modeling solute diffusion in the presence of pore-scale
TRANSCRIPT
Modeling Solute Diffusion in the Presence of Pore-Scale Heterogeneity
*
~ *O
Sean W. Fleming (? 2$*
Department of Geosciences “me~
4**104 Wilkinson Hall, Oregon State University, Corvallis, OR 97331-5506 ~ ,“’
e,.
current address:
Waterstone Environmental Hydrology and Engineering Inc.
165038” St., Suite 201E, Boulder, CO 80301
e-mail: [email protected]
* Roy Haggerty
Department of Geosciences
104 Wilkinson Hal!, Oregon State University, Corval]is, OR 97331-5506
e-mail: [email protected]. edu
telephone: (541) 737-1210
* corresponding author
Submitted toJournal of Contaminant Hydrology
August 10, 1999
key words: djjkrion model, n]dfinzfe, rafe-hmih?dnfas.rIranflcc Culeh, contaminam’ iranporfj
dolomile
:.....’” . ...’.. . . . . . . . -. ..,”,- -------- .’ .”-. . . . . . . . . ..- . . .. .
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Fleming & Hsrty, Modeling soluie diffusion in the presence of pore-scale l)ettmgeneity. ~
<
Modeling Solute Diffusion in the Presence of Pore-Scale Heterogeneity
Sean W. Flemingll ~ and Roy Haggertyl~s
ABSTRACT
A range of pore diffusivities, DA is implied by the high degree of pore-scale heterogeneity
obsemed in core samples of the Culebra (dolomite) Member of the Rustler Formation, NM. Earlier
tracer tests in the Culebra at the field-scale have confirmed significant heterogeneity in diffusion rate
coefficients (the combination of DPand matrix block size). In this study, expressions for solute
diffision in the presence of multiple simultaneous matrix diffki~~ities are presented and used to
model data from eight laboratory-scale difflision experiments performed on five Culebra samples.
lognormal distribution of Qois assumed within each of the lab samples. The estimated standard
deviation (ad) of ln(D,J within each sample ranges from () to 1, with most values lying between 0.5
and 1. The variability over all samples leads ro a combined ad in the range of 1.0 to 1.2, which
appears to be consistent with a best-fit statistical distribution of formation factor measurements for
similar Culebra samples. A comparison of our estim~tion results to other rock properties sugysts
that at the lab-scale, the geometric mean of D} increases with bulk porosity and the quantity of
macroscopic fefitures such as vugs and fractures. However, ad appears to be determined by
variability within such macroscc)pic features and/w- by micmpore-scale heterogeneity. In addition,
comparison of these experiments to those at larger spatial scales suggests that increasing sample
volume resu]~ ill Xl illcre~se ill ~d,
1Department of Geosciences, 104 Wilkinson Hall, Oregon State University, Conullis, OR 97330-55062 Now at: Waterstone Environmental Hydrolo~ and En&neering, Inc., 165[}3Sth St., Suite 201E,Boulder, CO 80301s Corresponding autho~ e-maii: ha~ertr(i?jjgeo. orst.edu,telephone: (541) 737-1210
A
1. INTRODUCTION
3
Mass transfer refers to the movement of a constituent from one phase or region to another
along a concen~tion gradient (e.g., WeLj el A, 1984). In the context of solutes in geologic materials,
this involves movement of solute from pores dominated by advection (advecuve porosity) into dead-
end pores dominated by diffusion (diffusive porosity) or onto sorption sites on the surface of grains.
This phenomenon has two well-known and significant implications for solute transport in
groundwater. First the rate of advective transport away from the source decreases through time as
@e solute spends less time in advective porosity and more time in diffusive porosity containing
stagnant water (e.g., Roi)ent et aL, 1986;Qtakoa’o~ arzdKahcct)i, 1993). This phenomenon partly
determines how far and how fast a solute will be transported, and is particukdy important to
evaluation of the environmental risk involved in geolo~-c disposal of nuclmr waste. Secondly,
remediation of a contaminated aquifer by pump-and-treat, soil vapor extraction, and bioremediation
techniques is slowed if mass tram fer from diffusive to advective porosity is rate-limited (Jfeid7erg et
aL, 1987; Bnaseaz and Rae, 1989; Gob< and O.x~, f 991; Gierke et a~, 1992;ArmslnJrg ezuL, 1994; Fry and
I~tok, 1994; Harvey et aL, 1994; Rak-ieott andA4iLir, 1994; Hager~ utzdGordck, 1995). Diffusion in
geologic media also has implications for petroleum migration (e.g., Mann, 7994), evaluation of
residual oil saturation in hydrocarbon resemoir studies (e.g., TonzicJet at., ?973; Dem.r arzdCarhJe,
?98G), determination of @Ar/3qAr cooling histories in geochronology studies (bwra el al., 1989;..
Lvera et aL, 7993), and evaluation of the relationship between fluid and melt inclusions and magma
chemistry (Qin et aL, 1992).
A simplifying assumption made in most treatments of nonequiiibrium mm transfer in
geologic media is that a single rate of sorpricm or diffision is present. The implication of this
assumption is that rock is homogeneous, or can be treated as effectively hc)rnogeneous at the spatial
scale of interest. However, most rock is not homogeneous at the spatial scales at which mms transfer
processes operate. In the context of lab-scale diffision, this heterogeneity may result from variability
Fleming & Haggmty, M~dclulg SOIUICd, ffmion i IIIC prcsrnce of p@re-sc#le hetemgrnt-i~, %
4
in the volume, size, and geometry of macroporosity or microporosity in aquifer particles and
aggre~tes of particles (Rao el u!., 1980; Pignfzlefb,1990; [P’oofief al., 1990; Bui/andROberls,1991; i%znon
ezd, ?992; ELmzon and RoLerff, 1994) and the external and internal geometry of small lenses of
impermeable material, and the proportion of this material (SJarke~ord, 1991). .4s these factors
control rates of matrix diffision, and as they are heterogeneous at the pore scale, difhsion should
also be heterogmeous at the pore scale. That is, in a given volume of rock multiple rates of
diffhsion occur simultaneously, and “multirate” mass transfer may therefore be needed to adequately
describe solute transport (e.g., Vik?erm-azx,1981; PeditandMU&, 1994; Haggeg andGoRLck, 199~.
The effects of multirate mass transfer have been meamred in lab-scale soil column experiments (e.g.,
BaJ and Robent, 1991; Grah~h! and Kkineidarn,1995; PeditatzdMilLr, 1995; Culveret aL, ?997; W’ertbet
aL, 1997; Hagerzj and GoreLck, 1998; Riigneret aL, 1999) and more recend y in integrated and
comprehensive field-scale studies of the transport properties of the Culek dolomite at the Waste
Isolation Pilot Plan t (WIPP) site (Meig.rwzdBeudteim,irznwicu<Hagtr~ et J, in review;MrKenna el al, in
mien].
The WIPP is a proposed rcpositoq fcjr defense-generated transumnics located about 2(I miles
northeast of Carlsbad, New hfexico (F@lre 1). The Culebra Member of the Rustler Formation, a
Permian evaporitic dolomite, is considered to be the most transmissive laterally continuous
hydrogeologic unit in the WIPP area and the most likely pathway of radionuclide transport from the
repository in the event of human intrusion (Hoi, 1997, A4eig~and BeaAz-m, in reie+. Due to a.
complex history of deposition, diagenesis, and frachlring (see JYol, 1997 for a re~’iew), the Cule&a
dolomite is characterized by a high degree of lithologic and structural heterogeneity (F@tre 2).
Detailed analysis of core has clearly revealed multiple scales of fracturing, along with spatially variable
amounts of vugg porosity and gypsum -fd~ingof \w<gsand fractures; pocn-ly cemented, silt-size
dolomite interbeds are also common (HoJ, 1997). The presence of multirate diffitsive mass transfer
in the Culebra, at both lab- and field-scdes, has been inferred from this observed heterogeneity (Hoti,
Fleming & HWAWY. Modeling solute diflision iI1 the pmsmce OfpOK-SC21Cl,ettrob~:,city, <
5
195’7),and its effects at the field-scale have been confirmed by suites of tracer tests at two locations
(Mez& and Beanbeim,in rwiez~;Hagerp et a[., in revieq McKenna et a[., in wieu~.
