Table Of ContentPREFACE
Water resources are of central importance to humankind. As the world population grows
and increasing incidences of pollution of surface and subsurface water resources are dis-
covered, prudent management and protection of these resources becomes increasingly
important. Natural systems are extraordinarily complicated, and the understanding and
management of these resources cannot be accomplished effectively using heuristic means.
In response to this problem, mathematical modeling has played a critical role in the in-
vestigation and management of water resources systems for the last three decades. The
reliance on such models continues to increase at a rapid rate, and the sophistication and re-
liability of such models is also increasing. Researchers from around the world are engaged
in the evolution of all aspects of modeling water resources systems, and these combined
efforts serve all of society.
A visionary international forum was established at Princeton in 1976 to bring together re-
searchers involved with computational methods in water resources, to foster interactions,
and to catalyze new research. This initial Finite Elements in Water Resources meeting
started a tradition of biennial meetings that have rotated between North America and
Europe ever since. As the field has evolved, so too has the name of this conference-which
is currently Computational Methods in Water Resources-reflecting a breadth of compu-
tational approaches currently being developed and applied for such problems. Confer-
ences have been hosted at" I. Princeton (1976), II. Imperial College London (1978), III.
University of Mississippi (1980), IV. University of Hanover (1982), V. University of Ver-
mont (1984), VI. Laboratorio National de Engenharia Civil of Portugal (1986), VII. Mas-
sachusetts Institute of Technology (1988), VIII. Giorgio Cini Foundation of Italy (1990),
IX. University of Colorado at Denver (1992), X. University of Heidelberg (1994), XI.
Mexican Institute of Water Technology (1996), XII. the Institute of Chemical Engineer-
ing & High Temperature Chemical Processes-Foundation for Research and Technology,
Greece (1998), XIII. The Faculty of Science of the University of Calgary, Canada (2000),
and XIV. Faculty of Civil Engineering & Geosciences at Delft University of Technology
(2002).
The XV International Conference on Computational Methods in Water Resources (CMWR
XV) was held in Chapel Hill, North Carolina, 13-17 June 2004. The conference was spon-
sored by the Department of Environmental Sciences and Engineering, School of Public
Health, The University of North Carolina at Chapel Hill. This two-volume set represents
the reviewed and edited proceedings of this meeting, including 651 papers. In addition,
many posters were presented at the meeting, which are not included in this formal written
record.
These collective works include contributions by many of the leading water resources re-
search groups from around the world. Broad in scope, these papers address numerous
aspects of water resources systems, ranging from the microscale to the field scale and
from the very fundamental to the most compelling and important of applications. Virtu-
ally all major classes of numerical methods for water resources problems are represented in
these proceedings, from the evolution of traditional approaches to the latest in methods of
recent invention. As has been traditional at past CMWR meetings, subsurface hydrology,
land surface hydrology, and surface water hydrology are well represented.
vi
The Organizing Committee acknowledges gratefully the participation of our distinguished
plenary and invited speakers: Shiyi Chen, George Christakos, Olaf A. Cirpka, Clint N.
Dawson, Gedeon Dagan, C.T. Kelley, Daniel R. Lynch, Dennis McLaughlin, Mario Putti,
and Thomas F. Russell. In addition we thank several people that assisted with the
organization of special sessions: Mark Bakker, Kathleen Fowler, Mohamed .S Ghidaoui,
Markus Hilpert, Dong-Sheng Jeng, Ling Li, and Alex .S Mayer.
We are especially grateful to Randall Goodman, Rebecca R. Lloyd, Lucinda .S Thompson,
and Christopher J. Windolph for their many contributions to this meeting, which required
many months of their time and considerable talents. In addition, Chandra Abhishek, De-
ona .N Johnson, Huina Li, James E. McClure, Chongxun Pan, Joseph A. Pedit, and Carl
P. Rupert, all members of Professor Miller's research group, made valuable contributions
to these proceedings and provided local technical assistance associated with the meeting.
