transport. The tracer undergoes both sorption and first-order loss. Sorption is via a linear Freundlich isotherm in tandem with rate-limited exchange (after Lapidus ...
WATER RESOURCES RESEARCH, VOL. 24, NO. 3, PAGES 343-350, MARCH
1988
Unidimensional Solute Transport Incorporating Equilibrium and Rate-Limited
1.
Isotherms
With
First-Order
Loss
Model Conceptualizations and Analytic Solutions KEITH
R. LASSEY
Institute of Nuclear Science,Lower Hurt, New Zealand
I derive analytic solutionsto the advection-dispersion equation for unidimensionalsolute or tracer
transport.The tracerundergoes both sorptionand first-order loss.Sorptionis via a linearFreundlich isothermin tandemwith rate-limitedexchange(afterLapidusand Amundson). An arbitrarytracerinput
is postulated, with pulseinputandstepinputspecializations givendetailedconsideration. Many distinct conceptual modelsof unidimensional transportare mathematically equivalentto this formulation,includinga modelof exchange betweenmobileand immobilewater,andincluding modelswith selective first-orderremovalprocesses. Ambiguities in interpreting modelsimulations of experimental data have two origins:morethan one conceptual modelis a contending simulator;a conceptual modelmay be mathematically overspecified with parameters. The implications of suchambiguities arediscussed.
ingnumerical results for a downstream positionL, thefollowvariables are moreuseful' Thispaperconsiders analyticalsolutionsto the unidi- ingdimensionless mensional dispersion-advection equationfor soluteor tracer X = x/L r = ut/RL (4) INTRODUCTION
transport
The two setsare related by
•-- PX
where C and Q are to within multipliersthe concentrations of mobileand immobilizedtracer along the flow path. It is con-
venient to specify a multiplierwhichnormalizes C to the level 0fmobiletracerin unit volume(macroscopic)of the bulk flow
r = PT
(5)
with P ---uL/D the P6cletnumberfor the problem,and RT --= *F= ut/L is the volumeratio of displacedfluid to pore occu-
pancyby mobilefluid.The"retardation factor"R is discussed later.
Equations (1)and(2) includefirst-order losswith a (dimensystem; thenormalization of Q depends uponthe conceptual sionless) removalrate A. Althoughit is presentedas a univermodel of the transportprocess,as is demonstrated in the next section.
Sorption is presumed to be kineticallycontrolled according to
sal removal(suchas radioactiveor microbial decayor irreversiblechemicaldegradation),the problem is not further
complicated by anyselective first-order removal.Thisis clarified in the next section,which examinesconceptualmodelsof
Pr
Q = K•C - (K2+ A)Q
(2)
dispersive flowmathematically equivalent to (1),{2). I employa generalnomenclature whichavoidsunnecessary specificity to anexperimental setting. Thespecies whose trans-
I referto (2) as a LA isothermafter Lapidusand Aroundson portisofinterest isthe"tracer." Morespecifically, it istermed [1952],whoapparentlyfirst appliedit to a dispersive prob- the'bdispersant" or "adsorbate" whenin the mobilephaseor lem.The merits of this and other isotherms are discussedin a when immobilized,with "dispersantlevel" C or "adsorbate survey by Travis and Etnier [ 1981].
