More than half of the Earth's population depends largely
or entirely on fractured or karst aquifers for their drinking water supply.
Both the characterization and modeling of these groundwater reservoirs are
therefore of worldwide concern. Artificial tracer testing is a widely used
method for the characterization of solute (including contaminant) transport
in groundwater. Tracer experiments consist of a two-step procedure: (1) introducing a conservative tracer-labeled solution into an aquifer, usually
through a sinkhole or a well, and (2) measuring the concentration
breakthrough curve (BTC) response(s) at one or several downstream monitoring
locations, usually spring(s) or pumping well(s). However, the modeling and
interpretation of tracer test responses can be a challenging task in some
cases, notably when the BTCs exhibit multiple local peaks and/or extensive
backward tailing. MFIT (Multi-Flow Inversion of Tracer
breakthrough curves) is a new open-source Windows-based computer package
for the analytical modeling of tracer BTCs. This software integrates four
transport models that are all capable of simulating single- or multiple-peak
and/or heavy-tailed BTCs. The four transport models are encapsulated in a
general multiflow modeling framework, which assumes that the spatial
heterogeneity of an aquifer can be approximated by a combination of
independent one-dimensional channels. Two of the MFIT transport models are
believed to be new, as they combine the multiflow approach and the
double-porosity concept, which is applied at the scale of the individual
channels. Another salient feature of MFIT is its compatibility and interface
with the advanced optimization tools of the PEST suite of programs. Hence,
MFIT is the first BTC fitting tool that allows for regularized inversion and
nonlinear analysis of the postcalibration uncertainty of model parameters.
Introduction
Artificial tracer testing is one of the most valuable methods for the
characterization of flow and solute transport in groundwater. Tracer
experiments consist of a two-step procedure: (1) introducing a known mass of
a tracer species into an aquifer, usually through a sinkhole or well, and (2) measuring the concentration breakthrough curve (BTC) response(s) at one or
several downstream monitoring locations, usually spring(s) or pumping
well(s). The analysis of a tracer BTC is best done by fitting a
model-computed time–concentration curve to the measured values. Although
spatially distributed numerical models (e.g., MODFLOW/MT3DMS or FEFLOW) can
be used for this purpose, simpler (i.e., spatially lumped) models are
generally used, at least in the early stages of tracer studies, either
because of time constraints or because of a lack of model input data. A
number of computer codes for BTC fitting have been developed in recent
decades: CATTI (Sauty et al., 1992), TRACI (Käss, 1998, 2004), OTIS
(Runkel, 1998), STANMOD (van Genuchten et al., 2012), TRAC (Gutierrez et
al., 2013), OM-MADE (Tinet et al., 2019), and OptSFDM (Gharasoo et al.,
2019). Note that STANMOD integrates a number of former codes, including the
widely used CXTFIT code developed by Parker and van Genuchten (1984) and
Toride et al. (1999). Despite the range of possibilities offered by these
programs, the fitting and interpretation of tracer BTCs remains a
challenging task in some cases, notably for BTCs exhibiting multiple local
peaks and extensive backward tailing. Such BTC shapes, which fall in the
general category of non-Fickian (or anomalous) transport (Berkowitz et al.,
2006; Neuman and Tartakovsky, 2009), are frequently observed in fractured
and karst aquifers (Tsang and Neretnieks, 1998; Streetly et al., 2002;
Massei et al., 2006; Loefgren et al., 2007; Goldscheider et al., 2008; Field
and Leij, 2012; Bertrand et al., 2015; Yang et al., 2019).
To the best of the author's knowledge, only the TRACI and OM-MADE programs
are able to simulate multimodal BTCs. Unfortunately, these two programs
suffer from some limitations both in terms of ease of use and with respect
to their modeling and calibration capabilities. For instance, the TRACI software
has not been maintained since 2004 and can only be used on physical or
virtual computers running Windows operating system versions from Windows 98
to Windows 7. Another drawback of TRACI is the inability of the inversion
(automated calibration) algorithm included in the software to handle
multimodal BTCs. Each local concentration peak must be sequentially fitted
through a manual (trial-and-error) calibration procedure. The OM-MADE
program was written as a Python script and has neither a graphical user
interface (GUI) nor inverse modeling functionality. The purpose of this
paper is to present a new open-source GUI-based software, named MFIT, that
aims to help in the interpretation of single- or multiple-peak and/or
heavy-tailed BTCs. MFIT stands for “Multi-Flow Inversion of Tracer
breakthrough curves”. The MFIT software integrates four transport models
that can be tested against field and laboratory tracer BTCs with the
assistance of the PEST automated calibration and uncertainty analysis
routines (Doherty, 2019a). In its current version, the scope of the software
is limited to tracer tests involving nonreactive tracer species and
performed in steady flow conditions. These assumptions are maintained
throughout the paper.
The remainder of this paper is organized as follows. Section 2 discusses the
possible origins of multiple peaks and long tails in tracer BTCs and
presents the conceptual and mathematical framework of the transport models
integrated in MFIT. The code implementation and coupling with PEST for
automated BTC fitting are discussed in Sect. 3. In Sect. 4, the accuracy
of MFIT-computed BTCs is verified against CXTFIT and TRACI simulations for
five test cases. An additional test is presented in Sect. 5 to assess the
reliability of a new multistart method that was specifically developed to
improve the automatic optimization of the model parameters. In Sect. 6, we
illustrate the use of the software by analyzing tracer BTCs obtained in the
karst aquifer of the Hydrogeological Experimental Site (HES) in Poitiers,
France. The summary and conclusions are presented in Sect. 7. A number of
acronyms, model abbreviations, and model parameters are employed throughout
this paper. Two glossaries, Tables A1 and A2, are provided in Appendix A for
easy reference.
Multimodal and heavy-tailed BTCs: causes and modeling
Under the already mentioned assumptions (nonreactive tracer, steady-state
flow), multimodal BTCs unequivocally indicate that a number of tracer-plume
splittings occurred somewhere between the injection site and the monitoring
point. Although injection artifacts may be involved in some cases (see Guvanasen and Guvanasen, 1987), tracer splitting most commonly
originates from the spreading (transverse dispersion) of the solute into
areas of contrasting flow velocities; see Moreno and Tsang (1991),
Siirila-Woodburn et al. (2015), and Boon et al. (2017). More precisely,
assuming a single-pulse tracer injection signal, multimodal BTCs reflect a
three-step process: (1) tracer spreading into different flowing or nonflowing
aquifer subdomains characterized by different transit or residence times, (2) tracer motion within each subdomain with little or no exchange between the
different subdomains, and (3) convergence (mixing) of the subtracer fluxes
somewhere upstream from, or at, the monitoring point. The different models
that have been proposed in the literature for simulating multimodal tracer
BTCs share a common “multiflow” approach initially proposed by Zuber (1974)
for the modeling of layered aquifers. In this approach, which is depicted in
Fig. 1, the flow system is described as a juxtaposition of a number of
one-dimensional (1-D) channels that are connected by a single common
diverging (splitting) node at the entrance to the system and a single common
converging (mixing) node at the outlet.