Cass T. Miller
Matthew W. Farthing
vii
Editors
Cass T. Miller
Matthew W. Farthing
William .G Gray
George F. Pinder
Organizing Committee
Clint .N Dawson, University of Texas
Matthew W. Farthing, University of North Carolina
William .G Gray, University of North Carolina
Rainer Helmig, University of Stuttgart
Cass T. Miller, University of North Carolina
Marc .B Parlange, Johns Hopkins University
George F. Pinder, University of Vermont
Francisco E. Werner, University of North Carolina
953
Characterization and sensitivity analysis of tracer breakthrough curves
T. Vogel ,a D. Bachmann ,a and J. KSngeter a
aInstitute of Hydraulic Engineering and Water Resources Management,
Aachen University, Mies-van-der-Rohe-Str. ,1 52056 Aachen, Germany
The objective of the proposed paper is to present a new approach to characterize tracer
breakthrough curves (BTCs) and to perform sensitivity analysis with respect to these
characteristics. This approach is motivated by the ongoing research in the framework of
the Aquifer Analogue Project where multi-scale field, laboratory and numerical experi-
ments are performed by four German partner universities (Karlsruhe, Tiibingen, Stuttgart
and Aachen) for a large range of length scales to provide a very detailed description of a
fractured permeable sandstone block (10 z 7 z 2 rn )3 in southern Germany.
1. INTRODUCTION
Within the framework of the Aquifer Analogue Project experiments and numerical
modeling have been performed since the mid-nineties in order to describe flow and trans-
port processes in fractured permeable sandstone. These investigations by four German
partner universities (Karlsruhe, Tiibingen, Stuttgart, and Aachen) on the laboratory and
field block scale are described in detail in .4
1.1. Motivation
Even though a lot of experiences and insights have been obtained, some open questions
remain unsettled. Among these open questions, two main aspects are motivating the
current research:
- Which are the most relevant processes and which processes may be negligible?
- Which sensitivities do measurements and models show?
Berkowitz 1 also points out that there is still a lack of knowledge concerning the
interpretation and extrapolation of information obtained from pump and tracer tests.
1.2. Outline
)ROFIDA(
Automatic Differentiation has been used to obtain a differentiated code of
the multi-continuum model STRAFE. By means of this model, characteristic parameters
describing the shape of tracer breakthrough curves such as gradients or arrival times
may be analyzed with respect to their dependency on certain input parameters (e.g.,
hydraulic conductivity, porosity, exchange parameters of the multi-c~ model).
This knowledge of sensitivities provides for a better understanding of transport processes
and may be used to optimize sampling design and numerical modeling (see Figure 1).
Sensitivity analysis shows, for example, that for the water-saturated case diffusion may
be neglected with respect to first arrival of tracer, whereas in the gas-saturated case
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Figure .1 Scheme of characterization and sensitivity analysis
diffusion is dominating compared to porosity and permeability.
The characteristics of tracer breakthrough curves and their sensitivities may then be
used as a decisive criteria to choose - if possible - the appropriate multi-continuum ap-
proach to be applied in order to interpret field block scale measurements.
2. CONCEPTUAL MODEL
2.1. Description of the conceptual model
Based on a conceptual one-dimensional model introduced by Jansen [6], a large number
of tracer breakthrough curves is calculated in order to develop an appropriate scheme to
interpret tracer breakthrough curves. The proposed model focuses on a simplification of
the sandstone block (near T/ibingen, Germany) with a transport distance of 10.0m.
The boundary conditions for the one-dimensional case study for identification of the
integral transport behavior are shown in Figure .2 The tracer infiltration is performed
locally at one point. At the beginning (t - to), the whole area is free of tracer and the
piezometric water level is 3.0m. The boundary conditions are held constant during the
simulation. A Dirichlet boundary condition is chosen for the transport simulation with
a relative concentration of 1.0 at the inlet. The flow field is stationary with a hydraulic
gradient of 0.5 %. The outlet is on the opposite side of the inlet.
2.2. Properties of the conceptualized aquifer
During the laboratory and field experiments three hydraulic active components could
be identified, a macro fracture system (fl), a micro fracture system (f2), and the host
matrix (m). The orientation of the fractures is assumed to be orthogonal, implying a
rectangular shape of the matrix blocks. The equivalent and exchange parameters used for
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Figure 2. Boundary conditions of the one-dimensional case study ]6[
the basic configuration of the multi continuum model are summarized in Tables 1 and 2.