level"Q, respectively. The "mobilefluid" is the dispersant-
In theaboveequations,• and r are dimensionless distance bearingfluidwhose"ttowpath"is througha porousor aggre-
{position alongtheflowtrajectory) anddimensionless elapsed time. Thereareseveralwaysto selectnormalizing lengthand timescales fromamongthe interstitialflow speedu, the hydrodynamic dispersion coefficient D, anda longitudinal length scale L relevantto the problemin hand.In developing the
gatedconfining medium termedthe"sorptive medium." The
mathematics I use u, D as scalesand thereby avoid encumbrance with the arbitrarylengthL. Thus
modelassumptions. Whilstgroundwater or surfacewaterway
whole,both fluid and solid,is termedthe "flow system."A
glossary ofnotation appears aftertheappendix. The usefulness of a model suchas (1), (2) as an interpretive
tooldepends on theconformity of theexperimental setting to
trajectories wouldrarelybe longitudinally homogeneous as themodelsupposes, suchflowsarenevertheless oftenmodeled • =- x r =---- t •3) in thisway,sometimes with the simplifying assumption of D RD equilibrium isotherm (•', t,'2-* ,:cin (2)),sometimes withthe Inproceeding to theasymptotes u/D-, 0 or cr,,or in provid- complication of axialdiffusion intothe surrounding matrix [e.g.,Thackston andSchnelle, 1970'Tanget al., 1981'Ma/1
II2
loszewskiet al., 1983' Maloszewskiand Zuber, 1982, 1985'
LeGrand-Marcq andLaurielout, 1985].Of particularimportis
Copyright 1988by theAmerican Geophysical Union.
the simulatedradionuclide leachingfrom hypotheticalnuclear
Paper number7W4844.
wastedepositories [e.g.,Neretnieks et al., 1982].A recent
,01143!397/88/007W-4844505.00 343
344
LASSEY'UNIDIMENSiONAL SOLUTE TRANSPORT, 1
reviewembracing solutetransportmodelsis suppliedby Niel-
At type2 sites (adsorbate levelQA),adsorbate dynarnically
competes for desorptionto the mobilefluid and irreversible into a sink (representing any first-orderremoval All solutionsto (1), (2) reported in this paper employ assimilation sites). Therespective transfer ratesare(1 - q)k., and boundaryconditions(bc's)for a semi-infiniteflow region;i.e., fromsuch of type2 the flow path is confinedto 0 < • < oo (with entranceat qk2, so that p = 1 -q and q are the proportions • = 0), and bc'sare imposedonly at •-, 0+, •. This math- adsorbate so transferred. ematicalidealizationcannot,in principle,be compatiblewith Thisassimilative sinkrepresents first-order lossselectiveI3 experimentsin laboratory columnsbecauseof their short fromonetracerreservoir. An alternative or supplementary processmight envisageassimilationdirectlyfrom lengths.In practicethis incompatibilityseemsto be of no conceptual consequence for sufficiently longcolumnsandfor modelsimu- the dispersant(or type 1 site)as describedby lations which incorporatethe correct (flux) concentrations [e.g.,van Genuchten and Parker, 1984; Parker, 1984]. SoluQt• = q'k•C i11} tions for finite-lengthbc'sare more complicated;somehave beenpresented in respectof equilibriumisotherms[e.g.,Bren- This doesnot further complicatethe mathematics. ner,1962;ClearyandAdrian,1973],but analogues for the LA The followingtransformationof (6)-(9) produces(1•3):
sen et aL [1986].
isotherm seem unavailable. Solutions for the semi-infinite flow
l•2
1l2
region are commonlyemployedto analyzebreakthrough curves(BTC's)fromlabora.tory columns[e.g.,Nkedi-Kizza et
=5
+ ^)
=
+ ^)
al., 1983; Nielsen et al., 1986]. p---l--q= CONCEPTUAL
(•C1 q- A)(•:2 4- A)
MODELS
(i2}
I considersomeconceptualmodelsmathematicallyequivalent to (1), (2) in order to illustratethe generalityof these
Q•t =
R(•c• + A)
equations. First,surfaceadsorptionmodelswithselective first-
/41
R=I+K
order tracer removal illustrate the mathematical equivalence
Q
a
of any first-orderremovalprocess.! then examinea popular As is well known [e.g., Hashimoto et al., 1964; Cameronand modelof tracerexchangebetweenmobileand immobilefluid KIute, !977'1,the Freundlichisotherm(7) is manifested onlyin zones.
a "retardation factor" R, which, in the absenceof kineticinflu-
ences,is just the ratio of the hostftuid'sadvectionrateto the
Model 1: SurfaceAdsorptionModels
Considera model of dispersiveflow characterizedby the equations
migration rate of solute. It is evident that the selective assimilation (nonzero q• is
mathematicallyequivalentto a universalremoval rate,along with rescaled transfer rates and adsorbate level. In general,the
D••.,c--u•.x
•-•(C+ S+ Q,t+ Qv) S = KdC
(6) relationshipbetweenQ,t and Q
dependsupon the detailed
kinetics assumed in the model.
