Varun Chaudhary

Mohan Sai

Kiran Nartu

Sriswaroop Dasari

Sai Sree

Meenakshi Varahabhatla

Abhishek Sharma

Madhavan Radhakrishnan

Srinivas Aditya Mantri

Stéphane Gorsse

Narendra B Dahotre

M.S.K K Y Nartu

S M Varahabhatla

R V Ramanujan

R Banerjee
email: raj.banerjee@unt.edu

Magnetic and mechanical properties of additively manufactured Alx(CoFeNi) complex concentrated alloys

Keywords: Complex concentrated alloys, additive manufacturing, microstructure, magnetic properties

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The development of novel magnetic materials has gained high interest in recent years, especially due to an urgent increase in demand for superior rotating electrical machines operating at high frequency, large torque and elevated service temperatures [START_REF] Silveyra | Soft magnetic materials for a sustainable and electrified world[END_REF][START_REF] Chaudhary | Accelerated and conventional development of magnetic high entropy alloys[END_REF][START_REF] Balakrishna | Design of soft magnetic materials[END_REF][START_REF] Chaudhary | Additive manufacturing of magnetic materials[END_REF]. Most commercial magnetic alloys are based on one or two principal elements alloyed with other minor elements and exhibit low efficiency for next generation high stress, elevated temperature, and large rotating frequency electrical machine applications. Such applications require both mechanical as well as magnetic properties.

There is significant interest in exploring such novel materials using the strategy of complex concentrated alloys (CCA) having multiple principal elements in high concentration [START_REF] Chaudhary | Accelerated and conventional development of magnetic high entropy alloys[END_REF][START_REF] Chaudhary | Additive manufacturing of magnetic materials[END_REF][START_REF] Lan | Highly complex magnetic behavior resulting from hierarchical phase separation in AlCo(Cr)FeNi high-entropy alloys[END_REF][START_REF] Han | Ultrastrong and Ductile Soft Magnetic High-Entropy Alloys via Coherent Ordered Nanoprecipitates[END_REF][START_REF] Fu | Exceptional combination of soft magnetic and mechanical properties in a heterostructured high-entropy composite[END_REF][START_REF] Han | A mechanically strong and ductile soft magnet with extremely low coercivity[END_REF]. A range of CCA have been processed using different techniques and some of them exhibited very interesting magnetic and mechanical properties. However, only a tiny fraction of the possible compositions of CCA has been explored. Tuning the length scale of the complex multiphase microstructures of selected CCA by thermal treatment can result in an optimum set of mechanical and functional properties. Therefore, further research must be performed to discover a novel CCA with a promising multi-property set.

Direct energy deposition (DED), an additive manufacturing (AM) technique, is an emerging method to rapidly explore multiple compositions through accelerated methodologies [START_REF] Teh | High throughput multi-property evaluation of additively manufactured Co-Fe-Ni materials libraries[END_REF][START_REF] Ahn | Directed Energy Deposition (DED) Process: State of the Art[END_REF][START_REF] Colorado | Sustainability of additive manufacturing: the circular economy of materials and environmental perspectives[END_REF]. It can also provide near net shape components, which are challenging to build using conventional techniques, such as casting. DED-AM has been used for printing several types of magnetic materials including, Ni-Fe-V, Fe-Ni-Mo, Co-xFe, Ni-xFe, Fe-Co-Ni, AlxCoCrFeNi, AlxCrCuFeNi2, etc [START_REF] Chaudhary | Additive manufacturing of magnetic materials[END_REF][START_REF] Teh | High throughput multi-property evaluation of additively manufactured Co-Fe-Ni materials libraries[END_REF][START_REF] Chaudhary | Additive manufacturing of functionally graded Co-Fe and Ni-Fe magnetic materials[END_REF][START_REF] Nartu | Magnetic and mechanical properties of an additively manufactured equiatomic CoFeNi complex concentrated alloy[END_REF][START_REF] Mikler | Laser Additive Manufacturing of Magnetic Materials[END_REF].

Conventional approaches of finding a promising material typically involve synthesis and characterization of a few compositions, are expensive, inefficient and time consuming. Instead, we used a thermodynamics-guided approach to reduce the experimental load to identifying promising materials. The primary objective of this study is to rapidly investigate the change in microstructural, magnetic and mechanical properties with Al content in AM processed Alx(CoFeNi) CCA, thus identifying promising compositions. Al alloying content in Alx(CoFeNi) (x = 0, 10, 30) was chosen such that three major phase fields in the pseudo-binary isopleth can be explored. High temperature magnetometry, in-situ high temperature X-ray diffraction (XRD), Transmission Electron Microscopy (TEM) were performed to determine the magnetic and structural properties at high temperatures.

Alx(FeCoNi) CCA were printed by an Optomec LENS (Laser Engineering Net Shaping) 750 system using Fe, Co, Ni and Al powders (purity >99.9%), with sizes ranging from 50 to 150 m. CCA of desired compositions were roller-mixed prior to the deposition. A 1500 W Nd: YAG laser producing near-infrared radiation at a wavelength of 1.064 m was used to deposit the CCA. The processing parameters were: 300 W laser power; 0.5 mm diameter laser beam on the sample surface; 12.7 mm/s laser scan speed; 0.254 mm vertical layer spacing; 0.381 mm hatch width with 90° rotation in the hatch direction between layers. These combinations of laser parameters provided an input energy fluence of 47.24 J/mm 2 . The input energy fluence/energy density was calculated using the expression E = P/V × D, where P is the laser power, V is the scan speed, and D is the laser beam diameter. The oxygen level in the glove box was maintained below 10 ppm to fabricate samples with dimensions of 25.4 mm × 25.4 mm ×25.4 mm. The deposited build was then separated from the steel substrate and subsequently sliced for microstructural and magnetic property measurements using a KENT USA (WSI-200) electric discharge machine (EDM).

The phase constitution was examined by X-ray diffraction (XRD) in a Bruker 8D Advance diffractometer (room temperature) and Bruker 8D Discover diffractometer (for in-situ high temperature measurements), using CuK (1.54 Å) radiation. TOPAS Software was used for Rietveld refinement of the XRD data. Room temperature and high temperature magnetic properties were measured using a physical property measurement system (PPMS) equipped with a vibrating sample magnetometer (VSM). The sweep rate for magnetization versus temperature measurements in the cases of both heating and cooling was fixed at 3 K/min. Figure 1 (a) shows a pseudo-binary isopleth for the Alx(CoFeNi) alloy system, with changing Al content, calculated using the TCHEA4 database in Thermocalc. Three compositions of Alx(CoFeNi) with x = 0, 10 and 30 were chosen based on the different phase fields they exhibit.

The isopleth predicts a single-phase FCC for x = 0 and x = 10, and B2 for x = 30, at a temperature of 1300 C. With decreasing temperature, there is a substantial change in the phase stability, especially for x = 10. The room temperature XRD data of CoFeNi (Al-0) and Al10(CoFeNi) (Al-10) exhibited a single-phase FCC phase, while Al30(CoFeNi) (Al-30) was a single B2 phase (Figure 1b). The rapid cooling rate (10 3 -10 5 C/sec) in direct energy deposition processes leads to the high temperature phase quenched in to room temperatures [START_REF] Gibson | Directed energy deposition processes[END_REF] T or M-T) from room temperature to 725 C in both heating and cooling modes at an applied magnetic field of 100 Oe. All three graphs of M vs. T did not follow the same path while heating and cooling. The most significant difference is in the case of Al-10, there is a steep decrease in the M value around 475°C-500°C due to the ferromagnetic to paramagnetic transition (Tc) of the FCC matrix. A re-entrant magnetic moment upon heating which begins at ~ 500 C, goes to a maximum at ~ 625 C can be clearly observed. This re-entrant magnetization in Al-10 possibly indicates the decomposition of the FCC matrix due to heating.

A rationalization of this anomalous change during the M-T cycle is discussed in later sections of this paper.

As printed samples exhibited a Hc of 10.9 Oe, 4.0 Oe and 14 Oe, Ms of 160 emu/g, 125.7 emu/g and 61.1 emu/g, Tc of >725 C, 475C and 397 C for x =0, 10 and 30 in Alx(CoFeNi) CCA, respectively. The variations in Ms, Hc and Tc with the alloy composition for as deposited and after the M-T heating/cooling cycle are plotted in figure 2 (d, e, andf). Interestingly, the most significant change in all these properties is observed for Al-10. Ms improved from 125.7 to 139.5 emu/g, Hc increased from 4 to 197.3 Oe, and Tc changed from 475C to 627 C. both as printed and after MT samples. For as printed samples, there is a clear trend of increasing Vickers microhardness with higher Al content in CoFeNi. Such a rise in microhardness with increasing Al content has been reported in other multicomponent systems, e.g., AlxCrCuFeNi2 [START_REF] Borkar | A combinatorial assessment of AlxCrCuFeNi2 (0 < x < 1.5) complex concentrated alloys: Microstructure, microhardness, and magnetic properties[END_REF], AlxCoCrFeNi [START_REF] Li | Evaluation of microstructure and mechanical property variations in AlxCoCrFeNi high entropy alloys produced by a highthroughput laser deposition method[END_REF][START_REF] Wang | Effects of Al addition on the microstructure and mechanical property of AlxCoCrFeNi high-entropy alloys[END_REF], primarily due to the change in microstructure from FCC to BCC/B2 as well as due to solid solution strengthening arising from multiple principle elements. After the M-T cycle, there is almost no change in microhardness values for Al-0 and Al-30. However, the heat-treated Al-10 sample exhibited ~3 times increase in hardness compared to the hardness for the as deposited Al-10 sample. The microhardness of 577 HV for heat treated Al10(CoFeNi) CCA is higher than those of AlxCrCuFeNi2 [START_REF] Borkar | A combinatorial assessment of AlxCrCuFeNi2 (0 < x < 1.5) complex concentrated alloys: Microstructure, microhardness, and magnetic properties[END_REF] and AlxCoCrFeNi [START_REF] Li | Evaluation of microstructure and mechanical property variations in AlxCoCrFeNi high entropy alloys produced by a highthroughput laser deposition method[END_REF][START_REF] Wang | Effects of Al addition on the microstructure and mechanical property of AlxCoCrFeNi high-entropy alloys[END_REF] CCA systems, irrespective of the Al content and processing route.

