Chemical Engineering Journal 221 (2013) 487–491

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Chemical Engineering Journal
journal homepage: www.elsevier.com/locate/cej

Analytical solutions of the one-dimensional advection–dispersion solutetransport equation subject to time-dependent boundary conditions
J.S. Pérez Guerrero a, E.M. Pontedeiro b, M.Th. van Genuchten c,d, T.H. Skaggs e,⇑aRadioactive Waste Division, Brazilian Nuclear Energy Commission, DIREJ/DRS/CNEN, Rio de Janeiro, Brazil
Department of Nuclear Engineering, POLI&COPPE, Federal University of Rio de Janeiro, UFRJ, Rio de Janeiro, BrazilcDepartment of Mechanical Engineering, COPPE/LTTC, Federal University of Rio de Janeiro, UFRJ, Rio de Janeiro, BrazildDepartment of Earth Sciences, Utrecht University, Utrecht, The Netherlandse
US Salinity Laboratory, USDA-ARS, Riverside, CA, USAbh i g h l i g h t s" We extend the Duhamel theorem to the case of advective–dispersive solute transport.
" Analytical formulas relate exact solutions to time-independent auxiliary solutions.
" Explicit analytical expressions are developed for selected particular cases.
" Results are compared with other speciﬁc solutions from the literature.

a r t i c l e

i n f oArticle history:
Received 4 October 2012
Received in revised form 11 January 2013
Accepted 30 January 2013
Available online 9 February 2013
Keywords:
Duhamel theorem
Analytical solution
Advection–dispersion equation
Solute transporta b s t r a c t
Analytical solutions of the advection–dispersion solute transport equation remain useful for a large number of applications in science and engineering. In this paper we extend the Duhamel theorem, originallyestablished for diffusion type problems, to the case of advective–dispersive transport subject to transient(time-dependent) boundary conditions. Generalized analytical formulas are established which relate theexact solutions to corresponding time-independent auxiliary solutions. Explicit analytical expressions
were developed for the instantaneous pulse problem formulated from the generalized Dirac delta function for situations with ﬁrst-type or third-type inlet boundary conditions of both ﬁnite and semi-inﬁnite
domains. The developed generalized equations were evaluated computationally against other speciﬁc
solutions available from the literature. Results showed the consistency of our expressions.
Published by Elsevier B.V.

1. Introduction
Analyses of many contaminant transport problems require theuse of mathematical models commensurate with the application.
While many ﬁeld problems require comprehensive numericalmodels simulating transient ﬂuid ﬂow and solute transport, manytransport problems may well be addressed using simpliﬁed oneor multi-dimensional analytical models. As pointed out in several
studies [1–5], analytical models are useful for providing initial orapproximate studies of alternative pollution scenarios, conducting
sensitivity analyses to investigate the effects of various parameters or processes on contaminant transport, extrapolating resultsover large times and spatial scales where numerical solutions
become impractical, serving as screening models, estimating⇑ Corresponding author. Address: US Salinity Laboratory, 450 W. Big Springs Rd.,
Riverside, CA 92507, USA. Tel.: +1 951 369 4853; fax: +1 951 342 4964.
E-mail address: Todd.Skaggs@ars.usda.gov (T.H. Skaggs).
1385-8947/$ - see front matter Published by Elsevier B.V.
http://dx.doi.org/10.1016/j.cej.2013.01.095

transport parameters from laboratory or well-deﬁned ﬁeld
experiments, providing benchmark solutions for more complex
transport processes that cannot be solved analytically, and forverifying more comprehensive numerical solutions of the governing transport equations.
The literature contains many analytical solutions for advection–
dispersion type transport problems in one and multiple
dimensions, including solutions for sequential decay chains, nonequilibrium transport problems and ﬁnite and inﬁnite domains(e.g. [2,6–15]). Most or all of these solutions are for boundary conditions that are constant in time or change in only a very simple
manner with time (e.g., step or exponential functions). By contrast,
very few analytical solutions are available when the boundary conditions are arbitrary functions of time. Still, it is known from theheat conduction literature that Duhamel’s theorem provides a convenient approach for developing solutions to heat conduction
problems with time-dependent boundary conditions, and/or fortime-dependent energy generation scenarios by utilizing the488