For the basic configuration all components influence the regional transport behavior
of the system. Since this study is meant to be conceptual, homogeneous properties are
assumed. For the chosen transport parameters a high connectivity is assumed.
Table 1
Equivalent parameters for the basic configuration
Equivalent parameter macro micro matrix
(fl) (f2) (m)
Permeability k [m ]2 5.00.10 -12 5.00-10 -13 5.00-10
-14
Specific storativity ~S [l/m] 1.00.10 -~ 1.00.10 -~ 1.00.10 -~
Porosity 5 ]-[ 1.00.10 -03 5.00.10 -~ 1.00.10 -~
Dispersivities: Ira]
- longitudinal 16 5.00-10 .02 1.00.10 -01 2.00-10 -01
- transversal horizontal ht~ 5.00 10 (cid:12)9 .o3 1.00 10 (cid:12)9 .o2 2.00 10 (cid:12)9 .o2
- transversal vertical 6t~ 2.00. 10 .o3 5.00. 10 .o3 1.00. 10 .o2
Coefficient of
Diffusion lo~D [m2/s] 8.00.10 -10 6.00.10 -10 4.00.10 -10
Relative volume 2t ]-[ 1.00-10 00+ 1.00.10 00+ 9.94.10 -~
2.3. Variation of model parameters
2.3.1. single continuum model (SPSP)
For the single continuum model, the parameters of permeability and porosity are varied
from 10 % to 190 % of the values indicated in Table 1 for the three different types of
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Table 2
Exchange parameters for the basic configuration
Exchange parameter macro / macro / micro /
micro matrix matrix
flf2 flm f2m
Shape of blocks cube cube cube
Ira
Penetration depth xam8 5.00" 10 -01 5.00-10 E-~ 2.50-10 -01
Specific
- surface ~o l/m 6.00.10 00+ 6.00-10 00+ 1.20.10 +~
- fracture surface (~w 1/rn 2.00.10 +~176 2.00.10 +~176 4.00.10 +~176
Geometric factor c - 5/3 5/3 5/3
Exchange parameter:
- flow QZ5 Ira~s 8.33.10 .06 8.33.10 .07 8.33.10 .07
- transport c~( rn2/s 5.00. 10 -12 6.67. 10 -11 6.67. 10 -11
Interface function
A(s)/Ao - 1 - (2 - V/g,~a~)S + )~a~.s2-/1( S 2
continua: the macro fracture system, the micro fracture system, and the matrix. Thus a
total number of 111 single continuum realizations is achieved.
2.3.2. double continuum model (DPDP)
The double continuum model's parameters, permeability, and porosity are varied for
both continua in the same way as for the single continuum approach. Additionally, the
exchange parameters (specific surface ft0, specific fracture surface f~w, and the exchange
coefficient &c) are changed within a range from 01 % to 091 % of the values indicated in
Table .2 This results in a total number of 381 double continuum simulations.
2.3.3. triple continuum model (TPTP)
Three different types of couplings (serial, selective, parallel) are investigated with dif-
ferent model parameters for each continuum. The permeability and porosity of each
continuum are varied from 01 % to 190 % of the values indicated in Table .1 This results
in a total number of 327 triple continuum realizations.
3. CHARACTERIZATION OF TRACER BREAKTHROUGH CURVES
3.1. Introduction
In order to better distinguish the types of tracer breakthrough curves, the normed
concentration is not plotted versus time s but versus the logarithm of time loglo(s).
Additionally, the time of first tracer arrival tinit the point in time when a relative
concentration of 0.1% is detected at the output is subtracted in order to achieve a
better comparability between the curves. This process is illustrated for different tracer
breakthrough curves in Figure .3 This means that for the following investigations the
957
time of first tracer arrival is always a characteristic number.
Figure .3 Influence of time-axis on the presentation of tracer breakthrough curves
3.2. Characterization by key figures
Different points in time x~t as certain characteristic key figures are used to describe a
tracer breakthrough curve. This idea results from soil mechanics where grading curves
are described by means of the unconformity coefficient U and the curvature number C,
as shown in equations (1) and (2).
