SUPERCOMPUTERS
SUBSURFACE
AND
SOLUTE
THEIR
USE IN MODELING
TRANSPORT
D. A. Barry
Centre for Water Research
University
of Western
Australia,
Nedlands
Abstract. Supercomputers
offer a wide range of pos- reducedto solutionsof algebraicsystemswhen they are
sibilitiesfor advancingknowledgeand solvingdifficult, implementedon computers,algorithmsfor linearalgebraic
real-worldproblems. Solutetransport-related
theoryand systemsof equationssuitable for vector and parallel
applicationshave long been an area of specialization machinesare discussedin some detail. Data analysis,
within hydrology. In this review, recentapplications
of necessaryboth for evaluatingemergingtheoriesand for
supercomputers
to solutetransport
problemsarepresented. more "routine" applications like determining model
is identifiedasanothergeneralareathatis ably
Theseapplications
arebasedmostlyon theassumption
of a parameters,
In particular,supercomputers
dilute solute, allowing the governing equationsto be handledby supercomputers.
decoupled. An applicationof supercomputers
is the offer a way to become less dependenton parametric
simulation of solute transport in large heterogeneous assumptionsand constraints,substituting,instead, raw
spatialdomainsby both direct solutionof the governing computationalpower. Visualization techniques and
facilitieshaveflourishedalongsidethe developequations
and by particle-tracking
methods.An impetus associated
for this researchhasbeen the rapid increasein predictive mentof supercomputer
centers.Theseoffer thepossibility
dispersionmodelsbasedon analyticalstochastictheory. for real-time viewing of solutetransportprocesses,both
Up until now, relatively few applicationshave incorpo- real and simulated,althoughlittle hasbeenproducedthus
ratedalgorithmsdesignedwith a targetedsupercomputer
in far. It is expectedthat many more researchersinvolved
mind, with theresultthat the machine'spotentialcannotbe with subsurface
solutetransportwill make use of superexploited fully. Since many numericalproblemsare computersin the future.
INTRODUCTION
ation. There are two basic paths by which computer
calculationalratescan be improved. In the first, ratesare
Hydrological uses of computers are increasing increasedby speedingup the passageof informationin the
monotonically
with the passage
of time. We are now into machine and the response time of the processors.
the third phase of the "computer revolution" where Theoretically, information cannot be transmittedfaster
"problemsare beingroutinelydefinedthatcouldnot even thanthe speedof light. Superconductors
offer a possible
have been conceivedwithout the existenceof computers" way to increasepresentlyavailable transmissionrates.
[Wallis, 1987].
Certainly, available supercomputers Reducingthe distances
betweenthe machinecomponents
continueon this"revolutionary"path. Thereare two rather is another,more obviousmethodof speedingup informaobvioustrendsin theuseof supercomputers.
First,thesize tion transfer. The circulararchitecture
of the Cray X-MP
of problemswhichbecomesfeasiblecontinuesto increase illustratesan approachof this type. This and other
by Seitzand Matisoo [1984], who
in magnitude. Here the word "size" refers not only to limitationsare discussed
computer-related
limitations,e.g., speed or number of indicatethatthelimitsof theavailabletechnology
maybe
processors,
amountof memory,input/outputand physical in sight.
The secondway to increasecomputationalrates is to
storagecapacity,but also to the complexitylevel of the
models investigated. Second, smaller, more routine perform more calculationsat the same instant. Vector
analysesare carriedout muchmorerapidlyandaccurately, machinesachieve their speed by operatingon vectors
sometimes in real time.
ratherthanscalars,whereasparallelmachinesincreasethe
Supercomputers
embodythe notion of speed. Unfor- numberof processors
in the computer.With this simple
tunately,it canbe difficult to realizehighratesof comput- statementit is apparentthat, on the one hand, a vector
Reviewsof Geophysics,
28, 3 / August1990
Copyright1990 by theAmericanGeophysical
Union.
8755-1209/90/90RG-00342
pages277- 295
Papernumber90RG00342
$05.00
277
'
278 ß Barry:SUPERCOMPUTERS
AND SOLUTETRANSPORT
28, 3 / REVIEWS
OF GEOPHYSICS
machineneedsto operateon vectorsmostof the time to speedupof 50 but vectorizesonly 90% of the code will
achievespeedscloseto its ratedmaximum. That is, the achievea speedupof only 8.5. Denning [1988] discusses
degreeto whichan algorithmwill vectorizedetermines
its Amdahl'slaw asit appliesto parallelprocessors.
Parallel machinesare affected by factorsapart from
suitablityfor runningon a vectorprocessor.On theother
hand, parallel machinesachieve the same result by simplythetotalnumberof operations.Boththetime taken
betweenprocessors
and the synchronizacompletingmanycomputational
taskssimultaneously
on to communicate
differentprocessors.The combinationof an army of tion of the computationaltasks need to be considered.
vector processorsproducesa hybrid, vector-parallel Adamsand Crockett[1984] presentmethodsto quantify
machine. For a given algorithm, then, the degree of thesetiming costsusinga linear systemexample. More
parallelism,
or concurrency
[Sharp,1987],achievable
in its generally,on parallel machinesthere are two ways by
implementation
is the overridingfactorin determining
its which speedupcan be enhanced. First, the problemis
computationrate. So far, only "course-grained"
paral- broken into independentsubproblems. Monte Carlo
lelism has been referred to. The operationsto be per- simulationsprovidean immediateexamplesuitedto this
formed, suchas matrix multiplication,can be separated treatment. Other examplesincludematrix multiplication
intoindependent
component
parts,eachof whichis carded or scalar products,both of which can be parallelized.
out independently,
"in parallel." Often,the instructions
to Second,the algorithmcanbe reorderedso thatthe degree
a machine central processingunit (CPU) (e.g., loads, of parallelismincreases.
stores,adds)can also be cardedout in parallel,yielding
This brief introductionindicates that realizing the
is often more
"fine-grained"parallelism. Very long instructionword potentialperformanceof a supercomputer
(VLIW) machines [Fisher, 1984] overlap processor than simply loading and runningan existingcomputer
instructionsthat can be carried out in parallel with code. The characteristics
of the machinebeing used
speedups
canbe
substantial
increases
in computation
rate. The advantageis dictatethe methodsby which substantial
that VLIW machinescan operatein scalarmode,relying obtained. Supercomputers
can be usedto providevery
on a sophisticatedcompiler to produce fine-grained substantial
gainsin solutetransportmodelingcapability
of the relativeimportance
parallelism in object code. There are, apparently,no and concomitantunderstanding
VLIW machines that are supercomputers,although of theprocesses
involved. Problemsinvolvinglarge-scale,
heterogeneous
domainsand complex chemicalreactions
minisupercomputers
of thistypeareavailable.
Simply transferringcodes from serial to vector or becomeopento analysis.However,the time takento mna
parallel machinesmay not result in very impressive problem is always a limiting factor to some degree.
speedups.Thispointis demonstrated
clearlyby Pelka and Thereforethe designof a computercode is worthy of
Peters [1986], who comparecomputation
ratesfor scalar serious consideration. General methods by which
canbe utilizedefficientlyare one focusof
and vectorizedfinite elementsolutionsof a groundwater supercomputers
flow model. Particularly for parallel machines,the thisreview. An overviewof transporttheoryfor a single
flow is presented
algorithmsusedmay turnout to be not very efficient. Not solutespeciesin a density-dependent
surprisingly,algorithmsfavoredon sequentialmachines first, followed by numericalapproaches
to solvingthe
tend to be those that minimize the number of arithmetic
Effect
of Portiol
Foctorizotion
operations
necessary
to completethe taskat hand. Vector
processorscarry out pipelined operations. Pipelining
102
refersto theprocessby whichthearithmeticoperations
are
brokendown into a numberof elementarysubcomponents.
Eachsubcomponent
operationdependson theresultsfrom
the previoussubcomponent.By operatingon vectors,the
calculationson each pair of operandscan be performed
simultaneously
in assemblyline fashion,hencethe term
pipeline. The natureof thepipeliningis suchthatfor some
time no restfitsare availablewhile the necessarysubcom5
ponentoperationsare completedfor the first time. This is
known as the start-uptime. It is clear that shortvectors
100
shouldbe avoidedas muchas possible,sincestart-uptime
0.2
0.4
0.6
0.8
1.0
0.0
is incurredfor every new pair of vectorsoperatedupon.
The effect of the degreeof vectorizationof an algorithm,
Froction
Vectorized
as givenby Amdahl'slaw [Amdahl,1976;Eramen,1987], Figure 1. Relative performance,or speedup,R/S, plottedon a
is presentedin Figure 1, whereit is shownthatthe relative logarithmicscale as a function of the fraction of the code
(or actual)performanceis a functionof the fractionof the vectorized.R is the realizedspeedandS is the machine'sscalar
algorithmthat vectorizes. The curve labels list possible speed.Threehypotheticalvectorprocessors
are considered,
with
speedup,i.e., the ratio of possiblevector speedto scalar the curve labels corresponding
to the value of R/S for fully
speedon the computer. A machinethat has a possible vectorized code.
28, 3 /REVIEWSOF GEOPHYSICS
governingequations. It is shownthat solutetransport
models are certainly candidatesfor numerical solution
usingsupercomputers.