Our objectives in this paper are (1) to formulate a mathematical mode] of diffusion in a rock
with variable diffhsivity (2) to solve the resulting mathematical equations; and (3) to estimate
dkrnbutions of dii%.tsivities from a suite of laboratory diffusion experiments performed upon the
Culebra dolomite.
2. DIFFU!30 N EXPERIMENTS A.ND DATA
Data obtained from static diffusion experiments conducted by Sandia National Laboratories
(Tidw-t4ez d, in$mw) were analyzed in this study. The x-ray imaging technique used in these
experiments is described in Tidwfl and Gh.cr (1994), and detxils of rhe sample selection, experimental
set up, and results are given in Cbnlr&in-Fmzr et al. (1997) and Tidvt# et al. (inp-e.w); a brief oven’iew is
given here. Five small (approximately 6 cm by 4 cm by 2.5 cm thick) rectangular blocks of Culebra
core were selected to represent a range of Culebm porosity types (see Figure 2 and Table 1). Because
of limitations on sample availability, samples with significant interpxticle porosity (silty dolomite,
which is not uncommon in the Culebra) could not be represented among the samples. The samples
were atmched to a constant-concentration tracer resenmir at one end and sealed along their other
edges to form no-flux boundaries. Constant concentration was maintained within l% of a target.
concentration by slow circulation in the tracer resen~oir. The tracer consisted of potassium iodide
(W), dissolved in a sodium chloride &laCl) solution simulating nmurally-occurring Culebra pore
fluids @rine). Each block was i]iitiaily saturated with brine. Tmcer was then allowed to diffuse from
the reservoir into the sample (k-diffision). .At the end of the in-diffusion experiment, x head
gradient was established across the block and tracer wm forced across it until the Kl concentration
within the pore space was equal to that in the reservoir (i.e., until the sample was complete] y flushed
with the I-Clsolution). The reservoirfluid was then replaced with pure NJICI solution, and the tracer
Fleming & H=q, MO&-Ii,Ig SOILIICdiffusion in tile prrsrIIce of pore-scale llemrowncity. .
6
within the block was permitted to diftlse hck out into the reservc)ir (out-diffusion). Slow
circulation of fluids through the reservoir was again performed to maintain concentrations within 10/o
of the target concentration.
X-ray images of each block were taken at intervals during the in- and out-diffkion
experiments (Tiduz# et al, inpw$. As the presence of the iodide ion alters the amount of x-ray -
energy transmitted through the sample, these images provide maps of porosity and of normalized K.1
concentration in the rock through time. The im~ges were then further processed to provide the
normalized mass uptake and recovery data used in our modeling (F@re 3).
Three important difficulties were encountered during the course of the experiments. Firs~
tracer concentration in the reservoir was accidental y varied during the earl kst phases of the in-
difksion study, resulting in a time-varying boundary condition. Second, sufficient dissolution of
pore-filling ~’psum in sample RC6-G occurred during the course of the experiment that its porosity
was significantly increased; this block was omitted from the out-difft]sion experiment. Lesser
gypsum dissolution also took place in sample 1333-FI. In both cases, this probably occurred while the
sample was being saturated with KI tracer in preparation for the out-diffusion phase (V. Tidwd,
pmsorzafmnmwzicalioz f5’98). Third, it proved impossible to fill]y saturate sample RC4-D with tracer,
so it was also omitted from the out-diffusion phase.
In addition to examining each data set separately, we combined the data from individual
samples to form bulk in- and out-difiision data sets. The bulk data are what would be obtained*
experimentally if all the samples were joined together in parallel along a singje resen’oir, and
normalized mass difksed in or out wm measured for all block at once using a singje x-ray irrnge.
This was done for the purpose of obtaini!lg x rough approximation to”(iiffusicm in a larger, mc]re
heterogeneous volume of Culebra. It is important to re-emphasize, however, that the samples
examined do not represent the full lithologic ~’ari~bilityu’ithin the Culebra. The bulk mms ratio
curve we obtained is the arithmetic mean of the individual mass ratio curves, weighted by the sizes of
the samples. Use of the bulk diffision data sets is intended to simulate a larger-scale scenario in
Fleming & H~ny, Mode]iIlg SOIUICdiffi,sio,, i,, I]le prcsm,cc oflxm..scalc l,c~crown.ity 7<
which an extended reservoir of solute (e.g., m advective flow pathway) encounters a variety of nmtrix
types, as represented by the different blocks of varying porosity scales and types Figure 2 and Table
1) used in the diffision experiments.
3. MATHEMATICAL AND CO DE DEVELOPMENT
Here we present expressions describing one-dimensional diffkion into or out of a rock matrix
in the presence. of both uniformity and heterogeneity in DP Consider a sample of Culebra dolomite
of bulk porosity, 6, bounded at one end by a reservoir of solute (potassium iodide in our case) with a
time-varying concentration, bounded at the other end by a no-flux boundaq, and having an arbitrary
but uniform initial solute concentration. Conventional, single-rate diffkion (i.e., a single diffkivity)
of ISl tracer into or out of the pore space of the sample is described by the diffusion equatioix
(Eqn. la)
where C is the concentration of solute within the pore space ~4/Ls, or dimensionless if normalized],
and DPis the pore diffisivity ~~/Tl, which is some fraction of the aqueous diffusivity of the solute in
water, D% ~-/Tl. The boundary and initial conditions are.
C(x=z,l)=c, (f) fEq?I. Ilq
3C(X=0,1)=0ax
C(x,l=o) = co
(Eq?l. Id
(Eq?L Id)
Fleming & H~rty, N40dcling solute diffusion i,l Ihe prcscncc o(pore-scak lIeIrrcgc-neity. . 8
where Co is the initial concentmtion in the sample ~/Ls]. A simple expression for DP is (e.g., Bear,
1972’):
where ~, the tortuosity [-], is defined as F/# (e.g., Be.a~/972); /is the straight-line distance from one
end of the pore to the other ~] and 4 is the actual length along the winding pore ~]; and P/.4~ <1.
DP maybe reduced if sorption occurs within the pore, or if restrictivity is significant (i.e., if the pore
radks is similar to the ionic or molecular radius of the solute, e.g., futterjieti e[uf., 1973). For a
complete review of the theory of dift%sion in rock, see OJkron and A~eref~tiek(199>~.
Taking the Laplace transforms of E9ns. I.a-c simplifies the diffusion equation to a second-
order nonhomogeneous ordinwy differential equation:
(32? p ~.-co—.—t%2 DP DP
where ~ is the Lapl:}ce tmnsform of concentr~tion within the pore space ‘~~d} is the Laplace
parameter. The boundag conditions m-enovx .
W(x = o)=o
ax
and:
C(p,x=1) = c, (p)
(Egn. 3a)
(Eqn.3b)
(Eqn.3.c)
Flcrnillg.9 Hs~. ~~~delillg SOIUICd!l-kion in IIW pre.tvm- ofpcm-scak lICICrCIWtIViIy.