Conceptual sketch of the (generic) multiflow modeling
approach, modified from Leibundgut et al. (2009).
In the Multi-Dispersion Model (MDM) proposed by Maloszewski et al. (1992)
and implemented in the TRACI software, the transport along each channel is
assumed to obey the one-dimensional (1-D) advection–dispersion equation
(ADE), and no mass exchange is allowed between different channels. In the
dual-advection dispersion equation (DADE) proposed by Field and Leij (2012), only two channels are considered. The tracer is transported by
advection and dispersion along each channel, and mass exchanges between the
two domains are possible. These exchanges are assumed to be governed by a
first-order process. The transport model implemented in the OM-MADE code can
be viewed as a generalization of the DADE model, where (i) a larger number
of channels can be used, (ii) each channel can be discretized to a number of
subelements with different hydraulic and transport properties, and (iii) some channels can be specified as nonflowing (stagnant) water volumes. Mass
exchanges between the different channels (either flowing or nonflowing) are
likewise modeled as a first-order process. As pointed out above, the
production of a multimodal BTC requires little or no exchange between the
subtransport domains; otherwise, the mixing of the mass fluxes would
rehomogenize the subtracer plumes. In accordance with this principle, small
exchange coefficient values must be used in the DADE and OM-MADE models for
simulating multimodal BTCs, and this approach makes these models converge
toward the MDM.
The interpretation of the long-tail behavior of a BTC may be more difficult
than that of multiple peaks, as different mechanisms can be involved. The
possible sources of extensive BTC tailing can be listed as follows: (i) tracer retention that produces a decaying boundary condition at the injection site; (ii) tracer splitting into well-separated flow paths and then downstream
convergence, mixing, and overlapping of the individual pathway responses; and
(iii) mass exchanges between flow domains characterized by different
transit or residence times. The above-listed processes are referred to below as
“injection decay”, “multiflow overlapping”, and “multiflow exchanges”,
respectively. The MDM can simulate long-tailed BTCs as a result of multiflow
overlapping. Multiflow exchanges are the core of the DADE model, and both
multiflow overlapping and multiflow exchanges can be combined in the OM-MADE
model. A number of other models have been proposed in the literature for
simulating unimodal long-tailed BTCs; see reviews in Bodin et al. (2003b), Neuman and Tartakovsky (2009), Zhang et al. (2009), and Dentz et al. (2011) and examples of recent works in Field and Leij (2014) and Labat and
Mangin (2015). The two most commonly used models for the analysis of
artificial tracer tests are the two-region nonequilibrium (2RNE) model by
Toride et al. (1993), implemented in the CXTFIT code, and the
Single-Fracture Dispersion Model (SFDM) by Maloszewski and Zuber (1990),
implemented in TRACI and OptSFDM software. Both the 2RNE model and SFDM
assume mass exchange between a single mobile (flowing) domain and a single
immobile domain. A key distinction between the 2RNE model and SFDM is the
formulation of mass exchange, which is described as a first-order process in
the 2RNE model (as in the DADE and OM-MADE models) and as a second-order
(diffusion) process in the SFDM.
As already noted, multimodal and long-tailed BTCs are typical of tracer
tests performed in fractured and karst aquifers. A common feature of both
aquifer types is the existence of low-hydraulic-resistance pathways provided
by the fractures and karst conduits (Tsang and Neretnieks, 1998; Worthington
and Ford, 2009). A generic multiflow modeling approach is therefore
intuitively appealing for the interpretation of tracer tests in fractured
and karst aquifers. Of course, the actual (and generally unknown) geometry
of the discrete flow network experienced by the tracer is likely more
complex than that depicted in Fig. 1. The channels are therefore not assumed
to represent individual fractures or karst conduits but are lumped submodels
of the main flow routes used by the tracer through the fractures or karst
conduit network. The four transport models integrated in the MFIT software
are based on the multiflow approach. The first model is a reimplementation
of the MDM. The second model is a variant of the MDM that assumes an
exponentially decaying injection of the tracer concentration at the inlet of
the flow system. In the third and fourth models, the double-porosity concept
(2RNE model and SFDM) is applied at the scale of the individual channels. It
is unclear whether this idea of combining multiflow and double-porosity
systems is new. In the TRACI software, it is technically possible to fit a
series of SFDM curves to a multimodal tracer BTC and then calculate the mean
combined model curve, but to the best of the author's knowledge, this method
has never been discussed or applied in the literature. A possible reason is
the increasing number of fitting parameters, which makes the inverse problem
more complicated. Among the challenges related to the inversion of a
multiflow model is the inherent problem of nonuniqueness (or equifinality).
A variety of parameter sets can yield nearly identical simulated BTCs,
because the change in the value of a parameter of a given channel can be
compensated by modifying at least one other parameter that pertains to this
same channel or the parameters of the other channels. This nonuniqueness
causes the inverse problem to be ill-posed in the sense of Hadamard (1902)
and requires the use of advanced optimization methods, such as
regularization, to make the inverse problem tractable (Tikhonov and Arsenin,
1977; Moore and Doherty, 2006; Zhou et al., 2014).
In this article, the combination of multiflow and double-porosity systems is
referred to as the multi-double porosity (MDP). The immobile domain
that is assigned to each flow channel is assumed to describe the porous rock
matrix in contact with the fractures or karst conduits and/or any other
stagnant water zones (e.g., pool volumes) adjacent to the main tracer
pathways. For each of the four MFIT models, the channels are assumed to be
independent of each other, i.e., no mass exchange is allowed between the
channels. Actually, this assumption is mathematically convenient rather than
physically motivated. As already indicated, the channels are abstractions of
the real main tracer pathways, which may cross (and therefore exchange
between) each other between the injection site and the monitoring point.
Assuming fully separated channels allows for analytical modeling of mass fluxes
in the multiflow system, and this approach makes the inversion of model
parameters computationally more efficient (see discussion in Sect. 3).
The governing equations of the transport models are given as follows. The
concentration at the outlet of a multiflow system as depicted in Fig. 1 can
be calculated from the mass flux balance as follows:
QC=∑j=1NQjCj,
where Q [L3 T-1] is the total system flow rate; C [M L-3] is the
outflow concentration; N is the number of flow channels; the subscript j
denotes the flow channel index; and Qj [L3 T-1] and
Cj [M L-3] are the flow rate and concentration in the jth channel,
respectively.
