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    Particle Swarm Optimization for inverse modeling of solute transport in fractured gneiss aquifer Ramadan Abdelaziz, Mauricio Zambrano-Bigiarini PII: DOI: Reference:

S0169-7722(14)00086-2 doi: 10.1016/j.jconhyd.2014.06.003 CONHYD 3012

To appear in:

Journal of Contaminant Hydrology

Received date: Revised date: Accepted date:

28 October 2013 3 June 2014 6 June 2014

Please cite this article as: Abdelaziz, Ramadan, Zambrano-Bigiarini, Mauricio, Particle Swarm Optimization for inverse modeling of solute transport in fractured gneiss aquifer, Journal of Contaminant Hydrology (2014), doi: 10.1016/j.jconhyd.2014.06.003

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Particle Swarm Optimization for inverse modeling of solute transport in fractured gneiss aquifer

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Ramadan Abdelaziza,∗, Mauricio Zambrano-Bigiarinib,1 a Institute

for Geology, Technische Universit¨ at Bergakademie Freiberg, Freiberg, Germany Resources Unit, Institute for Environment and Sustainability, Joint Research Centre, European Commission, Via E. Fermi 2749, TP261, 21027 Ispra (VA), Italy

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b Water

Abstract

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Particle swarm optimization (PSO) has received considerable attention as a global optimization technique from scientists of different disciplines around the world. In this article, we illustrate how to use PSO for inverse modeling of a

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coupled flow and transport groundwater model (MODFLOW2005-MT3DMS)

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in a fractured gneiss aquifer. In particular, the hydroPSO R package is used as optimization engine, because it has been specifically designed to calibrate environmental, hydrological and hydrogeological models. In addition, hydroPSO

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implements the latest Standard Particle Swarm Optimization algorithm (SPSO2011), with an adaptive random topology and rotational invariance constituting the main advancements over previous PSO versions. A tracer test conducted in the experimental field at TU Bergakademie Freiberg

(Germany) is used as case study. A double-porosity approach is used to simulate the solute transport in the fractured Gneiss aquifer. Tracer concentrations obtained with hydroPSO were in good agreement with its corresponding observations, as measured by a high value of the coefficient of determination and a low sum of squared residuals. Several graphical outputs automatically generated by hydroPSO provided useful insights to assess the quality of the calibration results. ∗ Corresponding

author Email addresses: [email protected] (Ramadan Abdelaziz), [email protected] (Mauricio Zambrano-Bigiarini) 1 Present address: Dept. of Environmental Engineering, Faculty of Environmental Sciences and EULA-Chile Centre, University of Concepci´ on, Concepci´ on, Chile.

Preprint submitted to Journal of Contaminant Hydrology

June 3, 2014

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It was found that hydroPSO required a small number of model runs to reach the region of the global optimum, and it proved to be both an effective and efficient optimization technique to calibrate the movement of solute transport

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over time in a fractured aquifer. In addition, the parallel feature of hydroPSO

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allowed to reduce the total computation time used in the inverse modeling process up to an eighth of the total time required without using that feature. This work provides a first attempt to demonstrate the capability and versatility of

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hydroPSO to work as an optimizer of a coupled flow and transport model for contaminant migration.

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Keywords: MODFLOW2005, MT3DMS, hydroPSO, calibration, inverse modeling, fractured aquifer, gneiss aquifer

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1. Introduction and Scope

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Over the last decades, groundwater has got a major role in sustaining the economy of certain regions, by meeting ever-increasing water demands (Shah, 2006). In particular, fractured aquifers are important water resources around

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the world (Dietrich et al., 2005), playing a significant role in the development of geothermal resources or the safe disposal of high-level radioactive waste (Hurter and Schellschmidt, 2003). Flow in a fractured granite and gneiss aquifer is highly heterogeneous over

a range of scales (Bonnet et al., 2001) and several uncertainties arise from the heterogeneous flow in the aquifer (Zhang, 2001; Neuman, 2005). This may have significant implications for water resources management, from borehole to catchment scale. In addition, a proper understanding of flow heterogeneity in the aquifer is of great importance for groundwater source protection and for our ability to predict the movement and fate of contaminants in the fractured gneiss (Krupka et al., 1999). Consequently, there is a need for better understanding the flow heterogeneity in fractured aquifers and to be able to predict the nature of this heterogeneity (Kr´ asn` y and Sharp, 2007). Unfortunately, very often the previous task has to be performed on the basis of a relatively sparse data set.