The foundationof thesenumerical
solutionsis solving linear systemsof equationsin an
efficientmanner. Modem algorithmsdevelopedfor vector
and parallel machines are presented. Data analysis
constitutesanotherarea where supercomputers
can be of
great benefit. A summaryof conceptuallysimple but
highly computationallyintensive,nonparametric
estimation proceduresis included. The number of studies
concerningvariousaspectsof solutetransportthat have
usedsupercomputers
is relativelylimited. There is little
doubt that supercomputer-based
researchendeavorswill
increase. Even consideringthe existingresearch,it is
apparent
thatvery littleefforthasbeendevotedto "pushing
the envelope"of supercomputer
performancein solute
transportapplications.
Barry:SUPERCOMPUTERS
AND SOLUTETRANSPORT
ß 279
nonisothermal, multiphase, multispecies, densitydependent solute transport in variably saturated
heterogeneousporous media. The parameterizafion
requirements
of sucha model could well exceedavailable
knowledge of the necessary consfitufive or chemical
relationships. Even with adequateresolutionof these
issues, the numerical solution of such a model is a
significanttask requiting,certainly,a supercomputer
for
results to be obtained in a reasonabletime scale, if at all.
Therehavebeenfew attemptsto developgeneralcodesfor
theseextreme, but relevant, conditions.
The densityof subsurfacewater is very often nearly
constant,or at leastit is assumedto be so. The density
varieswith changesin temperature,
pressure,or presence
of dissolvedchemicals. Density changescan become
noticeable
if thesequantifies
undergolargevariations.It is
possiblefor dissolvedsolutesto produceextremedensity
changes. In some countries, waste disposal in salt
formationsis under consideration. In such areas, and in
THEORY
OF MISCIBLE
FLUID
FLOW
the freshwater/seawater
interfacesof coastalregions,the
density dependenceof the fluid componentsmust be
Solutetransportproblemsmustalwaysbe consideredin considered
in any modelingof the system[Huyakornet al.,
the contextof a particularscale,e.g., a catchmentbasin, 1987; Hassanizadehand Leijnse, 1988; Herbert et al.,
part of a basin, a geologicalstructure,or a laboratory 1988].
apparatus. Many researchershave developedtheories
In groundwaterapplications,densityeffectsof thermal
applicableto the macroscale,
definedas beingmuchlarger variationsare typicallyignored. Likewise,the problemis
thanthe microscaleor pore scale[e.g., Gelhar et al., 1979; simplified further if only nondeformablemedia are
Chu and Sposito,1980, 1981; Gelhar and Axness,1983; considered. Both of theseeffects will be importantin
Winter et al., 1984; Dagan, 1982, 1984, 1987]. These somecases,and both are the subjectof readily available
foundationaltheoriesare not directlyrelevantto supercom- treatments. For example, the problem of heat and mass
puterapplications
exceptinasmuchastheyarenecessary
to transportis discussedby Bear [1972, section 10.7.2].
provide the governingmodelswhich must be solvedby Thermaleffectscannotbe ignored,however,if geologic
some means. It is worth noting, however, that the time scalesare considered.Bethke [ 1985] andBethke et al.
approachesused in deriving these theoriesare open to [1988] modelreactivechemicaltransportin sedimentary
debate. For example,Spositoet al. [1979] identifiedand basinsoverlongtime scalesmakinguseof a machinewith
reviewed three approachesused to derive foundational a novelparallel-vectorstructure[Kucket al., 1986].
theoriesleadingto the commonlyusedmacroscopic
solute
A thoroughderivationof the governingequationsfor a
transportequationapplicablein an isothermal,saturated singlesolutespeciesin a rigid, saturated
mediumis given
porousmedium:
by Hassanizadehand Leijnse [1988]. With suitable
assumptions
theyrecoverthe familiar forms:
Oc/Ot= V.(D'.Vc)- v.Vc
(1)
o•p
k
(2a)
n
-•
=
V.p•.(Vp
- pg)
wheret is time, D' is the solutedispersiontensor,v is the
solute velocity, and c is the solute concentrationin
&0
k
solution. Equation (1) is just the familiar convectionn-•- - •.(Vp- pg).Vc0
=V.(D.Vc0)(2b)
dispersionequation(CDE). The threeapproaches,
which
are basedexplicitlyon, or are tantamountto, averaging In obtainingtheseequations,Hassanizadeh
and Leijnse
operationsapplied at the microscale,are all subjectto [1988]haveassumed
thepresence
of a rigidporousmatrix
either mathematicalor physicalassumptions
which have and isothermalconditions.No productionor removalof
yet to be provedrigorously[Spositoet al, 1979].
soluteis accountedfor. The densityof the fluid is
Under appropriatecircumstances,
foundationaltheories dependenton both its solute mass fraction c0 and fluid
producemacroscopicequationslike (1) that, with some pressure
p. A suitablerelationship
betweenthesequanassumptions,are applicable in practice. Typically, a tifies is given by the densitystateequation[Bear and
numberof coupled,nonlinearequationsare reducedto a Verruijt, 1987, section3.2; Hassanizadehand Leijnse,
few, possiblydecoupled
linearequations
usingsimplifying 1988]:
assumptions.It is apparent,however,that complexfield
applicationscould require numericalmodelsdescribing
p(c0,p)/po: exp [yc0- [•(p - Po)]
(3)
280 ß Barry:SUPERCOMPUTERS
ANDSOLUTE
TRANSPORT
28, 3 / REVIEWS
OFGEOPHYSICS
Thecoefficients
), and[3in thisformulaarebestconsidered movement(approximately1 m in 120 days)was observed.
as fittingparameters
whichattainspecificvaluesfor given Identifiedas possiblecausesfor this downwardmovement
solutes.The dynamicviscosityof thefluid, •, will change were the presenceof a vertical componentin the local
with to as well:
groundwatervelocity field due, perhaps, to aquifer
g = m(to)go
(4)
recharge or, second, density differences between the
injectedplume and the native groundwater. The relative
magnitudeof theseeffectsis uncertainat thistime.
where re(to)is a functionto be specifiedfor the specific
SoluteTransport
solute. If go is definedto be the reference
dynamic Stochastic
viscositywhento= 0, thenm(0) = 1.
The
ease with
which
numerical
solutions of
the
nonlinearsystem(2) are obtaineddepends,amongother
things,on the dimensionality
and extentof the spatial
domainand the degreeof variabilityof the coefficients.
These factorsalone, in the contextof a practicalsimulation, make the use of a supercomputer
desirable. If the
systemis furthercomplicated
by, for example,a nonlinear
chemicalreactionor the coupledtransportof a numberof
solutes,then the need for a supercomputer
becomesmore
advantageous.
To show that (2) reducesto (1) as a specialcase,we
consider the definitions of concentration and mass fraction.
Solute
concentration
c isequivalent
toMs/Vr, andthemass
fraction
tois equivalent
toMs/Mr Thusc/to= M•Vr-- p,
or to = c/p. Figure 1 of Herbertet al. [1988] showsthat,
for a solutioncontainingup to 26% of saltby mass,
P/Po-- 1 + 0.2to
O<to<l
(3')
For to << 1 (dilutesolutions),
to-- c and!x -- go. Under
steadyflow conditions,(2b) thenreduces,approximately,
to (1). An alternativeexpressionto (3) or (3') for the
solutedensitystateequationis availablewhenthe volumes
of the componentsin the final solutionare additive
Recent, comprehensive reviews of the stochastic
approachesto water flow and solute transport in
groundwaterhave been publishedby Dagan [1986] and
Gelhar [1986]. The startingpoint is the recoupledform of
thewaterflow andsolutetransportequations,
meaningthat
a dilute solute is considered. The logarithm of the
hydraulic conductivity,In (K), is treated as a random
spatialfunctionwith known constantmean and spatially
stationarycovariancefunction. Stationaxityrefers to the
assumption
that the covariancefunctionis unchangedby
spatial translation. The low-order statistics of the
piezometrichead (i.e., fluid potential)and flow field can
then be calculated. With the assumptionof a constant
porositythe statisticsof the groundwatervelocity field
follow directly. The stochasticsolutetransportequationis
just (1) exceptthatv is now a randomspatialfunctionwith
known low-order statistical properties. The solute
concentration
c is the dependentvariablein (1), and hence
it is a randomfunctionas well. Theoreticalanalyseshave
focusedon ensembleaveraging,i.e., averagingover all
possiblerealizationsof the velocityfield in (1) to derivea
governing
equation
of (c), theensemble
meanconcentration [GelharandAxness,1983;Dagan, 1984, 1987, 1988;
Neuman et al., 1987]. Theories based on the stochastic
CDE essentiallytreatthe solutedrift velocityas a random
variable with assumedlow-order statisticalproperties.
They are, however,not withoutfundamentalproblems,as
(3')
discussedby Spositoet al. [1986]. These authorsnoted
P/P0= pS/[(1- to)pS+ toP0]
that each aquifer representsa single realization of the
processunderconsideration,
whereasthe results
Herbert et al. [1988, Figure 1] show(3') is likely to be a stochastic
goodapproximation
in practicalcases.However,it should of a stochasticanalysisrepresentthe statisticsfor the
be notedthat the dependence
of fluid densityon pressure ensemble of realizations. The results of the stochastic
CDE approaches,therefore, are not applicable to a
hasbeenneglectedin (3') and(3').
are made
The governingequations
canbe manipulated
into forms particularaquiferunlessadditionalassumptions
suitablefor numericalsolution[e.g.,Huyakornet al., 1987, [Dagan, 1984]. Numerical simulationsreinforcingthis
below.
