Mass Transport Analysis In The Near Field Of Geologic Repository

Transcription

1 UCB-NE-48 Dissraio for h dgr of Docor of Philosoph Mass Traspor Aalsis I Th Nar Fild Of Gologic posior Doo-Hu Lim Dparm of Nuclar Egirig Uivrsi of Califoria Brkl Dcmbr

2 This work was prformd as par of h projc suppord b h Japa Nuclar Ccl Dvlopm Isiu. Comms ad oificaio of a rrors i his rpor would b apprciad. Joohog Ah Pricipal Ivsigaor of h projc Dparm of Nuclar Egirig Uivrsi of Califoria Brkl CA 947 USA ah@uc.brkl.du ii

3 Tabl of Cos. INTODUCTION AND SUMMAY.... INTODUCTION.... STATE OF THE AT.... OBJECTIVES....4 SCOPE AND SUMMAY.... LITEATUE EVIEWS LITEATUE EVIEW FO MASS TANSPOT IN FACTUED OCK War Flow Modlig i Fracurd ock Solu Traspor Modlig i a Fracurd ock LITEATUE EVIEW FO FACTUE GEOMETY PAAMETES Fracur Siz..... Aprur..... Oriaio Agl Locaio Liar Frquc Spacig ad Numbr Dsi SUMMAY WATE FLOW ANALYSIS IN THE NEA FIELD OF GEOLOGIC EPOSITOY INTODUCTION MODEL SPACE FACTUE NETWOK GENEATION Graio of Fracur Sgms Flow-Barig Clusr FBC of Fracurs..... Idificaio of FBC....4 TANSFOMATION OF FBC INTO EQUIVALENT CONTINUUM Trasformaio of FBC io a Hrogous Coiuum Equival porosi Equival hdraulic coducivi Trasformaio of h FBC io a Homogous Coiuum GOVENING EQUATION AND SIDE CONDITIONS Soluio for Govrig Equaio b Fii Elm Mhod Eampl for suls of War Flow Aalsis Fii-Elm Msh Siz NUMEICAL ESULTS Fracur Nwork Graio ad Occurrc of FBC Effcs of Hrogi of NF o Flow a Drmiaio of h umbr of ralizaios Effc of Lgh disribuio Effcs of Aprur Corrlaio Effc of Modlig Approach for NF o Flow a iii

4 .7 CONCLUSIONS LIMITATIONS MATHEMATICAL MODEL FO WATE FLOW ANALYSIS IN THE NEA FIELD INTODUCTION MATHEMATICAL POBLEM AND SOLUTION OF POTENTIAL EQUATION STEAM FUNCTIONS COMPAISON BETWEEN THE ANALYTICAL AND NUMEICAL SOLUTIONS VEIFICATION OF ANALYTICAL SOLUTION FO POTENTIAL Govrig Equaios Boudar Codiios CONCLUSIONS TANSPOT ANALYSIS IN THE NEA FIELD OF GEOLOGIC EPOSITOY INTODUCTION EVIEW OF TOMPSON S TEATMENT Tompso s Modl adom Movm icludig Advcio Drivaio of 5.8 ad 5. from Sp Equaio for Cosa Vloci ad Diffusivi i a Ifii O-Dimsioal Mdium Drivaio of Sp Equaio from 5.8 ad 5. for Cosa Vloci ad Diffusivi i a Ifii O-Dimsioal Mdium Numrical Simulaio b Sp Equaio for Cosa Vloci ad Diffusivi i a Ifii O-Dimsioal Mdium Discussio GOVENING EQUATION AND SIDE CONDITIONS Eampl for rsuls of paricl raspor aalsis NUMEICAL ESULTS Effcs of Hrogi of NF ad Modlig Approach for NF o sidc Tim Effcs of Lgh Disribuio Effcs of Aprur Corrlaio Effcs of Modlig Approach for NF Effcs of Hrogi of NF ad Modlig Approach for NF o Mass Absorpio a CONCLUSIONS LIMITATIONS & FUTUE STUDIES EVALUATION OF COEFFICIENT C IN KOZENY-CAMAN EQUATION INTODUCTION TOTUOSITY EVALUATION OF TOTUOSITY Evaluaio of Toruosi b Pah Lgh Avrag Lgh of Pahs Avrag Lgh of Flow Pahs Evaluaio of Toruosi b Oriaio Agl NUMEICAL ESULTS AND OBSEVATIONS iv

5 6.4. Effc of h Siz of Fracur Lgh o Toruosi CONCLUSIONS & DISCUSSIONS FLOW EVALUATION WITH TOTUOSITY COECTION INTODUCTION FLOW ATE THOUGH A FACTUE NETWOK Modl Spac ad Boudar Codiios Discr Fracur Nwork DFN Approach Homogous Coiuum Approach NUMEICAL ESULTS CONCLUSIONS & DISCUSSIONS CONCLUSIONS... 4 EFEENCES... 6 APPENDIX A: ANDOM SAMPLING FO LOGNOMAL DISTIBUTION... APPENDIX B: PAAMETES FO OCCUENCE OF FBC... B. OIENTATION... B. LENGTH... B. APETUE... 4 APPENDIX C: EXAMPLE FO ASSEMBLING GLOBAL MATIX... 6 APPENDIX D: FEATUES OF FFDF CODE... 8 APPENDIX E: ANALOGY TO TANSPOT EQUATION... 9 APPENDIX F: IMAGINAY ABSOBING BOUNDAY FO TANSPOT... APPENDIX G: EXAMPLE FO IDENTIFICATION OF PATHS... APPENDIX H: POTENTIAL DISTIBUTION IN THE FACTUE NETWOK... 5 APPENDIX I: ANALYTICAL FACTUE FLOW MODEL... 7 v

6 Lis of Figurs Figur. a Schmaic viw of h Swdish KBS- rposior dsig showig h small hol i h caisr ad h locaio of h various scapr rous. Pah Q rlas dircl o h flowig war i a fracur irscig h dposiio hol. Pah Q rlas o h disurbd zo aroud h ul. Pah Q rlas o h ul backfill ad furhr o a fracur zo irscig h ul. Pah Q 4 rlas hrough h rock o a arb fracur or fracur zo [omro al. 995 a b; Gllig 997]. b Th comparmalizaio of h rposior; ach bo solid li is a comparm ad dashd lis ar fir subdivisios of h bos. Th arrows show h various pahwas followd b h spcis from h caisr o h flowig war. Drawig is o o scal. [omro al. 995 b].... Figur. a Cocpual modl of Japas HLW rposior. b Cocpual modl for uclid raspor i h EBS of h rposior. I b hpohical war-filld rgio is cosidrd o spara h virifid was ad h buffr [JNC Vol. p.v-6].... Figur. Schmaic diagram of h rasformaio of h flow-barig clusr FBC of fracurs. Local homogizaio ad uiformizaio ar dfid i Chapr. Shadd rgio rprss h flowig-barig rgio Figur. Probabili dsi fucios of logormal disribuios for lgh of fracur sgms for h crsalli rock a dph blow ha 5m basd o liraur rviw. Thr diffr disribuios L =. L =.m L =. L =.m ad L =. L =.m ar dscribd b h hick lis ad will b usd i his sud. A ara udr ach curv is ui.... Figur. Probabili dsi fucios for aprur basd o liraur rviw. Logormal disribuio is assumd. A disribuio wih L =.8 ad L = i.. ma ad sadard dviaio of m is dscribd b h hick li ad will b usd i his sud... Figur. Probabili dsi fucio for oriaio agl... 4 Figur.4 Probabili dsi fucio for locaio of h primar fracurs... 4 Figur.5 Probabili dsi fucio for locaio of h scodar fracurs... 5 Figur. Modl spac... 8 Figur. Drmiaio of coordias for o d of a fracur sgm... 9 Figur. Drmiaio of lgh l ad aprur b... 9 Figur.4 Drmiaio of oriaio... 9 Figur.5 Drmiaio of h locaio r lgh l ad aprur b of a scod fracur sgm Figur.6 Possibl wo cass for wo fracurs... Figur.7 Fracur work graio... Figur.8 Thr ralizaios of fracur work b h sam saisics.... Figur.9 Fracur clusrs. Thick lis rprs h FBC of h DFN whras h hi lis rprs h DEC i h DFN... Figur. Irscio bw wo fracur sgms. I a irscio is coud oc.... Figur. A ampl for h idificaio of h FBC i a fracur work.... Figur. FBC for h fracur work i Figur.... Figur. FBC for h fracur work i Figur Figur.4 Calculaio of a quival porosi for a riagular lm Figur.5 Cocps of h spcific surfac ara... 6 Figur.6 War-flowig rgio i.. FBC ad buffr i h modl... 7 Figur.7 Triagular lm i h domai Figur.8 Coordias of hr ods i a riagular lm b global coordia ssm... Figur.9 Triagular lms covrig h FBC ad buffr rgio.... Figur. Triagular msh graio o h FBC Figur. ad h buffr rgio.... vi

7 Figur. Darc vloci for h FBC i Figur Figur. Domai discrizaio b four diffr sizs of msh Figur. Th rlaioship bw h cocivi ad h umbr of fracurs for hr ralizaios I II ad III. Saisics for Cas i Tabl. is usd Figur.4 Frquc for h hrshold cocivi for hr cass i Tabl Figur.5 CDF for FBC occurrc. Labls ad idica h cass show i Tabl Figur.6 CDF of oal discharg a h our boudar for a modl wih a hrogous NF b h local homogizaio for ralizaios for 9 cass i Tabl Figur.7 Fracur works FBCs ad Fii lm msh ad Darc vloci disribuio for a slcd ralizaio for 9 cass Figur.8 Toal dischargs a h our boudar b h local homogizaio for 9 cass wih 9% cofidc irvals Figur.9 CDF of oal discharg a h our boudar for a modl wih a hrogous NF for cass A ad B corrlad ad ucorrlad Figur. Darc vloci disribuios i h modl spac boh for corrlad ad ucorrlad cass Figur. Compariso of h 9% cofidc irvals of oal dischargs a h our boudar bw a modl wih a homogizd NF b h uiformizaio hick lis ad a modl wih a hrogous NF b h local homogizaio hi lis for 9 cass Figur 4. Modl spac for h mahmaical modl Figur 4. Poial ad sram fucios for cass Figur 4. Domai discrizaio for umrical soluio b FFDF Figur 4.4 Aalical vs. umrical for cas: K /K = ad 44 pois Figur 4.5 Aalical vs. umrical for cas : K /K = ad 8 pois Figur 4.6 Aalical vs. umrical for cas 5: K /K = ad 56 pois Figur 4.7 Aalical vs Numrical for cas : K /K =. ad 44 pois Figur 4.8 Aalical vs Numrical for cas 4: K /K =. ad 8 pois Figur 4.9 Aalical vs. umrical for cas 6: K /K =. ad 56 pois Figur 4. Avrag diffrc ad umbr of pois Figur 5. Probabili dsi fucio of for cass Figur 5. Paricl raspor for diffr im sps ad for war vloci i Figur Figur 5. Tracks of all paricls for war vloci i Figur Figur 5.4 Tracks of all paricls for war vloci i Figur. for highr diffusio cofficis i h NF FBC ad o-fbc rgio ha Figur Figur 5.5 Cumulaiv disribuio fucios of h rsidc ims of 4 paricls Figur 5.6 Cumulaiv disribuio fucios of h rsidc ims for corrlad aprur for cas A Figur 5.7 Cumulaiv disribuio fucios of h rsidc ims for corrlad aprur for cas B Figur 5.8 Cumulaiv disribuio fucios of h rsidc ims of 4 paricls for h modl wih a hrogous NF b h local homogizaio solid li ad for h modl wih a homogizd NF b h uiformizaio dashd li for 9 cass Figur 5.9 Compariso of h 9% cofidc irvals of rsidc ims for h ir modl spac bw a modl wih a homogizd NF b h uiformizaio hick lis ad a modl wih a hrogous NF b h local homogizaio hi lis for 9 cass Figur 5. Fracioal absorpio ras a h our boudar for 4 paricls for cass A B ad C wih hrogous NF a~c ad homogizd NF d~f Figur 6. Modl spac for oruosi simaio Figur 6. a Fracur work graio i ui squar domai. b Ircocd fracurs cocig h lf ad righ boudaris. c Ircocd fracurs wih assigd labls a h irscios... 9 vii

8 Figur 6. Fracur work wih assigd h labl a h irscios ad boudar codiios for war flow Figur 6.4 Cumulaiv disribuio fucios CDFs of L ad * for l=. a ad l=.75 b ad l=.5 c for ralizaios Figur 6.5 Fracur work ircocd fracurs hick lis cocig h lf ad righ boudaris ad ircocd fracurs formig h acual flow pahs for a slcd ralizaio for ach cas Figur 7. Modl spac for flow ra calculaio Figur 7. A fracur work wih N f =7 for cosa lgh l=. aprur b= -4 m ad uiform locaio ad oriaio Figur 7. Disribuio of hdraulic had a h irscios of fracurs... Figur 7.4 CDFs of h flow ra b DFN solid ad homogous coiuum dashd approachs for ralizaios for four cass i Tabl Figur A. Polar coordias... Figur A. Logormal disribuio b Aalical soluio usig. ad b umrical soluio usig quaios A. A.... Figur B. Fracur work graio... Figur B. Fracur work graio wih cosa oriaio.... Figur B. Fracur work wih h cosa lgh of.m shor fracurs Figur B. 4 Fracur work wih h cosa lgh of.m log fracurs Figur B. 5 Fracur work wih cosa lgh.m ad oriaio Figur B. 6 Fracur work wih h cosa aprur... 5 Figur C. Simpl ampl for war flow aalsis Figur F. a Gap rgios shadd ad b h imagiar absorbig boudar... Figur F. Calculaio of h maimum hickss of h gap rgio h gap.... Figur G. Ircocd fracur work sam as Figur 6. c... Figur G. Idificaio of pahs from h lis i Tabl G Figur H. Simpl ampl of fracur work... 5 Figur H. A fracur work wih boudar codiios... 5 Figur I. A fracur work i Figur H. b... 7 Figur I. A fracur work which has wo ils Figur I. Equival arra of rsisors o h fracur work i Figur I. a ad h fracur work i Figur I. b. ij [ohm] dos h rsisac bw pois i ad j i lcrical circui Figur I. 4 Simplifid arra of rsisors from Figur I. b b Ohm s law for rsisors i paralll viii

9 Lis of Tabls Tabl. Combiaios of Modlig Approachs for War Flow ad Solu Traspor... 9 Tabl. Numbr of Fracurs Calculad wih Eprimal Daa for Frquc... 6 Tabl. Saisics of Fracur Gomr Paramrs... 7 Tabl. Saisics of Fracur Gomr Paramrs Usd o Gra Figur.7... Tabl. Msh Siz Effc o War Flow Calculaio... 6 Tabl. Saisics of h Fracur Gomr Paramrs for Thr Curvs i Figur Tabl.4 Saisics of h Fracur Gomr Paramrs for 9 cass... 8 Tabl.5 Saisical suls of Toal Dischargs a h Our Boudar ad a h Irfac for a Modl wih a Hrogous NF for Diffr Numbr of alizaios for Cas E i Tabl Tabl.6 Saisical suls of Toal Dischargs for alizaios a h Our Boudar ad a h Irfac for a Modl wih a Hrogous NF b h Local Homogizaio for 9 cass... 4 Tabl.7 Saisical suls of Toal Dischargs for alizaios a h Our Boudar ad a h Irfac for a Modl wih a Hrogous NF b h Local Homogizaio for Corrlad Aprur Cass Tabl.8 Toal Dischargs of a Paricular alizaio for Cass A ad B wih h Corrlad Aprur i Figur. b ad d Tabl.9 Toal Dischargs a h Our Boudar for a Modl wih a Hrogous NF ad a Modl wih a Homogizd NF for alizaios for 9 cass Tabl 4. Ipu Paramrs for Graphical prsaio of Aalical Soluio Tabl 4. Numrical Valus of Cofficis i Aalical Soluio Tabl 4. Ipu Paramrs for Compariso Tabl 4.4 Diffrc for K /K = Tabl 4.5 Diffrc for K /K = Tabl 4.6 Compariso of Toal Discharg a h Our Boudar ad a h Irfac Tabl 5. Ipu daa for umrical simulaio for cass Tabl 5. Compariso of Aalical ad Numrical Tabl 5. Compariso of Aalical ad Numrical Tabl 5.4 Saisics of sidc Tim for h Eir Modl Spac ad h Buffr gio for 9 Cass Tabl 5.5 Saisics of sidc Tim for h FBC ad No-FBC gios for 9 Cass Tabl 5.6 Numbr of Paricls Which Has Zro No-FBC sidc Tims... 8 Tabl 5.7 Saisics of sidc Tim for h Eir Modl Spac ad h Buffr gio for Corrlad Aprur for Cass A ad B... 8 Tabl 5.8 Saisics of sidc Tim for h FBC ad o-fbc gios for Corrlad Aprur for Cass A ad B... 8 Tabl 5.9 Saisics of sidc Tims for h Eir Modl Spac ad h Buffr gio for 9 Cass b h Uiformizaio Tabl 5. Saisics of FBC sidc Tims for 9 Cass b h Uiformizaio Tabl 5. Maimum Fracioal Absorpio a a h Our Boudar Tabl 6. Lis of Cocd Pois for Figur 6. c... 9 Tabl 6. Lis of Cocd Pois basd o Poial Disribuio i Figur Tabl 6. Saisics of h Fracur Gomr Paramrs for Toruosi Tabl 6.4 Saisics of L ad * Tabl 6.5 Mdia Valus of L ad * Tabl 7. Uiformizd Hdraulic Propris for h Fracur Nwork i Figur Tabl 7. Flow as hrough h Fracur Nwork i Figur i

10 Tabl 7. Paramrs for War Flow a... Tabl 7.4 Flow as b DFN Q DFN ad Homogous Coiuum Q HOMO Approachs for alizaios for 4 Cass i Tabl Tabl G. Lis of Cocd Pois for Figur G....

11 . Iroducio ad Summar. Iroducio A gological rposior is cosidrd as h mos favorabl opio for saf disposal of h High-Lvl radioaciv Was HLW. Th mai rols of h rposior ar o isola ad cofi h HLW ad o miiga h radiological ffc o h huma virom for a sufficil log priod of im. Th saf of h rposior ca b valuad basd o h amou of radiouclids rlasd from h was mari o h huma virom. Two rgios of irs hav b cosidrd for h prformac assssm of HLW rposior i.. h ar fild ad h far fild. Th poi of irs of his dissraio is mass raspor aalsis i h ar fild for a war-saurad HLW rposior. Th ar fild rgio is cosidrd [omro al. 995 a b; Gllig 997; JNC Vol. p.v-5] o cosis of h Egird Barrir Ssm EBS ad par of h surroudig hos rock which is calld h Nar-Fild ock NF * i his dissraio. Br characrizaio of h groudwar flow ad radiouclid raspor spciall i h ar fild is cruciall impora for improvig h prformac assssm of HLW rposior [Ah al. 996; Ah al. 998]. Procsss affcig h EBS prformac ar carrid b groudwar hrough h NF such as boi swllig was-mari dissoluio ad caisr corrosio. Th shor-livd radiouclids would dca wihi a shor disac from h was caisr [Ah al. 996] whras h cocraios of loglivd radiouclids ca build up i h war sram coaiig mulipl faild caisrs i h rposior [Ah al. 998]. adiouclids rlasd from h was-mari will b raspord b groudwar hrough h NF ad vuall o h huma virom. Prvious ar-fild modls [omro al. 995 a b; Gllig 997; JNC ] hav o ak io accou h hrogi of h NF for h radiouclid raspor aalsis. omro al. [995 a b] ad Gllig [997] rad h NF as a homogizd comparm rprsd b h avragd hdraulic ad raspor propris. JNC [ Vol. 4 p.v-48] calld h ar-fild rock h Ecavaio Disurbd Zo EDZ which is modld as a highl prmabl coiuum rprsd b avragd valus of hdraulic propris. Th NF grall has a low prmabili cp whr i is fracurd [Naioal sarch Coucil 996]. Th fracurs ar major coduis for groudwar flow ad radiouclid raspor. Th gomr of h fracurs i h NF ca affc sigifical h groudwar flow ad h radiouclid raspor hrough h rock [Naioal sarch Coucil 996; Lim al. a; Lim al. b; Lim al. ]. Thrfor h hrogi of h NF should b ak io accou i h simulaios of groudwar flow ad radiouclid raspor i h ar fild of HLW rposior.. Sa of h Ar Traspor i a fracurd rock is subsaiall diffr dpdig o whhr h pors ad fracurs i h rock ar saurad wih war or o. Whil h Uid Sas is cosidrig a HLW rposior abov h war abl i Yucca Mouai ma ohr couris such as Caada Filad Frac Japa Swd ad Swizrlad ar cosidrig cosrucios of HLW rposioris i a crsalli rock blow h war abl. I his dissraio h HLW rposior blow h war abl i h war-saurad rgio is cosidrd. Th prformac assssms of Swdish ad Japas HLW rposior cocps which ar pical for warsaurad HLW rposior ar summarizd i his scio. Th prformac of Swdish HLW rposior cocp was assssd b a igrad modl which cosiss of sub-modls for raspor i h ar-fild NUCTAN ad i h far-fild CHAND [Gllig 997]. Th arfild modl calculas h rlas ra of h radiouclids from h caisr [omro al. 995 a b]. Th farfild modl is basd o h chal work CN ** cocp ad is usd for a aalsis of radiouclid raspor o h huma virom. I h Swdish modl h ar-fild icluds a was caisr a boi-filld buffr ad a NF s Figur. a. Th ar-fild is rprsd as a umbr of comparms [omro al. 995 a b] s Figur. b. Avragd propris ar dfid for hs comparms. * NF is cosidrd o b fracurd i his dissraio. I is assumd ha war flows ol hrough h fracurs i h NF ad h rock mari is imprmabl. ** Dail is giv i Scio.

12 Pah-Q Q Tul Sad-boi Tul Sad-Boi Boi Q Disurbd Zo Damag hol Boi Pah-Q Caisr Dposiio hol Q Boi ock NF Chal Caisr Hol Pah-Q Ful Caisr wall ock NF Fracur zo Q 4 Pah-Q 4 Figur. a a Schmaic viw of h Swdish KBS- rposior dsig showig h small hol i h caisr ad h locaio of h various scapr rous. Pah Q rlas dircl o h flowig war i a fracur irscig h dposiio hol. Pah Q rlas o h disurbd zo aroud h ul. Pah Q rlas o h ul backfill ad furhr o a fracur zo irscig h ul. Pah Q 4 rlas hrough h rock o a arb fracur or fracur zo [omro al. 995 a b; Gllig 997]. b Th comparmalizaio of h rposior; ach bo solid li is a comparm ad dashd lis ar fir subdivisios of h bos. Th arrows show h various pahwas followd b h spcis from h caisr o h flowig war. Drawig is o o scal. [omro al. 995 b]. b adiouclids lakig ou hrough h damagd hol of a caisr diffus io h boi-filld buffr ad h migra hrough various pahwas io h flowig war i rock fracurs. Figur. shows four possibl pahwas. Bcaus h uclid raspor hrough o spcific pahwa is iflucd b h rlas i h ohr pahs h rlas should b calculad simulaousl for all pahs [Gllig 997]. Gllig [997] aks h followig hr sps o simula h groudwar flow ad radiouclid raspor from h was caisr o h huma virom. Firs h flow ra of groudwar is calculad b usig h farfild modl CHAND i h ssm icludig h ar fild. Th calculad flow ra is usd as ipu daa for h ar-fild raspor modl. Scod h radiouclid raspor i h ar fild is simulad b h ar-fild modl NUCTAN. Th oupu of h raspor aalsis i h ar fild is usd as ipu daa for h far-fild raspor. Third h radiouclid raspor i h far fild is simulad b CHAND. I h Swdish prformac assssm hrogi of h ar-fild for radiouclid raspor is rprsd b four major chals or pahs. Howvr ffcs of fracur work gomr i h NF ar o plicil cosidrd. Thus h hrogi du o discr fracur work has o b ak io accou for radiouclid raspor i h ar fild. JNC [] prformd h saf assssm for a cocpual HLW rposior for Japas gologic virom. Thir cocp is similar o Swdish cocp [omro al. 995 a b; Gllig 997]. Th ar fild cosiss of h EBS ad a limid volum of h surroudig hos rock which is calld h EDZ s Figur. a. A igrad modl is also dvlopd for h radiouclid raspor from h rposior o h huma virom. Firs h groudwar flow ra i h ar fild is calculad b usig h hr-dimsioal discr fracur work DFN * cod FracMa/Mafic. Th h EDZ is rad as a homogous coiuum whos quival hdraulic propr such as rasmissivi is obaid from a discr-fracur work modl. Scod h flow ra obaid b D DFN modl i h EDZ is usd as ipu daa for h uclid migraio aalsis i h ar fild. Th radiouclid raspor i h EBS is simulad b MESHNOTE which ras simplifid o-dimsioal gomr of a was caisr ad boi-filld buffr ad h homogizd EDZ s Figur. b. * Dail is giv i Scio.

13 adiouclids rlasd from h buffr ar assumd o b mid isal ad compll wih groudwar flowig hrough h EDZ. Toal uclid flu rs fracurs irscig h EDZ. Third h radiouclid raspor i h far fild is simulad b o-dimsioal muli-pahwa modl MATICS which aks io accou hrogi of h hos rock b a o-dimsioal plaar-fracur modl. A plaar fracur i h modl rprss a major chal of radiouclid raspor. War-coducig fracurs Fracur Hpohical war-filld rgio Ecavaio Disurbd Zo EDZ Nuclid migraio Buffr marial Virifid was Was Hos rock Figur. EDZ Boi-filld buffr EBS Nar-fild a EDZ Hos rock Buffr marial a Cocpual modl of Japas HLW rposior. b Cocpual modl for uclid raspor i h EBS of h rposior. I b hpohical war-filld rgio is cosidrd o spara h virifid was ad h buffr [JNC Vol. p.v-6]. As has b rviwd abov i Swdish ad Japas prformac assssm sudis h NF is rad as a homogous coiuum for radiouclid raspor aalsis whil h hrogi of NF was rcogizd for war flow aalsis ad implicil icorporad io hir ar-fild raspor modls. Howvr his simplificaio iroducd i prvious assssm sudis has o b vrifid o b cosrvaiv or accpabl. A ar-fild modl icorporaig h hrogi of h NF o ol for h groudwar flow aalsis bu also for h radiouclid raspor aalsis is dd o chck h validi of h simplificaio applid i h prvious sudis for h prformac assssm of HLW rposior.. Objcivs Th objcivs of his dissraio ar: o dvlop a ar-fild modl icorporaig h hrogi of h NF for h groudwar flow ad paricl raspor aalss for a hpohical war-saurad HLW rposior i a wo-dimsioal spac o ivsiga ffcs of h hrogi of h NF o h groudwar flow ra ad paricl raspor ad o ivsiga ffcs of h modlig approach for h NF o h groudwar flow ra ad paricl raspor..4 Scop ad Summar I his sud h ar fild cosiss of h was caisr h boi-filld buffr ad h ighborig surroudig hos rock. Th was caisr would fail du o corrosio. Th was caisr is assumd o cofi radioaciv marials for ars afr h mplacm i h rposior. I h ars afr mplacm h ha graio from h caisr will rapidl dcras ad h mpraur i h rposior will sl dow o h ambi mpraur [Ah al. 995]. Th im priod afr h mpraur has sld dow o h ambi is cosidrd i his sud. Boi is cosidrd as a basic marial for h buffr i h war-saurad HLW rposior. Th buffr is cosidrd o compsa h mchaical flucuaio aroud h EBS o ihabi h war praio io h was caisr ad o rard radiouclid raspor. I his sud w% boi is assumd as a buffr marial [JNC Vol. p.iv-68]. b

14 A o-was caisr cofiguraio s Figur. i a wo-dimsioal modl is cosidrd for h groudwar flow ad paricl raspor aalss i h ar fild i his sud. Th NF rgio wih h radius of 5m is cosidrd [JNC Vol. p.iv-4]. Chapr shows h liraur rviws for h modlig of h war flow ad solu raspor i fracurd rock. Saisical disribuios of fracur gomr paramrs ar also summarizd basd o liraur rviw. Basd o h liraur surv for h fracur gomr paramrs logormal disribuios ar assumd for h lgh ad aprur of fracurs. Th uiform ad poial disribuios ar assumd for h spaial disribuios of h primar ad scodar fracurs rspcivl *. Th uiform disribuio is assumd for h oriaio agl of fracurs. Chapr is a k chapr i his dissraio. A wo-dimsioal ar-fild modl is dvlopd for h owas caisr cofiguraio o simula h groudwar flow for a hpohical war-saurad HLW rposior. NF FBC DFN Firs approach Local homogizaio Scod approach Uiformizaio NF Saga rgio NF A hrogous coiuum rasformd from h FBC A homogous coiuum rasformd from h FBC Figur. Schmaic diagram of h rasformaio of h flow-barig clusr FBC of fracurs. Local homogizaio ad uiformizaio ar dfid i Chapr. Shadd rgio rprss h flowig-barig rgio. A Discr-Fracur Nwork DFN is grad i h NF basd o disribuio fucios of h fracur gomr paramrs b radom samplig i.. lgh aprur oriaio ad locaio. Th war-flowig fracur sgms i h DFN i h NF ar drmid b chckig h ircocivi of h fracurs. Th Flow-Barig Clusr FBC of fracurs is rasformd io a quival coiuum i wo diffr was ** s Figur.. Firs a rasformaio is applid o h FBC locall rsulig i a hrogous coiuum. Thus h NF is dividd io h flow-barig rgio i.. hrogous coiuum rasformd from h FBC ad h war-saga rgio. Scod h FBC is uiformizd ovr h ir NF. Th NF is rasformd io a * Dfiiios of h primar ad scodar fracurs ar giv i Scio..4 ** Th firs ad scod approachs ar calld h local homogizaio ad h uiformizaio rspcivl i his dissraio. Dails ar giv i Chapr. 4

