Claudia Cherubini
email: claudia.cherubini@lasalle-beauvais.fr

Concetta Giasi

Nicola Pastore
email: nicola.pastore.ing@gmail.com

Campione Di

Roccia Fratturata

A Scala

D I Banco

FISICO DI LABORATORIO PER ANALIZZARE DINAMICHE DI FLUSSO E TRASPORTO IN UN

Keywords: fractured media, laboratory experiments, non-darcian low, nonickian transport

RIASSUNTO

La conoscenza dei fenomeni di lusso e trasporto nelle rocce fratturate è molto importante al ine di ottimizzare strategie di boniica e di monitoraggio, effettuare analisi di rischio e gestire convenientemente le risorse idriche di tali acquiferi. Recentemente sia la comprensione delle modalità di circolazione delle acque sia la caratterizzazione e modellazione delle interazioni isico-chimiche all'interno di acquiferi fratturati stanno acquisendo sempre più importanza, specialmente per la necessità di pervenire ad una corretta gestione delle risorse idriche o anche per implementare corrette strategie di monitoraggio, o per garantire la qualità delle stesse, sia in termini di prevenzione da possibili inquinamenti che di recupero nel caso di contaminazione in atto. Talvolta l' approccio del mezzo poroso equivalente non è in grado di riprodurre le condizioni di lusso e trasporto in tali complesse formazioni. Questioni critiche emergenti per gli acquiferi fratturati sono: la validità della "legge cubica locale" che assume una relazione lineare tra portata e gradiente idraulico (legge di Darcy), nel caso si voglia descrivere con precisione le condizioni di lusso; l'uso dell'equazione classica di advezione-dispersione ove si voglia descrivere la propagazione del soluto.

La maggior parte degli studi sul trasporto attraverso fratture discrete sono tuttora basati su modelli di lusso molto semplici, cosa che ha limitato l' interpretazione delle curve di avanzamento (breakthrough curve) del soluto. I dati sperimentali ottenuti sotto condizioni controllate come quelle di laboratorio permettono di migliorare la comprensione delle isiche fondamentali del lusso di falda e del trasporto di contaminante nelle fratture.

In questo studio sono state effettuate prove idrauliche e di tracciamento su un campione di roccia fratturata di forma parallelepipeda (0.60×0.40×0.8m) creato artiicialmente. Sono stati misurati i volumi di acqua passanti attraverso differenti percorsi attraverso il campione fratturato per varie differenze di carico idraulico; inoltre sono state misurate le curve di avanzamento relative ad un tracciante salino immesso ad impulso per vari percorsi. I suddetti esperimenti sono inalizzati a comprendere le relazioni esistenti tra le condizioni al contorno applicate, la geometria del sistema ed i fenomeni di lusso e trasporto in atto.

I risultati sperimentali hanno mostrato evidenza di non linearità nel lusso e proili di concentrazione che non possono essere descritti da modelli di trasporto di soluto tradizionali.

SUMMARY

The knowledge of low and transport phenomena in fractured rocks is very important in hydrogeologic engineering in order to optimize clean up and monitoring strategies, to carry out risk assessment and to manage interventions in aquifers.

Recently, understanding, characterizing and modeling physical and chemical interactions within fractured aquifers has acquired increasing importance, especially with regard to the question of water resources development and groundwater contamination. Sometimes the equivalent porous medium approach fails to reproduce low and transport patterns in such complex geological formations. Critical emerging issues for fractured aquifers are the validity of the Darcian-type "local cubic law" which assumes a linear relationship between low rate and pressure gradient to accurately describe low patterns and of the classical advection-dispersion equation to describe the propagation of solute.

Most studies of transport through discrete fractures are still based on simpler low models which has limited the interpretation of solute breakthrough curves.

Experimental data obtained under controlled conditions such as in a laboratory allow to increase the understanding of the fundamental physics of luid low and solute transport in fractures.

In this study hydraulic and tracer tests on artiicially created fractured rock samples of parallelepiped (0.60×0.40×0.8m) shape have been carried out.

The volumes of water passing through different paths across the fractured sample for various hydraulic head differences and breakthrough curves for saline tracer pulse across different pathways have been measured.

The above experiments are aimed at understanding the relations existing between the applied boundary conditions, the geometry of the system and the occurring low and transport phenomena.

The experimental results have shown evidence of non linearity in low and concentration proiles that cannot be described by conventional solute transport models.

In fact, the classical advection-dispersion equation -used as a benchmark for comparison in a numerical model-poorly describes the experimental breakthrough curves of the tracer propagation.

INTRODUCTION

Numerical modeling in subterranean hydrology has become increasingly important in ields like petroleum exploitation, contaminant transport, geothermal energy and groundwater supply management.

In particular the modeling of transport in fractured media has been the subject of considerable recent research [START_REF] Roden J | Contaminant transport in fractured rocks with signiicant matrix permeability, using natural fracture geometries[END_REF][START_REF] Berkowitz | Characterizing low and transport in fractured geological media: A review[END_REF][START_REF] Cherubini | Bench scale laboratory tests to analyze non-linear low in fractured media[END_REF]wu et alii, 2010): solute transport models are aimed at predicting aquifer responses to be used, for instance, in remediation studies, waste disposal and management and in environmental risk assessment.

The fundamental equations familiar to groundwater hydrologists, Darcy's law-based groundwater low equation and the Advection-Dispersion Equation (ADE) are of limited usefulness when applied to inertial lows (high Reynolds numbers) and anomalous transport patterns which are commonly present in heterogeneous formations such as fractured aquifers (SukoP et alii, 2008).

In fact in real rock fractures, microscopic inertial phenomena can cause an extra macroscopic hydraulic loss (kløv, 2000) which deviates low from the linear relationship among pressure drop and low rate.

Microscopic inertial forces are represented by the formation of local vortices and the development of tortuous streamlines inside pores with increasing Reynolds numbers [START_REF] Barak A.Z | Comments on high velocity low in porous media by Hassanizadeh and Gray[END_REF].

Moreover, heterogeneity intervenes even in solute transport in that the equation that prescribes a linear relationship between the dispersive mass lux and the concentration gradient is not valid.

The aim of present work is to experimentally investigate the behavior of a nonlinear (non-Darcian) low regime and its impacts on solute propagation in a fractured formation.

A way for understanding and quantifying the migration of contaminants in groundwater systems [START_REF] Dagan | Subsurface low and transport: A stochastic approach[END_REF] is that of analyzing tracer transport. Numerous investigations have been carried out in the ield, in laboratory, together with numerical simulations [START_REF] Becker W. & Shapiro A.M | Tracer transport in fractured crystalline rock: evidence of non-diffusive breakthrough tailing[END_REF][START_REF] Sorensen I | Effects of pore volume transmissivity correlation on transport phenomena[END_REF]HaGGERty et alii, 2000;REimuS et alii, 2003, diEtRiCH et alii, 2005).

Tracer breakthrough curves from transport in highly heterogeneous media such as fractured media are distinctly different from those in homogeneous porous media.

Infatti, la classica equazione di advezione-dispersione, usata come benchmark per confronto in un modello numerico, non descrive in maniera suficientemente accurata le curve sperimentali relative alla propagazione del tracciante. Un' analisi comparativa dei risultati ha permesso da una parte di studiare il comportamento del lusso e del trasporto nel mezzo indagato e dall' altra di valutare possibili miglioramenti al setup sperimentale.

Parole chiave: mezzi fratturati, esperimenti di laboratorio, lusso non darciano, trasporto non-ickiano

INTRODUZIONE

La modellazione numerica in idrologia sotterranea sta acquisendo un ruolo sempre più rilevante in campi come l'estrazione petrolifera, il trasporto di contaminante, la geotermia e la gestione sostenibile delle risorse idriche sotterranee. In particolare la modellazione del trasporto nei mezzi fratturati è stata di recente oggetto di numerosi studi [START_REF] Roden J | Contaminant transport in fractured rocks with signiicant matrix permeability, using natural fracture geometries[END_REF][START_REF] Berkowitz | Characterizing low and transport in fractured geological media: A review[END_REF]CHERuBiNi & Pa-StoRE, 2010;wu et alii, 2010): i modelli di trasporto di soluto sono inalizzati a prevedere la risposta degli acquiferi per la progettazione di interventi di boniica, per lo smaltimento e la gestione dei riiuti e per l' elaborazione dell'analisi di rischio sanitario-ambientale.

Le equazioni costitutive del lusso di un luido nel mezzo poroso, ossia la legge di Darcy e la equazione di Advezione-Dispersione (ADE), sono di utilità limitata se applicate ai regimi di lusso inerziale (alti valori del numero di Reynolds) e alle condizioni anomale di trasporto che sono presenti solitamente nelle formazioni eterogenee ed anisotrope quali gli acquiferi fratturati (SukoP et alii, 2008).

Infatti nelle fratture reali in roccia, fenomeni inerziali microscopici possono causare una aggiuntiva perdita di carico macroscopica (kløv, 2000) che devia il lusso dalla condizione di linearità che lega caduta di pressione e portata. I fenomeni inerziali microscopici sono rappresentati dalla formazione di vortici locali e dallo sviluppo di percorsi tortuosi all' interno dei pori all' aumentare del numero di Reynolds [START_REF] Barak A.Z | Comments on high velocity low in porous media by Hassanizadeh and Gray[END_REF]. Inoltre, l'eterogeneità interviene anche nel trasporto di soluto in quanto l'equazione che descrive una relazione lineare tra il lusso di massa di soluto e il gradiente di concentrazione non è più valida.