Interestingly, the hardness value of this additively manufactured Al10(CoFeNi) CCA is higher than the hardness for a nearly identical composition Al0.3CoFeNi processed by conventional casting and superior to famous soft magnetic materials, e.g., permalloy (Fe15Ni80Mo5), permendur (Fe49Co49V2), etc [START_REF] Dasari | Highly tunable magnetic and mechanical properties in an Al0.3CoFeNi complex concentrated alloy[END_REF]. [START_REF] Houin | Relationship between Structure and Mechanical Properties of Pearlite between 0.2% and 0.8%C[END_REF] that developed during the heating/cooling cycle of the M-T analysis. A similar lamellar microstructure has been previously reported by Dasari et al. in conventionally fabricated and thermo-mechanically processed Al-(9 at%)CoFeNi [START_REF] Dasari | Hierarchical Eutectoid Nano-lamellar Decomposition in an Al0.3CoFeNi Complex Concentrated Alloy[END_REF]. On the other hand, such lamellar features are not observed in the other two compositions, i.e., Al-0 and Al-30 conditions that were subjected to similar heating/cooling cycles during the M-T run.

To understand the evolution of the secondary lamellar features, in-situ high temperature XRD was performed on the as printed Al-10 condition at different temperatures (27 C, 500 C, 600 C and 725C) during heating and cooling, the resultant plots are shown in figure 4(b). The maximum temperature of 725 C is the temperature limit of the instrument. At room temperature, the Al-10 condition is found to consist of a face centered cubic (FCC) phase (𝑓 =100%). As the temperature increased to 500 C, the formation of a BCC phase was observed. The diffraction peak intensity of BCC phase increases with higher temperature up to 725 C and remained constant during cooling to room temperature. However, a simulated pseudo-binary isopleth using Thermocalc predicts a three-phase field, FCC+B2+L12 for this alloy composition between 500C and 600C (refer to Fig. 1(a)). Since the (110) reflections observed in the XRD pattern (Fig. 4(b)) are allowed for both BCC and B2 phases, it is difficult to distinguish between them using a conventional XRD analysis. The change in phase fraction with increasing temperature is plotted in figure 4(c). Nearly 70% of the FCC matrix decomposed/transformed to the BCC/B2 phases. The backscattered SEM images along with Phase and IPF maps obtained from the EBSD scans performed on the Al-10 condition (post MT) are provided in Supplementary Figure S2. The phase map identifies the untransformed regions as FCC and the transformed regions, i.e., the lamellar microstructure, as single-phase BCC. The lamellae are too fine (nano-scale) to be resolved by SEM-EBSD. Therefore, TEM analysis was performed to identify the phases present within the nano-lamellar microstructure. The improvement in Ms observed before and after MT measurements (from 125.7 to 139.5 emu/g) could be associated with the additional phases formed during the heating cycle of the MT measurements. Atom probe tomography (APT) composition analysis in identical hierarchical FCC/L12 + BCC/B2 lamellar microstructure in conventionally processed and subsequently annealed Al-(9 at%)FeCoNi CCA revealed a higher percentage of ferromagnetic elements (Fe, Co and Ni) in the BCC phase compared to the FCC, L12 and B2 phases [START_REF] Dasari | Hierarchical Eutectoid Nano-lamellar Decomposition in an Al0.3CoFeNi Complex Concentrated Alloy[END_REF].

Therefore, the higher observed Ms value in the present study could be due to the formation of a signification fraction (70%) of the BCC+B2 phases (from the XRD analysis, Fig. 4(c)) in the Al-10 condition during the heating cycle of the M-T analysis.

Due to the higher Tc of the new B2 and BCC phases forming from the decomposition, the M value increases and reaches a maximum around 625 ℃ (Refer to Fig. 2(b)) during the heating cycle of the M-T analysis. Although the increase in BCC/B2 phase is observed until the maximum temperature of 725 C, the magnetic moment starts decreasing from 650 C due to the ferromagnetic to paramagnetic transition of the BCC/B2 phase. M-T measurements upon cooling did not show such a change in magnetic behaviour, and the major transition temperature shifted to higher values by ~ 150 C. The Curie temperature for the Al-10 alloy measured from the M-T behavior of the cooling cycle was found to be 627 °C. There is no structural transition in this alloy during the cooling cycle, since the decomposition of the FCC phase has already occurred during the heating cycle.

A substantial increase in the Hc can be attributed to the numerous interfaces arising from the FCC/L12 + BCC/B2 complex hierarchical microstructure. The coercivity of the single-phase FCC in the case of as deposited sample increased ~50 times due to the decomposition into the FCC/L12 + BCC/B2 heterostructure, after M-T measurements. The heterostructure with an abrupt chemical/phase change in the samples heat treated at 725 C increases the resistance to the movement of magnetic domain walls, leading to a higher value of coercivity compared to single-phase samples.

Conclusions

Using the example of a AlxCoFeNi CCA, this study presents a thermodynamics-guided approach for designing novel additively processed CCAs with a good balance of magnetic and mechanical properties. The example presented in this paper investigated the effect of Al content in the AlxCoFeNi CCA, on the microstructure, magnetic and mechanical properties. These properties were found to change multi-fold due to the composition and microstructural complexity. The addition of non-ferromagnetic Al progressively changes the phase stability from single phase FCC to lamellar FCC/L12 + BCC/B2 finally to a single B2 phase with attendant reductions in the overall Ms and Tc. These variations in phase stability were accurately predicted via thermodynamic modelling. Equiatomic CoFeNi exhibits a high saturation magnetisation (Ms) and high Curie temperature (Tc) but high coercivity (Hc) and poor mechanical properties. On the other hand, as printed Al10(CoFeNi) CCA exhibit low value of Hc, reasonably good Ms and Tc and a high mechanical hardness. The re-entrant magnetic moment at higher temperatures in Al10(CoFeNi) CCA was associated with a decomposition of the FCC phase to form a lamellar FCC/L12 + BCC/B2 microstructure, introducing BCC/B2 phases with higher concentrations of ferromagnetic elements. This thermodynamics-guided alloy design through additive manufacturing can be employed in developing materials with balanced magnetic and mechanical properties.

  Imaging and Electron backscatter diffraction (EBSD) analysis was performed in an FEI-Quanta Nova-Nano SEM 230. OIM™ software was used to produce the Inverse Pole Figure (IPF) maps and the Phase maps from the EBSD scans. TEM foils were prepared inside an FEI Nova 200 Nanolab dual-beam Focused ion beam (FIB/SEM) and were characterized using a FEI Tecnai G2 F20 ST S/TEM, a transmission electron microscope (TEM operating at 200 kV) outfitted with a (scanning transmission electron microscopy-energy dispersive spectroscopy) STEM-EDS detector.

  . The change in lattice parameters with increasing Al content in Alx(CoFeNi) is shown in supplementary Figure S1.

Figure 1 .

 1 Figure 1. (a) Psuedo-binary isopleth for varying Al in ternary CoFeNi system (b) Rietveld refinement of the X-ray diffraction patterns and (c) SEM backscatter images of Alx(CoFeNi) CCA.

Figure 2 (

 2 Figure 2 (a) shows the field dependence of magnetization at room temperature for as deposited Al-0, Al-10 and Al-30 alloys. The saturation magnetization (Ms) decreased from 160 to 61.1 emu/g with increasing Al content from 0 to 30 in Alx(CoFeNi) CCA. Such a decrease in Ms with increasing Al content can be rationalized by considering the magnetic behaviour of the elements present in Alx(CoFeNi) CCA. The paramagnetic element (Al) replaced the ferromagnetic elements (Co, Fe, Ni), reducing the effective ferromagnetic moment and therefore Ms.The Curie temperature is another important performance metric especially for high temperature applications. Figure2(b) shows the temperature dependence of the magnetic behaviour (M vs.

Figure 2 .

 2 Figure 2. Magnetic properties of Alx(CoFeNi) CCA at room temperature and elevated temperatures (a) Magnetization (M) as a function of applied magnetic field at room temperature (b) Magnetization versus temperature for heating and cooling at an applied field of 100 Oe. (c) M-H loops after M-T measurement shown in "b". (d) saturation magnetization (Ms), (e) coercivity (Hc) and (f) Curie temperature (Tc) with changing Al content in Alx(CoFeNi) CCA for as printed and after MT samples.

Figure 3

 3 Figure 3 represents the microhardness plot for varying compositions in Alx(CoFeNi) CCA for

Figure. 3

 3 Figure. 3 Variation in microhardness of as printed and after HT with changing the Al content in Alx(CoFeNi) CCA

Figure 4 .

 4 Figure 4. (a) Backscattered electron images of Alx(CoFeNi) CCAafter MT measurements. (b) The in-situ high temperature XRD results for Al10(CoFeNi) CCA in the range from room temperature to 725 C for both heating and cooling modes. (c) Variation in phase fraction as a function of temperature measured over heating and cooling cycles.

Figure 5 .

 5 Figure 5. For Al-10 condition samples: (a) SEM backscatter image showing the lamellar microstructure. (b) SAED pattern recorded along the [100] BCC Zone Axis from one of the alternating lamellae shown in Fig. 5(a). (c) DFTEM image, recorded using the {001} B2 superlattice reflection, outlined in red in Fig. 5(b). (d) SAED pattern recorded along the [110] FCC Zone Axis from the other alternating lamellae shown in Fig. 5(a). (e) DFTEM image, recorded using a {001} L12 superlattice reflection, outlined in green in Fig. 5(d). (f) SAED pattern obtained from the interface of the FCC/BCC lamellae showing the Nishiyama-Wasserman (N-W) orientation relationship between the phases.

Acknowledgments

This work is supported by AME Programmatic Fund by the Agency for Science, Technology and Research, Singapore under Grant No. A1898b0043. The authors acknowledge the infrastructure and support of Center for Agile & Adaptive and Additive Manufacturing (CAAAM) funded through State of Texas Appropriation #190405-105-805008-220 at the University of North Texas. The authors would also like to acknowledge Materials Research Facility (MRF) for access to advanced characterized techniques.


François Lehmann

Mohammad Mahdi Rajabi
email: mmrajabi@modares.ac.ir

Benjamin Belfort

Frederick Delay

Marwan Fahs

Philippe Ackerer

Anis Younes

Bayesian inversion of laboratory experiments of transport through limestone fractures

Introduction

It is estimated that up to a quarter of the world's population depends on karstic aquifers of mostly limestone nature, for water supply [START_REF] Ford | Karst hydrogeology and geomorphology[END_REF]. Fracture (or conduit) networks are generally considered the dominant flow and transport pathways in limestone media, because limestone matrix permeability and porosity are low, and secondary permeability is high [START_REF] Florea | Solute transport through laboratory-scale karstic aquifers[END_REF]. Water and contaminants travel at much greater speeds within the fracture networks compared with the surrounding porous limestone matrix [START_REF] Li | Contaminant sequestration in karstic aquifers: Experiments and quantification[END_REF], which renders limestone aquifers highly vulnerable to contamination [START_REF] Morales | Predicting travel times and transport characterization in karst conduits by analyzing tracerbreakthrough curves[END_REF]. In the past three decades, numerous studies have provided insights into flow dynamics and transport mechanisms in limestone fractured systems. These studies have mostly concluded that accurate simulation of water and contaminant movement in fractured limestone is complicated by the uncertainty of characterizing fracture properties [START_REF] Qian | Experimental study of turbulent unconfined groundwater flow in a single fracture[END_REF]. This uncertainty stems from the high spatial variability of fracture properties, and the lack of direct measurements methods [START_REF] Wang | A hybrid inverse method for hydraulic tomography in fractured and karstic media[END_REF].

