Spatial Up-scaling of the Retention by Matrix Diffusion

Working Repor 2006-101 Spaial Up-scaling of he Reenion by Mari Diffusion Ani Poeri November 2006 POSIVA OY FI-27160 OLKILUOTO FINLAND Tel +358-2-8372 31 Fa +358-2-8372 3709

Working Repor 2006-101 Spaial Up-scaling of he Reenion by Mari Diffusion Ani Poeri VTT Technical Research Cenre of Finland November 2006 Working Repors conain informaion on work in progress or pending compleion. The conclusions and viewpoins presened in he repor are hose of auhors and do no necessarily coincide wih hose of Posiva.

PREFACE This work has been carried ou as a par of he European research projec FUNMIG: Fundamenal processes of radionuclide migraion www.funmig.com. FUNMIG is a four year projec ha will be carried ou beween years 2005 and 2008. Paricipaion of he VTT o he FUNMIG projec is joinly funded by EU and Posiva Oy. The presen repor is an 18 h monh projec delivery PID4.6.1 of he work package 4.6. Work package 4.6 covers up-scaling processes of he reenion and ranspor processes in he crysalline rock.

ABSTRACT Fracured rocks are composed of porous bu impermeable rock mari and waer conducing fracures which are he main conduis for he groundwaer flow. The main characerisic of he fracured rock is he grea heerogeneiy in differen scales. The heerogeneous srucure of he fracured rock leads easily o preferenial flow pahs ha will govern boh flow and ranspor properies. This channelling of he flow is an imporan process ha needs o be aken ino accoun when he reenion in he fracured rock is modelled. A leas hree disinc flow environmens can be idenified in he fracured rock: channeling causing variable flow in he individual fracure planes ransmissiviy differences beween he fracures leading o preferenial flow pahs hrough he fracure nework and eensive fracure zones providing highly ransmissive connecion over long disances. The main challenge of he spaial up-scaling of he reenion properies is conneced o he descripion of he flow characerisics of he fracured rock. A he momen here is no general up-scaling mehod ha can be applied for he performance assessmen scale problems. The field eperience and modelling has shown ha he preferenial flow and ranspor pahs may run over long disances. Therefore i is imporan ha he upscaling mehod does no lack of long disance connecions which may be imporan in he field scale ranspor. The presen siuaion suppors recommendaion of he RETROCK projec 2005 o apply muliple approaches o ensure ha he esimaes are robus. Keywords: Mari diffusion groundwaer flow ranspor of solues performance assessmen

MATRIISIDIFFUUSION AIHEUTTAMAN PIDÄTTYMISEN ETÄISYYSRIIPPUVUUDEN LAAJENTAMINEN TIIVISTELMÄ Rakoillu kallio muodosuu huokoisesa kalliomariisisa ja veä johavisa raoisa. Pohjaveden viraus apahuu käyännössä pääasiasiassa pikin veä johavien rakojen muodosamia reiejä. Rakoilleen kallion merkiävimpiä piireiä on eri miakaavoissa havaiava heerogeenisuus. Heerogeenisuus johaa helposi veden virauksen kanavoiumiseen. Virauksen kanavoiuminen piää oaa huomioon arvioiaessa veden mukana kulkeuuvien aineiden pidäymis- ja kulkeuumisominaisuuksia eri virausreieillä. Rakoilleessa kalliossa voidaan unnisaa ainakin kolme erilaisa viraukseen vaikuavaa ilmiöä: virauksen kanavoiuminen yksiäisissä raoissa rakojen vedenjohokykyeroisa johuva hallisevien yksiäisen virausreiien muodosuminen rakoverkosojen läpi sekä yksiäise laaja rakovyöhykkee joka muodosava hyvin johavia yheyksiä pikien eäisyyksien yli. Suurin haase eri miakaavoissa apahuvan pohjaveden virauksen kuvauksessa liiyy virauksen kuvaukseen rakoilleessa väliaineessa. Tällä hekellä ei ole olemassa yleisä meneelmää jolla pienen miakaavan virausilmiö olisi mahdollisa oaa huomioon suuren miakaavan malleissa joia arviaan loppusijoiusilan oiminakykyanalyyseissa. Kenäukimukse ja mallinnus ova osoianee eä yksiäisiä hallisevia viraus- ja kulkeuumisreiejä voi muodosua pikilläkin eäisyyksillä. Tämän vuoksi onkin ärkeää eä miakaavan muuoksissa käyeävä meneelmä oava huomioon yksiäisen rakeneiden muodosama ison miakaavan yheyde. Loppusijoiusilan oiminakykyanalyysien kannala ämän heken ilanne ukee RETROCK projekin 2005 suosiusa jonka mukaan kulkeuumisominaisuuksien arvioinnin ulisi perusua useilla eri lähesymisavoilla saauihin uloksiin. Avainsana: Mariisidiffuusio pohjaveden viraus kulkeuuminen liuennee ainee oiminakykyanalyysi

1 TABLE OF CONTENTS PREFACE ABSTRACT TIIVISTELMÄ 1 INTRODUCTION... 2 2 FLOW IN FRACTURED ROCK... 3 2.1 Channelling... 3 2.2 Transpor... 4 3 RETENTION BY MATRIX DIFFUSION... 5 4 SPATIAL DEPENDENCY OF THE RETENTION PROPERTY... 10 5 PERSISTENCE OF THE FLOW ALONG THE PARTICLE PATHWAYS... 12 6 SPATIAL UP-SCALING USING CONTINUUM OR FRACTURE NETWORK MODELS... 15 6.1 Coninuum models... 15 6.2 Fracure nework models... 16 7 SPATIAL UP-SCALING USING STATISTICAL MEANS... 17 8 SPATIAL UP-SCALING USING STOCHASTIC CONTINUUM... 19 9 IMPLICATIONS FOR PA... 21 10 SUMMARY... 23 REFERENCES... 25