Lo scopo del presente lavoro è di investigare sperimentalmente il comportamento di un regime di lusso non lineare (non darciano) e i suoi impatti sulla propagazione del soluto in una formazione fratturata. Una maniera, infatti, per comprendere e quantiicare la migrazione dei contaminanti nei sistemi acquiferi [START_REF] Dagan | Subsurface low and transport: A stochastic approach[END_REF] è quella di analizzare il trasporto di traccianti. Numerosi studi sono stati condotti in campo ed in laboratorio, e sono state effettuate anche simulazioni numeriche [START_REF] Becker W. & Shapiro A.M | Tracer transport in fractured crystalline rock: evidence of non-diffusive breakthrough tailing[END_REF][START_REF] Sorensen I | Effects of pore volume transmissivity correlation on transport phenomena[END_REF]HaGGERty et alii, 2000;REimuS et alii, 2003, diEtRiCH et alii, 2005).

Le curve di avanzamento relative alla propagazione di traccianti nei mezzi altamente eterogenei come quelli fratturati sono molto differenti da quelle relative ai mezzi porosi omogenei.

Normalmente la forma della curva è fortemente non simmetrica con un arrivo iniziale precoce e la presenza di una lunga coda e spesso anche picchi multipli, come evidenziato nei test con traccianti condotti da numerosi autori [START_REF] Becker W. & Shapiro A.M | Tracer transport in fractured crystalline rock: evidence of non-diffusive breakthrough tailing[END_REF]HaGGERty et alii, 2000;REimuS et alii, 2003).

CaRdENaS et alii (2007) hanno simulato il lusso (con le equazioni di Navier-Stokes) e il trasporto di soluto in una frattura reale aperta con proilo di apertura molto irregolare. Il lusso era prettamente darciano tranne che a velocità media molto alta. Nonostante tutto, nella zona in cui l'apertura era più ampia, si evidenziava la presenza di un vortice pronunciato che era responsabile della natura non Fickiana del trasporto.

Boutt et alii (2006) hanno simulato il trasporto di sostanze colloidali in una singola frattura rugosa dove il lusso era macroscopicamente laminare. Hanno dimostrato che variazioni pronunciate nel proilo della frattura possono dare luogo a zone di ricircolo (zone di intrappolamento). Hanno inoltre evidenziato che fratture interconnesse fra loro possono creare percorsi preferenziali di lusso che diventano scorciatoie per il trasporto causando fenomeni di arrivo rapido nelle curve di breakthrough (BTC); d'altra parte fratture non interconnesse tra loro possono dare luogo a zone immobili che trattengono un certo quantitativo di soluto, rilasciato poi lentamente per diffusione, portando a lunghe code nelle BTC. Gli stessi autori hanno anche mostrato che meccanismi di lusso preferenziale (channelling) possono portare a picchi multipli nelle BTC relative a prove di tracciamento in singole fratture.

QiaN et alii (2011a,b) hanno effettuato esperimenti di laboratorio con il ine di analizzare il lusso e il trasporto in una singola frattura sotto condizioni di lusso non darciane. Il trasporto era non-Fickiano con arrivo rapido e lunghe code. Un modello Mobile-Immobile (MIM), il quale assume trasporto advettivo nel dominio di lusso ("mobile") e trasferimento di massa diffusivo tra i domini mobile e immobile, ha ittato sia i picchi che le code delle BTC osservate in maniera più precisa che il modello ADE classico.

I valori alti di dispersività trovati negli esperimenti hanno mostrato come la dispersione in condizioni di lusso non darciane sia molto più alta che in condizioni darciane. Come è stato evidenziato nella trattazione di QiaN et alii (2011a) i precedenti studi sul lusso e il trasporto nelle fratture hanno considerato lusso darciano. Invece, come hanno affermato alcuni studiosi (SHaRP & maiNi, 1972;[START_REF]Fundamental studies of luid low through a single fracture[END_REF]luNati et alii, 2003;QiaN et alii, 2005), il lusso nelle fratture può diventare non darciano anche per numeri di Reynolds relativamente bassi quando gli effetti inerziali non siano più trascurabili.

In letteratura sono riportate differenti leggi che tengono conto di relazioni non lineari tra velocità e gradiente di pressione.

Una estensione cubica della legge di Darcy che descrive la caduta di pressione in funzione della portata per bassi valori di velocità è l' equazione di "inerzia debole":

dove p [ML -1 T -2 ] è la pressione, k [L 2 ] è la permeabilità intrinseca, μ [ML -1 T -1 ] è la viscosità, ρ [ML -3 ] è la densità, v [LT -1 ] è la velocità e γ [L] è il fattore di inerzia debole.
Usually the shape of the breakthrough curve is highly nonsymmetrical with a fast rise at early times and very long tail at late times, and often, they consist of multiple peaks (Tsang & Tsang, 1993) as evidenced by many ield-scale tracer tests carried out in fractured formations [START_REF] Becker W. & Shapiro A.M | Tracer transport in fractured crystalline rock: evidence of non-diffusive breakthrough tailing[END_REF]HaGGERty et alii, 2000;REimuS et alii, 2003).

CaRdENaS et alii, (2007) simulated Navier-Stokes low and solute transport in a real open fracture with a very irregular fracture aperture. The low was mostly Darcian except at very high mean velocities.

Nevertheless, a pronounced eddy was present at the largest aperture zone and it was responsible for a non-Fickian nature of transport.

Boutt et alii, (2006) simulated colloid transport in a rough single fracture where low was macroscopically laminar. They demonstrated that sharp changes in fracture proile may cause zones of recirculation (trapping zones). Moreover they found that interconnected fracture zones may create preferential low pathways that become shortcuts for transport and lead to the early arrival of the breakthrough curves (BTCs); on the other side disconnected fractures may become immobile zones to retain or capture a certain amount of solute, leading to the long tail of the BTCs. Finally, they showed that preferential low (channeling) might lead to multiple peaks in the BTCs for solute transport in single fractures.

QiaN et alii, (2011a,b) carried out well-controlled laboratory experiments to investigate low and transport in a single fracture under non-Darcian low conditions. Non-Fickian transport was found to dominate with early arrival and long tails. A mobile-immobile (MIM) model, which assumes ADE transport in the mobile domain, and diffusion-driven mass transfer between the mobile and immobile domains, proved to it both peak and tails of the observed BTCs better than the classical ADE model.

The large dispersivity values found in the experiments showed that dispersion under non-Darcian low condition was much stronger than that under Darcian low.

As stated by QiaN et alii, (2011a) previous investigations of low and transport in fractures only considered Darcian low. However, as pointed out by several authors (SHaRP & maiNi, 1972;[START_REF]Fundamental studies of luid low through a single fracture[END_REF]lu-Nati et alii, 2003;QiaN et alii, 2005), low in fractures could become non-Darcian even at relatively small Reynolds numbers when inertial effects are no longer negligible.

In literature are reported different laws that account for nonlinear relations between velocity and pressure gradient.

A cubic extension of Darcy's law that describes pressure loss versus low rate for low low rates is the weak inertia equation:

where

p [ML -1 T -2 ] is the pressure, k [L 2 ] is the permeability, μ [ML -1 T -1 ] is the viscosity, ρ [ML -3 ] is the density, v [LT -1 ] is the velocity and γ [L]
is called the weak inertia factor.

(1)

In caso di numeri di Reynolds assai elevati (Re >> 1) le cadute di pressione passano da un regime di inerzia debole ad uno di inerzia forte, descritto dall' equazione di [START_REF] Forchheimer | Wasserbewegung durch Boden Zeit[END_REF], espressa come: dove β è il coeficiente di resistenza inerziale, o coeficiente non darciano.

Un'altra equazione usata comunemente per descrivere il lusso non darciano è l' equazione di Izbash o legge di potenza [START_REF]O iltracii V Kropnozernstom Materiale[END_REF]:

dove λ e n sono due coeficienti costanti e 1 ≤ n ≤ 2: Se n = 1 il lusso è darciano mentre se n = 2 il lusso è turbolento ampiamente sviluppato. Le equazioni di Forchheimer e di Izbash sono equivalenti se il termine lineare nell' equazione di Forchheimer è zero e n=2.

È possibile utilizzare un'equazione generale di tipo darciano (CHiN et alii, 2009) per descrivere tutti i regimi menzionati e per tener conto delle non linearità nella relazione tra gradiente idraulico e velocità apparente:

K eff [LT -1 ] è la conducibilità idraulica effettiva, mentre h [L] rap- presenta il carico idraulico totale.
La conducibilità idraulica effettiva può essere scritta come:

dove a [TL -1 ] e b [T 2 L -2 ] sono, rispettivamente, il coeficiente lineare e non lineare in termini di altezza idraulica e possono essere espressi:

Nello stesso modo la trasmissività effettiva per una frattura può essere espressa come una relazione tra portata speciica q (L 2 T -1 ) e gradiente idraulico:

a f e b f sono legati ad a e b:
dove w rappresenta l'apertura della frattura.

