid is thefraction of A within the liquid phase, and nz is the verticalcomponent of the fluid velocity. To compute M , the followingapproximations were made:

1. The integration in time was performed numerically fromthe 15 scans that followed the initial scan of the water-saturated sample, using time intervals Dt j, with j 5 1, z z z , 15.

2. The integration over A liquid was replaced by a summa-tion over discrete segments of A . Segments were defined byoverlaying the horizontal circular cross section of the flow cellwith a N 3 N grid, with N equal to 1, 2, 4, 9, and 17.

3. All 30 horizontal slices (N Þ 17) or the upper horizon-tal 15 slices only (N 5 17) were combined into a single crosssection, resulting in a vertical slice thickness of 0.436 mm (30 30.145 mm) or 0.218 mm (15 3 0.145 mm), respectively. Theposition of A was defined by the vertical center of the segments.

4. An approximation to the average vertical component ofthe fluid velocity nz for each segment was obtained by dividingthe vertical distance between inflow and A (20 mm) by the timerequired for the center of the pulse to travel this distance foreach specific segment of A . The center of the pulse was as-

Table 1. Mass Balance Results for Different GridResolutions

Number ofSegments N Voxels/Segment M , mg «, %

1 1,794,150 12.456 16.962 448,538 10.732 28.454 142,830* 10.513 29.919 28,830* 16.365 9.10

17 4096* 15.762 5.08

M, estimated total applied iodide mass; «, relative mass balance error.*For segments not intersecting the circular boundary.


sumed to correspond to the time at which the peak concentra-tion is observed at the respective segment.

Replacing the integration over A liquid by summation oversegments of A requires that concentration values of zero areassigned to all points not within the liquid phase. This is en-sured by using (12) to compute the values of c for all voxelspresent in each segment. Applying approximations (1) through(4), (13) is approximated by

M 5 OA

n z~n! A~n! OT

c# ~n , t! Dt j, (14)

where M denotes the approximation of M , n is the summationindex over all segments, A(n) is the nth segment’s area in thex-y plane, and c# (n , t) is the estimated mean c(t) value over all

voxels within the nth segment during time interval Dt j. Thesmallest segment size selected was 4096 voxels (N 5 17), sothat the standard error of the mean c value, due to photonnoise, was less than 1%. The value of nz for N 5 1 was 19.7mm h21. The difference to the average pore water velocity of13.7 mm h21 previously computed from known flow rate andporosity values also supports the interpretation of preferentialflow. Table 1 shows values of the estimated total iodide mass Mpredicted by (14) for the different selected grid sizes, as well asthe magnitude of the respective mass balance error « relativeto the true total solute mass value of 15 mg. Further reducingthe segment size (N . 17) did not result in a further reductionof the computed mass balance error of approximately 5%.

To illustrate the heterogeneity in solute breakthrough within

Figure 13. Estimated spatial distribution of cumulative solute breakthrough within the horizontal crosssection at 2 cm below inflow end for (a) N 5 2, (b) N 5 4, (c) N 5 9, and (d) N 5 17.


the flow cell, the spatial distribution of the estimated totaltransmitted mass of solute per area

M~n! 5 n z~n! A~n! OT

c# ~n , t! Dt j

is shown for decreasing segment sizes in Figure 13. The cor-responding spatial distribution of nz values for the 17 3 17 gridonly is plotted in Figure 14 together with the x-y locations ofthe 10 observed local breakthrough curves of Figure 10. Again,both Figures 13 and 14 clearly show the presence of a prefer-ential flow pattern. The local breakthrough curves in Figure10a correspond to the region of relatively high nz values (.20mm h21), those in Figure 10b correspond to the region ofrelatively low nz values (,20 mm h21), and the location of oneof the curves in Figure 10c is at the approximate boundarybetween both regions. Given this flow pattern, it was not sur-prising that the overall average concentration with respect tothe liquid phase and the breakthrough curve could not bepredicted by the standard convection-dispersion model. Fordispersivity values up to the bead size (0.5 mm) the standardmodel predicted a peak concentration value above 90 mg/mL,while the observed peak concentration was only about half ofthis value. The additional spreading of the solution pulse, rel-ative to the standard model, can most likely be attributed to

unstable flow during the displacement of water by the iodidesolution.

4. ConclusionsMicrofocus X-ray CT can provide local concentration mea-

surements with acceptable uncertainty for experiments involv-ing miscible displacement in porous media. After correctionfor drift, noise due to photon randomness was the dominantsource of uncertainty, resulting in an effective spatial resolu-tion of approximately 86 mm. As this represents a threefoldloss relative to the measured scanner resolution, extended ac-quisition times are clearly desirable to minimize noise by max-imizing the photon count. To avoid a loss in temporal resolu-tion, the flow must be sufficiently slow. A practical limit is setby the possible duration of the experiment, that is, the maxi-mum time of scanner access.

While the relationship between solute concentration andattenuation is linear for polychromatic as well as monochro-matic radiation, the beam hardening that is typical for indus-trial X-ray sources requires a calibration method that takesradial distance from the vertical object axis into account. It isproposed to base the calibration on sufficiently separated at-tenuation values for two phases that are present in the system

Figure 14. Distribution of the average vertical component of the fluid velocity within a horizontal crosssection located 2 cm below inflow. Numbers indicate local breakthrough observations at the spatial locationspresented in Figure 10.


over the entire range of r (water and glass). Filtering the X-raybeam before entering the object would have reduced the beamhardening effect thereby lessening the need for the additionalcalibration. However, beam filtering reduces photon fluxes andincreases scanning times, making transient transport experi-ments as presented here impossible. Inferring local fluid ve-locity from the respective peak-concentration travel time andusing concentration values predicted by the calibration re-sulted in an acceptable mass balance, provided a sufficientlyhigh spatial resolution was used.

Although we maintain our enthusiasm for application ofX-ray microtomography in porous media, we conclude that ifhigher spatial and temporal resolutions as obtained in thepresented experimental study are needed, a radiation sourcesuch as a synchrotron is needed. The associated higher photonfluxes will reduce acquisition time while increasing the signal-to-noise ratio and the spatial resolution. As the X-ray spectrumproduced by a synchrotron source can be controlled to producean almost monochromatic beam, corrections for complicationsas caused by beam hardening become unnecessary.

Acknowledgments. We are grateful to John Steude and Brett Si-mon of Scientific Measurement Systems, Austin, Texas, and to PatrickRoberson and Dan Schneberk of the Nondestructive Evaluation Divi-sion at Lawrence Livermore National Laboratory for their valuablesupport and assistance. Research was supported in part by USDA NRIproposal 95-37107-1602. We also acknowledge the valuable commentsby Harry Booltink, Peter Lehmann, and Hannes Fluhler, who reviewedthe manuscript.

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(Received March 29, 1999; revised December 28, 1999;accepted March 6, 2000.)


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