AppendixA; Hassanizadehand Leijnse,1988]. It is very pointarepresented
commonto ignorethe couplingbetween(2a) and (2b) for
The ensemble mean concentrationstaffsties, aptransportequationsimilarto (1),
dilute solutions. The questionof how dilute the solution proximately,a macroscale
has to be before this assumptionis valid is not easily exceptthat the dispersion
coefficientcomponents
are now
answered. Even for very dilute solutionsmoving in the functionsof time. For a coordinatesystemalignedwith
subsurfaceenvironmentthis assumptionmay need to be themeanflow direction(axis 1), SpositoandBarry [1987]
coefficients
consideredcarefully, particularlyin aquifers. Sudickyet derivedthe macrodispersion
t
al. [1983] and Freyberg [1986] analyzed nonreactive
solute transportexperimentsat the Borden site. These DMii(t)
=I I Sii(v:,
t')exp
(-•Dii•i) cos
(vvq
t')d}cdt'
0
i
(5)
experimentstrackedthe movementof plumesin a sandy,
unconfinedaquifer [MacFarlane et al., 1983]. In both
experimentsthe solutemassfractionof the injectedfluid whereSii(lc,t) is theFouriertransform
of thestationary
[Herbert et al., 1988]:
wasverysmall(<5 x 10-3),butappreciable
vertical
plume velocitycovariance
function
andDii arelaboratory
scale
28, 3 /REVIEWSOF GEOPHYSICS
Barry:SUPERCOMPUTERS
AND SOLUTETRANSPORT
ß 281
dispersion
coefficients.
Oncethefunction
Siiis specified, vectors,respectively,to be determined,and the coefficient
the macroscaledispersioncoefficientscan be evaluated
withoutdifficulty.
A major advantageof the theory lies in its practical
applicabilitywhensomebasicconditionsare satisfied.For
example,the authorscitedaboveassumethe presenceof a
large-scalevelocity field with a uniform, constantmean.
Becausethe theoryquantifiesthe effect of the local-scale
matricesare given by Huyakorn et al. [1987]. The finite
element equations(7) representa transformationof the
model(6) to a systemof algebraicequations.Note thatall
numericalsolutions
involvethesolutions
of linearsystems.
Perhapsthe mostcrucialpart of usinga supercomputer
efficientlyis solvingsuchsystems
efficiently.Because
of
the importanceof linearalgebraicsystemsin the solution
velocityvariationson solutedispersion,
one needs"only" of solutetransportmodels,varioussolutionmethodsare
to estimate the low-order statisticsof the hydraulic examined in detail below.
conductivityfield in order to predict macroscaledisperHavinga supercomputer
availablecaninfluence
even
sivity, and thus spread, of large-scalesolute plumes. the basicapproachto obtainingnumericalsolutions.The
Various researchers have investigated the value of
theoreticalpredictionsin the light of field experimental
data. The main body of analyticalresultsderived using
stochastictheory applies only for ideal tracers under
assumptionsthat may not be widely applicable. The
degreeto whichthe assumptions
canbe relaxedin practice
needsto be clearlydefined. Second,theeffectof chemical
and biologicalreactionsat the field scale is difficult to
determineanalytically. Some of theseissueshave been
investigated
usingsupercomputer
simulations,
asdiscussed
below.
Numerical
Solutions
finitedifference
method
applied
to theaboveproblem
would give a differentlinear systemof equationswhich
would also have to be solvednumerically. The trmite
element method, although more suited to irregular
domains,is usually time consumingto code. Finite
differencemethodsare more simpleto set up but not as
versatile.
A scientist must invest considerable time and
effortto setup a problemsothata high-accuracy
numerical solutionwith reducedCPU time and storagerequirements will result. The question which then arises is
whetherthis investmentis justified in the face of the
relativelycheap,highcomputation
ratesandlargememory
availableon a supercomputer.
Fully three-dimensional,transientnumerical solutions
of the completegoverningsolutetransportequationshave
notbeenthe subjectof manystudiesas the computational MATRIX EQUATIONS: MODERN SOLUTION
load is very large. Saltwater intrusionposesa serious METHODS
threat to groundwatersupplies in many coastal areas
[Newport,1977], and this haspromptedmodelingefforts,
In thissection,algorithmsleadingto the solutionof the
many of which have beenbasedon the "sharp-interface" linear systemAy = b (whereA is a known, often banded,
assumption. A variety of numericalsolutionshave been squarecoefficientmatrixof dimensionN, y is an unknown
presentedby Lee and Cheng [1974], Segolet al. [1975], vector, and b is a known vector) will be discussed.
Segol and Pinder [1976], Huyakornand Taylor [1977], Elements
of these
arrays
areidentified
by aij forthe
Taylor and Huyakorn [1978], Frind [1982a, b], and elements
of A, etc. The typeof supercomputer
available
Huyakornet al. [1987]. The Galerkinapproachis used will be the arbiterof the bestalgorithmto use. In partypically. For example,Huyakornet al. [1987] consider ticular,
algorithms
designed
for a singlevectorprocessor
the governingsystem
cannotbe expected
to performwell on a massively
parallel
machine. The literature devoted to numerical methods for
the solutionof linearsystems
of equations
on supercomputers
is
very
extensive,
and
only
a
small
overview
canbe
V-[K-(V•
+ c)]=Ss
-•-+nrl-• Pw
attemptedhere. Ortegaand Voigt [1985] reviewgeneral
applications
of vectorandparallelmachinesto the solution
•c
of
partial
differential
equations.For the purposes
of this
V-(D-Vc)-v-Vc=n•-• +q(c- Co) (6b)
section,we will refer frequentlyto the degreeof paralP/P0 = 1 + ec/c•
(6c) lelismof an algorithm,
meaningthenumberof operations
that can be carriedout at the sametime [Hockneyand
Applicationof the Galerkintechniquefor theseequations Jesshope,1981, section5.1.2]. On a vectormachinethisis
yields[Huyakornet al., 1987]
givenby themaximumvectorlengththatcanbeprocessed
at a singletime, andon a parallelmachineit is thenumber
(7a) of processors
operatingconcurrently.
A•j•j + B•j•- = F• jr = 1,..-, mt
•_• a• acpq (6a)
~ do= •
Eucj+ B•j•-
I = 1,..., mt
(7b)
Direct Methods
The most familiar direct method is Gaussian elimina-
form
wherethesubscripts
referto nodes,
mt is thetotalnumber tion,whichsolvesAy = b by reducingA to triangular
of nodes,•j andcj are unknownheadandconcentrationand thenproceeding
to arriveat the solutionby back
28, 3 / REVIEWSOF GEOPHYSICS
282 ß Barry:SUPERCOMPUTERS
AND SOLUTE
TRANSPORT
substitution. On vector machines it can be shown that
Gaussian elimination is not a suitable method of factoriza-
tion.
written as
k+l
k
•
k+l
k
Yi =Yi+w i - •aiiyi - •,•aii
yi a•i•
Parallel machines fare somewhat better, since for
j=l
each step in the reduction,each columnof A can be
handledindependently.Voigt [1977] detailsthe efficient
implementationof the factorizationphaseof Gaussian
eliminationin the Cray-1.
Ortegaand Voigt [1985] point out threedrawbacksto
usingfactorization
algorithmson parallelarrays: (1) the
degreeof parallelismis reducedat eachstep,(2) significant communication
betweenprocessors
is necessary,
and
(10)
j=l
i:I...,N
k>0
wherew is the relaxationparameterand k indexesiterations. The scheme,which divergesunless0 < w < 2
[Hager, 1988, section7-3], is usuallytermedsuccessive
over relaxation(SOR) for 1 < w < 2 or successiveunder
(3)it ma:9
be•difficult
toschedule
theworkefficienfiy.
An relaxation(SUR) if 0 < w < 1, while it reducesto the
algorithmif w = 1.
example
of drawback
3 is row pivoting,
i.e., swappingGauss-Seidel
Althoughthe Jacobimethodis eminentlysuitedto
rOwsin the matrix to avoidroundingerrorsinducedwhen
on parallelor vectormachines,it converges
dividingby smallnumbers.It is likely thatintroducing
a computation
time,themorerapidly
sequential
processsuchas a pivotingstrategywouldcause tooslowlytobeuseful.At thesame
convergent
SOR methoddoesnot presentany easymutes
substantial
programming
difficulties.
by parallelmethods.Asynchronous
SOR
Anotherapproachis to Usethe GiVensor Givens- to computation
Householderalgorithms [Young and Gregory, 1972, [Kung,1976;Baudet,1978;BarlowandEvans,1982]is an
section4.2] to tridiagonalize
A by a seriesof plane alternativefor parallelmachines. Basically,Jacobi-like
processors,
with
rotations. For parallelprocessors
the Givensalgorithmis iterationsare carriedout by independent
pickingup the mostrecentiteratesfrom a
slightly more effective [samehand Kuck, 1978]. The eachprocessor
therate
resultingtridiagonalsystems
canbe solvedby substitution commonmemory.Underfairly generalconditions
is not reducedwhen comparedwith the
usingthe Thomasalgorithm[Noye,1982]. The Thomas of convergence
algorithm,however,is manifestlyserial. For example,a usualserialform of thealgorithm.
An interestingiterative scheme,termed the quadrant
lowertriangular,uppertriangular(LU) factorization
needs
interlockingmethod, was developed by Evans and
to incorporate
operations
of the type
Hatzopoulos
[1979],Evanset al. [1981]andEvans[1982].
ThecoefficientmatrixA is setequalto WZ, where
Ui : di - f i ei-1/Ui-1 i: 2,' '', N
where the elementsof the upperand main diagonalsand
subdiagonals
aregivenby d/, ei, andf/, respectively,
and
theui arethediagonal
elements
of theuppermatrixof the
LU factorization.