EQUAZIONE DI ADVEZIONE-DISPERSIONE (ADE)

L'equazione costitutiva della propagazione del soluto in un mezzo poroso è l' equazione di conservazione della massa:

dove C [ML -3 ] è la concentrazione del soluto, v è la velocità effettiva, J è il lusso di massa e r è il fattore di scambio di massa.

L'equazione che tradizionalmente descrive il lusso dispersivo dei soluti in un mezzo poroso è un'estensione della legge di Fick: (HaS-SaNizadEH & lEijNSE, 1995):

In case of higher Reynolds numbers (Re >> 1) the pressure losses pass from a weak inertial to a strong inertial regime, described by the Forchheimer equation [START_REF] Forchheimer | Wasserbewegung durch Boden Zeit[END_REF], given by: where β is called the inertial resistance (coeficient), or non-Darcy coeficient.

Another commonly used non-Darcian low equation is the Izbash or power-law equation [START_REF]O iltracii V Kropnozernstom Materiale[END_REF]:

where λ and n are two constant coeficients and 1 ≤ n ≤ 2: If n = 1 the low is Darcian, whereas if n = 2 the low is fully developed turbulent. The Forchheimer and Izbash equations are equivalent when the linear term in the Forchheimer equation is equal to zero and n = 2.

A general Darcian-like relationship can be used (CHiN et alii, 2009) to describe all the mentioned low regimes and to account for nonlinearities in the relationship between hydraulic head gradient and apparent low velocity:

K eff [LT -1 ] is the effective hydraulic conductivity and h [L] is the total hydraulic head.
Effective hydraulic conductivity can be written as:

where a [TL -1 ] and b [T 2 L -2 ] are the linear and non linear coeficients in terms of hydraulic head and can be expressed as:

In same way the effective fracture transmissivity for a discontinuity can be deined and the relationship between speciic discharge q (L 2 T -1 ) and hydraulic head gradient can be written as:

ADVECTIVE-DISPERSIVE EQUATION (ADE)

The equation governing the spreading of solutes in aquifers is the equation of conservation of mass:

where C [ML -3 ] is the solute concentration, v is the mean low velocity, J is the dispersive mass lux and r is the rate of solute mass exchange.

The traditional equation that describes the dispersion lux of solutes in porous media is an extension of the Fick' s law of diffusion [START_REF] Hassanizadeh | On the transient Non-Fickian Dispersion Theory[END_REF]:

(2) (3) (4) (5) (6) (7) (8) (9) (10)
dove D [ML -2 ] è il tensore di Dispersione. Secondo questa equazione, il lusso di massa dispersivo di soluto è proporzionale al gradiente di concentrazione di soluto.

Combinando le due equazioni precedenti si ottiene l' Equazione di Advezione-Dispersione (ADE):

Un' assunzione alla base dell' ADE (SCHlumER et alii, 2001) è che la dispersione meccanica, come la diffusione molecolare possa essere descritta dalla legge di Fick. In altre parole, questa formulazione assume che il centro di massa del plume di contaminante avanzi con la velocità media (macroscopica) del luido, mentre il contaminante disperso intorno al centro di massa (a causa sia degli effetti meccanici e chimici) costituisca un processo puramente Fickiano (BERkowitz et alii, 2001). Tali assunzioni sono in realtà valide sotto condizioni molto restrittive. I dettagli su queste condizioni, le quali per dirla breve richiedono una conducibilità idraulica fortemente omogenea, sono trattati in BERkowitz & SCHER (2000).

La ADE spesso non è in grado di descrivere in maniera accurata i meccanismi di propagazione del soluto nel sottosuolo [START_REF] Cherubini | Perspective on theories of anomalous transport in heterogeneous media[END_REF].

Normalmente si incontrano due tipi di deviazioni dal comportamento ickiano [START_REF] Hassanizadeh | On the transient Non-Fickian Dispersion Theory[END_REF]. La prima riguarda la dipendenza della dispersività dalla scala di osservazione spaziale e/o temporale, come già analizzato da numerosi ricercatori (e.g. matHERoN & dE [START_REF] De | -Is transport in porous media always diffusive? A counterexample[END_REF][START_REF] Dagan | Flow and transport in porous media[END_REF]. La seconda invece riguarda la relazione tra il lusso di massa dispersivo J e il gradiente di concentrazione di soluto che diventa non lineare per elevati gradienti di concentrazione (HaSSaNizadEH & lEijNSE, 1995).

Per quanto riguarda la prima deviazione, se i modelli numerici utilizzano una discretizzazione che non è in grado di riprodurre l'eterogeneità del campo delle velocità, le simulazioni di trasporto a mezzo dell'ADE sono affette da errore (llERaR-mEza, 2009).

Infatti, la formulazione classica del trasporto di soluto sottostima signiicativamente il comportamento osservato, a lungo termine, di una breakthrough curve (BTC) in mezzi eterogenei. Il trasporto di soluto è condizionato dalla presenza di zone ad alta e bassa velocità, dove il soluto può essere "canalizzato" oppure ristagnare (llERaR-mEza, 2009). Per tener conto degli effetti dell'eterogeneità sono stati proposti modelli concettuali alternativi, come: Multi-rate Mass Transfer (HaGGERty & GoREliCk, 1995;CaRRERa et alii, 1998), Time-Dependent Macrodispersive (daGaN,1991;[START_REF] Rajaram | Plume scale-dependent dispersion in heterogeneous aquifers 2. Eulerian analysis and three-dimensional aquifers[END_REF]dENtz et alii, 2000), Continuous Time Random Walk (BERkowitz & SCHER, 1998) e inine Fractional Advection-Dispersion Transport (BENSoN et alii, 2000[START_REF] Metzler | The random walk's guide to anomalous diffusion: A fractional dynamics approach[END_REF]SCHumER et alii, 2003).

Per quanto riguarda la seconda tipologia di deviazione, una teoria recentemente sviluppata (HaSSaNizadEH & lEijiNSE, 1995) suggerisce una relazione non lineare tra il gradiente di concentrazione e il lusso di massa dispersivo, introducendo un nuovo parametro in aggiunta alla dispersività longitudinale e trasversale.

where D [ML -2 ] is the Dispersion tensor. According to this equation, the dispersive mass lux of a solute is proportional to the solute concentration gradient.

Combining the previous equations the well known Advection-Dispersion Equation (ADE) is obtained:

An underlying assumption of the ADE (SCHumER. et. alii, 2001) is that mechanical dispersion, like molecular diffusion, can be described by Fick's irst law.

In other words, this formulation assumes that the center of mass of the contaminant plume advances with the average ("macroscopic") luid velocity, while the contaminant spread about this center of mass (due to both mechanical and chemical effects) is a purely Fickian process (BERkowitz et alii,, 2001). These assumptions is in reality valid only under highly restrictive conditions. Details of these conditions, which in simple terms require a very high degree of homogeneity in the hydraulic conductivity, can be found in BERkowitz & SCHER (2000).

The classical ADE often fails to predict observed behavior of solute in the subsurface [START_REF] Cherubini | Perspective on theories of anomalous transport in heterogeneous media[END_REF].

Commonly, two kinds of deviations from the Fickian behavior are encountered [START_REF] Hassanizadeh | On the transient Non-Fickian Dispersion Theory[END_REF]. The irst regards the dependence of dispersivity values on the spatial and/or temporal scale of observation, investigated by many researchers (matHERoN aNd dE [START_REF] De | -Is transport in porous media always diffusive? A counterexample[END_REF][START_REF] Dagan | Flow and transport in porous media[END_REF].

The second kind of deviation regards the linear relationship between the dispersive mass lux J and the solute concentration gradient that becomes non linear for large concentration gradients (HaSSaNiza-dEH aNd lEijNSE, 1995).

Regarding the former deviation, as soon as the numerical models use a discretization that cannot reproduce this heterogeneity of the velocity ield, transport predictions by ADE fail (llERaR-mEza, 2009) In fact, the classical formulation of solute transport signiicantly underestimates the late-time behavior of breakthrough curves at observation locations in heterogeneous media. Solute transport is affected by the presence of high and low water velocity zones, where solute can be channelized or stagnant (llERaR-mEza, 2009).

Alternative conceptualizations have been proposed to account for the effects of this heterogeneity, such as: Multi-rate Mass Transfer (HaGGERty & GoREliCk, 1995;CaRRERa et alii, 1998) (

) 11
In questo studio sono state condotte una serie di prove di tracciamento in laboratorio per investigare il trasporto di soluto in un campione di roccia fratturata in cui è stata evidenziata sperimentalmente la presenza di condizioni di lusso non darciano (CHERuBiNi et alii, 2012). Lo scopo di tale lavoro è di mostrare come le curve di avanzamento (BTC) relative al trasporto di soluto misurate in condizioni non darciane presentano delle caratteristiche dificili da modellare utilizzando l' equazione di Advezione -Dispersione (ADE) convenzionale basata sulla legge di Fick.

Infatti tale modello non riproduce adeguatamente le code consistenti e i proili di concentrazione a volte bimodali che sono stati registrati nelle BTC.