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Traditionally, the heterogeneity of the fractured gneiss has been represented as an Equivalent Porous Medium (EPM) (e.g. Pankow et al., 1986; Scanlon et al., 2003). Flow transport in EPM assumes that diffusion rates are fast relative to

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the fractured groundwater velocity (Pankow et al., 1986). However, for media

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where the matrix blocks have a high porosity for fluid storage while the fractures have a high permeability for fluid transport, a Double-Porosity (DP) medium (Barenblatt et al. 1960) is a more suitable approach (Zhang and Sun, 2000).

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The DP assumes that the medium can be represented as a superposition of these two systems with different hydraulic and solute transport properties, in both of

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which the fractures are accounted for implicitly (e.g. Gerke and van Genuchten, 1993a,b; Vogel et al., 2000). The two pore systems interact by exchanging water and solute in response to pressure head and concentration gradients. In the

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dual-porosity model, the groundwater flow, as well as convection and dispersion

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of solutes, takes place only in the fractures, while diffusion of solutes between the fractures and matrix connects the two systems. The dual-porosity approach is selected in this work, instead of EPM, because it is considered to provide a better representation of the transport process in a fractured gneiss rock.

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Several aquifer properties are difficult to be directly measured at the correct

spatial scale. Therefore they have to be estimated through inverse modeling or a calibration procedure. Inverse modeling techniques are widely used in groundwater, being PEST (Doherty, 2005, 2013) and UCODE (Poeter et al., 2005) two of the most widespread inverse codes. PEST and UCODE perform inverse modelling by estimating parameter values that minimize a weighted least-squares objective function using non-linear regression (Hill and Tiedeman, 2006). However, the main drawback of all gradient-based techniques such as PEST and UCODE2005 are the dependency of the optimization results upon the initial guess about the solution (initial point of the optimization), which may result in finding a local optimum instead of the global one (Hsu et al., 2011). Particle Swarm Optimization (PSO) is a global optimization technique introduced by Eberhart and Kennedy in 1995, inspired by the social behavior 3

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of bird flocks looking for corn (Eberhart and Kennedy, 1995; Kennedy and Eberhart, 1995). PSO shares some similarities with genetic algorithms (Eberhart and Shi, 1998), but it has no presence of evolutionary operators such as

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crossover or selection. In the last decades, PSO has been applied in several

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fields (see Poli et al., 2007), e.g. in structural design (Schutte and Groenwold, 2005), geothermal heating planning (Ma et al., 2013; Beck et al., 2010), finance and economics (Huang et al., 2006), and image and video analysis applications

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(Slade et al., 2004; Wachowiak et al., 2004). In the water resources field, Sedki and Ouazar (2010) coupled MODFLOW2000/2005 with PSO for parameter es-

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timation purposes; Gill et al. (2006) used a multi-objective PSO to calibrate a well-known conceptual rainfall-runoff model; and Zambrano-Bigiarini and Rojas (2013) used a state-of-the art PSO version to calibrate a surface water model

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(SWAT-2005) and a groundwater model (MODFLOW-2005). However to the

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best of our knowledge, PSO has never been used to estimate the parameters of a coupled flow and transport model to simulate the transport of pollutants in a fractured gneiss aquifer. Therefore, this work provides a first attempt to demonstrate the effectiveness and efficiency of PSO for calibrating a coupled

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flow and transport model. It is worth to note that several implementations of PSO are available in lit-

erature, as OSTRICH (Matott, 2005) and squad/MADS (Vesselinov and Harp, 2012), but most of them implement the basic PSO configuration proposed in 1995. In this work we use hydroPSO (Zambrano-Bigiarini and Rojas, 2013), a model-independent R package that implements the latest Standard Particle Swarm Optimization 2011 (SPSO2011). In addition, hydroPSO was selected because it is a stand-alone software specifically designed for hydrological and environmental applications, with user-friendly plotting functions which facilitate the interpretation and assessment of the optimization results. hydroPSO is used here, for the first time, to estimate the parameters of a solute transport model in a gneiss fractured aquifer. The main objective of this paper is to test the capability and versatility of hydroPSO to work as an optimizer of a coupled flow and transport model for contaminant migration. 4

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2. Methodology 2.1. Study area