2 1 INTRODUCTION The presen repor is prepared as a par of he FUNMIG Projec Fundamenal processes of radionuclide migraion. FUNMIG is an inegraed projec wihin he 6h EU Framework Programme for Research and Technological Developmen. The main objecives of his projec are he fundamenal undersanding of radionuclide migraion processes in he geosphere he applicaion o performance assessmen and communicaion of he resuls. This work is a par of he work package 4.6 ha has one of he main objecives o sudy he up-scaling of differen individual processes o faciliae heir inegraion in PA oriened models. The wo main processes ha are normally considered responsible for he reenion of radionuclides in crysalline rocks mari diffusion and sorpion will be invesigaed for he up-scaling procedure developmen. The main objecive of he presen repor is o eamine spaial up-scaling mehodologies for he reenion caused by he mari diffusion. Reenion in he geosphere may have a significan role for he solue ranspor hrough he bedrock. Appropriae mehodology and ools are needed in he performance assessmen o deal wih he reenion in differen scales. Ouline of he presen repor is such ha firs he Secions 2 o 4 gives general descripion of he groundwaer flow in fracured rock mari diffusion process and defines he parameer groups ha conrol he reenion. Differen spaial up-scaling mehodologies of he mari diffusion are hen presened in he Secions 5 o 8. Finally implicaions of he differen up-scaling mehodologies for he performance assessmen are discussed in he Secion 9.

3 2 FLOW IN FRACTURED ROCK Among he oher opions sudies for deposiion of he nuclear wases concenrae o deep underground reposiory in crysalline fracured rock. Fracured rocks are composed of porous bu impermeable rock mari and waer conducing fracures. Fracures or he nework of fracures are he main conduis for he groundwaer flow. Fracure nework is also he main pahway for he ranspor of possible conaminans in groundwaer. The main characerisic of he fracured rock is he grea heerogeneiy in differen scales. The heerogeneiy is also a disincive feaure of he hydraulic properies of he fracured rock. In pracice he groundwaer flow akes place only hrough he fracures. Rock beween fracures he rock mari is porous and sauraed by he waer bu i does no paricipae o he process of he groundwaer flow. The heerogeneous srucure of he fracured rock leads o preferenial flow pahs ha will govern boh flow and ranspor properies hrough he rock mass. The preferenial flow pahs i.e. channelling of he flow is an imporan process ha needs o be aken ino accoun when he ranspor of solues is modelled. Below are discussed he main observaions of he groundwaer flow and channelling of he flow on he basis of he insiu eperimens and modelling. 2.1 Channelling Channelling of he flow and solue ranspor in fracured rock has been shown by models Bear e al. 1993; Nordqvis 1995 and in several in-siu eperimens Long and Billau 1987; Rasmuson and Nerenieks 1986; Nordqvis e al. 1996. A he Fanay-Augéres sie in France i was found ha only abou 0.1% of he fracures conribued o flow on a large scale Long and Billau 1987. I has also been found ha 90% of fluid flow can occur hrough 5-20% of he fracure plane Rasmuson and Nerenieks 1986 and many field eperimens indicae ha only a limied par of he volume wihin he fracures even less han 10% is open o fluid flow and solue ranspor Nordqvis e al. 1996. This affecs significanly flow and ranspor properies of he poenial ranspor pahs. Underlying reasons for he channeling follows from he basic srucure of he fracured rock. Sizes of fracures vary in scales ranging from microns o hundreds of kilomeres. Throughou he differen scales fracures have a significan effec on flow and ranspor processes Berkowiz 2002. The modelling and invesigaion of he fracured rocks is complicaed by he inheren properies of he fracuring e.g. here are indicaions ha properies of he fracured rock canno be homogenized in any scale. This follows from he observaions ha he fracured rock follows a scaling behaviour. For eample analysis of he field daa suggess ha fracure diameer disribuions follow a power law disribuion wih eponens varying from 1.3 o 2.1 Berkowiz and Adler 1998. Sahimi 1995 makes a conclusion ha fracure neworks are fracal objecs over he scale of observaion because i has been noiced ha macroscopically fracured rocks are barely conneced i.e. hey are close o he percolaion limi. Scaling behaviour of he fracuring is no he only feaure in he fracured rock ha causes channelling. A characerisic feaure of he fracured rock is he high degree of heerogeneiy of he flow and ranspor properies in differen scales. The flow and ranspor behaviour is governed by he fracure conduciviies. A broad disribuion of

4 he fracure conduciviies causes channelling Cacas e al. 1990a 1990b. Berkowiz 2002 even noes ha a geomerically well-conneced nework can ehibi sparse preferenial flow pahs and he nework appears o be near he percolaion hreshold if he disribuion of fracure conduciviies is sufficienly broad. I should also be remembered ha in addiion o he variabiliy beween he fracures here is also heerogeneiy wihin fracures. As i has been earlier noed i may be ha even less han 10% of he fracure is open o fluid flow. 2.2 Transpor Many in-siu racer eperimens in he fracured rock have been carried ou during recen years. Observaions of he eperimens reflec he flow characerisics of he fracured rock. Flow canno be described accuraely wihou considering disinc flow pahs. Descripion of he solue ranspor and reenion properies is even more dependen on he descripion of he preferenial flow pahs and dominaing flow channels. Tracer eperimens usually show fas iniial arrival imes more han one peak in he breakhrough curves and/or long ails and srong dependence on scale Becker and Shapiro 2000; Nordqvis e al. 1996; Tsang e al. 1991. Modelling has shown ha he preferenial flow and ranspor pahs run over long disances and reduce miing especially in case of high ransmissiviy variance Nordqvis e al. 1996. All hese observaions underline he cenral role of heerogeneiy of he fracured rock as a governing feaure of he flow and ranspor properies. The channellised naure of he solue ranspor in he fracured rock has also been sudied in he modelling work. In he case of no miing beween channels hey can be reaed as a bundle of one dimensional individual conduis Bear e al. 1993. The assumpion of he channelling and weak miing beween he channels i.e. represening he ranspor channels as a bundle of individual channels has been commonly applied in he performance assessmen modelling. In spie of he inensive work on in-siu racer eperimens here are unesed properies ha are imporan for he performance assessmen. Many ess ypically involve injecion and pumping raes which are considerably higher han flow raes occurring under naural gradien condiions and he spaial scales of he ess are also much smaller han is required in he performance assessmen. In pracice his canno be avoided because collecion of he racer in he in-siu eperimen is no possible unless he eperimenal flow field does no overcome he naural background flow field. For he same reason racer ess need o be carried ou in he well conducing hydraulic srucures and over shor disances alhough in he performance assessmen reenion of he radionuclides akes place mainly in he low ransmissive fracures. However erapolaion of he ranspor and reenion properies from he field measuremens o performance assessmen condiions need o based on proper process undersanding.