Thedependency
of ui onui-1precludes
vectorizationor parallelizationof this statement. VLIW
machines,on the other hand, perform very well in this
situation.An immediateway to parallelize(8) is to solveit
iterativelyin vectorform [Traub, 1973]. The convergence
rate of the iterationsincreaseswith increasingdiagonal
dominanceof the system. Parallel direct methodshave
beendevelopedaswell. Stone[1973]reformulates
(8) as
1
(11a)
i
(9)
and
whereui = q]qi-1. All theqi canbe foundin log2 N
iterations.
Bycomparison,
serial
machines
solving
(8)take
time proportionalto N.
Z•
Iterative Methods.
Iterativemethodstendto be more suitedto paralleland
vector machines than direct methods.
The Jacobi and
Gauss-Seidel methods are well-known examples of
iterative schemes.
Relaxed Gauss-Seidel iteration is
(11b)
28, 3 /REVIEWSOF GEOPHYSICS
Barry:SUPERCOMPUTERS
AND SOLUTETRANSPORTß 283
The triangularportionsof (11) are filled with elementsto
be determined.
These unknown elements are calculated
beginningwith the outermostelementsin the unknown
matrices.The algorithmproceedsby first assigning
Zli = au
ZNi= aNi
i = 1,. --,N
(12a)
Next, the2 x 2 linear systemsare solved:
1. Incrementk by 1, thencalculate
[•: r•r•/p•np•
r,+l = r, - •k Apk
^ 1 ----B-1 rk+ 1
rk+
0•k: rk+1rk+l
Pk+l = rk+l + O•kPk
i=2,...,N-
1
(12b)
•iN•NN3LWiN3
Ld'•J
2. Compute
Y,+I= Y,+ •,P,
3. If convergence
conditionis notsatisfied,go to step1.
If A is not symmetricand positive definite but is
Equationssimilar to (12) can be solvedto find the other
unknownelementsof W and Z. Eachring of elementscan
be calculatedindependently,
andthecomputation
is suited
to parallelprocessing.With theseoperationscompleted,
the originallinearsystembecomes
invertible,
thenpremultiplying
theequation
Ay = b by Ar
givesa systemin therequiredform. The PCG algorithm
theninvolvesmorecomputations,
however. An alternative
is the ORTHOMIN algorithm[Behie and Vinsome,1982].
Qualitatively,
theORTHOMIN algorithmis similarto the
onegivenabove,althoughthe computational
work load is
increasedbecauseof the nonsymmetric
coefficientmatrix
or
A. Details of the algorithm,relatedvectorizationissues,
Wz = b
(13b) andapplications
to petroleum
reservoirs
aregivenby Behie
Zy = z
(13c) and Forsyth[1983, 1984].
On vectormachines,PCG-typemethodsappearto be
Equation (13b) is solved for the intermediateunknown themostefficientavailablefor linearsystems
of equations
vectorz, whereupon(13c) is solvedfor y.
encountered
in subsurface
flow problems,as shownin a
WZy = b
(13a)
recentstudyby Meyer et al. [1989]. The choiceof
The Preconditioned
ConjugateGradientMethod
preconditioner
is related to the type of computerused.
The conjugategradientmethod(CG) wasfirst presented Meyeretal. [1989]compared
a number
of preconditioning
by Hestenesand Stiefel [1952]. The methodis basedon schemesimplementedon the Cray X-MP/48, a vector
the fact thatthe solutionof Ay = b minimizesthe quadratic machine,and the Alliant FX/8, an eight-processor
functional
yrAY - 2bryif A is symmetric
andpositive vector-parallelmachine. Overall, low-orderpolynomial
definite [Hager, 1988, section7-5], and the superscript
T preconditioning
[e.g.,Saad, 1985] was foundto give the
indicatestransposition. It is this functional which is bestresults.The largestproblemsolvedby Meyer et al.
minimizedto obtainthe unknownvectory. In the absence [1989] involved 980,000 unknownsand took apof rounding error the method convergesto the exact proximately
1 hourCPUtimeontheAlliant,witha factor
solutionfor y in N steps. It has been observedthat for of 15 speedup
expectedfor the Cmy. Ababouet al. [1988]
some matrix types a reasonableapproximationto the useda Cray-2 to performa large-scalethree-dimensional
solutionis obtainedafter relatively few steps,leading to flow simulation but did not use the PCG method. For this
the use Of the methodas an accelerationtechnique. The case,
Meyeret al. [1989]estimated
thatusingpolynominal
convergence
rate of CG dependson the conditionnumber preconditioning
withCG wouldhavereduced
thecomputof A. PieconditioningiS therefore used to increasethe ationtimeby an Order
of magnitude.
Thisexample
convergerice
rate [Hager, 1988, section7-5]. Concuset al. highlightsthe importance
of algorithmselectionfor
[1976]notethefollowingadvantages
of the method:
(1) large-scaleproblems.
not all of A needsto be storedin memoryat one time, (2)
unliker•laxationschemes,
e.g., (10), parameters
usedby Other Methods
the algorithmare automatically
determined,
and(3) some
Othersolutionmethods,althoughuseful,havenot been
restrictions
on A requiredby othermethodscanbe relaxed. discussed
in thissectionbecauseof spacelimitations.For
A vectorizedpreconditionedCG (PCG) algorithm was example, many problems lead to sparse coefficient
givenby Kershaw[1982].
matricesthatare solvedefficientlyby usingthe multigrid
The first step in the algorithmis to choosea precon- method[McCormick,1987]. Supercomputers
have been
ditioningmatrix B, relatedto A, suchthat B is symmetric appliedin a varietyof multigridapplications
[McCormick,
positivedefiniteandthe system
Bf = r is easilysolved. 1988]. Spectralsolutionmethodsexistas well [Voigtet
For example,B could be the tridiagonalpart of A. One
versionof thePCG algorithmis
0. Guess
Y0.Solve
B•0= b - Ay0= r0. LetP0= •0.
al., 1984]. Further information can be obtained from the
reviewof Ortegaand Voigt [1985] or the summaryof
HockneyandJesshope
[1988,chapter5].
284 ß Barry:SUPERCOMPUTERS
ANDSOLUTE
TRANSPORT
DATA
28, 3 /REVIEWS
OFGEOPHYSICS
ANALYSIS
computationallyintensivetechniquesare reviewedin the
following. In doingso,we echothe sentimentexpressed
in
New or updated solute transport models appear therecentarticleof TichelaarandRuff [1989], who aimed
regularlyin thehydrological
literature.Like anyscientific to "promote the use of resampling techniquesin
theory,modelsmustbe testedand evaluatedby compari- geophysics."
Presume that the model under consideration is a scalar
sonwith experimentaldatato ascertaintheir usefulness
in
(known)variablesE
practice.Two typesof testsareperformedcommonly.In functionf=f(E; P) of theindependent
parameters
P, where E is a matrixandP isa
the first, more rigorouscase, the model parametersare andunknown
determined
independently
of the experiment,if possible. vector
composed
of elements
pj. Theexperimental
data
Then, with knowledgeof the experimentalconditions,the
modelis appliedto predictsomeinitially unknownaspect
of the experimentalsystem.The secondcase,whichis not
a modeltest,is simplyto fit the modelto the databy some
procedurewhich determinesits parameters.If the model
fits well and the parameter values coincide with the
analyst's expectations, then the model has been
"validated." We refer to thesetwo possibilitiesas testing
by predictionandtestingby validation,respectively.
An exampleof the experimentaldifficultiesinvolvedin
consist
of nemeasurements,
f/(E/),i = 1, ßß-,ne.Def'me
a
scalar
"distance"
function
asH(e•,e2),wheree• ande2 are
locationsin the spaceof independent
variables.Usually,
wetakeH(ex,e2)= lex-e 2 12or H(ex,e2)= lex
- e21,wheretheformerchoiceis thesquare
of Euclidean
distance and the latter is the absolute value of the distance.
In these and many other cases, H satisfies some
"stationarity"
assumption,
i.e.,H = H•(r),meaning
thatH
is unchangedby translation. In fitting the model, one
minimizes the discrepancies
between the model and the
applyingtheorypredictively
is provided
by the Borden databy suitablechoiceof P; i.e., chooseP suchthat
aquifersolutetransportexperiment[Robertsand Mackay,
1986]. This extremely valuable and meticulouslyperformed experimentwas carded out in a hydrologically
d= •]H[f(E;P)- fi(Ei)1
(14)
i=1
uncomplicated
aquifer. Known quantifiesof soluteswere
injectedinto the aquifer,and their subsequent
movement is minimized.