15 homogous coiuum. Th ir NF is h flow-barig rgio. Th scod approach is similar o hos usd i h ar-fild modls i h Japas ad Swdish HLW rposior prformac assssm sudis as discussd i Scio.. Th prmabili for h rasformd coiuum is valuad b h Koz-Carma quaio. Th flow ra of groudwar is calculad b solvig a sad-sa poial quaio umricall b h fii lm mhod. Effcs of h hrogi of h NF ad ffcs of h modlig approach for h NF ar ivsigad for diffr ss of saisics of fracur gomr paramrs b comparig h groudwar flow ras bw h modl wih a hrogous NF b h local homogizaio ad h modl wih a homogizd NF b h uiformizaio. Th cocivi dfid as h raio of h oal umbr of irscios o h oal umbr of fracurs is usd as a masur ha prsss h dgr of hrogi of h NF. Numrical rsuls for ffcs of h hrogi of h NF s Figur.8 show ha h work wih a grar cocivi hibis a grar war flow ra ad ha a low cocivi givs a grar ucrai for war flow ra. Numrical rsuls for ffcs of h modlig approach of h NF s Figur. show ha h mdia valus of h war flow ras obaid b h wo approachs ar clos o ach ohr ha h flow ras obaid b h local homogizaio givs a grar ucrai ha a a larg cocivi wo approachs giv a similar rsul ad 4 ha h homogizd NF b uiformizaio dos o cssaril ovrsima h war flow ra compard o h hrogous NF b local homogizaio. Th las rsul implis ha h homogizd NF b uiformizaio ma o b applid o h war flow aalsis for a cosrvaiv prformac assssm. I Chapr 4 umrical rsuls obaid b h modl wih a homogizd NF dvlopd i Chapr ar bch-markd agais a aalical soluio. Aalical soluios for h sad-sa poial disribuio ad sram fucios ar obaid for h modl wih a homogizd NF. Wih h aalical soluio h war flow ra is valuad. Th umrical rsuls obaid b h modl dvlopd i Chapr ar compard wih hos obaid b aalical soluios. Th umrical rsuls for h valus of hdraulic poial diffr from hos b aalical soluio b a avrag diffrc lss ha.7 - s Tabl 4.5. Th umrical rsuls for h oal dischargs of war i h modl spac diffr from hos obaid b aalical soluios b a rlaiv diffrc lss ha.55% s Tabl 4.6. I Chapr 5 h radom walk rackig modl is dvlopd for paricl raspor i combiaio wih h flow modl which is dvlopd i Chapr. Th WTM is applid bcaus of such advaags as i a simpl algorihm ii as icorporaio wih h flow modl ad iii o umrical disprsio. Effcs of h hrogi of h NF ad ffcs of h modlig approach for h NF ar ivsigad for diffr ss of saisics of fracur gomr paramrs b comparig h paricl rsidc ims ad h mass absorpio ras a h our boudar bw h wo approachs as show i Figur. for rasformig a FBC io a coiuum. Numrical rsuls show ha h work wih a grar cocivi hibis a shorr rsidc im of paricls s Figur 5.9 ha h ffc of h lgh ad aprur corrlaio o h rsidc im of paricls i h ar-fild is gligibl s Figur 5.6 a ad Figur 5.7a ha h rsidc im of paricls for h homogizd NF is mail drmid b h boi-filld buffr rgio s Tabl 5.9 ad Tabl 5. 4 ha h homogizd NF b uiformizaio givs h cosrvaiv rsuls i.. udrsimaio of h rsidc im s Figur 5.9 ad ovrsimaio of h mass absorpio ra a h our boudar s Figur 5. compard o h hrogous NF b local homogizaio ad 5 ha b uiformizig h DFN ovr h ir NF h rsidc ims i h ir modl spac ad mass absorpio ra a h our boudar of h NF ar hardl affcd b h cocivi of fracurs s Figur 5.9 ad Figur 5. d~f. Th las wo rsuls impl ha h homogizd NF b uiformizaio ma b applid for a cosrvaiv prformac assssm bu o for rposior si slcio or dsig whr diffriaig o rock mdium o aohr is of primar irs. I Chapr 6 h coffici C i Koz-Carma quaio for prmabili is discussd. Th quaio has b usd i h prvious chaprs o sima h prmabili for h rasformd coiuum. Th quaio icluds a coffici C which ca b drmid b h oruois ad h shap of cross-scioal ara of h flow pahs ormal o h flow c. I Chapr C is assumd o b o. I Chapr 6 a sigl valu of h coffici C for h ir FBC is valuad basd o h fracur work gomr. If flow hrough a paralll-pla chal is assumd h valu for c is kow o b [Carma 956]. Toruosi ds o b simad b akig io accou fracur-work gomr bcaus h flow ra is simad b h rasformaio of h FBC io a coiuum i Chapr ad Chapr 5. Two formula for oruosi ar cosidrd i h fracur work. I o formula oruosi is dfid as L L/L whr L is h avrag lgh of h pahs hrough a fracur work ad L is h gomrical lgh of 5

16 h mdium [Bar 97]. I h ohr formula oruosi is dfid as <cos > [Fowlr al. 94; Carma 956] or * <cos *> whr is h oriaio agl of all fracurs icludd i h FBC s Figur.4 ad * is h oriaio agl of all fracurs formig h acual flow pahs i h FBC. Numrical rsuls show ha L givs h smalls valu amog h hr whras diffrc bw ad * is o sigifica s Figur 6.4 ad ha b usig isad of L cosrvaiv ovrsima ca b mad for h war flow ra i h fracurd rock. I Chapr 7 h homogous coiuum modl rasformd b h scod approach or uiformizaio wih h coffici C /c is compard wih h covioal DFN modl for h sam fracur work. Numrical rsuls show ha h homogous coiuum modl wih oruosi corrcio ovrsimas war flow ras compard wih hos b h DFN modl s Figur 7.4. I Chapr 8 h coclusios ha hav b drivd from his sud ar summarizd. 6

17 . Liraur viws Liraur rviws hav b mad i his chapr for h mass raspor modlig i fracurd rock ad h fracur gomr paramrs.. Liraur viw for Mass Traspor i Fracurd ock A sochasic approach has b mplod o rprs h hrogi of h fracurd rock du o a discr fracur work bcaus h masurm ad characrizaio of all fracurs ar oall impracical [Adrsso al. 987]. Thr ps of sochasic modls hav b usd o simula h war flow ad solu raspor i fracurd rock [Gir al. 99]: Discr Fracur Nwork DFN modl Sochasic Coiuum SC modl ad Chal Nwork CN modl... War Flow Modlig i Fracurd ock I h Discr Fracur Nwork DFN modl approach a discr fracur work is grad b Mo- Carlo samplig from giv disribuio fucios of fracur gomr paramrs. A sad-sa flow ad imprmabl rock mari ar usuall assumd. Th cubic law for h volumric war-flow ra hrough a paralllpla chal [Sow 965; Wilso 97; Wihrspoo al. 98; Log al. 98; Log 985] is usuall applid for flow i idividual fracurs. Two-dimsioal DFN modls wr dvlopd ad sd b various rsarchrs such as Log al. [98; 987] Schwarz al. [98] ad Adrsso al. [986]. Thr-dimsioal DFN modls wr also dvlopd [Log 985; Elsworh 986; Adrsso al. 987; Drshowiz al. 994]. Th major advaag of DFN modls is h plici rprsaio of h gomrical ad phsical propris of fracurs for a fracurd rock [Gir al. 99; Naioal sarch Coucil 996]. A fracur work of various scals ca b rad [Gir al. 99]. Th major disadvaags of DFN modls ar saisical fracur gomric daa spciall a dph ar difficul o obai [Naioal sarch Coucil 996] wh samplig locaios ar o wll disribud rapolaio is rquird [Gir al. 99] h modl ca b compl ad compuaioall isiv for ralisic fracur dsi [Naioal sarch Coucil 996; Liu al. ] ad 4 h modl o rla iformaio abou fracur gomr o h hdraulics of fracurd rock is i qusio [Numa 988]. Numa [988] applid h Sochasic Coiuum SC cocp o a hrogous fracurd rock. Th SC modl is basd o h rprsaio of hdraulic propris i rms of probabili modls ha dscrib h mdium as a radom fild. Th SC modls mak i compuaioall fasibl for larg-scal flow modlig bcaus h do o ak io accou h idividual fracur gomr o rprs h hrogi of h rock. I h SC modl howvr du o h wak couplig bw fracur gomr ad h hdraulic propris h raspor propris which ar ssiiv o h hrogi of h mdium ma b diffr from h DFN modl [Dvrsorp al. 99]. Th Chal Nwork CN modl has b dvlopd o icorpora a uv flow disribuio i a fracurd rock ad o rflc h sochasic apparac of aural propris [Gllig 997]. Th basis of h CN modl [Moro al. 99; Moro al. 997; Gllig 997] is ha flow hrough a fracur work is cofid o discr ffciv o-dimsioal pahwas kow as chals. I h CN modl a larg umbr of chals ar cosrucd i h D ssm o rprs a fracurd rock. Each chal has a hdraulic coducac ak sochasicall from a disribuio. Th flow ra is calculad b solvig h prssur disribuio i h ssm wih mass balac a ach irscio poi. For h flow modlig i h NF h DFN approach migh b appropria bcaus h fracurs rprsig h hrogi of h NF could corol h groudwar flow i h NF. Also difficulis for DFN approach such as limiaio of h fild daa ad larg-scal modlig would o b h cas for h ar-fild rgio of h HLW rposior bcaus rlaivl larg amou of fild daa will b availabl i h ar-fild ad bcaus h ar fild rgio aroud a was caisr is o so larg as h far fild [JNC Vol. p. V-5]... Solu Traspor Modlig i a Fracurd ock Th modls for solu raspor i a fracurd rock ar summarizd basd o liraur surv for h cass whr flow vloci profil i h rock is kow ad o affcd b h solu raspor. I h discr work modls such as h DFN ad CN modls solu raspor hrough a work of fracurs or pips was modld b rprsig h racr as a fii umbr of paricls i.. a paricl-followig 7

18 mhod * [Schwarz al. 98; obiso 984 Tsag al. 988; Yamashia al. 99; Cacas al. 99b; Drshowiz al. 995; Gllig 997; Moro al. 997]. A paricl-followig mhod has b dvlopd o simula h mass raspor hrough h discr fracur work [Schwarz al. 98; obiso 984]. I his mhod a larg umbr of paricls ar addd iiiall ihr o a sigl fracur a h upsram boudar of h ssm [Schwarz al. 98] or o h irscios alog a sid of h work [obiso 984]. Paricls mov b advcio hrough h fracur work basd o war vloci for idividual fracur sgms. A a irscio h dircio of paricls is drmid from probabiliis wighd accordig o h rlaiv quaiis of flow movig awa from h irscio wih assumpio of prfc miig. As a cosquc of h prfc miig a h irscios mass disprss as i movs b advcio hrough h work. Thus larg-scal disprsio is modld i his wa. Howvr h pariclfollowig mhod b Schwarz al. [98] ad obiso [984] dos o ak io accou h mari diffusio which is o of h mos impora mchaisms for rardaio of h radiouclids [Nriks 98; Ah 988]. Yamashia al. [99] iroducd a paricl-followig mhod icorporaig h mari diffusio for a sigl fracur modl wih a ifii porous rock mari. Th dscribd h ffc of mari diffusio i rms of h rsidc ims of paricls bcaus of h fac ha a paricl ma rsid i h mari for som im i addiio o is rsidc im i h war i h fracur. Th rsidc im of ach paricl is drmid b a probabili dsi fucio of h paricl rsidc im which is obaid b ormalizig h mass flu. This mhod is adopd b Moro al. [997] o simula h solu raspor i h CN modl. Drshowiz al. [995] dscribd a implmaio of a DFN modl i which h mari diffusio was modld b a probabilisic paricl-followig chiqu. Th solu cocraio is obaid aalicall from h o-dimsioal diffusio quaio wih a cosa ffciv diffusivi. Th h fracio of h umbr of paricls rsidig i h mari is calculad b igraig h oal mass rasfr ra across h fracur-mari boudar ad abls o valua h probabili of fidig a paricl i h mari ad i h fracur. Similar o h modl b Yamashia al. [99] h oal rsidc im of a paricl is calculad b summig h rsidc im i fracurs ad rsidc im i mari. For coiuum basd modls such as h SC modl wo basic approachs wr usd o simula h solu raspor: h Fii Diffrc Mhod FDM or Fii Elm Mhod FEM ad h adom Walk Trackig Mhod WTM. Th FDM or FEM is a umrical approach o solv a diffrial quaio b prssig h diffrial quaio as a s of liar quaios b domai discrizaio. Alhough his mhod is wll dvlopd for solvig h raspor quaio i a mahmaical ss hr ar disadvaags. Firs compuaioal work o solv h N q quaios pr im sp is isiv. Also h soluio ma iclud h pars of domai which vr ravrsd b solus [Tompso al. 987]. Scod h us of fii diffrc or fii lm causs umrical rrors du o h umrical disprsio ad/or oscillaio [Bar 987; Tompso al. 987 Kizlbach 988; Uffik 988]. Th umrical disprsio is causd b h approimaio of h firs-ordr drivaiv which ivolvs rror of h ordr of magiud of h scod-ordr drivaiv [Bar al. 987]. Th umrical disprsio is svr wh h advcio is domia P >~ ** [Huakor al. 98; Tompso al. 987]. Th umrical rror causd b h umrical disprsio ca b mad small b dcrasig h siz of msh [Bar al. 987; Huakor al. 98; Tompso al. 987; Kizlbach 988]. Howvr a fi msh is o alwas possibl [Huakor al. 98] ad somims h mos impracical opio [Tompso al. 987]. Various fii diffrc mhods o ovrcom h umrical disprsio hav b summarizd b Huakor al. [979]. Th WTM is a aalogu procss ha obs h rasi advciv-disprsiv raspor quaio. Advaags of his mhod ar as follows. Firs h algorihm is simpl ad ca asil b applid i larg hrdimsioal as wll as usaurad problm [Tompso al. 987; Kizlbach 988]. Also his mhod ca b asil combid wih a flow modl [Kizlbach 988]. Scod hr is o umrical disprsio sic h radom walk mhod is o a dirc umrical soluio of h diffrial quaio [Tompso al. 987; Kizlbach 988; Uffik 988]. Third h compuaioal ffor pr im sp is proporioal o o h umbr of ods bu o h umbr of paricls. Hc h sorag rquirm will b dramaicall rducd from a fii lm or fii diffrc mhod [Tompso al. 987]. * I som sudis [Schwarz al. 98; Yamashia al. 99; Drshowiz al. 995] i is calld a pariclrackig mhod. ** Pcl umbr P dfis h raio bw h ra of raspor b covcio o h ra of raspor b molcular diffusio [Bar 97]. 8

19 Thr ar disadvaags of WTM. A solu mass is rprsd as a larg collcio of paricls. I ordr o obai cosis ad rliabl rsuls a larg umbr of paricls ar rquird [Tompso al. 987]. Th saisical flucuaio of umrical rsuls ca b avoidd b icrasig h umbr of paricls. Howvr icrasig oal umbr of paricls dos o show h sam dgr as a improvm of rsuls bcaus his flucuaio is proporioal o h squar roo of h umbr of paricls [Kizlbach 988]. Th applicaio of sourc/sik codiios is complicad [Tompso al. 987] ad h prscribd oal flu boudar codiio is difficul o simula i WTM [Kizlbach 988]. Th WTM which is wll dvlopd i coiuum such as porous mdium has a difficul i applig o h fracurd rock rprsd b dual coiua [Liu al. ]. Tabl. shows h summar ad combiaio of h applicabl modlig approachs for h war flow ad solu raspor aalss basd o h liraur rviw. I prvious sudis if h work modls i.. DFN ad CN ar usd for war flow h paricl-followig mhod is usuall usd for h solu raspor as summarizd i Tabl.. O of h major diffrcs bw h DFN ad h CN is ha h ssm b h DFN approach coducs war ol if h ssm is prcolad whras h ssm b h CN approach alwas coducs war. This is bcaus h hdraulic coducac for h CN approach is drmid sochasicall o b h fracur gomris bu b assigig for ach chal from h disribuio. Th SC approach for war flow is suiabl for couplig wih FEM/FDM or wih WTM for solu raspor bu has a difficul i icorporaig mari diffusio bcaus fracur-gomr is o plicil cosidrd. Tabl. War flow Combiaios of Modlig Approachs for War Flow ad Solu Traspor FEM / FDM: causs svr umrical disprsio if advcio is domia P >~. No discussd i his sud Solu raspor WTM: dvlopd for coiuum modls. simpl algorihm as icorporaio o flow modls ad o umrical disprsio. Paricl-followig: dvlopd for work modls aks io accou h largscal disprsio du o h miig a fracur irscios. DFN Difficul o combi WTM wih Applicabl DFN wih mari diffusio SC Applicabl Applicabl No applicabl CN No discussd Difficul o combi WTM wih Applicabl i his sud CN wih mari diffusio I his sud o ak io accou h hrogi of h NF a discr fracur work is grad i h NF. A flow-barig fracur clusr is drmid from h fracur work. Thus h NF is dividd io h flow-barig rgio ad h saioar war rgio. To compar h modlig approachs h flow-barig clusr FBC of fracurs is rasformd io o ol a hrogous coiuum bu also a homogous coiuum. Th hdraulic propris of h rasformd coiuum ar drmid b avragig fracurs icludd i h flowbarig rgio. I h saioar war rgio mari diffusio of h solu is cosidrd o ak plac s Chapr 5. Thrfor h war-flow modl dvlopd i his sud is cosidrd o appl a Sochasic Coiuum SC approach i a ss ha hdraulic propris ar drmid b sochasicall disribud fracurs. Bu h major diffrc bw h prvious SC approach ad h prs o is ha h flow-barig rgio is firs drmid b chckig prcolaio of a fracur work ad ha h hdraulic propris ar drmid ovr ha flowbarig rgio. I should b od ha b cosrucig a coiuum his wa h hrogi rsulig from fracur work is icorporad ad uilizd. I followig scio h fracur gomr paramrs which characriz h discr fracurs i fracurd rock ar rviwd basd o h liraur surv.. Liraur viw for Fracur Gomr Paramrs A fracur is a mchaical discoiui or brak i h rock [Gir 99; Naioal sarch Coucil 996]. A discr fracur work ca rprs h hrogi of h fracurd rock. Saisical approachs ar mplod for fracur work graio [Log al. 98; Schwarz al. 98; Log 985; Adrsso al. 987; Dvrsorp al. 99; Bour al. 997; d Druz al. ]. 9

20 Fracur gomr paramrs ar h fracur siz or h lgh for a wo-dimsioal dscripio aprur oriaio ad locaio. Th saisical disribuios of h paramrs ar rviwd. Ohr paramrs for a fracur work such as frquc spacig ad umbr dsi of fracurs ar also rviwd... Fracur Siz Th siz of a fracur ca b characrizd diffrl basd o h shap of a fracur. For a wo-dimsioal li fracur h siz is rprsd b is lgh. For a hr-dimsioal disk-shapd fracur is radius rprss h siz. Th shap of a fracur ma b approimad b polgoal circular llipical or irrgular shap [Log 985; Gir al. 99]. Bachr al. [977] rviwd h liraur o lgh disribuio. Th lgh was rpord o var logormall or poiall. Log [985] miod ha rac lghs ar disribud ihr logormall or poiall. Log al. [987] chos a logormal disribuio for h lgh of fracurs for a grai rock a dph of m a Faa-Augèrs i Frac. Ma ad sadard dviaio of h fracur lgh rag bw.5m ad.4m ad.5m ad.m rspcivl s Figur.. Cacas al. [99a] miod ha h mpirical rac lgh disribuio could rasoabl b approimad b a logormal disribuio for highl fracurd grai 5m blow groud a Faa-Augèrs i Frac. Ma ad sadard dviaio of h rac lgh var from.74m o.6m ad from.m o.7m rspcivl s Figur.. Hsir al. [99] cosidrd h logormal ad poial disribuios for lgh of fracurs o sima prmabili i h fracur work. Odlig al. [99] obaid primall ha fracur lghs ar disribud logormall for m4m ara o h bddig surfac of a sadso bd of Dvoia i Norwa. Ma ad sadard dviaio of lghs i a log scal ar -.8 ad.98 rspcivl. Nimi al. [] sudid h crsalli rock which is h poial HLW rposior si i asr Filad a dph of m upo m. This si is cosidrd as a rprsaiv of a lowcoducivi fracurd rock. Th sizs of fracur ar disribud logormall wih ma valus ragig bw.65m ad.5m ad wih sadard dviaio ragig bw.45m ad.7m s Figur.. Dvrsrop al. [989] assumd a poial disribuio from h daa obsrvd i Sripa mi. Obaid ma fracur radius rags bw m ad m. Kulailak al. [99] obaid h primal daa from h vilaio drif grai rock a h Sripa mi i Swd. Th claimd ha a logormal disribuio is foud o b usuiabl for dscribig h disribuio of rac lgh. Th rac lgh of fracur could b dscribd b h gamma ad poial disribuios. Ma ad sadard dviaio of h obsrvd joi rac lgh rag bw.m ad.64m ad bw.m ad.47m rspcivl. Kulailak al. [996] claimd ha a gamma disribuio is h bs disribuio for joi rac lgh ad joi diamr for grai mass of a Prsiia ssm i Chia. Obsrvd ma valus of joi rac lgh rag bw.67m ad.55m. Obsrvd ma valus of joi diamr rag bw.48m ad 4m. Pusch [99] gav h primal valus of h rac lgh a Fisjö si Gissous gradiori ad Sripa mi. For h Fisjö si 5% of all fracur has lgh shorr ha m. 5% of all fracur varis from m o m. % of all fracur is largr ha 7m. For h Sripa mi 5% of all fracur is largr ha 4m. % of all fracur is largr ha 7m. Adrsso al. [99] obsrvd h rac lgh of fracurs varis from.m o m a h m48m cll i Fisjö si. I summar h disribuio of siz of fracurs is wll characrizd b h logormal disribuio [Bachr al. 977; Log 985; Log al. 987; Cacas al. 99a; Hsir al. 99; Odlig al. 99; Nimi al. ]. Th ormalizd logormal disribuio is dfid as l[ / ] L f p L L. L L whr L is h shap paramr ad L is h scal paramr. Th scal paramr L has h sam ui as h variabl. Th shap paramr L is dimsiolss. Th ma of h disribuio is dfid as ad is calculad b subsiuig. io. as f d. L p L /..

21 PDF 5 4 Cacas al.99a =.74m =.4m L =.55 L =.m Log al.987 =.44m =.m L =.97 L =.74m Curr sud Cas i Tabl. =.65m=.6m L =. L =.m Cacas al.99a =.6m =.7m L =.7 L =.455m Nimi al. =.65m =.7m L =.877 L =.44m Log al.987 =.5m =.5m L =.798 L =.86m Curr sud Cas i Tabl. =.m =.m L =. L =.m Nimi al. =.5m =.45m L =.94 L =.44m Curr sud Cas i Tabl. =.m =.m L =. L =.m Lgh [m] Figur. Probabili dsi fucios of logormal disribuios for lgh of fracur sgms for h crsalli rock a dph blow ha 5m basd o liraur rviw. Thr diffr disribuios L =. L =.m L =. L =.m ad L =. L =.m ar dscribd b h hick lis ad will b usd i his sud. A ara udr ach curv is ui. Th variac is dfid as ad is calculad b subsiuig. ad. io.4 as Th scal paramr L ca b wri b rarragig. as f d.4 p [p ]..5 L L L p /..5a L L B subsiuig.5a io.5 h shap paramr L is obaid i rms of ad as / l L..5b

22 Thus if ad ar giv h shap paramr L ad h scal paramr L ca b calculad as / l L ad p / L L..6 I his dissraio bcaus a wo-dimsioal modl is dvlopd lgh is rprsd as h siz of a fracur hrafr. Th shap L ad scal L paramrs of h logormal disribuio of fracur lgh ar assumd o b L =. or. ad L =.m or.m for h war flow ad paricl raspor aalss i his sud. L =.m ad L =. is quival o ma of.m b. ad sadard dviaio of.m b.5. Figur. shows h probabili dsi fucios of logormal disribuios for lgh of fracur sgms basd o h liraur rviw for h crsalli rock a dph blow 5m. I his sud hr diffr disribuios hick lis ar slcd basd o h dimsio diamr of m of h modl spac for h DFN ad will b usd for h war flow ad paricl raspor aalss... Aprur For aprur of fracurs wo diffr ps of aprurs ar usd i h liraurs: h mchaical aprur ad h hdraulic aprur [Naioal sarch Coucil 996]. Th mchaical aprur is dfid as a disac bw fracur walls. Th hdraulic aprur is calculad b usig a masurd war flow ra a prssur gradi ad h cubic law. Th hdraulic aprur was obsrvd o b approimal a ordr of magiud smallr ha h mchaical aprur [Log al. 987; Ch al. ]. Biachi al. [968] obsrvd ha aprurs i cors or wll logs wr disribud logormall. Sow [97] claimd ha a logormal disribuio was foud o dscrib h aprur disribuio i various fracurd rock. Bourk al. [985] drivd a logormal disribuio for h hdraulic aprur from prmabili ss i grai. Log al. [987] sad ha h shap of aprur disribuios was arl logormal for a grai rock a dph of m a Faa-Augèrs i Frac for h uclar was sorag rsarch. Mchaical aprurs wr obsrvd o b disribud logormall wih ma valus ragig bw 94m ad m ad sadard dviaio ragig bw 46m ad 4m s Figur.. Moro al. [988] assumd a logormal disribuio for aprur wih a ma valu ragig bw 56.m ad 45m. Tsag al. [988] usd a logormal disribuio for h aprur wih a ma valu of 8m ad a sadard dviaio ragig bw 8.7m ad m as show i Figur.. Kllr al. [999] showd ha aprurs ar disribud arl logormall for a aural fracurd grai cor. Th arihmic ma ad h sadard dviaio ar 85m ad 68m rspcivl. Gomric ma ad sadard dviaio ar 586m ad 864m rspcivl s Figur.. Gir [986] idicad ha aprurs follow a gamma disribuio for aural fracur i grai. Prak-ol [996] showd a approimal Gaussia disribuio wih ma valus ragig bw 4m ad 5m ad a maimum valu upo 7m for 4 iac biumious coal cors from a coal mi ad gas producig wll i h Sa Jua Basi. Oro al. [998] claimd ha a shifd Gaussia disribuio is appropria for aprur bcaus h claimd ha h logormal ad gamma disribuios hav o zro aprur whil i fracurs hr mus b coac poi havig zro aprur. Abli al. [985] sudid a sigl fracur i crsalli rock i h Sripa mi. Hdraulic aprurs calculad from h war rsidc im ad h flow ra of war rag bw m ad m. Pusch [99] gav h quival hdraulic aprur a 5m dph i h Sripa mi basd o 69 packr-sald borhol s. Aprurs rag bw m ad m. % of h aprurs rags bw m ad m. Larsso [997] masurd mchaical aprurs for 8 subscios of a sampl from Äspö Hard ock Laboraor a m blow sa lvl i Swd. A oal of aprur ar masurd for 8 scios wih mdia valus ragig bw 7m ad 7m. Ma ad sadard dviaio valus var from 7m ad 84m ad m ad m rspcivl. Ch al. [] obaid h mchaical aprur i a laboraor masurm for a grai sampl from h subsurfac a Olmpic Dam a miig si i Ausralia. Masurd mchaical aprurs varid from m o m. I summar h disribuio of aprur of fracurs is wll characrizd b h logormal disribuio [Biachi al. 968; Sow 97; Bourk al. 985; Log al. 987; Moro al. 988; Tsag al. 988; Kllr al. 999].

23 PDF Tsag al. 988 =8 m = m L =.5 L =4. -5 m.e+4 Tsag al. 988 =8 m =8.7 m L =. L = m.e+4 Log al. 987 = m =4 m L =.989 L =.9-4 m Kllr al. 999 =586 m =864 m L =.7 L =.9-4 m Curr sud = m = m L =.8 L =7.7-5 m Log al. 987 =94 m =46 m L =.469 L =.6-4 m Kllr al. 999 =85 m =68 m L =.7 L =6.5-4 m.e Aprur [m] Figur. Probabili dsi fucios for aprur basd o liraur rviw. Logormal disribuio is assumd. A disribuio wih L =.8 ad L = i.. ma ad sadard dviaio of m is dscribd b h hick li ad will b usd i his sud. I his sud aprurs ar sampld i wo diffr was. I h firs wa h corrlaio bw h lgh ad h aprur is o cosidrd. A valu of aprur is sampld basd o is ow saisics idpdl. Th probabili dsi fucio for h aprur of fracurs is assumd logormal basd o h liraur surv. Th shap L ad scal L paramrs of h logormal disribuio ar assumd o b L =.8 ad L = m i.. ma ad sadard dviaio of m. I h scod wa a valu of aprur is drmid basd o h obsrvd corrlaio bw h lgh ad h aprur of h fracur afr h valu of lgh is sampld. A rlaioship bw h lgh ad aprur of fracur is giv as [Oho al. 987] whr l [m] is h fracur lgh ad b [m] is h aprur. log l.7.7 log b.7 Numrical rsuls obaid b hs wo sampl mhods ar compard i Scio.6... Figur. shows h probabili dsi fucios for aprur basd o h liraur rviw cpig Oho al. [987]. Similar o h logormal disribuio of lgh i Figur. h disribuios dscribd b h ma ad sadard dviaio ar show i Figur.. A disribuio wih L =.8 ad L = m hick li i Figur. is slcd as a pical disribuio of aprur of h fracur sgms ad will b usd for h war flow ad paricl raspor aalss i his sud.

24 .. Oriaio Agl A oriaio agl of a fracur is dfid as h dircio of a vcor prpdicular o h fracur pla [Gir al. 99] or as h agl bw h borhol ad h fracur pol [Bachr al. 977; Log 985]. Dvrsrop al. [989] usd a Fishr disribuio for h oriaio agl of fracurs i Sripa mi. Cacas al. [99a] claimd ha h Fishr-vo-Miss ad uiform disribuios appard o b a saisfig sima of h mpirical disribuio of oriaio agl for h highl fracurd grai 5m blow groud a Faa-Augèrs i Frac. Odlig al. [99] usd a Vo Misss disribuio for a sadso bd. Nimi al. [] dscribd h oriaio of fracurs as a Fishr disribuio for h crsalli rock a dph blow ha m. Log al. [98] assumd a ormal disribuio wih ma valus of ad 6 ad variac of 5 ad rspcivl. Hsir al. [99] cosidrd h uiform ad ormal disribuios for h paramr sudis o sima prmabili i fracur work. Kulailak al. [996] claimd ha h availabl horical probabili disribuios wr foud o b isuffici o rprs h saisical disribuio of oriaio of fracurs. I summar h oriaio agl of fracurs for a grai rock a dph blow ha 5m is wll characrizd b h Fishr disribuio [Dvrsrop al. 989; Cacas al. 99a; Nimi al. ]. f Figur. Probabili dsi fucio for oriaio agl I his sud h oriaio of a fracur is dfid as a agl bw h fracur ad h horizoal ais of is ow coordia ssm. I is assumd ha h oriaio of a fracur sgm is disribud uiforml bw ad for simplici as follows. f..8 Figur. shows h probabili dsi fucio.8 for h oriaio agl...4 Locaio Fracur locaio rfrs o h spaial coordias of h cr of a fracur [Gir al. 99]. Log [985] claimd basd o fild daa ha h bs curr sima for h gomr of fracur locaio is radoml disribud i spac. A Poisso procss is applid o gra h cr of fracurs i spac [Dvrsorp al. 989; Cacas al 99a]. Kulailak al. [99] assumd a uiform radom disribuio for locaio of fracurs a h Sripa mi i Swd. fr r Figur.4 Probabili dsi fucio for locaio of h primar fracurs I his sud for h locaio of fracurs i h wo-dimsioal modl spac wo paramrs r mus b drmid. r is dfid as h disac bw h o d of a fracur ad a origi. is dfid as h agl bw h ais ad h li which cocs h o d of a fracur wih h origi. Two ps of fracurs ar 4

25 cosidrd. Th primar fracurs ar dfid as fracurs ha is bfor h disurbac du o rposior cosrucio ad ar assumd o disribu uiforml i h rgio bw ad < rsulig i r f r r..9 Figur.4 shows h probabili dsi fucio.9 for locaio of h primar fracurs. Th scodar fracurs ar dfid as fracurs ha ar addd as a rsul of h rposior cosrucio ad ar assumd o disribu poiall [Pusch al. 99; Gllig al. 997] f r r r.. Figur.5 shows h probabili dsi fucio. for locaio of h scodar fracurs. fr r Figur.5 Probabili dsi fucio for locaio of h scodar fracurs Th agl disribuio boh for h primar ad scodar fracurs is assumd o b uiform..5 Liar Frquc Spacig ad Numbr Dsi f.. Th liar frquc of fracurs is dfid as h umbr of fracurs irscd pr ui lgh of a li sampl i.. a borhol or sca-li [Pris al.976]. Spacig is dfid as h disac bw fracurs alog a li ad ca b obaid as h ivrs of h fracur frquc [Gir al. 99]. Numbr dsi of fracurs A is dfid as h umbr of fracurs pr ui ara A. Pris al. [976] ad Hudso al. [979] amid h disribuio of a umbr of fracurs wih spacig alog a sca-li. Th cocludd ha h valu of h spacig disribus poiall. Bachr al. [977] cocludd b liraur rviw ha h spacig disribus poiall or logormall. Kulailak al. [99] cocludd ha h poial fucio appars o b h mos favord o o rprs h saisical disribuio of h fracur spacig whil i som cass a logormal disribuio is slcd as h bs. Kulailak al. [996] usd h poial ad gamma disribuios for h fracur spacig for crsalli rock mass i Chia. Hudso al. [979] sad ha b a s of sca-li masurms ak i various dircios o a rock surfac h umbr dsi A is foud o b a squar of h liar frquc i.. A.. Howvr a cour ampl for. ca asil b raisd whr. is o valid. Lim al. [b] showd ha h umbr dsi b. ds o ovrsima h ara umbr dsi for a cas wih log fracur sgms whras a cas wih shor fracur sgms udrsimas. I mas ha h lgh disribuio of fracur sgms should b cosidrd for drmiig h ara umbr dsi. Hsir al. [99] givs a rlaioship bw A ad for a wo-dimsioal Poisso work of li fracur sgms wih radom oriaios giv b h probabili dsi fucio f. l A Ecos. whr l [m] is h avrag fracur lgh ad is h agl bw h pols of h fracurs ad h rfrc li. Ecos is h pcd valu of cos ad is dfid as 5