5 3 RETENTION BY MATRIX DIFFUSION The advecion-sorpion-mari diffusion equaion for one-dimensional sreamube can be derived in a following way. Firs we consider a case of a single immobile zone along he flow pah. The flow pah is bordered by sreamlines and i is characerised by a fied flow rae denoed by Q. Figure 1 shows a schemaic illusraion of he mobile and immobile pore spaces along he flow pah. We consider solue mass flu over a small conrol volume a he arbirary locaion of he flow pah as shown in he Figure 2. The whole flow pah can be envisaged o be composed of a series of successive conrol volumes. The flow pah is seleced so ha he molecular diffusion keeps solue concenraion well mied across he flow pah over he ranspor disance ha is considered. In pracice his means ha he widh i.e. flow rae Q of he flow pah is seleced so ha all solue paricles raveling hrough he flow pah lengh L will eperience he same average flow properies.

6 Figure 1. Schemaic illusraion of he sreamube mobile porespace and rock mari immobile porespace ne o he sreamube.

7 Figure 2. Sreamube and a conrol volume used o assess he mass balance equaion for he solue ranspor hrough he sreamube of Figure 1. Immobile zone is no shown in his figure o keep he picure more readable. Noe ha only one immobile zones is assumed a he op of he sreamube. The mass balance equaion wrien for he conrol volume in Figure 2 is m r r Ra = Q[ C f C f + d ] J ; z = 0 ea W d 1 where m is he solue mass in he conrol volume C f is he solue concenraion in he conrol volume and he flow pah sreamube is characerised by he widh W aperure 2b and flow rae Q and e r A is he ouer normal of he conrol volume a he inerface of he mobile and immobile pore spaces i.e. a he diffusion inerface. The facor R a incorporaes he ineracions ha can be described by insananeous equilibrium beween he mobile and immobile phases of he solue paricles equilibrium sorpion. All properies may depend on he locaion along he flow pah ecep Q ha is consan and defines he sreamube i.e. he heerogeneiy of he flow field along he flow pah is aken ino accoun. Diffusional mass ransfer beween immobile and mobile zones is described by he mass flu J r of Equaion 2. r J z D z C z r z m = e ez 2 where C m is he solue concenraion in he immobile pore space and D e is he effecive diffusion coefficien for he diffusion beween he immobile and mobile zones. In he immobile zone he solue mass ransfer is governed by he diffusion as indicaed by he Equaion 3 R 2 Cm z Cm z p = D p z 2 z 3

8 where D p is he pore diffusiviy and R p is he reardaion coefficien in he immobile zone. We require coninuiy of he solue concenraion over he inerface of he mobile and immobile zones. This means ha 0 C z C f m = =. 4 Noe ha Equaion 4 is simplified by saing C m =C f a z=0 insead of z=b. These are equivalen formulaions because i is assumed ha well mied condiions prevail in he fracure i.e. C f does no depend on z. In his case z measures he disance from he fracure wall no from he cenerline of he fracure. I is convenien o describe he condiion 4 by represening he solue concenraion in he immobile pore space according o he Equaions 5 and 6 C z g z C f m = 5 1 0 = = z g. 6 Diffusional mass ransfer o he immobile zone can now be represened by using he normalized concenraion gz e e z z z g z D z j r r =. 7 The diffusion mass flu of he Equaion 1 is wrien as z j C z J f r r =. 8 The mass balance Equaion 1 can hen be wrien using Equaions from 5 o 8 as 0 2 2 = = z j b C C b W Q C R f f f a. 9 In order o eamine some of he main properies of he ranspor equaion i is no necessary o specify he srucure and boundary condiions of he immobile zone i.e. i is no necessary o solve Equaion 3. The reason for his is ha some of he main feaures of he solue ranspor Equaion 9 can be oulined wihou specifying he eac funcional form of he normalized diffusion flu j. However we do need o ake some seps furher o solve he Equaion 9. Laplace ransform of he Equaion 9 in respec of he ime gives ~ ~ ~ ~ 0 2 2 = = z s j b s C s C b W Q s C s R f f f a 10 where he ilde denoes Laplace ransformaion and s is he variable of he Laplace ransformed domain. Equaion 10 can be solved for he iniial condiion of empy no racer rock mari and a sudden release of mass M 0 a he inle of he flow pah i.e. 0 0 0 0 ~ 0 0 0 = = = = = = = = C z C Q M s C Q M C f m f f δ 11