Breakthrough curves (BTCs), obtained from dye or salt tracer tests, are useful for the estimation of model parameters [START_REF] Jørgensen | Modeling of non-reactive solute transport in fractured clayey till during variable flow rate and time[END_REF][START_REF] Helmke | Simulating conservative tracers in fractured till under realistic timescales[END_REF] and/or the characterization of fracture properties (Goldscheider and Drew, 2014;[START_REF] Zhao | Laboratory investigation and simulation of breakthrough curves in karst conduits with pools[END_REF]. The approach often consists of employing an inverse method based on optimization or inference processes, to fit observed and model-simulated BTCs. Various fracture parameters, such as effective porosity, aperture, length, roughness, density, and network patterns can affect the shape of BTCs [START_REF] Mohammadi | The Effect of Hydraulic Gradient and Pattern of Conduit Systems on Tracing Tests: Bench-Scale Modeling[END_REF], and thus could be estimated from them. BTCs can also be employed for estimating contaminant transport parameters in fractured media, such as dispersivity, retardation and degradation, as well as the characterization of conduit-matrix interactions [START_REF] Goldscheider | Tracer tests in karst hydrogeology and speleology[END_REF]. Key inverse methods in this context include gradient-based minimization (e.g. [START_REF] Abdelaziz | Sensitivity analysis of transport modeling in a fractured gneiss aquifer[END_REF][START_REF] Hawkins | Predictive Inverse Model for Advective Heat Transfer in a Short-Circuited Fracture: Dimensional Analysis, Machine Learning, and Field Demonstration[END_REF] and sample-based Bayesian inference such as Markov chain Monte Carlo (MCMC) (e.g. [START_REF] Somogyvári | Synthetic fracture network characterization with transdimensional inversion[END_REF][START_REF] Zhou | Thermal experiments for fractured rock characterization: theoretical analysis and inverse modeling[END_REF].

However, estimating fracture parameters from BTCs is not a straightforward task [START_REF] Hartmann | Karst water resources in a changing world: Review of hydrological modeling approaches[END_REF]. On the one hand, the operation is prone to common inverse modeling challenges such as conceptual model uncertainty, instability, ill-posedness, non-convergence, non-unique and non-optimal solutions [START_REF] Wang | A hybrid inverse method for hydraulic tomography in fractured and karstic media[END_REF][START_REF] Rajabi | Model-data interaction in groundwater studies: Review of methods, applications and future directions[END_REF]. On the other hand, BTCs may exhibit asymmetry with long tails [START_REF] Massei | Interpreting tracer breakthrough tailing in a conduit-dominated karstic aquifer[END_REF][START_REF] Labat | Transfer function approach for artificial tracer test interpretation in karstic systems[END_REF][START_REF] Fiori | Power law breakthrough curve tailing in a fracture: The role of advection[END_REF] or multiple peaks due to factors such as the strong heterogeneity of limestone aquifers, the presence of pools along main fractures and multiple flow paths [START_REF] Hauns | Dispersion, retardation and scale effect in tracer breakthrough curves in karst conduits[END_REF][START_REF] Wang | Influence of dual conduit structure on solute transport in karst tracer tests: An experimental laboratory study[END_REF]. Capturing such complicated BTCs has proven problematic, which affects the choice of the modeling approach (Field and Leji, 2012). Partitioning models are often used to simulate such BTCs, but these models require a large number of parameters [START_REF] Zoghbi | Simple transport models for karst systems[END_REF]. Bearing in mind that not all parameters are of equal importance with respect to sensitivities of BTCs, determining the most significant factors so as to limit the number of parameters is necessary [START_REF] Li | An analysis of tracer flowback profiles to reduce uncertainty in fracture-network geometries[END_REF].

Different approaches have been explored for fracture parameter estimation from BTCs. Most previous studies in this context are based on 'hypothetical' examples (e.g. Somogyvári et al., 2017;[START_REF] Zhou | Thermal experiments for fractured rock characterization: theoretical analysis and inverse modeling[END_REF]. This is a key shortcoming, as hypothetical examples often lack the complexity of real-world problems, and are limited to what the chosen numerical modeling approach can reproduce. Moreover, the true values of fracture parameters are often not known at the field-scale, thus rendering method validation using field-scale data often impractical. As a consequence, bench-scale laboratory experiments under controlled conditions are important for testing the validity of these inversion methods [START_REF] Pastore | Fluid Flow, Mass, and Heat Transport Laboratory Experiments in Artificially Fractured Rock[END_REF]. The fact that there are few laboratory-based studies in this context can be mainly attributed to the difficulty of the experimental process, as laboratory models must reflect a reduction in the scale of the physical parameters and dimensions while maintaining similar forces [START_REF] Zhang | Laboratory simulation of groundwater hydraulic head in a karst aquifer system with conduit and fracture domains[END_REF], the latter often being represented by Reynolds, Froude and Peclet dimensionless numbers for flow and solute transport in fractured media (Palmer 1999;[START_REF] Florea | Solute transport through laboratory-scale karstic aquifers[END_REF].

Among the reported laboratory-scale studies, most have focused on the understanding of BTCs and the relevant flow and transport processes. Examples include the study of scale effects on BTCs by [START_REF] Hauns | Dispersion, retardation and scale effect in tracer breakthrough curves in karst conduits[END_REF], interaction and mass transfer between a single conduit and its bounding porous medium by [START_REF] Li | Laboratory simulation of solute transport and retention in a karst aquifer[END_REF], sequestration mechanisms including the exchange of water and solute between conduits and porous matrix by [START_REF] Li | Contaminant sequestration in karstic aquifers: Experiments and quantification[END_REF], empirical models of water hydraulic head under different hydrological conditions by [START_REF] Zhang | Laboratory simulation of groundwater hydraulic head in a karst aquifer system with conduit and fracture domains[END_REF], and the effect of variations in hydraulic gradient and conduit systems on BTCs by [START_REF] Mohammadi | The Effect of Hydraulic Gradient and Pattern of Conduit Systems on Tracing Tests: Bench-Scale Modeling[END_REF]. In these and other similar studies, analyzing flow and solute transport within a single or a twofracture system often forms the basis for studying complex fracture networks [START_REF] Liu | Experimental and numerical study of bimolecular reactive transport in a single rough-wall fracture[END_REF].

The estimation of limestone fracture parameters using laboratory-scale models for method analysis and validation often relies upon single or dual pipes made of silicon or PVC to model the fractured medium (Field and Leji, 2012;[START_REF] Zhao | Effects of flow rate variation on solute transport in a karst conduit with a pool[END_REF][START_REF] Wang | Influence of dual conduit structure on solute transport in karst tracer tests: An experimental laboratory study[END_REF]. These studies use nonlinear least-squares estimation or the MCMC method for the estimation of fracture parameters. [START_REF] Jørgensen | Modeling of non-reactive solute transport in fractured clayey till during variable flow rate and time[END_REF] for bromide transport through undisturbed soil columns containing a high number of macropores (fractures and biopores). In this study, the measured BCTs were used to assess the suitability of different modeling approaches for the simulation of flow and transport in clayey till. Likewise, [START_REF] Helmke | Simulating conservative tracers in fractured till under realistic timescales[END_REF] compared different modeling approaches to simulate solute transport through fractures in a laboratory column of till. Their results showed that the different modeling approaches employed could satisfactory reproduce the observed BTCs, which suggests that more elaborate models do not necessarily render more accurate results.

Realistic laboratory experiments have been conducted by

The simplest way for modeling flow and transport in fractured-porous media is based on the equivalent porous medium (EPM) approach in which the fractured porous medium is replaced by an equivalent porous medium with the same hydraulic conductivity [START_REF] Helmke | Simulating conservative tracers in fractured till under realistic timescales[END_REF]. However, the most common models for the simulation of flow and transport in fractured porous media are the discrete fracture and double porosity models [START_REF] Jørgensen | Modeling of non-reactive solute transport in fractured clayey till during variable flow rate and time[END_REF][START_REF] Helmke | Simulating conservative tracers in fractured till under realistic timescales[END_REF][START_REF] Nikan | Numerical approach for modeling fractal mobile/immobile transport model in porous and fractured media[END_REF]. Some sophisticated variants also exist such as the triple continuum models [START_REF] Wu | A triple-continuum approach for modeling flow and transport processes in fractured rock[END_REF], the multi-rate mass transfer models [START_REF] Guo | Adaptive Multirate Mass Transfer (aMMT) model: A new approach to upscale regional-scale transport under transient flow conditions[END_REF] or the fractal mobile/immobile models [START_REF] Nikan | Numerical approach for modeling fractal mobile/immobile transport model in porous and fractured media[END_REF].

The mobile/immobile (MIM) model is based on the dual porosity approach and considers the fractures as the mobile region and the matrix as the immobile region. Advection and dispersion occur exclusively in the mobile region, and the immobile region is a sink that stores the solute.

The exchange between the mobile and the immobile regions is often ruled by a first-order (FO) mass transfer coefficient approach [START_REF] Coats | Dead-end pore volume and dispersion in porous media[END_REF]. The resulting model has been widely used for flow and transport in fractured porous media because of its simplicity compared to discrete fracture models. Indeed, the latter require that the location, shape, orientation, size, aperture, and hydraulic and solute transport properties of each fracture to be explicitly specified as model inputs.

The objective of the present work is threefold. First, we aim at developing a novel experimental setup for the study of transport through limestone fractures using salt tracer BTCs. The proposed bench-scale setup includes multiple fractures, and is based on real limestone beams.

These features are novel compared to previous work. Second, we validate and analyze the results of an inverse method using laboratory-scale data obtained under controlled conditions. Third, we investigate the degree to which three transport models can reproduce the experimental results under different (slow, medium, and fast) flow conditions. The first transport model, named ADE, is based on the EPM approach and corresponds to the linear advection dispersion equation. The second model, named FOMIM (first-order mobile-immobile), is based on the MIM approach relying upon a linear first-order transfer function between mobile and immobile zones. For its part, the third model, named NLMIM (non-linear mobile-immobile), uses a nonlinear transfer function between the fractures and the matrix.

The structure of the paper is as follows. In Section 2 we describe the experimental setup, and then briefly explain the three transport models. Section 2 also includes a description of the inverse method employed. Then, the experimental BTCs and the simulation results of the BTCs are presented and analyzed in Section 3. The paper is complemented in Section 4 by a review of our main findings and prospects for future research.