26 / E cos cos f d..4 Sic varis bw ad / h probabili dsi fucio f for h oriaio is B subsiuig.5 io.4 Ecos is calculad as f /..5 / / / E cos cos d cos d..6 B subsiuig.6 io. h rlaioship bw A ad bcoms A..7 l Th umbr dsi of fracurs A ca b calculad b.7 if w kow h liar frquc ad h avrag lgh of h fracur sgm l. Eprimal daa for h liar frquc of h fracurs ar show i h firs colum i Tabl.. Th valus for show variaio origiad from diffr primal ss. Th scod ad hird colums show valus of A calculad b.7 wih h valus of i h firs colum ad h avrag fracur lgh l for.m ad.m rspcivl *. Usig A valus h oal umbr of fracurs i h NF rgio wih ara of - is obaid for l =.m ad.m i h fourh ad fifh colums rspcivl. I his sud h umbr of h scodar fracurs is assumd o b % of ha of h oal umbr of fracurs [Gllig 997]. Tabl. Numbr of Fracurs Calculad wih Eprimal Daa for Frquc. Fracur frquc masurd [/m] Fracur umbr dsi calculad b.7 A [/m ] Toal umbr of fracurs calculad [fracurs] [*] l =.m l =.m l =.m l =.m.~.8.9~..6~6.7 4~56 47~55 Chalk [*] Sadso [*].~6. 5.~4..7~.8 79~5 6~8 Sadso [*] Sadso [*4] Limso [*] 8.7~5.8.7~ ~8. ~67 4~4 Limso [*].8~7. 7.4~58..5~9.4 89~48 94~46 Mudso [*].6~6.9~9.4.~. 7~7 4~7 Grai [*5]. ~..~..~. 4~7 8~79 Grai [*6] ~ 5.7~.4 5.~.5 84~69 95~79 Grai [*7].6~.8.~.8.~.9 7~ ~7 Grai [*8]. ~ 4..~6.6.~. 4~497 8~66 Grai [*9] 5~.5 7.9~5..6~.8 59~665 97~888 Graiic giss [*] ~.6~4.7.5~.6 8~55 9~8 Gradiori [*7].5 ~ 6..4~9.4.8~. 78~7 59~7 Crsalli rocks [*] Crsalli rocks [*] [*] For h ara of h ar fild rock calculad b - wih =5m =m. [*] Pris al [*] Hudso al.979. [*4] Odlig al. 99. [*5] Dvrsrop al [*6] Cacas al. 99a. [*7] Pusch al. 99. [*8] Kulailak al. 99. [*9] Kulailak al [*] Boadu al.994. [*] JNC Vol. p.iii-. [*] Nimi al. ock * l =.m ad l =.m ar slcd as ma fracur lghs i Scio.. 6

27 . Summar I his sud h discr fracur work is grad o drmi war-flow pahs i h NF. Th flowbarig clusr is rasformd io a hrogous coiuum or a homogous coiuum. Th modlig approachs for h NF will b compard i Chapr for h groudwar flow ra ad i Chapr 5 for h paricl raspor. For paricl raspor modlig WTM is slcd bcaus i is suiabl o combi wih a sochasic coiuum Chapr 5. Basd o h liraur surv saisical disribuios of fracur gomr paramrs ar slcd i h prvious scio ad summarizd i Tabl.. Paramrs i Tabl. ar usd for h simulaio of h war flow aalsis i Chapr ad raspor aalsis i Chapr 5. Tabl. Saisics of Fracur Gomr Paramrs Locaio Fracur lgh Aprur Oriaio r Dis L L Dis L L Agl Dis Pri Sc Pri Sc [m] [m] Ui Ep Ui Ui LN. or.. or. LN Ui Dis: Disribuio; LN: Logormal; Ui: Uiform; Ep: Epoial Pri: Primar fracurs; Sc: Scodar fracurs; L : Shap paramr; L : Scal paramr 7

28 . War Flow Aalsis i h Nar Fild of Gologic posior. Iroducio I his chapr a wo-dimsioal modl for h groudwar flow is sablishd for h o-was caisr cofiguraio i h ar fild of a hpohical war-saurad gologic rposior. A ar-fild modl cosiss of a imprmabl was caisr a boi-filld buffr ad a ar-fild rock NF. Th war flow pah i h NF is firs rprsd b a Discr-Fracur Nwork DFN which is grad basd o h disribuio fucios of h fracur gomr paramrs summarizd i Scio.. Th flow-barig rgio i h DFN i h NF is drmid b chckig h ircocivi of h fracurs ad is rasformd io a hrogous coiuum or a homogous coiuum. Th spaial disribuio of h hdraulic poial is obaid b solvig a sad-sa poial quaio umricall b a fii lm mhod. Th objcivs of his chapr ar o dvlop a war flow modl icorporaig h hrogi of h NF wih h o-was caisr cofiguraio o ivsiga ffcs of h hrogi of h NF o h groudwar flow ra ad o ivsiga ffcs of h modlig approach for h NF o h groudwar flow ra. Modl Spac A ar-fild modl for h o-was caisr cofiguraio is dfid i a wo-dimsioal modl spac as show i Figur.. Th our circl wih radius rprss h modl spac boudar. Th middl circl wih radius rprss h irfac bw h boi-filld buffr rgio ad h NF. Th circl wih radius rprss h surfac of h was. Th war-saurad buffr rgio bw ad is assumd o b homogous ad isoropic. Th hdraulic coducivi ad h porosi i h buffr rgio ar assumd o b uiform. Th hdraulic coducivi for h buffr is assumd. -4 m/s [Vrbk al. 997]. Th porosi for h buffr is assumd. [PNC 99; Vrbk al. 997]. Modl spac boudar l b Fracur Was Buffr =.5m =.m Nar-fild rock =5.m Figur. Modl spac Th valus of ad hav b drmid as.5 m. m ad 5. m rspcivl basd o h prvious sudis [PNC 99; JNC Vol. IV-4].. Fracur Nwork Graio A wo-dimsioal discr fracur work is grad i h NF rgio bw ad i Figur.. I h -D modl a fracur sgm is rprsd as a li lm of lgh l ad aprur b as show i Figur.. Discr fracurs ar grad basd o giv saisics of h fracur gomr paramrs i Tabl. i Scio. b radom samplig *. Each fracur sgm is grad idpdl ad h suprimposd uil rquird umbr of fracur sgms is grad i h modl spac bw ad. * Dail abou h radom samplig for a logormal disribuio is giv i Appdi A 8

29 .. Graio of Fracur Sgms Dails abou h graio of fracur sgms ar giv blow. Two radom umbrs r i Figur. for coordias of o d of h fracur sgm ar grad bw ad basd o h disribuio fucios. For h primar fracurs o radom umbr is grad for r basd o h uiform disribuio fucio.9 i h rag bw ad. For h scodar fracurs o radom umbr is grad for r basd o h poial disribuio fucio.. For boh primar ad scodar fracurs o radom umbr for agl is grad basd o h uiform disribuio. masurd agais h -ais. r Figur. Drmiaio of coordias for o d of a fracur sgm O radom umbr for h lgh l of h fracur sgm is grad basd o h logormal disribuio.. For h lgh h diffr valus of h scal L ad shap L paramrs ca b assigd for h primar ad scodar fracurs. Th radom samplig for h logormal disribuio is show i Appdi A. r l ad b Figur. Drmiaio of lgh l ad aprur b Fracur Figur.4 Drmiaio of oriaio 9

30 r l ad b Figur.5 Drmiaio of h locaio r lgh l ad aprur b of a scod fracur sgm Figur.6 a Cas for irscio Possibl wo cass for wo fracurs b Cas for o irscio O radom umbr for h drmiaio of h aprur b of h fracur sgm is grad basd o h logormal disribuio.. Figur. shows possibl fracur sgms wih h assumpio of isoropic fracur for h locaio of o d of h fracur r lgh l ad aprur b. I Figur. ol 6 possibl fracur sgms ar show bcaus of h limiaio of graphical rprsaio bu hr ar a ifii umbr of possibl fracur sgms. 4 O radom umbr for h drmiaio of h oriaio agl masurd from h -ais of h coordia ssm is grad o slc o fracur sgm ou of ifii possibl fracur sgms basd o h uiform disribuio.8. I Figur.4 a hick li wih wo d pois rprss a slcd fracur sgm for h locaio of o d of h fracur r lgh l aprur b ad oriaio. Thi lis rprs possibl fracur sgms. Each fracur sgm is grad idpdl b procdurs o 4 ad h suprimposd i h modl spac. Figur.5 shows h locaio lgh ad aprur of a scod fracur sgm ha is suprimposd i h modl spac. Thr ar wo possibl cass dpdig o whhr wo fracur sgms irsc or o. Figur.6 implis ha isorop of fracur oriaio is a impora assumpio for drmiig irscio of fracurs. If fracurs ar orid aisoropicall spciall if orid i h sam dircio h possibili for fracurs o irsc is sigifical dcrasd s Appdi B. I his sud fracur irscio is crucial bcaus ol ircocd fracurs form a pah for war flow.

31 Figur.7 Fracur work graio Tabl. Saisics of Fracur Gomr Paramrs Usd o Gra Figur.7 Numbr of Fracur lgh Aprur Oriaio fracurs Ma SD Dis L L Ma SD Dis L L Agl Dis [m] [m] [m] [m] [m] [m] LN LN Uiform Dis: Disribuio; LN: Logormal; SD: Sadard Dviaio; L : Shap paramr; L : Scal paramr Figur.7 shows a ralizaio of fracur work usig paramrs show i Tabl.. Th valus of h ma ad h sadard dviaio boh for h lgh ad h aprur ar obaid b. ad.5 wih h scal paramr L ad h shap paramr L giv i Tabl.. Th disribuios of h lgh aprur ad oriaio of h scodar fracurs ar assumd o b sam as hos for h primar fracurs. For ach locaio of h fracurs h uiform disribuio.9 ad h poial disribuio. ar assumd for h primar ad scodar fracurs rspcivl. Figur.8 a b c Thr ralizaios of fracur work b h sam saisics. Figur.8 shows hr ralizaios of h fracur work basd o paramrs i Tabl.. Thr ralizaios a b ad c show diffr fracur works producd b diffr sds bcaus of radom samplig of fracur paramrs basd o h disribuio fucios. A subroui dvlopd b Prss al. [99] is usd as a radom umbr graor. A diffr sd will rsul i a diffr radom umbr squc. Sic h sam sd will alwas rur h sam radom squc a fracur work ca b r-producd b assigig h sam sd i h subroui.

32 .. Flow-Barig Clusr FBC of Fracurs Thr ar wo ps of fracur clusrs i a fracur work. I h firs p fracur clusrs coc h our boudar r= of h modl spac wih h irfac bw h buffr ad h NF r=. W call his p as a Flow-Barig Clusr FBC * of fracurs. Clusrs of h scod p ar hos ohr ha h FBC calld Dad-Ed Clusrs DECs of fracurs. Figur.9 shows a ampl of fracur clusrs. Fracurs icludd i h FBC ar rprsd b hick lis whras fracurs icludd i h DECs ar rprsd b hi lis. No ha h clusr dsigad as B i Figur.9 is also icludd i h FBC bcaus a war flow pah from h our boudar iss hrough h buffr rgio. I his sud i is assumd ha h rock mari is imprmabl ad war ca b supplid from h our boudar of h modl spac o h EBS ol hrough h FBC ad h buffr rgio. Our boudar Buffr B Irfac Figur.9 Fracur clusrs. Thick lis rprs h FBC of h DFN whras h hi lis rprs h DEC i h DFN... Idificaio of FBC Afr h graio of a fracur work Figur.7 a FBC is idifid b chckig h ircocivi of all fracurs. Irscio of fracurs is dfid as: wo fracur sgms cross ach ohr wih o commo irscio poi o d of h fracur sgm li o aohr sgm wo fracur sgms ovrlap parl. Figur. a shows h cass for irscio ad b shows h cass for o irscio. I Figur. b dashd lis rprs h sios of h fracur sgms. d a Irscio b No irscio Figur. a c b a d c d b Irscio bw wo fracur sgms. I a irscio is coud oc. c a b a b c d c b d * Hr i is implicil assumd ha war flows io ad ou of h modl spac hrough hos fracur sgms irscig h our boudar. Fracur sgms which irsc wih hs ar also assumd o bar war flow.

33 Numrical schms dvlopd b O ourk [99] ar usd i his sud for drmiig whhr wo fii lis irsc or o. Ipu paramrs for O ourk s schms ar coordias of 4 d pois of wo lis. Irfac Figur. Our boudar A ampl for h idificaio of h FBC i a fracur work. To illusra how o idif h FBC a ampl is giv i Figur. whr fracurs ar icludd. Th our boudar is h ourmos circl ad h irfac is h middl circl. Th sps for idifig h FBC ar as follows. Idif fracurs irscig wih h our boudar fracurs ad 9 i Figur.. Idif fracurs irscig wih h irfac fracurs ad i Figur.. Slc o amog h fracurs idifid i procdur. Chck all ohr fracurs whhr h irsc wih h slcd fracur or o. If irscio occurs fi h irscig fracur ad chck all ohr fracurs cp for hos wo fracurs. pa his procdur uil hr is o mor irscig fracurs. 4 pa for all fracurs irscig h our boudar idifid b sp. Afr a las o clusr of ircocd fracurs is idifid sp is rpad for h fracurs idifid b sp o iclud fracurs irscig h irfac. Irfac Figur. FBC for h fracur work i Figur. Our boudar To idif a FBC i Figur. l us sar wih fracur ad chck h irscio wih ohr fracurs. Ol fracur irscs wih fracur. Tak fracur ad chck h irscio wih ohr fracurs cp for fracurs ad. Ol fracur 4 irscs wih fracur. Fracur 4 irscs wih h irfac ad hr ar o mor irscig fracurs wih 4. Thrfor fracurs ad 4 ar ircocd ach ohr ad ar pars of FBC. pa his procdur for ohr fracurs idifid i sp. I his cas fracurs ad 9. For fracur hr is o irscig fracur wih fracur. Also fracur dos o irsc wih h irfac. Thus fracur is isolad ad is par of DEC. For fracur 9 ol fracur irscs wih fracur 9 ad fracur irscs wih h irfac. Thus fracurs 9 ad ar ircocd ad ar pars of FBC. Afr a las o clusr of ircocd fracurs is idifid sp is rpad for h fracurs idifid b sp o iclud fracurs irscig h irfac. Thrfor h FBC cosiss of fracurs ad. Figur. shows h FBC. Thick lis rprs fracurs icludd i h FBC.

34 Th FBC for h fracur work i Figur.7 is show i Figur.. Figur. FBC for h fracur work i Figur.7..4 Trasformaio of FBC io Equival Coiuum Th FBC idifid b ispcig h ircocivi of h fracurs i h NF is rasformd io a quival coiuum i wo diffr was. Firs a rasformaio is applid locall o h FBC which is calld h local homogizaio i his sud. B h local homogizaio h FBC cosisig of h discr fracurs is rasformd io a hrogous coiuum. Thus h NF is hrogous ad is dividd b h flow-barig rgio i.. hrogous coiuum rasformd from h FBC ad h war saga rgio. I a scod approach a global rasformaio is applid o h FBC o covr h ir NF which is calld h uiformizaio i his sud. B h uiformizaio h NF is rasformd io a homogous coiuum. Thus h ir NF is homogous ad is a flow-barig rgio..4. Trasformaio of FBC io a Hrogous Coiuum I his scio a mhod of rasformaio of h FBC io a hrogous coiuum b h local homogizaio is dvlopd. Firs h FBC is covrd b ma riagular lms. Th ir-cocd fracur sgms icludd i h FBC i a small riagular ara ar rasformd o a homogous isoropic porous mdium. Sigl valus of h porosi Scio.4.. ad hdraulic coducivi Scio.4.. ar obaid for ha ara. Bcaus riagular lms coai diffr umbrs of fracurs wih varig lgh ad aprur ach lm has diffr porosi ad hdraulic coducivi. Th hrogi of h FBC sill rmais afr h rasformaio. Thus h FBC is rasformd io a hrogous coiuum cosisig of riagular lms which hav diffr hdraulic propris. B h local homogizaio of h FBC o obais o ol h small scal variabili of h hdraulic coducivi ad h porosi isid h FBC bu also iformaio abou whr localizd flow chals is i h FBC. All his iformaio is icorporad ad uilizd i h flow aalsis..4.. Equival porosi Th porosi is dfid as h raio of h por volum o h oal volum por + solid. To formula h quival porosi for a riagular lm a ampl is cosidrd i Figur.4. Th figur shows four fracur sgms wih diffr lghs ad aprurs i a riagular lm. I his ampl fracur 4 is o icludd i h calculaio of h porosi bcaus i is o icludd i h FBC. Ol fracurs ad ar icludd i h FBC. Ol h par of h fracur wihi h lm is cosidrd. is h ara of h lm. l l ad l ar fracur lghs icludd i h lm. b b ad b ar fracur aprurs. Wih h assumpio ha h lm has a ui dph ormal o h pla of h quival porosi for lm ca b wri as l b l b l b.. 4

35 Ara b l l b 4 b l Figur.4 Calculaio of a quival porosi for a riagular lm. Th gralizd formula for h porosi for lm is N i l ib i. whr N dos h umbr of fracurs icludd i h FBC which pass hrough h lm..4.. Equival hdraulic coducivi Th hdraulic coducivi K [m/s] is a coffici ha dpds o boh mdium ad fluid propris. Th rlva fluid propris ar h dsi ad h viscosi w. Th rlva mdium propr is h prmabili k [m ]. For h quival prmabili k for lm h Koz-Carma formula [Bar 979] k C. M s is usd i his sud. Th formula icluds h quival porosi of h lm for h flow of war hrough h pors ad h spcific surfac ara of h lm M s which is dfid as h por surfac ara w b h flowig war pr ui volum of h porous mdium [Bar 979]. Coffici C is a fucio of h shap of cross-scioal ara ormal o flow ad h oruosi [Carma 956]. Th valu of coffici C is lss ha ui ad is a spacdpd fucio bcaus oruosi * porosi ad h spcific surfac ara var wihi h FBC rgio. I his chapr howvr i is assumd ha C=. Th coffici C will b discussd i Chapr 6 i dail. M s is calculad b h raio of h oal surfac ara of irscig fracurs i lm o h volum of solid i lm. N d l i d i M s N l i b i b i.4 [ l b ] d i i i whr d is a arbirar dph ormal o h lm ad h fracur as show i Figur.5. If aprur b i is much smallr ha l i or d ad <<.4 ca b simplifid as M s N l i i - [m ]..5 Th quival prmabili k for lm ca b calculad b. wih h quival porosi b. ad h spcific surfac ara M s b.5. Th quival hdraulic coducivi K for lm is prssd as * Dail abou oruosi is giv i Chapr 6. 5

36 K k g.6 whr is h dsi of war g is h cosa for gravi ad w is h viscosi of war. Ara w l d b l b l b Figur.5 Cocps of h spcific surfac ara.4. Trasformaio of h FBC io a Homogous Coiuum I his scio a mhod of rasformaio of h FBC io a homogous coiuum b h uiformizaio is dvlopd. B h uiformizaio h ir NF is rasformd io a homogous coiuum ad is a flow-barig rgio. Th uiformizaio givs a sigl valu of hdraulic propris for h ir NF whras h local homogizaio show i h prvious scio givs as ma diffr valus of hdraulic propris as h umbr of lms. Th procdur for h uiformizaio cosiss of h followig hr sps. A sigl valu of uiformizd porosi for h ir NF is calculad b N i l b i i A.7 whr N is h umbr of fracurs icludd i h FBC ad A is h ara of h ir NF rgio giv b A. A sigl valu of h uiformizd spcific surfac ara M s for h ir NF is calculad for b i is much smallr ha l i or d ad A as M s N l i i - [m ]. A.8 B usig obaid b.7 ad M obaid b.8 a sigl valu of uiformizd prmabili k s for h ir NF is calculad b Koz-Carma quaio k C..9 M s whr C is assumd o b. Wih.9 h uiformizd hdraulic coducivi K for h ir NF is calculad as K kg. w whr is h dsi of war g is h graviaioal acclraio ad w is h viscosi of war. 6

37 .5 Govrig Equaio ad Sid Codiios Figur.6 shows h flowig-barig domai which covrs h FBC ad h buffr rgio. Th govrig quaio of war flow i domai ca b obaid b h cosrvaio of mass [Bar 979] div q q q q i. whr [kg/m ] is h dsi of war ad q ad q ar Darc vloci compos [m/s] i ad dircios rspcivl. is h porosi. =.5[m] Flu= C =[m] - =[m] Irfac C C - =.5[m] Figur.6 War-flowig rgio i.. FBC ad buffr i h modl I h rcagular coordia ssm. ca b wri as For h im-idpd cas q q i.... For h icomprssibl cas is cosa. Wih h cosa ad. h cosrvaio quaio. bcoms q q..4 For a sad flow i a aisoropic hrogous mdium if pricipal dircios ar usd as h coordia ssm Darc s law.5 ad.6 applis [Bar 979]. q q K.5 K.6 whr is h poial had [m] ad K ad K ar h hdraulic coducivi [m/s] for wo pricipal dircios. Subsiuig Equaios.5 ad.6 io.4 ilds h govrig quaio for h sad-sa flow i a hrogous aisoropic mdium as 7

38 K K..7 Th poial had is o b foud i a wo-dimsioal soluio domai show i Figur.6. Two diffr kids of boudar codiios ar cosidrd. Alog h C boudar hik lis o h our boudar r= of h modl spac h poial had is prscribd b whr / [ m] o C.8. m. Th Darc vloci ormal o h C boudar hick li o h surfac of h mal caisr r= is prscribd zro or K K o C.9 whr ad ar h dircio cosis of h ouward ormal vcor. For h boudar of i h NF cp for C boudar as show i Figur.6 h poial had is ukow ad flu is prscribd as zro..5. Soluio for Govrig Equaio b Fii Elm Mhod To solv h govrig quaio.7 h fii lm mhod is applid. Triagular lms * ar usd o discriz h domai. Th mhod of wighd rsiduals is usd for obaiig a approima soluio. Th mhod of wighd rsiduals is summarizd accordig o h rfrc [Hubr 975] hrafr. If a approimad fild variabl is subsiud h lf-had sid of h govrig quaio.7 is appro o qual o zro. This o-zro valu is calld rsidual. Thus w hav appro appro K K i.. I h mhod of wighd rsiduals appro is drmid i such a wa ha h wighd avrag of h rsidual ovr h ir domai bcoms qual o zro or W i d i m. whr W i is a wigh fucio. Th domai is discrizd io m ods. B subsiuig. io. appro appro [ K ] [ K ] Wi d i m.. Th subscrip appro is droppd. is a approima soluio of h poial had ad is rprsd b m Wi i i i m. whr i is a ukow odal poial had a od i. Sic hr is o amp o choos W i so as o saisf h boudar codiios igraio b pars is do ad ilds * Ths lms ar h sam as hos cosidrd i Scio.4. for rasformaio s Figur.4 8

39 Wi [ K Wi K ] dd C [ K K ] Wi ds.4 whr variabl s is h arc lgh of h our boudar. Th righ-had sid igral rprss h wighd igral of h Darc ormal vloci o C. =.5[m] Flu= C =[m] - =[m] Irfac C C - =.5[m] Figur.7 Triagular lm i h domai. Equaio. is for h ir domai show shadd i Figur.7. Th govrig quaio.7 mus b saisfid a a poi i h domai. Thrfor.4 is valid for a collcio of pois dfiig riagular lms i h whol domai [Hubr 975]. I Figur.7 a arbirar riagular lm is show wih ods ad. Domai rprss h rgio wihi h riagular lm. B applig.4 o lm Wi [ K Wi K ] dd C [ K K ] Wds i.5 whr h id i idicas h od icludd i lm. Th righ-had sid of.5 has a o-zro valu ol if lm forms par of h C boudar. Noic ha i his cas K ad K ar h hdraulic coduciviis [m/s] i i h pricipal dircios. Followig. i.5 is wri as W i i i. i.6 Th locaio of a poi wihi h lm or o h lm s boudaris ca b dscribd as h aural coordias L L ad L. Ths coordias rag bw zro ad ui wihi h lm [Hubr 975]. O paricular coordia has a ui valu a o od of h lm ad zro valu a h ohr ods. Variaio bw ods is assumd liar. Th origial Carsia coordias of a poi i h lm ar liarl rlad o h aural coordias b h followig quaios [Hubr 975]: L.7 L L L..8 L L Hr ad ar h coordias of ods ad i lm s Figur.8. Th hird codiio rquirs h sum of h aural coordias o b ui. L L L..9 9

40 p p Figur.8 Coordias of hr ods i a riagular lm b global coordia ssm. Th soluios of.7 ad.8 ild h aural coordias i rms of h Carsia coordias. L. L. L. whr ara of riagl- -. ad for i ; j ; k..4 i j k k j i j k i k j Subscrip i.~.4 dos lm. W i is drmid such ha W i =L i. Th approima poial had i lm ca b valuad b.6 as L..5 L L Subsiuig.. ad. io.5 ad diffriaig i wih rspc o ad ilds [ ]{ }ad [ ]{ }.6 whr [ ]=[ ] [ ]=[ ] { }.7 ad [ ] {} ad [ ] {} ar scalar produc. Diffriaig.. ad. wih rspc o ad ilds L i i L i ad i..8 Subsiuig.6 ad.8 io h lf-had sid of.5 ad oig ha W i =L i ilds

41 i[ ] i[ ] dd[ K K ]{ } 4 4 C [ K whr i [ ] ad i [ ] ar mari for h riagular lm. K ] Lds i i.9 Whil K ad K ca var wihi h lm wo assumpios ar iroducd for simplici. Firs a isoropic mdium is assumd for h ir domai. Th valus of h hdraulic coducivi for h pricipal dircios ar idical K K K [Bar 979]. Scod b homogizaio i ach riagular lm Scio.4. h hdraulic coducivi of lm is assumd o b a cosa vrwhr i lm i.. K K i. Th sam assumpios ar mad for h porosi i lm i.. i. Homogizd hdraulic propris ca b diffr from lm o lm i domai. Thus hrogi of domai sill rmais. Bcaus dd quaio.9 for h lm is wri i a mari form as [ ]{ } { Q }.4 whr K [ ] 4 ad.4 { Q } K [ ] L ds i..4 i C Equaios.4.4 ad.4 appl o h lm. To formula h problm for h whol domai icludig h FBC ad h boi-filld buffr rgio is discrizd io riagular lms as show i Figur.9 whr a magifid riagular lm wih hr ods ad is also show. If ods ad i h aural coordia ssm corrspod o ad i h global umbrig s Appdi C h fiilm soluio for all lms wih h oal umbr of ods N N ad h oal umbr of lms N E i h domai is wri as [ ]{ } { Q}.4 whr N E [ ] [ ]..44 If i= or ad j= or h ar o zro. Ohrwis i j. Thus N N N N mari [ ] is wri as i j [ ].45 B.4

42 4 ] [ K.46 whr } { } { } { } { N N N Q Q Q Q Q Q Q Q E N N i Q i L ds K Q i C i i.5 A ampl for assmblig global mari.4 is giv i Appdi C. No flow rgio C C No flow rgio C C Figur.9 Triagular lms covrig h FBC ad buffr rgio.

43 .5. Eampl for suls of War Flow Aalsis Afr idifig a FBC Figur. from a fracur work Figur.7 h FBC i h NF is rasformd io a hrogous coiuum b h local homogizaio Scio.4. ad a homogous coiuum b h uiformizaio Scio.4.. For h local homogizaio riagular fii-lm mshs ar grad ovr h FBC ad h buffr rgio as show i Figur.. Hdraulic propris ar obaid for ach riagular lm. Figur. Triagular msh graio o h FBC Figur. ad h buffr rgio. A ar-fild modl wih a hrogous NF b h local homogizaio is dscribd i his scio. Wih obaid hdraulic propris ad h prscribd boudar codiios.8 ad.9 h valus of poial had a iral ods i h FBC ad a ods i h buffr icludig h irfac r= ad h ir boudar C r= ar calculad b solvig h mari quaio.4 umricall. A ods o h our boudar C whr h poial had is prscribd h corrspodig compo of h vcor {Q} o h righ sid of.4 is calculad. Th compo Q i [m /mr] of h vcor phsicall mas h volum of war flowig across h our boudar associad wih od i ormal o h boudar pr ui im hrough ui dph ormal o - pla. Posiiv Q i mas ha h dircio of flow a boudar od i is ouward whras gaiv mas iward. To solv h mari quaio.4 h cod dvlopd b Hubr [975] is usd. Upo covrsio from FOTAN o C som pars of h cod hav b modifid. Wih h obaid umrical rsul for h poial had a ods h poial had ca b umricall valuad a a locaio i h FBC ad h buffr b.. Wih hs umrical rsuls obaid b solvig h mari quaio.4 four quaiis of irs ca b calculad: h oal discharg a h our boudar h oal discharg a h irfac h rsidc im of war i h buffr rgio ad 4 h spaial disribuio of Darc vloci i h FBC. Th oal discharg Q [m /r] dfid as h volumric flow ra pr ui dph ormal o - pla a h our boudar r= is calculad b summig posiiv valus of Q i o h our boudar. B h mass balac assumd for h sablishm of h govrig quaio.7 h sum of gaiv valus of Q i should hav h sam magiud wih h sum of h posiiv valus. This has b cofirmd i h umrical ampls show blow. I h prs sud bcaus a sad sa is assumd oal dischargs io ad ou of h modl spac a h our boudar should b qual o ach ohr b mass balac. Th valus 996 kg/m for ad Ns/m for w ar usd for.6 ad. b assumig ha h ssm mpraur is C. 9.8 m/s for g is usd i.6 ad.. I ordr o dmosra ha umrical rsuls saisf h mass balac rquirm h FBC i Figur. is usd as a ampl. Th oal dischargs io ad ou of h modl spac ar calculad as.69 m /r for h FBC i Figur.. Th diffrc bw wo quaiis is of h ordr of - m /r which is ordrs of magiud smallr ha.69. For umrical rsuls show i his dissraio i has b cofirmd ha mass balac is acuall kp wih a similar ordr of magiud rror.