9 Equaion 12 gives soluion o Equaion 10 using iniial condiions of Equaion 11. The solue mass flu a he oule of he flow pah =L is ~ L L m L s Q W 2b W ~ & ~ j s z = 0 = C f s = Ep s Ra d d M 0 M 0 Q Q 0 0 L V W ~. 12 j s z = 0 = Ep s Ra d Q Q 0 I is noed from he Equaion 12 ha besides he advecive delay V/Q he soluion depends on he eniy ha is denoed here by he parameer u L ~ W j s z = 0 u = d. 13 Q 0 Equaions 12 and 13 are derived for a flow pah ha is bounded by he sreamlines i.e. a fied flow rae passes hrough all poins along he sreamube. Molecular diffusion keeps he concenraion of he solue paricles well mied inside he sreamube i.e. all solue paricles will eperience he same flow field when passing hrough he flow pah. However in he numerical simulaions i may no be possible o follow he sreamlines ha bound he flow pah. Insead numerical models usually provide samples of he flow properies averaged over he model elemen size. In his case Equaion 13 should be approimaed by equaion L ~ W' j s z = 0 u = d 14 Q' 0 where W /Q is he measure of local flow rae along he pahway. Accuracy of his approimaion depends on he sizes model elemens i.e. he scale of averaging. Preferably he numerical approimaion of he flow soluion should no average he flow field over much larger region han wha he solue paricles will inegrae along he flow pah.

10 4 SPATIAL DEPENDENCY OF THE RETENTION PROPERTY Spaial up-scaling is firs sudied for he case of homogeneous immobile zone properies along he flow pah homogeneous rock mari. In pracice his means ha Equaion 14 simplifies o u = ~ j s L W ' d Q' 0 15 ~ where j s describes he coupling o he immobile zones and he flow dependen par is represened by u F = L W ' d Q' 0. 16 Subscrip F in Equaion 16 denoes he flow dependen par of he parameer u. Equaion 16 saes ha he flow dependen par of he reenion propery u F is an inegral along he flow pah weighed by he inverse of he local 2D Darcy velociy in he fracure plane. Sricly Equaion 16 is wrien for he mass flu of he solue paricles bu in pracice ha is srongly correlaed o flu of he groundwaer flow in he ranspor channel. Equaion 16 indicaes ha he reenion propery depends on he lengh of he flow pah bu i is weighed by he local flow condiions. This means ha flow characerisics in differen spaial scales govern he spaial scaling of he reenion properies. There are many differen flow domains in he fracured rock ha are conneced o he size scale of he hydraulic feaures. Especially his is rue for he sparsely fracured rock and in case of wide fracure ransmissiviy disribuion. In pracice his means ha here is no averaging scale for flow in he fracured rock and ha individual conduis may have significan role for he overall flow and ranspor characerisics. A leas hree differen flow environmens can be idenified: channeling causing variable flow in he individual fracure planes ransmissiviy differences beween he fracures leading o preferenial flow pahs hrough he fracure nework and eensive fracure zones providing highly ransmissive connecion over long disances. In connecion wih he Equaion 16 his has wofold influences. Firs local flow rae is higher along he in-plane channels and along he highly ransmissive fracures and fracure zones. Secondly eensive ransmissive fracures and fracure zones provide shorer connecion i.e. shorer oal pah lengh. Boh of hese characerisics end o reduce he overall u F i.e. provide less reenion. Equaion 16 does no direcly indicae he resoluion required o deermine he local disribuion of he flow in he fracure planes. Equaion 16 has been derived by assuming well-mied condiions across he ranspor channel. A rigorous requiremen o he resoluion of he numerical model is ha well mied condiions should be assumed in each elemen of he model. In pracice his leads o unfeasibly fine discreisaion of he model. More accurae descripion of he model resoluion akes ino accoun he whole flow pahs. However his canno be assessed before he flow field is firs numerically solved. Soluions o his problem could be for eample calculaion of he

11 flow and ranspor properies using differen levels of discreisaion and comparison of he resuls or possible adapive refinemen of he mesh along he paricle flow pahs.

12 5 PERSISTENCE OF THE FLOW ALONG THE PARTICLE PATHWAYS I has been observed ha he flow properies in fracured rock end o persis along he flow pahs. As an eample of his persisence some modelling resuls are presened below. The resuls are based on he simulaions of he Task 6 of he Äspö Task Force on groundwaer flow and solue ranspor Poeri 2006. The objecive of he Task 6 has been o bridge beween he sie characerisaion and he performance assessmen models by evaluaing consraining power of he sie characerisaion eperimens o he relevan ranspor properies in he performance assessmen scale. Subasks Task 6D and Task 6E deal wih solue ranspor hrough a fracure nework under he sie characerisaion and performance assessmen condiions. The persisence of he flow properies along he flow pah is here eamined by sudying he behaviour of he u F. In Task 6 he parameer u F is defined as u F = i 2Li q i 17 where L i is he lengh of he flow pah segmen i and q i is he corresponding 2D Darcy velociy i.e. he local flow rae in he fracure of he flow pah segmen i. Figure 3 and Figure 4 show he accumulaion of he u F along he simulaed flow pahs for he sie characerisaion and performance assessmen model respecively. The main observaion from he resuls is ha he slope of he pah lengh vs. cumulaive u F curve is seep a he beginning and genly sloping a he end. This general rend of he pah lengh vs. cumulaive u F curves indicaes ha he flow rae hrough flow pahs increase as he flow pah proceeds. In pracice his is a resul of a process where flow pahs accumulae o larger well conducing feaures and here is only a small probabiliy for he ransiion back o he smaller feaures once he flow pah has enered a larger hydraulic feaure. This behaviour can also be represened by he persisen of he flow properies beween he successive segmens of he flow pah.

13 Figure 3. Accumulaion of he hydrological conrol of he reenion u F as a funcion of he pah lengh in case of sie characerisaion boundary condiions racer es beween wo boreholes. Solid lines indicae u F and he coloured circles a he background indicae he visied srucures from Poeri 2006.

14 Figure 4. Accumulaion of he u F along he flow pahs as a funcion of he pah lengh in case of he performance assessmen condiions. Differen colours indicae flow pahs o he successive conrol planes along he main flow direcion. The conrol planes are locaed a abou 10 m 50 m and 130 m from he source locaion.