The mathematical equations that have been used by Maloszewski et al. (1992)
in the MDM to describe the solute transport in each flow channel are the 1-D
ADE as follows:
∂Cj∂t=-uj∂Cj∂xj+Dj∂2Cj∂xj2,
and its analytical solution for the case of an instantaneous solute
injection in a semi-infinite medium with both injection and detection in
flux, which is expressed as (Kreft and Zuber, 1978)
Cj=mj2QjT0jπPejtT0j3exp-PejT0j4t1-tT0j2,
where t [T] is the time variable; xj [L] is the spatial coordinate along
the jth flow channel; uj [L T-1] and Dj [L2 T-1] are the
advection velocity and the dispersion coefficient, respectively;
mj [M] is the part of the solute mass that flows through the jth channel; and T0j [T] and Pej [–] are the mean transit time and Péclet
number, respectively, which are expressed as
T0j=Ljuj,Pej=ujLjDj,
where Lj [L] is the length of the jth pathway. Substituting Eq. (3) into
Eq. (1) yields
C=1Q∑j=1Nmj2T0jπPejtT0j3exp-PejT0j4t1-tT0j2.
The calibration of Eq. (6) against a tracer test BTC requires determination
of the total system flow rate Q; the number N of flow channels; and for each flow channel, the values of mj, T0j, and Pej. In this work, we generalize the above-described method by considering alternative models
for the transport in individual channels and substituting the related
analytical expressions of Cj into Eq. (1). The analytical transport
models that are considered are (i) the solution of Eq. (2) for the case of a
decaying injection boundary condition, (ii) the SFDM, and (iii) the 2RNE
model.
The analytical solution of Eq. (2) for the case of a decaying injection
boundary condition Cj(xj=0,t)=C0exp(-λjt) was derived by Marino (1974) and can be written in the
following form:
Cj=C02erfc1+γjtT0jPejT0j4texpγjPej2+erfc1-γjtT0jPejT0j4texp-γjPej2expPej21-1-γj2t2T0j,
where C0 [M L-3] is the initial (maximum) injection concentration at
the inflow boundary, λj [T-1] is the time decay constant,
and
γj=1-4Djλjuj2.
The SFDM developed by Maloszewski and Zuber (1990) describes solute
transport in a double-porosity fracture-matrix system. The considered
transport mechanisms are advection–dispersion in the fracture and diffusion
in the surrounding rock matrix. The fracture is idealized as a
parallel-plate channel, and the matrix diffusion is assumed to be unlimited,
i.e., not influenced by the fluxes from other fractures. The transport
equations can be written as follows:
∂Cj∂t=-uj∂Cj∂xj+Dj∂2Cj∂xj2+θpjDpjbj∂Cpj∂yjyj=bjfor0≤yj≤bj,∂Cpj∂t=Dpj∂2Cpj∂yj2forbj≤yj≤∞,
where Cj [M L-3] and Cpj [M L-3] are the solute concentrations
in the flow channel and in the rock matrix, respectively; θpj [–] is the matrix porosity; Dpj [L2 T-1] is the
molecular diffusion coefficient in the matrix; bj [L] is the
half-aperture of the flow channel; and yj [L] is the spatial coordinate
perpendicular to the channel extension. The solution to Eqs. (9) and (10)
for the case of an instantaneous injection is
Cj=mjβjPejT0j2πQj∫0texp-PejT0j-ξ24T0jξ-βj2ξ2t-ξξt-ξ3dξ,
where ξ [T] is the integration variable and βj [T-1/2]
is the so-called diffusion parameter defined as
βj=θpjDpj2bj.
Coats and Smith (1964) proposed a different mathematical formulation of
solute mass exchange between flowing and stagnant water regions in
double-porosity media, which is well known in the literature either as the
mobile–immobile (MIM) model or the 2RNE model as
θj∂Cj∂t+θimj∂Cimj∂t=θj-uj∂Cj∂xj+Dj∂2Cj∂xj2,θimj∂Cimj∂t=αjCj-Cimj,
where θj [–] and θimj [–] are the mobile and
immobile volumetric water contents, respectively; Cimj [M L-3] is the
concentration in the immobile domain; and αj [T-1] is a
first-order mass transfer coefficient. The two main differences with respect
to the SFDM are (i) the dual-domain formulation of the problem (mobile and
immobile regions are assumed to coexist at each point in space, and this
assumption differs from the parallel-plate channel geometry in the SFDM) and
that (ii) the solute mass exchange between mobile and immobile domains is
assumed to be governed by a first-order process, whereas the SFDM refers to
the second-order diffusion Eq. (10). Building on a general set of analytical
solutions developed by Toride et al. (1993), the solution of the 2RNE model
for the case of an instantaneous injection can be written as follows:
Cj=mjQj1T0j4πPejtT0j3exp-Pej1-tT0j24tT0j-ωjLjtT0j+ωjψjT0jLj3Pej4π1-ψj∫0ψjLjtT0j1τψjLjtT0j-τexp-PejψjLj-τ24ψjLjτ-ωjτψj-ωjψjLjtT0j-τ1-ψjI12ωjτψjLjtT0j-τψj1-ψjdτ,
where I1 is the modified Bessel function of the first kind, τ [L]
is the integration variable,
ψj=θjθj+θimj, andωj=αjθjuj.
It is notable that when ψj=1 and ωj=0, Eq. (15)
simplifies to Eq. (3). Table 1 summarizes the parameters of the four MFIT
transport models.
Parameters of the transport models integrated in the MFIT
software. The subscript j denotes a parameter that must be defined for
each flow channel. The parameters without this subscript are common to all
channels.
The four analytical models described in the previous section have been
implemented in C++ and compiled as executable Windows programs named
MDMi.exe (for MDM, instantaneous injection), MDMed.exe (for MDM,
exponentially decaying injection), MDP_SFDM.exe, and
MDP_2RNE.exe. The code and executable files are freely
available on the public Zenodo repository: 10.5281/zenodo.3470751. In the MDP_SFDM and
MDP_2RNE programs, the numerical evaluation of the integrals
in Eqs. (11) and (15) is performed using the QAG adaptive integration
routine from the GNU Scientific Library with a 61-point Gauss–Kronrod rule
and a relative error convergence criterion of 10-2. These four programs
can be run as console applications to solve a direct (forward) problem,
i.e., computing a series of time–concentration values for a given set of
model parameters. Both the input and output files are in ASCII format and
can be edited with any text editor program for pre- and/or postprocessing. A
convenient alternative is to use the MFIT software as a GUI for these
applications. The MFIT software has been developed using the C++ Builder
environment (Embarcadero RAD Studio 10.1 Berlin) and provides a GUI for (i) importation and graphic visualization of user-provided BTC data; (ii) parameterization, direct running, and graphical output of the analytical
transport models; (iii) inversion (automatic calibration) of model
parameters for optimal curve fitting; and (iv) assessment of the uncertainty
of calibrated parameter values.
The optimization and uncertainty analysis of the model parameters for a
given number of flow channels are carried out using PEST routines (Doherty,
2019a, b). The influence of the number of channels on the model fitting
performance can be analyzed once a series of calibrations has been performed
for a variety of channel numbers, as illustrated below. PEST is a public-domain model-independent program suite that has been widely used over the
past 2 decades, notably in the field of surface and subsurface hydrology
(e.g., Long, 2015; Woodward et al., 2016; Gaudard et al., 2017; Wang et al.,
2019). The theoretical framework and full range of capabilities of the PEST
software are well documented (Doherty et al., 2010; Doherty, 2015, 2019a, b) and are not repeated here. Only the concepts and methods that were
deemed to be the most relevant to the multiflow modeling approach and that
have been made accessible through the MFIT GUI software are briefly reviewed
below.