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The experimental area at the TU Bergakademie Freiberg Campus is located in Saxony, Germany, close to a polymetallic sulfide deposit (see Figure 1). The

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Inner Grey Gneiss is a met-granite (Tichomirowa, 2003), which is not only the most common rock type within the Freiberg region but also the oldest rock of

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the Ore Mountains. The density of the Freiberg Grey Gneiss is 2.645 kg m−3 as determined by the Pykonmeter method (Kranz and Dillenardt, 2010). The Freiberg test field contains six wells that were drilled for research and

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educational purposes. One of the wells was drilled to a depth of 100 m and 5 wells to a depth of 50 m. The upper part of the boreholes (3 to 5 m) is protected by a steel stand pipe; the deeper part was left as open borehole. A major

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horizontal fracture zone was determined to be 11 m below the ground surface

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by geophysical and hydraulic tests (Abdelaziz and Merkel, 2012; Abdelaziz et al., 2013), while Packer tests, geophysical logs, and tracer tests were conducted in the test field.

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Figure 1: Location of the test site in Freiberg and research wells (modified after Junghans and Tichomirowa 2009).

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2.2. Tracer Tests Tracers may be organic or inorganic constituents in water, small particles or isotopes (K¨ ass et al., 1998). They can be introduced into the environment by

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natural processes, human activities (e.g. nuclear weapon tests, industrial emis-

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sions) or by intention during a tracer experiment (Singhal and Gupta, 2010). Tracer tests are used in surface and groundwater to investigate flow and transport processes in heterogeneous porous media, in particular to estimate effective

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parameters describing the aquifer system (NRC, 1996; Ptak et al., 2004). Tracer tests may be used in a fractured aquifer to provide a better understanding of

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the connectivity in the fractured aquifer.

Several methods for carrying out tracer tests at field scale are available today, under natural or forced hydraulic gradient conditions (NRC, 1996; Ptak et al.,

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2004). In the natural gradient method a tracer is injected into the groundwa-

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ter and moves under natural hydraulic gradient conditions in an undisturbed groundwater flow field (NRC, 1996; Singhal and Gupta, 2010). In the forced gradient method the movement is induced by groundwater pumping or groundwater/tracer solute injection.

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Tracer tests can be carried out under steady or transient states, and the

interpretation of the test can be done through numerical or analytical analysis. The question that arises is whether there is any ideal tracer for porous and fractured aquifers. Actually, there is no ideal tracer substance for all sites. However, there is always an optimum tracer for a certain site, but it may not necessarily be suitable for another area. The objective of the test plays an important role in the selection of tracer substance, sampling scheme, number of tracers used, tracer properties, and to avoid any complexity in designing the tracer test. A tracer that fits for porous media do not necessarily fits for fractured aquifers, due to the double-porosity behavior of many aquifers (Davis et al., 1980). However, NaCl seems to be an appropriate tracer because of its low cost, conservative character, non-toxicity, and ease to be used and detected (Field, 2002). The natural gradient technique was used in this research to identify the 7

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Figure 2: Set up of the tracer test (no scale).

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transport parameters for the fractured zone. The advantage of this technique is its consistency with natural conditions while the disadvantage is the long duration of the experiment. Figure 2 shows the tracer test with NaCl performed

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in the fractured Gneiss aquifer, by using a two-packer system to inject the tracer in a single fracture zone. In our experiments, a tracer solution was injected under steady-state conditions, over a period of 6.5 hours into the observation well 4, while concentrations were recorded as depth-integrated breakthrough curve at the monitoring well 6 (see Figure 1). Further information and technical details about the tests and solute analysis are provided in Abdelaziz and Merkel (2012) and Abdelaziz et al. (2013). 2.3. Groundwater model The partial differential equations that govern the solute transport in groundwater are non-linear and have to be solved simultaneously. In this study, the finite-difference model MODFLOW (Harbaugh, 2005) is coupled to the second generation of the transport model MT3D, known as MT3DMS (Zheng and Wang, 1999; Zheng, 2006; Zheng et al., 2010) to implement a three dimensional

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groundwater flow and solute transport model in a double-porosity fractured aquifer.