15 6 SPATIAL UP-SCALING USING CONTINUUM OR FRACTURE NETWORK MODELS The wo main modelling approaches ha have been applied o assess he reenion properies in he performance assessmen are he coninuum models and fracure nework models cf. e.g. RETROCK 2005. Modelling of he reenion by mari diffusion is reaed quie differenly in hese wo models. 6.1 Coninuum models Coninuum models use average properies of he rock including fracures and rock blocks o describe he groundwaer flow. In pracice his means ha he varying flow field over he large numbers of inerconneced fracures is block wise replaced by an average conduciviy. Coninuum can include a sochasic componen i.e. in he sochasic coninuum model he block conduciviies are drawn from he saisical disribuions. Applicaion of he sochasic coninuum models is discussed more in deail in he Secion 8. Fracured bedrock is very heerogeneous and usually he large-scale zones need o be aken ino accoun by applying differen properies for hem. The main difference beween he fracure nework model and coninuum model is conneced o he reamen of he sochasic feaures. The benefis of he coninuum approach are ha he models are usually compuaionally cheaper so ha hey can cover he whole volume of ineres and assessmen of he ranspor pahs can be implemened easily. Transpor environmen of he paricle pahways is usually concepualised as imaginary one-dimensional sreamubes. The averaging of he ranspor properies Equaion 16 akes place in he sreamube and he averaging combines boh he flowing fracures and he immobile pore space wihin he rock mari. Sreamubes are usually reaed as a bundle of independen conduis. This means ha here is no miing beween he sreamubes i.e. he boundaries of he sreamubes are impermeable. Sreamubes are usually deermined by following paricle pahways hrough he model. Properies of he sreamubes are based on he informaion colleced from he paricle pahways e.g. he lengh and averaged 3D Darcy velociies along he pahway. The main problem wih coninuum models is conneced o he averaging of he ranspor properies over large volumes. I can be perceived ha in he coninuum model he upscaling of he reenion properies akes place already a he averaging scale of he model i.e. in he scale of he elemenary blocks of he model. The reenion propery u F Equaion 16 is evaluaed by assuming some simple srucure for he fracuring in he coninuum blocks. In pracice he srucure of he fracuring in he coninuum blocks is represened by a se of fracures where he flow is divided usually evenly beween he fracures. In his case he 2-D Darcy velociy of he Equaion 16 is replaced by he 3-D Darcy velociy of he coninuum block and he average flow weed surface per volume of rock. Equaion 16 indicaes ha he maimum reenion is obained by dividing he oal flow rae evenly beween he fracures. Therefore he underlying heerogeneiy of

16 he fracured rock implies ha he averaged esimae of he reenion propery may be overly opimisic. 6.2 Fracure nework models Fracure nework modelling ries o mimic he srucure of he fracured rock by eplicily incorporaing fracures o he model. The flow model is consruced so ha he groundwaer flow akes place hrough he inerconneced fracures. In many cases blocks of he rock mari beween he fracures are even no included o he flow model. Fracure nework models offer a good descripion of he groundwaer flow and ranspor properies by incorporaing huge amoun of deails o he model. In he ranspor applicaions he individual flow pahs are ried o represen as eplicily as possible based on daa on fracure frequency size orienaion and ransmissiviy. In order o have a meaningful conneciviy for he racer ranspor or performance assessmen applicaion i may urn ou ha raher small fracures need o be included o he model. Fracure nework models are sochasic models because i is no possible o deerminisically characerise all fracures in he level of deails ha is required in he ranspor calculaions. However he sochasic naure of he fracure nework models does no preclude addiion of he deerminisic feaures o he model and usually known large-scale hydraulic feaures are represened deerminisically in he model. Fracure nework model is a good ool o deermine e.g. he required flow field properies Equaion 16. Fracure nework model ries o be closely idenical wih he epeced physical environmen faced by he solue ranspor problem in fracured rock. Averaging of he flow and ranspor properies akes place in a differen scale han for he coninuum models. In he fracure nework model he hydraulic properies are usually averaged in he scale of he individual fracures no over blocks of fracures as i he case in he coninuum models. The main problems wih he fracure nework models are conneced o he level of deails ha are aken ino accoun in he model. The number of fracures in he model ha is usually direcly proporional o he volume of he modelling domain deermines he compuaional resources ha are required o solve he flow field. Presenly he number of fracures resrics he size of he model o some hundreds of meres.