CONFIGURAZIONE DEL MODELlO SPERIMENTALE

Gli esperimenti sono stati effettuati su un blocco di roccia carbonatica di forma parallelepipeda delle dimensioni di (0.6×0.4×0.08 = 0.0192 m 3 ) estratto dalla formazione del "Calcare di Altamura" rinvenibile nella parte centrale della regione Puglia. In Tab 1 sono riportate alcune caratteristiche isiche ed idrauliche del blocco. Mediante colpi di mazzetta da 5 kg è stato creato artiicialmente un sistema di fratture.

Successivamente, per ciascuna discontinuità, sono state digitalizzate mediante una fotocamera ad alta risoluzione le tracce sulle facce del blocco. Le immagini sono state poi rettiicate e calibrate rispetto a misure manuali effettuate con il calibro. Inine da queste immagini è stato possibile individuare il proilo medio e la distribuzione delle aperture meccaniche. In Fig. 1 è mostrata la distribuzione delle aperture meccaniche ottenuta sulla base di 16388 misurazioni. Il campione è stato sigillato mediante una resina epossidica trasparente (Fig 2b) e in corrispondenza delle estremità di ciascuna discontinuità è stato effettuato un foro circolare del diametro di 1 cm mediante l'utilizzo di un trapano a percussione. All'interno dei fori sono stati alloggiati dei raccordi idraulici 3/8''F-1/4' 'M'' saldati con il blocco di roccia mediante resina epossidica a rapido indurimento (Fig 2c -2d). Il blocco così sigillato è stato collegato ad un circuito idraulico (Fig. 2). L'acqua all'interno del blocco può In this study a series of controlled laboratory tracer test experiments have been carried out to investigate solute transport in a fractured rock sample where non-Darcian low conditions have been evidenced experimentally (CHERuBiNi et alii, 2012).

The purpose of this study is to show how breakthrough curves (BTC) of the solute transport measured under the non-Darcian low condition have some features that are dificult to explain using the Advection-Dispersion Equation (ADE) based on Fick' s law.

In fact this model fails to reproduce adequately the consistent tails and the sometime bimodal concentration proiles that have been registered in the BTCs.

EXPERIMENTAL SETUP

The experiments have been carried out on a block of carbonatic rock of parallelepiped shape (0.60×0.40×0.8 m = 0.0192 m 3 ) recovered from the 'Calcare di Altamura' formation which is located in central part of Apulia region (Southeastern Italy). In Tab. 1 are reported the bulk hydraulic parameters of limestone block. The fracture network has been made artiicially through 5 kg mallet blows.

Subsequently for each discontinuity fracture traces on the block surfaces have been digitalized by means of a high resolution digital camera.

The images have been then rectiied and calibrated with manual measurements carried out by means of a gauge.

Through these images it has been possible to obtain the average proile and the distribution of mechanical apertures.

In Fig. 1 it is showed the distribution of mechanical apertures obtained on the basis of 16388 sample data.

The surface of block sample has been sealed with transparent epoxy resin (Fig. 2b) and a hole of 1 cm diameter has been opened for each discontinuity in correspondence of the boundary of the block by means of a percussion drill.

Inside the holes hydraulic joints 3/8''F-1/4' 'M'' have been placed sealed with the block by means of rapid-hardening epoxy resin (Fig 2c -2d). muoversi, secondo diversi percorsi, in virtù della differenza di carico tra il serbatoio di monte, a cui è collegata la porta di entrata, e quello di valle, a cui è collegata la porta di uscita. I serbatoi di monte e di valle presentano le stesse dimensioni e sono di forma cilindrica con una sezione circolare di circa 0.1963 m 2 . Il carico all'interno dei due serbatoi non è mantenuto costante comunque la variazione di quest'ultimo risulta trascurabile relativamente alle modalità di esecuzione degli esperimenti. La portata istantanea che attraversa il blocco sigillato è misurata attraverso un velocimetro ad ultrasuoni posto in corrispondenza della porta di entrata. In corrispondenza della stessa è posizionato un sistema per l'immissione di tracciante (cloruro di sodio, NaCl) mentre, in corrispondenza della porta di uscita, è posizionata una cella di lusso predisposta per alloggiamento di una sonda multiparametrica del tipo IDRONAUT OCEAN SEVEN 304 CTD LOGGER con frequenza di campionamento di 8 Hz per la misura istantanea di pressione (dbar) del luido, temperatura (°C) e conducibilità elettrica (μS/cm) rispettivamente con la risoluzione di 0.0015%, 0.0006 °C, 0.1 μS/cm.

Prove idrauliche

Lo studio della dinamica del lusso d'acqua che attraversa il blocco è stato effettuato monitorando il movimento dell'acqua tra il serbatoio di monte e la cella di lusso avente una sezione circolare di circa 1.28×10 -4 m 2 priva della sonda multiparametrica. Al tempo t 0 le valvole 'a' e 'b' sono chiuse e l'altezza idraulica in condizioni idrostatiche misurabile in corrispondenza della cella di lusso è h 0 .

The sealed block sample is connected with a hydraulic circuit (Fig. 2). Water inside the block can move through different pathways according to the hydraulic head difference between the upstream tank connected to the inlet port and the downstream tank connected to the outlet port.

The upstream and downstream tank have the same dimensions and are of cylindrical shape with a circular cross section of 0.1963 m 2 . The hydraulic head inside the two tanks is not maintained constant but its variation is negligible relative to the modality of execution of the experiments.

The instantaneous low rate that lows across the block is measured by a ultrasonic low velocimeter placed in correspondence of the inlet port. In correspondence of the inlet port a system for the intake of tracer (sodium chloride, NaCl) is placed while in correspondence of the outlet port a multiparametric probe is arranged for the placement of a multiparametric probe IDRONAUT OCEAN SEVEN 304 CTD LOGGER with frequency of sampling 8 Hz for the instantaneous measurement of pressure (dbar), temperature (°C) and electric conductivity (μS/cm) respectively with resolution 0.0015%, 0.0006 °C and 0.1 μS/cm.

Hydraulic tests

The analysis of the low dynamics through the block regards the observation of water low from the upstream tank to the low cell which has a circular cross section of 1.28×10 -4 m 2 and has no multiparametric probe.

Initially at time t 0 , the valves 'a' and 'b' are closed and the hydrostatic head in low cell is equal to h 0 . The experiment begins with the L'esperimento inizia con l'apertura della valvola 'a' che viene richiusa quando l'altezza idraulica nella cella di lusso raggiunge l'altezza h 1 . Inine l'altezza idraulica della cella di lusso è riportata ad h 0 attraverso l'apertura della valvola 'b'. L'esperimento può essere ripetuto per differenti valori dell'altezza idraulica del serbatoio di monte e, per differenti conigurazioni delle porte di entrata ed uscita.

Per differenti valori di h c è stato misurato il tempo Δt = (t 1 -t 0 ) necessario per riempire la cella di lusso da h 0 ad h 1 .

Dal momento che la capacità del serbatoio di monte è di gran lunga maggiore rispetto alla capacità della cella di lusso è ragionevole ritenere che durante gli esperimenti l'altezza del serbatoio di monte rimane costante. Sotto questa ipotesi il lusso all'interno del sistema è governato dalla seguente espressione: opening of the valve 'a' which is reclosed when the hydraulic head in the low cell is equal to h 1 . Finally the hydraulic head in the low cell is reported to h 0 through the opening of the valve 'b'.

The experiment is repeated changing the hydraulic head of the upstream tank and for each coniguration of inlet -outlet ports.

For different values of h c the time Δt = (t 1 -t 0 ) required to ill the low cell from h 0 to h 1 has been registered.

As the capacity of the upstream tank is much higher than that of the low cell it is reasonable to assume that during the experiments the level of the upstream tank remains constant. Under this hypothesis the low inside the system is governed by the equation: Integrando l'equazione differenziale tra t = t 0 e t = t 1 e tra h = h 0 e h = h 1 si ottiene la seguente equazione: Mediante il ittaggio della relazione sperimentale tra Δt = t 1 -t 0 e h c è possibile ottenere la stima dei coeficienti A e B di ciascuna conigurazione di porte.

Prove di tracciamento

Lo studio della dinamica del trasporto di soluto che interessa il blocco è stata effettuata mediante un test di tracciamento utilizzando cloruro di sodio.

È stata imposta una differenza di carico iniziale tra i serbatoio di monte e valle pari a 0.30 m. Al tempo t = 0 la valvola 'b' è chiusa e l'altezza idrostatica all'interno del blocco è uguale a quella del serbatoio di monte. Al tempo t = 10 s viene aperta la valvola 'b' mentre al tempo t = 60 s vengono aperte contemporaneamente le valvole 'c' e 'd'. In questo modo viene immessa nel blocco una massa di soluto pari a 2×10 -4 kg. In corrispondenza della cella di lusso, nella quale è alloggiata la sonda multiparametrica, è possibile misurare la curva di restituzione del tracciante e l'altezza idraulica. Allo stesso tempo la portata d'ingresso nel sistema è misurata mediante il velocimetro ad ultrasuoni.

RISULTATI E DISCUSSIONE

Sono stati condotti differenti esperimenti per lo studio della dinamica di lusso e trasporto usando come ingresso la porta 1 e come uscita le porte 2 -3 -4 -5 -6 -7, in modo tale da poter analizzare differenti percorsi.