MODFLOW simulates both confined and unconfined flow in three-dimensional

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heterogeneous aquifer systems, while MT3DMS is a three-dimensional model

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that can simulate multi-species transport of solutes. MT3DS considers advection, dispersion/diffusion, and chemical reactions, by using spatially distributed hydrogeological conditions. MT3DS has been extensively tested, and is known

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to be robust, accurate, and computationally efficient. MT3DMS was selected to carry out a three dimensional simulation of the solute transport model, be-

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cause it is one of the few three-dimensional and publicly-available model codes implementing the dual-porosity approach directly. Both model codes can use several types of time-dependent boundary conditions and forcing terms, which

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are implemented with modular structures, allowing users to modify them for

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any particular application.

Based on the characterization of the study site (Abdelaziz et al., 2013; Abdelaziz and Merkel, 2012) and using the double-porosity approach, the fractured aquifer is represented as three main layers, where fractures have different hy-

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draulic and solute transport properties than the two matrix blocks (Figure 3). The implementation of the groundwater flow and solute transport model is based upon a three dimensional finite difference grid, considering the geological setting described in Abdelaziz et al. (2013) and Abdelaziz and Merkel (2012). Since the inverse problem is ill-posed and affected by model uncertainties, the parameters used to characterize the spatial distribution of hydrogeological parameters are estimated through calibration. Physical ranges used in the calibration process are based on prior expert knowledge, which is subject of ongoing research. Among the calibration parameters, equivalent hydraulic conductivity (HK ), porosity (p), immobile porosity (imp), horizontal transverse dispersivity ratio (TRPT ), and vertical transverse dispersivity ratio (TRPV ) are assumed to be unknown for both the matrix blocks and the fractured zone. Longitudinal dispersivity was assumed to a known parameter, with a value of 0.08 [m] (Abdelaziz and Merkel, 2012). Table 1 shows the set of 8 parameters selected to be 9

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estimated through inverse modeling and the physical ranges used for calibration, based on prior expert knowledge (Abdelaziz and Merkel, 2014).

Notwithstanding usually the Darcy’s law is not appropriate for modeling

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flow into fractures, the Reynolds number for both the matrix block and the

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fractured layer was within the range of validity of the Darcy’s law.

(a) Conceptual model used and parameters being calibrated.

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(b) Boundary conditions selected for the conceptual model. Figure 3: Conceptual model used to represent the groundwater flow and solute transport model in the fractured gneiss aquifer system. (a) calibration parameters. (b) boundary conditions.

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Table 1: Parameters of the flow and transport model to be estimated through inverse modeling.

ID

Calibration Range

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Units

Parameter Name

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Meaningful physical ranges used for calibration are based on prior expert knowledge.

Min

Hydraulic conductivity of the matrix blocks

[m day

Hydraulic conductivity of the fractured layer

[m day−1 ]

]

HK 2

9.00E-01

Max

Value

1.55E+00

1.02E+00

HK 3

3.70E+01

3.98E+01

P24

5.50E-02

6.70E-02

5.63E-02

5.00E+01

Porosity of the fractured layer

[-]

P3

8.70E-02

1.20E-01

1.06E-01

Horizontal transverse dispersivity ratio

[-]

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−1

Calibrated

TRPT

1.80E-01

2.10E-01

2.03E-01

[-]

TRPV

2.70E-01

3.10E-01

2.79E-01

[-]

imp24

1.80E-02

2.30E-02

2.04E-02

[-]

imp3

5.00E-02

6.00E-02

5.14E-02

[-]

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Porosity of the matrix blocks

Vertical transverse dispersivity ratio Immobile porosity of the matrix blocks

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Immobile porosity of the fractured layer

2.4. Inverse Modeling

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Parameter estimation is a challenging task for groundwater modelers due

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to the many uncertainties associated with the conceptual model, observations and model parameters (Carrera et al., 2005), which usually leads to an ill-posed problem (Yeh, 1986). In contrast to gradient-based techniques, such as the widespread PEST (Doherty, 2005, 2013) and UCODE (Poeter et al., 2005),

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global-search methods are able to identify parameter values that produce the best fit to observations and prior information, regardless of the degree of model nonlinearity and the presence of local minima (Hill and Tiedeman, 2006). Within the recently developed global optimization techniques, Particle Swarm