(1)
usoilmechanics __ D6o
Dlo
__ (2)
Dlo" o6D
These grading curves show similar shapes to a breakthrough curve. Transferred to
the tracer breakthrough curve , this means that the portion of mass corresponds to the
relative tracer breakthrough, whereas the grain size relates to time.
Similar key figures are chosen for the characterization of tracer breakthrough curves.
The shape of the lower part of the curve between tl and 05t is described by
t50
= (3)
tl
and
C3o m (4)
tl (cid:12)9 o5t
tl is chosen to describe the behavior at the beginning. The upper part of the tracer
breakthrough curve is described in a similar way with
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t99
U99= (5)
and 077( =
ts0" "99t (6)
The key figure U~ provides for information concerning the time spread, which is covered
by the tracer breakthrough curve during this period: the larger is for example 0sU the
larger is the time spread between the tracer breakthrough at t l and ts0. A number of one
would describe a vertical curve progression.
C~ characterizes the curvature of a tracer breakthrough curve. A large value of C~
points out a concave curvature of the tracer breakthrough curve whereas a small value
describes a convex shape. A definite declaration may only be made in consideration of
.0sU
In addition to the key figures mentioned above the characteristic number 7 proposed
by Lagendijk 7 is used. This number describes the relation of tracer breakthrough at
early and late points in time. Unlike described in 7, it is not referred to tl0 and 09t but
to t l and "99t
ts0 - tl
99t - tl" (7)
3.3. First results of characterization technique
The resulting tracer breakthrough curves from the investigations described in section 2
sum-up to 819. This large number of results is a solid basis to interpret and characterize
tracer breakthrough curves.
Preliminary results of the investigations of key figures are illustrated in Figure .4 The
key figure ~ in combination with 0sU allows for a very good distinction between different
model approaches. SPSP models as well as DPDP approaches may be separated by these
two key figures. For example a low value of 0sU separates tracer breakthrough curves of
models with a fast tracer breakthrough at the beginning such as SPSP models and double
continuum models (tim and f2m). Taking into account the key figure ,~- a more detailed
distinction between the approaches is possible. SPSP models may be characterized by a
value of ~- close to 0.5.
The plot of 99U versus 0sU (see Figure 4) allows to differentiate tracer breakthrough
curves with a large hydraulic contrast from tracer breakthrough curves with a minor
contrast. Therefore, the DPDP (tim) results may be separated from the others because
of their large values of 99U (higher than )51 and a low value of 0sU (less than ten).
4. SENSITIVITY ANALYSIS OF CHARACTERISTIC PARAMETERS
Within the context of this analysis, sensitivities describe the change of an output value
resulting from a variation of an input parameter. This knowledge of sensitivities may
allow for a better understanding of natural processes and improve analysis by numerical
models.
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Figure .4 Classification of tracer breakthrough curves by means of ~, [/50 and 99U
Different approaches to calculate sensitivities such as automatic differentiation, sym-
bolic differentiation, finite differences or the adjoint-state method are presented in litera-
ture (e.g., [5,8]).
For the presented investigations automatic differentiation is used to obtain sensitivities
of the characteristic key figures, as discussed in section .3
4.1. Automatic differentiation
Automatic differentiation is based on the fact that every function is executed on a com-
puter as a sequence of elementary operations. Applying the chain rule to these operations
over and over, the derivatives of a function are computed .]3[
An advantage of automatic differentiation is the exact calculation of sensitivities in-
dependent of the value of a variable as well as a relatively uncomplicated application to
numerical codes programmed in FORTRAN.
4.2. Implementation of sensitivity calculation of key figures
The numerical model STRAFE [2,6] is programmed with a one-dimensional world array.
This array contains all input data, intermediate results as well as model output results.
A pointer indicates the position where information is stored (see Figure 5). As input
variables ADIFOR requests for a definition of dependent (to be differentiated) and inde-
pendent variables. Additionally, the maximum number of independent variables has to
be assigned to pmax.
The structure of STRAFE with the one-dimensional world array causes all variables
(dependent and independent) to be found in one array. This would result in a multidi-
mensional field of the size z (cid:141) z. This field is the Jacobian matrix of the one-dimensional
world array:
