p. 1 

# PICK-U P DATE : BO X 

# 10 O F 

# 10 Ste p 2 : FO R ASR C INTERNA

L US E ONLY 

### (PAR T A o

f B )



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Ste p 1 : FO R ASR

C INTERNA L US E ONL Y A^~\ Are ther e more document s fo r thi s Docke t i n anothe r box ? /*YES% N O ''Fil l ou t Continuatio n Shee t an d g o t o Ste p 2 Interna l SFUN D Bo

x Numbe r 3 5 O F 28 3 Initials/Date : _LD S /_3-8-200 7

# - V /02^-N/<*5~-fH< NUREG/CR-333 2 f 

## ' ORNL-596 8

! 

# Radiological

Assessment A Textboo k

o n Environmenta l Dos e Analysi s -

Dat e Published : September 198 3 Edite db y Joh n E . Til l

an d H . Rober tMeye r 

# "TITt fety ReSearch Di

# ^on Oa*Hi Nr? ional

Laboratory ualcRidge, Tennessee . -- "" ' 

## Prepare d for fcfrtIntegratio n U 

## -S • NucWp 31 

## " « e 

## ? c t w 

## Regulatio n n 

## *l* Ff N B076 64- 1 

# 4 Transport

of Radionuclides in Groundwater ByR. B . CODELL * and

J . D . DUGUID * 1

4, 1 INTRODUCTIO N Groundwater

flow i s on e o

f th e likel y pathway s fo

r radionuclide s release d fro mwaste disposa

l areas . Groundwate r

transport i s als o amajor

pathwa y fo r certai n classe s o

f accidenta l an dnorma

l release s fro

mnuclea r facilitie s suc

h a s powe r plants , fue

l reprocessin g plants , arid minin g or . millin g operations . Th e primar y emphasi s o

f thi s chapte r

will b e "groundwate r

transport

o f radionuclide s usin g waste disposa l a s a n application . Concentration s o f radionuclide s tha t

coul dreac h th e biosphere an dth e resultin g consequence s mus t b e predicte dusin g scenarios.o f event s an dprocesse s tha t ar e possibl e bu t unlikel y t o occur

a t th e disposal " site . Estimation s o f groundwater

flo w and transpor t ar e importan t

i n assessin g th e performanc e o f adisposa

l syste m becaus e the y ar e probabl e migratio n pathway s between th e nuclea r wast e an d th e biosphere . Th e transport

o f radionuclide s throug h th e groun d ca n b e estimate db

y •th e us e of tracers , groundwate r

dating , mathematica l models , or . by a combination , o

f al l o f th e above . Chemica

l o r .radioactiv e tracer s ma

y b e deliberatel y introduce dt o th e groundwate r an dmonitore dthroug h well s fo r th e direc t determinatio n of

. groundwate r

velocit y and : transport . Alternatively , pollutant s no t

deliberatel y introduce dt o th e groundwater ma y als o b e traced . Groundwate r

datin g i s atechniqu e by whic h th e ag e o

f aregion o

f groundwater

ma y b e estimate dfro

mth e concentratio n o

f a n atmospheri c *U.S . Nuclea

r Regulator y Commission

, Hydrologi

c Engineerin

g Section , Washing

ton , D.C . Battell e Offic e o f Nationa l Wast e Termina l Storag e Integration , Germantown

, Md .4- 2 Radiologica l Assessmen

t radionuclid e i tcontains . Tritiu m release d sinc e th e beginnin g o f th e nuclea r er a ca n b e use d t o dat e wate r u p t o severa l decades . Wate r ma y b e date d usin g l4 C ove r longe r period s o f hundred s t o thousand s o f years . Direc t measurement s o f

