Springer-Verlag 2004
Stochastic Environmental Resea (2004) 18: 198 – 204
DOI 10.1007/s00477-003-0174-0
ORIGINAL PAPER
D. C.-F. Shih
Uncertainty analysis: one-dimensional radioactive nuclide
transport in a single fractured media
Abstract Uncertainty analysis of radioactive nuclide
transport for one-dimensional single fracture has been
studied. First order differential analysis is applied to
introduce analytical form of output expectation and
variance for contaminant transport equation, by
regarding uncertainty of dispersion coefficient and
retardation factor. Breakthrough curve of dimensionless
concentration is demonstrated by taking I-129 as
radioactive nuclide in fracture transport. It is possible to
pick up critical ranges in spatial and temporal domain
from the output variance. From the viewpoint of preliminary performance assessment for nuclear waste disposal the parameter importance in such system can be
substantially measured in the site characterization in
future.
Keywords Uncertainty analysis Æ Differential
analysis Æ Radioactive nuclide Æ Fracture Æ I-129
Introduction
For the radioactive waste geologic repository built in
hard rock, radioactive nuclide transport in fractures may
play an important role in the process of performance
assessment for the disposal site. For some critical
radioactive nuclides, time scale for performance assessment may be from a few hundred years up to millions of
year. Hard rock formation may be crushed into fracture
due to tectonic activities in an unexpected time period.
Therefore, the analysis of radioactive nuclide transport
D. C.-F. Shih
Ph.D., Civil Engineering
Institute of Nuclear Energy Research,
AEC, Taiwan Lungtan P.O. Box 3–7
Taiwan, Republic of China
Tel.: +886-2-82317717 ext. 5750
Fax: +886-3-4711411
E-mail: [email protected]
in fracture is one of the key studies in performance
assessment of waste isolation.
Many studies on solute transport in fractured media
have been attempted. These include analysis of transport
in a single fracture in which effects of the transport in the
fractures as well as interactions with or without porous
matrix are considered (e.g. Neretneiks 1980; Grisak and
Pickens 1980, 1981; Tang, Frind and Sudicky 1981;
Barker 1982; Sudicky and Frind, 1982, 1984; Berkowitz
et al. 1988; Shih et al. 2002).
A mean value, or expected value, is a center value of
parameter. Variance is a measure of deviation from its
mean value. One can detect variance to identify uncertainty of model or parameter. This study conduct differential analysis to derive the analytical variance of
dimensionless concentration of one-dimensional transport of single species radioactive nuclide with decay term
for pulse input source. A number of literature is available on analytical and numerical methods for uncertainty modeling (Soong 1973; Sabelfeld 1991; Romero
1998; Romero and Bankston 1998). The largest effort in
a differential analysis is usually the determination of the
partial derivatives appearing in model equations. A
number of specialized methods have been developed to
facilitate the calculation of these derivatives. The adjoint
techniques (Cacuci 1981) and Green function techniques
(Dougherty and Rabitz, 1979) can provide significant
computational savings in the calculation of partial
derivatives. For the performance assessment for radioactive waste disposal, site characterization will provide
supporting data in the analysis of contaminant transport
in fracture. Constant input of parameter will not be the
case in the most waste isolation research. For this stage,
simple and fast performance assessment and uncertainty
analysis needs to be adopted to screen out unsuitable
site. Rock matrix diffusion may be ignored in this
assessment phase (Shih et al. 2002). In this research, the
variance of model output is analytically derived by differential analysis and I-129 is then performed to demonstrate the uncertainty of contaminant transport in a
one-dimensional fracture system.
199
System definition
EðyÞ ffi yðp0 Þ þ
It assumes that the system model under consideration
can be represented by a function of the form
y ¼ f ðp1 ; p2 ; . . . ; pn Þ ¼ f ðpÞ
n
X
of ðp0 Þ
Eðpj pj0 Þ ¼ yðp0 Þ
opj
j¼1
ð8Þ
and
ð1Þ
n n X
n X
X
of ðp0 Þ 2
of ðp0 Þ
V ðpj Þ þ 2
opj
opj
j¼1
j¼1 k¼jþ1
of ðp0 Þ
Covðpj ; pk Þ
opk
where y: system prediction; p1, p2,..., pn : system input
parameters.