Minimizationof d can be achievedby
was monitoredby an extensivesamplingarray over a differentiation
of (14) with respect to the model
3-year period. By averagingthe experimentaldata over parameters, or
the vertical axis, Freyberg [1986] and Sudicky [1986]
foundthat the theoryof Dagan [1982, 1984]predictedthe
temporal growth of the tracer plume's second central
i=1
momentsreasonablywell. Analysesof this experiment
havehighlightedthe largeuncertainties
introducedby lack wherenp is the length
of P. The combination
of
of knowledgeof the initial configurationof the solutein parametersyielding a minimum d is obtainedby the
theaquifer[Barryet al., 1988;Naff et al., 1988].
solutionof thesimultaneous
equations:
Becausetesting by prediction is not often a viable
option, testing by validation becomesthe standardby
3d/3pj= 0 j = 1,..., nt,
(16)
which modelsarejudged. In a given casethe inclusionof
substantial
experimentalnoiseensuresthata large classof Equation (16) is, in general, a system of nonlinear
models fits the data equally well. This scenariowill equations. In this outline of regressionit has been
alwaysfavor moresimple,evenempiricalmodels,as there assumedthat the model outputis scalar. Extensionto a
is little point in invokinga complex,albeit more theoreti- vectorof predictions
followswithoutdifficulty.
cally soundmodel if it offers no advantagesover simpler
Traditionallinear regressionresultswhen the system
approaches.Thereforeexperimentaluncertaintiescan be (16) is linearin the elementsof P, the functionH in (15)
seento havea profound
effectonthedegree
to whicha measures the square of Euclidean distance, and the
new theoryis regardedasbeingan improvement.
distributionof predictionerrorssatisfiesthe normality
Scientistsand engineersfrequentlywish to characterize assumption.After fitting the model it is necessary
to
a transportprocessin the light of a selectedmodel. checka posteriori,amongotherthings,thatthe distribution
Mechanistically, this case is identical to testing by of residuals,
si (wheresi = H(f(E; P) -f/(E/)]),
validation in that the goal is to determinethe model's i = 1, ß ß., n,, is approximately
normal[CoxandSnell,
parameters.Nonlinearregressionis usedto fit the model 1968]. If this conditionis satisfied,it follows that one can
and so obtain estimatesof the parameters. Statistical calculate a covariance matrix of the residuals as well.
theoryprovidesa very elegantsolutionto this problemfor Lineartheorygivesbothan estimateof P andan indication
linear regressionwith normally distributedmeasurement of theaccuracyof thisestimate.
errors. Efron [1982, chapter1, 1988] suggeststhat the
constraints
imposedby parametricstatisticaltheorycanbe The Bootstrap
abandonedin an age of readily available, large-scale
What happenswhen the residualsdo not satisfythe
computersand inexpensivecomputation. Some of these normalityassumption
or whenthe modelis nonlinear?A
3d_ 3 •-/[f(E;
P)- f•(Ei)]j =1.. n•, (15)
28, 3 /REVIEWSOF GEOPHYSICS
Barry:SUPERCOMPUTERS
AND SOLUTETRANSPORTß 285
centeredat thoselocations. An interpolationprocedure
mustbe performedto estimateCoat unsampledlocations.
is to use the data set to define the distribution of residuals.
Again,thisinvolvesa nonlinearregression
procedure,and
in
principle,
the
nonlinear
regression
procedure
described
This empiricaldistributionis resampleda largenumberof
timesto createsyntheticdatasets. For eachresamplingan above, or any of the methodsdescribedby Yeh [1986]
estimateP* of P is obtainedusingsomenonlinearfitting could be used. This includesmethodsfrom spatial
methodsuchas that of Marquardt [1963]. The bootstrap statistics[Ripley, 1981], includinggeostafisticalmethods
estimate
of P is PB,the meanof theP*. The bootstrap [Journeland Huijbregts,1978]. However, as the aim is
simplyto estimatethe integralin (18), simplealternatives
estimate
ofthecovariance
matrix
Ca ofPais
basedon crossvalidationseemreasonable.One procedure
1 • (p,k_PB
)(P*k
- PB
)r (17) couldconsistof definingtheinterpolator:
Cz=n*-I
useful computer-based
method is the bootstrap. Efron
[1982, section5.7] outlinesthe method. In brief, the idea
k=l
wheren* is the numberof resamplings
andthe superscript
k identifiesthe variousP* for the resampleddata set. Of
•g [Hg
(x,xi, t,ti);Plc0
(xi, ti)
Co
(x,t) =
course,many variationsof this procedureare possible.
(19)
Note that the groundwaterliteraturecontainsexamplesof
•g [Hg
(x,xi,t,ti);el
preciselythismethod[e.g.,Yehand Wang,1987],although
i=1
the term bootstrappinghas not been usedas such. The
researchof Yeh and Wang [1987] was directed at
parameterizing
solutetransportmodels.Theseauthorscite whereH g is a "distance"functionof both spaceand time,
other, similar analyses. The role of supercomputers
in e.g., H. m H.(x•, x2, q, t2), and the function
parameterestimationis apparentfrom the observation
of
Yeh and Wang [1987] that the "intensivecomputational parametersin the vectorP. Elementsof P can be usedin
effort required by the Monte Carlo method...
may the definitionof Ho, e.g., to scalethe distancefunction.
prohibit[its] applicationto... largescaleproblems." A Cross
validation
th%n
consists
of choosing
P soasto
morecompletereview of modelparameterization
studiesin minimize
S•, where
nt
nt
groundwaterhydrology is given by Yeh [1986]. The
generalityof thesemethodsis that they can be appliedin
Scv:•Hr [Co
(Xj, tj) - •co(xi, t/)l
(20)
j=•
i=l
conjunctionwith many existing applications. Recent
examplesto which this commentappliesare theparameter
g(/-/g;
P)'generali[es/-/s'
toinclude
anumber
ofadjustable
estimation studies of Kool and Parker [1988], who
analyzedunsaturated
flow, andMishra andParker [1989],
who dealtwith both waterflow andsolutetransport.
Cross Validation
Crossvalidationhas been usedfrequentlyto estimate
spatialcovariancefunctions[e.g., Samperand Neurnan,
1989]. The methodhas at its base a very simple idea
whichcanbe appliedin manycontexts.For example,data
from the solutetransportexperimentat the Bordenaquifer
had to go througha multistepanalysisbeforethe desired
spatialstatisticswere obtained[Freyberg,1986; Barry et
al., 1988]. In the case of soluteplumesthe statisticsof
interestare often the plume's zeroth-, first-, and secondorderspatialmovements,
from whichonecanestimatethe
plumemass,meanpositionand variance. The calculation
of thesequantifiesamountsto a numberof triple integrations to be carried out numerically. Central spatial
momentsaredefinedby
1(t)I (Xl
gij•(t)
=M000
- gl)i(x2
- g2)/
ß(x3- !.t3)• Co(x, t) dx
If g[ttg(x,
x, t, t);P] = o%thenthefunction
in (19)will
reproducethe measureddata. The cross-validation
idea,
writtenas an equationin (20), is to try to predicteach
measurement
usingtherestof the dataset. The sumof the
prediction
errors,
S•, isminimized
toyieldP. Themethod
givenhereis to try anynumberof differentfunctions,
g, in
(19), selecting
the onethatgivesthe leastS•. Other
functionalformscouldbe usedin (19) insteadof thelinear
combinationsuggested.Note that g can be definedto
includeonly a limitedspatialor temporaldomainin (19).
A variationof thisprocedure
(calledthe"supersmoother"),
discussedby Efron [1988, section6], is to define the
functiong as a linearinterpolantand thento selectthe size
of thespatialdomainusinga criterionlike thatin (20).
Whena suitableg is chosen,
Coin (18) is replaced
by the
interpolantin (20), and the required integrationsare
performednumerically.The momentestimatesobtainedin
this mannerare point estimatesonly. It is possibleto
calculateestimatedvariancesof these point estimates,
againby someresamplingscheme. One suchschemeis
jackknifing.
(18)
lackknifing
whereMooo(t)
is theplumemassandCo(X,
t) = c(x,t)0(x,t).
Assume,as in (18), thatwe wishto derivea statistic,Q
from a dataset. The methodusedto calculateQ (e.g.,
to be biased.
number
nt of locations
or asaverages
oversmallvolumes maximumlikelihood)is knownor suspected
The functioncowill be knownor estimatedat only a finite
28, 3 / REVIEWSOF GEOPHYSICS
286 ß Barry:SUPERCOMPUTERS
AND SOLUTE
TRANSPORT
Jackknifing[Quenouille,1956;Gray and Schucany,1972] concentration
increments,
domainsof higherconcentration
is a techniqueto reducebias in the estimateof the statistic are surroundedby more surfacesthan areas of lower
When(mathematical)
lightraysarepassed
Q. Thedataset,consisting
of n•measurements,
isdivided concentrations.
into k groups,eachcontainingrn elements.Eachgroupis throughthe surfaces,they are diffused. Domainscontainremoved,in turn, and the statisticQ(/3, j = 1, ß ß., k, ing relativelyhigher concentrations
appearas being
corresponding
to Q is calculated.The jackknifedestimate relativelymoreopaque.Thereis completefreedomin how
thefinal imagesare vieweddueto variations
in thelight
Q•isthen
given
by
Q: kQ - (k - 1)•j
(21)
sourceandobservation
positions.
Somerenditionsof thisprocessappearin Plate 1. The
bromideplumeafter462 daysof traveltimein theaquifer
is viewedasmoving,approximately,
fromtheupperfight
k
to lowerleft portionsof Platela. Platelb providesa side
viewof theplume.Theimageis clearlyveryunsymmetrij=l
cal, the front and rear portionscorresponding
to the
verticallyaveraged
bimodaltwo-dimensional
plumesnoted
appears
to behighly
Thevariance
of Q?Var(Q), is estimated
from[Tukey,byFreyberg[1986]. Solutetransport
1958]
concentrated
alongverystratifiedlayerswithintheaquifer.
k
The difficultyin achievingcompleteplumesamplingis
Var
(Qj)=
k-k 1•[• _Q(0/)]2 (22)revealedby the laggingedge of the plume apparently
j=l
extending
beyondthe imageintoan unsampled
area. Plate
Mosteller and Tukey [1968, sectionE2] showthat under lc displaysthe plume movingdirectlytowardthe viewthatsolutetransport
is dominated
by
certainconditions,
confidence
intervals
for Q•,, the point.Thesuggestion
populationparameterbeingestimated,
canbe derivedfrom the morehighlypermeableregionswithin the aquiferis
where
reinforced.