Materials and Methods

Laboratory Experiment

A fractured porous medium is constructed at the laboratory scale using a column of inner Three experiments were performed under fast, medium, and slow flow velocities through the column corresponding to injected flow rates of in Q = 20.0, 2.0, and 0.2 ml/min, respectively.

Each experiment was repeated three times.

Mathematical Models

Three models are investigated to simulate transport through the column. The first model assumes an EPM, thus the transport through the column can be ruled by the following linear ADE:

dc = 6 cm Lc = 15 cm (a) (b)   CC D UC t z z z             (1) where C [-] is the non-dimensional (normalized) concentration, Lm D U d   [L 2 T -1 ] is the dispersion coefficient, U [LT -1 ] is the fluid velocity, m d is the molecular diffusion coefficient
assigned a small value of 5×10 -10 m 2 /s (as a common value in the literature), and

L  [L] is the dispersivity coefficient.
The second transport model FOMIM is based on the MIM approach and assumes a first-order mass transfer between the fractures and the matrix:

    0 m im m m im m im C C C UC D t t z z z C CC t                           (2)
where m C and im C stand for the (normalized) concentrations in the mobile and immobile phases, respectively,  [T -1 ] is the first-order mass transfer coefficient between the fractures and the matrix [START_REF] Coats | Dead-end pore volume and dispersion in porous media[END_REF]. It is worth noting that the coefficient  can be dependent on the mean flow velocity, as shown in [START_REF] Brissaud | Transferts dispersifs dans une tranche verticale de milieu aquifere[END_REF] and [START_REF] Herr | Experimental studies of mass transport in porous media with local heterogeneities[END_REF].

The third transport model uses the MIM approach but assumes a nonlinear transfer function between the mobile and the immobile continuums as:

  2 1 im m C t C       (3)
where  [T -1 ] and  [-] are parameters of the nonlinear transfer function.

For the three transport models, initial conditions correspond to a zero solute concentration inside the column. The lower boundary    corresponds to a null concentration gradient.

Numerical Solutions

The transport equation for both models is solved using the Discontinuous Galerkin (DG) method. Usually, transport at small scales under pressure gradients (e.g., laboratory-scale experiments) is advection-dominated, making the differential transport equations to become hyperbolic [START_REF] Godlewski | Numerical approximation of hyperbolic systems of conservation laws[END_REF]. Those are characterized by moving sharp fronts that classical numerical methods (such as finite element or finite volume methods) fail to capture, with the result of solutions plagued by non-physical oscillations and/or numerical diffusion [START_REF] Huyakorn | Computational Methods in Subsurface Flow[END_REF]. For its part, the DG method leads to a robust and accurate numerical scheme for problems involving sharp fronts [START_REF] Tu | A slope limiting procedure in discontinuous Galerkin finite element method for gasdynamics applications[END_REF]. The DG method yields a highly-resolved scheme, which maintains the local mass conservation of finite volume (FV) methods but also allows for high-order approximations [START_REF] Kirby | Local time stepping and a posteriori error estimates for flow and transport in porous media[END_REF]. When applied to hyperbolic systems, the DG method is clearly superior to finite element methods [START_REF] Siegel | Solution of the advection-diffusion equation using a combination of discontinuous and mixed finite elements[END_REF][START_REF] Arnold | Unified analysis of discontinuous Galerkin methods for elliptic problems[END_REF].

The transport equation is solved with the explicit upwind DG scheme. The stability of the scheme, is ensured by the use of a suitable slope limiter to damp out the spurious oscillations produced close to discontinuities or strong gradients of the approximate solution [START_REF] Younes | An efficient geometric approach to solve the slope limiting problem with the discontinuous Galerkin method on unstructured triangles[END_REF]. The obtained scheme is "total variation diminishing" and stable for Courant-Friedrichs-Lewy (CFL) numbers equal or less than 0.5.

We use a linear discontinuous approximation of the concentration at each element. The degreesof-freedom are the mean and the first spatial derivative of the concentration inside each element.

This choice allows for an easy and efficient combination between the DG method for advection and the FV method for dispersion without resorting to any operator splitting [START_REF] Younes | Solving the advection-dispersion equation with discontinuous Galerkin and multipoint flux approximation methods on unstructured meshes[END_REF]. For the nonlinear transport problem (model NLMIM), a Newton-Raphson algorithm is employed with a precision of 10 -7 . The time step length is managed using the Richardson extrapolation, that is an adaptive time-stepping procedure based on the posteriori error estimate [START_REF] Fahs | On the efficiency of the direct substitution approach for reactive transport problems in porous media[END_REF].

Bayesian Parameter Inference and Model Selection

The vector of observations mes y consists of the outflow EC measured at different times, for the three (fast, medium, and slow) experiments assembled within the same "synthetic" transport scenario of successive mass transfer at diverse velocities. All the parameters of the ADE model of Eq. ( 1), the FOMIM transport model of Eq. ( 2) and the NLMIM model of Eq. ( 3) are assumed unknowns, and sought by inverse modeling. In the case of the ADE transport model, the vector of unknown parameters is the second and third columns of Table 1.

  1 2 3 , , , L U U U   ξ
Regarding the FOMIM transport model, the coefficient  is considered as varying with the fluid velocity in the column [START_REF] Herr | Experimental studies of mass transport in porous media with local heterogeneities[END_REF], thus resulting in three sought coefficients (one per transport experiment). The vector of unknowns becomes

  1 2 3 1 2 3
, , , , , ,

L U U U      ξ .
For the last NLMIM transport model, two cases are investigated. In the first case, the vector of unknown parameters is

  1 2 3 , , , , , L U U U     ξ
where  and  are the transfer coefficients, assumed to be similar for all three experiments. In the second case, the coefficient  is considered as varying with the fluid velocity in the column, thus resulting in a vector of unknowns as

  1 2 3 1 2 3
, , , , , , ,

L U U U       ξ .
The parameters are estimated via a Bayesian approach combining the prior parameter information with the observations to determine the posterior probability distribution functions (PDFs) of model parameters. We rely upon the Markov chain Monte Carlo technique (MCMC) which has also been widely used by several authors in Hydrogeology (e.g. [START_REF] Rajabi | Efficient fuzzy Bayesian inference algorithms for incorporating expert knowledge in parameter estimation[END_REF][START_REF] Moreira | Bayesian estimation of the hydraulic and solute transport properties of a small-scale unsaturated soil column[END_REF][START_REF] Younes | Global sensitivity analysis and Bayesian parameter inference for solute transport in porous media colonized by biofilms[END_REF][START_REF] Linde | On uncertainty quantification in hydrogeology and hydrogeophysics[END_REF][START_REF] Wang | Influence of dual conduit structure on solute transport in karst tracer tests: An experimental laboratory study[END_REF].

MCMC generates random sequences of parameter sets, which converge asymptotically toward the target distribution. The statistical measures (e.g., mean and standard deviation) from the obtained distributions can then be used to estimate the mean parameter values and their confidence intervals to characterize parameter uncertainty. Using the Bayes theorem, the posterior density function of the parameters conditioned onto observations is as follows [START_REF] Gelman | Bayesian data analysis[END_REF]: In this work, the prior distributions for all parameters are assumed independent from one another and uniform. Further, large prior intervals are chosen for all parameters due to lack of knowledge of model parameter values. Error measurements of the output concentrations are assumed to be normally and independently distributed with zero mean and standard deviation equal to 0.02

     
C  
(for normalized concentrations bounded between 0 and 1).

Setting the calibration problem in a Bayesian framework yields the following posterior PDF:

    2 2 C C N mes C C SS p | exp        ξ ξy ( 5
)
where

          2 1 C N kk C mes mod k SS C C   
ξξ is the sum of the squares of differences between the observed and predicted concentrations,

 :         1 11 1 i i i mes H i i i mes p | q min , p | q         ξ y ξ ξ ξ y ξ ξ (6)
In this work, we use the DREAM(ZS) MCMC sampler [START_REF] Vrugt | A Shuffled Complex Evolution Metropolis algorithm for optimization and uncertainty assessment of hydrologic model parameters[END_REF] with three parallel chains. The results are considered stationary if the chains are not auto-correlated and if the Gelman and Ruban (1992) criterion is verified   The Bayesian information criterion (BIC), used for the selection of the best model is defined as [START_REF] Schwarz | Estimating the dimension of a model[END_REF]:

      ln | ln MAP mes C BIC p ξ y +K N 2 ( 8
)
where K is the total number of estimated parameters and   MAP mes p| ξy is the parameter density function evaluated at the MAP estimate [START_REF] Schöniger | Model selection on solid ground: Rigorous comparison of nine ways to evaluate Bayesian model evidence[END_REF]. Since each error has a noncorrelated (with the others) normal distribution:  

, C N  2 0
, it follows that [START_REF] Zhu | Comparison of six statistical approaches in the selection of appropriate fish growth models[END_REF]:

      mod | exp C mes MAP N ii MAP mes i C C CC p ξy             ξ 2 2 1 1 2 2 (9)

Results and Discussion

Experimental BTCs

The measured breakthrough curves at the outlet are depicted in Fig. 2 for the various experiments. Each of the fast (Fig. 2a), medium (Fig. 2b), and slow (Fig. 2c) flow rate experiment has been repeated three times. The results of Fig. 2 show a very good reproducibility of the experiments (repeating three times the same experiment results in almost similar breakthrough curves). It is worth noting that the breakthrough curves in Fig. 2 show a significant tail for the slow flow rate experiment (Fig. 2c). The tail is moderate with the medium flow experiment (Fig. 2b) and almost inexistent with the fast flow rate experiment (Fig. 2a).

Parameter Estimation for the ADE Model

For the ADE model, the MCMC sampler reaches convergence after 15,000 model runs. The results of Bayesian inversion are illustrated in Fig. 3. The diagonal plots show the posterior parameter distributions, whereas the off-diagonal plots display the pairwise correlations within the MCMC samples. Given the round-shaped patch delineated by pairwise correlations, there is almost no correlation between the four parameters, and all parameters show bell-shaped posterior distributions in the diagonal plots of Fig. 3. The results of the Bayesian calibration (transport parameters in Table 1) show that the four parameters are well-estimated with narrow confidence intervals. Fig. 4 depicts the measured output concentrations as well as the calibrated concentrations and the predictive uncertainty. In this figure, the green region represents the total predictive uncertainty which accounts for both the parametric uncertainty and measurement errors. Fig. 4a shows that the first experiment with the fast flow rate is well reproduced by the ADE model. Nevertheless, discrepancies between simulated and observed output concentrations can be observed for the second experiment (Fig. 4b). These discrepancies increase especially for the last experiment (Fig. 4c) dealing with the slow flow rate, with noticeable errors mainly in the leaching phase. The tails observed on the measured breakthrough curves for the medium-and slow-velocity experiments cannot be reproduced by the ADE transport model. The sum of the squared differences between the observed and predicted concentrations using parameters corresponding to the MAP and the BIC values are given in the Table 2.