Optimization (PSO, Kennedy and Eberhart 1995; Eberhart and Kennedy 1995) has received a surge of attention given its flexibility, ease of implementation, and efficiency (Poli et al., 2007), and it was selected to calibrate the coupled MODFLOW-MT3DMS solute transport model in the gneiss fractured aquifer. PSO is a population-based technique which shares some similarities with genetic algorithms (GA), but without crossover or selection (Eberhart and Shi, 1998). In PSO, a ”swarm” of candidate solutions (called ”particles” in PSO terminology) flies around the user-defined parameter space looking for the best model performance. The position of each particle is updated taking into account its own best ”position” and the best position in its neighborhood. 11

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Since 1995, a plethora of variants of the canonical PSO algorithm have appeared in the literature (see e.g. Poli et al., 2007), aimed at improving its performance. In particular, the Standard Particle Swarm Optimization 2011

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(SPSO-2011) is a major improvement over previous PSO versions (Clerc, 2012),

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with an adaptive random topology and rotational invariance constituting the main advancements (Clerc, 2012; Zambrano-Bigiarini et al., 2013). For each

the velocity is updated as follows,

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~ ti , ||Gr ~ ti − X ~ t ||) and particle, a random point is defined in the hypersphere H(Gr i

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~ it ||) − X ~ it ~it + H(Gr ~ ti , ||Gr ~ ti − X ~ t+1 = ω V V i

(1)

while the particle’s position is updated following equation (2).

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~ t+1 = X ~t +V ~ t+1 X i i i

(2)

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~ t+1 are the previous and new particle velocities, respectively; ~ t and V where V i i ~ t and X ~ t+1 are the previous and new position of each particle, respectively; X i i i = 1, 2, . . . , N , with N equal to the swarm size, and t = 1, 2, . . . , T , with T equal

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to the maximum number of iterations. ω is the inertia weight that controls the impact of the previous velocity of the particle on its current one. At each iteration the swarm radius (δ t ) is computed as the median Euclidean

~ and any given particle distance between the position of the swarm optimum (G) ~ i ), as follow (modified after Evers and Ghalia (2009)): (X → − → − δ t = mediank X ti − G t k

(3)

where k · k denotes the Euclidean norm. Let diam(Ψ) = kranged (Ψ)k be the t diameter of the search space, then the normalised swarm radius (δnorm ) is used

as a measure of convergence, indicating how close particles are to each other: t δnorm =

δt diam(Ψ)

(4)

Further details about SPSO-2011 can be found on Clerc (2012) and ZambranoBigiarini et al. (2013). 12

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SPSO-2011 was run using the open source implementation provided by the hydroPSO R package v0.3-3 (Zambrano-Bigiarini, 2014; Rojas and ZambranoBigiarini, 2012; Zambrano-Bigiarini and Rojas, 2013), which is a model- and

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platform-independent optimization software specifically designed for environ-

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mental applications. To run hydroPSO with MODFLOW-MT3DMS only a few basic text files have to be prepared by the user, with no need of template or instruction files as in PEST or UCODE. Two (problem-specific) ASCII files are:

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(i) ParamFiles.txt, which defines the names and location of the parameters to be calibrated and, (ii) ParamRanges.txt, which provides meaningful parameter

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ranges for the calibration. In addition, the name of the executable model code and instructions for reading model outputs have to be provided in a separated R script (R Core Team, 2013), which is easily adapted from an existing template

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provided with hydroPSO (Rojas and Zambrano-Bigiarini, 2012). For this work, a

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batch file was used for running in sequence MODFLOW and MT3DMS, while a short FORTRAN code was written to speed up the reading of parameters being optimized (preproc.exe). Figure 4 describes the main files used for coupling the MODFLOW and MT3DMS model with hydroPSO.

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For optimization, the sum of the squared residuals (SSR, Equation 5) was

selected to measure the goodness-of-fit (GoF) value between observed concentrations (Cio ) and its corresponding model outputs (Cis ) at each time step. After some preliminary trials, the maximum number of iterations T was set to 35 and the number of particles was set to 8 (i.e., 280 model runs), while other PSO arguments were kept in the default values provided by hydroPSO. Parallel computations were activated in order to accelerate the optimization process in an Intelr CoreTM i7-875K machine (four cores and 8 threads), 2.93 GHz.

SSR =

n X

(Cis − Cio )

i=1

13

2

(5)

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Figure 4: Flow chart showing the functions and files involved in the optimization the groundwater flow and solute transport model. *.xxx stands by all the input files with extension xxx. In particular, *.ins are all the instruction files, *.mto are all the files with simulated

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concentrations, while *.dsp, *.btn, and *.rct represent the setting up the transport model, and *.PVAL is the input file with flow parameters.