44 Similar o h oal discharg a h our boudar h oal discharg a h irfac r= is calculad b summig posiiv valus of Q i o h irfac. Q i a h ods o h irfac ar calculad as follows. Bcaus h valus of h poial had a h irfac r= ar alrad obaid for h calculaio of Q i a h our boudar a similar mari quaio ca b sablishd for h buffr rgio wih h hdraulic propris i h buffr rgio ad h prscribd o flu boudar codiio a h surfac of h was caisr r =. I h sablishd mari quaio h righ-sid vcor {Q} coais ukow compos which ar a h ods o h irfac. B solvig his mari quaio umricall h ukow Q i a h ods o h irfac ar calculad Figur. Darc vloci for h FBC i Figur.. Th rsidc im of war i h rgio of irs ca b calculad b dividig h war volum [m /m] of h rgio pr ui dph b h oal discharg [m /r] io ad ou of h rgio. B.6 h gradi of h poial had i h -dircio ad i h -dircio ar obaid umricall for lm. Wih h hdraulic coducivi K obaid b.6 ad / ad / b.6 for lm h Darc vloci compos q ad q i ad dircios for lm ar umricall calculad b.5 ad.6 rspcivl. Th Darc vloci has a ui of [m of war/m of mdium im] ad is dfid as h volum of war flowig pr ui im hrough a ui cross-scioal ara ormal o h dircio of flow. Figur. shows a umrical rsul for Darc vloci disribuio obaid for h FBC i Figur.. I his figur Darc vloci i h buffr rgio cao b idifid du o vr small magiud compard o ha i h FBC i.. abou 9 ordr diffrc. A arrow sars a h cr of wigh of ach lm. Th lgh of a arrow idicas h magiud of Darc vloci whil h dircio of h arrow idicas h flow dircio a ha poi. Th magiud of h Darc vloci is obaid b q q for lm. Th flow dircio a h cr q of wigh for lm is obaid b agl f a. For q > h flow dircio is f masurd agais q h posiiv ais whras h flow dircio is f masurd agais h gaiv ais for q <. Valu of agl icrass cour-clockwis from h ais. To obai h saisics of oal dischargs calculaios should b rpad for ma ralizaios of fracur works. Calculaio im dpds o h umbr of ralizaios h umbr of fracurs h umbr of fii lms ad h usr s compuig ssm. A compur cod FFDF Fii-lm Flow ad raspor cod for Discr-Fracur works has b dvlopd for h procsss dscribd i Scio. Scio.4 ad Scio.5. Th cod also icluds saisical rouis for ma ralizaios. Th faurs of FFDF ar dscribd i Appdi D. 4

45 .5. Fii-Elm Msh Siz Th siz of fii-lm msh i.. riagular lms should b drmid b cosidrig h umrical rrors du o fii lm approimaio ad h siz of h rprsaiv lmar volum EV [Bar 97] for h quival porous mdium. From h viwpoi umrical rrors ca b dimiishd b dcrasig h siz of lms or b icrasig h umbr of lms [Hubr 975]. From h viwpoi lms mus hav sufficil larg volums o dfi h EV. Two viwpois ad giv opposi ffcs o umrical rsuls. A umrical prim has b prformd o drmi propr siz of lms b cosidrig wo viwpois. Four diffr sizs of mshs ar cosidrd. Figur. shows h domai discrizaio b four diffr sizs of mshs for h sam fracur work i Figur.. a Cas b Cas c Cas d Cas 4 Figur. Domai discrizaio b four diffr sizs of msh. 5

46 Cas Tabl. Cas Msh Siz Effc o War Flow Calculaio Numbr of ods* Numbr of lms* Toal discharg a h our boudar [m /r] Normalizd CPU im wih sc 68.65E E E E+ 7.5 *: Valus for h NF ad h buffr rgios Th valus of oal discharg a h our boudar ar calculad wih four diffr sizs of fii-lm mshs for h sam fracur work. Numrical rsuls ar summarizd i Tabl.. A prsoal compur wih Pium III-5MHz ad 8M-mmor is usd o masur CPU im. Th msh siz usd i cas i Tabl. Figur. c ca b slcd bcaus h CPU im icrass sigifical bw cass ad 4 whras rlaivl small diffrc i h umrical rsul for h oal discharg a h our boudar is obsrvd. Amog four cass cas i Tabl. is h bs choic wih rspc o h viwpoi. Now h viwpoi is cosidrd for h cas. A hpohsis giv b Mahro [967] is adopd i his sud. Mahro [967] claimd ha if global hdraulic coducivi K global lis bw arihmic ma K a ad gomric ma K g of h hdraulic coduciviis masurd i som small scal i.. K K K.5 g global h h mdium of h small scal ca b rad as a quival coiuum. Daga [986] ad Cacas al. [99a] usd his hpohsis o drmi h propr siz of masurms for hir modls. For cas i Tabl. K global for h NF is obaid as.7-5 m/s b. applig h uiformizaio Scio.4.. K g = m/s ad K a = m/s for h NF ar also calculad from h valus of h hdraulic coducivi for all riagular lms icludd i h FBC which ar obaid b.6 for ach lm applig h local homogizaio Scio.4.. Th rsuls show ha cas i Tabl. saisf h iquali.5. Thus riagular lms usd i cas i Tabl. ca b rad as a quival coiuum ad will b usd for war flow aalsis i his chapr ad paricl raspor aalsis i Chapr 5..6 Numrical suls I his scio umrical ploraios ar mad b FFDF o ivsiga: fracur work graio ad occurrc of FBC ffcs of h hrogi of NF o war flow ra ad ffcs of h modlig approach for NF o war flow ra..6. Fracur Nwork Graio ad Occurrc of FBC Thr diffr ss of saisics of h fracur gomr paramrs ar cosidrd for graio of fracur works i h NF as show i Tabl. basd o Tabl.. Th spaial disribuios giv i Tabl. ar usd for h locaio of fracurs i.. r i Figur.. Th disribuios of h lgh aprur ad oriaio of h scodar fracurs ar assumd o b sam as hos for h primar fracurs. Th umbr of h scodar fracurs is assumd o b % of ha of h oal umbr of fracurs [Gllig 997]. Tabl. Saisics of h Fracur Gomr Paramrs for Thr Curvs i Figur.5. Fracur lgh l Aprur b Oriaio Ma [m] SD [m] Dis L [m] L Ma [m] SD [m] Dis L [m] L Agl Dis.65.6 LN LN Uiform.. LN LN Uiform.. LN LN Uiform Dis: Disribuio; LN: Logormal; SD: Sadard dviaio; L : Shap paramr; L : Scal paramr a 6

47 Cocivi Cocivi Th cocivi is usd as a masur o characriz h fracurd mdia i his sud. Th cocivi is dfid as h raio of h oal umbr of irscios o h oal umbr of fracurs i his sud. For wo irscd fracurs h cocivi is calculad as.5 i his sud sic h umbr of irscios is coud as ad h umbr of fracurs is I : N f =5 I II II : N f =6 III III : N f =56 II III I Figur Numbr of fracurs a 5 5 Numbr of fracurs Th rlaioship bw h cocivi ad h umbr of fracurs for hr ralizaios I II ad III. Saisics for Cas i Tabl. is usd. b I Figur. h rlaioship bw h cocivi ad h umbr of fracurs is illusrad for h saisics of fracur paramrs of cas i Tabl.. Thr diffr ralizaios I II ad III ar show i Figur.. N f rprss h umbr of fracurs a which h firs FBC occurs. Figur. a ad b show h rlaioship for wo diffr rags of h umbr of fracurs from o ad from o rspcivl. Whvr a fracur is addd i h modl spac h cocivi is calculad ad prcolaio i.. occurrc of h firs FBC is chckd. I is show i Figur. ha h umbr of fracurs for h firs FBC o occur is qui diffr from ralizaio o ralizaio. I ralizaio I h firs irscio occurs wh h fifh fracur is addd. Thus h cocivi is zro uil h 5h fracur is addd. I icrass o. = irscio/5 fracurs a h addiio of h 5h fracur. Th cocivi bw h 6h ad 8h fracurs dcrass bcaus h addd fracurs do o irsc ohrs. Th cocivi icrass o 4/9 a h addiio of h 9h fracur bcaus i irscs wih iig fracurs. For hr ralizaios h firs FBC occurs a h 5h 6h ad 56h fracurs rspcivl ad h corrspodig valus of cocivi ar..88 ad.7 rspcivl. Th cocivi a which h FBC occurs for h firs im is calld h hrshold cocivi i his sud. Figur.4 a b ad c show h frquc of h hrshold cocivi for cass ad i Tabl. rspcivl. 5 ralizaios hav b mad for ach cas show i Tabl.. I Figur.4 a h hrshold cocivi is obsrvd o b zro for 9 ou of 5 ralizaios. This mas ha h prcolaio occurs wihou irscio of fracurs. I his zro hrshold cocivi cas ol o fracur cocs h our boudar wih h irfac wihou a irscio wih ohr fracurs. Cumulaiv disribuio fucios CDFs of h hrshold cocivi for hr cass i Tabl. ar show i Figur.5. Cas is characrizd as shor fracurs wih a larg sadard dviaio. Log fracurs ca b grad du o h larg sadard dviaio. For cas log fracurs wih h small sadard dviaio ar assumd. So fracurs hav lgh clos o h ma lgh. For cas shor fracurs wih h small sadard dviaio ar assumd. Fracurs ar shor ad clos o ma lgh du o h small sadard dviaio. Du o h 9 ralizaios for which h hrshold cocivi was obsrvd o b zro i Figur.4 a CDF of cas is.8 a h zro cocivi. A h hrshold cocivi of. CDF bcoms ui. I mas ha FBC mus is if h cocivi is grar ha. for 5 ralizaios. For cas bcaus log fracurs ~. m ar grad prcolaio occurs a smallr cocivi ha for cas whr shor fracurs ar grad. CDF rachs ui a cocivi of.7 ad. for cass ad rspcivl. 7

48 CDF Frquc Frquc Frquc 8 Cas 8 Cas 8 Cas Thrshold cocivi Thrshold cocivi a b c Figur.4 Frquc for h hrshold cocivi for hr cass i Tabl Thrshold cocivi.8 Cas Cocivi.7 Cocivi..6 Cocivi..4 Cas. Cas Thrshold cocivi Figur.5 CDF for FBC occurrc. Labls ad idica h cass show i Tabl..6. Effcs of Hrogi of NF o Flow a Ni cass show i Tabl.4 hav b slcd o rprs h hrogi of NF basd o h CDF of h hrshold cocivi i Figur.5. Thr cass ar s for ach curv i Figur.5. For ampl cass A B ad C ar drmid b h curv i Figur.5. Th valus of cocivi ar s as.. ad 6. rspcivl. Bcaus a h cocivi of. h probabili ha h FBC occurs is i cass A B ad C a FBC will b grad. For o ralizaio fracurs ar grad basd o saisical disribuios of h fracur gomr paramrs. Whvr a fracur is addd i h modl spac h cocivi is calculad b FFDF cod. Fracurs ar addd uil h prdrmid cocivi i... for cas A is rachd. Th oal dischargs Q [m /r] a h our boudar r= ad a h irfac r= ar calculad b FFDF cod. Tabl.4 Saisics of h Fracur Gomr Paramrs for 9 cass Cas i Lgh l Oriaio Tabl. Cas Cocivi Ma SD L L Dis Agl Dis [m] [m] [m] A. B C 6. D.7 E..... LN - Uiform F 6. G. H..... I 6. Dis: Disribuio; LN: Logormal; SD: Sadard dviaio 8

49 CDF CDF CDF.6.. Drmiaio of h umbr of ralizaios Bcaus a saisical approach has b applid umrical rsuls should b prssd o b a sigl umbr bu b a rag of valus. I ordr o drmi h umbr of ralizaios for saisicall maigful rsuls h oal dischargs a h our boudar ad a h irfac hav b calculad for fiv diffr umbrs of ralizaios N r = 5 5 ad. Tabl.5 Saisical suls of Toal Dischargs a h Our Boudar ad a h Irfac for a Modl wih a Hrogous NF for Diffr Numbr of alizaios for Cas E i Tabl.4. Toal discharg a h our boudar [m /r] Toal discharg a h irfac [m /r] N r Ma Sadard dviaio Ma Sadard dviaio 7.5E+.E+.5E-7.6E E+ 9.8E+ 4.88E-7.8E-7.88E+.8E+ 4.9E-7.9E E+ 9.97E+ 4.9E-7.E E+ 5.59E+ 4.E-7.99E-7 Tabl.5 shows h oal dischargs a h our boudar ad a h irfac for fiv diffr umbrs of ralizaios for cas E i Tabl.4. Th diffrcs of h oal dischargs a h our boudar ad a h irfac for 5 diffr umbrs of ralizaios ar o sigifica. I his sud h umbr of ralizaios for h war flow ad paricl raspor aalss is slcd as..6.. Effc of Lgh disribuio To ivsiga ffcs of h hrogi of h NF war flow aalsis has b prformd for ralizaios for slcd 9 cass show i Tabl.4 for a modl wih a hrogous NF b h local homogizaio. I his scio aprur of a fracur sgm is grad basd o h giv logormal disribuio as show i Tabl.. Figur.6 a b ad c show h CDFs of h oal discharg a h our boudar for cass A B ad C cass D E ad F ad cass G H ad I rspcivl. Figur.7 shows a grad work FBC riagular msh graio ovr h FBC ad buffr ad Darc vloci disribuio for a ralizaio which givs h mos probabl oal discharg of war idicad b dos i Figur.6 for ach cas A B C 4 Q [m /r] Figur D E F 4 Q [m /r] G H I 4 Q [m /r] a Cass A B ad C b Cass D E ad F c Cass G H ad I CDF of oal discharg a h our boudar for a modl wih a hrogous NF b h local homogizaio for ralizaios for 9 cass i Tabl.4. For cas A i Figur.6 a h oal discharg a h our boudar rags widl from 6.5 o.97 [m /r] bcaus of irrgular shap of FBC du o h larg sadard dviaio of h fracur lgh. As show i Figur.7 h work is spars bcaus h cocivi is as small as.. B icrasig h cocivi o. Cas B ad o 6. Cas C wih h sam fracur lgh disribuio h work bcoms br cocd ad covrs a grar ara of h modl spac. As show i Figur.6 a h oal dischargs for cas B ad C ar disribud mor arrowl ha ha of cas A. Figur.6 a also shows ha h work wih a grar cocivi has a grar oal discharg bcaus a largr amou of war ca flow hrough a mdium wih mor fracurs. Sam rds ar obsrvd for ohr cass i Figur.6 b ad c. L us compar cass C F ad I wih h cocivi 6. i Figur.6. Du o h larg sadard dviaio of h lgh disribuio for cas C h smallr umbr of fracurs is rquird o obai h cocivi of 6.. Toal dischargs a h our boudar for cas C ar smallr ha hos of cas I bcaus h work for cas I

50 coais mor fracurs ha hos of cas C. Du o h small sadard dviaio of h lgh disribuio for cas I h work cosiss of shor fracurs of arl qual lgh. Th oal discharg a h our boudar for cas I is disribud mor arrowl bw. ad. m /r. This is cofirmd b h spr slop for cas I ha ha for cas C i Figur.6 a ad c. Du o h small sadard dviaio or shap facor of h lgh disribuios boh for cass F ad I h works cosis of arl qual lgh fracurs. Mor fracurs ar rquird o obai h cocivi of 6. for cas I ha cas F du o a smallr ma lgh for cas I. As a rsul oal dischargs for cas I ar largr ha ha of cas F du o h fac ha h work coais mor fracurs for cas I ha for cas F. Th oal dischargs a h our boudar for cas I rag mor arrowl ha hos of cas F. A spr slop for cas I is also obsrvd ha ha for cas F i Figur.6 b ad c. Tabl.6 Saisical suls of Toal Dischargs for alizaios a h Our Boudar ad a h Irfac for a Modl wih a Hrogous NF b h Local Homogizaio for 9 cass Toal discharg a h our Toal discharg a h irfac War ma rsidc Cas boudar [m /r] [m /r] im i buffr [r] Mdia Ma Sadard Mdia Ma Sadard dviaio dviaio A.65E+.56E+.85E+.58E-7 4.E-7.56E-7.7E+6 B 7.48E+ 8.87E+ 5.E+.4E-7.66E-7.7E-7.7E+6 C.4E+.5E+ 5.76E+.87E-7.88E-7 6.5E-8.88E+6 D.E+.95E+.8E+.5E-7.69E-7.75E-7.5E+6 E.E+.88E+.8E+.9E-7 4.9E-7.9E-7.98E+6 F 8.E+.E+ 7.96E+.4E-7.57E-7.4E-7.E+6 G.95E+.58E+.5E+.99E-7.4E-7.E-7 4.7E+6 H 6.76E+ 8.8E+ 5.E+.68E-7.76E E-8.E+6 I.58E+.65E+.49E+.6E-7.4E E-8.54E+6 Tabl.6 shows h rsuls of h oal dischargs a h our boudar ad a h irfac for a modl wih a hrogous NF for ralizaios for 9 cass. Th mdia ma ad sadard dviaio ar giv i Tabl.6. Th oal dischargs a h irfac ar abou 9 ordrs of magiud smallr ha hos a h our boudar Tabl.6. I mas ha h war flow i h buffr rgio is vr slow. This is du o h assumd small hdraulic coducivi. -4 m/s for h buffr. To cofirm h slow war flow i h buffr h war rsidc im i h buffr is calculad as follows. Th volum of h pors i h clidrical buffr pr ui high wih h our radius of m h ir radius of.5 m ad h porosi of. is calculad as.8 m. Th oal volumric war flow ra [m /r] hrough h buffr is calculad b muliplig h oal discharg Q [m /r] a b ui dph [m]. Th rsidc im is obaid b dividig.8 m b h volumric flow ra Q [m /r]. Th rsuls ar show i h righ-mos colum of Tabl.6. I Figur.8 h rlaioship bw h oal discharg a h our boudar for a modl wih a hrogous NF b h local homogizaio ad h cocivi is show basd o h umrical rsuls for h i cass. Mark i Figur.8 rprss h mdia of h oal dischargs a h our boudar for ralizaios. A rror bar for h 9% cofidc irval is show. Th lowr limis ad h uppr limis ar h valus a h cumulaiv probabili of.5 ad.95 rspcivl i Figur.6 a b ad c. Obsrvaios from Figur.8 ar as follows. Firs compar hr cass wih h sam lgh saisics show i Tabl. i.. compar cass A B wih C cass D E wih F ad cass G H wih I. As h cocivi icrass h oal discharg icrass bcaus war ca flow hrough mor fracurs. For h sam lgh saisics a grar cocivi is achivd b addig mor fracurs. Scod h rror bar of oal dischargs a h our boudar dcrass as h cocivi icrass. Wih mor fracurs h FBC bcoms mor homogous rsulig i a arrowr disribuio of h oal discharg. Third cas E has a widr rror bar of h oal discharg ha hos of cass B ad H. Alhough cas E has h sam cocivi as cas B ad H h umbr of fracurs o obai h cocivi of. is smallr ha ohrs du o h graio of logr fracurs. Bcaus h FBC cosiss of log fracurs aras whr o fracurs is occur isid h FBC. Du o hs aras for cas E h mdium is mor hrogous ha ohrs wih h sam cocivi. Thrfor h rror bar is widr ha ha of ohrs. Sam obsrvaio ca b mad for cas F i compariso wih cass C ad I. 4

51 A B C 4

52 D E F 4

53 Figur.7 G H I Fracur works FBCs ad Fii lm msh ad Darc vloci disribuio for a slcd ralizaio for 9 cass. 4

54 CDF CDF Toal discharg a h our boudar [m/r].e+4.e+.e+.e+ A D G B E H C F I.E Cocivi Figur.8 Toal dischargs a h our boudar b h local homogizaio for 9 cass wih 9% cofidc irvals.6.. Effcs of Aprur Corrlaio Aprur of a fracur sgm is grad basd o h giv logormal disribuio i h prvious scio. I his scio aprur is drmid basd o h rlaioship.7 bw h lgh ad aprur [Oho al. 987] afr h valu for h lgh is sampld. Ohr paramrs ad disribuios ar kp h sam as show i Tabls.4. Cass A ad B i Tabl.4 hav b slcd bcaus fracur lgh has a wid variaio du o h grar sadard dviaio.6 m ha hos assumd i cass D E F G H ad I. Tabl.7 Saisical suls of Toal Dischargs for alizaios a h Our Boudar ad a h Irfac for a Modl wih a Hrogous NF b h Local Homogizaio for Corrlad Aprur Cass Cas Toal discharg a h our boudar [m /r] Toal discharg a h irfac [m /r] Mdia Ma Sadard Mdia Ma Sadard dviaio dviaio A 5.4E+5 4.8E+6.5E E-7 5.7E-7.79E-7.65E+.56E+.85E+.58E-7 4.E-7.56E-7 B.6E+6 4.7E E+6 5.9E-7 6.6E-7.99E E+ 8.87E+ 5.E+.4E-7.66E-7.7E-7 Th valus i parhss ar for ucorrlad cas sam as i Tabl Ucorrlad Corrlad Cas A Q [m /r] a Cas A corrlad ad ucorrlad aprur Figur Ucorrlad War ma rsidc im i buffr [r].46e+6.7e+6.5e+6.7e+6 Corrlad Cas B Q [m /r] b Cas B corrlad ad ucorrlad aprur CDF of oal discharg a h our boudar for a modl wih a hrogous NF for cass A ad B corrlad ad ucorrlad 44

55 Tabl.8 Toal Dischargs of a Paricular alizaio for Cass A ad B wih h Corrlad Aprur i Figur. b ad d Cas Toal discharg a r= [m /r] Toal discharg a r= [m /r] A.6E+5 5.E-7 B.E+6 8.6E a Ucorrlad aprur for Cas A b Corrlad aprur for Cas A Figur c Ucorrlad aprur for Cas B d Corrlad aprur for Cas B Darc vloci disribuios i h modl spac boh for corrlad ad ucorrlad cass Toal dischargs for a modl wih a hrogous NF for h cass wih aprur-lgh corrlaio ar show i Tabl.7. Figur.9 shows h CDF of h oal discharg a h our boudar for a modl wih a hrogous NF b h local homogizaio boh for h cass wih or wihou h lgh-aprur corrlaio for cass A ad B. I Figur.9 wo disic diffrcs ar obsrvd. Firs h oal discharg a h our boudar for h corrlad cas is grar b abou o 4 ordrs of magiud ha hos for h ucorrlad cas. This diffrc ca b plaid as follows. Abou ~ ordrs of magiud widr valus of aprur ca b grad b h corrlad cas rsulig i abou o ordr grar porosiis ha hos of h ucorrlad cas. Accordig o. h prmabili k is proporioal o k~ /M s. Bcaus M s i.5 is idpd of h aprur prmabili is proporioal o h porosi cubd. Thus abou ordr of magiud grar porosi rsuls i abou ordrs of magiud grar prmabili or hdraulic coducivi ad hus oal dischargs. Scod h rag of oal discharg disribuio for h corrlad cas is widr ha ha for h ucorrlad cas. Bcaus log fracurs hav grar aprurs i h corrlad cas war flows mail hrough a fw log fracurs whras war flow i h ucorrlad cas is mor avragd ovr h FBC. Du o his diffrc h 45

56 Toal discharg a h our boudar [m/r] flow bcoms mor hrogous for h corrlad cas. Thus oal dischargs a h our boudar for h corrlad cas ar disribud mor widl ha hos of h ucorrlad cas. Figur. dpics h Darc vloci disribuios boh for h corrlad ad ucorrlad cass for a paricular ralizaio for cass A ad B show i Figur.7. Tabl.8 shows h oal dischargs of a paricular ralizaio for cass A ad B wih h corrlad aprur i Figur. b ad d..6. Effc of Modlig Approach for NF o Flow a I his scio o ivsiga ffcs of h modlig approach for h NF war flow aalsis has b prformd for a modl wih a homogizd NF b h uiformizaio. Aprur of a fracur sgm is grad basd o h giv logormal disribuio as show i Tabl.. Tabl.9 Cas Toal Dischargs a h Our Boudar for a Modl wih a Hrogous NF ad a Modl wih a Homogizd NF for alizaios for 9 cass Toal discharg a h our boudar for a modl wih a hrogous NF b h local homogizaio [m /r] Sam as valus i Tabl.6 Mdia Ma Sadard Toal discharg a h our boudar for a modl wih a homogizd NF b h uiformizaio [m /r] Mdia Ma Sadard dviaio dviaio A.65E+.56E+.85E+.97E+.6E+ 8.7E+ B 7.48E+ 8.87E+ 5.E+ 6.E+ 6.7E+.9E+ C.4E+.5E+ 5.76E+.7E+.4E+.5E+ D.E+.95E+.8E+.55E+.77E+ 8.E+ E.E+.88E+.8E+.95E+.5E+.E+ F 8.E+.E+ 7.96E+ 6.4E+ 6.49E+.46E+ G.95E+.58E+.5E+ 4.98E+ 5.6E+.44E+ H 6.76E+ 8.8E+ 5.E+ 7.57E+ 7.67E+.7E+ I.58E+.65E+.49E+.6E+.6E+.5E+ Toal dischargs a h our boudar for a modl wih a homogizd NF b h uiformizaio hav b calculad for ralizaios for i cass i Tabl.4. Tabl.9 shows h oal dischargs a h our boudar boh for a modl wih a hrogous NF b h local homogizaio ad for a modl wih a homogizd NF b h uiformizaio for ralizaios for 9 cass..e+4.e+ A D G.E+.E+ B E H C F I.E Cocivi Figur. Compariso of h 9% cofidc irvals of oal dischargs a h our boudar bw a modl wih a homogizd NF b h uiformizaio hick lis ad a modl wih a hrogous NF b h local homogizaio hi lis for 9 cass. 46

57 Figur. shows h compariso of h 9% cofidc irvals of oal dischargs a h our boudar bw a modl wih a homogizd NF hick lis ad a modl wih a hrogous NF hi lis for 9 cass. Smbols ad rprs h mdia of h oal dischargs a h our boudar for ralizaios for ach cas for a hrogous NF ad for a homogizd NF rspcivl. Characrisics for ach cas for a homogizd NF show i Figur. ar similar o hos for a hrogous NF as show i Figur.8. Obsrvaios from h compariso of h Figur. wih Figur.8 ar as follows. Firs h oal discharg a h our boudar for a homogizd NF disribus mor arrowl ha hos for a hrogous NF. This is bcaus hdraulic propris for a hrogous NF ar spaiall disribud ovr h FBC i h domai whras h ar sigl valus ovr h ir NF for a homogizd NF afr rasformaio b h uiformizaio. Scod h rags of h oal discharg disribuios for ach cas for a homogizd NF ar wihi h rags obaid for a hrogous NF. Th mdia valus ar vr clos o ach ohr..7 Coclusios A wo-dimsioal ar-fild modl icorporaig h hrogi of h ar-fild rock NF for h groudwar flow for h o-was caisr cofiguraio has b dvlopd for h im-idpd icomprssibl Darc s flow hrough a fracur work. A flow-barig clusr FBC of fracurs is racd from h discr fracur work DFN b chckig h ircocivi of fracurs. Th FBC is rasformd io a hrogous coiuum b h local homogizaio ad a homogous coiuum b h uiformizaio. Th Koz-Carma quaio is usd o calcula h prmabili for a rasformd coiuum. A fii lm soluio has b drivd for h spaial disribuio of h hdraulic poial i h FBC. A compur cod FFDF has b dvlopd. laioship bw h hrshold cocivi ad h umbr of fracurs has b umricall plord. For a giv s of disribuio fucios for fracur gomr paramrs a hrshold cocivi bod which a fracur work i NF alwas coais a las o FBC has b drmid. Th FBC is cosidrd o b mor hrogous if h valu of h cocivi is closr o h hrshold cocivi. As h cocivi icrass h FBC bcoms lss hrogous. Effcs of h hrogi of h NF hav b ivsigad b prformig h war flow aalsis for diffr ss of saisics of fracur gomr paramrs. Numrical rsuls show ha: A work wih a grar cocivi hibis a grar oal discharg bcaus a largr amou of war ca flow hrough h mdium wih mor fracurs s Figur.8. Th oal discharg a h our boudar rags mor arrowl as h cocivi of h work icrass s Figur.8. For wo works wih h sam cocivi h oal discharg rags mor widl wih h work cosisig of log fracurs ha wih h work cosisig of shor fracurs s Figur.8. Th rsidc im of war i h buffr rgio is rml log du o h assumd low hdraulic coducivi for h boi K=. -4 m/s compard wih ha i NF s Tabl.6. Effcs of modlig approachs for h NF ar ivsigad b comparig h groudwar flow ras bw h modl wih a hrogous NF b local homogizaio ad h modl wih a homogizd NF b uiformizaio. Numrical rsuls show ha: Th oal discharg a h our boudar obaid for a homogizd NF disribus mor arrowl ha ha obaid for a hrogous NF s Figur.. Th homogizd NF dos o cssaril ovrsima h oal discharg of war s Figur.. Th mdia valus of h oal dischargs obaid for h homogizd NF ad h hrogous NF ar clos o ach ohr s Figur.. For h NF hosig a gologic rposior a small cocivi or highl hrogous rock is dsirabl. For highl hrogous fracurd rock h homogizd NF cao rproduc high ad low ails of h flow ra disribuio ad dos o cssaril ovrsima h war flow ra. Thus h homogous mdium b h uiformizaio ma o b a appropria approach o simula h war flow i highl fracurd rock..8 Limiaios Th curr modl for h war flow aalsis has followig limiaios: 47

58 A wo-dimsioal aalsis ds o udrsima h prmabili bcaus fracurs o cocd o h work i h pla of aalsis ma b cocd i h hird dimsio [Log 985]. Procsss affcig h EBS b groudwar i.. corrosio boi swllig ha grad b was ar glcd. Effcs of mulipl caisrs [Ah al. 998; Kawasaki al. ] ar o cosidrd. Coffici C i h Koz-Carma quaio icludig oruosi corrcio is glcd b assumig C=. Coffici C will b discussd i Chapr 6 i dail. 48

59 4. Mahmaical Modl for War Flow Aalsis i h Nar Fild 4. Iroducio This chapr sablishs a mahmaical modl o pariall vrif h flow modl dvlopd i Chapr. A mahmaical modl cosiss of a imprmabl was caisr a homogous buffr ad a homogizd NF b h uiformizaio Scio.4.. A aalical soluio of h sad-sa poial disribuio is formulad for h mahmaical modl. Wih aalicall obaid poial disribuio i h modl spac a disribuio of h sram fucios ca b obaid aalicall. Th h war flow ra hrough h modl spac is calculad from h disribuio of sram fucios ad hdraulic coducivi of h rgio of irs. Th comparisos of h rsuls obaid b umrical modl wih hos obaid b aalical soluios hav b mad for h som spcific cass. Th objcivs of his chapr ar o formula aalical soluios of h poial disribuio ad sram fucios for h mahmaical modl ad o vrif pariall a umrical modl dvlopd i Chapr b h mahmaical modl. 4. Mahmaical Problm ad Soluio of Poial Equaio Th modl spac = + i Figur 4. cosiss of wo homogous sub-domais wih K =cos ad wih K =cos. Th variabls usd for obaiig h aalical soluios for h poial had ad h sram fucios ar dimsiolss [ r K K ]=. Th poial had is prscribd arbiraril as f a r=. Th flu is prscribd zro a r=. =f r r K r K Figur 4. Modl spac for h mahmaical modl. saisfig = i cao b solvd bcaus K is discoiuous alog r=. Thrfor h problm is dcomposd io wo sub-problms doig h poial i b ad i b. Simulaous soluios ar h sough for ad saisfig h cosidrd parial diffrial quaio i hir rspciv sub-domais ad spcifid boudar codiios a r= for ad r= for [Bar 979]. Govrig quaios ar r r i r 4. r r ad r r i r. 4. r r 49

60 Th forms of h soluios for 4. ad 4. ar highl dpd o ha of f which is a priodic picwis coiuous ad picwis diffriabl fucio [Hildbrad 95]. For ampl if f is jus a cosa h h soluio should b agl idpd. If f is h fucio of cosi h h soluio is also rspodd wih h fucio of cosi. Th poial had has b prscribd a r= arbiraril as No flu boudar is assumd a r= r. 4. f a r. 4.4 A h irfac r= h lvaio had ad prssur had ar h sam wh a poi o h irfac is approachd from boh sids [Bar 979]. a. 4.5 Coiui of flu across h irfac is prssd hrough h compos of h spcific discharg ormal r dircio i polar coordias o h irfac [Bar 979]. K K a r. 4.6 r r Gral soluios for 4. ad 4. ar show i 4.7 ad 4.8 rspcivl [Hildbrad 95]. A B lr C D a r b r cos c r d r r si r = A B C ar cosa D E F lr G H r f r cos g r h r r si r = E F G H ar cosa. I ordr ha r b sigl-valud i h domai h followig mus b ak [Hildbrad 95] D H. 4.9 If D ad H ar o qual o zro hr ar o uiqu soluio bcaus givs diffr valus for = + whr is a arbirar agl bw ad ad is h igr. Subsiuig 4.9 io givs Usig h BC 4. a b lr a r b r cos c r d r r si 4. r = a A C b B C f lr r f r cos g r h r r si 4. r = E G f F G. f l f cos g h si f 4. Usig h BC 4.4 5

61 5 si cos ] [ d c b a b 4. Usig h BC 4.5 si cos l si cos l h g f f d c b a b a 4.4 Usig h BC 4.6. si cos ] [ si cos ] [ h g f f K d c b a b K 4.5 Equaig hr cofficis b b a ad d c of cosa si cos rspcivl of righ-had sid ad zro of lf-had sid i 4. giv ad 4.8 for. b 4.6 a b 4.7 c d. 4.8 Equaig wo cofficis of si cos of lf-had sid ad righ-had sid i 4.4 givs ad 4. rspcivl. l f a 4.9 a b f 4. g c d h. 4. Equaig hr cofficis of cosa si cos of lf-had sid ad righ-had sid i 4.5 givs ad 4.4 rspcivl. f 4. f K K b K K a 4. K h K d g K K c ad rigoomric sris wih Fourir cofficis givs [Hildbrad 95] l d f f 4.5 cos d f f 4.6 si d f h g. 4.7

62 Now hr ar ukow a b f a b c d f g ad h ad quaios. To simplif h quaios subsiuig 4. io 4.9 ad 4.5 givs rspcivl a 4.8 d 4.9 f whr is obaid b igraig f wih rspc o from o ad dividig b. quaios ar summarizd as follow. f b b 4.6 a 4.7 d c 4.8 b a 4. h d c g 4. f 4. Ka K Kb K f 4. Kc Kg Kd K h 4.4 f f cos d g h f si d a 4.8 d. 4.9 f To obai from 4.6 f should b rplacd b h form of. For his subsiuio of 4.7 io 4. givs f a a. 4. Subsiuio of 4.7 ad 4. io 4. givs rlaioship bw a ad. K K K a 4. K K a K K K whr K. 4. K K Accordig o h gomr of h modl i Figur 4. is largr ha. Thus variabl i 4. is alwas largr ha zro. Subsiuio of 4. io 4. givs f ha is h ol fucio of. 5

