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1994, Water Resources Research

Water Resources Research

Equivalence of bicontinuum and second-order transport in heterogeneous soils and aquifers2000 •

The mathematical equivalence of a closed second-order transport equation for reactive solutes in saturated heterogeneous porous media (e.g., soils and aquifers) and a two-region, mobile-mobile, formulation is demonstrated in two ways: (1) by averaging the bicontinuum equations and (2) by transforming the second-order equation to canonical form. The derivation of this equivalence is limited to media with heterogeneities in a

Water Resources Research

Boundary Conditions for Convergent Radial Tracer Tests and Effect of Well Bore Mixing Volume1996 •

Water Resources Research

Significance of porosity variability to transport in heterogeneous porous media1998 •

Water Resources Research

Stochastic analysis of solute transport in heterogeneous aquifers subject to spatiotemporal random recharge1999 •

Journal of Hydraulic Engineering

Analytical Solutions for Advection and Advection-Diffusion Equations with Spatially Variable Coefficients1997 •

Water Resources Research

Comment on “An efficient numerical solution of the transient storage equations for solute transport in small streams” by R. L. Runkel and S. C. Chapra1994 •

A new method is proposed to simulate groundwater age directly, by use of an advection-dispersion transport equation with a distributed zero-order source of unit (1) strength, corresponding to the rate of aging. The dependent variable in the governing equation is the mean age, a mass- weighted average age. The governing equation is derived from residence- time-distribution concepts for the case of steady flow. For the more general case of transient flow, a transient governing equation for age is derived from mass-conservation principles applied to conceptual 'age mass.' The age mass is the product of the water mass and its age, and age mass is assumed to be conserved during mixing. Boundary conditions include zero age mass flux across all noflow and inflow boundaries trod no age mass dispersive flux across outflow boundaries. For transient-flow conditions, the initial distribution of age must be known. The solution of the governing transport equation yields the spatial distribution of the mean groundwater age and includes diffusion, dispersion, mixing, and exchange processes that typically are considered only through tracer-specific solute transport simulation. Traditional methods have relied on advective transport to predict point values of groundwater travel time and age. The proposed method retains the simplicity and tracer-independence of advection-only models, but incorporates the effects of dispersion and mixing on volume- averaged age. Example simulations of age in two idealized regional aquifer systems, one homogeneous and the other layered, demonstrate the agreement between the proposed method and traditional particle-tracking approaches and illustrate use of the proposed method to determine the effects of diffusion, dispersion, and mixing on groundwater age.

Water Resources Research

A conservative semi-Lagrangian transport model for rivers with transient storage zones2001 •