J.S. Pérez Guerrero et al. / Chemical Engineering Journal 221 (2013) 487–491

solution to the same problem with time-independent energy
sources [16,17].
The objective of the present work is to apply Duhamel’s theorem to one-dimensional advection–dispersion solute transport
problems with boundary conditions that are dependent upon time.
A general framework is provided to obtain the desired solutions
from known solutions of cases having time-independent boundary
conditions. The equations developed in this paper hence may be
used to extend existing solutions to situations where the boundary
conditions are time-dependent.

3.1. General Duhamel solution

2. Problem formulation

orContaminant transport in homogeneous porous media is generally modeled by assuming a constant average transport velocity,
linear equilibrium sorption, and ﬁrst-order decay. For a ﬁnite (oflength L0) or semi-inﬁnite medium, this problem can be formulated as:R@cðx; tÞ
¼ Lcðx; tÞ
@t

ð1aÞ

where L is a linear operator of the form:

@
@
m l
@x
@x2

@ Uðx; t; sÞ¼ LUðx; t; sÞ@tð5ÞUðx; 0; sÞ ¼ 0

ð6ÞUð0; t; sÞ ¼ gðsÞ

ð7aÞ

D

@ Uð0; t; sÞþ mUð0; t; sÞ ¼ mgðsÞ@x@ UðL0 ; t; sÞ
¼0
@x

ð7bÞ

ð8cÞ

or

@ Uð1; t; sÞ
¼0
@x

ð1bÞ
cðx; tÞ ¼

which is to be solved here subject to the following initial andboundary conditions:cðx; 0Þ ¼ 0

ð2Þ

cð0; tÞ ¼ gðtÞ

ð3aÞ

or

@cð0; tÞþ mcð0; tÞ ¼ mgðtÞ
D
@x

ð3bÞ

@cðL0 ; tÞ
¼0
@x

ð4aÞ

ð8dÞ

@
@t

Z t

Uðx; t  s; sÞdsð9Þ

0The proof of the extended Duhamel’s theorem for the advection–
dispersion equation is not shown here, but can be easily obtainedby applying Laplace transforms to Eqs. (1–4) and (5–8) and by considering the deﬁnition of the generalized convolution as deﬁned by
Bartels and Churchill [21]. Because the initial condition (Eq. 6) iszero, one obtains

cðx; tÞ ¼

Z t0@
Uðx; t  s; sÞ ds
@t

ð10Þ

We now adopt the following substitution:or@cð1; tÞ
¼0
@x

R

Duhamel’s theorem relates c(x, t) of Eqs. (1–4) to the solution U(x, t;s) of Eqs. (5–8) by means of the following integral expression:

2

LD

Let U(x, t; s) [M L3], be the auxiliary solution of the following
problem where s is a parameter (but not the time variable):

ð4bÞ

where c(x, t) is the dimensional concentration [M L3] as a functionof distance x [L] and time t [T], g(t) is some known time-dependentfunction representing the inlet concentration [M L3], R is a constant retardation coefﬁcient [], m is a constant average pore watervelocity [L T1], l is a constant ﬁrst-order decay constant [T1], andD is a constant dispersion coefﬁcient [L2 T1].
Eqs. (3a) and (3b) describe two possible formulations of the inlet boundary condition at x = 0, generally referred to as ﬁrst-type(or Dirichlet) and third-type (or Cauchy) boundary conditions,
respectively. Boundary condition (4a) for a ﬁnite transport domain
is often referred to as the Danckwerts outlet condition [18]. The
physical basis and applicability of these conditions are discussed
in great detail elsewhere [8,19,20].