Per quanto riguarda gli esperimenti sulla dinamica di lusso l'altezza di controllo del serbatoio di monte è stata fatta variare entro il range 0.17 -1.37 m e, all'interno di detto range, è stata osservata una portata entro l'intervallo 3.08×10 -7 -2.99×10 -5 m 3 /s. Tutti gli esperimenti effettuati mostrano un comportamento non lineare del lusso che può essere ben descritto da una relazione del tipo di Forchheimer (2). In Fig. 4 è mostrata la metodologia di stima dei coeficienti A e B.

Integrating the differential equation between t = t 0 and t = t 1 and h = h 0 and h = h 1 the following expression is obtained:

By itting the experimental relation between Δt = t 1 -t 0 and h c it is possible to obtain an estimation of the coeficients A and B for each coniguration of ports.

Tracer tests

The study of solute transport dynamics through the block has been carried out by means of a tracer test using sodium chloride.

The initial hydraulic head difference imposed between the upstream and downstream tank is equal to 0.30 m. At t = 0 the valve 'b' is closed and the hydrostatic head inside the block is equal to the downstream tank. At t = 10 s the valve 'b' is opened while at time t = 60 s the valves 'c' e 'd' are opened at the same time. In this way a mass of solute equal to 2×10 -4 kg is put into the block.

In correspondence of the low cell in which the multiparametric probe is located it is possible to measure the breakthrough curve of the tracer and the hydraulic head; in the meanwhile the low rate entering the system is measured by means of ultrasonic velocimeter.

RISULTS AND DISCUSSION

Different experiments have been carried out for the study of low and transport dynamics using the port 1 as inlet and the ports 2 -3 -4 -5 -6 -7 as outlet, in such a way as to analyze different pathways.

As far as the hydraulic tests the control head of the upstream tank has been varied in the range 0.17 -1.37 m and among this range an average low rate of 3.08×10 -7 -2.99×10 -5 m 3 /s has been observed.

All the carried out experiments show a nonlinear low behavior that can be accurately described by Forchheimer law. In Fig. 4 a method for the estimation of coeficients A and B is shown.

As far as tracer tests an average low rate in the range 7.64×10 -7 -1.003×10-5 m 3 /s has been observed, while the solute residence time ( 14) Fig. 4 -Procedura di ittaggio per il percorso 1-2. La linea marcata rappresenta le relazioni senza l'interferenza del circuito idraulico -Fitting procedure for 1 -2 path.

Marked lines represent the relationships without the hydraulic circuit interference

Per quanto riguarda gli esperimenti sulla dinamica di trasporto è stata osservata una portata variabile tra 7.64×10 -7 -1.003×10 -5 m 3 /s mentre il tempo di residenza del soluto all'interno del blocco, ricavato come rapporto tra il momento del 1° ordine e quello di ordine 0 delle curve di restituzione, è stato osservato compreso tra 18.8 -471.57 s.

Se la geometria del sistema fosse semplice si potrebbero rappresentare le dinamiche di lusso e trasporto mediante modelli analitici. Nel caso in esame è stato necessario implementare un modello numerico tridimensionale ad elementi initi che riproducesse adeguatamente la geometria del network di fratture. Nello speciico è stata ricostruita la geometria del network di fratture e delle porte mediante modellazione nurbs usando il software GMSH. Successivamente la geometria ricreata è stata importata mediante un ile di interscambio di tipo STL nel software COMSOL Multiphysics per le successive simulazioni numeriche sia del lusso che del trasporto (CHERuBiNi et alii, 2012).

Per quanto riguarda il lusso (Fig. 5a), per ciascun percorso indagato sono state condotte differenti simulazioni in regime stazionario variando il differenziale di altezza idraulica tra la porta di ingresso e la porta di uscita. Per ciascuna conigurazione e per ciascuna simulazione numerica è stata confrontata la portata sperimentale (Q obs ), ricavata utilizzando l'espressione polinomiale, con quella simulata (Q sim ) ricavata effettuando l'integrale di linea sulla porta di uscita della portata speciica simulata. L' obiettivo era quello di trovare i parametri di Forchheimer equivalenti associabili a ciascun percorso che minimizzassero la differenza tra Q obs e Q sim . In Tab. 2 sono riportati, per i percorsi investigati, i coeficienti A e B, la permeabilità k e il termine inside the block, obtained as the ratio of the 1st order to the 0th order temporal moment of the BTC curves, varies in the range 18.8 471.57 s.

If the geometry of the system was simple it could be possible to represent low and transport dynamics by means of analytical models.

In the case study it has been necessary to implement a three dimensional inite element numerical model that reproduces adequately the geometry of the network of fractures.

Specifically, the geometry of the network of fractures has been reconstructed by means of nurbs modeling using GMSH software. Successively the geometry has been imported by means of an interchange STL file in COMSOL Multiphysics software for numerical simulations of flow and transport (Cherubini et alii, 2012).

As far as low (Fig. 5a), for each investigated pathway different steady state simulations have been carried out varying the hydraulic head difference between inlet and outlet ports.

For each coniguration of ports and for each numerical simulation the experimental low rate (Q obs ) obtained by means of the polynomial expression and the simulated one (Q sim ), obtained by means of the line integral on the outlet port of the simulated speciic low rate, have been compared.

The aim is to ind the equivalent Forchheimer parameters associable to each pathway that minimize the difference between Q obs and Q sim .

In Tab. 2 are reported for the investigated pathways the coefi- cients A and B, the permeability k and the inertial term β as well as the equivalent aperture w eq obtained from k assuming valid the cubic law (moRENo et alii, 1985).

On the basis of the obtained results from the steady state low model a transport model has been implemented based on a transient ADE-R approach (Fig 5b). In particular in correspondence of the inlet and the outlet ports the measured values of low rate and hydraulic head have been imposed; the tracer intake has been simulated by means of the imposition of a concentration function assigned on the inlet port calibrated on the basis of the characteristics of the intake system.

In Fig. 6 is shown for each pathway the comparison between the observed and simulated breakthrough curves. The dispersion coeficient D [L 2 T -1 ] and the retardation coeficient R, to be considered as a corrective coeficient to apply to the low ield calculated on the basis of the hydraulic tests, have been calibrated on the basis of the tracer irst arrival and the irst peak.

All the measured BTCs exhibit the same trend, which is a non-Gaussian distribution with a long tail.

The ADE model has done a poor job in explaining the solute transport as it fails to it the peak concentration and the tail. inerziale β nonché l'apertura equivalente w eq ottenuta a partire da k supposta valida la legge cubica (moRENo et alii, 1985).

Sulla base dei risultati ottenuti dal modello di lusso in regime stazionario è stato implementato un modello di trasporto basato su un modello advettivo dispersivo con ritardo (ADE-R) in regime transitorio (Fig 5b). In particolar modo, in corrispondenza della porta di entrata e di uscita, sono state imposte, rispettivamente, la portata e l'altezza idraulica sperimentali registrate durante la prova di tracciamento, mentre l'immissione del tracciante è stata simulata mediante l'imposizione di una funzione di concentrazione, imposta sulla porta di entrata, tarata sulla base delle caratteristiche del sistema di immissione. In Fig. 6 è mostrato, per ciascun percorso, il confronto tra le curve di restituzione osservate e quelle simulate.

Il coeficiente di dispersione D [L 2 T -1 ] e il coeficiente di ritardo R, da intendersi quest'ultimo come un coeficiente correttivo da applicare al campo di velocità stimato sulla base delle prove idrauliche, sono stati calibrati sulla base del primo arrivo e del primo picco.

Tutte le BTC misurate mostrano una distribuzione non Gaussiana con una lunga coda. L'approccio ADE non è stato in grado di descrivere compiutamente la dinamica del trasporto in quanto non ha ittato bene il picco e la coda di ciascuna curva di avanzamento. This failure is most clearly evidenced by the inding of "scaledependent dispersion": contrary to the fundamental assumptions underlying use of the ADE, dispersivity is not constant, and the very nature of the dispersive transport seems to change as a function of time or distance traveled by the contaminant (GElHaR et alii, 1992). This scale-dependent behavior is what we shall refer to as "non-Fickian" transport.

Evidence of non-Fickian behavior lies in the often observed long late time tails in measured breakthrough curves which have been attributed to: (1) the diffusive exchange of solutes between the lowing zone in the main low channel and immobile luid zones near the walls (RavEN et alii., 1988), (2) solute diffusion into rock matrix [START_REF] Neretnieks I | A note on fracture low dispersion on mechanism in the ground[END_REF]laPCEviC et alii, 1999), (3) channeling or tortuous low due to aperture variations [START_REF] Neretnieks I | A note on fracture low dispersion on mechanism in the ground[END_REF]moRENo et alii,1985;tSaNG aNd tSaNG, 1987), and (4) sorption of solutes on the fracture walls (NEREtNiEkS et alii, 1982).

CONCLUSIONS

In this study hydraulic and tracer tests on artiicially created fractured rock samples have shown evidence of non linearity in low and concentration proiles that cannot be described by conventional solute transport models.