17 7 SPATIAL UP-SCALING USING STATISTICAL MEANS Painer and Cvekovic 2005 have developed an up-scaling mehod ha is based on he applicaion of he saisical means. The moivaion for heir work has been ha he convenional advecion-dispersion descripion is no able o predic he behaviour observed in he field eperimens. The mehod also provides a way o predic racer ranspor over longer disances ha can be prediced by he advecion-dispersion concepualisaion. Advecion-dispersion represenaion suis beer for highly and uniformly fracured rock. Programs of he geological disposal of he nuclear wase are more ineresed in sparsely fracured rock and i is noe clear ha advecion-dispersion models are suiable for he modelling of he sparsely fracured rock especially for he esimaion of he reenion properies as i has been noed in he Secion 6.1. The approach of Painer and Cvekovic 2005 employs sie-specific fracure nework modelling as a ool o assess he saisical properies of he ranspor pahways for a sub domain of he sie scale domain. The simulaed saisics is hen applied for he assessmen of he ranspor over longer pahways. An imporan implici assumpion is ha he fracure nework model used in he simulaion is also represenaive for he fracure nework in he larger scale or in oher words ha he assumpion of he saisical homogeneiy over a larger volume of he rock is valid for he modelled sie. Painer and Cvekovic 2005 approach is based on a Lagrangian represenaion of he ranspor in fracured rock. The impulse response funcion for he advecion-reenion process can be represened as s Bgs ˆ ˆ γ s; τ B = ep τ 18 l where γˆ l is he Laplace ransform of he response funcion τ is he waer residence ime along he rajecory ĝ is he Laplace ransform of he memory funcion for he reenion process and B is a cumulaive reaciviy parameer ha inegraes reenion properies along he rajecory. Subscrip l indicaes he lengh of he flow pah. The funcion γ l can also be inerpreed as he condiional ime-dependen discharge a a monioring boundary locaed a a disance of l from a source of he Dirac δ-funcion inpu for given values of τ and B. I can be noed ha Equaion 18 is idenical wih Equaion 12 wih an ecepion ha he memory funcion ĝ is wrien in more general form in he Equaion 18. In he Equaion 12 i is consruced for he diffusive mass ransfer beween he immobile and mobile zones. The parameer B in Equaion 18 is proporional o he u Equaion 15 i.e. i is also proporional o he u F. Calculaion of he τ and B saisics is based on he simulaed rajecories of he racer paricles. Each rajecory is discreised o a number of jumps wih each jump corresponding o ransi hrough an individual fracure. Calculaion of he τ and B values for a paricle pahway is carried ou by accumulaing he τ and B values of he discreised jumps for each jump he { l B} riple is recorded. I is observed ha τ N B consiues a sochasic process ha governs he τ and B saisics. { l N l } l 0 > Painer and Cvekovic 2005 claim ha fracure nework simulaions have shown ha here is a correlaion in he ranspor properies beween he successive segmens along he rajecory. A paricle ha is in a high-velociy segmen is more likely o indicae

18 high-velociy in he subsequen segmen due o conservaion of flu a he fracure inersecions. In he framework of he Equaion 12 his can also be reasoned by he conservaion of he mass; larger and more ransmissive hydraulic feaures collec flow from he smaller ones. In pracice his means increasing Darcy velociy along he visied fracures and he corresponding correlaion beween he successive segmens of he flow pah. Painer and Cvekovic 2005 call he sochasic process ha governs he flow properies along he paricle pahways as a Markov-direced random walk MDRW. According o Painer and Cvekovic 2005 he correlaion beween successive segmens along he paricle rajecory is an imporan conrol on he breakhrough curves. I needs o be aken ino accoun in he random walk models of he ranspor hrough a fracure nework. A grea benefi of he approach presened by Painer and Cvekovic 2005 is ha he fracure nework simulaions are performed only o deermine he disribuion for he variables τ and B in a fracure nework ha is large enough o be saisically represenaive for he sie scale fracuring. Muliple reenion processes can be incorporaed o he model based on he disribuion fτb and he memory funcion g no addiional fracure nework simulaions are required. The approach does no eiher require volume averaging and oher assumpions ha are inheren in he coninuum approach.

19 8 SPATIAL UP-SCALING USING STOCHASTIC CONTINUUM Sochasic coninuum models are used o describe he groundwaer flow when a more deailed descripion of he heerogeneous hydraulic properies is required han wha can be represened by he coninuum model. This is he saring poin of he up-scaling mehod applied by Öhman e al 2005. They use he fracure nework model ha is based on he geological and hydraulic daa o obain he flow and ranspor saisics a some suppor scale. The suppor scale is seleced so ha flow bu no necessarily ranspor can be represened by means of a coninuum. Sochasic coninuum flow simulaion is applied in combinaion wih paricle racking o model large-scale ranspor. Deailed fracure scale properies are ransferred o he regional scale using he block scale descripion. Öhman e al. 2005 summarises he approach by he following seps: 1. Hydraulic characerisics of large number of fracure nework realizaions are sudied o deermine coninuum ensors of he hydraulic conduciviy a he block scale. I is hen used as a suppor scale in a regional-scale sochasic coninuum model. 2. Transpor properies are deermined from he same block-scale fracure neworks as were used o deermine he coninuum conduciviy ensors. A large number of paricles are released wihin he fracure nework blocks and disribuions of he paricle ransi imes are measured and colleced as probabiliy disribuions. Disribuions of he paricle ransi imes a he block scale will be hen used direcly in he regional-scale simulaions. Simulaions performed by Öhman e al. 2005 show ha he ransi imes hrough he fracure neworks are clearly correlaed o heir block conduciviy. Low conduciviy seems o indicae long ransi imes; however he spread of he simulaed ransi imes is wide for all block scale conduciviies. 3. Regional-scale flow field is simulaed wih a sochasic coninuum model. This model is discreised a he suppor scale of he sep 1 and he disribuion of upscaled conduciviy deermined in sep 1 is used as inpu. 4. Regional-scale ranspor is modelled by sampling block scale paricle ransi ime from he previously obained block-scale ransi ime disribuions. Each up-scaled coninuum block conduciviy value is linked o is own ransi ime disribuion. This can be carried ou easily since seps 1 and 2 are conduced for he same nework realizaions represening he same block. To obain he proper paricle ransi ime a each sep in he regional-scale model he sampled blockscale ransi imes need o be scaled according o he local ambien hydraulic gradien. Öhman e al. 2005 pay special aenion o channelling when esing he mehodology. This is carried ou by applying differen sraegies o he sampling of he paricle ransi imes for wo neighbouring blocks as a paricle moves from one block elemen o he ne. In many cases he paricle ransi imes in wo neighbouring blocks are assumed o be independen i.e. no correlaed. I is also possible o assume some correlaion beween he successive elemens. The correlaion means ha paricles raveling along a fas pah in one block would also ravel along a fas pah in he ne. Öhman e al.