PEST is based on a gradient optimization method and, as such, requires the
derivatives of model outputs with respect to the adjustable model parameters
to be calculated in each iteration for implementing the Jacobian
(sensitivity) matrix. As pointed out by Doherty (2015), the accuracy of
these derivative calculations is critical to the performance of the PEST
optimization algorithm. In the MFIT program suite, most of the model partial
derivatives are calculated analytically and externally provided to PEST.
This approach ensures both the accuracy and speed of this part of the
optimization process. Less straightforward partial derivative expressions
were derived using MAPLE and exported as C code using the MAPLE code
generation routine. The partial derivative functions were implemented in the
MDMi, MDMed, MDP_SFDM, and MDP_2RNE programs
and are processed during the PEST system calls to these programs by
providing an optional “/d” command-line argument to the program name. In a
few cases, however, the partial derivatives cannot be calculated
analytically, as they involve undefined limits. Such is the case for the
derivatives of Eq. (15) with respect to the parameters ψj,
Lj, and T0j. In these cases, the partial derivatives are
computed by PEST using finite differences.
The calibration of a multiflow transport model against a tracer BTC is
hampered by two well-known issues in inverse modeling: (i) model
nonlinearity and (ii) solution nonuniqueness. Both issues may cause
numerical instabilities that can prevent the inversion algorithm from
converging to the optimal solution. PEST includes two regularization methods
that can be used either individually or together to guide the optimization
process. The singular value decomposition (SVD) method subtracts parameter
combinations for which the tracer BTC is uninformative. The inversion is
conducted on the basis of a reduced set of orthogonal linear combinations of
the model parameters rather than attempting to estimate the parameters
individually. The Tikhonov regularization method provides a different but
complementary strategy, where the information content of the tracer BTC is
supplemented with expert knowledge pertaining to the model parameters. When
using Tikhonov regularization, the objective function that is minimized by
PEST is defined as the sum of two terms. The first term is the “measurement
objective function” and is defined as the sum of the squared weighted
differences between the real tracer BTC and the model-simulated curve. The
second term is referred to as the “regularization objective function” and
acts as a penalty function for deviations from some preferred parameter
conditions. Two Tikhonov regularization options have been implemented in
MFIT. The first option, referred to as “preferred homogeneity”, promotes a
solution of minimum variance for the model parameters pertaining to the
different channels. In the second option, referred to as “preferred value”,
the optimization process seeks the solution that is the closest to some
prior estimates of the model parameters.
Unfortunately, neither SVD nor Tikhonov regularization can guarantee that
the PEST optimization algorithm will converge to the global optimal solution
in the parameter space. Where local minima exist in the objective function,
which is the rule rather than the exception with nonlinear models, the
optimization process may become trapped and fail to identify existing better
solutions (Singh et al., 2012; Espinet and Shoemaker, 2013; Abdelaziz et
al., 2019). A central issue in this case is the sensitivity to initial
parameter values, i.e., different initial parameter sets may lead to
different optimized solutions. Global optimization methods have been
proposed in the literature to overcome this issue; see Arsenault et
al. (2014) for a review and comparison of various algorithms. The PEST
program suite includes two such global optimizers based on the SCE-UA method
(Duan et al., 1992) and the CMA-ES method (Hansen and Ostermeier, 2001). The
corresponding programs are named SCEUA_P and
CMAES_P, respectively. It must be noted, however, that global
optimization methods suffer from their own drawbacks, including sensitivity
to tuning parameters and low computational efficiency. An alternative
strategy to improve the chances of convergence toward the global optimum
with gradient-based methods is the “multistart” approach, which consists of
repeating the optimization process starting from different initial parameter
value sets (Skahill and Doherty, 2006; Piotrowski and Napiorkowski, 2011).
Such a strategy has been implemented in the MFIT software. The key principle
of the proposed algorithm is that rather than conducting the optimization
for a fixed number N of channels only, a series of automatic tracer BTC
fittings is performed for a decreasing number of channels ranging from
Nmax to 1. The main steps of the MFIT multistart algorithm are detailed
as follows:
The first optimization is performed by considering the maximum number of flow channels Nmax. The initial transport parameters are automatically tuned by MFIT to obtain Nmax well-separated concentration peaks. For this goal, the tracer BTC is first analyzed to determine the times T5 and T95, which are defined asT5=maxT5th,1.1×Tmin,T95=minT95th,0.9×Tmax,where T5th and T95th are the earliest and latest times at
which the concentration values are above and below 5 % of the maximum
concentration value, respectively, and Tmin and Tmax are the minimum
and maximum time values of the user-provided BTC, respectively. The mean
travel times, T0j, are then uniformly distributed between the times
T5 and T95. Next, the initial Péclet number, Pej, of each channel
is calculated asPej=15NmaxT0jT95-T52.Equation (20) is based on a semiempirical relationship between the standard
deviation of travel times for transport by advection and dispersion, σj=T0j2/Pej (see Bodin et al., 2003a; Eq. 10), and the time span of the jth concentration peak, which is on the order
of 6σj. The constraint of well-separated concentration peaks
may be formulated as 6σjNmax≪(T95-T5), which is verified by Eq. (20). The initial values of the
other transport parameters in Eqs. (7), (11), and (15) are chosen to
minimize the tailing effect due to noninstantaneous injection or solute mass
exchange between flowing and stagnant water regions as follows: γj=0.1, βj=0.001, ψj=0.9, and
ωj=0.05.
Once the optimization has been performed for the Nmax channel model, the
next step is to optimize the transport parameters for Nmax-1 channels.
The multistart optimization approach begins here as not only one but
Nmax optimizations are performed in this step. The initial parameter
values for the Nmax-1 channels are initialized from the previously
optimized Nmax channel solution by sequentially removing one of the
channels. Only the solution corresponding to the lowest sum of the squared
weighted differences between the tracer BTC and model-simulated curve is
retained.
This procedure is repeated up to the single-channel solution. The total
number of PEST optimizations is Nmax(Nmax+1)/2.
Calling the multistart algorithm has been made optional in MFIT, as this
algorithm significantly increases the computational cost and running time of
the optimization process. However, experience has shown that the multistart
approach can truly improve the model fit results and can be worth the effort
in many circumstances. A comparison between optimizations conducted by the
PEST multistart algorithm and the global SCE-UA and CMA-ES methods was
conducted in this study and is discussed in Sect. 6.