Solving Eqn. 3.a subject to Eqns. 3.11and c by the method of undetermined coefficients gives:
[1——C(p,x) = c] (P)-:
9
(Egn. 4)
The Oh spatial moment of the concentration profile gives a measure of the solute mass, h4(~, within
the slab at time, &
M(f) = e A Jc(x,l)dx (Eqn.5)
r-o
where A is the cross-sectional mea of rock exposed to the solute resemoir, and which is required if
C(X;tJ is given conventional (three-dimensional) units of mass/unit volume. Taking advantage of the
fact that the Laplace transform is a linear operator, substituting Eqn. 4 into Eqn. 5, integrating and
recognizing the definition of the hyperbolic tmgent gives the following expression for diffisive mass
uptake or release in the presence of uniform DPsubject to the boundmy and initial conditions
encountered in the static dlffision experiments: .
fw(r, Dp) = (oA)L-’Il=lanh(’m)+(Eqn. 6)
where the symbol L-J indicates the inverse Lapkce tram form. Eqn. 6 and the inverse Laplace
transform of Eqn. 4 are generally useful solutions to the diffusion equaticm, m they can
Fleming & Hqyq. hfodelin~ solute d[ffusio,] III the presence of pwc-sc.alc lIctcrogt-\Iciq. 1()
accommodate any initial concentmtion and a constant or time-vmying boundary ccmcentmtion. For
notational convenience larer in the cleriv~tion, we define i14’(f,DP)thus] y:
A4’((,Dp)=M(f, Dp)/e (Eqn. 7)
Now consider a model of pore-scaJe heterogeneity k u’hich porosity in the rock matrix
consists of many l-D, non-intersecting pores. The pore diffusivity is constant along the lengdl of
each pore but may be different between neighboring pores. Changes in pore dift%sivity are due, at a
rninikn~ to differing tortuosity and cross-sectional area of the pores. Although the pores do in fact
intersecq the assumption of many l-D, non-intersecting pores allows us to retain the simplicity of 1-
D diffkion while incorporating the significant effects-of variable diffusivity. Approaches similar to
this have previously been successfully used to accor-mnodnte the effects of wuiable diffusi~~ityupon
So]ute transport (e.g., &fHi@um etd, ~997; &“.ger/y d (%dl”k, 1998; l]d#yly et u~, ik rwiev).
The total mass diffised into the pcxe space at a gi~’en time is the sum of the contributions to
diffusive mass uptake or release of the different values of DP present in the rock. For the case of a
continuous distribution of 111,total maw uptake or rele~se for the sample is described by the
following expression (Ffemi~g,f998):
M(f) = e ]fc(DJM’(LD,)dD,a
(Eqn. 8)
where the integration limits, a and 12represent the minimum and maximum possible values of DPand
may be taken in general to be () and m, andJ@P) is a densiy function (DFj which describes the
relative frequencies of different values of D? in terms of the prcqmrtion c}fthe total pc)rosiry in which
those values of DPare operative:
[1- f3Dfc(Dp)=-& —, 0’ (Eqn. 9)
Fleming & HWWIV. ~~OdKIIIWSOIUICdiffusicm in the prc-smlce of p~re-scale I)ctcrogelwity 11
where 9r#3 is the ratio of the porosity constituted by pores with a particular pore diffusivity, DA to
the total porosity of the sample. It is important to note that the use of a DF does not imply that DP
is stochastic. Rather, the fi_mctionJ@P) is simply a means of weighting the contribution, A4’(LDP),to
total mass uptake or release, M(z), made by a particular pore diffusivity, Dp, on the basis of tke
relative (volumetric) prevalence of pores with that value of DP The tidl derivation is given in FLvning
(199/7),but note that Eqn. 8 is very similar in form to an expression presented in the chemical
engineering literature over a quarter of a century ago which quantified diffhsive mass uptake in the
presence of a continuous distribution of ~grainsizes (IWherz and L-JIghLrz,1977).
The functionJ@P) maybe any continuous distribution. A number of general considerations
suggest that a lognormal distribution may be appropriate (Hu~ger~ u)lJ Gonvkk, f~98). Discrete
distributions and alternative continuous DFs have also been used, such as gynma and piece-wise
linear distributions (see Ho.!lwheckel u1., 1999). However, fommtion factor datz from the Culebra
dolomite support the assumption of a lognormal DF (Figure 4), and previous analyses of tracer tests
performed in the Culebra obtained excellent results using a lognormal distribution (Haggero et al, in
fevieu<McKenna et A., in rw&+).
We normalize A4(f)bv the mfisimum total solute mass present in the rock, A4T. In the crise of
diffision into the rock, this is the mass at I = m. Fcx difftlsicm out of the rock, this is the mass at r =
O. This maximum total mass is given by:
iW~=@AIC~ (Eqn. 10)
where the 1is the length of the rock sl;lb as mezsured away from the resenwir and C= is the initial
sample concentration out-diffusion (Co)and the late-time sample concentration for in-diffusion
(C(GCOJI. If normalized conce~ltrxtions are used, C~ = 1.
Making the appropriate substitutions forjf-@P) and A4’(i,DP)into Eqn. 8 and normalizing by
Eqn. 10 yields:
c, (p) -~
\m’ ‘anh(’w)+dDp
J
(Egn. 11)
where ~ and ad are the mean and standard deviation, respectively, of h@P).
A FORTIL4N77 code was written to so]ve Eqn. 11 for both forward simulation and
parameter estimation. The purpose of the code was to obtain estimates of the values of ~ and
ad from the diffk.ion data previously discussed. Note that a distribution of diffisivities interpreted
from data under the assumption of nonintersecting pores in a parallel arrangement, aligned
approximately nomud to the surface of the sample, should be considered a lumped distribution
representing what may be diffusion in parallel and series in a more complex geometty (Ha~.ger~yad
Gunzkk, f$%’). Several subroutines from the IMSL (h~ternational Mathematics and Statistics
Libraries) Version 3.0 package u’ere used. These subroutines use the de. Hoog algorithm to
numerically evaluate the inverse Laplace mmsform (deHoog e[ al., 19S2),Gauss -Krrn~rod rules to
perform numerical integration, and a modified h\-ell}lerg-h4arqtlardt algorithm (Mtwqwd, 1963) and
a fmite-di fference Jacobian to solve the non] inear least-squares c)ptimization problem. The lower and
upper limits of integmtion were set to the 1 x 10-7and (1 - 1xl O-q percentile values of D*
respectively, for a gi~’en f.ldand ~.+ For kge ~,dues of the Lap]ace parameter, p, the ccmlp]ex vahe of
the hyperbolic tangent in Eqn. 11 was set to [1,0] to overcome numerical difficulties (Ffimit.y, 19%’).
Capability to use Eqns. 6 and 10 to estimate a conventional , singje-rate diffkivity from the
normalized mass data was also incorporated into the code.
...... ..- . . .. . . ... .. ....... . .....---- . . ...... ..... . . . :. --, .--.’ . . .
Log-space root mean square error @4SE) wm used m a measure of fit, as log-transformed
data shows a greater sensitivity to the late-time, h’ mass ratios at which multirate effects are b~eatest
(e.g., IWJven and Loughh, 1971; fi~ggertyutldGoruftk, 1998). Similar concerns led us to convert the
in-diffusion mass ratio curves, which are large-valued at late times, to:
(Eqn. 12)
which is small-valued at late times. This approach u’as also used by lUdwetZutzdLwghilz(1971).
When the data is then log-transformed, small relative changes in the raw late-time mass ratio data
give rise to significaptl y larger relative changes in the converted and transfomled values.
An approximation of parameter estimation statistics are calculated using the Jacobian, or
sensitivig, matrix. This is a )1by ?Itmatrix, where His the number of pm-arneters to be estimated (1
for a single-rate model, 2 for multirate) and t)t is the number of data points in the mass ratio cumes.
The entries of the Jacobian, JX consist of the deti\~i~ti\~eof the calculated mass ratio at the i:btime,
Mi/MT, with RSPt2Ct to thejb pmXmeter,A (e.g., ~mpnwl ud ~“OJ& 19$7; J%Jwejv et u~.,7996):
{/)h4iA4T
Jv =zp j
(Eqn.13)
The covariance rrultrix is a rZby n matrix estinmted by tding the inverse of the square of the Jzcobian.
and multiplying each entry by the variance of the random error in the observed data (e.g., KtzopzLzn
and VOJJ,1987). As we have no direct measure of the error in the data, we follow the common
approach [e.g., Bani, 1974, p. 178] of using the mean square error between the data and the calculated
cumes as a surro~te for the replicate vmiance. The square roots of the entries along the diagonal of
the covariance m~trix give, in turn, the standard cie~~iations(08(X,confidence intel~’ids) of nomdy-
distributed error about the best-fit values cjf the estinmted parameters. C)ur cc)de returned estimates
Fleming & Hawq, Nfocleling SOIUIrdiffltsic,n III the presence ofpore-scale IIeterogrm.,ty, 14
of the arithmetic value of M but of the narurd ]o~rithrns of DP and ~d, so in arithmetic space the
confidence internals are symmetric about M but asymmetric about Dp and ~d.