All BTCs from the experiments exhibit a non-Gaussian distribution with a long tail and demonstrates the non-Fickian nature of transport. The ADE approach implemented on a low model calibrated on the hydraulic tests has done a poor job in explaining the solute transport as it fails to it the peak concentration and the tail.

The carried out study has allowed moreover to evaluate possible improvements to the experimental setup about the system for solute intake, the techniques for characterizing and preparing the rock sample.

Laboratory experiments give the advantage of improving the understanding of physical mechanisms under relatively well-controlled conditions, since the dependence of a physical process on different parameters can be tested and modelled.

Nevertheless, laboratory tests are characterized by relatively small temporal and spatial scales that pose the question on the representativity and transferability of laboratory data to the real system (up-scaling).

Therefore further developments of this study would be investigating the migration processes at a spatial scale larger than the lab one up to the ield tests.

Moreover, in order to be able to generalize the obtained results in terms of solute transport, different tracer tests will be carried out for each path varying the low rate in order to evaluate the possible interactions between the form of the tracer breakthrough curve and the low regime.

Ciò è attribuibile alla dispersione che dipende dalla scala di studio: contrariamente alle assunzioni alla base dell' utilizzo dell' ADE, la dispersività non è costante e la vera natura del trasporto dispersivo varia in funzione del tempo e della distanza percorsa dal contaminante (GElHaR et alii, 1992). Questo comportamento dipendente dalla scala è responsabile della natura non-Fickiana del trasporto.

Evidenza di comportamento non-Fickiano è riscontrabile nella presenza di lunghe code nelle curve di avanzamento che sono state attribuite a: 1) processi di scambio diffusivo tra le zone a lusso più rapido al centro della frattura e le zone di ristagno vicino alle pareti (RavEN et alii, 1988), 2) diffusione del soluto nella matrice rocciosa [START_REF] Neretnieks I | A note on fracture low dispersion on mechanism in the ground[END_REF]laPCEviC et alii, 1999), (3) fenomeni di canalizzazione o lusso tortuoso dovuto alle variazioni di apertura delle fratture [START_REF] Neretnieks I | A note on fracture low dispersion on mechanism in the ground[END_REF]moRENo et alii, 1985;tSaNG & tSaNG, 1987), e (4) fenomeni di adsorbimento di soluto sulle pareti delle fratture (NEREtNiEkS et alii, 1982).

CONCLUSIONI

In questo studio le prove idrauliche e i test di tracciamento condotti su un campione di roccia fratturato artiicialmente hanno mostrato evidenza di fenomeni di lusso non lineare e proili di concentrazione che non possono essere descritti dai tradizionali modelli di trasporto di soluto.

Tutte le BTC misurate mostrano una distribuzione non Gaussiana con una lunga coda e dimostrano la natura non-Fickiana del trasporto. L'approccio ADE implementato su un modello di lusso tarato sulle prove idrauliche effettuate non è stato in grado di descrivere la dinamica del trasporto in quanto non ha ittato bene il picco e la coda di ciascuna curva di avanzamento.

Lo studio condotto ha permesso inoltre di valutare possibili miglioramenti all' apparato sperimentale circa il dispositivo di immissione del tracciante, le tecniche di caratterizzazione e le modalità di confezionamento del blocco di roccia.

Gli esperimenti di laboratorio hanno il vantaggio di migliorare la comprensione dei processi isici di lusso e trasporto sotto condizioni relativamente controllate, poiché è possibile testare, e successivamente modellare, la dipendenza del processo isico da differenti parametri Di contro i test di laboratorio sono caratterizzati da una scala spazio-temporale limitata il che pone la questione della rappresentatività e della trasferibilità dei dati ottenuti in laboratorio a sistemi reali (upscaling).

Pertanto sviluppi futuri di questo studio potranno riguardare l'analisi dei fenomeni di migrazione a scala spaziale maggiore di quella di laboratorio ino all' esecuzione di prove di campo.

Inoltre, per poter generalizzare i risultati ottenuti in termini di trasporto di soluto, saranno condotte per ciascun percorso diverse prove di tracciamento al variare della portata di delusso al ine di valutare le possibili interazioni tra la forma della curva di restituzione del tracciante e il regime di moto.

  a f and b f are related to a and b:where w represents fracture aperture.

  , Time-Dependent Macrodispersive [daGaN (1991);[START_REF] Rajaram | Plume scale-dependent dispersion in heterogeneous aquifers 2. Eulerian analysis and three-dimensional aquifers[END_REF] dENtz et alii, (2000)], Continuous Time Random Walk[START_REF] Berkowitz | Ther role of probabilistic approaches ti transport theory in heterogeneous media[END_REF]) and Fractional Advection-Dispersion Transport [BENSoN et alii, (2000[START_REF] Metzler | The random walk's guide to anomalous diffusion: A fractional dynamics approach[END_REF];SCHumER et alii, (2003)].As far as the latter deviation, a recently developed theory(HaS- SaNizadEH & lEijiNSE, 1995) suggests a nonlinear relation between the concentration gradient and dispersive mass lux, introducing a new parameter in addition to the longitudinal and transversal dispersivities.

Tab. 1 -

 1 Fig. 1 -Distribuzione delle aperture meccaniche delle discontinuità ricavata sulla base di 13688 misurazioni -Mechanical aperture distribution obtained from 13688 sample data

Fig. 2

 2 Fig. 2 -a) discontinuità artiiciali create mediante colpi di mazzetta da 5 kg; b) colata con resina epossidica; c) esempio di foro in corrispondenza delle estremità delle discontinuità;. d) inserimento delle porte per la connessione al circuito idraulico -a) artiicial discontinuities produced by means of 5 kg mallet blows; b) epoxy resin casting; c) example of a hole in correspondence of the edges of the discontinuities; d) insertion of ports for the connection to hydraulic circuit

  dove A 1 [L 2 ] e h [L] sono, rispettivamente, l'area della sezione e l'altezza idraulica della cella di lusso; h c [L] è l'altezza idraulica del serbatoio di monte; Δh è la perdita di carico all' interno del circuito che si ricava dalla legge di Chezy. Combinando insieme quest' ultima (CHERuBiNi & PaStoRE, 2012) e l' espressione polinomiale Δh = A • Q + B • Q 2 (dove A e B sono, rispettivamente, i coeficienti di perdite di carico lineare e non lineare relativi al blocco sigillato), la conduttanza idraulica Γ [L 2 T -1 ] relativa all'intero sistema può essere espressa come: dove R c rappresenta il coeficiente di perdite di carico relativo al circuito.

  where A 1 [L2] and h [L] are respectively the section area and the hydraulic head of the low cell; h c[L] is the hydraulic head of upstream tank; Δh is the hydraulic loss inside the circuit that can be expressed by means of Chezy law.Combining the latter (CHERuBiNi & PaStoRE, 2012) with the polynomial expression Δh = A • Q + B • Q 2 (where A and B are the linear and nonlinear hydraulic loss coeficients of the sealed block), the hydraulic conductance term Γ [L 2 T -1 of the whole system can be expressed as:where R c represents the loss coeficient of the hydraulic circuit.

Fig. 3 -

 3 Fig. 3 -Diagramma schematico del setup sperimentale -Schematic diagram of the experimental setup

Fig. 5

 5 Fig. 5-a) Distribuzione dell'altezza idraulica per il percorso attivo 1-5 per una differenza di carico tra la porta di entrata e di uscita di 0.7 m. b) distribuzione della concentrazione normalizzata rispetto alla massa iniettata per il percorso 1-5 al tempo t=15 s. -a )Hydraulic head distribution for active path 1-5 for a difference of 0.7 m of hydraulic head between inlet and aoutlet port b) normalized concentration distribution for active path 1-5 at time t=15 s

Italian Journal of EngineeringGeology and Environment, 1 (2013)© Sapienza Università Editrice www.ijege.uniroma1.it

C. CHERUBINI, C. GIASI & N. PASTORE


Alain Pirovano

Fabien Garcia

José Radzik

Garcia Fabien

Capacity Dimensioning for Aeronautical Communications in North Atlantic Corridor

In the context of the Internet everywhere paradigm, aircraft passengers expect to get connectivity during flights. Several solutions based either on cellular networks in continental area or on satellite links have been designed and even deployed for some of them. But to face the increasing number of users and expected services, a migration to future satellite system such as Inmarsat's Ka band Global Xpress system have been planned. The considered geographical area of the present study is the north Atlantic corridor. In this service zone, the specific structure of aeronautical traffic must be taken into account. NAT (North Atlantic Tracks) are oceanic routes between North America and Europe that are periodically defined considering weather and wind conditions. This explains why the exact locations of the tracks change daily according to weather and also traffic demands. On each continent, specific air traffic control is provided in order to manage entrance and movement along these tracks. The proposed study makes the assumption of a flexible global coverage provided by multi beam Ka band satellites that operate in geosynchronous orbit. Aircraft flying in the north Atlantic corridor use the system in order to offer in-flight connectivity for airline passengers. Doing so, passengers may access common Internet services, namely web browsing, electronic mail, or file transfer. More specific applications for pilots and companies are also taken in consideration enhanced with new services such as real time black box data streaming. In order to assess induced traffic load and instantaneous required capacities in each spot beam, a simulation model has been specifically developed by ENAC and ISAE with Opnet Modeler software.