63 5 f. 4. Subsiuio of 4. io 4.6 givs. cos d f 4.4 whr. 4.5 I his cas ad alwas largr ha zro sic > > > as show i Figur 4.. is obaid b igraig cos f wih rspc o from o ad dividig b. To obai h soluio for g from 4.7 h sam procdurs 4.~4.5 ar applid. Subsiuig 4.8 io 4. givs g c c h. 4.6 Subsiuio of 4.8 ad 4.6 io 4.4 givs rlaioship bw g ad c. c c K K K K K g 4.7 whr K K K K K K. 4.8 Subsiuio of 4.7 io 4.6 givs h ha is h ol fucio of g. g h. 4.9 Subsiuio of 4.9 io 4.7 givs g. si d f g 4.4 whr. 4.4 g is obaid b igraig si f wih rspc o from o ad dividig b. Now h obaid aalical soluio is summarizd as follow. Th vrificaio of h soluio is giv i Scio 4.5. si cos r r d r c b r r a a r 4.4 si cos r r h r g r f r r 4.4 whr d f a 4.44

64 cos a f d 4.45 cos b f d 4.46 si c f d 4.47 si d f d 4.48 d 4.49 f cos f d 4.5 cos f f d 4.5 si g f d 4.5 si h f d 4.5 K K K K K K whr. 4. Sram Fucios From h obaid poial fucios i h sram fucios i ca b drmid b 4.55 i i i i i 4.56 r r r r whr i= rprss h rgio <r< ad i= rprss h rgio <r<. Thus from 4.4 ad 4.56 h sram fucio i h rgio <r< ca b obaid as follows. Diffriaig 4.4 wih rspc o r ad ad rlaig o 4.56 ilds r r 4.57 a cos si r br cr dr r a si cos r br cr dr r Igraig 4.57 wih rspc o givs. 4.58

65 whr r is h arbirar fucio of r. a r b r si c r d r cos r To drmi r diffriaig 4.59 wih rspc o r ilds r ar br si cr dr Compariso of 4.58 wih 4.6 givs Thrfor from 4.59 ad d r cos. 4.6 dr d r or r C. 4.6 dr a r b r si c r d r r cos C 4.6 r C is cosa. Followig h abov procdur h sram fucio i rgio <r< is r f r si g r h r r cos C 4.6 r C is cosa. Th cosas C =C = i 4.6 ad 4.6 ar ak sic diffrc bw wo sram-lis givs phsicall maigful rsul. Thus sram fucios bcom ar br si cr dr cos r r 4.64 r fr si gr hr cos r r Whil variabls hav b assumd o b dimsiolss o obai h aalical soluios h acuall hav dimsios as show i Tabl 4.. For h graphical rprsaio spcial cass wih f= cos for = ar assumd as show i Tabl 4. whr [m] is h ui poial had. Tabl 4. Ipu Paramrs for Graphical prsaio of Aalical Soluio. Cas K [m/s] K [m/s] K /K [m] [m] [m] f [m] cos cos cos Th aalical soluios of h poial fucios 4.4 ad 4.4 ad sram fucios 4.64 ad 4.65 for cass i Tabl 4. ar b r ar cos r 4.66 r 55

66 [m] [m] [m] f r r cos r 4.67 r b r ar si r 4.68 r f r r si r r Cofficis i 4.66~4.69 ar calculad b 4.44~4.55 usig h paramrs i Tabl 4. ad ar show i Tabl 4.. Ohr cofficis a c d g ad h ar qual o zro for h paramrs i Tabl 4.. Tabl 4. Numrical Valus of Cofficis i Aalical Soluio. Cas a b [m ] f [m ] Figur 4. shows h poial ad sram fucios for cass. I Figur 4. vrical lis rprs quipoial lis ad horizoal lis rprs h sramlis. If phsical propr K is cosidrd as h hdraulic coducivi [m/s] h cas rprss o homogous mdium wih K =K cas rprss ha rgio is mor prmabl ha rgio K >K ad cas rprss ha rgio is lss prmabl ha rgio K <K [m] [m] [m] a K /K =. cas b K /K =. cas c K /K =. cas Figur 4. Poial ad sram fucios for cass. 4.4 Compariso bw h Aalical ad Numrical Soluios For h purpos of compariso bw h aalical soluio 4.4 ad 4.4 ad h umrical soluio b FFDF cod for a modl wih a homogizd NF spcial cass for f= -cos/ ar cosidrd whr = [m]. Assumd valus for ach variabl ar show i Tabl 4.. Figur 4. a b ad c show h domai discrizaio for a modl wih a homogizd NF b 44 8 ad 56 pois rspcivl. Th siz of msh usd i cas c i Figur 4. is h sam as ha usd i war flow aalsis i Chapr. Hr ol wo aalical soluios for all cass is. O is for K /K = K =. m/s ad K =. m/s ad h ohr is for K /K =. K =. m/s ad K =. m/s. Ohr paramrs ad f ar assumd o b idical for all cass. Aalical soluios for cass ad 5 ar obaid b 4.4 ad 4.4 as 56

67 Poi 4 /=.5 Poi = Poi 7 = Poi Poi /=.5 a 44 pois b 8 pois c 56 pois Figur 4. Domai discrizaio for umrical soluio b FFDF 57

68 .9 r.5.995r cos r r r.5.995r cos r. r ad 4.7 ar idical bcaus K is qual o K. Aalical soluios for cass 4 ad 6 ar.957 r r cos r r r r cos r. r 4.7 For all cass h absolu diffrc is simad as 4.74 aal um whr aal is h poial obaid from h aalical soluio ad um is h poial obaid umricall b FFDF. Figur 4.4 shows h valus of poial obaid b h aalical soluio li ad hos obaid umricall b FFDF do a all 44 pois for cas. I his cas h avrag diffrc which is obaid b avragig diffrcs b 4.74 arihmicall for 44 pois is.6 -. Th uppr boud of diffrc which is h maimum diffrc for 44 pois is Th lowr boud of diffrc which is miimum diffrc for 44 pois is zro. Figur 4.5 shows h rsuls for cas. I his cas h avrag diffrc for 8 pois is Th uppr ad lowr bouds of diffrc ar ad zro rspcivl. Figur 4.6 shows h rsuls for cas 5. I his cas h avrag diffrc for 56 pois is Th uppr ad h lowr bouds of diffrc ar ad zro rspcivl. For K /K qual o. cass 4 ad 6 rsuls ar show i Figur 4.7 Figur 4.8 ad Figur 4.9 for cass 4 ad 6 rspcivl. Tabl 4.4 ad Tabl 4.5 show h summar of diffrc for K /K = ad K /K =.. Accordig o hs abls diffrc bw h aalical soluio ad h umrical soluio dcrass as h umbr of pois icrass Figur 4.. For all cass h diffrcs for h aalical ad umrical soluios b 4.74 a all pois ar lss ha Diffrcs for K /K =. ar largr ha hos for K /K =. bcaus of diffr phsical propris bw wo rgios for K /K =.. Alhough diffrcs for K /K =. ar largr ha ha for K /K =. agrm bw h aalical soluio ad h umrical soluio for boh cass is good ough < Tabl 4. Ipu Paramrs for Compariso Cas Pois K /K [m] [m] [m] f [m] cos/ Tabl 4.4 Diffrc for K /K = Cas Avrag Uppr boud of Lowr boud of.6e- 8.66E- 7.E E E-4 5.8E- Tabl 4.5 Diffrc for K /K =. Cas Avrag Uppr boud of Lowr boud of.46e-.88e- 4.75E-.8E- 6.7E- 9.84E- 58

69 Poial [m] Poial [m] Poial [m]..8 Aalical Numrical.6.4. K/K= Poi umbr Figur 4.4 Aalical vs. umrical for cas: K /K = ad 44 pois...8 Aalical Numrical Poi umbr K/K= Figur 4.5 Aalical vs. umrical for cas : K /K = ad 8 pois...8 Aalical Numrical Poi umbr K/K= Figur 4.6 Aalical vs. umrical for cas 5: K /K = ad 56 pois. 59

70 Poial [m] Poial [m] Poial [m[..8 Aalical Numrical.6.4. K/K= Poi umbr Figur 4.7 Aalical vs Numrical for cas : K /K =. ad 44 pois...8 Aalical Numrical.6.4. K/K= Poi umbr Figur 4.8 Aalical vs Numrical for cas 4: K /K =. ad 8 pois...8 Aalical Numrical Poi umbr K/K=. Figur 4.9 Aalical vs. umrical for cas 6: K /K =. ad 56 pois. 6

71 Avrag Diffrc - - K /K = K /K =. Figur Numbr of pois Avrag diffrc ad umbr of pois. Tabl 4.6 shows h compariso of h oal discharg [m -war/m-mdiums] a h our boudar ad h irfac bw h aalical soluio ad h umrical soluio. Toal discharg i rms of volum of war pr ui widh ormal o h r- pla pr ui im bw wo sramlis is qual o h diffrc i h sram fucios corrspodig o hs lis [Bar 979]. Th sram fucios a rgios ad for K /K =. ar obaid b usig h poial fucios 4.7 ad 4.7 ad h rlaioship bw h poial fucio ad h sram fucio 4.56 as.9 r.995r si r 4.75 r ad.9 r.995r si r r For K /K =. h sram fucios ar obaid b ad 4.56 as ad.957 r.5979r si r 4.77 r.867 r.9669 r si r r Th oal discharg a h our boudar r= ca b aalicall valuad b 4.76 for cass ad 5 ad 4.78 for cass 4 ad 6. Th oal discharg a h irfac r= ca b aalicall valuad b 4.75 for cass ad 5 ad 4.77 for cass 4 ad 6. For ampl h oal discharg a h our boudar for cas ad 5 h maimum a r= =5 =/ ad miimum a r= =5 =/ valus of h sram fucio ar calculad b 4.76 as.495 m ad m rspcivl. Thus h oal discharg a h our boudar for cass ad 5 is obaid b subracig h miimum valu m from h maimum valu.495 m ad muliplig h rsula.99 m b h hdraulic coducivi K =. m/s rsulig i.99 [m /s]. Tabl 4.6 Cas Compariso of Toal Discharg a h Our Boudar ad a h Irfac. Toal discharg a r= [m /s] Aalical Toal discharg a r= [m /s] Numrical % Diffrc Toal discharg a r= [m /s] Aalical Toal discharg a r= [m /s] Numrical % Diffrc 9.5E E E- 9.67E-.6.75E-.7E E E E E E- 9.E-..8E-.7E E E-.55 6

72 Accordig o Tabl 4.6 a diffrc bw h oal dischargs obaid aalicall ad umricall is dcrasd as h umbr of poi icrass. For ach cas i Tabl 4.6 h amou of war rs from h our rgio o h ir rgio i rms of prcag ca b calculad as h raio of h oal discharg a h irfac r= o h oal discharg a h our boudar r=. For cass 5 ad 6 h raios of amou of war flowig io h ir rgio from h our rgio ar 8% ad % rspcivl. Th highr raio 8% for cas 5 K /K = ha ha % for cas 6 K /K =. is pcd rsul sic K >K for cas 6. Cass ad 5 K /K = hav sam rd of war flow as show i Figur 4. a. Figur 4. c shows h rd of war flow for cass 4 ad 6 K /K = Vrificaio of Aalical Soluio for Poial 4.5. Govrig Equaios To vrif h aalical soluio for poial h aalical soluios 4.4 ad 4.4 should saisf h govrig quaios 4. ad 4.. Accordig o h obaid soluios a o h icludig ad 4.44 o 4.55 ar cosa. r r Subsiuio of 4.4 io h firs rm r r ad h scod rm i 4. ild h r r sam magiud bu opposi sig as show i 4.79 ad 4.8 rspcivl. r r r a cos si r br cr dr 4.79 r r r a r b r Subsiuio of 4.79 ad 4.8 io 4. givs zro. cos c r d r si. 4.8 r r. 4.8 r r Thrfor h soluio of h 4.4 saisfis h govrig quaio 4.. Similarl h soluio for 4.4 also saisfis h govrig quaio Boudar Codiios Th boudar codiio BC prscribd a r= is f as show i 4.. Subsiuig r= io 4.4 givs f cos g h si. 4.8 Equaio 4. for f wih f = 4. givs f cos g h si f 4.8 Sic h righ sid of 4.8 is qual o h righ sid of 4.8 h soluio saisfis h BC 4.. Th boudar codiio prscribd as r= is / givs a a b cos c d si r as show i 4.4. Subsiuig r= io Diffriaig h righ-had sid of 4.84 wih rspc o r givs zro bcaus h righ-had sid of 4.84 is idpd of r. Thrfor h soluio saisfis BC

73 6 Th boudar codiio prscribd as r= is as show i 4.5. Subsiuig r= io givs si cos d c b a a 4.85 si cos h g f Subsiuig 4.44 ~ 4.55 io 4.85 ad 4.86 rspcivl givs si si cos cos d f d f d f 4.87 si si cos cos d f d f d f 4.88 Th soluios saisf BC 4.5 bcaus h righ-had sid of 4.87 ad 4.88 ar idical. Coiui of flu prscribd a r= is r K r K / / as show i 4.6. A r= h lf-had sid of h BC 4.6 bcoms si cos d c b a K r K 4.89 A r= h righ-had sid of BC 4.6 bcoms si cos h g f K r K 4.9 Subsiuig 4.44 ~ 4.55 io h 4.89 ad 4.9 givs rspcivl ]si ][ si [ ]cos ][ cos [ d f K d f K r K 4.9 ]si ][ si [ ]cos ][ cos [ d f K d f K r K 4.9 If 4.9 is saisfid 4.9 is idical o 4.9 ad h h soluio saisfis BC 4.6. K K 4.9

74 K K K whr K. K K Th lf-had sid of 4.9 givs Th righ-had sid of 4.9 givs K K K K Sic 4.94 ad 4.95 ar idical so ar 4.9 ad 4.9. Thrfor h soluio saisfis BC 4.6. Bcaus h soluios of h ad i ad 4.4 rspcivl saisf govrig quaios 4. ad 4. ad h boudar codiios ad 4.6 h ar corrc. 4.6 Coclusios Numrical soluio obaid b h umrical modl wih a homogizd NF b h uiformizaio dvlopd i Chapr is bch-markd agais a aalical soluio. For h siz of msh i Figur 4. c usd for war flow aalsis i Chapr h umrical rsuls for h valus of hdraulic poial diffr from hos b aalical soluio b a avrag diffrc lss ha.7 -. Th umrical rsuls for h oal dischargs a h our boudar ad h irfac diffr from hos obaid b aalical soluios b a rlaiv diffrc lss ha.55%. 64

75 5. Traspor Aalsis i h Nar Fild of Gologic posior 5. Iroducio I his chapr a ar-fild modl for h paricl raspor simulaio is dvlopd basd o h flow modl which is dvlopd i Chapr. Paricl raspor is modld b a adom-walk Trackig Mhod WTM [Tompso al. 987] which is basd o h spaial disribuio of h flow obaid b h war flow aalsis i Chapr. Paricl movm ovr discr im sps is govrd b a radom walk algorihm for h advcio ad h molcular diffusio. Th rsidc im of paricls is obaid b obsrvig racks of ach paricl. Th mass absorpio ra a h our boudar is also obaid b couig h umbr of paricls big absorbd a h boudar pr ui im. Th objcivs of his chapr ar o dvlop a modl for paricl raspor simulaio basd o h flow modl dvlopd i Chapr o ivsiga ffcs of h hrogi of h NF o h paricl raspor ad o ivsiga ffcs of h modlig approach for h NF o h paricl raspor. I his chapr firs h WTM b Tompso al. [987] for h paricl raspor simulaio i h coiuum is discussd. This discussio was iiiad b Chambré []. Th dvlopm show i Scio 5. is basd o h discussios wih Chambré []. 5. viw of Tompso s Tram 5.. Tompso s Modl Tompso al. [987] dvlopd h raspor modl b usig a adom-walk Trackig Mhod WTM o ivsiga h solu movm hrough larg hr dimsioal hrogous flow ssms i saurad ad usaurad porous mdia. I ordr o obai h mass disribuio of a larg umbr of paricls corrspodig o ha giv b h rasi raspor quaio h sp quaio which govrs h paricl movms ovr discr im sps is obaid b adopig h Io assumpio; i.. a paricl movm a h im sp is govrd b h propris of h curr im sp bfor movm. Th sp quaios ar obaid b usig a aalog bw h Io-Fokkr-Plack quaio [Hak 98] ad h rasi advcio-disprsio quaio such ha h Io-Fokkr-Plack quaio wih h sp quaio is quival o h rasi raspor quaio. For a wo-dimsioal saurad flow ssm h sp quaio for h im-dpd spac-dpd war vloci ad cosa molcular diffusio coffici wihou mchaical disprsio i a im irval was giv b Tompso al. [987] as follows: v X Z D 5. v X Z D whr ad ar radom variabls ad ad ar drmiisic paramrs. X is a posiio vcor of a paricl a im ad X ad X ar h compos of h war por vloci vcor a im i ad v v dircios rspcivl. D is h diffusio coffici icludig mdium oruosi ad is cosa i im ad spac i 5.. Z ad Z ar idpd radom umbrs a im saisfig h moms <Z >= <Z >= ad <Z >= <Z >= rspcivl *. Th bracks < > do h avrag valu of. Z D ad Z D do h ormalizd radom displacms i ad dircios rspcivl **. * I is ol cssar for Z o hav h saisical qualiis such as <Z>= ad <Z >=. Thus ma kids of disribuios migh b implmd. A ormal disribuio wih ma of ad variac of is o possibl disribuio for Z saisfig wo saisical qualiis [Tompso al. 987]. ** I was dfid b Tompso al. [987] wihou daild plaaio. Th rm ormalizd radom displacm ca b irprd as a ormalizd radom displacm b h sadard dviaio of h displacm s Scio

76 For h radom displacm rms i h sp quaio 5. Tompso al. [987] did o driv h radom rms Z ad Z aalicall bu assumd a priori som probabili disribuios wihou a jusificaio. Thus a mahmaical basis of h sp quaio ds o b sudid. I h followig a sp quaio is drmid aalicall for a cas wih cosa vloci ad diffusivi. I rvals som rsricios o h sp quaio usd b Tompso al. [987]. 5.. adom Movm icludig Advcio I h flowig war wih a cosa war vloci v i h -coordia movms of paricls ar drmid b h wo forc compos a drmiisic forc ad a saisical radom forc. Gravi is glcd. I his scio advcio ad molcular diffusio rfr o h drmiisic forc ad h saisical radom forc rspcivl. Th mahmaical framwork for Browia moio s up b Eisi [95] * is usd i h followig ad summarizd blow. Howvr Eisi [95] did o dal wih h problm icludig h advcio i his sud. Sic a displacm i -coordia b advcio i im is whr is a radom variabl ad ad v ar drmiisic paramrs. v a w radom variabl is dfid as v 5. To driv a sp quaio b usig h hor of Browia movm for a giv spcial cas is firs cosidrd ad h is obaid b chagig h variabl usig 5.. A displacm i -coordia b h saisical radom forc i im irval ca b obaid b h hor [Eisi 95]. Thr assumpios ar mad. Each paricl cus a movm which is idpd of all ohr paricls. Th movms of o ad h sam paricl afr diffr irvals of im mus b cosidrd as muuall idpd procsss so log as hs irvals of ims big chos ar o oo small. Th im irval is o b vr small compard wih h obsrvd irval of im bu vrhlss of such a magiud ha h movms cud b a paricl i wo coscuiv irvals of im ar o b cosidrd as muuall idpd phoma. I Scio 5. is assumd o b cosa ad h obsrvaio im afr m sps of im irval is m i.. =m. I a irval of im h -coordia of h sigl paricl will icras b whr has a diffr valu posiiv or gaiv for ach paricl. L b a probabili of a paricl o b displacd bw o +d a a irval. Th disribuio fucio is ormalizd as d 5. ol diffrs from zro for vr small valus of ad fulfils h codiio such ha h probabiliis a ad - ar sam. = L fd b h umbr of paricls locad bw ad +d a im. Th h umbr of paricls locad bw ad +d a h im + i.. f+d ca b calculad from h umbr of paricls a im as follows. For a crai posiio f+ dos h umbr of paricls locad a + a im. B h dfiiio of f+ dos h umbr of paricls displacd from + a im o i a im irval. Sic varis from - o h oal umbr of paricls displacd from a posiio a im o b displacm i i.. f+ is obaid b igraig f+ wih rspc o. Thus f+d is obaid as f d d f d. 5.5 Sic is assumd o b vr small compard wih f+ ca b padd i powr of o h firs ordr of as * Eisi [96] is usd as a rfrc for Eisi [95]. 66

77 f f f. 5.6 Furhr f+ ca b padd i powr of o h scod ordr of as Subsiuig 5.6 ad 5.7 io 5.5 ilds f f f f. 5.7 f f f f f d d d ca b wri as f f d 5.9 bcaus d b 5. ad d b L whr D is a cosa. Subsiuig 5. io 5.9 ilds d D 5. f D f. 5. If 5. is cosidrd as h o-dimsioal raspor quaio for diffusio h D ca b irprd as h diffusio coffici. Cosidr h cas whr N b paricls ar isrd isaaousl a = iiiall = io a ifii odimsioal mdium; N 5. f b whr A = hr ar o paricls i... f. 5.4 Th soluio for h diffusio quaio 5. subjc o h sid codiios 5. ad 5.4 is giv b 5.5. f N 4D b 4D. 5.5 I should b rmarkd ha 5.5 is spcificall applicabl o a ifii mdium i viw of h boudar codiio 5.4. f i 5.5 dos h profil of paricl disribuio i posiio ad im. Furhrmor f ca b irprd as h probabili of h paricl displacd i h im o a rgio bw ad +d sic h rlaiv frquc of h displacms pricd i succssiv irvals of im for a sigl paricl will likwis b giv b f i 5.5 [Eisi 96 os b Fürh]. Thus h probabili w ha h paricl has b displacd i h im o a rgio bw ad +d ca b wri as 67

78 f w N 4D 4D whr is a radom variabl ad is a drmiisic paramr. b 5.6 Th avrag displacm b diffusio of a paricl i im from h iiial i.. = posiio is dfid as Subsiuig 5.6 io h righ-had sid of 5.7 ilds B sig / 4D r w d w d 4D D h righ-had sid of 5.8 bcoms zro. d. 5.8 Thus 4 D 4D d 4D r r dr whr 5. is a applicaio of h scod quaio of 5. i.. <>=. Th avrag squar displacm dfid as b diffusio of a paricl i im from h iiial posiio is Subsiuig 5.6 io h righ-had sid of 5. ilds B sig w d w d 4D / 4D r h righ-had sid of 5. bcoms D d. 5. Thus 4 D 4D 4D d r r 4D dr D. 5. D 5.4 whr 5.4 is a applicaio of 5. i.. < >=D. I summar ad ar obaid as 5. D. 5.4 B h rlaioship bw ad as dfid i 5. w <> ad < > ca b obaid from w i 5.6 <> i 5. ad < > i 5.4 rspcivl. Bcaus wd=w d ad d=d from 5. h probabili w ha h paricl has b displacd i h im o a rgio bw ad +d ca b obaid from w b subsiuig 5. io 5.6 as 68

79 w 4D v 4D whr is a radom variabl ad v ad ar drmiisic paramrs. 5.5 <> dos h avrag displacm b diffusio ad advcio of a paricl i im from h iiial posiio ad < > dos h avrag squar displacm b diffusio ad advcio of a paricl i im from h iiial posiio. <> ad < > ca b obaid b subsiuig 5. io 5. ad 5.4 rspcivl as follows. Subsiuig 5. io h lf-had sid of 5. ilds Th lf-had sid of 5.6 ca b calculad as v. 5.6 v v v. 5.7 B 5.7 ad 5.6 is obaid as v 5.8 whr 5.8 shows ha h radom movm big avragd is drmiisic. Subsiuig 5. io lf-had sid of 5.4 ilds v D. 5.9 Th lf-had sid of 5.9 ca b calculad as B 5.9 ad 5. v v v v v. 5. Subsiuig 5.8 io 5. ilds v v D. 5. v D. 5. As show i 5. h ma of radom variabl is giv b h sum of wo drmiisic rms v ad D. 5.8 ad 5. ar h basic rlaios bw ad ad bw ad rspcivl. Usig 5.8 ad 5. h variac of i.. Var ca b calculad as follow. Var is dfid as Var. 5. Var is obaid b subsiuig 5.8 ad 5. io h righ-had sid of 5. as Var D Drivaio of 5.8 ad 5. from Sp Equaio for Cosa Vloci ad Diffusivi i a Ifii O-Dimsioal Mdium Th sp quaio i -ais giv b Tompso al. [987] for cosa v ad D ca b wri as v Z D 5.5 whr + ad ar radom variabls ad ar drmiisic paramrs. Z is a idpd radom variabl saisfig h wo propris [Tompso al. 987] 69

80 Z 5.6 ad Z. 5.7 Th sp quaio for ach im irval i.. = m- ca b wri b 5.5 as v Z v Z D D m m v Z m m v Z 5.8 m m D D whr is a cosa Z i dos a ralizaio of h radom variabl Z for i= m. A locaio of a paricl i h -coordia a im which dos m sps of irval i.. =m ca b obaid for = b summig h m quaios i 5.8 as m v D. 5.9 Z i i B akig avrag of 5.9 h avrag displacm of a paricl i im from h iiial posiio is prssd as v v D D m i m i Z Z i i. 5.4 Sic m i m Z i Z i i is giv b h rul of addiio of mas [Pima 99]. B bcoms v is h sam as 5.8. Thus h aalicall obaid rlaio bw ad i.. v has b drivd from h sp quaio 5.5. Similar o h abov procdurs for h avrag squar displacm of a paricl i im from h iiial posiio 5.4 is prssd as is obaid b akig avrag of squar of 5.9 as m v D. 5.4 Z i i m v D Z v D Z Sic v D ad ar cosa 5.44 ca b wri usig 5.4 as i m i v D Z v D Z i i Sic Z i dos a ralizaio of h radom variabl Z for i= m ad Z i is assumd o b idpd [Tompso al. 987] 5.45 ca b wri as m i m i i i 7

81 m v D Z v D Z 5.46 i i m i i whr m i m Z i Z 5.47 i i whr 5.47 dos h rul of addiio of variacs [Pima 99] for <Z>=. B bcoms m v D Z i i Sic Z as giv i ca b wri usig =m as v D 5.49 whr m m Zi m. 5.5 i i 5.49 is h sam as 5.. Thus h aalicall obaid rlaio bw ad i.. v D has b drivd from h sp quaio 5.5. I summar h wo aalical formula v ad v D hav b drivd from h sp quaio 5.5 for cosa v ad D giv b Tompso al. [987] Drivaio of Sp Equaio from 5.8 ad 5. for Cosa Vloci ad Diffusivi i a Ifii O-Dimsioal Mdium Th moms of radom variabl ar obaid aalicall i h Scio 5.. as v 5.8 v D 5. Var D 5.4 whr is a radom variabl is drmiisic paramr. 5.4 is obaid from h 5.8 ad 5. as show i 5.. Sic h posiio of a paricl a im + i.. + is drmid b h displacm of h paricl i a im irval i.. ˆ from ha of paricl a im i.. a radom variabl + is dfid as whr ad ˆ ar idpd radom variabls. ˆ 5.5 Sic ad ar drmiisic paramrs h ma ad variac of + ar obaid b subsiuig =+ io 5.8 ad 5.4 rspcivl as Subsiuig 5.5 io h lf-had sid of 5.5 ilds v [ ] 5.5 Var D[ ]. 5.5 B h rul of addiio of mas [Pima 99] 5.54 ca b wri as ˆ v [ ]

82 ˆ v [ ] or ˆ v [ ] Th ma of ˆ is obaid b subsiuig 5.8 io 5.55 as Subsiuig 5.5 io h lf-had sid of 5.5 ilds ˆ v Var ˆ D[ ] If ad ˆ ar assumd o b idpd radom variabls 5.57 ca b wri b usig rul of addiio of variacs [Pima 99] as Var Var ˆ D [ ] or Var ˆ D [ ] Var Th variac of ˆ is obaid b subsiuig 5.4 io 5.58 as Var ˆ D Th radom variabl ˆ dos h displacm of a paricl i a im irval ad has h ma v ad variac D. From a giv radom variabl ˆ a w radom variabl * ca b obaid b h rasformaio [Krszig 99] as whr a ad b ar cosa. * a ˆ b a 5.6 I h paricular cas whr a / Var ˆ ad b ˆ / Var ˆ h radom variabl * is dfid as Z ˆ ˆ * Z 5.6 Var ˆ ad is calld h sadardizd radom variabl [Krszig 99] bcaus Z has ma ad variac i.. <Z>= ad <Z >=. 5.6 ca b wri i rms of ˆ as Subsiuig 5.56 ad 5.59 io 5.6 ilds whr <Z>= ad <Z >=. ˆ ˆ Z Var ˆ. 5.6 Subsiuig 5.6 io h scod rm of righ-had sid of 5.5 ilds whr <Z>= ad <Z >=. ˆ v Z D 5.6 v Z D 5.64 I summar h sp quaio 5.64 i -ais for cosa v ad D has b drivd from h aalical formula v i 5.8 ad v D i 5.. 7

83 I viw of rsuls of Scio 5.. ad Scio 5..4 h rlaios v i 5.8 ad v D i 5. form cssar ad suffici codiios of h sp quaio [Tompso al. 987] wih cosa v o b valid Numrical Simulaio b Sp Equaio for Cosa Vloci ad Diffusivi i a Ifii O-Dimsioal Mdium I his scio i is ivsigad umricall if h rsulig disribuios of a larg umbr of paricls movd b h sp quaio 5.5 ar quival o hos obaid aalicall b 5.5 whr w i 5.5 dos h probabili dsi fucio of h radom variabl *. For h aalical soluio quaios 5.8 ad 5. ar usd o chck h agrm. v 5.8 v D 5. ad v Z D 5.5 whr Z is assumd o b a idpd radom umbr ** wih ma of zro ad variac of D [m /r] is a diffusio coffici ad is cosa i spac ad i im. Iiiall a sigl paricl is locad a =. Th paricl movs accordig o 5.5 for ach im irval o a giv obsrvaio im =m. Paricl movms for ach im irval dpd ol o h curr im ad ar o affcd b h hisor of h curr im. To gra h radom umbr Z i 5.5 umricall h probabili dsi fucio PDF is rquird. Sic ol moms of Z ar giv b Tompso al. [987] i his umrical simulaio a ormal disribuio wih ma ad variac is assumd for h PDF of Z. Sic =m h locaio of h paricl j a im i.. j ca b prssd b summig 5.5 for = m- as show i 5.9 as m j v DZi i pa h abov simulaio N ims o simula radom walk of N paricls. Sic movms of N paricls ar idpd of ohr paricls a avrag displacm of a paricl i im from h iiial posiio um is obaid umricall b avragig arihmicall j ovr N as j N m um v DZi N j i A avrag squar displacm of a paricl i im from h iiial posiio is obaid um umricall b avragig arihmicall j ovr N as j N m um v DZi N j i For h umrical simulaio cass ar cosidrd as show i Tabl 5.. For all cass h valu for h diffusio coffici i fr war D=.[m /r] is usd. I cas molcular diffusio is h domia mchaism for raspor P << [Bar 97]. I cas advcio ad molcular diffusio ar compiiv mchaisms.4<p <5 [Bar 97]. I cas advcio is h domia mchaism P > ~ [Tompso al. 987]. j * I h Appdi E i is show aalicall ha h sp quaio 5.5 saisfis h raspor quaio b chckig if h probabili dsi fucio w i 5.5 saisfis h rasi raspor quaio. Also wo dimsioal problm of radom walk icludig advcio i h - coordia ssm is cosidrd i his appdi. ** S Appdi A for radom samplig for logormal disribuio. 7