th e migratio n o f

radionuclide s release d from naturall y occurrin g uraniu m an d thoriu m or e bodie s ca n b e use d a s a clos e analo g t o man-mad e radioactiv e wast e disposa l situations . Usin g groundwate r flo w an d transpor tmodel s provide s a mean s t o calculat e th e expecte d concentration s o f radionuclide s followin g releas e t o th e environment . Wher e groundwate r contaminatio n o f well s o r surfac e wate r bodie s suc h a s lakes , streams , o r rivers occurs , th e radionuclid e concentration s ar e use d i n othe r pathwa y model s t o calculat e th e consequence s o f th e release . Pathwa y model s consis to f surfac e wate r transportmodel s an d biologica l pathwa y models , whic h i n tur n provid e th e basi s fo r dos e calculations . Dose s t o human s aris e throug h th e contaminatio n o f drinkin g wate r an d foo d an d from contaminate d surface s suc h a s floo d plain s an d beaches . Thi s chapte r discusse s curren tpractic e i n groundwate r flo w and transpor t modelin g a s wel l a s dat a requirement s an d possibl e misuse s o f models . A se to f analytica l model s i s presente d alon g with illustration s o f thei r use . 4.1. 1 Type s o f

Groundwater Assessment s . ; , . ' 4.1.1. 1 Geologi c Isolatio n o f High-Leve l Wast e Actua l test s and demonstration s o f th e behavio r pf

a high-leve l wast e (HLW ) repository syste m canno tb e perfprme d oye r t}i e operationa l lifetim e o f th e repository . Therefore , w e mus trel y o n mathematica l model s fo r performanc e assessments , usin g dat a collecte d ove r comparativel y shor tperiod s o f time , t o predic tth e long-ter m performanc e o f th e system. ' Thi s i s th e onl y mean s b y whic h th e cumulativ e effect s o f change s i n th e propertie s o f th e respositor y system , th e effect s o f variou s desig n feature s o f th e repository , and th e effect s o f

th e repository o n th e environmen tca n b e analyzed . Performanc e assessmen tno tonl y provide s thi s typ e o f analysi s bu tals o provide s informatio n tha ti s usefu l i n guidin g researc h an d developmen tactivitie s i n sit e selection , repositor y design , an d wast e packag e design . Performanc e assessmen ttreat s concept s tha tca n b e quantified, rtha tis , failur e analysi s an d consequenc e assessmen t(Klingsber g an d Dugui d 1980) . A n assessmen to f

th e long-ter m performanc e o f a repository analyze s th e event s an d processe s tha tcoul d releas e radionuclide s fro m th e wast e an d th e phenomen a tha tmigh ttranspor t• radionuclide s t o th e biosphere . Thes e phenomen a ma y b e roughl y classifie d a s thos e tha toccu r i n th e nea r fiel d (wher e wast e and repositor y phenomen a dominate ) and thos e tha toccu r i n th e fa r fiel d (a ta greate r distanc e fro m th e repository wher e natural phenomen a dominate) . Althoug h thes e tw o region s ar e no tseparate d b y a precisel y define d Radionuclide s i n Groundwate

r 4- 3 boundary , th e distinctio n i s usefu l becaus e th e physica l an d chemica l effect s o f hea tan d radiatio n fro m th e wast e ar e limite d t o th e nea r fiel d (Klingsber g an d Dugui d 1980) . Differen tmethod s pf analysi s ar e therefor e appropriat e fo r th e tw o regions . Near-fiel d analysi s studie s th e combine d effect s o f heat , radiation , repositor y desig n an d construction , an d th e wast e package . Far-fiel d analysi s studie s th e effect s o f

event s tha taris e fro m natura l phenomena , and from potentia l huma n action s afte r th e repositor y ha s bee n sealed . Thes e far-fiel d phenomen a usuall y appea r i n th e geospher e an d th e biospher e outsid e o f th e repository . Both near-fiel d an d far-fiel d performanc e mus tb e considere d i n determinin g ho w wel l th e natura l an d th e man-mad e component s o f th e disposa l syste m mee tth e criteri a fo r repositor y performance . 4.1.1. 2 Shallo w Lan d Buria l Fo r near-surfac e disposa l suc h a s shallo w lan d burial , th e analysi s o f syste m performanc e i s simila r t o tha to f isolatio n o f