20 2005 sudies four differen alernaives for he correlaion successive ransi imes persisence i No correlaion indicaing ha he ranspor channel does no coninue in he ne block. ii Perfec correlaion indicaing ha he ranspor channel coninues hrough ou he model. iii Two neighbouring blocks correlaed so ha ransi ime may change by a given amoun. This case represens a dispersion process. iv Perfec correlaion over a given number of blocks persisen correlaion. Öhman e al. 2005 conclude ha he mos promising resuls gave applicaion of he persisen correlaion. The sudy by Öhman e al. 2005 does no direcly deal wih reenion by he mari diffusion i.e. hey do no eamine properies of he u F Equaion 12 or oher equivalen reenion propery. However he mehodology can be direcly applicable o assessmen of he reenion properies.

21 9 IMPLICATIONS FOR PA The RETROCK projec 2005 noes ha recen PAs for fracured rock have used geosphere reenion more as a complemenary safey funcion han as a main one. Bu sill also in hese PAs reenion in he geosphere becomes imporan when he wase canisers are broken and he radionuclides sar o spread. In ha siuaion one of he roles of he geosphere is o reard and aenuae he flu of radionuclides ha are released from a reposiory. Assessmen of he reenion of he radionuclides needs o be carried ou for a large volume of he rock ha conains he whole reposiory and he bedrock beween he reposiory and he poenial discharge locaions on he ground surface. In pracice he arge volume of he rock mass can be several hundreds of meers verically and a few kilomeres horizonally. Esimaion of he reenion properies for a so large volume requires special modelling echniques. The differen upscaling mehods may give suiable ools o deal wih he long release pahs. The main challenge in spaial up-scaling of he mari reenion properies is coupled o he flow characerisics of he fracured rock. Individual fracures along he flow pah conribue o he overall reenion by a proporion ha is depends on he size of he fracure weighed by he inverse of he flow rae Equaion 16. If he flow condiions along he flow pah do no vary considerably hen he up-scaling is quie sraighforward i.e. he reenion propery increases linearly as he spaial scale i.e. he lengh of he flow pah increases. However he fracured rock is highly heerogeneous and comprises of muliple scales ha have disinc flow properies. Differen up-scaling mehods can be compared using needs of he PA as a basis: Coninuum model is based on he up-scaling of he ranspor properies already a he level of he building block of he model. In pracice his leads o represenaion of he fracure nework by an averaged coninuum block. Earlier safey assessmen calculaions have been based on he sreamube concep ha employs similar kind of averaging of he fracure nework over large spaial scales. RETROCK 2005 noes on his approach ha applicaion of i will be difficul o defend on he concepual grounds. Reason for his is ha he underlying scale of averaging gives clearly oo opimisic esimae of he reenion properies because he inernal heerogeneiy of he fracure nework is no aken ino accoun. Applicaion of he fracure nework modelling is sraighforward bu he compuaional requiremens of he fracure nework modelling are very demanding. Fracure nework modelling is he only one of he sudied approaches ha is able o ake ino accoun he muli-scale srucure of he fracured rock. From he ranspor and reenion poin of view i is imporan ha he large-scale srucures ha may provide fas connecion over large disances are incorporaed o he model. Saisical up-scaling mehod is compuaionally very feasible. I is based on he saisical descripion of he fracure nework reenion properies in one scale and applicaion of sochasic processes for erapolaion in larger scales. I is based on he fracure nework saisics ha means i is able o avoid he volume averaging of he equivalen coninuum models. The descripion of he reenion properies is realisic a leas in he scales ha are close he scale of he

22 underlying fracure nework model. Uncerainies of his mehod increase when he difference beween he scales used o derive he saisics and he scale of he up-scaled ranspor pahs is large. The approach relies on assumpion of he saisical homogeneiy of he fracuring over he whole up-scaled lengh scale. However he saisics of he underlying fracure nework ha is used o derive he saisics canno conain accurae informaion on he larger scale srucures ha may evenually provide fas connecion over large disances. Parly his is can be compensaed by defining a correlaion beween successive segmens along he paricle rajecory. Painer and Cvekovic 2005 noe ha he correlaion is very imporan and i need o be aken ino accoun in he random walk models of he ranspor in fracured rock. Applicaion of he sochasic coninuum model is based on he averaging of he properies in he seleced suppor scale. The mehod presened in he presen repor differs from he up-scaling by he coninuum model so ha he ranspor properies are no based on he averaged properies on he suppor scale bu on he library of simulaion resuls derived from he suppor scale fracure nework. Also in his mehod i is possible o correlae he properies for he successive blocks and in fac i was found ha his ype of persisency in he flow pah properies gave bes resuls. This mehod lacks he large-scale srucures ha can be imporan for he reenion properies along he longer upscaled ranspor pahs. However in principal i is possible o incorporae deerminisic srucures ino he sochasic coninuum. The field eperience and modelling has shown ha he preferenial flow and ranspor pahs may run over long disances. I is imporan he up-scaling mehod does no lack of long disance connecions which may be imporan in he field scale ranspor. Difficulies in all sudied up-scaling mehods suppors he recommendaion of he RETROCK projec 2005 o apply muliple approaches o ensure ha he esimaes are robus.