Parameter Estimation for the FOMIM Model

The FOMIM includes a mobile and an immobile phases with a linear transfer function between phases. Three transfer coefficients  are calibrated corresponding to the fast, medium and slow flow rate experiments. The three transfer coefficients have prior uniform distributions over large intervals (between 0 and 100, see Table 3). The MCMC sampler reaches convergence after approximately 20,000 model runs. Fig. 5 shows the posterior parameter distributions and the pairwise correlations in the MCMC sample. The results of Fig. 5 show that there is no correlation between the seven parameters. Further, five parameters are well estimated corresponding to the three velocities, the dispersion coefficient and the last transfer coefficient (that of the slow flow rate experiment). All these five parameters have bell-shaped posterior distributions in the diagonal plots of Fig. 5. The posterior distributions of the transfer coefficients corresponding to the fast and medium flow rate experiments given in the diagonal plots of Fig. 5 show almost uniform distributions. The results of the Bayesian calibration given in Table 3 show that the three velocities are estimated with small confidence intervals, although the mean estimated values are significantly different from the estimated values using the ADE model (Table 1). The posterior confidence intervals of the transfer coefficients of the fast and medium flow rate experiments are almost similar to their prior intervals. This shows that these two parameters are insensitive (they have almost no influence on the BTCs). The transfer coefficient of the slow flow rate experiment is very sensitive since it's posterior interval is strongly reduced compared to the prior interval and as a consequence, the parameter is well identified.

Compared to the ADE model, the sum of the squares of differences between the observed and predicted concentrations for the three experiments with the FOMIM model reduces by only 8%

and the BIC by only 12% (Table 1). The measured breakthrough curves as well as the calibrated concentrations and the predictive uncertainty obtained with the FOMIM model are depicted in whereas, the BTC of the medium flow rate experiment (Fig. 6b) is less well reproduced, especially the tail of the curve. For the slow flow rate experiment, compared to the ADE model, the FOMIM model allows to better reproduce the tail of the measured BTC. However it fails to well reproduce the first sharp front. As a consequence, the overall fitting of this experiment is not satisfactory and Fig. 6c shows strong discrepancies between measured and observed BTCs.

Parameter Estimation for the NLMIM Model

The nonlinear model includes an immobile phase and two associated transfer parameters (  and  , see Eq. 3) in addition to the single mobile phase of the linear model. The transfer parameters are a priori assumed uniformly distributed over the large intervals (between 0 and 50, see Table 4). Two scenarios are investigated. In the first, the transfer coefficient  is assumed constant for the three flow rate experiments, whereas in the second scenario,  may depend on the mean flow velocity (as for the parameter  with the FOMIM model).

In the case considering that the transfer parameter should be constant, irrespective of the velocity in the mobile phase, the MCMC sampler reaches convergence after approximately 25,000 model runs. Fig. 7 shows the posterior parameter distributions and the pairwise correlations in the MCMC sample. The results of the Bayesian calibration given in Table 4 show that the six parameters are well estimated with narrow confidence intervals. Fig. 8 shows measured breakthrough curves compared with calibrated concentrations, and the predictive uncertainty obtained with the NLMIM model. In Fig. 8b, the second experiment with the medium flow rate is very well reproduced by the numerical model, with a tail of the breakthrough curve well reproduced by the NLMIM model compared with rough results from the ADE and FOMIM models (see Fig. 4b and Fig. 6b).

However, the first experiment, with the fast flow rate (Fig. 8a), is not as well reproduced as by the ADE model (Fig. 4a) because the NLMIM model generates a short concentration tail not observed on the measured concentrations. In opposition, the third experiment with the slow flow rate (Fig. 8c) is better reproduced by the NLMIM model than the ADE and FOMIM models (Fig. 4c and Fig. 6c). Nevertheless, the tail of the simulated breakthrough in Fig. 8c is spread enough over time, but there is not enough mass to mimic the high concentration values observed in the experimental breakthrough curve. With a NLMIM model correctly simulating medium velocity breakthroughs, overestimating late concentration values at high fluid velocity, and underestimating late concentrations at small velocity, the model is still performing better than the ADE and FOMIM models. The feature is exemplified by the squared differences between the observed and predicted concentrations at the MAP; compared to the ADE model, it is reduced by 40%, and the BIC is significantly decreased from 9173 to 2862 (Table 2).

Parameter Estimation for the NLMIM Model with Variable Coefficient

In this case, the coefficient  is assumed to vary according to the fluid velocity in the column.

The nonlinear model is therefore calibrated using three transfer coefficients 1 ).

The results of the Bayesian calibration given in Table 5 show that the eight parameters are well estimated with narrow confidence intervals. Fig. 10 shows that the calibrated model correctly fits data from the measured breakthrough curves for the three velocities investigated by the experiments, with uncertainties on simulations (gray areas in Fig. 10) tightly bounding the observed concentration breakthroughs. The overall quality of the estimation has been significantly improved, with a squared difference between the observed and predicted concentrations at the MAP 6 times smaller than that of the previous nonlinear model with fixed  . The BIC is also significantly reduced from 2862 to -4557 (Table 2). Inversion of the three transport models is performed using Bayesian MCMC inference. Key findings of the current study can be summarized as follows:

1. The proposed multi-fracture experimental setup can produce asymmetric BTCs that are consistent with past field-and lab-scale studies.

  beads and small parallelepipedshaped limestone beams. The limestone used for the experiment is mainly composed of dolomite (CaMg(CO3)2). The porous medium is composed of silicon dioxide beads (SiLibeads Glass beads Type S N°4503, Sigmund Lindner) of 0.75 to 1.00 mm in diameter with a bulk density of 3 1510 kg m . Small parallelepiped-shaped limestone beams of 1 cm 2 section and 4 to 5 cm length are placed inside the porous medium as shown in Fig.1a.The column shell (Fig.1b) is a Plexiglas tube with the top and bottom end-plate assemblies held together with 3 threaded rods. Each end-plate assembly has one access port in its center, plate to support the porous medium and distribute the flow over the entire column section. At the top and bottom end plates, two electrical conductivity (EC) meters were used to continuously record (at 10 s time steps) the inlet and outlet concentrations. The two EC meters were calibrated against standard solutions, which allows for accurate (and reproducible) measurements of the EC. The injected aqueous solution is made of deionized water supplemented with sodium chloride (NaCl) at 2g/l. This mixture allows for obtaining a solution with a significant sensitivity of EC to the concentration, while density variations remain negligible. The column is mounted vertically with flow from bottom to top assured by a peristaltic pump. The flow rate was regularly controlled by weighing the cumulative outflow.

Figure 1 .

 1 Figure 1. The laboratory experiment setup: (a) Plexiglas flow cell (inner diameter

  0 z  corresponds to a Dirichlet condition with the normalized concentration fixed to one during solute injection, and to zero during clean water injection. The upper boundary condition   z L

  where 12 , UU and 3 U are the fluid velocity for the fast, medium and slow flow rate experiments, respectively, and L  is the dispersivity, assumed to be independent of the fluid velocities in all three experiments. Initially, the parameters of the model are assumed uniformly distributed over the large intervals reported in

  is the likelihood function measuring how well the model outputs are in agreement with the observations mes y , and   ξ p is the prior PDF of ξ , which encapsulates any prior knowledge about the unknown parameters.

Fig. 2 .

 2 Fig. 2. Breakthrough curves for fast (a), medium (b), and slow (c) flow rates through the column, each

Fig. 3 .

 3 Fig. 3. MCMC solution for the ADE model. The diagonal plots represent the posterior distribution of the parameters. The off-diagonal scatterplots represent the correlations in the MCMC draws.

Fig. 4 .

 4 Fig. 4. Observed concentrations (dots), calibrated solution corresponding to the MAP (lines), and predictive uncertainty (grey band) with the ADE model for the fast (a), medium (b), and slow (c) flow rate experiments.

Fig. 5 .

 5 Fig. 5. MCMC solution for the FOMIM model. The diagonal plots represent the posterior distribution of the parameters. The off-diagonal scatterplots represent the correlations in the MCMC draws.

Fig. 6 .

 6 Fig. 6. Observed concentrations (dots), calibrated solution corresponding to the MAP (lines), and predictive uncertainty (grey band) with the FOMIM model for the fast (a), medium (b), and slow (c) flow rate experiments.

Fig. 6 .

 6 Fig. 6. The results of the fast and slow flow rate experiments are similar to those of the ADE

Fig. 7 .

 7 Fig. 7. MCMC solution for the NLMIM model. The diagonal plots represent the posterior distribution of the parameters. The off-diagonal scatterplots represent the correlations in the MCMC draws.

Fig. 8 .

 8 Fig. 8. Observed concentrations (dots), calibrated solution corresponding to the MAP (lines), and predictive uncertainty (grey band) with the NLMIM model for the fast (a), medium (b), and slow (c) flow rate experiments.

Fig. 9 .

 9 Fig. 9. Observed concentrations (symbols), calibrated solution corresponding to the MAP (lines), and predictive uncertainty (grey band) with the NLMIM model and variable coefficients  for the three experiments.

Fig. 10 .

 10 Fig. 10. Observed concentrations (dots), calibrated solution corresponding to the MAP (lines), and predictive uncertainty (grey band) with the NLMIM model and variable coefficients  for the fast (a), medium (b), and slow (c) flow rate experiments.

2.

  All the unknown parameters of the ADE model can be well-estimated with narrow confidence intervals. With respect to state estimation, the ADE model can reproduce with fair precision the BTC obtained under fast flow conditions but fails to reproduce the tails of the BTCs observed with the medium and slow flow rate experiments. 3. The FOMIN model improves the tailing of the BTCs but significant discrepancies remain between simulated and measured concentrations. The first-order transfer coefficients of the fast and medium flow rate experiment are insensitive and cannot be estimated from the BTCs contrarily to the transfer coefficient of the slow flow rate experiment which is well identified.4. The NLMIM model with a constant (kinetic) coefficient  has better performance in simulating the medium flow BTC, but performs worse for the fast flow conditions as it overestimates the concentration values characterizing the tail of the BTC.5. The nonlinear model with a velocity-dependent  coefficient is the only model correctly capturing the experimental BTCs under all three conditions of fast, medium, and slow flow rates. All its parameters are well estimated with narrow confidence intervals.The experimental results presented in this paper could be used to validate and analyze other inversion and data assimilation methods applicable to fracture parameter estimation. Future studies may consider sequential Bayesian filtering and smoothing methods, which have been less considered in this context. The proposed experimental setup can serve as a basic platform to investigate less-explored phenomena related to transport in limestone fractures, such as nonaqueous phase and colloid transport.