84 Probabili dsi fucio of Probabili dsi fucio of Probabili dsi fucio of Tabl 5. Ipu daa for umrical simulaio for cass Cass v [m/r] D [m /r] P =v L/D For L=m Tabl 5. Compariso of Aalical ad Numrical Cas Cas Cas <> <> um [%] <> <> um [%] <> <> um [%] E-. 7.5E E E E E-4 [<>-<> um ]/ <> Tabl 5. Compariso of Aalical ad Numrical Cas Cas Cas < > < > um [%] < > < > um [%] < > < > um [%] E E E E E E E E E-4 [< >-< > um ]/ < > = r Cas D=. [m /r] v =. [m/r] Cas D=. [m /r] v =. [m/r] = r = r Cas D=. [m /r] v =. [m/r]..... Figur 5. = r =5 r um.. =5 r = r =5 r = r 4 a Cas b Cas c Cas Probabili dsi fucio of for cass For all cass = [r] ad N= 7 ar usd. For ach cas valus of ad um ar obaid um umricall b 5.66 ad 5.67 rspcivl for = 5 ad [r]. Figur 5. shows h umricall obaid probabili dsi fucio of for cass. Tabl 5. ad Tabl 5. show h comparisos bw b 5.8 ad b 5.66 ad um b 5. ad b 5.67 rspcivl. I Tabl 5. is dfid i h fooo of h abl * ad dos h prc diffrc bw b 5.8 ad umb I Tabl 5. is dfid i h fooo of h abl ad dos h prc diffrc bw b 5. ad b um I Figur 5. corrspodig valus of -coordia o h paks of ach curv do h ma valus of for im. For cas i Figur 5.a sic h molcular diffusio is h domia raspor mchaism h * ad i Tabls 5. ad 5. ar calculad basd o h valus of <> um ad < > um bfor rucaio. 74

85 umricall obaid ma valus for diffr ims ar clos o zro as show i Tabl 5.. Paricl displacm from h ma valus is primaril drmid b D as giv i 5.4. For cas i Figur 5.c sic h advcio is domia h ma valus for diffr ims ar apar b v as giv i 5.8 ad paricls ar displacd aroud h ma valus. For cas sic h advcio ad h molcular diffusio ar compiiv mchaisms bhavior of h paricl movms is i bw wo cass ad. Thr is a diffrc bw umricall obaid disribuios i Figur 5. ad aalical formula prdics ha wih a vr shor im priod afr = som paricls ca mov vr fas ad is a >>. I umrical simulaio howvr a radom umbr is hardl grad i h ail par of h disribuio. Thus umrical simulaio cao gra h aalical formula prfcl. As show i Tabl 5. ad Tabl 5. bcaus umricall calculad Z ad Z ar o ad rspcivl * h prc diffrcs i.. um um ad bw aalical soluios ad umrical soluios ar obsrvd. For h compariso of as show i Tabl 5. h prc diffrc is rlaivl larg for cas a = r i.. =6.6% ad i dcrass as v icrass or as icrass. For cas rlaivl larg diffrcs occur bcaus h molcular diffusio i.. saisical radom rm which causs h saisical rror is domia. For cas howvr h diffrcs ar gligibl small sic h advcio i.. drmiisic rm is domia. For cas whr h advcio ad molcular diffusio ar compig mchaisms for raspor prc diffrcs ar gligibl small ad li i bw wo rm cass ad. For h compariso of as show i Tabl 5. prc diffrcs hav similar rds as ad ar gligibl small for all cass. Th diffrcs ad ca b dimiishd ihr b graig br radom umbr Z.g. <Z> um ad <Z > um or b icrasig h umbr of ralizaios i.. N. Howvr icrasig N dos o show h sam dgr as a improvm of rsuls bcaus h umrical flucuaio is proporioal o h squar roo of N [Kizlbach 988]. As show i Tabl 5. ad Tabl 5. h valus of ad um obaid umricall b h um sp quaio 5.5 for cosa v ad D ad hos obaid aalicall b 5.8 ad 5. rspcivl agr wll wih gligibl small diffrcs. Thus h aalical formula 5.8 ad 5. suppor h sp quaio 5.5 for cosa v Discussio Th sp quaio for cosa vloci ad diffusivi giv b Tompso al. [987] has a srucur of a paricular form of h rasformaio of radom variabl i.. sadardizd radom variabl Z. Th sadardizd radom variabl Z is usful for h ormal disribuio **. I h sp quaio howvr <Z>= ad <Z >= hav o spcial maig bcaus h probabili dsi fucio of Z is o drmid. This implis ha h sp quaio is o uiqu ad is a paricular form *** of h rasformaio of h radom variabl ˆ which dos h displacm of paricls i a im irval. I Scio 5. a mahmaical basis of h sp quaio has b dvlopd for cosa vloci ad diffusivi. Basd o h hor of h Browia movm [Eisi 95] h aalical formula <> ad < > hav b drivd Scio 5... I has b show umricall ha h aalical formula suppor h umrical soluio obaid b h sp quaio wih gligibl small diffrcs Scio I has b show aalicall ha h aalical formula <> ad < > ar drivd from h sp quaio for cosa vloci ad diffusivi Scio 5.. ad ha h sp quaio for cosa vloci ad diffusivi is drivd from h * <Z>= ad <Z >= ar applid i 5.5. ** Th sadardizd radom variabl Z is pracicall usful for ormal disribuio bcaus h poial rm i h probabili dsi fucio of h ormal disribuio cao b igrad b o of h mhods of calculus [Krszig 99]. Thus b dfiig Z=X-/ whr ad ar ma ad sadard dviaio of X rspcivl h disribuio fucio of X ca b calculad usig h abulad valus for disribuio fucio of h Z wih ma ad sadard dviaio i.. <Z>= ad <Z >= [Krszig 99]. *** Thr ar ifiil ma possibiliis for dfiig a w radom variabl for giv X.g. Z=X-/ Z a =X- / Z b =X-/ c. For Z a wo moms ar <Z a >= ad <Z a >=/4. Thus X ca b wri i rms of o ol Z bu also Z a Z b or c. as X=+Z X=+Z a or X=+Z b c. 75

86 aalical formula <> ad < > Scio Thrfor h aalical formula ar boh cssar ad suffici codiios for h sp quaio for cosa vloci ad diffusivi. 5. Govrig Equaio ad Sid Codiios To obai h sp quaio 5. Tompso al. [987] did o ak io accou h boudar codiios ad implicil assumd a ifii domai. As show i Scio 5. i is ssial o kow abou h domai for radom walks for drmiig h sp quaio. Paricularl h siz of h mdium ad h boudar codiios mus b kow o drmi a propr sp quaio. For ampl o driv h sp quaio 5.5 i his sud followig codiios ar usd. i Isaaous sourc ipu ii ifii mdium boudar codiio iii spac idpd war vloci ad diffusivi ad iv o dimsioal domai. If a crai codiio i i~iv is chagd h sp quaio should b chagd accordigl. Thus rigorousl spakig h sp quaio 5.5 obaid for a ifii domai wih cosa v ad D is o applicabl o a hrogous fii mdium. For mor rigorous ram a propr sp quaio should b obaid. I his sud h sp quaio 5.5 is applid wihou a jusificaio o ach fii riagular lm wih cosa vloci ad diffusivi. Sic movms of paricls i -coordia ar idpd of hos i - coordia h sp quaio i -coordia ca b obaid similar o h sp quaio i -coordia as show i Paricl movm ovr discr im sps is govrd b h followig radom walk algorihms which ar icorporad i h FFDF cod for h advcio ad h molcular diffusio for h boi-filld buffr FBC ad o-fbc rgios v v Z Z D D 5.68 whr Db D Df D for buffr for FBC for No - FBC whr + is h locaio of a paricl i dircio a im + Z ad Z ar assumd o b ormall disribud idpd radom umbrs a im wih ma of zro ad variac of. v ad v ar h uiform war vlociis i lm for h buffr * ad FBC rgios i ad dircios rspcivl ad ar obaid from h war flow aalsis as v =q / ad v =q / whr q ad q ar h Darc vloci compos i lm i ad dircios rspcivl. is h porosi i lm. I h No-FBC rgio i h NF v ad v ar assumd o b zro. D b D f ad D ar h ffciv molcular diffusio coffici icludig oruosi for h boi-filld buffr rgio h FBC rgio ad h o-fbc rgios rspcivl. D b D f ad D ar assumd o b cosa i im ad spac ovr h whol buffr FBC ad o-fbc rgios rspcivl. As a iiial sourc codiio isaaous ipu sourc codiio is applid o h surfac of h caisr. To simula his codiio N b paricls ar iiiall locad uiforml aroud h surfac of h was caisr wih a small disac awa from i for ach ralizaio of DFN ad h rsulig war flow vloci profil. Bcaus N diffr ralizaios will b simulad for war flow vloci disribuios or DFN for a giv s of saisics h movms of oal of N b N paricls ar cosidrd for h paricl raspor aalsis. Two boudar codiios ar prscribd i his aalsis. A absorbig boudar codiio ** is prscribd a h our boudar r=. A rflcig boudar codiio is prscribd a h surfac of h was caisr r=. A paricl ca mov io a rgio h boi-filld buffr FBC ad o-fbc bfor big absorbd a h our boudar. * Th homogous isoropic boi-filld buffr is cosidrd. Th boi swllig b war upak is glcd. ** Whvr a paricl arrivs a h boudar i is o cosidrd a mor. 76

87 5.. Eampl for rsuls of paricl raspor aalsis Paricl movm ovr h discr im sps is govrd b radom walk algorihm Th uiform war vlociis v ad v i lm i ad dircios ar obaid from h Darc vloci compos q ad q ad porosi as v = q / ad v = q /. I h No-FBC rgio v ad v ar assumd o b zro. Th ffciv molcular diffusio coffici icludig oruosi for h boi-filld buffr D b FBC D f ad o-fbc D rgios ar prscribd i his paricular ampl as D b =. - [m /r] ad D f =D =. -4 [m /r] [PNC 99; Ah 999]. Tim sp i 5.68 should b drmid proprl basd o h hdraulic ad raspor propris for ach rgio. For h diffusio domia rgios such as h buffr ad h o-fbc rgios h im sp is drmid b D / L d whr D=D b for h buffr ad D=D for h o-fbc rgio. Th characrisic lgh L d is - i.. hickss of h buffr for h buffr ad is assumd o b - /H i.. siz of a msh i radial dircio for h o-fbc rgio. H is h umbr of ods i h radial dircio bw h irfac r= ad h our boudar r=. For ampl H is for h rd figurs from h op for ach cas i Figur.7. rprss ha paricls pass hrough a lm wih / sp o mak sur ha h lm is sampld. Ths im sps for h buffr ad o-fbc rgios ar uchagd whil a paricl is i hos rgios for all ralizaios. Du o ragig war vloci ad molcular diffusio coffici a im sp for h FBC rgio should b drmid for ach lm for ach ralizaio boh b war vloci as v / L ad b molcular diffusio as D / L whr f v v v L is assumd o b - /H. Smallr bw im sp obaid b war vloci ad im sp obaid b diffusivi is usd for ach lm o mak sur ha paricls pric a las / im sps i ha lm. I his sud =.5 ad H= ar prscribd o drmi h im sp. As h iiial sourc codiio a isaaous ipu sourc codiio is applid o h surfac of h caisr. paricls ar iiiall locad uiforml a small disac.m awa from h caisr for ach ralizaio of war flow vloci profil as show i Figur 5. a. Th Darc vloci profil show i Figur. is cosidrd i his ampl. Two boudar codiios ar prscribd. A absorbig boudar codiio a h our boudar r= =5m ad a rflcig boudar codiio a h surfac of h was caisr r= =.5m. For h purpos of umrical calculaio acual absorbig boudar codiio is prscribd o a r= bu a r=. Imagiar absorbig boudar * is cosidrd a r= wih radius of 4.98m o avoid h umrical rror du o gap rgios which ar crad b riagular lm discrizaio i a circular domai ad ar ihr FBC or No-FBC rgios. Figur 5. a b ad c show h locaio of h paricls for paricular im sps of zro ad. Th phsical im for a paricl i h modl ca b obaid b muliplig h umbr of im sp ad h magiud of ach im sp of h paricl. I Figur 5. b bcaus all paricls rmai i h buffr rgio phsical im for all paricls is h sam ad is calculad as. I Figur 5. c howvr ach paricl has is ow im hisor bcaus im sp for ach paricl is diffr dpdig o is locaio i.. buffr o-fbc ad lms i FBC. Th rsidc im is obaid for all paricls b obsrvig h daild racks of ach paricl. Figur 5. shows h racks of all paricls uil h ar absorbd a h imagiar absorbig boudar r=. Paricls ca mov awhr i h buffr rgio bcaus h molcular diffusio is domia P ~ -6 mchaism i h buffr. Paricls mov alog h sramlis i h FBC rgio sic h advcio is domia P ~ 8. Sharp chags ar obsrvd i h racks of paricls i Figur 5. du o h uiform war vloci for ach lm bu diffr from lm o lm. Mor sigifica sharp chags ar obsrvd i h rgios clos o o-fbc rgio ad clos o h our boudar. Figur 5.4 a b ad c show h racks of all paricls for h sam ipu daa as Figur 5. ad Figur 5. cp for h highr diffusio coffici as D f =D =. m /r a D f =D =. m /r b ad D f =D =. 4 m /r c rspcivl. I Figur 5.4 c h diffusio is impora raspor mchaism as wll as h advcio whras h advcio is domia i Figur 5.4 a. * Dail abou h imagiar absorbig boudar is giv i Appdi F 77

88 a im sp b im sp c im sp Figur 5. Paricl raspor for diffr im sps ad for war vloci i Figur. Figur 5. Tracks of all paricls for war vloci i Figur. Figur 5.4 a. m /r P~ b. m /r P~ c. 4 m /r P~ Tracks of all paricls for war vloci i Figur. for highr diffusio cofficis i h NF FBC ad o-fbc rgio ha Figur Numrical rsuls Paricl raspor aalsis has b prformd o op of h flow modl for ralizaios for 9 cass show i Tabl.4. Effciv molcular diffusio coffici icludig oruosi for h boi-filld buffr FBC ad o- FBC rgios ar prscribd as D b =. - D f =. - ad D =. -4 m /r rspcivl [Nriks 98; PNC 99; Ah 999]. =.5 ad H= ar also usd. 78

89 I his scio umrical ploraios ar mad b FFDF o ivsiga ffcs of h hrogi of h NF ad ffcs of h modlig approach for h NF o h rsidc im of paricls ad h mass absorpio ra * a h our boudar Effcs of Hrogi of NF ad Modlig Approach for NF o sidc Tim Th rsidc im which is dfid as h im for a paricl o ak uil i arrivs a h imagiar absorbig our boudar r= is obaid b obsrvig h daild racks of ach paricl Effcs of Lgh Disribuio Figur 5.5 shows h CDFs of h rsidc ims of 4 paricls for a hrogous NF for 9 cass basd o h war vloci profils obaid i Scio.6... Figur 5.5 a ad i show h CDFs of h rsidc ims for h ir modl spac for cass A B ad C cass D E ad F ad cass G H ad I rspcivl. Th rsidc im for h ir modl spac is simpl dod as h rsidc im i his sud. Figur 5.5 b f ad j ad c g ad k ad d h ad l shows h rsidc ims for h buffr FBC ad No-FBC rgios rspcivl. Th rsidc ims for h buffr FBC ad No-FBC rgios do h buffr rsidc im FBC rsidc im ad No-FBC rsidc im rspcivl. Th mdia ma ad sadard dviaio of h rsidc ims ad h buffr rsidc ims ar show i Tabl 5.4. Th mdia ma ad sadard dviaio of h FBC ad No-FBC rsidc ims ar show i Tabl 5.5. Tabl 5.4 Saisics of sidc Tim for h Eir Modl Spac ad h Buffr gio for 9 Cass sidc im [r] Buffr rsidc im [r] Cas Mdia Ma Sadard dviaio Mdia Ma Sadard dviaio A.E+ 5.7E+.E+ 5.68E+ 7.85E+ 6.9E+ B 6.7E+ 9.E+.E+ 5.5E+ 6.6E+ 5.4E+ C 4.9E+ 6.45E+ 5.E+ 4.85E+ 6.E+ 5.5E+ D.6E+ 8.77E+.8E+ 6.4E+ 8.8E+ 7.96E+ E.4E+.4E+.4E+ 5.E+ 7.E+ 5.98E+ F 5.99E+ 8.5E+ 8.7E+ 5.E+ 6.94E+ 5.77E+ G 7.E+.8E+.75E+ 5.5E+ 6.7E+ 5.47E+ H 5.45E+ 7.84E+ 8.7E+ 4.95E+ 6.59E+ 5.4E+ I 4.89E+ 6.4E+ 5.E+ 4.85E+ 6.5E+ 5.4E+ Tabl 5.5 Saisics of sidc Tim for h FBC ad No-FBC gios for 9 Cass FBC rsidc im [r] No-FBC rsidc im [r] Cas Mdia Ma Sadard dviaio Mdia Ma Sadard dviaio A 7.58E-4.56E-.9E+.9E+ 4.9E+.E+ B.7E-4.E-4.98E-4.E+.7E+ 8.49E+ C.7E-4.7E-4 5.E-5.E+.4E+.8E+ D.45E-.4E+ 8.7E+.5E+ 7.9E+.6E+ E.E-4.74E-.6E- 6.E+.6E+.88E+ F.46E-4.84E-4.47E-4.E+.58E+ 5.6E+ G.6E-4.5E- 8.7E-.E+ 5.E+.67E+ H.58E-4.8E-4.E-4.E+.5E+ 6.4E+ I 9.8E-5.4E-4.4E-5.E+ 6.E- 9.E+ * Mass absorpio ra a h our boudar is usd as a comparabl quai o mass rlas ra a h boudar. S Scio 5.4. for mor dails. 79

90 CDF CDF CDF CDF CDF CDF CDF CDF CDF CDF CDF CDF sidc ims i h ir modl spac a sidc im[r] A B C sidc im[r] D E F i sidc im[r] G H I b Buffr rsidc im[r] A B C f sidc ims i h buffr Buffr rsidc im[r] D E F j Buffr rsidc im[r] G H I c FBC rsidc im[r] A B C g sidc ims i h FBC FBC rsidc im[r] D E F k FBC rsidc im[r] G H I sidc ims i h No-FBC d No FBC sidc im [r] A B C h No FBC sidc im [r] D E F l No FBC sidc im [r] G H I Figur 5.5 Cumulaiv disribuio fucios of h rsidc ims of 4 paricls. 8

91 Th rsidc ims of 4 paricls for 9 cass ar calculad b obsrvig h daild racks of ach paricl. Paricls ar iiiall locad uiforml aroud h surfac of h was caisr wih a small disac.m awa from h caisr. War flow ra i h buffr rgio is vr small as show i Tabl.6 for all 9 cass. Th warsaurad boi-filld buffr rgio is assumd o b homogous isoopic wih h ffciv molcular diffusivi of. - m /r. Du o a vr small Pcl umbr P ~ -6 i h buffr rgio for all ralizaios for all 9 cass molcular diffusio is domia for raspor. For cass G H ad I i Figur 5.5 j h diffrcs of h buffr rsidc ims for hs cass ar gligibl small. This is bcaus h o-fbc rgios for hs cass occup small aras i h NF du o wllcocd fracur work as show i Figur.7 ad hus paricls ou of h buffr rgio mov alog h sramli wihou r-rig h buffr rgio. Howvr for ohr cass A~F as show i Figurs b ad f paricls gig ou of h buffr rgio ca r-r h buffr rgio b molcular diffusio from h o-fbc ad h FBC rgio du o slow advcio. Ths paricls which r-rd h buffr rgio mak h diffrc of h buffr rsidc ims for cass from A o F. Paricls rlasd from h buffr rgio mov ihr o FBC or o No-FBC rgio b molcular diffusio. For cass B C F G H ad I h works ar wll cocd s Figur.7 so hus hav a grar FBC rgio which covrs mos of h modl spac. For hs cass mos paricls ou of h buffr mov alog h sramli aroud h buffr o h our boudar wihou rig h o-fbc rgio. Ths paricls hav zro o-fbc rsidc im as show i Tabl 5.6. Th FBC rsidc ims rag arrowl from -5 o - r for cass B C F G H ad I. Tabl 5.6 Numbr of Paricls Which Has Zro No-FBC sidc Tims Cas Numbr of paricls which has zro No-FBC rsidc ims for 4 paricls for ach cas A 86 B 6659 C 9676 D 4 E 77 F 755 G 568 H 88 I 988 For cass A D ad E a grar o-fbc rgio iss du o o wll-cocd works s Figur.7. Paricls i h o-fbc rgio ca mov io h FBC rgio b diffusio. If paricls mov io h FBC whr h war vloci is vr small * h ak a rlaivl log im o arriv a h our boudar rsulig i h wid variaio of h FBC rsidc ims as show i Figur 5.5 c ad g. Th FBC rsidc ims for hs cass ar ragd widl from -5 o ~ r. If h paricl dos o mov io h FBC rgio bfor i gs ou of h modl spac zro FBC rsidc im occurs. ou of 4 paricls hav zro FBC rsidc im for cas A ad ou of 4 paricls for cas D. This is possibl wh h paricl movs io h No-FBC rgio dircl from h buffr rgio ad is absorbd a h our boudar wihou goig io h FBC rgio. Zro FBC rsidc im is o obsrvd for cass B C E F G H ad I. If a paricl movs io h FBC rgio dircl from h buffr rgio ad was absorbd a h our boudar zro o-fbc rsidc im occurs. Th umbr of paricls which has zro o-fbc rsidc im is show i Tabl 5.6. Du o h occurrc of zro o-fbc rsidc im h CDFs of h o-fbc rsidc ims rag o from h zro bu from h raio of h umbr of paricls whos hav zro o-fbc rsidc im o h 4 paricls as show i Figur 5.5 d h ad l. Th rsidc im of a paricl i h ir modl spac is obaid b summig h rsidc ims i h buffr FBC ad o-fbc rgios. Th rsidc ims for 4 paricls ar summarizd i Tabl 5.4 ad Tabl 5.5. Th FBC rsidc im shows ha h mdia valus rag bw -4 r ad - r whras h rsidc im i h ir modl spac has h mdia valus ragig bw 5 r ad r. Thus h coribuio of * Th FBC rgio spciall adjac o h o-fbc rgio usuall has a rml low war vloci. 8

92 CDF CDF CDF CDF FBC rgio o h rsidc im i h ir modl spac is gligibl. O h ohr had rsidc ims i h buffr ad i h o-fbc ar comparabl o h rsidc im. For a mdium wih a grar FBC rgio du o wll-cocd fracur work such as cass B C F G H ad I h rsidc im is mail drmid b h buffr rsidc im. For a mdium wih a smallr FBC rgio du o o wll-cocd fracur work such as cass A D ad E h rsidc im is mail drmid b h o-fbc rsidc im Effcs of Aprur Corrlaio Paricl raspor has b simulad basd o h war vloci profils obaid i Scio.6.. for a modl wih a hrogous NF for cass A ad B wih h lgh-aprur corrlaio.7. Figur 5.6 ad Figur 5.7 show h CDFs of h rsidc ims of 4 paricls boh for h cass wih ad wihou h lgh-aprur corrlaio for cass A ad B rspcivl. I Figur 5.6 ad Figur 5.7 h dashd lis labld as corrlad rprs h CDF for h corrlad aprur wih h lgh of fracur b.7. Th solid lis i h figurs labld as ucorrlad ar h sam as i Figur 5.5 a~d. For ucorrlad cas aprur of a fracur sgm is grad basd o h giv logormal disribuio as show i Tabl.. Th mdia ma ad sadard dviaio of h rsidc ims ad h buffr rsidc ims boh for h cass wih ad wihou h lgh-aprur corrlaio for cass A ad B ar show i Tabl 5.7. Th mdia ma ad sadard dviaio of h FBC ad No-FBC rsidc ims ar show i Tabl sidc im[r] Cas A a Eir modl spac Ucorrlad Corrlad Cas A 4 5 Buffr rsidc im[r] b Buffr Ucorrlad Corrlad Figur Cas A FBC rsidc im[r] Ucorrlad Corrlad No FBC sidc im [r] c FBC d No-FBC Cas A Ucorrlad Corrlad Cumulaiv disribuio fucios of h rsidc ims for corrlad aprur for cas A 8

93 CDF CDF CDF CDF Tabl 5.7 Saisics of sidc Tim for h Eir Modl Spac ad h Buffr gio for Corrlad Aprur for Cass A ad B sidc im [r] Buffr rsidc im [r] Cas Mdia Ma Sadard dviaio Mdia Ma Sadard dviaio A.6E+.E+ 6.9E+ 5.7E+.4E+.E+ 5.9E+ 5.68E+ 8.6E+ 7.85E+ 7.8E+ 6.9E+ B 6.4E+ 6.7E+ 8.98E+ 9.E+.6E+.E+ 5.5E+ 5.5E+ 6.98E+ 6.6E+ 5.69E+ 5.4E+ Th valus i parhss ar for ucorrlad cas sam as hos i Tabl 5.4 Tabl 5.8 Saisics of sidc Tim for h FBC ad o-fbc gios for Corrlad Aprur for Cass A ad B Cas FBC rsidc im [r] No-FBC rsidc im [r] Mdia Ma Sadard dviaio Mdia Ma Sadard dviaio A.97E E-4.57E-.56E-.9E+.9E+.76E+.9E+ 5.49E+ 4.9E+.4E+.E+ B 4.96E-6.7E-4.6E-5.E-4 7.8E-5.98E-4.E+.E+.E+.7E+.49E+ 8.49E+ Th valus i parhss ar for ucorrlad cas sam as i Tabl 5.5 Figur sidc im[r] Cas B a Eir modl spac Ucorrlad Corrlad Cas B FBC rsidc im[r] Ucorrlad Corrlad Cas B Buffr rsidc im[r] b Buffr No FBC sidc im [r] Ucorrlad Corrlad Cas B Ucorrlad Corrlad 4 5 c FBC d No-FBC Cumulaiv disribuio fucios of h rsidc ims for corrlad aprur for cas B

94 As show i Figur 5.6 b ad Figur 5.7 b h diffrc of h buffr rsidc ims bw h cas wih h lgh-aprur corrlaio ad h cas wihou h corrlaio is gligibl bcaus h fracur works ar idical cp for aprurs boh for h cass wih ad wihou h lgh-aprur corrlaio for ach ralizaio. As show i Figur 5.6 c ad Figur 5.7 c h FBC rsidc im for h corrlad cas is smallr ha ha for h ucorrlad cas du o h highr war vloci for h corrlad cas ha ha for h ucorrlad cas. I Figur 5.6 c h FBC rsidc im for h corrlad cas rags widl from o.4 r ha ha for h ucorrlad cas as from.6-5 o.9 r. This is bcaus h flow bcoms mor hrogous for h corrlad cas whr h war flows hrough a fw log fracurs s Scio.6... Th diffrc of h rsidc ims i h ir modl spac boh for h cass wih ad wihou h lghaprur corrlaio is gligibl for cass A ad B bcaus h coribuio of h FBC rsidc im is gligibl Effcs of Modlig Approach for NF To obsrv ffcs of modlig approach for h NF paricls raspor has also b simulad for a homogizd NF b h uiformizaio for ralizaios for i cass i Tabl.4. Th mdia ma ad sadard dviaio of h rsidc ims for h ir modl spac buffr rgio ad FBC rgios ar show i Tabl 5.9 ad Tabl 5.. I h homogizd NF b h uiformizaio h ir NF is cosidrd as a FBC rgio ad is rprsd b sigl valus of hdraulic propris. Thus o-fbc rgio has b smard wih FBC rgio. Tabl 5.9 Saisics of sidc Tims for h Eir Modl Spac ad h Buffr gio for 9 Cass b h Uiformizaio sidc im [r] Buffr rsidc im [r] Cas Mdia Ma Sadard dviaio Mdia Ma Sadard dviaio A 4.49E+ 5.79E+ 4.56E+ 4.49E+ 5.79E+ 4.56E+ B 4.4E+ 5.78E+ 4.64E+ 4.4E+ 5.78E+ 4.64E+ C 4.49E+ 5.86E+ 4.65E+ 4.49E+ 5.86E+ 4.65E+ D 4.46E+ 5.8E+ 4.65E+ 4.46E+ 5.8E+ 4.65E+ E 4.46E+ 5.8E+ 4.56E+ 4.46E+ 5.8E+ 4.56E+ F 4.5E+ 5.8E+ 4.55E+ 4.5E+ 5.8E+ 4.5E+ G 4.49E+ 5.8E+ 4.5E+ 4.49E+ 5.8E+ 4.5E+ H 4.46E+ 5.75E+ 4.5E+ 4.46E+ 5.75E+ 4.5E+ I 4.56E+ 5.87E+ 4.6E+ 4.56E+ 5.87E+ 4.6E+ Tabl 5. Saisics of FBC sidc Tims for 9 Cass b h Uiformizaio Cas FBC rsidc im [r] Mdia Ma Sadard dviaio A 6.65E-5 6.9E-5.7E-5 B 6.4E-5 6.9E-5.6E-5 C 6.44E E E-6 D 6.4E E-5.94E-5 E 6.4E-5 6.6E-5.54E-5 F 6.4E-5 6.4E-5.E-5 G 6.4E-5 6.5E-5 8.9E-6 H 6.4E E-5 8.E-6 I 6.7E-5 6.4E E-6 84

95 CDF CDF CDF CDF CDF CDF CDF CDF CDF Cas A Hrogous Homogous Cas B Hrogous Homogous sidc im[r] sidc im[r] sidc im[r] a b c Cas D Hrogous Homogous Cas E Hrogous Homogous Cas C Hrogous Homogous sidc im[r] sidc im[r] sidc im[r] d f Cas F Hrogous Homogous Figur 5.8 Cas G Hrogous Homogous 4 5 sidc im[r] sidc im[r] 85 Cas H Hrogous Homogous Cas I Hrogous Homogous 4 5 sidc im[r] g h i Cumulaiv disribuio fucios of h rsidc ims of 4 paricls for h modl wih a hrogous NF b h local homogizaio solid li ad for h modl wih a homogizd NF b h uiformizaio dashd li for 9 cass Figur 5.8 shows h CDFs of h rsidc im of 4 paricls boh for h modl wih a hrogous NF solid li ad for h modl wih a homogizd NF dashd li for 9 cass. Th rsidc ims for h homogizd NF ar alwas smallr ha hos for h hrogous NF as show i Figur 5.8. For a modl wih a homogizd NF h rsidc im disribuios ar mail drmid b h buffr rgio bcaus h o-fbc rgio dos o is ad h coribuio of h FBC rsidc im o h rsidc im is gligibl du o h small magiud of h FBC rsidc im wih mdia ad mdia of ordr of -5 r. Thus h rsidc im disribuios for a modl wih a homogizd NF is h almos sam for all 9 cass as show i Figur 5.8. Th diffrc i h rsidc im bw h hrogous NF ad h homogizd NF is gligibl for cass C F H ad I. For cass A D ad E h diffrc of h rsidc im is sigifica bcaus h No- FBC covrs a grar ara of NF. Figur 5.9 shows h compariso of h 9% cofidc irvals of rsidc ims for h ir modl spac bw a modl wih a homogizd NF b h uiformizaio ad a modl wih a hrogous NF b h local homogizaio for ralizaios for 9 cass. Smbols ad rprs h mdia of h oal dischargs a h our boudar for ralizaios for ach cas for a hrogous NF ad for a homogizd NF rspcivl. Obsrvaios from h compariso of h Figur 5.9 ar as follows.