Because of the nonuniqueness of the inverse problem, some uncertainties may
be associated with the PEST-optimized model parameter values. A nonlinear
analysis method has been implemented in MFIT for the assessment of
postcalibration parameter uncertainty. The method is essentially similar to
that described by Fang et al. (2019) and relies on the use of the PREDUNC7
and RANDPAR utilities documented in the PEST manual (Doherty, 2019b). The
algorithm can be described by the following steps: (1) compute a linear
approximation to the posterior parameter covariance matrix using PREDUNC7;
(2) sample the posterior parameter covariance matrix and generate multiple
calibration-constrained random parameter sets with RANDPAR; (3) recalibrate
each parameter set with PEST up to achieving a level of fit fairly similar
to the original calibration result (a tolerance of +5 % for the
measurement objective function is allowed by MFIT); and (4) compute
histograms of the recalibrated parameter values. The following two
assumptions underlie this method: (i) the upper and lower parameter bounds
specified by the user for the PEST inversion reflect the prior (expert
knowledge) parameter uncertainty, and (ii) the model parameters are
statistically independent from a prior point of view. This second assumption
is relaxed through the recalibration process.
Input parameters for the five verification tests.
TestParametersValues1 (single flow channel,Flow rate Q10 m3 h-1ADE, instantaneousInjected mass m20 ginjection)Mean transit time T0200 hPéclet number Pe22 (single flow channel,Mean transit time T070 hADE, exponentiallyPéclet number Pe10decaying injection)Initial (maximum) injection concentration C08.0×10-3 mg L-1Gamma coefficient γ0.93 (single flow channel,Q, m, T0, PeSame as test 1SFDM)Diffusion parameter β0.04 h-1/24 (single flow channel,Q, m, T0, Pesame as test 12RNE)Length of the flow channel L1000 mFraction of mobile water ψ0.7Omega coefficient ω0.1 m-15 (two channels,Total system flow rate Q10 m3 h-1MDM-ADE)Mass flowing through the first channel m112 gMass flowing through the second channel m28 gMean transit time in the first channel T01170 hMean transit time in the second channel T02300 hPéclet number in the first channel Pe115Péclet number in the second channel Pe280Code verification
The robustness of the PEST inversion program has been demonstrated in a
number of studies (see Anderson et al., 2015; and Hunt et al., 2019)
and is not reassessed here. The purpose of this section is to assess the
accuracy of MFIT direct simulations through five synthetic test cases. Tests
1 and 2 address the case of a single flow channel described as a
single-porosity medium in which the transport is governed by
advection–dispersion. An instantaneous injection of the tracer is assumed in
test 1, whereas test 2 addresses the case of an exponentially decaying
concentration at the inlet. A double-porosity medium and single flow channel are assumed in tests 3 and 4, which conform to the assumptions of the SFDM and
2RNE model, respectively. In test 5, the tracer is transported by
advection–dispersion in a multiflow system composed of two channels. This
scenario corresponds to the MDM. The input parameters for the five test
cases are listed in Table 2. The BTCs simulated by MFIT for tests 1, 2, and 4
are compared to those obtained by CXTFIT. The MFIT simulations for tests 3
and 5 are compared against those obtained by TRACI. As shown in Fig. 2, very
good agreement was obtained in each case.
Comparison among MFIT, CXTFIT, and TRACI simulations for
test 1 (single flow channel, ADE, instantaneous injection), test 2 (single
flow channel, ADE, exponentially decaying injection), test 3 (single flow
channel, SFDM), test 4 (single flow channel, 2RNE), and test 5 (two
channels, MDM-ADE).
Assessment of the multistart optimization method
The purpose of this section is to assess the automatic multistart method
described in Sect. 3 using a new synthetic test case. A multimodal BTC
that corresponds to three channels has been simulated using the MDMi program
with the parameters listed in Table 3. A “blind” inversion of this BTC has
been performed using the automatic multistart method with a maximum number
of flow channels Nmax=6. The only model parameter that has been
fixed prior to the inversion process was the total flow rate Q to simplify
the post-comparison of the inverted mass values in each channel with the
“true” mass values. Otherwise, a degree of freedom would persist for the
pairs of the optimized Q and mj values, i.e., multiplying or dividing
these parameters by the same constant would yield the same BTCs; refer to
Eq. (6). The parameters mj, T0j, and Pej of the different
flow channels were optimized with virtually no upper and lower bound
constraints (minimum and maximum allowed parameter values of 1.0×10-10 and 1.0×10+10, respectively). As shown in Fig. 3,
the inverted BTCs that correspond to N=3, 4, 5, and 6 channels overlap
perfectly with each other and with the original simulated BTC; as shown
in Table 4, the optimized values for the parameters of the three-channel model
are equal to the true parameter values.
Model parameters that correspond to the multimodal
simulated BTC in Fig. 3.
Inversion of the three-channel-simulated BTC using the
automatic multistart method with Nmax=6. The inverted BTCs that
correspond to N=3, 4, 5, and 6 channels overlap perfectly with each other
and the original simulated BTC.
Optimized model parameters that correspond to the inverted
BTCs in Fig. 3.
N123456m1 (g)21.1110.7910.002.792.662.66m2 (g)–9.546.007.197.357.35m3 (g)––4.006.025.995.91m4 (g)–––4.002.582.62m5 (g)––––1.421.45m6 (g)–––––0.02T01 (h)239.36155.82150.00126.17151.55151.31T02 (h)–302.91250.00158.91149.47149.60T03 (h)––350.00250.00249.97249.30T04 (h)–––350.01349.48347.68T05 (h)––––350.84351.08T06 (h)–––––405.58Pe16.7217.5220.0024.2219.8019.92Pe2–27.5550.0022.1820.0720.01Pe3––100.0049.9450.0450.62Pe4–––100.0098.4588.42Pe5––––102.68120.27Pe6–––––442.13Application example: analysis of tracer BTCs from the Hydrogeological Experimental Site in Poitiers, France
The HES is a field research facility operated by the University of Poitiers,
France. The facility consists of 32 wells that have been drilled within an
overall area of 0.2 km2 (Fig. 4) and fully penetrate a
100 m thick confined limestone aquifer. The interwell flow and transport
connectivity have been shown to be mainly related to karst conduits, 0.01–3 m in diameter, that develop preferentially within specific
lithostratigraphic horizons interbedded with nonkarstified limestone units.
The karstified layers may contribute to the connectivity from one well to
another either directly (e.g., the wells intersect with the same karst
network in a single layer) or indirectly (e.g., the wells intersect with
different karst network layers that are interconnected by either a third
well or a subvertical fracture); see Audouin et al. (2008) and Chatelier et
al. (2011).
A large number of pumping test experiments have been conducted at the HES
since 2002. As discussed in a number of studies (Delay et al., 2007, 2011; Riva et al., 2009; Bodin et al., 2012; Sanchez-Vila et al.,
2016; Le Coz et al., 2017), the drawdown responses exhibit complex
behaviors, which are likely due to the strong aquifer heterogeneity induced
predominantly by the presence of karst features. In addition to the pumping
test experiments, a number of cross-borehole tracer tests have been
performed at the HES since 2011. The standard experimental protocol of HES
tracer experiments can be summarized as follows:
Starting a pumping experiment and waiting for the establishment of a
pseudo-steady-state flow regime (i.e., stabilization of interwell piezometric
head gradients) is the first step, which typically takes approximately 6 h at the HES.
