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.

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.

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:

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:

The analytical solution of Eq. (2) for the case of a decaying injection
boundary condition

Parameters of the transport models integrated in the MFIT
software. The subscript

The four analytical models described in the previous section have been
implemented in C

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

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

The first optimization is performed by considering the maximum number of flow channels

Once the optimization has been performed for the

This procedure is repeated up to the single-channel solution. The total
number of PEST optimizations is

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

Input parameters for the five verification tests.

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).

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

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

Optimized model parameters that correspond to the inverted BTCs in Fig. 3.

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 km

Locations of wells at the HES in Poitiers, France. Map data are from © Google.

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

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

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 (

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.

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,

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

According to the PHI(

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

The Pareto curves in Fig. 6 indicate that the

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.

Acronyms and model abbreviations utilized in the text.

List of model parameters.

The source codes of the MFIT program
suite version 1.0.0 are available from

The supplement related to this article is available online at:

The author declares that there is no conflict of interest.

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.

This research was supported by the French
National Observatory H

This paper was edited by Jeffrey Neal and reviewed by two anonymous referees.