96 sidc im for ir modl spac [r].e+4.e+ B E H C F I.E+.E+.E+ A D G C ocivi Figur 5.9 Compariso of h 9% cofidc irvals of rsidc ims for h ir modl spac bw a modl wih a homogizd NF b h uiformizaio hick lis ad a modl wih a hrogous NF b h local homogizaio hi lis for 9 cass. Firs for h work wih h sam fracur gomr paramrs such as cass A B ad C a grar cocivi hibis a shorr rsidc im of paricls bcaus of highr flow ra ad smallr war-saga rgios which caus h dla of h paricl movm b molcular diffusio. Scod h rsidc ims for h ir modl spac for a homogizd NF ar shorr ad rag arrowl ha hos for a hrogous NF. Third h rsidc ims for h homogizd NF for all cass ar almos idical whras hos for h hrogous NF ar srogl dpd o h hrogi of h NF. This rsul occurs du o followig rasos. For h homogizd NF bcaus h o-fbc rgio dos o is ad h coribuio of h FBC rsidc im o h rsidc im for h ir modl spac is gligibl h rsidc im i h ir modl spac is mail drmid b h buffr rgio which is idical for all cass ad is idpd o h hrogi of h NF. Fourh a diffrc bw h local homogizaio ad h uiformizaio dcrass as h cocivi icrass. For cass C F ad I wih cocivi of 6. h local homogizaio ad uiformizaio giv similar rsuls Effcs of Hrogi of NF ad Modlig Approach for NF o Mass Absorpio a I som prvious prformac assssm sudis h mass rlas ra of radiouclids was obaid as o of h oupus from h raspor aalsis i h ar fild ad usd as ipu daa for h far-fild raspor [Gllig 997; JNC Vol. p.v-5]. I his sud as a comparabl quai o h paricl rlas ra a h our boudar paricl absorpio ra [paricls/im] a h boudar is cosidrd. Paricl absorpio ra is obaid b couig h umbr of paricls big absorbd a h boudar pr ui im. For h compariso h fracioal absorpio ra [r - ] a h our boudar is calculad b ormalizig h paricl absorpio ra b h oal umbr of paricls. Figur 5. shows h fracioal absorpio ra a h our boudar for a modl wih a hrogous NF b h local homogizaio a~c ad a modl wih a homogizd NF b h uiformizaio d~f for 4 paricls of cass A B ad C. Cass A B ad C hav h sam saisics of h fracur gomr paramrs as show i Tabl. ad Tabl.4 cp for h cocivi. For cas A h work is spars du o h small cocivi as. ad has irrgular shap of FBC du o h larg sadard dviaio of h fracur lgh. B icrasig h cocivi o. cas B ad o 6. cas C for h sam fracur gomr paramrs h work bcoms br cocd ad covrs a grar ara of h NF. 86

97 Fracioal absorpio ra [r - ] Fracioal absorpio ra [r - ] Fracioal absorpio ra [r - ] Fracioal absorpio ra [r - ] Fracioal absorpio ra [r - ] Fracioal absorpio ra [r - ] Tabl 5. Cas Maimum Fracioal Absorpio a a h Our Boudar Modl wih a hrogous NF Maimum fracioal absorpio ra [r - ] Tim for maimum fracioal absorpio ra [r] Modl wih a homogizd NF Maimum fracioal absorpio ra [r - ] Tim for maimum fracioal absorpio ra [r] A 4.6E E- 7.8 B.E- 8..8E-.5 C.6E E- 6.5 Bcaus fracioal absorpio ra a h our boudar is idical o h probabili dsi fucio PDF of h paricls big absorbd a h our boudar pr ui im i.. diffrial forms of CDFs of h rsidc ims i Figur 5.8 wih rspc o h im daild plaaios i prvious scios ar valid for Figur 5.. Tabl 5. shows h maimum fracioal absorpio ra ad im wh h maimum fracioal absorpio occurs a h our boudar boh for hrogous NF ad homogizd NF for cass A B ad C. Figur 5. a b ad c show ha h srog dpdc of h mass absorpio ra a h our boudar o h cocivi. For h modl wih a hrogous NF h mass of radiouclid absorbd a h our boudar icrass as h cocivi icrass for h sam fracur gomr paramrs. For cas A abou 4 ims grar mass is absorbd a h our boudar wih shorr rsidc im wih a homogizd NF Figur 5. d ha ha wih a hrogous NF Figur 5. a. For cas C howvr h modls wih h homogizd NF ad h modl wih h hrogous NF ild h similar mass absorpio ra a h our boudar. Figur 5. d ad f show ha h mass absorpio ra a h our boudar b h homogizd NF is almos idical for cass whras h srog dpdc o cocivi is obsrvd for h hrogous NF as show i Figur 5. a b ad c.. Cas A Hrogous NF. Cas B Hrogous NF. Cas C Hrogous NF cocivi=. cocivi=. cocivi= Tim [r] 4 5 Tim [r] 4 5 Tim [r] a b c Cas A Homogizd NF cocivi=.. Cas B Homogizd NF cocivi=.. Cas C Homogizd NF cocivi= Figur 5. Tim [r] 4 5 Tim [r] 4 5 d f Tim [r] Fracioal absorpio ras a h our boudar for 4 paricls for cass A B ad C wih hrogous NF a~c ad homogizd NF d~f 87

98 5.5 Coclusios A modl for h paricl raspor aalsis has b sablishd for h o-was caisr cofiguraio basd o h flow modl dvlopd i Chapr. A mahmaical basis of h radom walk for cosa war vloci ad diffusivi has b dvlopd. Paricl raspor is simulad b a radom walk rackig mhod WTM for h advcio ad molcular diffusio. Subrouis for paricl raspor hav b implmd io FFDF cod. Effcs of h hrogi of h NF i.. cocivi o h rsidc im ad h mass absorpio ra ar ivsigad for a modl wih a hrogous NF b local homogizaio b prformig h paricl raspor aalsis for diffr ss of saisics of fracur gomr paramrs. Numrical rsuls show ha: Th rsidc ims of paricls i h flow-barig rgio i.. FBC war-saga rgio i.. No- FBC ad h ir modl spac disribu mor widl as h cocivi of h fracur work bcoms smallr s Figur 5.5. Th rsidc im i h ir modl spac is drmid mail b ha i h buffr rgio for a work wih a larg cocivi. If h cocivi is small h rsidc i h war-saga rgio also bcoms comparabl o ha i h buffr s Figur 5.5. A work wih a grar cocivi hibis a shorr rsidc im of paricls ad a highr mass absorpio ra a h our boudar bcaus of a highr flow ra ad smallr war-saga rgios s Figur 5.9 ad Figur 5.. A diffrc i h rsidc ims boh for h cass wih ad wihou h lgh-aprur corrlaio is gligibl bcaus h coribuio of h rsidc im i h flow-barig rgio is gligibl small s Figur 5.6 ad Figur 5.7. Effcs of modlig approachs for h NF hav b ivsigad b comparig h rsidc im of paricls ad h mass absorpio ra a h our boudar bw h hrogous NF b local homogizaio ad h homogizd NF b uiformizaio. Numrical rsuls show ha: Th rsidc im obaid for a modl wih a homogizd NF is smallr ad rags mor arrowl ha ha obaid for a modl wih a hrogous NF s Figur 5.9. Th fracioal mass absorpio ra for h homogizd NF is largr ad disribus mor arrowl ha ha for h hrogous NF s Figur 5.. Diffrc i h rsidc im ad h fracioal mass absorpio ra bw h homogizd NF ad h hrogous NF bcoms grar as h cocivi bcoms smallr s Figur 5.9 ad Figur 5.. Th homogizd NF ilds almos sam rsidc im ad h absorpio ra of paricls for diffr characrisics of h NF i.. fracur gomr paramrs cocivi whras hos obaid for a hrogous NF ar ssiiv o h characrisics of h NF s Figur 5.9 ad Figur 5.. For h homogizd NF h rsidc im of paricls is mail drmid b h boi-filld buffr rgio s Tabl 5.9. Th homogizd NF cao rproduc h log ail of disribuio for rsidc im o h larg valu sid spciall for low cocivi cass. Howvr h homogizd NF givs cosrvaiv rsuls i.. udrsimaio of h rsidc im ad ovrsimaio of h mass absorpio ra. For a cosrvaiv saf assssm usig a homogizd NF assumpio ma b jusifid b his fac. Whil h ar-fild modl raig h hrogous NF as a homogous coiuum ilds cosrvaiv rsuls for paricl raspor i migh o b usd for si slcio bcaus i ca o diffria cadida rocks. 5.6 Limiaios & Fuur Sudis I his dissraio was dissoluio sorpio ad radioaciv dca ar glcd for h raspor aalsis. Cass i his sud ca b cosidrd as radiouclid wih log half-lif ad low sorpio disribuio coffici such as I-9. Afr failur of h caisr ars uclids i h was hav b assumd o b rlasd isaaousl b coacig wih groudwar ad o b disribud homogousl aroud h caisr wih a small disac.m awa from i. A isaaous puls ipu has b usd as h iiial sourc codiio. Th isaaous puls ipu codiio is usd i his aalsis bcaus h rsidc im for a mor gralizd rlas 88

99 modl ca b obaid b covoluio. For HLW rposior prformac assssm ralisic codiios such as cogru rlas ad solubili-limi rlas should b icorporad. This sud shows ha h aalical formula v ad v D drivd i Scio 5.. ar boh cssar ad suffici codiios for h sp quaio ol for cosa vloci ad diffusivi. Furhr sud is rquird for h spac ad im dpd war vloci ad diffusivi. I his chapr for h simulaio of h paricl raspor h sp quaio obaid for a ifii domai wih cosa v ad D has b applid wihou a jusificaio o h curr o-was caisr modl which cosiss of a larg umbr of fii riagular lms. A propr sp quaio is rquird for mor rigorous ram i h fuur. Th msh siz affcs h drmiaio of h war-flowig rgio i.. FBC ad war-saga rgio i.. No-FBC for a modl wih a hrogous NF. For h sam fracur work h flow-flowig rgio or warsaga rgio is chagd if h diffr msh siz applis. Thus w d furhr sudis o ivsiga h ffc of h fii lm msh o h paricl raspor i h fuur. 89

100 6. Evaluaio of Coffici C i Koz-Carma Equaio 6. Iroducio Th coffici C i Koz-Carma quaio. for h local homogizaio i Scio.4. ad.9 for h uiformizaio i Scio.4. is assumd o b ui i h umrical aalsis for h war flow Chapr. I his chapr h coffici C is valuad. If C is uiform ovr h modl spac ad idical for boh ad dircios h C is facord ou i.7. I such a cas h poial disribuio obaid i Chapr applis for a cosa valu of C. Howvr bcaus h valu of h war discharg is valuad b.5 ad.6 h war discharg is proporioal o h valu of C. Accordig o Carma [956] h coffici C is dpd o h shap of cross-scioal ara ormal o flow c ad h oruosi. O of h was o prss h coffici C is giv b Carma [956] as whr is a oruosi facor ad is giv b C / c 6. L 6. L whr L ad L ar dfid b Carma [956] as h lgh of h s-pic ad h avrag lgh of h pah of a sramli hrough h por spac rspcivl. Th valus of C ar alwas lss ha ui bcaus sic L L ad c > [Carma 956]. For a mdium wih a rcagular cross-scioal ara i.. a paralll sli h valu of c is [Carma 956]. Toruosi is rlad o h gomrical shap of h pah for war flow. I his chapr h oruosi is quaiaivl simad b usig daild iformaio abou h war flow pahs hrough h fracur work. I his sud usig a sigl valu of oruosi obaid for h ir FBC a sigl valu of coffici C is calculad b Toruosi Toruosi has b dfid as a raio bw h avrag lgh of h pah of a sramli L ad h lgh of s pic L. Bar [97] dfid h oruosi as L/ L whil Koz [97] dfid h oruosi as L/ L ad Carma [97; 956] dfid h oruosi as L /L. Koz [97] obaid h vloci v k hrough h graular bd basd o appar pah L alog h sramli b solvig h Navir-Soks quaios for all chals hrough a cross scio ormal o flow. Th vloci v k [m/s] is dfid as h quoi of h amou of war dischargs o h oal cross scioal ara. Koz assumd ha h graular bd is quival o a sram ub wih oal cross scioal ara cosisig of a group of paralll chals wih a quival op cross scio. I ordr o ak io accou h acual complicad pah of lgh L Koz iroducd h rducd prssur gradi rm corrcd wih L/L. Carma [97] modifid h vloci v k drivd b Koz [97] as v k L/L bcaus h acual pah pursud b a lm of h fluid is oruous. This wl iroducd rm L/L b Carma ad h rducd prssur gradi rm L/L b Koz ild h dfiiio 6. L/L. Th rm L /L is calld h oruosi facor b Carma [956]. Bar [97] plaid h oruosi facor b Carma [97] b iroducig h ffcs of oruosi i wo was. Th rm L/L iroducd b Koz [97] is du o h ifluc of oruosi o h drivig forc. Aohr rm L/L iroducd b Carma [97] is du o h ffc of oruosi o h vloci. Carma [97] foud ha h avrag agl bw h sramli ad h -ais is av =45 from h moio of a colord sramli hrough a bd of larg glass sphrs. Carma [97] adopd h avrag agl av of 45 o h dircio of flow ad hc usd L / L cos /. Howvr bcaus h agl bw h sramli av ad -ais could o hav a cosa valu bu varid from posiio o posiio i h porous mdium [Fowlr al. 94] Carma [956] claimd ha h corrc avrag for L/L should o b cos av bu <cos > as L L cos θ 9 6.

101 whr <cos > dos h avrag valu of cos ak ovr all valus of local agl i h porous mdium. I should b od ha 6. was obaid for h icomprssibl fluid ad viscous flow hrough a block of porous mdium of cosa cross-scioal ara. I his sud 6. is usd o sima h valu of oruosi i a fracur work s Scio Evaluaio of Toruosi Toruosi for a fracur work is simad i wo diffr was. Firs oruosi is calculad b h dfiiio of h oruosi giv i 6. which is basd o h pah lgh for war flow Scio 6... Scod oruosi is calculad b 6. which is basd o h oriaio agl of h flow pahs Scio 6... I h followig scios a sigl valu of oruosi ovr h ir fracur work is cosidrd. 6.. Evaluaio of Toruosi b Pah Lgh Toruosi b pah lgh i.. L is dfid basd o 6. as L L L 6.4 whr L dos h avrag lgh of h pahs hrough a fracur work * ad L dos h mdium lgh i prssur or poial applid dircio.g. -ais. To simpl h calculaio a squar modl spac is cosidrd. A wo-dimsioal squar L L modl spac Figur 6. is usd for h valuaio of h oruosi. A fracur sgm is dfid as a li lm wih lgh l ad aprur b which is h sam as i Chapr s Figur.. Th lf sid of h squar =-L/ ad righ sid of h squar =L/ ar calld h lf ad righ boudaris rspcivl i his chapr. L is s o b L = for simplici wihou losig grali. -L/L/ L L/L/ Lf boudar =-L/ igh boudar =L/ -L/-L/ L/-L/ Figur 6. Modl spac for oruosi simaio Avrag Lgh of Pahs Fracurs ar grad basd o giv saisics of h fracur gomr paramrs b radom samplig. For h purpos of discussio i his scio h uiform disribuio is assumd for h saisical disribuios of h fracur paramrs i.. locaio oriaio ~ ad aprur -4 ~ -. Lgh of a fracur sgm is s o b l=. For a discr fracur work graio i h modl spac ach fracur is grad idpdl ad h suprimposd i h modl spac. Whvr a fracur is addd i h modl spac h ircocivi of h fracurs ad h cocio of h ircocd fracurs wih h wo boudaris lf ad righ boudaris ar * I should b od ha L i 6.4 which is dfid i a fracur work is diffr from L i 6. which is dfid i a porous mdium. 9

102 chckd. If h firs cocio occurs bw h wo boudaris hrough h ircocd fracurs o mor fracur is addd Figur 6. Tabl 6. a b c a Fracur work graio i ui squar domai. b Ircocd fracurs cocig h lf ad righ boudaris. c Ircocd fracurs wih assigd labls a h irscios. Lis of Cocd Pois for Figur 6. c Pois Cocd pois Figur 6. a shows a fracur work graio i h ui squar modl. I Figur 6. a h firs cocio occurs a h 7 h fracur. Th ircocd fracurs hick lis will b cosidrd for h oruosi valuaio. Figur 6. b shows h ircocd fracurs cocig h lf ad righ boudaris. Figur 6. b is uilizd o masur a avrag lgh L of h pahs i h modl spac. A pah will b dfid blow. Th procdurs of masurig h avrag lgh of h pahs ar as follows. Each irscio of fracurs icludig h irscio of h fracur wih h lf ad righ boudaris is labld Figur 6.c. Irscio poi is simpl calld poi i his chapr. All pahs cocig h lf ad righ boudaris hrough h ircocd fracurs ar idifid s Appdi G. A pah is dfid as ss of ircocd fracurs bw wo pois o h lf ad righ boudaris rquirig ha ach poi icludd i h s should b visid a mos oc. 8 pahs ar idifid basd o h lis of cocd pois show i Tabl 6. s Appdi G for dail. Th idifid 8 pahs ar as follows ad A lgh of a pah is calculad b summig h disacs bw ighborig pois cosisig of a pah. For ampl a lgh of a pah is obaid b summig h disacs bw ad 6 6 ad 7 7 ad 9 ad 9 ad. Similarl lghs of h 8 pahs miod i ar obaid as ad.68 rspcivl. 4 A avrag lgh of pahs L is calculad b h arihmic ma of h lghs of h all idifid pahs. For h cas i Figur 6. c L is.5. 5 A valu of oruosi L/L for h fracur work i Figur 6. is calculad as.578 usig L =.5 ad L=. Procdurs o 5 should b rpad o sablish h saisics of oruosi Avrag Lgh of Flow Pahs Th pahs i h prvious scios ar idifid b gomrical cosidraio. Th gomrical pahs ar o cssaril acual flow pahs. Acual flow would occur i h dircio from a high-poial poi o a low- 9

103 poial poi. I his scio h flow pahs will b idifid for h fracur work b solvig h poial disribuio a h irscios of fracurs s Appdi H. No flow P L = P = Figur 6. No flow Fracur work wih assigd h labl a h irscios ad boudar codiios for war flow. As boudar codiios cosa prssurs ar prscribd as P L = a h lf =-.5 boudar ad as P = a h righ =.5 boudar. No flow boudar codiios ar prscribd a h op =.5 ad h boom =-.5 of h ui squar domai. Poial disribuio of h fracur work i Figur 6. is obaid b solvig h mass balac quaios of h flow ra a ach irscio. Th poials a h pois o h lf ad righ boudaris ar giv as P = P =P =P 4 =P 5 =. Poials a h irscios ar umricall obaid as P 6 =.97E- P 7 =.4E- P 8 =.E- P 9 =7.57E- P =6.E- P =.E- [kg/ms ]. For h procdurs o idif h flow pahs hrough h ircocd fracurs giv i h prvious scio o mor rul is addd. Bcaus h war ca flow from a highr poial o a lowr poial cocd pois i h d colum i Tabl 6. should hav lowr valus of poial ha hos of h pois i h s colum i Tabl 6.. B his poial cosidraio as a addiioal rul o idif a flow pah a lis of cocd pois is simplifid as show i Tabl 6.. As a rsul of cosidrig h poial 6 flow pahs for Figur 6. ar idifid basd o Tabl 6. as ad Two pahs ad which ar idifid i prvious scio b gomrical cosidraio ar limiad from h flow pahs. Th avrag lgh of pahs L is calculad for hs 6 flow pahs as.48. Th valu of oruosi L/L is calculad as.64 usig L =.48 ad L=. Tabl 6. Lis of Cocd Pois basd o Poial Disribuio i Figur 6. Pois Cocd pois Evaluaio of Toruosi b Oriaio Agl I his scio oruosi is valuad b oriaio agl of fracurs. Two formula ar cosidrd. Firs oruosi is valuad b oriaio agl of all fracurs icludd i h ircocd fracurs which coc h lf ad righ boudaris. Scod h oruosi * is valuad b oriaio agl * of fracurs which form h acual flow pahs hrough h fracur work. 9

104 is dfid [Fowlr al. 94; Carma 956] as cos θ 6.5 θ whr <cos > dos h avrag valu of cos ak ovr valus of local oriaio agl of all fracurs icludd i h ircocd fracurs which coc h lf ad righ boudaris. * is dfid as cos θ* 6.6 θ * whr <cos *> dos h avrag valu of cos * ak ovr valus of local oriaio agl * of all fracurs which form h acual flow pahs hrough h fracur work. I his sud fracurs which form h acual flow pahs ar idifid from h ircocd fracurs as follows. Firs h fracurs whos hav ol o irscio poi ar idifid. Scod h fracurs whos hav h sam valus of poial for all irscio pois o hos fracurs ar idifid. Bcaus war cao flow hrough h fracurs idifid i h firs ad scod procdurs fracurs formig h acual flow pahs ar drmid b cludig hos fracurs idifid i h firs ad scod procdurs. Th is calculad for h sam fracur work as usd for obaiig L whr is h oriaio of h fracurs icludd i h ircocd fracurs hick lis i Figur 6.b ad c. A valu of is.64 for h work i Figur 6. c. I Figur 6.c quals o * sic all ircocd fracurs form h acual flow pahs. 6.4 Numrical suls ad Obsrvaios For h sam fracur work h valu of orousi ovr h fracur work is calculad b hr diffr formula giv i his scio i.. L i 6.4 i 6.5 ad * i Effc of h Siz of Fracur Lgh o Toruosi Thr cass ar slcd o calcula h oruosi hrough a fracur work i h ui squar domai as show i Tabl 6.. For cas h firs cocio bw h lf =-.5 ad righ =.5 boudar hrough h ircocd fracurs ca occur wih fw log fracurs. For cas du o h giv shor lgh of fracur l=.5m mor fracurs ar rquird o occur h firs cocio ha ha for cas. Cas is slcd as l=.75 o s h ffc i bw l=. ad l=.5. To s h ffc of h siz of fracur lgh o h oruosi ohr fracur gomr paramrs ar kp h sam for hs cass. Figur 6.4 shows h cumulaiv disribuio fucios CDFs of oruosiis L ad * for ralizaios for h cass. Th ma ad sadard dviaio S.D. of L ad * for ralizaios ar show i Tabl 6.4. Th mdia valus of L ad * for ralizaios ar show i Tabl 6.5. Figur 6.5 shows h fracur works ircocd fracurs hick lis cocig h lf ad righ boudaris ad ircocd fracurs form acual flow pahs for a slcd ralizaio a h mos probabl oruosi of L idicad b do i Figur 6.4 from ralizaios for ach cas. A smallr valu of L mas ha h flow pah is mor oruous s 6.4. For oruosi grar ha.6 i Figur 6.4 diffrcs amog h curvs ar o sigifica. For oruosi smallr ha.4 h diffrcs amog curvs ar mor sigifica as h lgh l of fracur sgms dcras. Th mdia valus of for h cass i Figur 6.4 ar approimal.5 i ad.495 for cass ad rspcivl. This is bcaus h coiuous uiform disribuio is assumd for s Tabl 6.. Th for h ir squar domai ca b calculad for a giv coiuous uiform disribuio of oriaio as cos cos d Th mdia valus of * i Figur 6.4 ar ad.5 for cass ad rspcivl. Th valus of ad * ar idical if all ircocd fracurs form h acual flow pahs hrough h fracur work. A diffrc bw ad * i.. =.4 ad * =.56 for a ralizaio of cas i Figur 6.5 a coms from h fac ha som of fracurs icludd i h ircocd fracurs ar o icludd i h fracurs formig 94

105 CDF CDF CDF h acual flow pahs hrough h work. Howvr h diffrcs bw ad * ar o sigifica for cass as show i Figur 6.4. Amog h hr formula for oruosi * givs h largs mdia valus for hr diffr fracur lghs. Th diffrc bw ad * is o sigifica. L givs h smalls valu amog h hr which maks h war flow ra smalls. Bcaus L is cosidrd o b h mos rigorous dfiiio amog hr L should b usd. Howvr i mor ralisic siuaio whr housads o hudrds of housads of fracurs ar icludd calculaig h avrag pah lgh is im cosumig. For a pracical applicaio whr cosrvaiv simaio of war flow ras d o b obaid or * ma b usd. Tabl 6. Saisics of h Fracur Gomr Paramrs for Toruosi Cas Lgh l [m] Aprur b [m] Oriaio ag Disribuio ag Disribuio. -4 ~ - Uiform ~ Uiform.75-4 ~ - Uiform ~ Uiform.5-4 ~ - Uiform ~ Uiform Tabl 6.4 Saisics of L ad * L * Cas Ma Sadard dviaio Ma Sadard dviaio Ma Sadard dviaio 4.69E-.79E- 5.7E-.8E- 5.64E-.E- 4.8E-.87E- 5.E-.9E- 5.E-.6E-.48E-.5E- 5.9E- 8.59E- 5.4E- 9.4E- Tabl 6.5 Mdia Valus of L ad * Cas Mdia L * 4.6E- 5.7E- 5.6E-.7E- 4.96E- 5.5E-.4E- 4.95E- 5.E L * l= Toruosi Figur L * l= Toruosi L * l= Toruosi a l=. b l=.75 c l=.5 Cumulaiv disribuio fucios CDFs of L ad * for l=. a ad l=.75 b ad l=.5 c for ralizaios. 95

106 Fracur work graio Ircocd fracurs cocig h lf ad righ boudaris =.4 =.475 =.488 Ircocd fracurs formig acual flow pahs * =.56 * =.5 * =.468 a cas b cas c cas Figur 6.5 Fracur work ircocd fracurs hick lis cocig h lf ad righ boudaris ad ircocd fracurs formig h acual flow pahs for a slcd ralizaio for ach cas. 96

107 6.5 Coclusios & Discussios Th valus of h coffici C i Koz-Carma quaio for h ir FBC i h squar modl spac hav b calculad b hr diffr formula i.. L /c /c ad * /c. Th valu of c is assumd o b i his sud. Th valus of h oruosi ar calculad b h hr formula as L L/L <cos > ad * <cos *>. Amog hr L is h mos dirc formula. A umrical schm o calcula h avrag of h pah lgh of h discr fracur work has b dvlopd. ad * hav also b calculad b usig oriaio agls of fracurs icludd i h FBC racd from h DFN. Th horical valu of for a uiform agl disribuio bw ad has b obaid o b.5. Numrical rsuls also show ha h valu is clos o.5. Comparig h hr formula L givs h smalls valu whras diffrc bw ad * is o sigifica. Bcaus givs cosrvaiv ovrsima for oruosi wih which h war flow ra i h fracurd rock is ovrsimad ma b usd for a prformac assssm modlig. Amog hr diffr mhods for calculaio of oruosi is mos pracical i rms of umrical ffors bcaus ca b calculad immdial afr idifig h ircocd fracurs i.. FBC. For * addiioal umrical work.g. poial valus a h irscios of fracurs i his sud is rquird o idifig h fracurs formig acual flow pahs from h discr fracur work. For L h poial valus a h irscios of h fracurs icludd i h FBC mus b calculad o idif h flow pahs. Also idificaio of all flow pahs hrough h FBC rquirs h compuaioal ffors. For a ssm coais larg umbr of ircocd fracurs h mhods for L ad * rquir cosidrabl amou * of compuaio. Th avrag valu of h coffici C is calculad b /c =.5/=.67. I Chapr i has b assumd ha C=. Thus wih oruosi corrcio h war flow ra hrough FBC i h NF bcoms abou /6 of h valu obaid i Chapr. Alhough L ad * ar o usd o valua h coffici C i his sud i should b od i ha L is h mos dirc mhod o valua h oruosi for h fracur work ad ii ha * is mor ralisic ha bcaus h ol oriaio agls * of fracurs formig h acual flow pahs ar cosidrd. * For L h calculaio im is ovr a da wih Pium III-5 prsoal compur if h umbr of irscio pois is largr ha. For h calculaio of * i his sud fracurs which form h acual flow pahs should b idifid. For h idificaio poial valus a irscios of fracurs ar usd i his sud. To calcula poial valus a h irscios N umbr of irscio pois liar quaios should b solvd. 97

108 7. Flow Evaluaio wih Toruosi Corrcio 7. Iroducio I h flow modl dvlopd i Chapr a flow-barig clusr FBC racd from a discr-fracur work DFN has b rasformd io a hrogous coiuum b h local homogizaio Scio.4. ad a homogous coiuum b h uiformizaio Scio.4.. I Chapr 6 oruosi corrcio o h prmabili for h ir FBC has b discussd ad a sigl valu of C was obaid for h FBC. I has b cocludd ha for a cosrvaiv ovrsima of war flow ras ma b usd. I his chapr h war flow aalsis hrough a fracur work is mad i wo diffr was i.. a DFN modl ad a homogous coiuum modl *. I h DFN modl war flow ra hrough h fracur work is calculad b solvig mass cosrvaio quaios a irscio pois of fracurs [Log al. 98; Schwarz al. 98; obiso 984; Log 985; Adrsso al. 987; Dvrsorp al. 99]. I h homogous coiuum modl dvlopd i Scio.4. oruosi ** b h oriaio agl is icorporad i h coffici C /c. suls from hs approachs ar compard. 7. Flow a hrough a Fracur Nwork 7.. Modl Spac ad Boudar Codiios A wo dimsioal ui squar domai is dfid as show i Figur 7.. A li lm rprss a fracur wih lgh l ad aprur b. A fracur is grad basd o giv saisics of h fracur gomr paramrs. For a fracur work graio oal umbr of fracurs N f is prscribd i his chapr. Figur 7. shows a fracur work wih N f =7 for cosa lgh l=. m cosa aprur b= -4 m ad uiform oriaio ~ of fracurs. To calcula h flow ra i h modl spac boudar codiios ar prscribd as show i Figur 7.. Cosa had boudar codiios ar prscribd a h lf boudar =-.5 as =.[m] ad a h righ boudar =.5 as = [m]. No flow boudar codiios ar prscribd a h op =.5 ad h boom =-.5 boudaris. No flow Lf boudar = -.5 igh boudar =.5 =. [m] = [m] No flow.5-.5 Figur 7. Modl spac for flow ra calculaio * I his chapr a hrogous coiuum b local homogizaio is o cosidrd bcaus h coffici C has b obaid o for h local riagular lms bu for h ir NF i Chapr 6. ** Toruosi is o rquird o calcula h flow hrough a work of paralll fracurs. I his sud s Chaprs ad 5 howvr bcaus h flow ra is obaid b rasformaio of h FBC io a coiuum usig Koz- Carma quaio for h prmabili oruosi is dd. 98

109 Figur 7. A fracur work wih N f =7 for cosa lgh l=. aprur b= -4 m ad uiform locaio ad oriaio. 7.. Discr Fracur Nwork DFN Approach War flow hrough h fracur work has b modld b h DFN approachs [Log al. 98; Schwarz al. 98; obiso 984; Log 985; Elsworh 986; Adrsso al.986; Adrsso al. 987; Log al.987; Drshowiz al. 994]. I h DFN approachs idividual fracurs ar cosidrd o drmi h flow ra of war. A fracur is modld as paralll-pla war coducig chal. War vloci hrough a cross-scio ormal o flow is obaid b solvig h Navir-Soks quaios for a im-idpd lamiar flow i a viscous icomprssibl fluid. Th war flow ra ormal o flow hrough a fracur paralll pla of a lgh L wih a diffrc of hdraulic had is calculad approimal as h produc of h avrag vloci of war hrough h fracur wih aprur b ad h cross-scioal ara bw ormal o flow whr w is h widh of a fracur ormal o - pla. Th flow ra hrough idividual fracur for w=[m] is wri as: Q f gb w 7. L whr w Ns/m for C is h viscosi of war 996 kg/m for C is h dsi of war g9.8 m/s is h cosa for gravi b is h aprur of a fracur ad [m] is h hdraulic had. Hdraulic had is h sum of h lvaio had z ad h prssur had P/g whr P [kg/ms ] is h prssur. P z 7. g I Figur 7. h hdraulic hads a pois ad ar prscribd as = =. ad = [m]. Th hdraulic hads a 8 iral pois from 4 o ar ukow. Th disribuio of h hdraulic had a all irscios is calculad b solvig h N h umbr of irscios of fracurs i.. N=8 i Figur 7. liar quaios wih assumpio of mass balac a irscios of fracurs s Appdi H *. Th valus of ukow hdraulic hads i Figur 7. ar calculad umricall as 4 =5.5E- 5 =7.7E- 6 =6.84E- 7 =8.897E- 8 =4.4E- 9 =.849E- =5.4E- ad =4.6E- [m]. Th a flow ra hrough ach fracur is calculad b 7. wih hs obaid valus of. For ampl a flow ra from pois 9 o Q f9- is calculad as Q gb [ m / ] 7. w L 9 f 9 s * To calcula h war flow ra hrough h fracur work h hdraulic or poial had a h irscios of fracurs is o cssaril dd. I Appdi I h war flow ra is calculad for a simpl fracur work wihou calculaig h valus of poial had a h irscios of fracurs [Chambré ]. 99