Uðx; t; sÞ ¼ uðx; tÞgðsÞ

ð11Þ

in which u(x, t) is dimensionless [] and g(s) has the same dimension as before for the concentration [M L3]. Substituting Eq. (11)into Eqs. (5) through (8) leads toR@ uðx; tÞ¼ Luðx; tÞ
@t

ð12Þ

uðx; 0Þ ¼ 0

ð13Þuð0; tÞ ¼ 1

ð14aÞ

or

D

@ uð0; tÞþ muð0; tÞ ¼ m@x@ uðL0 ; tÞ
¼0
@x

ð14bÞ

ð15aÞ

or
3. Solutions using Duhamel’s theorem
Our solution follows the methodology outlined by Ozisik [16]for heat conduction problems. We ﬁrst give the general solution
for any arbitrary transient function of the inlet concentration,g(t), and then present the solutions for two special cases: an
instantaneous pulse described by the Dirac delta function (Section 3.2) and a ﬁnite pulse input function (Section 3.3).

@ uð1; tÞ
¼0
@x

ð15bÞ

whereas c(x, t) given by Eq. (10) becomes:

cðx; tÞ ¼

Z t
0

gðsÞ

@
uðx; t  sÞds
@t

ð16Þ

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J.S. Pérez Guerrero et al. / Chemical Engineering Journal 221 (2013) 487–491

By noting that @t@ uðx; t  sÞ ¼  @@s uðx; t  sÞ, the last equation also
can alternatively be written as:

Z t@uðx; t  sÞds
@
s
0
Z t
dgðsÞ
¼ gð0Þuðx; tÞ þuðx; t  sÞdsds0cðx; tÞ ¼ 

The auxiliary solution u(x, t) is given by Case C6 of van Genuchten and Alves [23]. Differentiation versus time gives (e.g., Skaggs
and Leij [20]; Table 6.1–3):

gðsÞ

ð17Þ

3.2. Special case: instantaneous pulse
Assume that g(t) is an instantaneous pulse of the form g(t) = m0d(t), where m0 [M L3 T] is the amount of injected mass over a certain cross-sectional area divided by the volumetric water ﬂux
through that same area, and d(t) is the generalized Dirac delta
function [T1]. Eq. (16) then reduces tocðx; tÞ ¼ m0

Z t
0

dðsÞ

@
uðx; t  sÞds
@t

ð18Þ

Using the property [22]:Z

4.2. Instantaneous pulse, third type inlet BC; semi-ﬁnite domainhiðt mRxÞ2lt
cðx; tÞ m exp  4DRt  R
¼
pﬃﬃﬃﬃ
C0
pðDRtÞ1=2"#
m2 expðmDx  lRtÞtm þ Rxerfc

2DR2ðDRtÞ1=2

ð24Þ

4.3. Instantaneous pulse, ﬁrst-type inlet BC; ﬁnite domain
The auxiliary solution u(x, t) is given by Case C7 of van Genuchten and Alves [23], which if differentiated versus time leads immediately to"#!
1cðx; tÞ X
m2 b2m D
mx lt m2 t b2m Dt
 
 2
¼
E12 ðbm ; xÞþexp
C0
2D R 4DR
4DR L20 R
L0 R
m¼1

ð25aÞ
dðn  rÞFðnÞdn ¼ FðrÞð19Þ
E12 ðbm ; xÞ ¼

Eq. (18) simpliﬁes tocðx; tÞ ¼ m0

@ uðx; tÞ
@t



ð20Þ



2bm sin bLm0x
 L 2 mL0þ 2D0
b2m þ m2D

in which the eigenvalues bm are roots of the following transcendental equation:

mL0

3.3. Special case: Finite pulse

bm cotðbm Þ þ

In this case, the function g(t) is deﬁned as a ﬁnite pulse of theform

4.4. Instantaneous pulse, third type inlet BC; ﬁnite domaingðtÞ ¼

f ðtÞ 0 < t 6 t00

t > t0

ð21Þ

Using Eqs. (17) and (21), and separating the integral at the discontinuity (t = t0) into two parts leads to(cðx; tÞ ¼