3. Results and Discussion One of the most common ways to measure the performance of any model

is by plotting the simulated values against its corresponding observations. Figures 5a and 5b show breakthrough curves and a scatter plot, respectively, for the best model output obtained with hydroPSO and the observed concentrations at observation well 6. Figure 5b depicts a good correspondence between the monitored solute concentration and their corresponding best simulated values, with a small over-estimation of low and high concentration values, and a slight underestimation of medium concentrations. In addition, the coefficient of determination (R2 ) is equal to 0.933, indicating that a high proportion of the variance of the response variable is explained by the model. 14

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Figure 6 shows the evolution of the global optimum (best model performance for a given iteration, i.e., smallest SSR) and the normalized swarm radius (δnorm , a measure of swarm spread on the search space) against the number of itera-

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tions. It is clear that both the global optimum and the normalized swarm radius

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decrease with increasing number of iterations, indicating that most of the particles converged into a small region of the solution space. Moreover, Figure 6 shows that only 5 iterations (i.e., 5 x 40 = 200 model runs) were required for

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finding the region of the global optimum, and the remaining iterations were just used for refining the search. The optimum value found for the sum of the

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squared residuals was 2.97E+03 [mg2 l−2 ].

Figure 5a shows that the breakthrough curve is positively skewed with a rapid rise to the peak, which lies after the mean tracer transit time. The long

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tail on the right is assumed to be due to dead end pores and retention of the

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tracer in the fractured rock (Abdelaziz et al., 2013). The long tail may also be linked to low hydraulic conductivity in some parts of the fractured rock. In addition, solute transport in the matrix block plays a major role on the diffusive transport of the tracer. Immobile porosity has a major impact on the late arrival

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of the tracer, but only a minor effect on the peak of the breakthrough curve. Further information on the analysis of the breakthrough curve can be found in (Abdelaziz et al., 2013). Boxplots in Figure 7a are useful non-parametric diagrams for summarizing

the statistical distribution of the sampled values. The top and bottom of the box show the first and third quartiles, respectively, while the horizontal line inside the box represent the second quartile (the median). The notches extend to +/ − 1.58 · IQR/sqrt(n), where IQR is the interquartile range and n is the number of points. Finally, points outside the notches are considered to be outliers. Dotty plots in Figure 7b show the parameter values against its corresponding goodness-of-fit value (SSR) obtained during the optimization. They are useful for identifying parameter ranges that produce the best model performance or with roughly the same model performance (equifinallity, Beven and Binley 1992). Visual inspection of Figure 7a shows that for 5 out of 8 15

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parameters (HK 2, HK3, P3, imp24 and imp3 ) the optimum value found during the optimization coincides with the median of all the sampled values, while for TRPT and TRPV those values were very close, confirming that most of the

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particles converged into a small region of the solution space. On the other hand,

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Figure 7b shows that the optimum values found for parameters HK 3 and P3 are really well defined, whereas HK 2, imp24 and imp3 show a wider region around their optimum.

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Histograms in Figure 8a are a graphical representation of sampled frequencies for each parameter, and are used to provide an estimate of the probability

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density function thereof. In addition, empirical cumulative distribution functions (ECDFs) in Figure 8b are used to estimate the true underlying cumulative distribution function (CDF) of the sampled points. Both figures confirm that

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parameters P3 and imp24 follow a near-normal distribution, while only imp3

for calibration.

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shows a sampled distribution highly skewed towards the lower boundary used

Figure 9 shows (quasi)-three dimensional dotty plots, which summarize interactions among parameters by projecting the SSR response surface onto the

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parameter space (for different pairs of parameters). In this figure low model performance is represented with dark-red color while high model performance is shown with light-blue color. In general, it can be seen that particles are spread all over the parameter space, indicating a good exploratory capability of PSO. However, regions with bad model performance have a low density of points while regions with better model performance are more densely sampled, confirming the good ability to exploitation of PSO. This figure shows that the optimum values found for HK 3 and P3 define a narrow range of the parameter space with high model performance. On the other hand, the coupled flow and transport model can achieve a good model performance for a wide range of values of imp24, confirming that this parameter is not well identified. Finally, a correlation matrix among parameters values and model performance (SSR) is shown in Figure 10. The panel above the diagonal shows the numeric (absolute) value of the Pearson’s product moment correlation coef16