WATER RESOURCES RESEARCH, VOL. 30, NO. 11, PAGES 3233-3235,NOVEMBER 1994
Commenton "A space-timeaccuratemethodfor solvingsolute
transportproblems"by S. G. Li, F. Ruan, and D. McLaughlin
Christopher
Zoppou
Australian
CapitalTerritoryElectricityandWater,Canberra
JohnH. Knight
Centre
for EnvironmentalMechanics,CSIRO, Canberra
in which c(x, t) is the solute concentration,D(x) is the
diffusioncoefficient,K(x) is the first-orderdecaycoefficient,
Analyticaltechniquesfor the solutionof the advection- u(x) is the fluidvelocity,S(x, t) is the sink/sourceterm, x is
diffusion
equationare generallyrestrictedto simpleprob- the spatialcoordinate
(0 -< x -< i), t is the time(t > 0), and
lemswith constantcoefficients.Practicalproblemsusually I is the lengthof the computationaldomain.
involve variable velocity and diffusion coefficients. The
It will be shownthat this equationis, with certainrestricsolutionschemedescribed by Li et al. [1992], based on the tions,mathematicallyequivalentto the conservativeform of
Laplacetransform,was developedwith the capabilityto the advection-diffusionequation.
solvethe advection-diffusionequationwith spatiallyvariable
Considerthe advection-diffusionequationwritten in concoefficients.
The Laplace transform is used to evaluate the servative form
temporal
derivativein the advection-diffusion
equationanalytically,therebyeliminatingthe effectsof the time deriva+
--(x) ......
o
(2)
tiveon accuracyand stability. Becausethe temporalderivOt
Ox
Ox
Ox
ativeis evaluatedusing the Laplace transform, there is no
needfor the time-stepping that is associatedwith more where the sink/source and reaction terms have been netraditionaltechniques for the solution of the advection- glected.
diffusion
equation.Therefore it is a potentially more efficient
An analytical solution to (2) will be given for the case
technique.
when the velocity, u(x), is taken to be a linear function of
Theauthorsdo not provide numericalsimulationsinvolv- distance,suchthat u(x) = UoX,andthe diffusioncoefficient
ing the spatially variable coefficient advection-diffusion a quadratic
function
of distance,
D(x) = Dox2, whereDo
Introduction
Oc(x,t)
O<c(x,t)u(x)•
OtD
Oc(x,t)
)
equation.A reasonfor this might be that there are very few
analyticalsolutionsto the advection-diffusionequation with
spatiallyvariable coefficients.
The purpose of this discussionis to present what we
believeto be a new analyticalsolutionto a particularform of
and u0 are constantsand 1 < x < I. Equation (2) now
becomes
Oc(x, t)
•+
Ot
theadvection-diffusion
equationwith spatiallyvariablecoefficients
and to highlighta potential mass conservation Expanding,
O(c(x, t)x)
uo
Ox
0( Oc(x,t).)=
0
-OOTx
x
problem that could be encountered with the formulation
Oc(x, t)
usedby Li et al. The analytical solutioncan be used to Oc(x, t)
+ (UoX- 2D0x)
validateschemesfor solvingthe advection-diffusion
equa- • at
Ox
tionwithspatiallyvariablecoefficients.
The efficiencyof the
method
proposedby Li et al. is demonstratedby comparing
O2c(x, t)
= D0x2 • Ox
itsperformancefor the solutionof the advection-diffusion
2 - UoC(
X,t) (3)
equation
with spatiallyvariablecoefficientswith that of a
well-known
time-stepping
scheme.
The last term in this equation is essentialfor the conserva-
tion of mass.The additionalterm, 2Dxo, in the advection
ConservativeForm of the Advection-Diffusion
term only affectsthe distributionof the mass.
Comparing(3) with (1) reveals that the form of (1) is
sufficiently
general,as it accommodatesdifferentcoefficients
Li et al. describea numericalscheme,based on the
(with
differing
physicalinterpretation)of (3). It follows from
Laplace
transform,
for the solution
of thefollowing
formof
thespatiallyvariablecoefficientadvection-diffusion
equa- the derivationof (3) that any expressioncanbe usedfor u(x)
and D(x) providedthat the concentrationprofile is smooth
tion:
and differentiable.It is possibletherefore to ensurethat (1)
conservesmass if in (1)
Equation
Oc(x,
t) Oc(x,
t) 0( Oc(x,
t).)
--------+
Ot
u(x)
Ox
=--
Ox
D(x)
-K(x)c(x,
Ox
t)+ S(x, t)
Published
in1994
bytheAmerican
Geophysical
Union.
Paper
number
94WR01492.
Ou
K( x) = -+ Kp(x)
Ox
(•)
whereKp(x) is the first-orderdecaycoefficient
of the
physicalprocessbeingmodeled.
3233
3234
ZOPPOU AND KNIGHT: COMMENTARY
If, in (1), the reaction and sink/sourceterms were ne-
120.
• ........--•.ta•
(•,•
7),
•,t6
o
glected,a naivemodelermightuse
Oc(x,t)
oc(x,t)
o2c(x,t)
Ot (UoX' Ox=DOx2
• Ox
-------+
2Dox)
2
(4)
The majordifferencebetweenthis equationand (3) is the