TO= (QJ- Qt,) [Var(Qj)]-•/2
whereTQhasa t distribution
withk-
(23)
findingof Sudicky[1986] that the ratio of the verticaland
horizontalcorrelationlength scalesof the hydraulic
at theBordensitesuggests
thattheaquiferis
1 degrees
of conductivity
freedom. A detaileddescriptionof the constructionof
confidence
intervalsusingthisapproach
is givenby Barry
and Sposito[1990].
Visualization
These observations are consistent with the
stronglyanisotropicand, indeed, reflects the lenticular
geologicalstructure.Thereis a clearindication,also,that
a mobile-immobileregion model might provide an
adequaterepresentation
of solutebreakthrough
in this
system,asalreadyfoundby GolzandRoberts[1986].
Improvements
in datavisualizationhavekeptpacewith
Theimagesin Plate1 arebuta fewof a largegroup.In
developmentsin computing hardware. Plotters and termsof computational
effort,thetotalCPU timerequired
associatedsoftware are standard equipment. Much to produceeachimagerangedfrom 10 rain to 10 hourson
specializedhardwareand software is available also. A aneight-processor
AlliantX-8 minisupercomputer.
powerful combination for real-time simulation is the
combinationof supercomputer
computational
ratesand a
dedicatedgraphicsworkstation.An exampleutilizingthis SOLUTE TRANSPORT ON SUPERCOMPUTERS:
technology
wasundertaken
aspartof a moregeneralstudy APPLICATIONS
on solute transport. Three-dimensionalvisualizationsof
thebromidetracerplumefrom the Bordensiteexperiment
Subsurfacehydrology researchersare utilizing the
[Roberts and Mackay, 1986] were generatedusing increasinglyavailablesupercomputer
time and resources.
facilities at the National Center for SupercomputingThe extentof supercomputer
use,however,is notextensive
Applications
(NCSA). Briefly,theraw data,consisting
of at the presenttime. An overviewof recentapplications
solute concentrationmeasurementsat given spatial follows.
locationsfor the 14 plume samplings,were fitted to an
interpolationfunctionusing the NCSA's Cray X-MP/48 ParticleTracking
computer following the cross-validationprocedure
Nomeactivesolute transportcan be modeled quite
describedabove. Full detailsare given by Barry and effectively by using particle-trackingtechniques. All
Sposito
[1990].Dense
(4-8x 104points),
regular
grids
of approaches
to dateare basedon the dilutesoluteassumpinterpolated
concentration
datawereproduced,
andsurface tion;i.e., thewaterflow modelis solvedseparately
to give
contours--three-dimensional
versions
of
contour
the groundwater
velocityfield. This velocityis thenused
lines--were computed using the "marching cubes" to provide the deterministic motion of the "solute"
algorithm [Lorensen and Cline, 1987]. The contour particlesin the simulation, with an additional random
surfaces
areallowedto be semitransparent
andsoallowthe componentproviding the dispersivemechanism. The
passageof light. Sincecontoursare formedat regular theoretical
basisof the methodis the Langevinequation.
Barry:SUPERCOMPUTERS
AND SOLUTE
TRANSPORT
ß 287
28 3 /REVIEWS OF GEOPHYSICS
-6.314
27.179
t = 462
59.849
59.849
x2
-2.155
-7.277
x3
6.467
-2.155
-6.314
27.179
59.849
x2
Plate 1. Three-dimensionalimagesof the bromidetracerin the
Bordenaquiferexperiment.In all images,the upperline of data
gives the elapsedtime (in days) sincesoluteinjection,and the
coordinates(in meters) of the soluteplume boundingbox in a
Cartesian
system
alignedwiththemeangroundwater
flow(thex•
axis). Thex3 axisis positiveverticallyupwardfromtheground
form of this stochastic differential
equationis [Gardiner,1983,section4.1]
dz/dt- a(z, t) + b(z, t) •(t)
6.467
-6.314
27.179
The one-dimensional
-7.277
x3
x2
xl
(24)
-7.277
x3
6.467
-2.155
surface.(a) The plumeviewedfromabove,with thex• axis
passingdiagonallyfrom the upper right to the lower left of the
viewpoint. (b) Side view of the plume. The meanflow is from
left to right. (c) Frontal view of the plume. In this case the
plumeis movingdirectlytowardthe viewpoint.
wherea andb areknownfunctions
and•(t) is a stochastic
forcing function idealizedas white noise. As it stands,
(24) hasno physicalmeaningbecausewhite noiseis delta
functioncorrelated[van Kampen, 1981, chapterVIII]. If,
288 ß Barry:SUPERCOMPUTERS
AND SOLUTETRANSPORT
however, we follow the It6 interpretation,then (24) is
equivalentto
28, 3 / REVIEWSOF GEOPHYSICS
upto107nodes.
Simulations
ofthis
kind
arepossible
only
on supercomputers.
Another example of the particle-tracking
method,
althoughnot supercomputer
based,was presented
by
t+At
andDOll[1989]. Theseauthors
developed
a
z(t+At)- z(t)=a[z(t),
t]At+b[z(t),
t] I •(t')dt' (25) Illangasekare
t
where At is a finite time interval. Clearly, different
realizationsof •(t) will influencethe locationof the
particle under consideration. Under suitableconditions,
(25) is equivalentto the Fokker-Plankor Kolmogorov
forwardequation[CoxandMiller, 1965,chapter5]'
•}P
1 •}2(b2p)
-3T - •
•z2
-
•(aP)
Oz
(26)
P(z, t) is the probabilitydensityfunctionof the positionof
any particle whose movementis controlledby (25).
Conversely,the distributionof a "large" number of
particleswill be describedby (26) [Barry and Sposito,
1988]. With appropriatechoicesof the functionsa and b,
(26) is equivalentto the equationdescribingthe transport
of a nonreactivesoluteunderthe assumption
that density
effectscanbe ignored. A commonproblemwith particletrackingmethodsis the noisysolutedistributions
resulting
from the simulations. Distributionsare derived by
assigninga solutemassto eachparticleandaveragingover
a specifieddomain size. The noise is proportionalto
modelof reactivesolutetransportbasedon the discrete
kernel[Illangasekare
andMorel-Seytoux,
1978]numerical
solutionof the verticallyaveragedflow equation. A
particle-tracking
schemewasimplemented
to modelsolute
transport.FollowingGarderet al. [1964],soluteparticles
were moved deterministically
throughthe systemin
response
to the localdrivinggroundwater
velocity. The
concentration
in eachcomputational
cell was determined
by arealaveraging
assuming
theparticlescoveredan area
ratherthana pointlocation.Withineachcell an explicit
finitedifferencesolutionof the solutetransport
equation
gave concentration
changesto a numberof processes,
including dispersion,sorption, and mobile-immobile
regionexchange.The arealaveraging
scheme
wasvery
effectivein removingthe problem of noisy solute
concentrations.
Numerical Solutions
Conventional
solutions
basedon the governing
equa-
tionsof flow and solutetransportare morecommon. It
seemsthat most applications,includingthosediscussed
below, use the assumptionof a dilute solute,so that the
1IN'm, where
N' is thenumber
of particles.
Themost governingequations
maybetreatedseparately.
straightforwardway to reduce noise is to increaseN',
Perhaps
the mostelaborate
solutetransport
modelyet
although smoothingor filtering of the concentration developedand implemented
on a supercomputer
is the
predictionsmay be more economicalcomputationally. DYNAMIX code[Liu and Narasimhan,1989a]. This is a
Similar equationscan be derived for multidimensional multidimensional, multispecies, reactive chemical
formsof (24). Thussolutions
to (26) canbe obtainedby transport model that includes the transformation
using a Monte Carlo approachapplied to (25). Each mechanisms:
acid-base
aqueous
complexation,
oxidationparticlein the simulations
movesindependently,
so (25) is reduction,precipitation-dissolution,
and kinetic mineral
well suitedto evaluationby a supercomputer.
Indeed,the dissolution.
The coderepresents
theinfluence
of superdegree of parallelism can be at least the number of computers
in enablingmodelingeffortsthatmoveaway
particlesin the simulation. Computationally,particle fromthesimple
distribution
coefficient
(Kd)approach
to
tracking is algebraicallyuncomplicated
and extremely modeling adsorption/desorption
reactions [Liu and
repetitive,makingfor an easilyvectorizedcode.
Narasimhan,
1989a].Thisversionof DYNAMIX updates
A recentstudyusingparticletrackingis thatreported
by the modelof Narasimhan
et al. [1986]. The transport
Tompsonet al. [1988]. Theseauthorsreview in detail the modelwill handlelargenumbers
of chemical
components,
theory outlined above, including the multidimensional species,andphases.The numericalsolutionof the model
formsof thefunctions
a andb in (24) whichareappropri- requiressolvinga set of multidimensional
transport
ate for simulationsof solutetransportfor both saturated equations,
onefor eachchemicalcomponent,
anda setof
andvariablysaturated
flows. The simulations
of Tompson thermodynamic
equilibriumchemistryequations. An
et al. [1988] wereaimedat evaluatingstochastic
theories, explicit finite difference scheme is used to solve the
as discussedelsewherein this review. To that end,flow in transport
equations.The thermodynamic
equations
forma
a heterogeneous
domain was generatedby using the nonlinearalgebraicsystemsolvedby usingthe Newtonlarge-scaleflow simulator of Ababou et al. [1988], Raphsonmethod. Within each elementof the numerical
developedfor the Cray-2 supercomputer.Ababouet al. grid,completemixingof the aqueous
chemicalspecies
is
[ 1988]usedthe stronglyimplicitprocedure,
a lessefficient assumed.
forerunner of the PCG method, to solve their linear
systems
of equations[Meyeret al., 1989]. Tompson
et al.