4. RESULTS
Suites of forward simulations demonstrating the effects of variation in M and ad, and
comparing single-rate diffusion to multirate diffusion, are given in F@re 5. Plots of the static
diffusion data and of curves calculated using best-fit single-rate and mulrirate models are given for
two examples in Figure 6. Table 2 summarizes the results for all experiments, giving the best-fit
value of Dp for each sample and for kl k diffusion obtained using a single-rate model, as well as the
corresponding best-fit multimte parameters. Larger ~ values indicate faster diffusion, on average, for
a given block length and larger ad values indicate the presence of a wider range of DP (i.e., more
significant multirate effects). RMSE values m-ealso given , as are estinY.ltes of 95’?/o confidence
intervals for Dp M, and ad.
While the quality of the fits of the calculated curves to the data is generally good, these
matches are inferior to those obrained from analysis of ficlc{-scale CLdcbra tmcer tests using a
multirate mass tram fer model (Htigy-y c! d. it~revicu.~.MdGmw et al, itl mien.) and to those obtxined
from other laboratory data where multirate mass transfer was investiwlted (e.g., lP’erfAet u1.,1997;
Haggerty and Gort&k, 1998). This is attributed to a combination of the quality of our data and model
assumptions that may not be fully did. The scztter app:lrent in Fi<gures3 and 6 may result from
some combination of fzilure to maintain a conscult-concen traticm boundary condition, gypsum
dissolution, and a Izck of precision in the s-ray imaging technique, which is still under develc)pment.
Our assumption of constant diffusit~ity along the pore, though also assumed in other studies of this
type, is likely a poor assumption for a small sample of the CLdcbra dc)lomite, which is high]y
heterogeneous at all spatial scales. The assumptions of purely one-dimensional diffusicm and of a
lognorrnal distribution of diffusivity are potenti;d]y in error, particularly in samples thzt may be
dominated by a small number of porosity types (for example, fractures and intrmgranuk porosity).
The model used in this paper is best applied when the m’emging volume is large relative to the scale
of heterosneity; this condition is probably not met within some of the individual samples we
examined.
Since D% increases somewhat with concentration (see Cff.mkr,1997), variation in tracer
concentration within the samples with time might have led to changes in DPover the course of the
experiment. However, Cm-h (1976) showed that concentration-dependence of D% has a negligible
effect on concentration profiles and breakthrough curves in systems simik to the static dt ffision
experiments. As the mass ratio curves used in OLMana$sis are integrated concentration curves,
concenwation-dependent Dw should not be a significant source of error. Moreover, we were
permitted to test. whether this was an important effect by noting that the average tracer concentration
within a given sample was significantly higher during the out-diffusion phase than during in-
diffision. If concentration-dependence of D%was present and Imd a significant effect Lipon the
mass curves, then the out-diffusion experiments would ‘giveconsistently larger diffusi~’ities than
tihose obtained with the same sample from the in-diffbsicm experiments. This was not observed
~able 2).
Comparison of D,q of the M tracer to the best-fit distribution of pore diffusivities (for the
multirate model) and the best-fit Dp (for the single-rate model) might provide a means of confirming
the reasonableness of the parameter estimates, because Dp can never exceed D~$ J-+owever, three,
considerations show that such comparisons are bc)th prohlemztic and unnecessary. First, alrhough
DP and Mare well-constrained, the confidence intends about the best-fit values are nonetheless
sufficiently wide (largely due to data scatter) to accommodate any reasonable value of DW for this
system. Second, judging the entire distribution against D“. presumes that the tails of the distribution
should be reliable. However, the lognorrnal distribution is only an approximation to the true
distribution of diffusivities in the rock. Moreover, the sensitivity of any experiment to diffi]sion rates
in the tails of the distributions is lower than thxt to rates of diffusion closer to the mean (Hug<c~ e~
Fleming & Has+gerty. hfocieliIIg solute dilhioII in the prrsc.i]ce of pore-scale I,ctrrc>gewiry, 1()
A., it~nvz”w).Third, D% is unknown for these experiments, which ccmsisted of a quatemav diffusion
system containing 1%+, Cl-, K+, and 1-ions. Some other species might also hxve been present, such
as calcium and SU1fate ions produced by ,g~psurn dissolution. The complexities of mul ticomponenr
diffusion are discussed in detail in Cm-LT (19Y7) and F.lemitg (1W8); essentially, Dw must be
determined experimentally, which has not been done for this diffusion system.
5. DISCUSS ONI
~.1 Sinrrie-Rare versus Multimte Mode]s
The 9570 confidence ranges in ad for P133-H, RC1-.A, and RC4-D do not include O Table 2).
It is also ciear from the RAKELi~ and RA{SEsRvalues that the mul tir.lte model prrx’ides a quanrifmbly
superior fit. .Multirate diffusion, therefore, occurs in these smnplcs. In contrxst, consen>ative
consideration of the in-diffusion and out-diffhsirm results mken together su<<qeststhar mul tirate
diffision was not detected in samples RC2-B and RC6-G.
Multiple simultaneous diffisivities are significant at the lab-scale in spite of the relatively small
estimated values of ad and the fact that two of the five samples can nc)t be prcn~en to shc)w mu]timte
effects from our modeling of this data. There are three reasons for this. First, the amumed
. distribution is lognormal, so a relatively small ~d represents significtilt ~“arixbilityin DP A cr,lvalue of
0.5, for example, indicates almost ml order of magnitude difference between the 0.02.5 and 0.975
qua.ntile diffkivities. This difference increases to a filctc)r of 50 for a O(Iof 1.
Second, while the ~’isibleeffects of variable diffhsi~’ity upon the mass mtio cun~es generally do
not appear to be large for small Values of o~ (Fi<gure()), single-rate and mu] tirate cunrcs diverge at
later times ~igure 5; see ako pu!lhrm ut~dbgiuftl, ~971; Hug;t+ u}zd~od’”k. ~~~/i’). This k
consistent with the finding by Cw/n@kZ md RAv-f.r (199 S) that even small ~’ariahilitv in di ffusivity,
may have significant implications for long-telrn solute transport.
Third, we consemmively rejected the hypothesis that a range of diffusivity is present in a given
sample if we could not prove that it was. ~f the 95% confidence range in estimated ad included or
very nearly included O, and if the RMSE vaiues did not show that the multirate model provided a
significantly better fit than the single-rate model, it was concluded that multirate diffusion um not
present. However, these tests do not indicate that a single-rate model is preferred, either.- Any ad
range which included O also included infiniv, the inversion statistics in such a case indicate that it is
impossible to determine from the available data whether multi rate effects are present in that sample.
This is due to a high degree of insensitivity of RMSE, over the dynamic range of the data, for a given
value of ~ (which is rexonab}y well-estimated in all cases), and in the presence of scatter in the data,
to the vaiue of ad . ~f a sin$e-r~ie model was ckdy superior, the rnultirate inversion would return a
value of cd near Owith narrow confidence bounds. Data collection to later times (i.e., over a greater
dynamic range) might facilitate far more reli:lble estimates C)fad (Figure 5).