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ficient between paired samples plus its statistical significance as stars. The bottom panel shows bivariate scatterplots between each column and row of the matrix, with a fitted line obtained using locally-weighted polynomial regression

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(loess, Cleveland 1979). On the diagonal, a histogram of each variable sampled

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during the optimization. This figure shows that the highest (linear) correlation between the measure of model performance (SSR) is obtained for the hydraulic conductivity (HK 3 ) and immobile porosity (imp3 ) of the fractured layer, fol-

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lowed by the hydraulic conductivity of the matrix blocks HK 2. In addition, a high (linear) correlation is observed between the hydraulic conductivity of the

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matrix blocks HK 2 and imp3 ; and the porosity of the matrix blocks P24 and the vertical transverse dispersivity ratio TRPV. The same figure shows that nonlinear relations are the dominating behavior between most of the calibrated

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parameters, e.g. HK 2 vs P24, HK 2 vs imp3, and HK 3 vs imP3.

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The transport of contaminants in the fractured aquifer is represented in MT3DMS by a convection-dispersion reaction (CDR) equation, whose solution is affected by several well-known problems of oscillation and artificial dispersion.

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In particular, even small negative concentration values may lead to nonlinear amplification of the reaction source/sink term of the coupled reactive-transport model, resulting in physically erroneous and unstable results (Herrera and Valocchi, 2006). In this work, the advective term of the CDR equation was solved by using the Generalized Conjugate-Gradient (GCG) method, with upstream thirdorder total-variation-diminishing (TVD) scheme, because it helps to diminish the artificial oscillations. The initial transport step size (DT 0) was specified as zero in the Basic Transport Package (BTN). Therefore, for each model run DT 0 was automatically computed by MT3DMS to meet various stability constraints that are solution dependent, using a maximum Courant number equal to 1.0. In spite of using the GCG-TVD solver, oscillation and artificial dispersion problems were detected in our simulations, posing an additional difficulty to allow hydroPSO to find a (near-)optimum parameter set. For that reason, we suggest to other global optimization users to define -a priori- a small value for DT 0, in

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order to avoid instability during the optimization. Comparing the results shown in this section with previous work done by Abdelaziz et al. (2013), where Lattice Boltzmann was implemented for the same

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case, we found that the use of hydroPSO provided a great improvement in the

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simulated concentration values. Therefore, using a global optimizer as hydroPSO proved to be important for simulating the coupled flow and solute transport mechanism. This approach opens promising possibilities for similar groundwater

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applications.

All the optimization was run taking advantage of 8 threads available in

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the Intelr CoreTM i7-875K machine used for the experiments. A single model run of the coupled flow and transport model takes 14 minutes, and therefore the 280 model runs involved in the optimization (8 particles x 35 itera-

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tions) would have taken 2.72 days (65.33 h or 3920 min). However, setting

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parallel="parallelWin" and nnodes=8 in hydroPSO reduced the total computation time up to 8.19 h, what represents (roughly) an eighth of the total time required without using the parallel argument. Therefore, hydroPSO allowed a significant decrease of the total computation time required for the inverse

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modeling process.

4. Conclusions

Despite the challenges involved in the representation of fractured rocks due to

the heterogeneity and uncertainty in the fracture properties, the double-domain approach is still capable of making a reasonable and accurate prediction of the plume distribution. The hydroPSO R package was successfully applied to calibrate a coupled groundwater flow and solute transport model for conservative tracer tests. hydroPSO proved to be both an efficient and effective tool for inverse modeling of the coupled MODFLOW-MT3DMS model, allowing a quick assessment of the calibration results. The main conclusions and recommendations are as follows:

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• A high value of the coefficient of determination (R2 = 0.93) and relatively low sum of squared residuals (2.97E+03 [mg2 l−2 ]) indicate that simulated values obtained after calibration matched the corresponding observed con-

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centrations quite well.

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• The parallel feature of hydroPSO allowed to reduce the total computation time used in the inverse modeling process up to an eighth of the total

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time required without using the parallel argument. • Solver packages for flow (MODFLOW) and solute transport (MT3DMS)

in the stability thereof.

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should be choosing carefully because their set up plays an important role

• The initial transport step size (DT0) of MT3DMS is also an important

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small as possible.