110 whr L -9 is h disac bw pois ad 9 ad obaid umricall as.86e- [m]. Th posiiv quai mas ha war flow from irscio 9 o irscio Figur 7. Disribuio of hdraulic had a h irscios of fracurs Flow ras bw ad 7 ad ad 7 ar calculad b 7. as Q f-7 =8.48E-8 [m /s] wih L -7 =.4E- [m] ad Q f-7 =.758E-8 [m /s] wih L -7 =6.87E- [m]. Th sum of h flow ras hrough idividual fracurs which irsc h boudar ilds h oal flow ra of war a ha boudar Q DFN. Q DFN =Q f9- =.4E-7 [m /s] is h oal ouflow ra a h righ boudar. Q DFN =Q f-7 +Q f-7 =.4E-7 [m /s] is h oal iflow ra a h lf boudar. Th umrical rsuls which show h qual amou of iflow ad ouflow ra cofirm ha h mass cosrvaio of h modl spac. I should b od ha h oruosi is o rquird o calcula h flow hrough a work of paralll fracurs as show i Figur Homogous Coiuum Approach War flow hrough h fracur work has b aalzd b rasformaio of h DFN io a coiuum i Chapr. For a homogous coiuum b h uiformizaio i Scio.4. sigl valus of h porosi h spcific surfac ara M s [ / m] h prmabili k [ m ] ad h hdraulic coducivi K [ m/ s] ar obaid for h ir FBC i Figur 7. b ad. rspcivl. Ara A i.7 ad.8 dos h ara of h ir NF rgio. Sic h ir squar modl spac i Figur 7. rprss h NF A= [m ] is usd i his scio. I ordr o ak io accou h ffc of oruous flow pahs s Chapr 6 hrough h fracur work h coffici C i.9 is calculad wih h oruosi b oriaio agl of fracurs as cos C 7.4 c c whr <cos > dos h avrag valu of cos ak ovr all valus of local oriaio agl of fracurs icludd i h ircocd fracurs which coc h lf ad righ boudaris. c is [Carma 956]. Tabl 7. Uiformizd Hdraulic Propris for h Fracur Nwork i Figur 7.. M s [ / m] cos C k [ m ] K [ m / s] Q HOMO [ m / s] 5.555E-4.E+.68E-.E-.4E-.77E-6.77E-7 w = [Ns/m ] =996 [kg/m ] g=9.8 [m/s ] For h calculaio of h hdraulic coducivi for h ir FBC b. h ssm mpraur of C is assumd. Numricall obaid hdraulic propris for h ir fracur work i Figur 7. ar show i Tabl 7.. I his abl <cos > i h hird colum is obaid b avragig cos ak ovr all valus of local

111 oriaio agl of sv fracurs i Figur 7.. A flow ra hrough a homogous domai Q HOMO [m /s] i.. homogous coiuum is calculad approimal b Darc s law as: Q HOMO [ m 7 / s] K A.77 [ m / s] L whr K.77 [ m/ s] A [ m ] L Numrical suls Tabl 7. shows h flow ras hrough h fracur work i Figur 7. b DFN Q DFN ad homogous coiuum Q HOMO. I Tabl 7. Q HOMO is obsrvd o b largr ha Q DFN. War flow ras hrough h fracur works ar calculad for four cass i Tabl 7. for ralizaios. For ach ralizaio of DFN war flow ras ar calculad b h DFN approach i Scio 7.. ad b h homogous coiuum approach i Scio 7... Tabl 7. Flow as hrough h Fracur Nwork i Figur 7. Q DFN.4E-7 Flow ra [m /s] Q HOMO.77E-7 For cas i Tabl 7. fracurs wih a cosa lgh l=[m] ad aprur b= -4 [m] ar grad i h modl spac. Th oriaio agls of fracurs ar disribud uiforml bw ad. For cas o ivsiga h ffcs of h umbr of fracurs h sam fracur gomr paramrs ar usd as cas cp for h umbr of fracurs. For cas o ivsiga h ffcs of h lgh of fracurs h sam fracur gomr paramrs ar usd as cas cp for h lgh of fracurs. A half ims shorr lgh of fracurs for cas ha ha for cas is prscribd. For cas 4 o ivsiga h ffcs of disribuio of lgh a logormal disribuio wih ma ad sadard dviaio of.5 i.. L =.8 ad L =.54 b.6 is cosidrd. Tabl 7. Cass Paramrs for War Flow a Numbr of Fracurs N f Lgh l [m] Aprur b [m] Oriaio ag Disribuio. -4 ~ Uiform ~ Uiform ~ Uiform 4 6 Logormal disribuio wih L =8.E- ad L [m]=.54e- -4 ~ Uiform Tabl 7.4 Flow as b DFN Q DFN ad Homogous Coiuum Q HOMO Approachs for alizaios for 4 Cass i Tabl 7.. Q DFN [m /s] Q HOMO [m /s] Cas Mdia Ma Sadard dviaio Mdia Ma Sadard dviaio 7.5E-7 7.6E-7.8E-7.E-6.9E-6.44E-7.77E-6.78E-6.78E-7.8E-6.9E-6.97E-7 6.8E-7 6.7E-7.44E-7.E-6.E-6.4E E E-7.85E-7.9E-6.E-6.6E-7

112 CDF CDF CDF CDF Tabl 7.4 show h mdia ma ad sadard dviaio of h flow ra b DFN Q DFN ad homogous coiuum Q HOMO approachs for ralizaios for 4 cass i Tabl 7.. Figur 7.4 shows h cumulaiv disribuio fucios of h flow ra b DFN solid ad homogous coiuum dashd approachs for ralizaios for four cass i Tabl 7.. For cas a diffrc i mdia valus bw h flow ra b DFN Q DFN ad h flow ra b homogous coiuum Q HOMO is.%. Cass ad rprs h works wih h sam gomr of fracurs bu diffr umbr of fracurs. B icrasig h umbr of fracurs from N f = o N f =6 h flow ra icrass as h umbr of fracurs is icrasd ad a discrpac bw h mdia valus of Q DFN ad Q HOMO ar dcrasd from 46.% o.%. For cas a diffrc i mdia valus bw h flow ra b DFN Q DFN ad h flow ra b homogous coiuum Q HOMO is 9.%. Cass ad rprs h works wih h sam umbr of fracurs bu diffr siz of fracur lgh. B dcrasig h siz of fracur lgh l= o l=.5 a discrpac bw h mdia valus of Q DFN ad Q HOMO ar icrasd from.% o 9.%. For cas 4 a diffrc i mdia valus bw h flow ra b DFN Q DFN ad h flow ra b homogous coiuum Q HOMO is 94.%. Cass ad 4 rprs h works wih h sam gomr of fracurs bu diffr disribuio of lgh. B graig h largr ad smallr fracur lghs ha is ma valu i logormal disribuio flow ras rag mor widl ha hos for h work wih a cosa lgh of fracurs DFN Homogous Cas N f = l= b= Cas N f = 6 l= b= -4 DFN Homogous Flow ra [m /s] Flow ra [m /s] a Cas b Cas DFN Homogous Cas N f = 6 l=.5 b= DFN Homogous Cas 4 Lgh: Logormal Ma=.5 SD=.5 N f = 6 b= -4 Figur Flow ra [m /s] Flow ra [m /s] c Cas d Cas 4 CDFs of h flow ra b DFN solid ad homogous coiuum dashd approachs for ralizaios for four cass i Tabl 7..

113 7.4 Coclusios & Discussios I his chapr h war flow aalsis hrough a fracur work has b mad b a discr-fracur work DFN modl ad b a homogous coiuum modl. For h homogous coiuum modl h oruosi corrcio has b icorporad i h coffici C /c i h prmabili as show i.. For h sam fracur work h flow ras obaid b wo modls ar compard. Numrical obsrvaios ar summarizd for h sam fracur gomr paramrs as follows: As h umbr of fracurs icrass h war flow ra icrass ad a diffrc bw h mdia valus of Q DFN ad Q HOMO dcrass. As h siz of fracur lgh dcrass h war flow ra dcrass ad a diffrc bw h mdia valus of Q DFN ad Q HOMO icrass. As h lgh of fracurs varis from is ma valu war flow ras rag mor widl. For all cass cosidrd i his sud h homogous coiuum approach ilds h ovrsimad war flow ras compard wih hos b h DFN approach. Th curr sud could jusif h cosrvaism of h modl raig h discr fracur work as a homogous coiuum. I his chapr paricl raspor has o b simulad bcaus h fracioal mass absorpio ra for h homogous coiuum i.. uiformizd NF is lss dpd o h dgr of h hrogi of h NF i.. cocivi s Figur 5.. Howvr h rsidc im of paricls i h uiformizd NF could b diffr bcaus h rsidc im i h NF dcrass or war flow ra icrass as h cocivi icrass. For h rasformaio of h FBC io a coiuum ol homogous coiuum b h uiformizaio is cosidrd i his chapr bcaus h coffici C wih oruosi corrcio is assumd o b uiform for h ir FBC s Chapr 6.

114 8. Coclusios I h ar-fild rock NF ircocd fracurs ar major coduis for groudwar flow ad radiouclid raspor. Hrogi of h raspor pah i h NF ca sigifical affc rsuls for HLW rposior prformac assssm. Howvr i h prvious ar-fild modls for war-saurad HLW rposior prformac assssms h hrogi of h NF was o icludd i h radiouclid raspor aalsis. Th objcivs of his dissraio ar o dvlop a ar-fild modl icorporaig h hrogi of h NF i h groudwar flow ad paricl raspor aalss ad o ivsiga ffcs of h hrogi of h NF ad ffcs of h modlig approachs for h NF o h groudwar flow ra ad paricl raspor i h ar fild. A hpohical simplifid rgio icludig o caisr h buffr ad h ar-fild rock is cosidrd. A wo-dimsioal modl for h groudwar flow simulaio has b dvlopd for h im-idpd icomprssibl Darc s flow. To icorpora h hrogi of h NF plicil a discr fracur work is grad i h NF. A flow-barig clusr FBC of fracurs is racd from h fracur work b chckig h ircocivi of fracurs. Th FBC is rasformd io a hrogous coiuum ad a homogous coiuum o compar h modlig approachs for h NF. Th Koz-Carma quaio is usd for prmabili. A fii lm soluio has b drivd for h spaial disribuio of h hdraulic poial i h FBC. Th umrical modl dvlopd for flow aalsis has b bch-markd agais a mahmaical modl b comparig h rsuls b h umrical soluio wih hos b h aalical soluio for h giv spcial cass. A modl for h paricl raspor simulaio has b sablishd basd o h flow modl b applig h radom walk rackig mhod. A compur cod FFDF has b dvlopd for h war flow ad h paricl raspor. Effcs of h hrogi of h NF ad ffcs of h modlig approach for h NF hav b ivsigad for diffr ss of saisics of fracur gomr paramrs b comparig h groudwar flow ras h paricl rsidc ims ad mass absorpio ras bw h modl wih a hrogous NF b locall homogizig h DFN i h NF ad h modl wih a homogizd NF b uiformizig h DFN ovr h ir NF. Th coffici C i Koz-Carma quaio for h prmabili has b valuad b calculaig oruosi for h fracur work usig hr formula L ad *. Th homogous coiuum modl rasformd from h discr fracurs b uiformizaio icorporaig h coffici C /c wih oruosi corrcio has b compard wih h covioal DFN modl for h sam fracur work. For h prformac assssm of h war-saurad HLW rposior h curr sud shows ha: For war flow aalsis i fracurd rock h NF as a homogous coiuum is o a appropria approach bcaus i dos o cssaril ild a cosrvaiv war flow ra. Th simplificaio i.. h NF was rad as h homogous coiuum iroducd i prvious ar-fild modls for radiouclid raspor is jusifid b h fac ha a homogous coiuum ilds cosrvaiv rsuls i.. udrsimas h rsidc im ad ovrsimas h mass absorpio ra compard o h modls icorporaig h hrogi of h NF i.. DFN hrogous coiuum. If h NF is rad as a homogous coiuum for radiouclid raspor h boi-filld buffr plas a impora rol as a barrir for h rlas of radiouclids ad h ffcs of h hrogi of h NF o h radiouclid raspor ar gligibl. To icorpora h oruois corrcio o h prmabili could b usd for a prformac assssm modlig bcaus cosrvaivl ovrsimas h oruosi ad hus h war flow ra i h NF is ovrsimad ad rquirs h las compuaioal ffors amog hr mhods. Th lgh-aprur corrlaio which is cosidrd o is bu which is vr difficul o obai b fild masurms could b glcd for h prformac assssm bcaus h ffc of h lghaprur corrlaio o h raspor i h ar-fild is gligibl. For h purpos of h si slcio i which prformac of various kids of hos rocks ds o b diffriad h curr sud shows ha: For a sparsl fracurd rock a homogous coiuum b uiformizig h rock is o a appropria approach o modl h war flow ad paricl raspor bcaus i cao proprl rproduc high ad low ails of h flow ras h rsidc im of paricls ad h mass absorpio ras whras h hrogous coiuum b locall homogizig h rock ca gra hos ails proprl ad 4

115 i wips ou h diffrcs i h rsidc im ad h absorpio ra or rlas ra of solu amog diffr rocks. 5

116 EFEENCES Abli H. I. Nriks S. Tubra ad L. Moro Migraio i a sigl fracur: Eprimal rsuls ad valuaio fial rpor Sripa projc Tch. p. 85- Swd. Nucl. Ful ad Was Maagm Co. SKB Sockholm Ma 985. Ah J. Mass rasfr ad raspor of radiouclids i fracurd porous rock Ph.D. Dissraio Uivrsi of Califoria Brkl 988. Ah J. T. Ikda T. Oh T. Kao Y. Sakamoo T. Chiba M. Tsukamoo S. Nakaama S. Nagasaki K. Bao ad T. Fujia Quaiaiv prformac allocaio of muli-barrir ssm for high-lvl radioaciv was disposal Joural of Aomic Erg Soci of Japa Ah J. P. L. Chambré B. Ligh D. obrs ad J. Vrbk adiouclid raspor i disurbd zo bw gird ad aural barrirs of dp gologic disposal of high-lvl radioaciv wass Dparm of Nuclar Egirig U.C. Brkl UCB-NE Ah J. S. Kuo D. A. obrs ad P. L. Chambré las of radiouclids from mulipl caisrs i a high lvl was rposior Dparm of Nuclar Egirig U.C. Brkl UCB-NE Ah J. Prlimiar assssm of h ffcs of ATW ssm applicaio o YM rposior prformac Iraioal cofrc o fuur uclar ssms Global 99 Womig 999. Adrsso J. ad. Thuvik Prdicig mass raspor i discr fracur works wih h aid of gomrical fild daa War sourcs sarch Adrsso J. ad B. Dvrsorp Codiioal Simulaios of Fluid Flow i Thr Dimsioal Nworks of Discr Fracurs War sourcs sarch Adrsso J.. Nordqvis G. Nbrg J. Smlli ad S. Tiré Hdrogological codiios i h Fisjö ara: Compilaio of daa ad cocpual modl SKB Tchical rpor Bachr G. B. N. A. La ad H. H. Eisi Saisical dscripio of rock propris ad samplig Proc. of h 8 h U.S. Smp. o ock Mchaics Amrica Isiu of Miig Egirs 977. Bar J. Damics of fluids i porous mdia Amrica Elsvir Publishig Co. 97. Bar J. Hdraulics of groudwar McGraw-Hill Nw York 979. Bar J ad A. Vrruij Modlig groudwar flow ad pollua: Wih compur programs for sampl cass Boso 987. Biachi L. ad D. T. Sow Prmabili of crsalli rock irprd from masurd oriaios ad aprurs of fracurs Aals of h Arid Zo Boadu. F. K. ad L. T. Log Th fracal characr of fracur spacig ad QD I. J. of ock Mch. & Mi. Sci. & Gomch. Absr Bour O. ad P. Dav Cocivi of radom faul works followig a powr law faul lgh disribuio War sourcs sarch Bourk P. J. E. M. Duac M. J. Hah ad D. D. Hodgkiso Fracur hdrolog rlva o radiouclid raspor AEE p. 44 Aomic Erg s. Esab. Harwll Uid Kigdom 985. Bo. G. E. P. ad M. E. Mullr A No o h graio of radom ormal dvias Aals of Mahmaical Saisics Bral P. B. L. Fo. ad L. E. Schrag A Guid o simulaio Sprigr-Vrlag Nw York 98. 6

117 Cacas M. C. E. Ldou G. d Marsil B. Tilli A. Barbrau E. Durad B. Fuga ad P. Paudcrf Modlig fracur flow wih a sochasic discr fracur work: Calibraio ad validaio.. Th flow modl War sourcs sarch a. Cacas M. C. E. Ldou G. d Marsil A. Barbrau P. Calmls B. Gaillard ad. Margria Modlig fracur flow wih a sochasic discr fracur work: Calibraio ad validaio.. Th raspor modl War sourcs sarch b. Carma P. C. Fluid flow hrough graular bds Trasacios of h Isiuio of Chmical Egirs Carma P. C. Flow of gass hrough porous mdia Acadmic prss Nw York 956. Chambré P. L. Lcurs i smiar NE98- Dparm of Nuclar Egirig U.C. Brkl. Chambré P. L. Lcurs i smiar NE98- Dparm of Nuclar Egirig U.C. Brkl. Ch Z. S. P. Naraa Z. Yag ad S. S. ahma A primal ivsigaio of hdraulic bhaviour of fracurs ad jois i graiic rock I. J. of ock Mch. Ad Miig Sci Daga G. Saisical hor of groudwar flow ad raspor: por o laboraor laboraor o formaio ad formaio o rgioal scal War sourcs sarch 9 S-4S 986. d Druz J.. P. Dav ad O. Bour Hdraulic propris of wo-dimsioal radom fracur works followig a powr law lgh disribuio:. Effciv cocivi War sourcs sarch Drshowiz W. G. L J. Gir S. Hichcock ad P. LaPoi FracMa iraciv discr faur daa aalsis gomric modlig ad ploraio simulaio Usr documaio. Vrsio.4 Goldr Associas Ic. Sal WA 994. Drshowiz W. ad I. Millr Dual porosi fracur flow ad raspor Gophsical sarch Lrs Dvrsorp B. ad J. Adrsso Applicaio of h discr fracur work cocp wih fild daa: Possibiliis of modl calibraio ad validaio War sourcs sarch Dvrsorp B. J. Adrsso ad W. Nordqvis Discr fracur work irpraio of fild racr migraio i sparsl fracurd rock War sourcs sarch Eisi. A. Ivsigaios o h hor of h Browia movm Edid wih os b. Fürh Traslad b A.D. Cowpr Mhu & Co. LTD Lodo 96. Elsworh D. A modl o valua h rasi hdraulic rspos of hr-dimsioal sparsl fracurd rock masss War sourcs sarch Fowlr J. L. ad K. L. Hrl Flow of a gas hrough porous mdia Joural of Applid Phsics Gir J. E. C-L Alsso L Hässlr ad A Babdrrahma Discr fracur modlig of h Fisjö rock mass: Phas SKB Tchical rpor T Gir S. Morphologi comporm hdromcaiqu d u fracur aurll das u grai sous corai ormal Docoral hsis Uiv. d Orlas Frac 986. Gllig B. Dvlopm ad applicaios of h chal work modl for simulaios of flow ad solu raspor i fracurd rock Docoral Thsis oal Isiu of Tcholog Sockholm 997. Hak H. Advacd srgics: Isabili hirarchis of slf-orgaizig ssms ad dvics Sprigr-Vrlag 98. 7

118 Hsir K. ad J. C. S. Log Aalical prssios for h prmabili of radom wo-dimsioal Poisso fracur works basd o rgular laic prcolaio ad quival mdia horis Joural of Gophsical sarch 95B Hildbrad F. B. Advacd calculus for girs Pric-Hall Nw York 95. Hudso J. A. ad S. D. Pris Discoiuiis ad rock mass gomr I. J. of ock Mch. & Mi. Sci. & Gomch. Absr Huakor P. ad K. Nikuha Soluio of rasi raspor quaio usig a upsram fii lm schm Appl. Mah. Modllig Huakor P. ad G. F. Pidr Compuaioal mhod i subsurfac flow Acadmic prss 98. Hubr K. H. Th fii lm mhod for gir Joh Wil & Sos 975. Japa Nuclar Ccl Dvlopm Isiu JNC Scod progrss rpor o rsarch ad dvlopm for h gologic disposal of HLW i Japa H: Projc o sablish sciific ad chical basis for HLW disposal i Japa JNC TN4 -~5. Kawasaki D. J. Ah ad P. L. Chambré Aalsis o radiouclid raspor i a rposior wih mulipl caisrs Procdigs of h 9 h Iraioal High-Lvl adioaciv Was Maagm Cofrc IHLWM April 9- Ma Las Vgas. Kllr A. A. P.V. obrs ad M. J. Blu Effc of fracur aprur variaios o h disprsio of coamias War sourcs sarch Kizlbach W. Th radom walk mhod i pollua raspor simulaio Groudwar flow ad quali modlig Koz J. Cocrig capillar coducio of war i h soil: is spag ad us i irrigaio raslad ad did b W.F. Sridick ad C.M. Davis oal Acadm of Scic Via Procdigs Class I Vol Krszig E. Advacd girig mahmaics 7 h diio Joh Wil & Sos ic. 99. Kulailak P. H. S. W. D. N. Wahugala ad O. Sphasso Joi work modlig wih a validaio rcis i sripa mi Swd I J. ock Mch. Mi. Sci. & Gomch. Absr Kulailak P. H. S. W. J. Ch J. Tg X. Shufag ad G. Pa Discoiui gomr characrizaio i a ul clos o h proposd prma shiplock ara of h hr gorgs dam si i chia I. J. of ock Mch. & Mi. Sci. & Gomch. Absr Larsso E. Groudwar flow hrough a aural fracur: Flow prims ad umrical modlig SKB Tchical por Lim D. J. Ah ad P.L. Chambré Flow aalsis i h ar fild of gologic rposior Procdigs of h 9 h Iraioal High-Lvl adioaciv Was Maagm Cofrc IHLWM April 9- Ma Las Vgas a. Lim D. J. Ah ad P. L. Chambré Aalsis of war flow hrough a fracur work i h ar fild Dparm of Nuclar Egirig U.C. Brkl UCB-NE-47 b. Lim D. J. Ah ad P. L. Chambré A modl for war flow ad paricl raspor i fracurd mdia Iraioal Groudwar Smposium March 5-8 Brkl CA. Liu H. H. G. S. Bodvarsso ad L. Pa Drmiaio of paricl rasfr i radom walk paricl mhods for fracurd porous mdia War sourcs sarch Log J. C. S. J. S. mr C.. Wilso ad P. A. Wihrspoo Porous mdia quival for works of discoiuous fracurs War sourcs sarch

119 Log J. C. S. Vrificaio ad characrizaio of coiuum bhavior of fracurd rock a AECL udrgroud rsarch laboraor Tchical por LBL Log J. C. S. ad D. M. Billau From fild daa o fracur work modlig: a ampl icorporaig spaial srucur War sourcs sarch Mahro G. Eléms pour u héori ds miliu poru Masso Ed. Paris 967. Moro L. Y. W. Tsag C. F. Tsag F. V. Hal ad I. Nriks Flow ad racr raspor i a sigl fracur: A sochasic modl ad is rlaio o som fild obsrvaios War sourcs sarch Moro L. ad I. Nriks Fluid flow ad solu raspor i a work of chals: Joural of Coamia Hdrolog Moro L. B Gllig ad I. Nriks Solu raspor i fracurd mdia: h impora mchaisms for prformac assssm Joural of Coamia Hdrolog Naioal sarch Coucil ock fracur ad fluid flow: Comporar udrsadig ad applicaios Naioal Acadm Prss Washigo D.C Nriks I Diffusio i h rock mari: a impora facor i radiouclid rardaio? Joural of Gophsical sarch 85B Numa S. P. Sochasic coiuum rprsaio of fracurd rock prmabili as a alraiv o h EV ad fracur work cocps Groudwar Flow ad Quali Modllig -6 Commi of U.S.A. Mmbrs of JAM. Tucso Ariz Nimi A. K. Koio ad A.Kuusla-Lahi Hdraulic characrizaio ad upscalig of fracur works basd o mulipl-scal wll s daa War sourcs sarch Odlig N. E. ad I. Wbma A. Coducac msh approach o h prmabili of aural ad simulad fracur pars War sourcs sarch Oho H. ad K. Kojima A Sud o h ffc of h scal of fracur o groudwar flow cosidrig fracal hor Procdigs of h 7 h Japa smposium o ock Mchaics Toko 987. Oro A.P. ad B. Brkowiz Flow i rock fracurs: Th local cubic law assumpio ramid War sourcs sarch O ourk J. Compuaioal gomr i C Cambridg Uivrsi Prss Nw York 99. Pima J. Probabili Sprigr-Vrlag Ic. Nw York 99. Powr acor ad Nuclar Ful Dvlopm Corporaio PNC sarch ad dvlopm o gological disposal of high-lvl radioaciv was PNC TN Prss W. H. S. A. Tukolsk W. T. Vrlig ad B. P. Flar Numrical rcips i C d diio Cambridg Uivrsi Prss Nw York 99. Pris S. D. ad J. A. Hudso Discoiui spacig i rock I. J. of ock Mch. Ad Miig Sci Pusch. adiouclid raspor pahs i h arfild: A KBS- cocp sud SKB Tchical rpor Pusch. ad. Safors Th zo of disurbac aroud blasd ul a dph I. J. of ock Mch. & Mi. Sci. & Gomch. Absr Prak-Nol L. J. Th sismic rspos of fracurs ad h irrlaios amog fracur propris I. J. of ock Mch. & Mi. Sci. & Gomch. Absr

120 obiso P. C. Cocivi flow ad raspor i work modls of fracurd mdia Ph.D. Thsis S. Cahri s Collg Oford Uivrsi 984. omro L. L. Moro ad I. Nriks Fas mulipl-pah modl o calcula radiouclid rlas from h ar fild of a rposior Nuclar Tcholog a. omro L. L. Moro ad I. Nriks Th fas mulipl-pah NUCTAN modl: rlas from a rposior Nuclar Tcholog b. Calculaig h radiouclid Schwarz F. W. L. Smih ad A. S. Crow A sochasic aalsis of macroscopic disprsio i fracurd mdia War sourcs sarch Sow D. T. A paralll pla modl of fracurd prmabl mdia Ph.D. Thsis Uivrsi of Califoria Brkl 965. Sow D. T. Th frquc ad aprurs of fracurs i rock I. J. ock. Mch. Mi. Sci Tompso A. F. B. E. G. Vomvoris ad L.W. Glhar Numrical simulaio of solu raspor i radoml hrogous porous mdia: Moivaio modl dvlopm ad applicaio Lawrc Livrmor Naioal Laboraor UCID Tsag Y. W. C. F. Tsag I. Nriks ad L. Moro Flow ad racr raspor i fracurd mdia: a variabl aprur chal modl ad is propris War sourcs sarch Uffik J. G. M Modlig of solu raspor wih h radom walk Groudwar flow ad quali modlig Vrbk J. J. Ah ad P. L. Chambré Log-rm bhavior of buffr marials i gologic rposioris for highlvl wass Dparm of Nuclar Egirig U.C. Brkl UCB-NE Wilso C. A ivsigaio of lamiar flow i fracurd porous rocks Ph.D. Thsis Uivrsi of Califoria Brkl 97. Wihrspoo P. A. J. S. Y. Wag K. Iwai ad J. E. Gal Validi of cubic law for fluid flow i a dformabl rock fracur War sourcs sarch Yamashia ad H. Kimura Paricl-rackig chiqu for uclid dca chai raspor i fracurd porous mdia Joural of Nuclar Scic ad Tcholog

121 Appdi A: adom Samplig for Logormal Disribuio Thr is o ac mhod o gra a sigl ormal or logormal varia from a sigl uiform varia [Bral al. 98]. Th Bo-Mullr mhod [Bo al. 958] is usd i a compur cod FFDF * o gra a pair of logormall disribud radom umbrs from a pair of uiforml disribud radom umbr bw ad. Th logormal disribuio. of variabl ca b wri as a ormal disribuio fz of variabl z for zro ma ad ui sadard dviaio z f z p - z A. l / L z L L A. L whr L is h shap paramr ad L is h scal paramr of h logormal disribuio. Th joi probabili dsi fucios PDF of wo idpd variabls z ad z is h produc of h idividual PDFs [Bo al. 958]. f z z z / z / z z / f z f z. A. B covrsio from h Carsia coordias wih z ad z as o h polar coordias r ad s Figur A. h joi PDF i A. is wri as [Bral al. 98] r r / f r θ. A.4 z z =r si r z z z =r cos Figur A. Polar coordias Th absc of i h righ had sid of A.4 implis ha agl is uiforml disribud bw ad which ca b grad b radom umbr wih U whr U is h uiforml grad radom umbr bw ad. B igraio of A.4 for agl bw ad givs h PDF which is ol fucio of r as Cumulaiv disribuio fucio CDF Fr ca b obaid as r / f r r. A.5 r / r / F r d. A.6 * Faurs of FFDF ar giv i Appdi D

122 PDF Usig dirc ivrsio of h CDF [Prss al. 99] h dsird dvia fr ca b obaid from a uiform dvia U bw ad as r l U l U. A.7 Now z ad z ca b obaid b uiforml disribud agl as U ad h rlaioship bw wo coordias show i Figur A. z r cos l U cosu A.8 z r si l U siu. A.9 Usig A.8 ad A.9 wo idpd radom umbrs z ad z which ar disribud ormall ar grad from h wo uiform radom umbrs U ad U bw ad. B h dfiiio of A. wo logormall disribud idpd radom umbrs ad also ca b obaid from h wo uiform radom umbrs bw ad L Lz whr L Lz whr p[ L L L L ad U U p[ L L L L ad U U l U cosu ] l U siu ] A. A. Equaios A. ad A. ar uilizd o gra logormall disribud radom umbrs i FFDF cod. Figur A. shows logormal disribuios for L =.m ad L =. obaid b aalical prssio. ad b umrical soluio b A. ad A.. 5 sampls hav b mad i Figur A...5 Aalical Numrical Lgh of fracurs Figur A. Logormal disribuio b Aalical soluio usig. ad b umrical soluio usig quaios A. A.

123 Appdi B: Paramrs for Occurrc of FBC I his sud h prcolaio of h ssm i.. whhr a FBC occurs i h ssm or o is crucial bcaus war ca flow io h buffr rgio ol hrough h FBC du o h assumpio of imprmabl rock mari. Th amou of war supplid o h buffr rgio dpds o fracur gomr paramrs. I his scio impora paramrs for prcolaio ar ivsigad. B simpl umrical simulaios oriaio is foud o b h mos impora paramr for prcolaio. B. Oriaio Figur B. shows a paricular ralizaio for a fracur work graio for ss of saisics of fracur gomr paramrs i Tabl.. To s h ffc of oriaio o prcolaio a cosa valu of h oriaio =/4 is assumd isad of isorop uiform disribuio from o. Disribuios for ohr paramrs ad hir valus ar h sam as giv i Tabl.. Figur B. Fracur work graio Figur B. shows a fracur work graio wih h cosa oriaio =/4. No irscio occurs i his cas whras h fracur work wih uiform oriaio i Figur B. ilds 97 irscios. This rsul idicas ha oriaio primaril drmis prcolaio. Figur B. B. Lgh Fracur work graio wih cosa oriaio. To s h ffc of fracur lgh o prcolaio a cosa lgh boh for h primar ad scodar fracurs ar assumd. Ohr paramrs ar h sam as giv i Tabl.. Figur B. shows a fracur work wih h cosa lgh of.m shor fracurs ad Figur B.4 shows a fracur work wih h cosa lgh of. m log fracurs. 8 irscios occur for h cas wih shor fracurs i Figur B. whras 4986 irscios occur for h cas wih log fracurs i Figur B.4. Prcolaio occurs ol for h log-fracur cas.

124 Noic ha boh cass ar for isoropic oriaio. If h cosa oriaio.5 is assumd for h cas wih h log fracurs o irscio occurs as show i Figur B.5. This rsul implis ha h oriaio is mor impora ha h lgh. Figur B. Fracur work wih h cosa lgh of.m shor fracurs. Figur B. 4 Fracur work wih h cosa lgh of.m log fracurs. Figur B. 5 B. Aprur Fracur work wih cosa lgh.m ad oriaio.5. Grall irscio ca occur mor likl wih a grar aprur ha wih a smallr aprur. To s h ffc of fracur aprur o prcolaio a cosa aprur of -4 m is assumd whil ohr paramrs ar h sam giv i Tabl.. Figur B.6 shows h fracur work grad wih h cosa aprur of -4 m. 4