(7) Model 2' ExchangeBetweenMobile and Immobile
)-•QA= klC--k2Q,• 8t
Qv = qkzQA
(8)
Fluid Zones
A modelproposed by Deans[1963] and CoatsandSmith [1964],andfurtherdeveloped by Wierengaandothers [ran
(9) Genuchten and Wierenga,1976,1977;Gaudetet al., 1977;De
in which 0 '+•[o(yt) dr d(x,y)=
=e-•+x•o •e-•+•[•(xu)du t52t
postulate f= 0,•/0 (or, equivalently, R = 0i,•/0•,•), leaving/• (rather than Kin) as the appropriately averageddistribution coefficient.With fm + f•,• and Km/K•mno longerspecified,there is greater flexibility in interpreting an empirical dependenceof K upon flow parameters.
The difficulty and ambiguity in empirically parameterizing laboratory-sourceddata suggeststhat prospectiveanalysesof field data might be quite forlorn. Nevertheless,field tracer data do need quantitative interpretation, even if the interpretive model has obvious imperfection at the outset. And models based on (!), (2) or on (19) have found favor. Analyses of groundwater data by Maloszewski and Zuber [1982] and by Maloszewskiet al. [1983] indicate that a weighting function derived from a dispersion-advectionmodel {actually the FL (40) as weighting function (44)) was marginally superior to other contenders examined. Thackston and Schnelle [1970'] and LeGrand-Marcq and Laudelout [1985] found the MIF model a usefulinterpretivetool for streamflowdata.
e-{•+t)[o{yt)dt
Interrelationships
J{x, y) + K{x, y) = I
(53•
J(x, y)= K(y, x) + e-{•'+Y•[o(Xy) Small-argumentexpansions
J(x,y)=e-• + e-{x+y) •• •yn• Xm •=1
K(x, y)=e-{•+•'• • • n=l
m=l
m=O
m• '
Neumann expansions
J(x,y)=e-{'+Y> •
y•r•(xy)
•=o
K(x,y) = e- •+ y• • x•-l•(xy)'
'
LASSEY' UNiDIMENSIONAL SOLUTE TRANSPORT, 1
349 tration of dispersantand companion
Asymptotes K(x, co} -- J{ :c, y) = 0
adsorbate.
(56)
Superscripts NOTATION
P, S (on C, Q, •P, (I)) specificto pulse,stepinput. FL (on C,Q) specializationin the Freundlich limit
Dimensions are in squarebracketswith L, T, G denoting dimensions of length,time,and tracermeasure.AbsentdimenAcknowled.qments. I acknowledgeuseful discussionswith colsions imply a dimensionless quantity. A definingequation leagues Mike Stewart(INS), BrentClothier!Plant Physiology Divinumberis shown where appropriate.The glossaryis in- sion,DSIR) and FreemanCook (NZ Soil Bureau,DSIR). exhaustive and excludesterms specificto an individualconREFERENCES
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b,/• attritionfactors dueto loss,(48). C(•, r)
dispersantlevel (per volume of flow
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dispersantconcentration,(21), (22)'
[GL- 3]. Co(r) "input stream"of injectedtracer,(25)' [GIœ3]. D hydrodynamic dispersioncoefficient' [L2T- •]. H(r)
Heaviside function, (26).
I,•(z), I,•(Z) modifiedBesselfunction,(31). J{x, y), K{x, y) Goldstein'sfunction, appendix. L length scale' [L]. M
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(Received March 2, 1987; revisedOctober 26, 1987; acceptedNovember 4, 1987.)