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(indirect) parameter to stabilize hydroPSO, and it should be chosen as

Acknowledgements

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The authors thank the associate editor and two anonymous reviewers whose

constructive comments significantly improved the original manuscript.

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ECDF of P24

0

0.6 0

0.2

0.4

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Probability

0.8

Q50 sim: 1.017

ECDF of HK_3

1

1

ECDF of HK_2

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red line indicates the optimum value found for each parameter.

0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

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0.021

0.050

imp24

0.054

0.058

imP3

(b) ECDFs of parameter values. Horizontal gray dotted lines represent a cumulative probability equal to 0.5 (median of the distribution). Vertical gray dotted lines represent a cumulative probability of 0.5 (its value is shown in the upper part of each figure). Figure 8: Graphical summary of parameter values sampled during the optimization. (a) Histograms. (b) Empirical cumulative distribution functions (ECDFs).

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RI SC

P3

0.060

0.10

1.0 1.2 1.4

HK_2

HK_2

0.065

0.021 0.020

0.060

0.019

P3

0.022

MA

1.0 1.2 1.4

P24

imp24

0.055

1.0 1.2 1.4

HK_2

HK_2

0.12 0.11 0.10

D

0.055 40

45

40

HK_3

0.019

40

45

0.10

0.055

HK_3

imp24

0.060 0.058 0.056 0.054 0.052 0.050

0.10

P3

0.12

imP3

0.020

0.11

0.065

0.055

40

0.021 0.020 0.019 0.12

0.065

0.022 0.021 0.020 0.019 0.055

0.065

P24

SSR

0.022

● ● ● ● ● ●

0.021 0.020 0.019 0.050 0.056

P3

45

HK_3

P24

0.022

0.10

45

0.060 0.058 0.056 0.054 0.052 0.050

P24

imp24

P3

0.021

AC CE P

imp24

0.12

0.022

0.060 0.058 0.056 0.054 0.052 0.050

HK_3

TE

HK_2

imp24

1.0 1.2 1.4

0.060 0.058 0.056 0.054 0.052 0.050

1.0 1.2 1.4

imP3

40

NU

P24

HK_3

45

0.11

imP3

0.12 0.065

imP3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

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(2.97e+03,3.25e+03] (3.25e+03,4.44e+03] (4.44e+03,1.22e+04] (1.22e+04,1.62e+04] (1.62e+04,2.99e+04] (2.99e+04,5.47e+04]

imP3

Figure 9: Model performance (SSR) projected onto the parameter space (for different pairs of parameters). Low model performance is represented with dark-red color while high model performance is shown with light-blue color. Parameter names are described in Table 1.

31

RI 0.0926

***

0.146

38

42

0.111

0.0484

*

0.329

MA

1.5

***

0.410

0.0797

0.0583

***

*** 0.244

D

0.105

***

*** 0.396

*

TRPT

0.118

0.0666

***

TE

0.090

*

0.333

*

0.0784

0.140

0.143

0.0796

0.244

0.179

0.0875

0.0579

0.180

0.31

*** 0.561

0.357

P3

TRPV

***

**

0.27

AC CE P

0.29

0.410

0.354

***

0.137

***

0.474

**

0.156

*

0.0712

0.120

0.050 0.054 0.058

1.3

** 0.182

P24

0.31

.

50 46

HK_3

0.29

1.1

0.296

0.27

0.9

***

0.328

0.120

0.056

***

0.105

0.062

0.090

0.210

50

0.195

46

SC

42

NU

38

HK_2

0.019

0.021

imp24

0.050 0.054 0.058

imP3

***

SSR

40000

0.524

10000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

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0.9

1.1

1.3

1.5

0.056

0.062

0.180

0.195

0.210

0.019

0.021

10000

40000

Figure 10: Correlation matrix between parameters and model performance (SSR). Horizontal and vertical axes show the physical range used for the calibration of each parameter.

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Graphical Abstract

D

TE

AC CE P

PT

RI

SC

NU

MA

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT

Highlights:

PT

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NU MA

·

D

·

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Coupled groundwater flow and solute transport model in a fractured gneiss aquifer. A double-porosity approach is used to simulate the solute transport. Standard Particle Swarm Optimization 2011 (SPSO-2011) is used for inverse modeling. SPSO-2011 proved to be both efficient and effective to calibrate the coupled model. The parallel feature of hydroPSO greatly reduced the total optimization time.

AC CE P

·