Table 1 .

 1 Prior intervals, estimated mean values and confidence intervals (CIs) for the parameters of the ADE model.

		Prior intervals		Estimations
	Parameter	Lower bound Upper bound	Mean	Standard deviation	95% Confidence Interval
	1 U [cm/s]	0.03	0.1	0.0352	1.1 10 -4	(0.0350-0.0354)
	2 U [cm/s]	0.003	0.01	0.0039	1.23 10 -5	(0.00385-0.00390)
	3 U [cm/s]	0.0003	0.001	4.42 10 -4	1.56 10 -6	(0.000439-0.000445)
	 [cm]	0.0	3.0	1.13	0.017	(1.096-1.163)
	L

Table 2 .

 2 The total number of estimated parameters (K), the squared differences between the observed and predicted (using parameters of MAP) concentrations, and BIC for the ADE, FOMIM and NLMIM models.

	Modeling Approach	K	C  N i 1 		mod ii mes CC 	  MAP ξ	 2	BIC
	ADE	4			6.12			9173.96
	FOMIM (variable  )	7			5.66			8044.77
	NLMIM-Scenario 1	6			3.59			2862.83
	NLMIM-Scenario 2 (variable  )	8			0.6162			-4557.80

Table 3 .

 3 Prior intervals, estimated mean values and confidence intervals (CIs) for the parameters of the FOMIM model.

		Prior intervals		Estimations
	Parameters	Lower bound Upper bound	Mean	Standard deviation	95% Confidence Interval
	1 U [cm/s]	0.03	0.1	7.24 10 -2	2.14 10 -4	(7.19 10 -2 -7.28 10 -2 )
	2 U [cm/s]	0.003	0.01	8.0 10 -3	2.51 10 -5	(7.94 10 -3 -8.05 10 -3 )
	3 U [cm/s]	0.0003	0.001	8.73 10 -4	4.13 10 -6	(8.6 10 -4 -8.81 10 -4 )
	1  [s -1 ]	0.0	100.0	56.59	25.33	(7 -100)
	1  [s -1 ]	0.0	100.0	48.12	29.48	(0 -100)
	1  [s -1 ]	0.0	100.0	1.68 10 -4	7.8 10 -6	(1.52 10 -4 -1.83 10 -4 )
	 [cm]	0.0	3.0	0.87	0.015	(0.84 -0.90)
	L

Table 4 .

 4 Prior intervals and estimated mean values and confidence intervals (CIs) for the parameters of the NLMIM model.

		Prior intervals		Estimations
	Parameters	Lower bound Upper bound	Mean	Standard deviation	95% Confidence Interval
	1 U [cm/s]	0.03	0.1	4.59 10 -2	4.08 10 -4	(4.51 10 -2 -4.67 10 -2 )
	2 U [cm/s]	0.003	0.01	4.93 10 -3	4.17 10 -5	(4.85 10 -3 -5.02 10 -3 )
	3 U [cm/s]	0.0003	0.001	5.48 10 -4	4.59 10 -6	(5.39 10 -4 -5.57 10 -4 )
	 [s -1 ]	0.0	50.0	2.77	8.4 10 -2	(2.61 -2.94)
	 [-]	0.0	50.0	7.17	4.15 10 -1	(6.36 -7.99)
	 [cm]	0.0	3.0	1.21	1.62 10 -2	(1.18 -1.24)

Table 5 .

 5 Prior intervals and estimated mean values and confidence intervals (CIs) for the parameters of the NLMIM model with variable coefficients  .

		Prior intervals		Estimations
	Parameters	Lower bound Upper bound	Mean	Standard deviation	95% Confidence Interval
	1 U [cm/s]	0.03	0.1	3.63 10 -2	1.69 10 -4	(3.6 10 -2 -3.67 10 -2 )
	2 U [cm/s]	0.003	0.01	4.81 10 -3	3.19 10 -5	(4.74 10 -3 -4.87 10 -3 )
	3 U [cm/s]	0.0003	0.001	1.01 10 -5	2.2 10 -5	(9.68 10 -4 -1.05 10 -3 )
	1  [s -1 ]	0.0	50.0	2.23 10 -1	4.64 10 -2	(1.32 10 -1 -3.14 10 -1 )
	2  [s -1 ]	0.0	50.0	2.77	1.08 10 -1	(2.56 -2.98)
	3  [s -1 ]	0.0	50.0	14.7	3.81 10 -1	(13.9-15.4)
	 [-]	9.11	4.39 10 -1	9.11	4.39 10 -1	(8.25 -9.97)
	 [cm]	0.0	3.0	1.0	1.49 10 -2	(0.97 -1.03)


Christina Reid

Virginie Privas-Bréauté

Docteur Ès

Etudes Irlandaises Lyon

How to "Reshape the Contours of Northern Irish Drama"? The Neo-Brechtian Use of Music in The Belle of The Belfast City (1989) by

delicate and spirited, and her command of an urgent, free-flowing theatrical form. "(2008: 393). The influence of Brechtian theory in Reid's piece is thus here under consideration. The Brechtian use of music becomes a device for the playwright to redefine the contours of Northern Irish drama and provide an aesthetic -as well as a political and social -perspective on the Northern Irish situation of the end of the 20 th century. This is how Reid's play becomes neo-Brechtian as shall be studied. In his introduction to the Methuen Anthology of Irish Theatre, Patrick Lonergan explains that Reid is indeed concerned with the redefinition of national identities for her people but also for theatre: [Reid] challenges and transgresses the boundaries that define national identity. In doing so, [she] push [es] against the conventions of theatrical form: by asking what it means to be Irish, [she] [is] also exploring how identities can be represented theatrically. Along the way, [she] crosses and redraws many other borders. The dramatic merges with different literary forms and other media, from storytelling to music to cinema. (LONERGAN 2008: vii) For Stewart Parker, another Northern Irish playwright contemporary of Reid, Brecht was indeed one of the colossi "of the European theatre in this age […] [to] offer a formidable choice and challenge to those playwrights caught in their historical wake" (DAWE et al. 2008: 23).

Since Brecht asserted that music played an important part in his epic theory, did not merely illustrate events but was rather a dramatic necessity, an anti-illusionist device, the use of music and songs in Reid's play is under scrutiny in this paper. Before tackling the effect of distanciation created by Reid's songs as recommended by Brecht, I will first highlight how they contribute to fragment the play, fragmentation being another element part of the epic.

Fragmentation

The Belle of The Belfast City revolves around Dolly, the matriarch who used to be a singer in a concert hall in Belfast in the 1940s. She is here welcoming, for the first time in Belfast, her 18 year-old granddaughter, born in England from a Black father and named Belle after her grand mother's then scene name. Twenty-two songs of different contents, lengths, rhymes, rhythms and genres are strewn over the play. They are folk songs, ballads, nursery rhymes, Sunday school chants, negro-spirituals. The title, which not only recalls the nickname of the singer Dolly but is also melodious, sets the tone. Through the repetition of the defining article "the", we are invited to understand that Belle is not only the Belle of Belfast. There is specifically only one Belle and only one Belfast. The repetition of "Bel" in Belle and Belfast also creates music from the very beginning. Yet at the same time as it conveys some kind of articulation through the preposition "of" it entails an idea of rupture and heralds the Brechtian resonances of the play. Similarly, through the name "Belle" ambiguously referring either to Dolly when she was young or to her granddaughter, Reid creates a tension between continuity and discontinuity, all the more so as from time to time, depending on the songs she sings, Dolly becomes "Belle" again, visibly rejuvenating on stage.

Fragmentation within the play recalls the then situation of Belfast, one of disharmony and fragmentation. Similarly, the starting point of Brechtian epic realism is a fragmented reality conveyed through several principles including music (VALENTIN 2000: 197). Brecht resorted to songs particularly to give a fragmented aspect to his play 1 . He gave the example of The Three Penny Opera as a real piece of innovation through the strict isolation of musical elements. Even though The Belle's stage directions do not put forward any particular light effects, on-stage orchestra, screen on which the titles of the songs are written as did Brecht 2 , there are changes when the characters sing. The title of Reid's play is extracted from a line found in the chorus of the first song, I'll tell me Ma, a children's song from the 19 th century which can be adapted -and indeed it has been adapted in The Belle -to other cities and situations: Dolly (sings) I'll tell me ma when I go home The boys won't leave the girls alone They pulled my hair they stole my comb Well that's all right till I go home She is a handsome she is pretty She is the Belle of the Belfast City She is courtin' One Two Three Please won't you tell me who is she (TBBC 3 ,179). Dolly, 77, sings the play open as if talking about her own childhood. She is then joined by other members of her family, Vi, Rose and Janet, and she will resume singing it alone, later, to close the selfsame scene of act 1. From the beginning, this song, which she had started singing in the present of the play, marks fragmentation since when it is then taken up by the others, the playwright mentions that the year is 1958. Furthermore, whereas Dolly was sitting as an old woman at the beginning of the song, when it is sung by the members of her family, "[she] joins in and becomes an agile woman of 49" (TBBC, 179). There is a rupture between the present and the past triggered by the singing of this song. Not only does this song frame this very first scene, fragmenting the text visually, cut by the playwright who gives directions in between the first part and the following verses of the song but it also ruptures the chronology of time. The changing of rhymes between song n°1 and the different verses of n° 2 4 and 3 5 also create discontinuity. This first device highlights the notion of fragmentation as stated by Brecht in his definition of epic drama based on montage, variety show, and biography 6 . Besides, Jeske Berg explains in Bertolt Brecht, L'Homme et son oeuvre, that in 1926, Brecht advocated the idea of a dramatic piece as a series of independent elements which interacted, completed or contradicted one another impeding a fluid, homogeneous action (BERG & JESKE 1999: 107). Brecht ultimately sought to make music, scenery, lighting, costume, acting and singing all convey the same impression of anti-illusion.

Along the Brechtian theory then, there should be no linearity, each piece has to be on its own, and there should also be no unique centre of perspective since reality is fragmentary. Songs in The Belle convey this impression of a fragmented play as well as they offer a plurality of voices, of perspectives. For instance, song n°18 is sung by Dolly who takes up the role of another character they knew back in the 1950s, Naomi: We have the double perspective of Dolly and her daughters on Naomi, but also that of Naomi who is remembered as she was perceived by Dolly and her daughters back then. Again, the absence of chronological linearity stands out through this embedding of points of view. There is also a rupture in the fluidity in term of space. All the characters pretend they are at the chapel where Naomi, being the piano player and wife of the reverend of the parish, used to meet them on Sundays for cult.