Performing flow log measurements in the candidate injection well to identify the main inflow and outflow levels along the well bore is the second step.
Connecting a series of 2.5 m length and 1.5 cm inner diameter PVC pipes in the injection well, from the ground down to the tracer injection depth (usually chosen to be as close as possible to a main outflow level) is the third step. The pipeline is terminated by a 5 cm length screened cap that ensures a horizontal outflow of the tracer solution in the injection well.
Injecting a tracer solution (typically 2 L of Uranine solution at 1 g L-1) in the pipe and flushing with 40 L of clean groundwater is the fourth step. The total duration of an injection is typically less than 3 min.
Monitoring the tracer BTC at the pumped well using a flow-through
fluorometer (Albillia GGUN-FL22) connected to a branch pipe extending from
the discharge line at ground level is the last step. The fluorometer is periodically
calibrated in the laboratory with solutions of 10 and 100 µg L-1.
To date, more than 70 cross-well tracer experiments have been performed at
the HES. The purpose here is not to interpret each of these experiments but
to pick a few examples for illustrating the application of the MFIT
software. The selected data correspond to three tracer experiments that were
performed in 2016 and 2017 using well M22 as the pumped well and M16, MP6,
and P2 as injection wells. Figure 5 shows the experimental BTCs and a
collection of calibrated MFIT curves for different numbers of channels. The
selected experiments were chosen for their representativeness of the BTC
shapes observed at the HES, which exhibit either a single peak followed by a
more or less pronounced tailing, e.g., P2-M22; overlapping double-peak
responses, e.g., M16-M22; or well-marked multimodal responses, e.g.,
MP6-M22. The mass recovery ratios for these three tracer experiments were 58 %, 79 %, and 60 %, respectively. Note that these recovery data
cannot be included in the model, because the flow structure assumption that
underlies the multiflow approach (Fig. 1) implies that all the mass that
enters the system flows out after a certain lapse of time. The same holds
for any single- or double-porosity modeling approach based on a 1-D flow
assumption. For tracer tests that are performed in steady state conditions
and involve nonreactive tracers, an incomplete recovery of the injected
mass indicates a diverging flow structure between the injection site and the
monitoring point. Unfortunately, no additional information can be obtained
about this flow divergence from the tracer data only. Therefore, the total
mass in a multiflow model must be consistent with the recovered tracer mass
rather than the injected mass.
The model fit results shown in Fig. 5 were obtained using the multistart
method discussed in Sect. 3 and only SVD as a regularization tool for the
inversion. None of the model parameters were fixed, and all were optimized
within realistic upper and lower limits. The optimized parameter values and
their composite sensitivities at the end of the optimization process are
provided in the Supplement (Table S1). Unsurprisingly, the model parameters
that influence the spreading of transit or residence times in the individual
flow channels, while accounting for different processes (Pe, γ, β, ψ, and ω) are sensitive to the number of channels.
For instance, when comparing single- with multiple-channel models, the
former requires lower Pe values to compensate for the coarser description of
the flow system heterogeneity (recall that the dispersion coefficient
integrated in the Péclet number reflects the unresolved variability of the
flow velocity below the modeling scale). The same observation holds when
comparing single- and double-porosity models with the same number of flow
channels, i.e., the Pe values of single-porosity models are lower than the
Pe values of double-porosity models, because part of the spreading of
transit or residence times in the latter case is implicitly captured by solute
mass exchanges between the mobile and immobile domains. A noticeable
exception is the diffusion parameter β of the SFDM model, whose
values are mostly around 1.0×10-3 h-1/2. This value
corresponds to the upper bound of the optimization range set for this
parameter, which is based on a matrix porosity of 30 %, a molecular
diffusion coefficient of 1.0×10-9 m2 s-1, and a
flow-channel aperture of 1.0×10-2 m. Beta values larger than
1.0×10-3 h-1/2 would be physically unrealistic. The
fact that the beta value is limited by its upper bound during the
optimization process indicates that the SFDM model is not suitable for
describing the HES tracer experiments, as further discussed below. All other
parameters have converged to values far from their optimization bounds.
Inversion solutions of three tracer BTCs for different
numbers of channels. Some model curves are hardly distinguishable, as they
perfectly overlap (refer to the text and Fig. 6).
Best-fitting performance of the multiflow models achieved
using PEST with the multistart optimization approach and using global
optimizers. N is the number of channels in the models, P is the number of
optimized parameters, and PHI is the sum of the squared weighted differences
between the tracer BTCs and the model-fitted curves.
Beyond what can be visually inferred from Fig. 5, the assessment of the
relative fitting performance of the different models can be analyzed through
the evolution of the measurement objective function, hereafter named PHI,
with respect to the number, N, of channels and/or the number, P, of optimized
model parameters. Figure 6 displays the PHI(N) and PHI(P) curves summarizing the
best-fitting results achieved with the multistart PEST optimization and the
SCEUA_P and CMAES_P global optimization
routines. A number of observations can be made from this figure. As a first
remark, the SCEUA curves for the two MDP models are missing in Fig. 6. The
reason is that the SCEUA_P program has no “forgive error”
capability, i.e., if a set of trial parameters causes the numerical
evaluation of Eq. (11) or Eq. (15) to crash, the optimization process is
stopped instead of moving to a new set of parameter values. Such a
forgiveness option is available in the PEST and CMAES_P
programs. The next observations that can be made from Fig. 6 are that the
CMAES and SCEUA curves are (i) more irregular, (ii) always above or equal to
their PEST-computed counterparts, and (iii) do not always follow the
expected decreasing trend in the PHI value (meaning a better model fit) as
the number of channels rises, as depicted by the PEST curves. However, it
must be mentioned that the number of optimization runs was much greater for
the CMAES_P and SCEUA_P programs, and various
optimization options were tested (e.g., changing the upper and lower
parameter bounds and log-transformation of parameters). The CMAES and SCEUA
curves shown in Fig. 6 are actually the “best results” obtained after
several days of computation time. It is clear that the multistart PEST
optimization method performs better in each case.
Postcalibration uncertainty of model parameter values for
the inversion of the M16-M22 tracer BTC by the MDMi model with 1, 2, and 3
flow channels.
The PHI curves obtained by PEST can be viewed as Pareto curves, illustrating
the trade-off between the model fitting quality and the number of channels or
the number of calibration parameters. It must be noted that since no
Tikhonov regularization was used in this illustration example, the model
inversion results for higher N values are likely affected by overfitting.
More reliable parameter values could be obtained by adding Tikhonov
regularization constraints to the optimization process.