omission
of the lasttermin (3), whichis necessary
for the
loo.
••,ml•
•o.
conservation of mass. This is a nonconservative form of the
advection-diffusion
equation,and caremustbe exercisedin
40.
implementing
the algorithmproposed
by Li et al. so that
conservation of mass is not violated.
20.
0•
'AnalyticalSolutionto a SpatiallyVariable
Advection-Diffusion
Equation
.
ß0
5.
10.
15.
20.
Ananalytical
solution
to (3)canbeobtained
using
Laplace
1. Concentration
profilefor the spatiallyvariable
transforms.
Thisis achievedwith the changeof variables, Figure
advection-diffusion coefficient.
y = In (x) (see,for example,Hildebrand[1962,p. 13]).
Equation (3) becomes
oc(x, t)
•+
Ot
Oc(x,t)
o2c(x,t)
(uo- Do)•=
Do•Oy
Oy2
UoC(X,
t)
Alsoillustrated
inFigure1arethesimulated
profiles
using
(3),(4),(5), and(6). Thefiniteanalytic/Laplace
timemethod
(5) proposed
by Li et al. was usedto solve(3) and(4).In
contrast
to Li eta!., whousedthewell-known
Crump[1976]
whichisin theformof anadvection-diffusion
equation
with algorithm,the more robustnumericalLaplaceinversion
constant coefficientsand a first-order reaction term. Condevelopedby de Hoog et al. [1982]was used.
sidera problemwith the following
initialandboundary Theresultsfor the numericalinversionof (6) areindistinconditions:
c(x, 0) = 0, c(1, t) = Coandc(o•,t) = 0, which guishablefrom the exact results, which indicate that the
is similarto the problem
usedby Li et al. TheLaplace concentration
profiledecaysexponentially.The resultsobtained
forthesolution
of(3)usingthefiniteanalytic/Laplace
time
scheme
are
more
accurate than the results obtained
d•(y, s)
d26(y,s)
(s + Uo)•(y, s) + (Uo- Do)
usingthe well-knownthird-orderHolly and Preissmann
=
[1977]scheme,
whichis a time-stepping
scheme.
To satisfy
in whichif(y, s) is the Laplacetransformof the solute theCourantcriterion,500time stepswererequiredforthe
of (5).Thisscheme
required
approximately
10times
concentration
in theLaplacespaces. Solving
for 5(y, s), solution
the computational
time requiredby the finite analytic/
transformof (5) is
•(y, s) =-
Laplacetimescheme
for the solutionof (3).
co((uo
-Do)
y
s
exp
2D 0
Conclusions
D•/2 4Dø + s+ u0
(6)
The solution of the nonconservative form of the advec-
tion-diffusion
equation,equation(4), producederroneous
Performing
theinverse
Laplacetransform,
(3) thenhasthe results.Massis not conserved,and the predictedprofile
followinganalyticalsolution
bearsno resemblance
to the analyticalsolution.
The solutionmethodproposedby Li et al. doesnot
provide
anysignificant
advantages
overexisting
methods
for
the solution
of the advection-diffusion
equationwithconstantcoefficients.
However,forproblems
involving
spatially
variable
coefficients,
it hassignificant
advantages
overconschemes
providedcareis exercised
+exp Do cftc• 2(Dot)
1/2
(7) ventionaltime-stepping
c(x,
t)=-•- erfc 2(D0t)
•/2
(Uo
In
(x))[ln(x)+t(uo+Do)])
to ensure conservation of mass.
in whicherfcis thecomplementary
errorfunction.
Theanalyticalsolutionpresented
canbe usedto validate
numerical
schemes
for solvingthe advection-diffusion
equa-
Hypothetical
Example
tionwithvariable
coefficients.
Theanalytical
solution
canbe
readilyextended
to multidimensional
problems.
Consider
anexample
whereu0 = 1.0,D0 = 0.005,1 _<
x _<I = 20,andCo= 100,which
corresponds
toa range
of
References
PecletnumbersbetweenPe = 40 and 2, wherePe =
K. S.,Numerical
inversion
of Laplace
transforms
using
a
uoAx/D
o andAx = 0.2. Theexactconcentration
profile, Crump,
givenby (7) at t = 2.5 for thisproblem,is illustrated
in
Figure 1.
Fourier
series
approximation,
j. Assoc.
Cornput.
Mach.,23(1),
89-96, 1976.
de Hoog,F. R., J. H. Knight,andA. N. Stokes,An improved
ZOPPOUAND KNIGHT: COMMENTARY
3235
method
for numerical
inversionof Laplacetransforms,
SIAM J.
J. H. Knight(corresponding
author),CSIRO Centrefor EnvironI-lildebrand,
F. B., AdvancedCalculus
for Applications,
Prentice- mentalMechanics,GPO Box 821, Canberra,ACT 2601, Australia.
(email:kni•ht@enmech.csiro.au)
HaH,EnglewoodCliffs, N.J., 1962.
$ci. Star. Cornput.,3(3), 357-366, 1982.
C. Zoppou,ACT ElectricityandWater, GPO Box 366,Canberra,
Holly,
F. M., andA. Preissmann,
Accurate
calculation
oftransport ACT
2601, Australia.
intwo-dimensions,
J. Hydraul.Eng.Div. Am. Soc.Civ.Eng.,
IO$(HYII), 1259-1277,1977.
Li, S. G., F. Ruan, and D. McLaughlin,A space-time
accurate
method
for solvingsolutetransportproblems,Water Resour.
Res.,28(9), 2297-2306, 1992.
(ReceivedNovember2, 1993;accepted
June8, 1994.)