Another element in the songs hindering fluidity in the play is conveyed through the repetition of lines, such as in song n°9, another children's song:

Our Queen can birl her leg Birl her leg, birl her leg Our Queen can birl her leg Birl her leg leg leg (TBBC,203) This repetition gives the impression that it is difficult to proceed, to move on. Continuity is further hampered by the fact that the songs tackle different themes and are not only sung by Dolly but also by her daughters, her niece, her nephew, and even her grand-daughter.

Lisa Fitzpatrick explains about Tea in a China Cup that Reid "uses music, an episodic structure, and minimal properties, and typically divides the stage into different spaces, depicting different times and spaces simultaneously to create juxtapositions and to allow actions to comment upon each other " (2005: 329). This idea can be taken up for The Belle in so far as juxtaposition, another Brechtian element called montage and conferring a fragmented dimension within a play, is here conveyed through songs. Even if most of the time, several pages, several dialogues or even several decades, separate two songs, there are occasions when two songs, of different themes and contents, of different genres, are juxtaposed. This is the case for songs n° 16 and 17. Dolly first sings a nursery rhyme while the rally is about to reach her street, then she pulls a party popper, Belle hands her a drink (only one line of stage directions to separate the two songs) and she resumes singing a historic and political song this time:

Dolly (sings) Vote vote for Maggie Thatcher

In comes Belle at the door, io For Belle is the one that'll have a bit of fun And we don't want Maggie anymore, io. (TBBC,224).

Like song n°1, this song has been adapted to the situation since the original characters mentioned were Harry Midgley -the sitting labour MP candidate for the 1938 elections in Ireland -instead of Maggie Thatcher on the one hand, and James Collins -the Nationalist candidate opposing Midgley in the constituency of Dock in Belfast -in lieu of Belle 7 on the other hand.

Kim H. Kowalke, in "Brecht and Music: Theory and Practice", for The Cambridge Companion to Brecht, further explains that "even a solo performance is a kind of montage, a combination of mimetic immediacy and diegetic distancing, a composite of dissected realities" (2006: 254). Indeed, many a time does Dolly sing alone as she did when she was on stage 40 years earlier and was renowned for being the Belle of the Belfast city. Her numerous breaks from the present reality lead Jack, her nephew, to call her a crazy senile old person.

An experimental play as were Brecht's, Reid's The Belle of the Belfast City resorts to songs to interrupt dialogues. They are not introduced and do not illustrate an event; they are part of the plot. When he wrote on experimental drama, Brecht advised modern playwrights to continuously propose new effects (VALENTIN 2000: 312). That is why Reid has many of her songs interrupted too. For instance, Dolly's song n°4 is suspended by her daughter Vi's refusal to know the end of the line for decency: Dolly (sings) There was an oul woman down Donegall Street, who went to the doctor' cause she couldn't… Vi: Mother! (TBBC, 183)

We may think that this old woman is Dolly now in the present as if she suddenly realised she has been ageing.

Social Gestus

As songs are devices to interrupt the play and highlight its episodic structure, they belong to what Brechtian theory calls social gestus. For Brecht, Gest is not supposed to mean gesticulation: it is not a matter of explanatory or emphatic movements of the hands, but of overall attitudes. A language is gestic when it is grounded in a gest and conveys particular attitudes adopted by the speaker towards other men. The musician sees this as an artistic principle. From this, we may judge the political value of the musical score" (BRECHT 1964: 104).

On gestic music, Brecht further explains "it is that music which allows the actor to exhibit certain basic gests on the stage. So called 'cheap' music, particularly that of the cabaret and the operetta, has for some time been a sort of gestic music" (BRECHT 1964: 87). Drama like the opera may mirror the contemporary social issues artistically. In The Belle, [Reid's] "Brechtian dramaturgical style reflects upon the shaping of the characters by their environment" (FITZPATRICK 2005: 329) in a musical-like fashion, not only through the leading theme of the cabaret itself, but also, through the songs in the play, such as the role of women in Northern Ireland -all the more so as she tackles social problems. Reid explores the "relationship between women and community, the writing of women into histories from which they have traditionally been excluded, the links between sexual repression and silence, and the mother-daughter relationship" (FITZPATRICK 2005: 332). Carla McDonough, in "I've never been just me: Rethinking Women's Positions in the Plays of Christina Reid", explains that "Reid's plays examine how religious ideology affects a woman's upbringing. In this ideology, woman is the temptress, Eve, the fallen woman" (2000: 184). One of the main examples shedding light on the problematic situation of women is dramatized by the character of Janet. In act 1 scene 3, Janet, psychologically imprisoned by Jack, her brother, and Peter, her husband, stands centre stage in between them. While Jack quotes from St Paul's epistle to the Corinthians, chapter 7, "it is good for a man not to touch a woman", Peter sings her a love song (song n° 10): They both end their replies by saying they love her. Janet then realizes that for both these men, women are seen alike: "a devil [-Jack -] and a saint [-Peter -] are the same thing. Afraid of women. Afraid they will tempt you. Afraid we won't" (TBBC, 209). To Liza Fitzpatrick, in this scene both men express their ideal images of femininity: in Peter's case, the woman on the pedestal, the Angel in the house; for Jack, the virgin sister, sexually unattainable. In each symbolic dimension, the woman is denied agency and a voice." (331). Some critics, like Laura Kane, have pointed out the fact that "some women in Belfast still suffer from a lack of material opportunity that prevents them from articulating their own identities and reforming their citizenships" (KANE 2005: 40). Therefore we may see songs in this play as possible weapons for women to reach freedom, reconstruct their identity far from patriarchy, while men remain trapped in the myths they were raised to believe. Furthermore the main character, Dolly, is a former cabaret star leading an unusual life for a woman at the beginning of the twentieth century. Her songs were both "politically and religiously irreverent" (MCDONOUGH 2000: 187), which was provocative. This first theme, emphasized by the use of music, becomes a social gestus.

Mary Trotter also overtly refers to Brechtian social gestus through another song in the play, the last (song n°22 8 ), sung by Dolly's granddaughter at the end of the play, after Dolly who suffered a heart attack cannot talk and, as a consequence, cannot sing anymore. Trotter gives the following analysis:

This Belle of the Belfast City sings a contemporary political ballad. Since this song, Ballad to a traditional refrain, by Maurice James Craig, is an ironic politicization of a jingoistic song tradition, Reid ends the play with two translations. Craig's song translates the literary/ music genre of the sectarian ballad into a call for the end of sectarian violence. She also semiotically translates Dolly's now silenced voice into the body of her half-Irish, half African American granddaughter. A song protesting Belfast sectarianism sung by a body with both Irish and African American identity markers becomes a Brechtian gestus, a call to the audience to think critically abut the historical conditions which have led to the constructions of identities in Ireland and around the world. (2000: 175).

In playing his role, the Brechtian performer must convey a social message to the audience. This task might be facilitated by songs if we believe Kowalke who explains that "gestic music could articulate that which the text does not make explicit and thereby provide a sub-text ready-made for the performer" (2006: 251).

Distanciation

In Reid's drama, songs do not only interrupt the stories for aesthetic innovation but also, as Brechtian theory points out, to criticize the contemporary world at the light of the past. This is part of what Brecht called "Verfremdungseffekt". This effect translated in English by "distanciation" for lack of better word is much present in The Belle. As Diderik Roll Hansen says of one of Reid's previous plays, "we move swiftly from scene to scene, quite often in defiance of chronological time" (2008: 393). Likewise in The Belle, even if the time of the play is 1986, flashbacks are legion. Dolly seems to regret her past life in cabarets. She lives in the past as the old photos on her wall illustrate and as the children's songs she sings let us know. Through this play, Reid seems to raise our "awareness of how the past can both anchor and restrain us" (LONERGAN 2008: xi). If song n°1 takes the performers back to 1958, there are other numerous songs bridging the gap between the past and the present. Many traditional Irish songs have been adapted to present situations like song n° 11:

Dolly sits cradling Janet's head in her lap. She strokes Janet's hair and sings. Dolly I know where I'm goin' And I know who's goin' with me I know who I love But the dear knows who I'll marry I'll wear gowns of silk And shoes of fine green leather Ribbons for my hair And a ring for every finger. I know who is sick And I know who is sorry I know who I've kissed But God knows who I'll marry. (TBBC,211) The characters are called to take some distance from their present at the light of their past; but it is not Reid's only goal. Still along her decision to create anti-illusionist effects, she has Belle, in the present, become a spectator of the past of her family. This is how Reid writes: Through this anti-illusionist disruptive effect in which Belle -in 1986watches her family sing in 1959, the Brechtian performer is asked to take some distance from her role. She becomes a spectator in turn. This leads the audience to be challenged too in their role as spectators. They are asked to take some distance from what they are watching. Once again, there is an embedding effect: the audience watches a performer who is watching a scene on stage. But on this occasion, reality is challenged as well. In fact, the performer watches the past. In his plays, as Llewellyn-Jones explains in Contemporary Irish Drama and Cultural Identity, "Brecht draws upon devices that, from time to time, distance the reader through the exposure of the creative processes thus challenging 'realism' not only through a deliberate lack of closure but through representation of competing and contradictory discourses " (2002: 45). Thanks to music and songs, Reid orchestrates a reconciliation of impossible elements: the past and the present. Moreover music, along Brecht's epic theory, is designed to comment on and explain what is happening on the stage and in reality; it is not particularly meant to arouse emotions; it should lead to a social reaction. Music is thus distanced from its primary aesthetic role too and is endowed with a social mission.

History and the past are privileged places of dramatisation in which historic periods mentioned are linked to the present. The fact Belle watches the past on stage is part of a process of historicisation, which is a fundamental means of distanciation. The play therefore puts to the fore the many alternatives which could have happened. What is historicized in Reid's play here according to Patrick Lonergan is "the resultant sense of anger and anxiety" entailed by "the implementation of the Anglo-Irish agreement, now regarded as an important step in the development of the peace process, then seen as a betrayal by the Unionists by the British government at that time" (2008: xi). For Brecht the effect of distanciation generated by music enabled to link history and politics too; this is how music may be used as a political weapon in his plays. (LECOUVEY 2009: 20).