high-leve l wast e with tw o majo r exceptions : (1 ) groundwate r flo w an d transpor tmodel s mus tfrequentl y conside r th e unsaturate d zon e an d (2 ) th e waste s ar e no thea tproducing . Th e analysi s proceed s i n muc h th e sam e fashio n a s fo r high-leve l wast e an d include s th e developmen to f a sourc e ter m throug h corrosio n o r breachin g o f wast e containers , definin g a n , appropriat e leac h rat e fo r th e wast e form , developin g a syste m releas e scenario , an d calculatin g groundwate r flo w and radionuclid e transpor tfo r us e i n th e biospher e model s an d dos e code s (Aiken s etal . 1979) . 4.1.1. 3 Uraniu m Minin g an d Millin g Ther e ar e severa l potentia l groundwate r contaminatio n problem s associate d with th e mining , milling , an d wast e disposa l operation s necessar y t o produc e uraniu m fuel s (USNR C 1979 ; Shepar d and Cherry 1980) . , Th e greates twaterborn e contaminatio n hazar d t o groundwate r i s th e seepag e fro m tailing s pond s resultin g fro m conventiona l millin g procedures . Th e wast e strea m contain s abou thal f solid s an d hal f

water , whic h i s usuall y dispose d o f

i n pond s (tailing s ponds ) forme d behin d earth o r rubbl e dams . Tailing s ar e sometime s reburie d i n th e or e pits . Aci d leac h mill s ar e th e mos t prevalenttype . Tailing s pond s usuall y receiv e highl y acidi c (p H 0.5-2 ) wate r an d tailings , buti n som e case s tailing s ar e firs tneutralized . Th e waste s fro m th e tailing s pond s diffe r mos tfro m othe r form s o f nuclea r wast e becaus e o f thei r uniqu e chemistry . I n acidi c tailings , mos to f th e radioactiv e an d othe r chemica l waste s ar e i n th e dissolve d state . Acidi c waste s are sometime s neutralize d t o reduc e th e solubilit y o f pollutants , bu ti n som e case s th e waste s slowl y becom e apidi c becaus e o f oxidatio n o f

pyrit e (iro n sulfide) . Th e behavio r o f th e radioactiv e contaminant s varie s fro m very simpl e t o very complex . Probabl y th e mos tradiologicall y significan tradioactiv e wast e4- 4 Radiologica l Assessmen t

component

present i s radium , which ha s afairl y simpl e Chemistr y sinc e i t exist s onl y i n + 2 valenc e state . Uraniu man dsevera l othe r radioactiv e wast e compound s behav e i

n amuch mor

e complicate dfashion , sinc e they ma y exis t i n severa l differen t valenc e state s an dfor

mcomple x compound s (Land a 1980) . . ' , Th e solubilit

y o f al l o f th e contaminant s i s high ; fo r low-p H condition s an d decrease s markedl y at

highe r pH . Neutralizatio n by carbonates , such a s limestone , eithe r

adde ddeliberatel y p r

encountere din th e environment , however , ca

n mobiliz e uraniu min th e for mo f soluble , carbonat e complexes . Uraniu mma

y als o b e mobilize di

n a n oxidizing environment

o r by certai n organi c chemical s i

n groundwater

(Shephar dan dCherr

y 1980) . 4.1.1. 4 Nuclea

r Powe r Plan tAccident s Postulate daccidenta l release s p f radioactivit

y t o th e groundwate r

pathwa y hav e been evaluate dfo

r awid e rang e of

nuclea r facilitie s eithe r

fo r generi c site s o r in actua l reacto r

licensin g reviews . Th e accidental release s considere d rang e fro msmal l leak s fro

mcontaminate dwater

stream s i

n nuclea r plant s to major release s cause dby acore meltdow n acciden t (USNR C 1975 ; USNR C 1978 ; Niemczyk