According to the PHI(N) curves shown in Fig. 6, the MDMi model and MDP-SFDM
perform similarly for the three tracer tests, and the related PHI(N) curves
are hardly differentiable. This result was expected, as the short duration
of the HES tracer tests, typically from a few hours to a few days, makes the
matrix diffusion process unlikely to be significant. Assuming exponential
decaying (MDMed) instead of instantaneous (MDMi) injection gives slightly
better fitting results for a low number of channels but provides no benefit
for a moderate-to-high number of channels. According to the PHI(N) curves,
the fitting performance of the MDP-2RNE model seems significantly better
than that of the three other models. However, this observation must be
counterbalanced by the larger number of calibration parameters in the
MDP-2RNE model (see Table 1). A two-channel MDP-2RNE model involves 13 parameters, which corresponds to the number of parameters in a four-channel
MDMi model. The PHI(P) curves shown in Fig. 6 provide a fairer assessment of
the fitting performance of the different models. According to these curves,
the MDP-2RNE model performs slightly better than the MDMi model for the
P2-M22 tracer test (single peak slightly tailed BTC), almost equally well
for the M16-M22 tracer test (overlapping double peaks), and worse for the
MP6-M22 tracer test (well-marked multimodal BTC). It must be appreciated
that these two models should not be opposed to each other. Both models
likely provide an equally valid description of the tracer transport in the
HES aquifer while relying on different conceptualizations of the medium
heterogeneity.
Postcalibration uncertainty of model parameter values for
the inversion of the M16-M22 tracer BTC by the MDP-2RNE model with 1, 2, and
3 flow channels. A logarithmic scale has been employed for Pe due to a wider
range of values than shown in Fig. 7.
The Pareto curves in Fig. 6 indicate that the final choice of a model, if one is to be
made, relies on a trade-off between the desired fitting accuracy and the
desired degree of simplification or complexity with respect to the model
structure (number of channels and/or number of model parameters). Beyond
this subjective (expert) decision, which may depend on the goal of the
study, and therefore will not be discussed further in the present
application case, uncertainty remains in the inverted model parameters as a
consequence of the nonuniqueness of the inverse problem. This uncertainty is
related to both the equifinality of the model parameters, which is partly
due to the multiflow framework structure, and the measurement noise in the
tracer BTCs. Figures 7 and 8 illustrate the postcalibration uncertainty
analysis capabilities of MFIT via an assessment of the MDMi and MDP-2RNE
model fittings of the M16-M22 tracer BTC with 1, 2, and 3 flow channels.
Owing to the balance between the Q and mj terms in the model equation
(Eqs. 6 and 15), at least one of these parameters must be fixed to
assess the uncertainty of the other parameters. Here, the value of Q was set
to 25 m3 h-1, which ensures the consistency of the model against
the recovered tracer mass that was independently calculated from the
experimental data (refer to Table S1). Following the PEST optimization of
the different model parameters, 500 calibration-constrained parameter fields
were stochastically generated and recalibrated by PEST. Depending on the
model (MDMi or MDP-2RNE) and number of flow channels, between 483 and 500
recalibration runs successfully achieved a level of fit that is fairly
similar (i.e., within a tolerance of +5 % for the PHI value; refer to
Sect. 3) to that associated with the original calibration parameter field.
The histograms shown in Figs. 7 and 8 were constructed from these
recalibrated parameter sets and illustrate the multitude of parameter
combinations that are equally good for a given number of flow channels, in
terms of fitting the M16-M22 tracer BTC. As shown in these figures, the
confidence intervals are quite narrow for most parameters but tend to widen
as the number of channels increases, which reflects the equifinality of the
multiflow modeling approach. Although not shown here, it has been
established that the tailed behaviors of the parameters Lj and ωj in Fig. 8 are due to a partial correlation between these two
parameters (refer to Eq. 15), i.e., fixing the value of one parameter
prior to the inversion drastically reduces the uncertainty of the other
parameter. As previously discussed, the higher Pe values in Fig. 8 compared
to Fig. 7 are due to the fact that the distribution of the transit or residence
times with the 2RNE model is primarily controlled by the solute mass
exchanges between the mobile and immobile domains.
Summary and conclusions
Multiple flow path transport is likely the rule rather than the exception in
most transport problems in fractured and karst aquifers. The main aim of
this paper was to present a new curve-fitting tool for the analytical
modeling of BTCs from tracer tests performed in such media. The MFIT
software is a free open-source Windows-based GUI that provides access to
four multiflow transport models. The multiflow approach assumes that the
transport from the injection site to the monitoring point takes place in a
number of independent 1-D channels. The channels are not assumed to
represent individual fractures or karst conduits but are lumped submodels of
the main flow routes used by the tracer through the fractures or karst conduit
network. The multiflow modeling framework allows for the simulation of
multimodal BTCs, which are frequently observed in fractured and karst
aquifers. Two of the MFIT transport models combine the multiflow framework
and the double-porosity concept, which is applied at the scale of the
individual channels. This modeling approach, which has been named MDP, is
believed to be new and versatile for the fitting of BTCs with multiple local
peaks and/or extensive backward tailing. The accuracy of the MFIT-computed
BTCs was verified against two other well-accepted simulation tools for five
synthetic test cases.
An important feature of MFIT is its compatibility and interface with the
advanced calibration tools of the PEST suite of programs. Hence, MFIT is the
first BTC fitting tool that allows for regularized inversion and nonlinear
analysis of the postcalibration uncertainty of model parameters. Given the
nonlinearity of the MFIT model equations, an original multistart algorithm
was implemented to maximize the chances for PEST to converge to the global
optimal solution in the parameter space during a BTC fitting procedure. The
main drawback of the multistart optimization method is that the processing
time can be long (up to a few hours) if a large number of channels is
assumed in the model. Time reduction for this method is one of the
development perspectives of the MFIT code, as the multistart process is
computationally parallelizable. Other development perspectives are the
management of more complex injection signals, e.g., described as multiple
steps, and the implementation of additional analytical transport models for
the simulation of reactive transport processes.
Three tracer test BTCs from the HES in Poitiers, France, were used for
illustrating the application of the MFIT software. An analysis of the Pareto
curves between the model fitting quality and the number of model calibration
parameters suggests that the MDMi and MDP-2RNE models are the most
appropriate for the interpretation of HES tracer tests. This preliminary
result needs to be refined or confirmed by the analysis of additional HES
tracer BTCs.
Glossary
Acronyms and model abbreviations utilized in the text.