Work by TidwY Cld. (zkp-m) detected vmi:dility in diffi]si~~ityin samples RC2-B and RC6-G
(for example, RC2-B contains a fmcmre), whereas our mode]ing did not. Tidvvl efd (it~pnx.j
anaiyzed concentrations along parallel transects into the samples, wherein we analyzed the rn:lss in
the whole sample through time. This whole-sample mas measurement is influenced by mulrirate
diffusion in approximately the same manner as a tracer test aicmg a fracture through the rock would
be influenced, insofar m a tracer test mmples all of the diff~lsi~~itiessimultmeously. In addition, both#
whole-sample mass measurements and solute tracer tests are most strongly influenced by small
amounts of rock with Iou’er-than-m’erage di ftisivities (e.g., Cmmi//gLvzmd Rdzti.r, 1998), whereas
the fracture con~lined in RC2-B hzs a higher-than avera<gedi ffusivity. The results obtained using
whole-sample mass measurements are thus more amenable tc) ccm~parison with previous, field-scaie
results. It is also important to recall tklr our results do not preclude the presence of multi-ate effects
in these two samples, m explained previcmsly.
The bulk diffusion data clearly shc)u’ multirate effects, as expected (see Table 2). In addition,
some of the key characteristics of the bulk diffusion cm be assembled frcxn ciiffhsion in the
individual samples. Using in-diffusion as an example, the geometric mean of the estimated DPvalues
for the individual blocks (2.27 x 10-6mz/hr) is very simik to the estimated geometric mean of the
lognormal distribution for the bulk data (2.26x 10-6 mz/hr). In addition, the minimum estimated
value of DP for an individual block (RC4-D, 9.57 x 10-7mz/hr) corresponds roughly to the 20*
percentile in the lognormal CDF of diffusivities (9.96 x 10-7mz/hr). Slmik-ly, the maximum
estimated value of DP for an individual block (B33-H, 4.78 x 10-Gm~/hr) corresponds roughly to the
80dIpercentile in the lognonnal CDF of diffusivities (5.16 x 10-6m~/hr). Of course, neither the five
single-rate DPvalues nor the five lognormal distributions from the individual slabs combine
additively, in a rigorous sense, to form z new lognormal distribution corresponding to the hulk
experiment. However, it is clear that the estimated Iognormal distribution estimated solely from the .
bulk diffhsion data provides a reasonable representation of the range of pore diffbsivities known to
exist within the individual slabs. A~in we note that the bulk di ffusion data do not represent the full
Iithologic variability within the Ctdebra, and therefore our results should not be taken to represent
the full variability of diffusivity within the Culebra.
5.2 Diffhsiviw and Formation Factor
As diffusivity is inversely proportional to formation factor, the distribution of one should be
similar to that of the other. Unfortunately, calculation of absc)lute D4 values from formation factor.
data may be unreliable and it is therefore difficult to directly convert a CDF of formaticm fhctors to a
CDF of DP However, the standard deviarion of the best-fit lognmmal distribution of formtion
factors determined from all 21 Culebra samples (see figure 4 and caption} shcdd be similar to ad
estimated from the bulk static diffusion experiments. CTfor the fommion f~ctors was found to be
approximately 0.7, which is similm to the values found for bulk diffilsion data. This reasonably close
correspondence serves as supporting evidence for our interpretation of the diffusion experiments,
and raises the possibility that formation filctnrs - which are much easier and cheaper to obtain than
laboratory diffusion data - mi<ghtprovide an indirect but reasonably reliable means of estimating
pore-scale heterogeneity.
5.3 Otie r Imolicat ions:
It is usefi.d to compare diffusion parameters with conventional descriptions of geologic media.
Our ability to determine firm correlations is currently limited by the small number of samples and the
krge confidence bounds on ~& Nonetheless, it appears that ~ increases with bulk porosity and the
relative presence of certain microscopically visible features (fracturin~ vugginess, and gypsum-
fillina, whereas ~a does not appear to correlate with these parameters or the number of different
types of features present in a given sample (Ffemhg. 7998). Greater interconnectedness of the pore
space due to increased porosity and increased presence of kmgwporosity features likely leads to faster
average rates of diffusion relative to dead-end pores in which solqte becomes trapped (e.g., DyL!!r(i~e/z
and Cmg, 1989). Gypsum occurs in the Culebra within vugs and fracmres (Ho1, 1997),and may
sirnpl y be an indicator of interconnected fracture and vwg porosity. The apparent lack of correlation
between ad and macroscopiczdly visible features su<ggeststhzt ad may be controlled by variability in
pore geometry within these features and/or by micropore-scale (intergmnular or intercrystal]ine)
porosity variability.
Another motivation for laboratory-scale studies was to evaluate whether spatial scaling
da&XIShipS might eXkt. (h?2pafk)ll Of best-fit ~d Values from eXperilnellts at dlree spatial scales
(individual samples, bulk diffusion, and the results of previous field-scde tracer tests) suggests that ad
increases with sampling ~’olume (Ffcni~{<.199X). In our work on sample volumes in the range of 60
to 70 cmJ, the estimzted value of ~d is Oto 1. The bulk diffusion darz lun’e 2 combined sample
volume of 3.3 x 10? cmJ and have an estimated u~ of 1 to 1.2. S@e-well injection-withal raud tests
conducted at the same location as the samples were t.ken from [Me&.ratzdBea//ieti,itznwieu]had
sample volumes of 2 x 10? m3 to 3 x 10~m3. l-f~ggc~ ‘eiu~.[~~zrw~cu.jestimated ~d on diffisicm rate
coefficients in these tests in the range of 2.5 to 7. Two-well tracer tests conducted at the same site
[kL-Kemzaet d., h retie~] on slightly larger sample volumes had estimated G, values of 5.5 [McKetzme~
al, in reu>u~,although a single-rare model also provided a good representation of the data. We note
that the estimates of ad at the field-scale also incorpomte the effects of ~m-iability in block-size in
addition to variability in Dp
The basic physical reasons for increasing o~ with sample volume are fairly clear. In general,
geologic materials are heterogeneous in most respects, and a greater range in the property of interest
is commonly found if a larger region is considered. As cd is a measure of heterogeneity in Dp withk
the volume of rock sampled, it may increxe as that sampling volume increases and a wider range of
DP is thus encountered. However, at least three conditions preclude determiwltion of a quantitative
relationship between spatial scale and mass transfer at this rime: (1) the small number of sampling
volumes for which Culebra n-nss transfer data are available (2) the samples used in the laboratory
diffusion experiments do not ftdly reflect the range of lithologies in the (ldebra; (3) the uncertainty
in block-size ~’ariabilit-yat the field-scale, the effects of which cannot currently he scpamted from
variability in DP
6, CO NCLUSIONS
..
An expression for solute diffkion in the presence of multiple simul ta.neous matrix
diffisivities, Dp was developed. This was then used to model data from eight labomtcmy-scale
diffusion experiments performed on five core samples (approximately 5 cm by 4 cm by 2.5 cm) of
the Culebra Member of the Rustler Fomxltiolr, a heterogeneous, fracrured, e~-aporitic dolomite. A
]ogyormal distribution of DOwas ;lssumed.
Our main conclusion are: (1) The estimated scmdard de~~iation (o,]) of ln(DJ) within each
Culebra dolomite sample ranges from O to 1. Nlost values lie herween 0.5 wxi 1, implying at least half
an order of magnitude variation in DPwithin each small sample; and (2) The ~wiability over ail
samples leads to a combined Crdin the range of 1.0 to 1.2, which appears to be consistent with both
diffision results from the individual samples and a best-fit statistical distribution of formation factor
measurements for similar Culebra samples. The high observed degree of variability in mass transfer
rates within even such small volumes as these is of considerable significance to groundwater
remediation and geologic waste disposal issues.
Inferences with respect to geologic controls upon multirate mass transfer parameters should
be regarded as tentative at this time due to limited daw availzl)ility. Nonetheless, important
prelirninaq conclusions in this regard, which should serve as a focus for finure investigations, include
the following (3) Comparison of our estimation results to other rock properties suggests tha~ at the
lab-scale, the geometric mean of DPincreases with bulk porosity and the prevalence of macroscopic
porosity features such as fractures and vugs. This is attributed to increases in the interconnectedness
of the pore space; and (4) Comparison of our results with experiments at larger spatial scales suggests
that increasing sample volume resdts in an increase in ad. The VdUe of cd does not appear to be
correlated to bulk porosity or the prevalence of macroscopic porosity features, but may be
determined by variability within such macroscopic features and/or by micropore-scale heterogeneity.
ACKNOWLEDGEMENTS
We wish to thnk V. Tidwell, L. C. Meigs, R. M. HoIt, S. A. h4cKenna, A. R. Lzppin, and C. F.
Harvey for their comments and su=estions. We also appreciate the logistical support of M. J.
Kelley. This work was funded by Sandia Cmpcmition, a Lockheed Martin company, fc)r the United
States Department of Energy under Ccmtract DE-.AC04-94,%L85[100.
Fleming & H~vrIY. F.fwlcling SCIILIIeciIlhIs&:I III tlIe prLwIIce oflmrr-scalr lICWmLTn&Iy ‘)7. -.
REFERENCES
Armstron~ J. E., Frind, E. O., and McC1cllan, R. D. 1994. hlonequilibriurn mms transfer betweenthe vapor, aqueous, and solid phases in unsaturated soils during vapor extraction. Water Resour. Res.30(2), 355-368.
.Ball, W. P., and Roberts, l?. V. 1991. Long-term sorption of }lalogenated orgw;c chemicals by aquifermaterial. 2. Intraparticle diffusion. Eviron. Sci. Technol. 25(7), 1237-1249.
Bard, Y. 1974. Nonlinear Parameter Estimation. Academic Press. New York, NY. 341 p..
Bear,J. 1972. Dynamics of Fluids in Porous Media. Dover. New York, NY. 764p.
Brusseau, M. L., and Rae, P. S. C. 1989. Sorption nonideality during organic contaminant transport inporous media. Crit. Rev. Environ. Control. 19(1), 33-99.
Christian-Fre~, T. L., Tldwell, V. C., and Meigs, L. C. 1997. Technical memcmndunx Results ofexperimental methodology development for visualizing and quantifying matrix di ffhsion in theCulebra Dolomite. NH184. Geohydrology Dept., Sandia National Laboratories. Albuquerque, Nhl,28 p.
Culver, T. B., Hall isey, S. P., %hoo, D., Deitsch, J. J., and Smith, J. A. 1997. h40deling the resorptionof organic contaminants from long- ternl contaminated soil using dis tributed mass tram fer rates.Environ. Sci. Technol. 31(6), 1581-1588.
Cunningham, J. A., and Roberts, P. V. 1998. Use of temporal moments to investigyre the effects ofnonuniform grain-size distribution on the transport of sorbing solutes. Water Rescmr. Res. 34(6),1415-1425.
Cunninghm, ]. A., Werth, C. J., Reinhard, h4., and Roberts, P. V. 1997. Effects of grain-scale masstransfer on the transport of volzrile organics through sediments. 1. h40del development. WarerResour. Res. 33(12), 2713-2726.
Cussler, E. L. 1976. hlulticornponent Diffusion. Elsevier. New York, NY. 176 p.
Cussler, E. L. 1997. Diffusion: hks Transfer in Fluid Systems. Cambridge University Press. New ..York, NY. 580 p.
Deans, H. A., and Carlisle, C. T. 1986. Single-well tracer test in complex pen-e systems. Society ofPetroleum Engineers/U.S. Department of Energy SPE/DOE 14886.61-68.
de Hoo& F. R,, Knight, J. H., and Stc]lies, A. N. 1982. An imprcn~ed method for numerical inversionof Lzplace transforms. SLAMJour. Sci. Stat. Comput. 3(3), X57-.366.
Dykhuizen, R. C., and C;uiey, \X1.H. 1989. .%1an;llysis of solute diffhsion in the Culebra Dolomite.SAND89-0750. %ndia National Labomtories. Albuquerque, NL4. 50p.
Flemina S. W. 1998. Single and Multiple Rates of Nonequilibrium Diffusive hhss Transfer at theLaboratory, Field, and Regional Scales in the Culebra h4ember of the Rustler Formation, NewMexico. M.S. thesis, Oregon Sr~te Uni~’ersi~7.Corval]is, OR. 172 p.
.
. .. ... .... . . .
Fry, V. A., and Istok, J. D. 1994. Effects of rate-limited resorption on the feasibility of in situbioremediarion. Water Resour. Res. 30(S), 2413-2422.
Gierke, J. S., Hutzler, N. J., and McKenzie, D. B. 1992. Vapor transport in u&.aturmed soil columns:Implications for vapor extraction. Water Resour. Res. 28(2), 323-335.
Goltz, M. N., and Oxley, hi. E. 1991. Analytical modeling of aquifer decontamination by pumpingwhen transport is affected by rate-limited sorption. Warcr Resour. Res. 27(-!), 547-556.
Grathwohl, P., and Kleineidam, S. 1995. Impact of heterogeneous zquifer materials on sorpti~ncapacities and sorption dynamics of or~mic contaminants. Groundwater C@ity: Remediarion andProtection. IAHS. Wallingford, Oxfordshire, England. 79-86.
Ha~rty, R., and Gorelick, S. M. 1995. Multiple-rate mass transfer for modeling difiision andsurface reactions k media with pore-scale heterogeneity. Water Resour. F@. 31(10), 2383-2400.
Hagyx-ry, R., and Gorelick, S. M. 1998. Modeling mass transfer processes in soil columns with pore-scale heterogeneity. Soil Sci. Sot. Am. J., 62(l), 62-74.
Haggm-ty, R., and Harvey, C. F. 1997. .4 comparkon of estinmed mass transfer rate coefficients withthe time-scales of the experiments from which they were determined. EOS Trans., 78(46), F293.
Haggerty, R., Fleminq S. W., Meigs, L. C., and McKenna, S. .4. In re~~iew.Convergent-flow tracertests in a fractured dolomite, 3, Analysis of mass transfer in single-well illjectit)ll-~vitl~dfil~~’:11tests.~later Resour. Res..
Hamlon, T. C., Semprini, L., and Roberts, P. V. 1992. Simulating groundwater solute transport usinglaboratory-based sorption parameters. J. Em’iron. Eng. 118(.5), 666-689.
Harmon, T. C., and Roberts, P. V. 1994. Comparison of htrapilrtde sorpti{m and resorption ratesfor a halogenated alkene in a sandy aquifer material. En~~iron. .$ci. Tcchnol. 28(9), 1650-1660.
Harvey, C. F., Haggmy, R., and Gorelick, S. M. 1994. Aquifer remediation: .4 method for estimatingmass transfer rate coefficients and an evaluation of pulsed pumping. Water Resour. Res. 30(7), 1979-1991.
Harvey, J. W., Wagner, B. J., and Bencala, K. E. 1996. E~duating the rcliahility of the stream tracerapproach to characterize stream-subsurface water exchange. Water Resour. Res. 32(8), 2441-2451.
Hollenbeck, K. J., Harvey, C. F., Hag-gerty, R., and Werth, C. J. 1999. .4 method for estimatingdistributions of nmss transfer rate coefficients with application to purging and batch experiments. J.Contain. Hydrol. 37(3-4), 367-388.
HoIt, R. M. 1997. Conceptual model for transport processes in the Culebra Dolomite Member,Rustler Formation. SA14D97-01 94. %ndia !+tional Lakratories, .41bucplrxque, NM.
Kelley, V. -4., and %ulnier, G. J., Jr. 199(I. Cm-e zuu-ilysesfor selected samples from the Cu]ebtilDolomite at the \i~mte Isc]laticm Pilot Plant site. S.IND9(}-7011. %ndia Nari(mal Lahomtc)ries,Albuquerque, NM.
Knopman, D. S., and Voss, C. I. 1987. Behavior of sensitivities in the one-dimensional advection-dispersion equation: Implications for parameter estimation and sampling design. Water Resrmr. Res.23(2), 253-272.
Love~ O. M., Richter, F. hf., and Harrison, T. M. 1989. The 40Ar/3qAr thermochronometry forslowly cooled samples having a distribution of diffusion domain sizes. J. Geophys. Res. 94(B12),17,917-17,935.
Lovers, O. M., Heizler, h~. T., and Harrison, T. M. 1993. Argon diffusion donmins in K-feldspar II:Kinetic properties of MH-10. Contrib. Mineral. Petrol. 113(3), 381-393.
Mann, U. 1994. An integrated approach to the study of primary petroleum migmtion. In Geofluids:Origin, Migration and Evolution of Fluids in Sedimentary Basins, J. Parnel] , ed. Geological, SocietySpecial Publication No. 78. The Geolo@l .Society, London.
Marquard~ D. W. 1963. An algorithm for least-squares estimation of nonlinear parameters. SIAM J..Indust. Appl. Math. 11(2), 431-441.
McKennz S. A., Meigs, L. C., and H+zgxty, R. In review. Convergent-flow tracer tests in a fractureddolomite, 4, Double porosity, multiple mass-transfer rate processes in two-well converwnt flow tests.Water Resour. Res.
Meigs, L. C., zmd Beauheirn, R. L. In review. Tracer tests in a fractured dolomite, 1, Experimentaldesign and observed tracer recoveries. Wilter Resour. Res..
Ohlsson, Y., and Neretnieks, I. 1995. Literature survey of matrix diffusion theory and ofexperiments and data including natural analogues. SKB Technical Report 95-12. Swedish I+clearFuel and Waste Mamgement Co. Stockholm. 89p.
Pedi~ J. A., and Miller, C. T. 1994. Heterogeneous sorpticm processes in subsurfilce systems, 1,Model formulations and applications- Environ. Sci. Technol. 2S(12), 2(}94-2104.
Pedit, J. A., and Miller, C. T. 1995. Heterogeneous sorption processes in subsurface systems, 2,Diffision modeling approaches. Environ. .$ci. Technol. 29~, 1766-1772.
Pignatello, J. J. 1990. Slowly reversible sorption of aliplxltic halocarhons in soils, 2, Mechanisticaspects. Environ. Toxicol. Chem. 9(9), 1117-1126.
Qin, Z., Lu, F., and Anderson, A. T., Jr, 1992. Diffhsive reequilibration of melt and fluid inclusions.Am. Mineral.. 77(5-6), 565-576.
Quinodoz, H. A. M., and Valocchi, A. J. 1993. Stochastic analysis of the transport of kineticallysorbing solutes in aguifers with randomly heterc)geneous hydraulic conductivity. Water Resour. Res.29(9), 3227-3240.
Rabideau, A. J., and hfiller, C. T. 1994. Two-dimensional mcdeling of aquifer remediaticm influencedby sorption nonequilibrium and hydraulic conducti~ity heterc)geneity. Water Resour. Res. 30(5),1457-1470.
Rae, P. S. C., Rolston, D. E., Jessup, R. E., and Da~~idson,]. M. 1980. Solute transport in a~egltedporous media Theoretical experimental euduation. Scil Sci. Sc)c. Am.]. 44(6), 1139-1146.
25
Roberts, P. V., Goltz, M. N., and MacIGy, D. L4. 19S6. .4 nztural ~gradient experiment on solutetransport in a sand aquifer. 3. Retardation estimates and mms balances for or.gpic solutes. \\hterResour. Res. 22(13), 2047-2058.
Rugner, H., Kleineidam, S., and Grathwohl, P. 1999. Long term sorption kinetics of phenanthrene inaquifer materials. Environ. Sci. Technol. 33(10), 1645-1651.
Ruthven, D. M., and Loughlin, K. F. 1971. The effect of cvstalline shape and size distribution ondift%sion measurements in molecular sieves. Chem. Eng. Sci. 26(5), 577-584.
Satterfield, C. N., Colton, C. K., and Pitcher, W. H., Jr. 1973. Restricted diffhsion in liquids withinfine pores. AIChEJ. 19(3), 628-635.
Shackelford, C. D. 1991. Laboratory diffusion testing for waste disposal - A review. J. Contain.Hydrol. 7(3), 177-217.
Steinberg S. M., Pignatello, J. J., and %vhney, B. L. 1987. Persistence of 1,2-dibromoethane in soils:Entrapment in intraparticle micropores. Environ. Sci. Technol. 21 (12), 1201-1208.
Tidwell, V. C., and Gks, R. J. 1994. Y ray and visible light transmission for laboratory .nmasurememof two-dimensional saturation fields in thin-slab systems. \Vatcr Rmour. Res., 30(1 1), 2873-2882.
Tidwell, V. C., Meigs, L. C., Christian-Frem-, T., and 130ney, C. Lf. In press. Effects of spatiallyheterogmeous porosity on matrix diffision as investiwted by X-ray absorption imaging. J. ContanI.Hydrol.
Tomich, J. F., Dalton, R. L.,Jr., Deans, H. A., and Shallenberger, L. K. 1973. Single-well tracermethod to mea..ure residual oil saturation. J. Petrol. Technol., 25(2), 211-218.
Villermaux, J. 1981. Theory of linear chrcmmtography. Percolation Pr&cesses: Thecxy andApplications. A. E. Rodrigues and D. Tondeur, eds. Sijthoff & Noordhoff. Rockrille, MD. NATOASI Ser. E. 33,83-140.
Welty, J. R., Wicks, C. E., and Wilson, R. E, 1984. Fundamentals of h40mentum, Heat, and MassTransfer. John Wiley and Sons. New York, NY. 803 p.
Werth, <C.J., Cunningham, J. .4., Roberts, P. V., and Reinhard, M. 1997. Effects of grain-scale masstransfer on t he transport of volatile organics through sediments. 2. Column resul K. Wirer Resour.Res. 33(12), 2727-2740.
Wood, W. W., Kmemer, T. F., and Heron P. P., J r. 1990. Intragmnuk- difthion: .4n impormntmechanism influencing solute transport in cl;lstic ;Iquifers? Science. 247(49.50), 1569-1.572.
TABLE CAPTIONS
Table k Brief descriptions of Culebm dolomite samples used in the s~ltic diffision experiments,including bulk porosity (adapted from Ctkz>/iun-Fnwre[ al., 1997 and Tidwtl e[ uL,i~zpess).
Table 2 Results for single-rate md multirate pmarneter estimations. D? is the best-fit pore
diffusivity estimated using a conventional single-rate model. Values of M and ad are best-fitparameters of the lognomul distribution of pore diffusivity used in the multimte inversion. The value
M maybe compared to ln(DP). C@ity of the fits to the dam we described by the log-space rootmean square errors for both single-rote (SR) and multirate (MR) inversions. Out-diffisionexperiments were not conducted for samples RC4-D and RC6-G.
I ‘“’“’””-’“’””’””’’”’”””””:~....... .,-.
Fkvnmg & Haggerty, Modeling .olutr difkiou in ilw presm)ce of pore-scale Iwtcrobm]wiry.
FIGURE CAP TIONS
Figure 1: Location map.
Figure 2: Scales and types of Culebra matrix porosity (from Hoi, 1997).
Figure 3: Mass uptake and recovery data. Figure 3.2: in-diffusion. Vertical axis gives the value of 1-
M@/M(l=m). Figure 3.b: out-diffusion. Vertical axis gives Al(~/M(t=O). Out-diffbsion experimentswere not conducted for sarnpies RC4-D and RC6-G.
Figure 4: Empirical and best-fit CDFS of formation factor. D~ta is for Culebra samples analyzed byTerraTek (HoL, 1997). Samples were taken from the H-19 hydropad, primarily borehole H-19 b4 (RBuuba”m,~cnona~mmmI{)uca~btz,1997); samples used in static difksion experiments were taken fromtbe same location. Best-fit lognorrnal CDF was determined by manual calibmtion and has ageomernc mean of 4.5 (corresponding to a formation factor of 90) and a standard devimion of 0.7.Earlier work at other locations at the \WPP site by Ke&yand SaAier (7990) also suggests thatformation factors in the Culebra dolomite are lognormall~7 distributed. Formation factors arecommonly taken to be inversely proportional to diffusivity (e.g.,Bear, 1972; Ke@ atzdSaw$zier,1990),so reasonably good quality of the fit su~sts that use of a lognormal distribution to approximate thetrue distribution of DP in the Culebra is appropriate. Calculations of DP from formation factors tendto be unreliable, due to the dependence of fommion factor measurements on the properties of theions present in tAe pore fluid and uncertainty in the precise form of the relationship bemveenformation factor and DP (see Kelly atzd.fawkierj 1990 for discussion). However, the standard
deviation of the lognormal distribution of fommtion C.lctors should be a good approximation to ~d inthe distribution of Ilk
Figure 5: Sensitivity analysis. Forward simulations were perfomled for the case of in-diffision.
Verdc~ a..is on both plots gives the value of 1-A4(r)/M(/=m). Figure 5.2: model responses for three
values of W; holding c$rJconstant at 1. Units of pd are hrl. Figure 5.LI:model responses for three
values of ad, holding ~ constant at –13; note that ~d=() corresponds to a conventional single-rate
model. The effects of variation in ad are far more prcmcmnccd at late times and for small values of
the mass ratio, relative to the effects c)fva:iation in pd.
Figure 6: Mass ratio datz md model responses. Best-fit curves were calculated using both aconventional single diffusivity (single-rate model) and a Iognormal distribution of diffusivities(multirate model). Figure 6.x in-diffusion fc)i~an~ple RC1-A. Vertical axis gives the value of 1-
M#)/M(t=@. Figure 6.b: out-diffision for bulk data. Vertical axis gives the value of A4(j)/A4@=O).
Sample Q L Description
(cm)B33-H 0.15 5.5 .4bundant open \-ugsdirectly interconnected and/or
interconnected by fractures; some qpsurn-filling of rugs. h4ajorporosiw tvpes = interc~stalline. vu,~ . and fractures.
RC1-A 0.13 4.9 Relativelyuniform; single faint lasnin~ minor micro~-ugsand<micro fmctures. hl~jor porositv t-vpes = intercwstdline.
RC2-B 0.09 6.6 Distinct fracture spanning half the length of the s.arnple. Majorporosi~ tvpes = Intercqstdline and fractures.
RC4-D 0.09 5.8 Faint laminations; minor vuggy porosity sc~ttered throubfiou~minor microfmctures. hfajor porosiw tvpes = intercm>stalline.
RC6-G 0.12 6.2 Abundant large (0.025 to 2.0 cm) vugs; abundant fractures;
considerable gypsum-filling of fractures and vugs. h4ajor porositytvpes = intercmstdline, fmctures. and vugs.
Table 1
.
.
Test DP [mZ/hr] J&d* 2G ad RMSESR ~SE&m
inpp) i 20 h(~d) k ~~
range range range
B33-H in 4.78E-06 .1 ~.() * o.~~~ 0.598 0.730 0.510-1~~ y I)J30 -0.514k 0.0610
3.80E-06, 6.02E-06” -11.8, -12.2 0.563,0.636B33-H out ~.64&06 -11.7 * 3.14 0.946 5.78 3.01
-12.821.15 -0.055522.128.74E-07, 8.72E-06 -8.56, -14.8 0.11+7.88
RC1-A in 1.Wq)(j -13.5 k 0.3-$4 0.938 Q34j 0.169
-13.6 k0.196 -0.0-/29 k 0.8921.00E-06, 1.51E-06 -13.2, -13.8 0.393.2.34
RC1-A out 4.96E-07 -14.5 &0.276 0.560 i3~18 0.163-14.5 ~0.178 -0.580 k 1.73
4.2U3-07, 6.03E-07 -14.2, -14.8 0.0997,3.16RC2-B in 1.49E-06 -13.4 k 0.334 0.666 0.161 0.157
-13.4 do.195 -0.407 i2.081.~jE-06, 1.ME46 -13.1.-13.7 0.0832.5.33
RC2-B out 1.30E-06 -13.5 * 0.240 0.00590 o.~7j 0774
-13.6 Y0.161 -5.13 Y 222E+&1.06E-06, 1.46E-06 -13.3, -13.7 0.()(), m
RC4-D in 9.75E-07 -13.9 k0.622 0.988 ()~ql 0.739
-73.8 ? 0.340 -0.0121 h2.067.23E-07, 1.43E-06 -13.3 .-14.5 Q.126,7.75
RC6-G in 3.66E-06 -12.5 * 0.496 0.0369 0.516 0.517-12,j k 0~7-f -3.30 Y 472
2.83E-06, 4.90E-06 -12.0, -13.0 0, m
Bulk in 2.15E-06 -13.0 f 0.194 0.976 0.?53 0.152-13.1zf O.198 -0.0243 k 0.0178
1.6fiE-06, 2.49E-06 -128, -13.2 0.959,0.994Bulk out 1.15E-06 -13.2 * 0..5.52 1.~[) 1.[)2 0.49.5
-13.7 f0.496 O.18.?&0.6526.84E-07, 1.84E-06 -12.6, -13.8 0.6?5 ~ 30- , -..
-,
Table 2
. . . .. .. . ., . .. .
L
#
Newt.4exico
/Albuquerque
i+
WIPP
Carisbad ~ )r..
-..\
u
. Locations of SWiwTracer Tests
WIPP SheBoundary
r
,,,:7si’Ov-, ..*.. .s
● H-19
I H-no
o 1 2 miI
1 I
o 2 km
-SW”<
Figure 1
. .... . . -,, .,, . .. ... . . . . . .,—..— . . ., -. ... .
Fleming & H~rty. Modeiing solute d, ffusion in dIe im.sel}ce of pore-scale Iwterogclwity
1.5m
1.0
0.5
0.0
1.5 \
Porosity Types:
fracturesVugsintercrystallineinterparticleintraparticle
Figure ‘2
10.3 m
0.2
0.1
0.0
0.3
I60 mm
40
20
0
60
I50 pm
25
0
50
31
.
TRI-6!15.5i4.0
.
1
Mass Ratio
[-1
0.1
Figure S.a
E
1
0
‘e
‘s
0
0
. 333-I-I● RC1-Aa RC2-B. RC4-D. RC6G
.0
.’●
●
✎
0.
.
.
G
o
c’
10 100
time [hr]
1000
Mass Ratio
H
1:
0.1 -
E
0
0
0..
0●
.
_---l0 B33-I-1 o 0● RC1-.4 ..0 RC?-B o
60.00
0.01 I
10 100 1000
time fllr]
32
. .. . .. . .
1
CDF [-]
c
..
...
.
.
10 100 1000
formotion fnctor [-]
Figure 4
33
Fleming & Haggc~, AfOdcling solute diffusiot, in tile prcsmtce of pore-scale hctc.ro~,,rily. ..
Mass Ratio
[-1
o. I
i
1~1
\\
\
\
\
\
0.1
Mass Ratio
[-1
— - Jld=-lz
— pd. -13
. . . . . yd=-l~
10
\
100 1000
—- Gd= o
Csd= 1
. . . . . Gd= ~
\
\
\
\0.01
10 100 1000
time Plr]
Figure 5.b
34
..
.
. ... .. .. .. . ... . .. .. . .. . .--, .. .. .. . . . . .. . .. .. . .
FIeming & H~rty, Modehg soluIe diffusiO!l in the prcw,cc of pore.scak lwterogcnci(y. ..
1.0
0.9
0.8
0.7
0.6
Mass Ratio 0.5
[-10.4
0.3
0.2
0 ●.
\
\
\
L-Jmullira[e
— - single-rate. Olxcrved
1 10 100 1000,.
time [hr]
Figure G.a
1
Mass Ratio
[-1
0.1
-.. .●
✎
✎
✎
✎
✎
✼
— multira[e
‘- single-rate
● ob served
, i10 100 1000
time (hrl
36
Fig=re ~.b