Rt
f ð0Þuðx; tÞ þ 0 uðx; t  sÞ dfdðssÞ ds
t 6 t0
R t0
df ðsÞ
f ð0Þuðx; tÞ  f ðt 0 Þuðx; t  t 0 Þ þ 0 uðx; t  sÞ ds ds t > t 0hi2ðt mRxÞ2lt
cðx; tÞ DR x exp  4DRt  R
¼
pﬃﬃﬃﬃ
C0
2 pðDRtÞ3=2

ð23Þ

ð26Þ

ð27aÞ
 
 i
L0
bm cos bLm0x þ m2D
sin bLm0xh L 2 mL ih 2 mL 2 i
0b2m þ m2D
þ D0 bm þ 2D0

2mL0
bm
D

4. Speciﬁc solutions for the instantaneous pulse

The auxiliary solution u(x, t) is given by Case C5 of van Genuchten and Alves [23]. Differentiation of that solution with respect totime as indicated by Eq. (20) leads to the well-known solution (e.g.,
Skaggs and Leij [20]; Table 6.1–3):

¼0

!
!
1
cðx; tÞ X
m2 b2m D
mx lt m2 t b2m Dt
exp
 
 2
¼
E32 ðbm ; xÞþC0
2D R 4DR
4DR L20 R
L0 R
m¼1

E32 ðbm ; xÞ ¼

4.1. Instantaneous pulse, ﬁrst type inlet BC; semi-inﬁnite domain

2D

The auxiliary solution u(x, t) is now given by Case C8 of van
Genuchten and Alves [23]. Differentiation versus time of that solution gives

ð22Þ

In the following we give solutions for the case of an instantaneous pulse for a solute transport scenario described by Eqs. (1–4). The expressions were developed using Eq. (20), together withequations compiled by van Genuchten and Alves [23] serving as
the auxiliary solution, u(x, t), of Eqs. (12–15) for the various typesof boundary conditions (BCs). The solutions are relatively routine,
but shown here to illustrate the implementation of Duhamel’s theorem. More involved examples are given in Section 5.

ð25bÞ

h

ð27bÞ

in which the eigenvalues bm are obtained from:bm cotðbm Þ 

b2m D mL0þ¼0
mL0 4D

ð28Þ

5. Example applications
The solutions listed in the previous section are relatively standard and could be obtained easily without application of Duhamel’s
theorem. Here we show two additional solutions implemented
within a more general Duhamel framework, and compare them withpreviously developed exact solutions.
5.1. Example 1: Exponential inlet distribution; decay
We solve the transport problem for a semi-inﬁnite medium
subject to a third-type inlet boundary condition. The governing
equations are given by Eqs. (1) and (2) subject to Eqs. (3b) and(4b), along with the following data set: g(t) = Ca + Cb exp(kt),
R = 1, l = 0.3 day1, D = 0.7 m2/day, v = 0.3 m/day, Ca = 1 kg/m3,

490

J.S. Pérez Guerrero et al. / Chemical Engineering Journal 221 (2013) 487–491

Table 1
Concentration values c(x, t) at t = 0.1 and t = 1.0 day (example 1).
xc(x, 0.1) Eq. (16)c(x, 0.1) Eq. (17)c(x, 0.1) C14, Eq. (29)c(x, 1) Eq. (16)c(x, 1) Eq. (17)c(x, 1) C14, Eq. (29)

0
1
2
3
4
5
6
7
8
9
10

0.345747–4.14732E-11 i
0.00129972
1.11677E-8
1.14162E-16
1.12897E-27
9.93261E-42
7.46131E-59
4.67686E-79
2.41191E-102
1.01396E-128
3.45271E-158

0.345747–2.95659E-28 i
0.00129972
1.11677E-8
1.14162E-16
1.12897E-27
9.93261E-42
7.46131E-59
4.67686E-79
2.41191E-102
1.01396E-128
3.45271E-158

0.345747
0.00129972
1.11677E-8
1.14162E-16
1.12897E-27
9.93261E-42
7.46131E-59
4.67686E-79
2.41191E-102
1.01396E-128
3.45271E-158