1981) . Consideratio n give

n to . nuclea

r powe r plan t acciden t

release s t o groundwate r

differ s fro

mthos e fo

r high

- an d low-leve

l waste disposa

l or othe r fue l cycl e problem s in severa l importan t

respects : (1 ) Th e risk o f contaminatio n wouldexis t only fo r th e lifetime o

f th e plant . Administrativ e control s wouldb e in effect durin g thi s period , s o mitigativ e measure s coul d presumabl y b e take n shoul dan accidenta l releas e occur ; (2 ) Th e isotope s o f importanc e i

n nuclea r powe r plan t liqui dpathway accident s ar e generall y thos e wit h hig h dos e factor s and/o r half-live s of

year s t o ten s o

f years , notabl y 3 H , 134 Cs , ,37 Cs , 89 Sr , ^Sr , an d I06 Ru . Unlike nuclea r waste , long-lived radionuclides , actinides , an dtransuranic s hav e been show n t o b e o f muc h lowe

r importanc e (USNR C 1978) ; (3 ) Fo r agive n event , consequence s o

f radioactiv e releas e t o th e groundwate r pathway 1 typicall y presen t muc h smaller

risk s than releas e t o th e airborn e pathway . Thes e consequence s shoul dno t b e neglecte di

n , citin g studie s for

nuclea r powe r plants , however . 

### 4. 2 TYPE S O

F GROUNDWATE

RMODEL S 4.2. 1 Groundwate

r Models fo

r Low-Leve

l Wast e Th e assessmen

t o f alow-leve

l waste burial , groun drequire s thre e type s o f models : (1 ) model s t o determin e th e portio n o f th e radioactiv e sourc e released , if infiltratin g water

contact s th e waste , (2 ) mathematica l model s i n term s o f measurabl e hydrologi c parameter s tha t predic t th e migration.o f

radionuclide s fro mth e sourc e t o location s accessibl e t o th e public , an d(3 ) model s fo r Radionuclide s i n Groundwate r 4- 5

determinin g th e potential radiatio n dos e usin g th e radionuclid e concentration s tha t reac h accessibl e locations . I n thi s chapte r only item2 wil

l b e discussed . Dos e assessment s ar e coveredin othe r section s o

f thi s document . Sourc e ter m definition s def

y simpl e explanatio n an d canno t b e adequatel y coveredi n thi s report . Althoug h difficult

t o analyze , th e nea

r fieldi s a s importan t

an d complex fo r shallo w lan ddisposa

l a s fo r deep geologica

l disposal . Mos t

o f th e actio n i s i n th e disposa

l trenc h an ddetermine s what i s availabl e fo

r future groundwate r transport . Determinin g th e wate r balanc e an dth e amoun t o f water infiltratin g i s difficult . Determinin g th e leachin g an dreleas e fro mchemicall y an d physically heterogeneou s waste s suc h a s low-leve l waste s i s eve

n mor e difficult . T o date , no model adequatel y addresse s th e problemo f modelin g th e near-fiel d environmen

t fo r shallo w lan dburial . Th e calculatio n o

f transport o f radionuclide s fro

mshallo w lan dburia l site s i s complicate db y th e waste frequentl y bein g leache din th e unsaturate dzone , andth e movemen

t o f waste t o th e wate r

tabl e mus t

conside r bot h flo w an d transport throug h thi s zone . Here , in th e simples t case , flo w an dtransport

ma y b e assume dt o b e downwar di n on e dimension . Wate r

flo w rate s fro

mawater balanc e ca

n b e use dt o approximat e th e unsaturate dflo w use da s inpu t

t o th e radionuclid e transport

model . Radionuclid e transport i n th e unsaturate dzone ca n als o b e assume dt o b e downwar dan don e dimensional . Thes e simplifying assumption s can b e usedt o calculat e th e concentration s o f radionuclide s bein g release dt o th e wate r

tabl e a s afunctio n o

f time . More rigorous calculation ? ca n als o b e don e usin g two - an dthree-dimensional model s o f unsaturate dflov