23 10 SUMMARY Fracured rocks are composed of porous bu impermeable rock mari and waer conducing fracures which are he main conduis for he groundwaer flow. In fracured rock he fracure nework is he main pahway for he ranspor of possible conaminans of he groundwaer. The main characerisic of he fracured rock is he grea heerogeneiy in differen scales. The heerogeneous srucure of he fracured rock leads easily o preferenial flow pahs ha will govern boh flow and ranspor properies. This channelling of he flow is an imporan process ha needs o be aken ino accoun when he reenion in he fracured rock is modelled. A leas hree disinc flow environmens can be idenified in he fracured rock: channeling causing variable flow in he individual fracure planes ransmissiviy differences beween he fracures leading o preferenial flow pahs hrough he fracure nework and eensive fracure zones providing highly ransmissive connecion over long disances. Large and ransmissive hydraulic feaures have an imporan role in he flow and ranspor properies of he fracured rock. The flow pahs end o accumulae o larger feaures which also carry a larger flow rae. In he modelled paricle racking simulaions his behaviour is refleced as persisence in he flow properies along he flow pahs. The main challenge of he spaial up-scaling of he reenion properies is conneced o he descripion of he flow characerisics of he fracured rock. Conribuion of he individual fracures o he reenion is proporional o he size of he fracure weighed by he inverse of he local flow rae i.e. he flow rae affecs direcly he imporance of he fracures o he overall reenion. In he radiional coninuum model up-scaling akes place when he properies of he fracure nework are averaged o he propery of he coninuum blocks. However he sraighforward averaging of he flow may lead o overly opimisic reenion properies for he flow pahs. Therefore i has been found ha i is difficul o defend his approach on he concepual grounds. I is also possible o up-scale based on he sochasic coninuum model and applying he corresponding fracure nework descripion of he ranspor properies in he suppor scale of he sochasic coninuum model. This approach may lack some of he sochasic large-scale srucures ha can be imporan for he reenion properies along he long ranspor pahs. However i should be possible o incorporae also he deerminisic srucures o he sochasic coninuum. An up-scaling mehod has been developed ha is based on he saisical descripion of he fracure nework properies in an appropriae saisically represenaive scale. As his approach is based on he fracure nework saisics i is able o avoid he volume averaging of he equivalen coninuum models. However his approach relies on he assumpion of saisical homogeneiy of he fracuring i.e. ha he small-scale fracure nework descripion is valid over much larger volumes. This means ha i is difficul o incorporae eensive fracure zones ino his approach.

24 Fracure nework modelling is he only one of he sudied approaches ha is able o ake ino accoun he muli-scale srucure of he fracured rock and direcly deermine he reenion properies of he flow pahs. Fracure nework modelling provides a sraighforward way o direcly simulae he up-scaling of he ranspor properies along he preferenial flow pahs hrough he fracure nework and also o incorporae he channelling of he flow in he individual fracure planes o he model. However i is compuaionally very demanding o direcly simulae all hese scales. A presen i is no possible o direcly apply fracure nework modelling o a PA-scale problem and ake ino accoun all differen levels of heerogeneiy channels in fracure planes preferenial flow pahs and eensive fracure zones. The final conclusion is ha a he momen here is no general up-scaling mehod ha can be applied for he performance assessmen scale problems. The field eperience and modelling has shown ha he preferenial flow and ranspor pahs may run over long disances. Therefore i is imporan ha he up-scaling mehod does no lack of long disance connecions which may be imporan in he field scale ranspor. The presen siuaion suppors recommendaion of he RETROCK projec 2005 o apply muliple approaches o ensure ha he esimaes are robus.

25 REFERENCES Bear J. Tsang C.-F. and Marsily G. de Ediors 1993. Flow and conaminan ranspor in fracures rock. Academic Press. Becker M.W. and Shapiro A.M. 2000. Tracer ranspor in fracured crysalline rock: evidence of nondiffusive breakhrough ailing. Waer Resources Research 2000 367:1677 1686. Berkowiz B Adler PM 1998. Sereological analysis of fracure nework srucure in geological formaions. J Geophys Res 1998 103 B7:15339 60. Berkowiz B. 2002. Characerizing flow and ranspor in fracured geological media: A review. Advances in Waer Resources 2002 25:861 884. Cacas M.C. de Marsily G. Barbreau A. Calmels P. Gaillard B. and Margria R. 1990b. Modeling fracure flow wih a sochasic discree fracure nework: calibraion and validaion 2. The ranspor model. Waer Resources Research 1990 263:491 500. Cacas M.C. Ledou E. de Marsily G. Tillie B. Barbreau A. Durand E. Feuga B. and Peaudecerf P. 1990a. Modeling fracure flow wih a sochasic discree fracure nework: calibraion and validaion 1. The flow model. Waer Resources Research 1990 263:479 89. Long J.C.S. and Billau D.M. 1987. From field daa o fracure nework modeling: an eample incorporaing spaial srucure. Waer Resources Research 1987 23:1201 16. Nordqvis A.W. Tsang Y.W. Tsang C.F. Dversorp B. and Andersson J. 1996. Effecs of high variance of fracure ransmissiviy on ranspor and sorpion a differen scales in a discree model for fracured rocks. J Conam Hydrol 1996 22:39 66. Nordqvis A. 1995. Discree modeling of solue ranspor in rock wih variable aperure fracures. Docoral Thesis Division of Hydraulic Engineering Deparmen of Civil and Environmenal Engineering. Royal Insiue of Technology Sockholm Sweden. Painer S. and Cvekovic V. 2005. Upscaling discree fracure nework simulaions: An alernaive o coninuum ranspor models. Waer Resources Research 2005 412 W02002 doi:10.1029/2004wr003682. Rasmuson A. and Nerenieks I. 1986. Radionuclide ranspor in fas channels in crysalline rock. Waer Resources Research 1986 22:1247 56. RETROCK 2005. Treamen of radionuclide ranspor in geosphere wihin safey assessmens RETROCK. Final Repor. European Commission. Conrac No FIKW- CT-2001-20201. June 2005. Repor EUR 21230 EN. Sahimi M. 1995. Flow and ranspor in porous media and fracured rock: from classical mehods o modern approaches. Weinheim Germany; VCH 1995.

26 Tsang C.F. Tsang Y.W. and Hale F.V. 1991. Tracer ranspor in fracures: Analysis of field daa based on a variable-aperure channel model. Waer Resources Research 1991 2712:3095-3106. Öhman J. Niemi A. and Tsang C.-F. 2005 A regional-scale paricle-racking mehod for nonsaionary fracured media Waer Resources Research 2005 413 W03016 doi:10.1029/2004wr003498.