Water resources research

Transient flow to open drains: Comparison of linearized solutions with and without the Dupuit assumption1994 •

Water Resources Research

Numerical solutions for dispersion in porous mediums1967 •

Water Resources Research

Stochastic analysis of one-dimensional transport of kinetically adsorbing solutes in chemically heterogeneous aquifers1997 •

Water Resources Research

Estimation of reservoir properties using transient pressure data: An asymptotic approach2000 •

Water Resources Research

Models of water transport in the soil-plant system: A review1981 •

Water Resources Research

A new derivation of the Taylor-Aris Theory of solute dispersion in a capillary1983 •

Water Resources Research

A mathematical formulation for reactive transport that eliminates mineral concentrations1998 •

Water Resources Research

Infiltration of a Liquid Front in an Unsaturated, Fractured Porous Medium1991 •

Water Resources Research

A numerical dual-porosity model with semianalytical treatment of fracture/matrix flow1993 •

Water Resources Research

A Groundwater Mass Transport and Equilibrium Chemistry Model for Multicomponent Systems1985 •

Water Resources Research

Galerkin Finite Element Procedure for analyzing flow through random media1978 •

Water Resources Research

Mass arrival of sorptive solute in heterogeneous porous media1990 •

Water Resources Research

Analysis of longitudinal dispersion in unsaturated flow: 1. The analytical method1981 •

Journal of Geophysical Research

A kinetic theory for the drift-kink instability1997 •

Water Resources Research

Solving three-dimensional hexahedral finite element groundwater models by preconditioned conjugate gradient methods1994 •

1990 •

Water Resources Research

Renormalization group analysis of macrodispersion in a directed random flow1997 •

Journal of Geophysical Research

A simulation of biological processes in the equatorial Pacific Warm Pool at 165°E1999 •

Water Resources Research

Partitioning tracer transport in a hydrogeochemically heterogeneous aquifer2001 •

Water Resources Research

Monte Carlo studies of flow and transport in fractal conductivity fields: Comparison with stochastic perturbation theory1997 •

Water Resources Research

An approximate solution for one-dimensional absorption in unsaturated porous media1989 •

Water Resources Research

A coupled inversion of pressure and surface displacement2001 •

Water Resources Research

Exact Solutions for Water Infiltration With an Arbitrary Surface Flux or Nonlinear Solute Adsorption1991 •

Siam Journal on Applied Mathematics

Image Quantization Using Reaction-Diffusion Equations2006 •

Water Resources Research

One-dimensional stochastic analysis in leaky aquifers subject to random leakage1994 •

Water Resources Research

Efficient simulation of single species and multispecies transport in groundwater with local adaptive grid refinement1994 •

Journal of Geophysical Research

Discrete angle radiative transfer: 1. Scaling and similarity, universality and diffusion1990 •

Journal of Geophysical Research

Rapid inversion of two- and three-dimensional magnetotelluric data1991 •

Journal of Geophysical Research

Time-dependent magnetic annihilation at a stagnation point1993 •

Journal of Geophysical Research

Mantle circulation with partial shallow return flow: Effects on stresses in oceanic plates and topography of the sea floor1978 •

Journal of Geophysical Research

Ekman layers and two-dimensional frontogenesis in the upper ocean2000 •

1993 •

Water Resources Research

A Monte Carlo assessment of Eulerian flow and transport perturbation models1998 •

Journal of Geophysical Research

Superposition model for multiple plumes and jets predicting end effects1996 •

Journal of Geophysical Research

On the performance of numerical solvers for a chemistry submodel in three-dimensional air quality models; 1. Box model simulations2001 •

Water Resources Research

Reliability analysis of contaminant transport in saturated porous media1994 •

Journal of Geophysical Research

Ocean carbon transport in a box-diffusion versus a general circulation model1997 •