For a real world that mathematical model can be
evaluated, for example, contaminant transport
V ðyÞ ffi
c ¼ cðt; x; y; z; pÞ
where E, V and Cov denote expected value, variance and
covariance.
Expected value is represented as a center value for a
group of parameter while variance is a measure of
deviation from its center. In the nature, parameter variance can be propagated to system output from Eq. (9).
The resultant variance of output response is dependent
on the degree of variance and their covariance of the
system inputs and the system input parameters.
ð2Þ
where t: temporal variable; x, y, z: spatial variable; p:
[p1, p2,..., pn], system parameters.
The technique of most uncertainty analysis procedure
is independent of the form of the system describe in Eq.
(1) or (2). However, the implementation of the differential analysis is closely tied to the specific form in such
equations, especially for its differential form.
Theory
ð9Þ
Solution of contaminant transport
It assumes that basic statistical model of system
parameters are constituted by its expected value and
variance. The expected value, or mean estimator can be
denoted as
m
X
Eðpj Þ ¼ m1
pj i
ð3Þ
i¼1
Variance and covariance estimator are
m
X
ðpj i Eðpj ÞÞ2
V ðpj Þ ¼ ðm 1Þ1
ð4Þ
i¼1
and
Covðpj ; pk Þ ¼ ðm 1Þ1
m
X
ðpj Eðpj ÞÞðpk Eðpk ÞÞ :
The one-dimensional transport of a single species
radioactive nuclide with decay term in a single fracture
(Fig. 1) can be described by (Bear 1972; Tang et al.
1981):
oc v oc D o2 c
þ
þ kc ¼ 0; for 0 z 1;
ð10Þ
ot R oz R oz2
where c : concentration (M/L3); t : time (T); z : onedimensional coordinate of fracture (L); v : groundwater
velocity (L/T); R : retardation factor; D : dispersion
coefficient(L2/T); k : radioactive decay constant (T )1), k
= ln (2)/t1/2 ; t1/2 : half life of the radioactive nuclide (T).
For a single pulse source input (Fig. 2), the initial and
boundary condition are
i¼1
ð5Þ
where m: sample size.
Differential analysis is based on using Taylor series to
approximate the system model under consideration. The
base values, ranges and distribution are selected for the
input variables pj, j = 1, 2,..., n. The base value can be
represented by the vector
p0 ¼ ½p10 ; p20 ; . . . ; Pn0 ð6Þ
The Taylor series approximation to y is developed by
using restriction of first-order terms, the approximation
has the form
n
X
of ðp0 Þ
yðpÞ ffi yðp0 Þ þ
ðpj pj0 Þ
ð7Þ
opj
j¼1
In essence, uncertainty analysis is to evaluate variance
propagation of input parameters to system response.
The expected value and variance of y can be estimated
by
Fig. 1 Fracture system (After Shih et al., 2002)
200
What has been termed the retardation factor, R, is given
(Fetter 1999, p.123) by
Bd
Kd
ð18Þ
h
where Bd: bulk density (M/L3); h: volume moisture content or porosity; Kd: distribution coefficient
(L/M).
R¼1þ
Uncertainty analysis
Fig. 2 Single pulse input source
cðz; 0Þ ¼ 0
ð11Þ
and
cðz; tÞ ¼ c0 ðuðt aÞ uðt bÞÞ;
z ¼ 0 and a t b
ð12Þ
where uðt aÞ and uðt bÞ are unit step function at
t ¼ a and t ¼ b, respectively.
By using Laplace transform and its inverter, the
solution of dimensionless concentration is obtained
(Shih et al. 2002),
"
#
Zb
vbz
cðz; tÞ
ðvbzÞ2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp
¼ expðvzÞ
c0
4ðt uÞ
3
a 2 pðt uÞ
exp ðk þ b2 Þðt uÞ du ;
ð13Þ
where
v ¼ v=2D ;
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
b ¼ 4DR=v2 ;
ð14Þ
ð15Þ
Expanding Eq. (13) by Eq. (14) and (15). Alternate form
of Eq. (13) is
qffiffiffi
R 2 R
Zb
Dz
cðz; tÞ
v
Dz
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp
z
¼ exp
c0
2D
4ðt uÞ
3
a 2 pðt uÞ
v2
ðt uÞ du ;
ð16Þ
exp k þ
4DR
where dimensionless concentration is dependent on v, D,
R, t, z, a, b and k.