[1988] did not discussthe specificalgorithmdesignor
computerusedin their simulations.Their code,however,
is designed
to trackmanythousands
of particles
ongridsof
Exampleproblems
solvedby DYNAMIX aregivenby
Liu andNarasimhan
[1989b].Afterfavorable
comparison
of DYNAMIX
to the one-dimensionalcodes of Walsh et
al. [1984] and Carnahan [1986], Liu and Narasimhan
[1989b]performed
a simulation
of solutetransport
froma
28, 3 /REVIEWSOF GEOPHYSICS
Barry:SUPERCOMPUTERS
AND SOLUTETRANSPORT
ß 289
domain solutionfrom a single set of Laplace domain
and arsenicwere included. Their 30-year simulation solutionvalues. Sudicky[1989b]extendedthe application
periodshowed
thatbothchemicals
werestrongly
retarded of the methodto the caseof fracturedporousformations.
in the subsurface. The total CPU time needed was 3.4 In the case of the traditionalfinite differenceapproximahourson a Cray X-MP/14. The code does not take tion of the temporalderivative,numericaldifficultiesarise
advantageof the Cmy's vector speed capabilities.because of the large differencesbetween the rapid
transportin the fracturesand much
Rewritingcomputationally
intensive
portions
of thecode advection-dominated
slower
diffusional
flux
into the porousmatrix. Sudicky
to allow vectorizationwould likely lead to substantial
[1989b]
combined
the
Laplace transformmethod with
improvements
in theCPU timeused.
ORTHOMIN
.acceleration
for the rapid solutionof the
A model that explicitly makesuse of supercomputer
derivedfromapplication
of
characteristics
in algorithmselectionwas developedby algebraicsystemof equations
Chiang[1985](cf. Chianget al. [1989]). The codewas the Galerkinprocedure. The combinedprocedurewas
developedto model pumping-induced
flow and con- shownto be very efficientand accurate. A 30,651-node
taminanttransportin a two-dimensional
domain. The simulationof downwardseepageof solutein a medium
fracturestook 72 s on a Cmy
modelusesa modifiedmethod-of-characteristics
procedure containingseveralthousand
contaminationsite. The reactive trace elementsselenium
to solvethe governing
equations.Examplesimulations X-MP/24 to obtain solute distributionsvalid for all times
especiallysuitedto vector
includedtransport
in domains
havinga randomhydraulic of interest.By usingalgorithms
Sudickyhasdeveloped
a codecapableof
conductivity
field. Realizations
of thelatterweredevel- supercomputers,
verylarge,field scaletransport
problems.
opedby usingthe turningbandsmethod[Mantoglou
and simulating
Wilson, 1982]. The name "turning bands"denotesan
Stochastic
SoluteTransport
Theory
algorithmthatgenerates
correlated
randomvariateson a Evaluating
As mentionedalready,the data base from the Borden
groupof linespassing
through
a point. A two-or threeexperiment
hasbeenusedto evaluatethe
dimensional
randomfield, with a predefinedcorrelation aquifertransport
structure,is derivedfrom a seriesof orthogonalprojections predictionsof stochasticsolute transporttheory. Both
fromthegroupof lines. Tompson
et al. [1989]reporton Freyberg[1986] andNaff et al. [1988] comparedtheoretithemethod's
performance
andefficiency.Chiang[1985] cal predictionsof the vertically averagedtracer plume
selectedthe PCG algorithm to provide the necessary dispersivitywith thoseestimatedfrom experimentaldata
numericalsolutionto the linear systemof equations. It and foundexcellentagreement.Garabedianet al. [1988]
was concludedthat the code developedwas efficient, analyzed reactive solute data from field experiments
accurate,and capableof modelingtransportin systems conductedat Chalk River, Ontario, and Cape Cod,
Massachusetts,
usingtheorydevelopedon the assumption
withrapidlyvaryinghydraulic
conductivity
fields.
Frind et al. [1989] reporta three-dimensional
simula- that both the flow field and the solute retardation factor are
tion of transportof organicsolutesundergoing
aerobic random.Again,onlytheplumevariancewasevaluated.
Dagan
biodegradation.They model a saturatedgroundwater Followingthe successof the two-dimensional
systemin whichthe solutetransport
process
is described [1984] model in predictingthe spreadof the vertically
by a nonlinear,coupledset of equations,
one for the avemgecltracerplume at the Bordensite, Barry et al.
contaminantand one for dissolvedoxygen. The dual- [1988] checkedwhetheror not the actualplumeconcentra-
Monodequation[Bordenand Bedient,1986] is usedto tions conformed with ensemble mean concentrations. The
model the growth of the microbialpopulation. By theorywas used to predict soluteplumesby using the
plumefrom a numberof differentsamplings
as
formulating
the numerical
modelto produce
a symmetric measured
coefficientmatrix, the PCG solutionmethodcouldbe used initial conditionsand then comparingthe predictedand
effectively.
A 24 x 103 nodesimulation
over155time measuredplumesat later times. The data analysisand
steps(300days)tookabout2 hoursof CPU timeona Cray generationof the predicted plumes, which involved
The authors concluded that comparable integratingthe initial conditionnumerically,were carded
two-dimensional simulations were to be avoided, as out usinga vectorizecl
codeon a Cmy X-MP/48. It was
oxygenavailabilitywas markedlyaffectedby problem foundthat the modelpredictedthe measuredplumequite
accuratelyon relatively short time scales, becoming
dimensionality.
Sudicky[1989a] considered
the transportof a dilute progressivelyworse with increasing elapsed time.
necessaryin the data analysis
solutein a steady,nonuniform,
two-dimensional
flowfield. However,the assumptions
of model-predicted
and
The spatialderivativeswere discretizecl
by using the madeequivocalany comparisons
Galerkinfiniteelementprocedure.Insteadof usinga finite measuredplumes.
The ensembleapproachhas led to elegantanalytical
difference
procedure
for the temporalderivative,
Laplace
by Dagan [1987], they cannotbe
transforms
were used. The Laplace-transformed
problem results,but as discussed
aremet. A demonstration
was shown to not suffer from the numerical dispersion appliedunlesscertainconditions
of
the
possible
differences
between
separate plume
problemsassociated
with other methods. The Crump
realizations
is
given
by
Frind
et
al.
[1987].
Theseauthors
[1976] algorithmfor numericalinversionof the Laplace-
X-MP/24.
transformed solution can be used to calculate the time
use a two-dimensionalvertical slice model of an aquifer
290 ß Barry:SUPERCOMPUTERS
AND SOLUTETRANSPORT
28, 3 / REVIEWSOF GEOPHYSICS
with a hydraulic
conductivity
fieldthatreproduced
the the directionof flow (and layering). Generationof the
estimatedlow-orderstatisticsof the Bordenaquifer. The
random hydraulic conductivityfield was producedby
synthetic
conductivity
fieldinvolveda significant
computational Cost, as it involved the numerical solution of a
usingtheturning
bands
method.Thelocalwaterfluxes, nonlinear
algebraic
system
of equations.
Theperfectly
and hencevelocities,were computedusingthe "principal
direction technique" [Frind, 1982c; Frind and Pinder,
1982]. The computedvelocitiesin turn were usedin the
solutionof the solutetransportequation.Two simulations
stratified
aquifer,although
certainlyunrealistic
physically,
is useful becauseit tends to representa "worst case"
scenario
for ensemble
averaging.An interesting
resultof
the simulations
was that the concentration
uncertainty
wereperformed,
oneof which
used
a spatial
gridof 106 tends to increase with decreasingensemblemean
elements. It was clearly demonstrated
that local variations concentration.
in the local hydraulic conductivity dominate plume
Black[1988]extended
theone-dimensional
approach
to
movement,particularlyat early times. The short-term a verticalslice,two-dimensional
model.Theturningbands
behavior can have persistent,long-term effects. The methodwas usedto generate
realizations
of the In (K)
simulation of Frind et al. [1987], which used 10 hours of
CPUtimeona CrayX-MP/48,
revealed
theformation
of
field. Theturningbandsmethodrequires
thegeneration
of
numerous one-dimensional correlated fields. Unlike most
two subplumesdue to the initial placementof the solute previousinvestigators
who usedthe methodof Shinozuka
sourcein a regionof rapidlyvaryingconductivity. The andJan [1972]for thistask,BlackandFreyburg[1990]
initialverticaldimension
of theplumewasonlya factorof present an alternative dalled the "matrix-factorization
3.5 timesthe verticalcorrelationscaleof the In (K) field. moving average" method. The method, which was
As discussed
by Dagan [1987], the initial condition implemented
on a CrayX-MP/48,wasshownto be both
dimensionmust be much greaterthan the characteristic
length scale of the conductivityheterogeneityfor the
ensembleapproachto be valid. That the factorof 3.5 used
by Frind et al. [1987] was insufficient for ensemble
averagingto apply indicatesthe importanceof considerationsof scalein the applicationof the analyticaltheory.
more efficient and more accurate than that of Shinozuka
andJan [1972]. Aftergeneration
of thetwo-dimensional
correlatedhydraulicconductivityfield, steadyflow
conditions
wereproduced.Particletrackingwasthenused
to simulatethetransport
of a soluteplume. It wasshown
that as the "verticalmultiplicity,"i.e., the ratio of the
Other,similarsimulations
reportedby Sudicky
et al. [1990] verticallengthdimensionof the systemto the vertical
demonstratethat in many cases,lessdramaticresultsare
obtained. For example, Sudickyet al. [1990] discuss
simulationsin which the solute plume approachesa
Gaussian
distribution,
aspredicted
by stochastic
theory.