In Northern Ireland, the political issue at that time was also triggered off by segregation from the Protestants towards the Catholics to cut a long story short. But Reid seems to have chosen not to talk about religious segregation as bluntly as that. On the contrary, she displaces the subject by having her marginal characters being either Black (Belle) or deaf and mute (Davy). Effectively, Reid seems to be interested in neither class nor religious conflicts as Brecht was, but rather in male-female oppositions. Her characters are alienated, estranged; alienation -or rather defamiliarisation 9 , as should rather been said, being another possible translation for the "V-effekt". This alienation is also connected with music as Davy, mute, exemplifies: he cannot sing, yet he can tap out rhythms (act 1). Another example is brought about by the song opening the second act. This song is turned into a poem since Dolly "recites" it:

Dolly (recites)

In a mean abode on the Shankill road Lived a man called William bloat He had a wife, the curse of his life, Who continually got his goat. So one day at dawn, with her nightdress on, He cut her bloody throat. With a razor gash he settled her hash, Oh never was crime so quick, But the steady drip on the pillow slip Of her lifeblood made him sick, And the pool of gore on the bedroom floor Grew clotted cold and thick. And yet he was glad that he'd done what he had, And she lay there stiff and still. But a sudden awe on the angry law Struck his soul with an icy chill. So to finish the fun so well begun, He resolved himself to kill. Then he took the sheet off his wife's cold feet, And twisted it into a rope. And he hanged himself from the pantry shelf. 'Twas an easy end, let's hope. In the face of death with his latest breath, He solemnly cursed the Pope. But the strangest turn to the whole concern Is only just beginnin'. He went to hell but his wife got well, And she's still alive and sinnin', For the razor blade was German-made, But the sheet was Irish linen. (TBBC,21) This engenders a rupture in the pattern of the play: it is the only time a poem is recited. Reid is here close to songs as Brecht wanted to have them: "poems sung" (DORT 1960: 57). The ironic contents of this poem also points to racism in a way -another theme that permeates Reid's playsince it draws a parallel between the poor quality of a German razor and that of hard-wearing Irish linen.

When he elaborated his theory, Brecht had in mind that each element had to comment in a different way. By contradicting expectations and by juxtaposing two contrary moods it could create conflict in the mind of the spectators and entice them to reconcile the two elements to find coherence. Henceforth the term "dialectical" emerged. Brecht viewed theatre as part of an enlightenment project in which discordant jarring elements (music v. text or commentary by actors, for example) would explore ideas and man's contradictory nature. If we go further, in The Belle Reid's goal is to go beyond the mere divide between Protestants and Catholics. The songs she has chosen shed light to other opposing notions which come against the characters' dialogues. For instance, songs n°2, 7 and 16 oppose the outside and the inside worlds, song n°14 frailty against solidity, songs n°3 and 6 nature and culture and song n°5 magic against reality. All these themes punctuate the play so as to show that the world can eventually be transformed. Likewise, while Dolly sings song n°1 Davy taps out its rhythm as if to echo her voice or reply to her. This is a conversation between two people talking different languages but understanding one another; this illustrates what Brecht meant when he suggested that each element had to be on its own but constituted a total coherent objective. The contrast between the gentle content of a song sung with a harsh voice is further relevant in an analysis of a Brechtian play. This is taken up by Reid when she has Jack sing a love song (song n°15) reluctantly only to please her aunt and cousins -and at the risk of being ridiculed since the girls dress him up with unfitting clothes, a situation he profoundly dislikes. Reid writes: "Jack becomes increasingly angry and humiliated as the song progresses" (TBBC, 217). He finally "runs out" (TBBC, 219). The way the actor playing Jack must sing this song is not overtly mentioned in the stage directions but the stage director will undoubtedly play on this.

Through the character of Davy who cannot speak Reid abides by the idea that "music has more impact than words" (KOWALKE 2006: 248), a sentence coined by Kurt Weil, one of the German musicians who collaborated with Bertolt Brecht on many pieces. This issue of impact is really interesting in the context of a Brechtian play, all the more so as the spectator is asked to be involved in the play. Indeed, Bernard Banoun in an article on Brecht and music explains that in Brechtian theory, the poietic function (that of the creator and the processes of creation) is not much present as far as music is concerned, the immanent function (the musical analysis, the technique and the piece itself), secondary; what prevails in fact is the esthesic function, that which links the piece of art with the addressee (BANOUN 2008: 336). So the didactic dimension of The Belle needs to be tackled. After rupturing the naturalist expectations of the audience, Reid breaks down the fourth wall when her characters directly address the audience through their songs in the manner of a cabaret (DOUBLE 1996: 47). Therefore the imperative mode used in song n°16 is unequivocal: Yet, and this is crucial for Brecht, epic plays are not designed to stimulate or arouse emotions, they must examine them. The audience has to think about the music and the songs they listen to. Art had to entail criticism and reflexion for Brecht (LECOUVEY 2009: 19). Art had to teach something to the audience; it had to be didactic and entertain the public. For Francine Maier-Schaeffer, in her book Bertolt Brecht, didactic plays "must abolish the separation between the stage and the audience, between the actors and the spectators, between professionals and amateurs. […] Didactic plays have become to be considered as musical genres nowadays. " (2003: 81). That is why The Belle becomes a didactic Brechtian play with all this music and its songs against religion, against politics, enabling to understand the present, provoking the spectators and asking them to participate in the play. The most didactic song in The Belle, that which stirs the mind of the audience the most effectively, might be the song which sets a jarring parallel between joy and war (song n°13). It is triggered by the word "party" in Belle's preceding line "She votes for the Unionist Party to keep the Republican Party out." (TBBC, 214) but is sung by Dolly: Dolly (sings) Will you come to our wee party will you come? Bring your own bread and butter and a bun You can bring a cup of tea You can come along with me Will you come to our wee party will you come? Will you come to Abyssinia will you come? Bring your own ammunition and a gun Mussolini will be there firing bullets in the air Will you come to Abyssinia will you come? (TBBC, 214) For Bernard Banoun, Brecht was so worried about being clear and unequivocal that music for him was tightly linked to meaning. He did not want music to be played on its own, for its own sake. Therefore he conjugated it with external elements such as drama and words. He had a heteronomous conception of music whose final goal was persuasion (BANOUN 2008: 336).

Dolly (sings)
If songs fragment Reid's play, like Brecht, the message she conveys is not fragmented. She aims at teaching her audiences and inspiring them on the social issues of the 1980s. Yet, her goal is not only social and political; it is first and foremost aesthetic. Reid is not interested in class-conflicts and she addresses Catholics and Protestants alike, she is more concerned with conflicts of sexual genres and racism; that is why The Belle of the Belfast City is neo-Brechtian rather than Brechtian. This play started with a song by Dolly about Belle, as we may understand in the enfolding of the play, and finishes with a song by Belle about segregation in Belfast. This echoes the cycle of violence, the idea that there might be no way out, a repetitive pattern that the structure of Reid's play aims to deconstruct. Giving voice to women in such a male-dominated context, and involving an Irish African English character, she paves the way for a post-colonial interpretation of The Belle along the definition that Julia Kristeva gave of post-colonialism awareness:

A paradoxical community is emerging, made up of foreigners who are reconciled with themselves to the extent that they recognize themselves as foreigners. The multinational society would thus be the consequence of an extreme individualism, but conscious of its discontents and limits, knowing only indomitable people ready-to-help themselves in their weakness, a weakness whose other name is our radical strangeness. (1991: 195).

Indeed, Reid has stated many a time that she resists labels, and that she writes plays open to interpretation (MAGUIRE 2006: 163). If we think, with Brecht that music made possible the poetic theatre (VALENTIN 2000: 702), then Reid is not only a playwright, she is also a poetess. BRECHT Bertolt, (1964), On Theatre, edited and translated by John

Bibliography

  Sound of a train. A British customs officer walks on. Belle watches as Dolly, Vi, Rose and Janet dress up from the box. They assume position as if on a train. They eat the dulse and yellowman. The year is 1959. (Dolly is 50; Vi 30; Rose 9; Janet 9.) Dolly (sings) At the oul Lammas Fair Boys Were you ever there Were you ever at the fair at Ballycastleo Did you treat your MaryAnn To some dulse and yellowman At the oul Lammas fair at Ballycastleo (TBBC, 201).

  In and out the windows In and out the windows In and out the windows As you have done before. Stand and face your partner Stand and face your partner Stand and face your partner As you have done before.(TBBC, 223)

  Primary playREID Christina, (1997), Plays 1: Tea in a China Cup, Did you hear the one about the Irish Man?, Joyriders, The Belle of the Belfast City, My Name, Shall I tell you my Name?, Clowns, London, Methuen Drama. Referential books BERG Gunter & Wolfgang JESKE, (1999), Bertolt Brecht, L'Homme et son oeuvre, Paris, Éditions L'Arche.

Dolly (sings badly) and mimes Naomi Standaloft playing the piano with exaggerated gestures. Sound of a piano. As Dolly sings, she becomes Naomi. Janet and Rose join in as ten year olds. They are torn between giggling and fascination at Naomi's elaborate playing.

	Dolly/Naomi (sings)
	Climb climb up Sunshine Mountain
	Singing as we go.
	Climb climb up Sunshine Mountain
	Faces all aglow.
	Turn turn your back on Satan
	Look up to the sky
	Climb climb up sunshine Mountain
	You and I.
	She shouts "All together now!' and Rose and Janet sing the song and
	march around the room. As they finish the preacher Isaac Standaloft
	walks on. […] Naomi heralds his arrival with a fanfare on the piano…
	(TBBC, 232).

  Janet is situated centre stage between them. She looks from one to the other. Peter I love you. Come back to me. Jack I love you. Come back to me. (TBBC, 209).

	Peter (sings)
	Green gravel, green gravel
	Your grass is so green you're the fairest young damsel
	That ever I've seen
	Green gravel, green gravel
	Your true lover's dead.
	So I've sent you a letter
	To turn round your head
	I washed her and I dressed her
	And I robed her in silk
	And I wrote down her name
	With a glass pen and ink.
	Green gravel green gravel
	Your grass is so green
	You're the fairest young damsel
	That ever I've seen.
	Jack (quoting from Saint Paul)
	[…]

Let them all come as they will For it's Joe Horner she loves still. (TBBC, 182) 6 "Le théâtre épique est un théâtre fondé sur le montage, la revue, un théâtre biographique aussi" (VALENTIN 2000: 228). Effectively, The Belle of the Belfast City relates the past life of cabaret star Dolly, juxtaposes her past and her present lives, and is somehow biographical. 7 On http://www.bbc.co.uk/northernireland/ashorthistory/archive/intro239.shtml; a short history of Ireland.

8 Dolly stares at the concert hall poster. Belle sings. Belle Red brick in the suburbs, white horse on the wall Italian marble in the city hall; O stranger from England, why stand so aghast? May the Lord in his mercy be kind to Belfast. This jewel that houses our hopes and our fears Was knocked up from the swamp in the last hundred years; But the last shall be first and the first shall be last; May the Lord in his mercy be kind to Belfast. We swore by King William, there'd never be seen An all Ireland parliament at College Green So to Stormont we're nailing the flag to the mast. May the Lord in his mercy be kind to Belfast. O the bricks they will bleed and the rain it will weep And the damp Lagan fog lulls the city to sleep. It's to Hell with the future, and live on the past May the Lord in his mercy be kind to Belfast (TBBC,250) 9 This is all the more telling as Jack does not want to recognize Belle as part of his family because, she is different physically.