0.636578
0.239872
0.0533083
0.00658916
0.000436546
1.51316E-05
2.69961E-7
2.45109E-9
1.12344E-11
2.58411E-14
2.96977E-17

0.636578
0.239872
0.0533083
0.00658916
0.000436546
1.51316E-05
2.69961E-7
2.45109E-9
1.12344E-11
2.58411E-14
2.96977E-17

0.636578
0.239872
0.0533083
0.00658916
0.000436546
0.0000151316
2.69961E-7
2.45109E-9
1.12344E-11
2.58411E-14
2.96977E-17ðm  wÞx
Rx  wt
erfc pﬃﬃﬃﬃﬃﬃﬃﬃ
exp
2D
wþm
2 DRt


m
ðm þ wÞx
Rx þ wt
erfc pﬃﬃﬃﬃﬃﬃﬃﬃ
exp

2D
wm
2 DRt

m2
m x ltRx þ mtþ
erfc pﬃﬃﬃﬃﬃﬃﬃﬃ
exp
DR2Dðl  kRÞ
2 DRt

Bðx; tÞ ¼ ekt



m

(
"
#
rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
1Rx  mtm2 t
ðRx  mtÞ2
erfc pﬃﬃﬃﬃﬃﬃﬃﬃ þ
exp 
2
pDR
4DRt
2 DRt



2
1
mx m tmx
Rx þ mt1þ þ
exp
erfc pﬃﬃﬃﬃﬃﬃﬃﬃ
2
D DR
D
2 DRt

ð31Þ

Eðx; tÞ ¼ ekt

ð32Þin which

Fig. 1. Outlet concentration for a ﬁnite domain (Eq. (25)) and semi-inﬁnite (Eq.
(23)) domain. Also, the concentration computed for a ﬁnite domain with thevelocity reversed (‘‘ﬂow against dispersion’’).
31

Cb = 2 kg/m , and k = 1 day . The analytical solution for this problem corresponds to the Case C14 in van Genuchten and Alves [23].
Eqs. (16) or (17) can be used together with auxiliary solution C8of van Genuchten and Alves [23] to obtain the desired solution.
Although closed-form expressions can be obtained using appropriate integral transform, Laplace transform or other techniques, we
obtain the results here by direct numerical integration. A large
number of numerical integration routines could be used for thispurpose. In our study we used the very ﬂexible NIntegrate subroutine from the Mathematica software package [24]. This numericalternative for the integral is appropriate when the exact analyticalsolution cannot be derived, or when the resulting closed-form
expressions are very difﬁcult to evaluate (e.g., by containing integral themselves).
Table 1 shows that results obtained with Eqs. (16) and (17) are
the same as the closed form solution given by Case C14 van
Genuchten and Alves [23]. For completeness we restate that solution here:cðx; tÞ ¼



C a Aðx; tÞ þ C b Bðx; tÞC o Aðx; tÞ þ C b Eðx; tÞl – kRl ¼ kR

u¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
m2 þ 4lD

ð33Þ

w¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
m2 þ 4Dðl  kRÞ

ð34Þ

5.2. Example 2: instantaneous pulse; ﬂow in the negative x direction(m < 0)Recently Ziskind et al. [25] obtained an analytical solution of theadvection–dispersion equation with a decay term for a ﬁnite onedimensional domain with an instantaneous pulse boundary condition modeled using the generalized Dirac delta function. They
speciﬁed the following initial conditions and boundary conditions:

ð29Þ

where

Aðx; tÞ ¼ðm  uÞx
Rx  ut
erfc pﬃﬃﬃﬃﬃﬃﬃﬃ
2D
uþm
2 DRt


m
ðm þ uÞx
Rx þ ut
erfc pﬃﬃﬃﬃﬃﬃﬃﬃ
exp

2D
um
2 DRt

m2
m x ltRx þ mterfc pﬃﬃﬃﬃﬃﬃﬃﬃ
exp
þ
D
R
2l D
2 DRt

m

exp



ð30Þ

Fig. 2. Re-analysis of transport data presented in Fig. 5 of Ziskind et al. [25]. The
model calculations and parameter estimates in [25] were obtained with the velocity
reversed (dashed lines). The solid lines show the model ﬁt obtained with thevelocity speciﬁed in the correct direction.