The dispersion coefficient can be calculated from
equation (Fetter 1999, p.54)
D ¼ va þ D ;
ð17Þ
where aL is groundwater dispersivity in fracture(L), and
D* represents molecular diffusion coefficient (L2/T).
Bd
R ¼ 1 þ Kd
ð18Þ
h
It is assumed that dispersion coefficient and retardation factor have uncertainty due to its measurement
from the site characterization. First, rewrite Eq. (16)
as
qffiffiffi
R
Zb
cðz; tÞ
v
Dz
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
f ðpÞ ¼
¼ expð zÞ
c0
2D
3
a 2 pðt uÞ
R 2 Dz
v2
exp k þ
ðt uÞ du ð19Þ
;
exp
4ðt uÞ
4DR
where p = [R, D].
Accordingly, Eq. (8) and (19), we obtain
qffiffiffiffiffiffiffiffi
" EðRÞ 2 #
EðRÞ
Zb
EðDÞ z
c
v
EðDÞz
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp
E
z
¼ exp
c0
2EðDÞ
4ðt uÞ
3
a 2 pðt uÞ
v2
ðt uÞ du ; ð20Þ
exp k þ
4EðDÞEðRÞ
This is an exact form of deterministic analysis, if we have
constant expect value of D and R.
By using differential rule and Leibnitz’s rule (Spiegel,
1968), output variance from Eq. (9) and (19) can be
derived,
c
V
¼ n2 V ðDÞ þ 2ngCovðD; RÞ þ g2 V ðRÞ
ð21Þ
c0
where
qffiffiffi
R 2 R
Zb
Dz
v
vz
Dz
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp
n ¼ 2 exp
pffiffiffi
2D
2D
4
ðt u Þ
3
a 2 p ðt uÞ
vz v2
ðt uÞ du þ exp
exp k þ
2D
4DR
q
ffiffi
ffi
R
Zb " pffiffiffi 1 3
Rð 2 D 2 Þz
Dz
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
þ
pffiffiffi
pffiffiffi
3
2
p
2
p
ðt
uÞ
ðt uÞ3
a
#
Rz2
v2 ðt uÞ
þ
4D2 R
4D2 ðt uÞ
R 2 Dz
v2
exp k þ
ðt uÞ du ð22Þ
exp
4ðt uÞ
4DR
201
and
qffiffiffi
R
2p1ffiffiffiffiffi
z
Dz
RD
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
g ¼ exp
pffiffiffi
pffiffiffi
2D
2 p ðt uÞ3 2 p ðt uÞ3
a
#
z2
v2 ðt uÞ
þ
2
4DR2
4D ðt uÞ
R 2 Dz
v2
exp ðk þ
Þðt uÞ du : ð23Þ
exp
4ðt uÞ
4DR
vz Zb "
The output variance is then dependent on input variance
and system parameters.
Analysis and discussion
In this study, we choose Iodine 129 (I-129) as a single
radioactive nuclide member. It is a fission production
with 0.150 MeV for beta and 0.04 MeV for gamma
emissions (Shirley et al. 1996; Stewart 1985). Considering granite as potential host rock, it is regarded as critical nuclide due to its higher solubility and lower
absorption, and long half-life (1.57 · 107 y). For SKB
WP-Cave project, Moreno et al. (1989, pp. 16–17)
reported that BdKd of I-129 for granitic rock could be
taken as 0.01. For an open fracture, we take h = 1 by
according to EPRI’s choice (Zou and Apted 1996).