, andtransport . However , considerin g th e uncertainty i n th e magnitud e o

f thL^f l sourc e term , thes e sophisticate d calculation s ar e warrante donly whe n alarg e ^ amount o f dat ai s availabl e and/o r detaile dresult s ar e required . Belo w th e wate r tabl e eithe r

numerical or

analytica l model s o

f two - o r three-dimensional groundwater

flo w an dradionuclid e transpor t ca n b e usedt o calculat e th e concentratio n o

f radionuclide s release da t

location s accessibl e t o th e public . 4.2. 2 Groundwate

r Models fo

r High-Leve

l Wast e Repositories 4.2.2. 1 Far-Field Perfonnanc e Althoug h ther e ar e significant difference s i

n th e stat e o

f development

an d verificatio n o

f different far-fiel dmodels , thes e model s ar e sufficiently wel l advance dt o b e use din assessment s o f repositor y performanc e a t

eithe r generi c o j specifi c sites . I n general , th e procedur e fo r calculatin g th e far-fiel d effect s o f arepositor

y breac h i s show

n in Fig . 4.1 . Afte r th e scenario s tha t

hav e t o b e modele dhav e been identified , th e next ste p in aperformanc e assessmen t i s t o predic t

thei r consequences . Whethe r

th e scenario s wil

l actuall y occu

r canno t b e predicte dwit

h complete certainty , bu t
r o O l4- 6 Radiologica l Assessmen

t ORNL-OW G 82-1440 5 RELEASE . SCENARI O 

# I SOURC E TER M 

# I TRANSPOR T • THROUG H GEOSPHER E 

# I TRANSPOR T .THROUGH, , BIOSPHER E 

# I DOS E CALpULATION S Figur e 4.1 . Element

s o f far-fiel d ris k assessment. Source : Klihgsberg , C , an dDuguid , J . 1980 . Status of

Technology for

Isolating ffigti-Level

Radioactive Waste in Geologic Repositories, U.S . Departmen t o f Energy , Tech . Inf

. Cent. , Oa k Ridge , Tenn . ' wher e possibl e th e probabilitie s o f thei r occurrence s , ar e estimated . Probabilitie s are , however , highly uncertai n fo r the , event s tha t

hav e occurre d in th e regio n aroun darepositor y sit e only afe

w time s in geologi c histor y o r tha t ma y no t hav e occurre dther e a t

all . Fp r thi s reason , th e assessmen t o f repositor y performanc e relie s heavily o n prediction s o

f th e consequence s o

f scenario s rathe r

tha n o n prediction s o

f thei r probabilities . Th e sourc e ter

mdescribe s th e waste at al l times ; i t

specifie s th e radionuclide s present

an dth e physica

l an dchemica

l condition s o

f th e waste . Th e radionuclid e concentration s a t th e tim e o

f th e breac h ca

n b e calculate d fro mthei r original concentration s i

n th e waste . Sourc e termevaluation , i s highly site-specific , dependin g o n such factor s a s th e chemistr

y o f th e waste , th e hos t

rock , an dth e groundwater . Also , interactio n between natura l an dman-mad e component s ca n pla y a n importan t Radionuclide s i n Groundwate

r 4- 7 rol e in definin g releas e mechanisms . Furthe r discussio n o

f thi s importan t aspec t o f waste-migratio n modelin g i s beyond th e scop e o f thi s boo

k (Klingsberg an dDugui d1980) . Onc e th e sourc e ter

mha s been defined , transport throug h th e geospher

e i s determine dby modelin g flui dflo w int o an dawa

y fro marepository . Th e outpu t fro mth e geospher

e transpor t

code s i s apredictio n o

f th e radionuclid e concentration s reachin g th e biosphere a s afunctio n o f time . A s contaminant s move wit h groundwater , the y ma

y b e sorbe dan dthu s retarde dby th e rock s throug h whic h the y pass . Th e part s o f th e contaminan t