J.S. Pérez Guerrero et al. / Chemical Engineering Journal 221 (2013) 487–491

491

cðx; 0Þ ¼ 0

ð35Þ

Acknowledgments@cðx ¼ 0; tÞ
¼0
@ x

ð36Þ

The authors acknowledge partial support by the Brazilian National Research Council, CNPq, as well as by CAPES-Brazil.

cðx ¼ l ¼ L0 ; tÞ ¼ m0 dðtÞð37Þ

Referencesin which x is represents the longitudinal variable used by [25]. The
spatial coordinates x and x (the latter being the longitudinal position used in our study) are related by x ¼ L0  x. With this relationship the boundary conditions given by Eqs. (36) and (37) areequivalent to Eqs. (4a) and (3a), respectively.
The analytical solution of the above problem is given Eqs. (25)and (26) in Section 4.3, where the velocity values must be considered negative in the equations. It is easily veriﬁed that our analytical solution is the same as the solution obtained by [25]. However,
the index of summation in our Eq. (25) starts with ‘‘m = 1’’ and notwith ‘‘m = 0’’ as speciﬁed in Eq. (21) of Ziskind et al. [25].
Numerical evaluations were made for the following data set
used in [25]: R = 1, l = 3  104 s1, D = 7  106 m2/s, v = 3 105 m/s, L0 = 0.23 m, m0 = MT/Q with MT = 8  1013 kg, andQ = 8.57  108 m3/s. Fig. 1 compares the solutions for both a ﬁnite
domain (Eq. 25) and a semi-inﬁnite-domain (Eq. (23)). The numberof summed terms used in Eq. (25) was N = 20. Some results presented in [25] were apparently computed, whether intentionallyor not, with the velocity reversed, i.e. v < 0 in our formulation orv > 0 in [25]. Such an arrangement, with ﬂow directed toward thecontaminant source boundary, has been termed ‘‘ﬂow against dispersion’’ [26] in the hydrology literature. Fig. 1 also shows results obtained for Eq. (25) with the velocity reversed (v = 3  105 m/s),
along with a few points digitized from a curve presented in Fig. 2of [25]. Results obtained using the solution developed by [25] were
exactly the same as obtained with our Eq. (25), provided the summation is assumed to start with ‘‘m = 1’’.
Fig. 2 presents a re-analysis of bacteriophage transport datafrom [27] which were analyzed in Fig. 5 of [25]. The dashed lines
are a reproduction of the modeling results presented in Fig. 5 of[25]. The dashed lines were computed using Eq. (25) with the Dand l values estimated in [25] (given in Fig. 2), the velocity reversed, and the other model parameter values the same as given
above for Fig. 1. The solid lines show the optimal model ﬁt withvelocity speciﬁed in the correct direction, where the indicated D
and l values were obtained from a new least-squares ﬁt of Eq.
(25) to the data. Fig. 2 shows that the obtained ﬁtted model parameters and model agreement with the data differ considerably fromthose reported in [25]. The model ﬁt for pH = 8 is particularly good.
6. Conclusion
Applications of the Duhamel Theorem were extended to advection–dispersion type solute transport problems in porous media
when the boundary conditions are time-dependent. Solutions wereobtained from solutions of corresponding non-time dependent
auxiliary solutions using very generalized formulas. In this paper
we illustrated the approach for relatively standard cases withknown exact solutions. In particular, we emphasized cases of
instantaneous and ﬁnite pulse applications to transport problems
in ﬁnite and semi-inﬁnite domains. The developed generalized formulas obtained duplicated several speciﬁc solutions available in
the literature. In this paper we showed applicability to relatively
simple one-dimensional equilibrium transport problems, including
decay. In future work we intend to show that the general Duhamel
approach is equally applicable to more complicated scenarios oftransport in multi-dimensional media of solutes subject to nonequilibrium sorption or sequential decay chain reactions.

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