Acronym ormodel nameDescriptionReferenceADEAdvection–dispersion equationZheng and Bennett (2002)BTCBreakthrough curveCATTIComputer Aided Tracer Test Interpretation: a computer program for tracer BTC fittingSauty et al. (1992)CMA-ESCovariance Matrix Adaptation – Evolution Strategy: a global optimization algorithmHansen and Ostermeier (2001)CMAES_PPEST-compatible program that implements the CMA-ES methodDoherty (2019a)CXTFITComputer program for tracer BTC fittingToride et al. (1999)DADEDual-advection dispersion equationField and Leij (2012)FEFLOWFinite Element FLOW model; a simulation package for flow, heat, and mass transport in groundwaterDiersch (2014)GUIGraphical user interfaceHESHydrogeological Experimental Site in Poitiers, FranceAudouin et al. (2008)MDMMulti-Dispersion ModelMaloszewski et al. (1992)MDMedComputer program that implements the Multi-Dispersion Model and assumes a noninstantaneous injection (exponentially decaying concentration) at the inlet of the flow systemThis articleMDMiComputer program that implements the Multi-Dispersion Model and assumes an instantaneous injection of tracer at the inlet of the flow systemThis articleMDPMulti-double porosity: a combination of multiflow and double-porosity modelsThis articleMDP_SFDMComputer program that implements the MDP approach, where the mass exchanges between the mobile and immobile domains are modeled as a second-order (diffusion) processThis articleMDP_2RNEComputer program that implements the MDP approach, where the mass exchanges between the mobile and immobile domains are modeled as a first-order processThis articleMFITMulti-Flow Inversion of Tracer breakthrough curves: a GUI for the MDMi, MDMed, MDP_SFDM, MDP_2RNE, and PEST programs.This articleMIMMobile-Immobile ModelCoats and Smith (1964)MODFLOWMODular three-dimensional groundwater FLOW model: a computer code developed by the U.S. Geological Survey that numerically solves the groundwater flow equationLangevin et al. (2017)MT3DMSModular Three-Dimensional MultiSpecies transport model: a numerical code to simulate solute transport in groundwaterZheng et al. (2012)OM-MADEOne-dimensional Model for Multiple Advection, Dispersion, and storage in Exchanging zones: a Python script to simulate solute transport in multiflow systems with possible mass exchanges between the flow channelsTinet et al. (2019)OptSFDMComputer program for tracer BTC fitting based on the SFDM modelGharasoo et al. (2019)OTISOne-dimensional Transport with Inflow and Storage: a numerical code to simulate solute transport in streams and riversRunkel (1998)PESTParameter ESTimation: a collection of computer programs for model-independent parameter estimation and uncertainty analysisDoherty (2019a)SCE-UAShuffled Complex Evolution method – University of Arizona: a global optimization algorithmDuan et al. (1992)SCEUA_PPEST compatible program that implements the SCE-UA methodDoherty (2019a)SFDMSingle-Fracture Dispersion ModelMaloszewski and Zuber (1990)STANMODSTudio of ANalytical MODels: a collection of computer programs for tracer BTC fittingvan Genuchten et al. (2012)SVDSingular value decompositionDoherty (2015)TRACComputer program for tracer BTC fittingGutierrez et al. (2013)TRACIComputer program for tracer BTC fittingKäss (2004)1-DOne-dimensional2RNETwo-region nonequilibrium equationToride et al. (1993)
List of model parameters.
ParameterDescriptionDimensionSpecific model*bjHalf-aperture of the jth flow channelLMDP-SFDMCjConcentration in the jth flow channelM L-3CpjConcentration in the immobile domain assigned to the jth channelM L-3MDP-SFDMCimjConcentration in the immobile domain assigned to the jth channelM L-3MDP-2RNEC0Initial (maximum) concentration at the inflow boundary for an exponentially decaying injection concentrationM L-3MDMedDjDispersion coefficient in the jth flow channelL2 T-1DpjMolecular diffusion coefficient in the immobile domain assigned to the jth channelL2 T-1MDP-SFDMLjLength of the jth flow channelLmjPart of the solute mass flowing through the jth channelMMDMi, MDP-SFDM, MDP-2RNENNumber of flow channels–NmaxMaximum number of flow channels–PejPéclet number in the jth channel–PNumber of optimized model parameters–PHIMeasurement objective function (sum of the squared weighted differences between the tracer BTCs and the model-fitted curves)M2 L-6QTotal system flow rateL3 T-1QjFlow rate in the jth channelL3 T-1tTime variableTTminMinimum time value of the user-provided BTCTTmaxMaximum time value of the user-provided BTCTT5T5 time, Eq. (18)TT5thEarliest time at which the concentration values exceed 5 % of the maximum concentration valueTT95T95 time, Eq. (19)TT95thLatest time at which the concentration values exceed 5 % of the maximum concentration valueTT0jMean transit time in the jth channelTujAdvection velocity in the jth flow channelL T-1xjSpatial coordinate along the jth flow channelLyjSpatial coordinate perpendicular to the jth flow channelLαjFirst-order mass transfer coefficient between the mobile and immobile domains assigned to the jth channelT-1MDP-2RNEβjDiffusion parameter in the jth flow channel, Eq. (12)T-1/2MDP-SFDMγjGamma coefficient in the jth flow channel, Eq. (8)–MDMedθjVolumetric water content of the mobile domain assigned to the jth channel–MDP-2RNEθimjVolumetric water content of the immobile domain assigned to the jth channel–MDP-2RNEλjTime decay constant that controls the exponentially decaying release of tracer in the jth channelT-1MDMedξIntegration variable, Eq. (11)TMDP-SFDMσjStandard deviation of travel times for transport by advection and dispersion in the jth channelTτIntegration variable, Eq. (15)LMDP-2RNEψjFraction of mobile water in the jth channel, Eq. (16)–MDP-2RNEωjOmega coefficient in the jth flow channel, Eq. (17)L-1MDP-2RNE
* An empty box means that the parameter is employed in all the models.
Code and data availability
The source codes of the MFIT program
suite version 1.0.0 are available from https://doi.org/10.5281/zenodo.3470751 (Bodin, 2020) under the terms of the CeCILL Free
Software License Agreement v2.1 (https://spdx.org/licenses/CECILL-2.1.html, last access:
24 June 2020). An “EXE” installation package compiled with Inno Setup
(http://www.jrsoftware.org/isinfo.php, last access: 24 June 2020) and a user's guide are provided along with the source codes.
The following numerical libraries are required for the compilation of the
MFIT suite of codes: Boost (https://www.boost.org/, last
access: 24 June 2020), GSL-GNU (https://www.gnu.org/software/gsl/, last access: 24 June 2020) and
Spline (https://github.com/ttk592/spline, last access: 24 June 2020). The PEST program package is also required for running MFIT.
PEST is distributed by default using the MFIT software installer or can be
independently downloaded from http://www.pesthomepage.org/Downloads.php (last access: 24 June 2020). The data of the HES tracer experiments processed in Sect. 6 of this
study are available from the H+ database (http://hplus.ore.fr/en/poitiers/data-poitiers, last access: 24 June 2020, de Dreuzy et al., 2020) with registration of a free account.
The supplement related to this article is available online at: https://doi.org/10.5194/gmd-13-2905-2020-supplement.
Competing interests
The author declares that there is no conflict of interest.
Acknowledgements
The HES field research facility is managed by
Gilles Porel. The experimental protocol used for the HES tracer experiments
was developed by Benoit Nauleau and Gilles Porel. The assistance of Benoit
Nauleau, Gilles Porel, and Denis Paquet in conducting the tracer experiments
is gratefully acknowledged. The author would like to thank the two anonymous
reviewers for their valuable comments and feedback, which helped with improving
the article.
Financial support
This research was supported by the French
National Observatory H+, the European Union (ERDF), and Région
Nouvelle Aquitaine.
Review statement
This paper was edited by Jeffrey Neal and reviewed by two anonymous referees.
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