Introduction, motivations and objective

In the current aeronautical context new services for civil aircraft passengers are expected and represent an increasing interest. For instance, airlines want to provide their customers Internet access and hence web browsing, mailing or even access to corporate VPNs.... All these services represent a new type of air-ground communications which should be based on relative high capacity air ground links. The presented study aims at dimensioning satellite capacity over the North Atlantic corridor for aeronautical communications based on a Ka-band multibeam satellite. Two beam layouts are considered. The first one corresponds to INMARSAT-5 F2 satellite: 89 spot beams cover the Earth surface as seen from the satellite; the orbital position is 55° West. A second configuration is defined with a limited coverage over the North Atlantic and narrow beams (aperture of 0.6°).

As in any multibeam satellite system, the distribution of the potential users over the service zone is of tremendous importance for capacity dimensioning. The context of aeronautical communications over the North Atlantic is however highly specific as aircraft follow predefined trajectories. Because there is a lack of radar coverage in oceanic area, predefined routes are periodically computed ensuring that aircraft are separated over the ocean. The minimum separations between aircraft flying between the altitudes of 28,500 and 42,000 feet in NATS airspace is about 50 NM for the lateral separation, 5 to 10 minutes for the longitudinal separation, and 1000 feet for the vertical separation. In order to ensure flight safety, each flying aircraft must report in as it crosses various known waypoints along each track. These reports enable the Oceanic Controllers to maintain separation between aircraft. These reports can be made to controllers via a satellite communications link or via High Frequency (HF) radios. Furthermore, the routes are aligned in order to maximize tail winds and minimize any head winds impact on the aircraft with the aim to provide efficient flight by reducing flight time and fuel consumption.

Hence, based on an assessment of air traffic density in the geographic area considering accurate aircraft position derived from real positions, we study the dynamic occupation of each beam during one day. These more realistic trajectories based on NATs are computed using Eurocontrol data which give aircraft reporting points for each flight. Between successive reporting points, intermediate aircraft positions are interpolated. The first advantage of this approach is that the aircraft trajectories are based on real positions rather than using great circle distances between the departure and the arrival airports.

The specific NAT structure and daily air traffic management leads to a large imbalance between data traffic loads in satellite beams. In order to assess the amount of generated traffic in beams, mainly on the return link, a simulation platform has been developed using Opnet Modeler software. It relies on wireless network cabin models based on a complete TCP/IP protocol stack simulation. The model is composed of aircraft on-board systems based on Wi-Fi access point connected to ground through Ka band satellite links in order to offer an Internet access to passengers.

System description and simulation tools

Communication systems for Aircraft passengers

The first systems that have been developed to provide Internet access to passengers used satellites. For instance, from 2004 to 2006 the Connexion-by-Boeing [START_REF] Jones | Connexion by Boeing SM-broadband satellite communication system for mobile platforms[END_REF] system was installed on about 150 aircraft. But mainly for economic reasons, the service was stopped in 2006. Since 2008, ARINC proposes another solution, named Oi for Onboard Internet [START_REF]ARINC Onboard Internet Website[END_REF]. Oi uses Inmarsat satellites with the SwiftBroadband service that offers throughput up to 432 kbps. In the same year, Panasonic offers a new solution called eXConnect [START_REF]Panasonic Website[END_REF] based on Intelsat satellites. OnAir solution enables passengers to access Internet through their laptop, tablet or smartphone during flights. Currently, OnAir uses Inmarsat's SwiftBroadband service operating in the L band with a limited bandwidth of 864 kbit/s. Another way of providing In-Flight Internet is to use a direct link to the ground, these solutions are obviously limited to continental areas (Gogo Inflight, Wi-SKY7 …).

Considered satellite communication systems

More recently, in order to accommodate a wide range of multimedia services, new multibeam satellite systems have been deployed with the aim to provide high links capacity to end users. In this study, we have mainly concerned by the definition of the spot beams over the North Atlantic region. Two configurations have been investigated. Figure 1 presents the two configurations in the standard (θ, φ) coordinates.

Figure 1 -Beam layout: INMARSAT V-F2 (55° W), hypothetical satellite (18° W)

The first one corresponds to the INMARSAT-5 F2 satellite. From data provided by the INMARSAT and Boeing companies (http://www.igx.com/introduction_to_satellites, http://www.boeing.com/boeing/ defense-space/space/bss/factsheets/702/Inmarsat-5/Inmarsat-5.page), the satellite covers the entire viewable Earth region with 89 equally distributed beams, additional capacity is provided by six steerable spot beams. The second configuration introduces a hypothetical satellite specifically dedicated to the North Atlantic coverage. The beam layout is defined by a regular beam pattern as seen from the satellite, the aperture is 0.6°. The resulting projection of beams on the Earth surface is illustrated in Figure 2.

North Atlantic aeronautical service zone

The North Atlantic Airspace is a zone going roughly form 10°N to the North Pole and from 60°W to 10°W. It is divided in 5 areas named after the Air Traffic Control (ATC) centre responsible for enforcing regulations in them: Reykjavik, Shanwick, Gander, New York and Santa Maria. As [START_REF]European and North Atlantic Office of ICAO[END_REF] states it: "The North Atlantic is the busiest oceanic airspace in the world. In 2011 approximately 450,000 flights crossed the North Atlantic." Due to this aircraft density and the lack of radar equipment on this zone, the ATC must enforce strict separation rules between aircraft by assigning routes to them and having each aircraft reports its position periodically (generally one report every hour).

Most of the traffic in the North Atlantic Airspace is organised in two major flows: a westbound flow departing Europe in the morning and an eastbound flow departing North America in the evening (UTC time). The first flow crosses the 30°W longitude between 11:30 and 19:00 (UTC) and the second flow crosses the same longitude between 01:00 and 08:00 (UTC). This Organised Track System (OTS) is the result of trying to satisfy passenger demands and airport noise reduction considering the time zone differences. Furthermore, it allows easy separation of westbound and eastbound flights by spreading them in different time slots. The set of westbound and eastbound tracks available for flying is published daily by ATC centres (Shanwick and Gander respectively). It is chosen so as to allow a maximum number of aircraft to follow them while guaranteeing the most economic flight conditions (considering winds and aircraft preferred altitude). Finally it is to be noted that those tracks are not mandatory, and the International Civil Aviation Organisation (ICAO) consider that half of the aircraft follow them as of 2012 [START_REF]European and North Atlantic Office of ICAO[END_REF], aircraft not following the tracks must nevertheless expect less than optimal flight altitude and frequent rerouting to ensure aircraft separation rules within the NAT.

a) Aircraft positions

In order to assess the necessary capacity for satellite beams on the NAT airspace, aircraft position have to be known. In this study, we used Eurocontrol data available from the OneSky website. Data has been gathered for several relevant days of traffic and then filtered so as to keep only air traffic relevant to the zone of interest. Finally, the position have been interpolated between the report points available so as to heighten time resolution in our study before applying a final stage of filtering to remove aircraft positions that lie out of the coverage of the satellite beams.

Data provided by Eurocontrol (OneSky website) contains aircraft position gathered through radar and periodic reports made by aircraft. In this second case, positions are known approximately every hour, which is not sufficient to infer satellite beam occupation as aircraft cover large distance in such a time period. It was thus decided to interpolate between known positions using a simple algorithm:

successive positions further apart than 100NM are interpolated using great circle arcs; loxodromy is used (i.e. constant heading route) for positions closer than 100NM. Aircraft speed is considered constant in between known positions, which is consistent with ATC rules in NAT airspace where aircraft are required to keep a constant velocity. This approach differs from the one taken in other studies where flight trajectories are approximated by great circle arcs between departure and destination airports as in [START_REF] Medina | A Geographic Routing Strategy for North Atlantic In-Flight Internet Access Via Airborne Mesh Networking[END_REF] and [START_REF] Medina | North Atlantic Inflight Internet Connectivity via Airborne Mesh Networking[END_REF]. Figure 3 shows, for a flight between Paris and New York, both the trajectory derived from the aircraft reporting points from our algorithm (in red) and the great circle arc trajectory. Considering the size of a beam satellite in our work, this clearly demonstrates the advantage of the first approach in order to avoid an important bias in our statistical study. The provided data include all commercial flights; however our focus is on passenger transcontinental flights at the exclusion of local and freight traffic. Several filters are implemented: first, a filter on the aircraft type as been applied so as to keep only aircraft that are technically able to carry satellite communication terminals; and then a filter is passed on aircraft crossing the 30°W longitude meridian to ensure that only the aircraft that actually cross the north Atlantic will be processed.

The interpolation procedure described in the previous section was then applied to all remaining aircraft trajectories. Finally the trajectories where pruned of all positions lying west of 90°W, east of 10°E, south of 23.5°N and north of 70°N. The satellite should always be seen with an elevation no less than 10°. This last condition was important so as to remove aircraft flying out of the zone where they would be able to catch the satellite signal.

Table 1 -Aircraft count statistics

It is to be noted that the number of aircraft flying over the NAT airspace doesn't scale in the same way as the total number of aircraft, which is consistent with the claim that this airspace is the densest in terms of aircraft flying through it.

d)

Simulation model

The considered system has been modelled using OPNET Modeler software (Opnet technologies Inc, www.opnet.com). The simulation paradigm is discrete event simulation. The model makes use of the incorporated libraries, mainly for the TCP/IP protocol stack, and specific modules.