Sudickyet al. [1990]go on to considerthetransport
of a
biodegradable
solutein a heterogeneous
porousmedium.
correlationscaleof the In (K) field, increases,the mean
distancetraversedby the plumeincreases
becauseof the
reducedinfluenceof low-conductivity
regions. Also,
underthesecircumstances
the concentration
uncertainty
tends to decrease.
In thiscasethereare threenonlineargoverningequations,
DISCUSSION
one each for the solute,the availableoxygen,and the CONCLUDING
microbial population [MacQuarrie et al., 1990]. The
behaviorof theplumeis quitedifferent.If thebiodegrada- No doubtothersupercomputer
applications,
apartfrom
tion rate is large enough,then the plume will tend to thosediscussed
here,areworthyof mention.On theother
decrease
in size. For thisreason,theplumewill encounter, hand,thetopicscoveredserveto illustrate
possibilities
for
and be influenced by, only a small portion of the supercomputerapplications. Also, the mechanismsfor
groundwater velocity field. Under these conditions, realizingsome degreeof the availablecomputational
plumes resulting from different realizations will not performance
in the solutionof algebraicsystemsof
necessarily
convergevery rapidlyto thosepredictedby equations
havebeendiscussed
in somedetail. In general,
ensembleaveragingof the governingequations. More choosing
algorithms
suitedto theclassof computer
being
comprehensive
simulations
presented
by MacQuarrieand usedcanwellhavea largepayofffor relativelylittleeffort.
Sudicky[1990] showin detail that biodegradable
solute Researchers
withaccess
to supercomputers
naturallywant
plumes evolve differently than plumes composedof to makeuseof existingcodesasrapidlyaspossible.It has
nonreactive
solutes.In particular,it is suggested
thatthe beenpointed
outalreadythatalthough
codes
maymnwith
scaledependence
of the plumedispersivity,
whichis well littleor no modification
on differentmachines,
including
documentedfor field scale transportof tracer solutes supercomputers,
the speedup
in execution
timesexpected
[Gelhar, 1986,Figure 12], doesnot applyin the caseof maynotoccurwithoutmodifying
thecode.Thereis ample
biodegradable,
organicsoluteplumes.
evidencethatcodesoptimizedfor scalarmachineswill not
The appropriateness
of ensemble averaging was achievehigh computational
rates on currentmachines.
investigatedalso by Black and Freyberg [1987], who The vast majority of scientificcodesare written in the
simulated
solutetransport
of an idealtracerin a perfectly FORTRAN language. "Standard"FORTRAN 77 is not
stratifiedaquiferconsisting
of homogeneous
layers. The well suitedto vectoror parallel machines,and manufachydraulicconductivity
was assumed
to vary transverse
to turers have introduced extensions to allow machine
28, 3 /REVIEWSOF GEOPHYSICS
Barry:SUPERCOMPUTERS
AND SOLUTETRANSPORTß 29!
Dagan [1986], in his review of stochasticflow and
featuresto be exploited. For example,Pelka and Peters
[1986] demonstrate the usefulness of the Cray's solu•tetransporttheory,notedthe vigorousgrowthin the
GATHER/SCA•R
featurefor vectorizingfinite element numberof researchpapersdevotedto stochastic
modeling
solutionsof the groundwaterflow equation. The research of groundwaterflow publishedin Water Resources
of Sudickyandcoworkerscitedaboveandthatof Meyer et Research. In retrospect,this surge in publicationsis
al. [1989] demonstratethe very substantialspeedupsthat entirelyreasonable
giventhe applicabilityof thetheoretical
can be gainedin a wide classof transportproblems. The approachto modeling fluid transportin heterogeneous
developmentof more sophisticated
compilerswill reduce permeable formations. Supercomputersoffer another
the need for special programmingof codes. In the potentiallyvaluable avenue for the understandingand
meantime, if researchersare to realize supercomputer resolutionof solutetransportproblems. Their rapidly
computation
rates,theywill profitby (1) choosingsuitable increasinguse in future investigationsis therefore a
algorithmsand (2) rewritingportionsof their code. On plausibleexpectation.
vector machineslike Crays, availabletimings show that
linearsystemsof equationsare mostefficientlysolvedby NOTATION
preconditioned
conjugategradient-type
algorithms.If an
a(x, t) knownfunctionin theLangevinequation.
existingcode is being transferredto a supercomputer,
a
A
squarecoefficientmatrix.
very usefulexerciseis to time thevarioustasksperformed.
b(x,
t)
It oftenhappensthat mostof the totalCPU time is devoted
knownfunctionin theLangevinequation.
b
vectorof knownquantifies.
to performingonly small sectionsof the code. Large
c
soluteconcentration,
M L-3.
speedupswill result by making sure that these small
concentration corresponding to maximum
sections
vectorizeasmuchaspossible.
density,
M L-3.
Besides fast computationrates, supercomputers
are
often linked with other facilities or have other features that
co
make possible new ways of going about research.
Accessiblememory on supercomputers
is generallyvast
comparedwith that of usualmainframes. Visualization
makespossiblewaysof viewingprocesses
or resultsthat
cannotbe achievedotherwise. A typicalcaseis that of a
graphics workstation connected to a supercomputer.
Results of a simulation can then be viewed directly.
Alternatively,slidesor moviescanbe generated.
In this review an attempthasbeen madeto link aspects
of solute transport-relatedresearchand modeling that
appear, on the surface, to link some rather disparate
Co
<c>
d
sum of residuals.
D
solute
dispersion
tensor,
L2T4.
coefficientalongthe i axis,i = 1, 2, 3,
Dii dispersion
L2 Tq.
DMii macrodispersion
coefficient
alongthei axis,i =
1, 2, 3, LZT-•.
reduced
solute
dispersion
tensor
(=D/n),L2T4.
location vector,L.
army of independent
variables.
arbitraryscal• function.
scientific threads. The common denominator,of course,is
supercomputer
applications. This situationis just as it
shouldbe: solutetransportencompasses
both the physics
of flow in heterogeneous
media and complex chemical
reactionsandmixingprocesses
over multipletemporaland
spatialscales.Diversemathematicaltoolsandanalysesare
necessaryto elucidatethe behaviorof specifiedsystems.
The role of a supercomputer
shouldbe as anotherspecialized instrumentthatcanbe calleduponas necessary.It is
emphasized
that thereare a wide varietyof circumstances
in which supercomputers
will aid in yielding valuable
insight. One instancewhere this has occurredis the
macroscalebehavior of a biodegradable,organic solute
plume. MacQuarrieandSudicky[1990]reportsupercomputer simulationsthat illustratethe significantdifferences
betweenthistype of plumeandthe well-studiedcaseof an
ideal tracer plume. In the latter case,both field experiments
and numerical
simulations
indicate
that
gravitational
acceleration
vector(magnitude
g) L
T-2.
H
distance function.
generalized"distance"function.
distance
functiondependent
on separation
only.
K
scalar
version
ofk, L 2.
permeability
tensor,
L2.
hydraulic
conductivity,
L T2.
K
hydraulic
conductivity
tensor,
L Tq.
k
k
function
in(4)tobeSpecified
fora givensolute.
theijkthrawspatial
moment,
M Li Lj Lk.
mass of salt.
Mr
Ile
Il l
under
known conditionsa reasonablyaccurate macroscopic
descriptionof the plume is possible. Biodegradation
effectivelylimitsthe spatialdomainsampledby theplume
with the result that a macroscaledescriptiondoes not
appeartenableon realistictime scales.
initialconcentration,
M L-3.
cO,M L-3.
ensemble
meanof c,M L-a.
totalmass(soluteplussolven0.
porosityof theporousmedium.
numberof experimentalmeasurements.
number of locations.
numberof parameters.
dimension of A.
N'
number of particles in a particle-tracking
simulation.
p
P0
thermodynamic
pressure
of thefluid,M L-• T -2.
reference
pressure,
M L-• T-2.
292 ß Barry:SUPERCOMPUTERS
AND SOLUTE
TRANSPORT
P
P
p,
P.
q
q
probabilitydensityfunction.
vectorof parameters.
IRIS 3130 workstation.Time on the Cray wasprovidedundera
grantfromthe AcademicComputingCenterof theUniversityof
estimate of P.
California, Riverside.
elementof P,j = 1,. ß., n.
GarrisonSpositowas the editorin chargeof this paper. He
thanksE.A. Sudickyand an anonymous
reviewerfor servingas
volumetric
flow
rate
offlt•id
sources
orsinks
per technical
unitvolumeof porous
medium,
T-•.
fitfidfluxvector,L T4.
arbitrarystatistic.
populationparameter.
relativeperformance.
residualvectorin thePeG algorithm.
temporaryresidualvector.
scalarspeed.
sumof squares.
Scv cross-validated
Sii Fourier transformof the stationarysolutedrift
velocitycovariancefunctionalongthei axis,i =
1,2, 3,L5T-2.
specific
storafivity,
L-1.
v
volumetric water content.
wavenumbervectorin Fourierdomain,L-1.
elementof •, L-1.
fluiddynamic
viscosity,
M L-1 T-1.
meanplumepositionalongtheith axis,L.
theijkthcentral
moment,
Li L• Lt•.
reference
dynamic
viscosity,
M L-1 T-1.
white noiseprocess.
po
referees.
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