Groundwater velocity is chosen as 10)8 (m/s) for the
intermediate order for fracture in SFR, Finnsjon and
Stripa site (Pusch 1990, pp. 28–30). Molecular diffusion
coefficient and dispersivity are selected as 5 · 10)14 m2/s
and 1.0 m, respectively (Moreno et al. 1989). Summarized data are shown in Table 1. Figures 3, 4 demonstrate the mean value of dimensionless concentration
distribution and its contour in such case. It shows that
the higher mean values, e.g. 10)1, are presented in the
Fig. 3 Breakthrough curve of dimensionless concentration for I-129
Table 1 Data used for uncertainty analysis of I-129 contaminant transport
Item or parameter
Content or value
Input source type
Start time
of input source, a
End time of input
source, b
Groundwater
velocity, v
Molecular diffusion
coefficient, D*
Dispersivity, aL
Dispersion coefficient, D
Bulk density · distribution
coefficient, BdKd
Porosity in opened fracture, h
Retardation factor, R
Input variance for D
Input variance for R
Input covariance for D and R
Pulse
0 (y)
3 (y)
1.0 · 10)8(m/s); 3.15 ·
10)1 (m/y)
5.0 · 10)10 (cm2/s); 1.58
· 10)6 (m2/y)
1 (m)
3.15 · 10)1 (m2/y)
0.01
1.0
1.01
0.01 or 0.0
0.01 or 0.0
0.01 or 0.0
Fig. 4 Contour of dimensionless concentrations for I-129
early time for z <10 m. The distinguish distribution for
dimensionless concentration (c/c0) less than 10)2 is
demonstrated before 300 y. A distinctive trend for E(c/
c0) <10)2 are distributed from 30–100 yr at z =20 m to
250–350 y at z =100 m. The variance of dimensionless
concentration decrease sharply from 10)3 to 10)20 before
300 y for z <10 m (Fig. 5). For the case with D = 10)2
and R = 10)2 (case 1), at the position z = 50 and 100 m,
the profile of output variance are the same with the
mean value. Peak of variance is approximately near
10)4. For the case with covariance (case 2) or with
202
variance of single parameter (case 3 or 4) the output
variances before 60 y present distinctive fluctuations
(Figs. 6–8). The cases are well mapped in spatial and
temporal domain (Figs. 9–12). Again, the trend for E(c/
c0) <10)5 are distributed from 25–120 y at z = 20 m to
220–400 y at z = 100 m. It is the same with the mean
value of dimensionless concentration.
The advantage of differential analysis is that the
mean and variance of the system can be rationally
evaluated by the variance or covariance of parameter in
an analytical system. This study, I-129 is demonstrated
Fig. 7 Case 3: variance of dimensionless concentration for I-129
Fig. 5 Case 1: variance of dimensionless concentration for I-129
Fig. 8 Case 4: variance of dimensionless concentration for I-129
Fig. 6 Case 2: variance of dimensionless concentration for I-129
to explain the uncertainty of dimensionless concentration influenced by dispersion coefficient and retardation
factor for a one-dimensional fracture transport system.
From the viewpoint of hydrogeological process dispersion coefficient is somewhat related to the groundwater
velocity. In order to have a whole picture for understanding uncertainty propagation in the system the
study considers dispersion coefficient and retardation
factor as uncertainty inputs for the purpose of demonstration. The research will be improved to an application
of real work if more data from site characterization are
provided in the future.
203
Fig. 9 Case 1: variance contour of dimensionless concentration for
I-129
Fig. 10 Case 2: variance contour of dimensionless concentration for
I-129
Conclusion
Uncertainty analysis of radioactive nuclide transport
for one-dimensional single fracture has been studied
using differential analysis. Breakthrough curve of
dimensionless concentration of I-129 is demonstrated
using published parameter in the world. By regarding
uncertainty of dispersion coefficient and retardation
Fig. 11 Case 3: variance contour of dimensionless concentration for
I-129
Fig. 12 Case 4: variance contour of dimensionless concentration for
I-129
factor, first order differential analysis is conducted to
derive analytical form of output expected value and
variance for contaminant transport equation. As illustration case, maximum output variance of dimensionless concentration will be confined under 10)3. From
the distribution of output variance, we can pick up
critical ranges or points in spatial and temporal domain. Uncertainty of model output can be well demonstrated and mapped from input parameter. The
204
study is helpful to demonstrate uncertainty assessment
for performance assessment of radioactive nuclide
transport in a one-dimensional fracture system.
Acknowledgments This research is supported by the Institute of
Nuclear Energy Research, AEC, TAIWAN, Republic of China.
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