The top-level simulation model is presented in Figure 4 (left). Several aircraft flying in north Atlantic corridor are modelled. The considered aircraft of a single simulation belong to one beam. Each aircraft is assumed to have at least one hundred passengers using their laptop in order to access to Internet resources.

Figure 4 -OPNET Modeler simulation model

This model is composed of different components: several aircraft (TCP/IP end users, aircraft router providing Wifi wireless access to end passengers' nodes, a satellite terminal) and a ground subnet (ground router linked with a server which answers to passengers requests, a satellite terminal for Earth gateway connectivity)

The Figure 4 (right) shows a part of the aircraft cabin model based on a Boeing 777 cabin plan with one hundred users, the aircraft router, and the satellite terminal. The set of applications used by the end users can be chosen. Based on applications provided by the Opnet libraries, two types of end users have been defined: "Classical users" (web browsing and simple mail with applications) and "Advanced users (web browsing, mail with attached files and voice over IP). The settings of applications parameters are given in annex.

Capacity Dimensioning in North Atlantic Corridor

In this part, we estimate the cumulated traffic generated by all the passengers of the aircraft flying in a same beam at one time. This assessment is obtained using the previously described model developed with OPNET Modeler. Considering the two satellite configurations, we have computed the maximum, minimum, and mean number of aircraft in a beam during one day. Considering these results, the cumulated traffic generated by the aircraft is obtained through simulations. We have considered four cumulated throughput types in each simulation:

• in each aircraft, the cumulated throughput generated by the passengers applications between the aircraft router and the aircraft satellite terminal (AC_GWToSatTerm, satellite return link) and the cumulated throughput generated by ground servers for the aircraft passengers between the aircraft satellite terminal and the aircraft router (AC_SatTermToGW, satellite forward link)

• on the ground network, the cumulated throughput generated by the passengers applications of all aircraft between the ground satellite terminal and the ground router (Ground_SatTermToGW, satellite return link) and the cumulated throughput generated by the ground servers between the ground router and the ground satellite terminal (Ground_GWToSatTerm, satellite forward link)

Beam occupation

In order to assess the necessary capacity for the satellite system, we had to compute the number of aircraft present in each beam with respect to time. Results of these measurements for the day of 2012 June 29 are shown in Figure 5 in the case of the INMARSAT-5 beam layout.

Table 2 -Beam aircraft occupation, INMARSAT-5 configuration

Table 3 shows the beam occupation for the proposed alternative satellite coverage (10 most loaded beams (in average) for the 29 th of June 2012). Here again we observed that the data reflected the flight rules for NAT airspace. As expected the maximum number of aircraft per beam is proportionally less than for the Inmarsat coverage.

Expected throughputs per beam

As explained in the previous part, we provide here the simulation results we obtained. Firstly, in order to partially validate the models and underline some trends, we present in Figure 6 the cumulated throughput observed for 5 aircraft considering two different profiles users. As expected, in the case of classical users, the cumulated throughputs are asymmetrical. The cumulated throughput from one aircraft is about 50 Kbit/s and the cumulated throughput to aircraft is about 110 Kbits/s. On the ground link, the aggregated throughputs for the 5 aircraft show the same trend: about 250 Kbit/s from the aircraft to the ground and about 580 Kbit/s from the ground to the aircraft.

In the Figure 6 (right), we can see that when the set of applications is enhanced with file transfer and foremost voice over IP, the cumulated throughputs are less asymmetrical. Considering one aircraft, the outcoming and incoming cumulated throughputs are quite similar that is about 600 Kbit/s. And the aggregated throughputs on the ground link for the 5 aircraft are about 2.5 Mbit/s. We can also deduce from these first results that the cumulated traffic properties between the aircraft and the ground should not allow obtaining a noticeable statistical multiplexing gain: the cumulated throughput on the ground link is expected to increase linearly with the number of aircraft as confirmed by simulation (Figure 7). The needed capacity in the heaviest loaded beam is about 7.9 Mbits/s from the aircraft to the ground and 18.6 Mbit/s from the ground to the aircraft in the case of 'classical' users. Yet again, we can see the asymmetric behaviour of classical Internet data flows. And in the case of advanced users, the needed capacity is about 76.5 Mbits/s from the aircraft to the ground and 85.4 Mbit/s from the ground to the aircraft. In the heaviest loaded beam considering 'classical' users, the needed capacity is about 2.2 Mbits/s from the aircraft to the ground and 5.2 Mbit/s from the ground to the aircraft. With 'advanced' users, the needed capacity reaches 21.5 Mbit/s from the aircraft to the ground and 24 Mbit/s from the ground to the aircraft. In this beam configuration, each one of these needed capacities is more than three times smaller.

Conclusions

In this paper multibeam satellite systems have been considered and the needed capacity has been assessed in order to provide Internet access to aircraft passengers in north Atlantic corridor. Two configurations have been investigated, the first one corresponds to the INMARSAT-5 F2 satellite and the second one makes the assumption of a satellite specifically dedicated to the North Atlantic coverage. In both configuration, an innovative approach allowing to assess accurate 4D aircraft positions is proposed in order to compute beam occupation and hence the needed capacity to provide Internet access to aircraft passengers. This latter results are obtained using a specific simulation model, and show that in both satellite configurations the needed capacity matches with the one which could be provided in future multibeam satellite systems.

Figure 2 -

 2 Figure 2 -Satellite spot beam coverage on North Atlantic zone: INMARSAT-V F2 (left), hypothetical satellite (right)

Figure 3 -

 3 Figure 3 -Trajectory estimation of a real flight between Paris and New York b) Data filtering

Figure 5 -

 5 Figure 5 -Number of aircraft in the 4 most loaded beams according to time for 2012 June 29When going east, the successive beams are numbered 3, 10 and 17 (beam 7 being north of 3 and 10). The traces shape clearly results from the flight rules described in section 2.3 (two flows, one going west in the morning, and one going east in the evening). Table2presents the statistics for the 4 most loaded beams.

Figure 6 -Figure 7 -

 67 Figure 6 -Cumulated throughput (5 aircraft, classical users left, advanced users right)

  For statistical analysis, three days from 2012 and 2013 have been analysed for our study. The first one, June 29 2012, was the most loaded day for 2012. The second, October 10 2013, was a medium to heavy load day. Finally the last day, January 1 2013 was the least loaded day for 2012 and 2013. Results for number of aircraft are show in Table1. First column gives the total number of flights recorded by Eurocontrol. The second and third columns correspond to the number of aircraft after filtering by aircraft type and 30°W crossing and by trajectory pruning respectively. Last column gives the maximum number of aircraft flying simultaneously on a given day over the considered area.

	c)	Aircraft number assessment
			Total flights in data Flights after first filter Flights after second filter Peek Instantaneous Aircraft Count
		June 29 2012	32680	1352	1346	482
		October 10 2012	30323	1227	1224	444
		January 1 2013	15324	905	900	325

  Table 2 presents the statistics for the 4 most loaded beams.

	Beam	2012 June 29		2012 October 05		2013 January 1
		Average	Max	Average	Max	Average	Max
	3	46.44	150	41.84	165	27.98	95
	7	23.48	94	20.70	61	11.92	53
	10	43.75	160	44.04	144	30.54	101
	17	42.81	153	41.97	156	29.63	117

Table 3 -Beam aircraft occupation, hypothetical satellite

 3

	Beam	2012 June 29
		Average	Max
	152	8.30	36
	28	8.05	45
	35	7.86	39
	22	7.72	40
	43	7.17	29
	134	6.79	37
	69	6.66	31
	116	5.97	28
	29	5.72	24
	143	5.44	29

Table 4

 4

				AC_GWToSatTerm AC_SatTermToGW Ground_SatTermToGW Ground_GWToSatTerm
		One	1	47,99	106,62	47,98	116,64
	Classic users	Mean 43,78	47,99	106,62	2151,34	5081,30
		Max 160	47,99	106,62	7862,37	18570,30
		One	1	569,07	687,62	618,49	687,73
	Adv. users	Mean 43,78	569,07	687,62	20933,98	23354,18
		Max 160	569,07	687,62	76506,08	85351,04

presents the results obtained for the heavy day, tables for the two others days are given in annex.

Table 4 -Cumulated Throughputs with beam configuration 1 (june 29 2012)

 4

Table 5

 5 gives the simulations results with the hypothetical satellite beam configuration more specifically dedicated to cover the north Atlantic corridor.

				AC_GWToSatTerm	AC_SatTermToGW	Ground_SatTermToGW	Ground_GWToSatTerm
		One	1	47,99	106,62	47,98	116,64
	Classic users	Mean 8,05	47,99	106,62	395,58	934,32
		Max	45	47,99	106,62	2211,29	5222,90
		One	1	569,07	687,62	618,49	687,73
	Adv. users	Mean 8,05	569,07	687,62	3849,21	4294,22
		Max	45	569,07	687,62	21517,34	24004,98

Table 5 -Expected Cumulated Throughputs with beam configuration 2

 5

Annex -Applications parameters settings