transport model s tha t describ e sorptio n generall y assume equilibriu mbetween th e concentratio n o f contaminant s in th e flui dan dth e concentratio n o n th e rock surfaces . Becaus e model s o

f flo w throug h fracture s ten dt o b e specifi c t o particular type s o f fractur e systems , they , ar e les s universall y applicabl e tha n porou s medi amodel s (Dugui dan dLe e 1977) . Ther e ar e tw o basi c problem s fo r th e modelin g o

f materia l transport

in fracture dmedia . On e proble mi s t o assembl e
, .sufficien t dat at o b e able't o adequatel y describ e th e hydrolog y o f th e far-fiel d regio n surroundin g arepositor y site . Th e determinatio n o f effectiv e permeabilitie s an dfractur e connections , i s difficult , an daconsiderabl e amount o f researc h remain s befor e reliabl e method s will b e available . Th e second proble mi s t o understan dth e sorptio n proces s i n th e fracture drocks ; althoug h sorptio n i s effectiv e in porou s rocks , i t might b e muc h les s effectiv e i n fracture drocks . Successful modelin g o f flo w i n fracture daquifer s i s extremel y limited . Modelin g o

f radionuclid e transport

i n fracture daquifer s i s nonexistent a t th e present

time . Curren t

model s fo

r porou s medi aar e bein g usedwit h equivalent . formulation s an dconservativ e assumption s t o establis h bound s o n

. th e effect s o

f flo w throug h fracture dmedi a(Klingsberg an dDugui d1980) . 4.2.2. 2 Near-Fiel dPerformanc e Model s fo

r assessin g th e performanc e o f ahigh-leve l waste repositor y i n th e nea r fieldmus t

tak e int o accoun t

mechanical stresses , hea t flow , chemica

l 1 interactions , an dradiation-induce dphysical-chemical processes . Al l o f thes e phenomena , in additio n t o th e propertie s o f th e hos t

rock , affect

th e enyironment

o f th e emplace dwaste . Th e followin

g section s discus s thre e principal type s o f model s require d fo

r near-fiel danalysis : heat transfe r models , thermomechanical models , an d model s o f physica l an dchemical interaction s amon g th e emplace d waste , th e component s o

f th e wast e package , an dth e hos t

rock . Heat Transfer Models. Therma l model s base do n physica l law s provide a

n accurat e portraya l o

f heat flo w an d change s in temperature . Th e experimental result s t o dat e sugges t

tha t prediction s o

f temperatur e withi n afe

w percen t o f measure dvalue s can b e achieved . Consideratio n o f hea t transfe r i s importan t

, becaus e temperatur e gradient s can b e adrivin g forc e i n groundwate r

flow .4- 8 Radiologica

l Assessmen

t Ove r 4 0 suc h model s ar e identifie da s useful in th e studie s o f wast e disposa l (SA I 1979) . Thermomechanical

Models. Th e model s ar e base do n relation s derive d fro mlaw s o f physic s an dfunctiona

l relationship s between stres s an dstrai n tha t ar e obtaine dfro

mlaboratory tests . Becaus e repositor y rock s ar e inhomogeneou s an dma

y b e fractured , generi c functiona l law s ar e more ' difficult t o obtai n fo

r rock s tha n fo

r mos t othe r constructio n materials . Thermomechanical code s ar e currentl y bein g used , t o analyz e th e uplif t an dsubsidence , roomstabilit y an drat e o

f closure , hol e stabilit

y an drat e o

f closure , caniste r

movement , pilla

r stability , thermomechanical effect s on ' groundwater

flow , stresse s an dstrain s a t critica l location s i

n th e rock

mass , an dmechanical failure o

f th e roc k mass . These , phenomen aar c rriost importan t t o flo w i n fracture drocks . Curren t researc h emphasi s i s o

n th e relationshi p between permeabilit

y an dchange'i n functiona l geometr y du e t o stres s (SiA

I Chemical Models. T o predic t

th e near-fiel dbehavior o f arepositor

y require s analyse s o

f th e interaction s betwee n the , tertiplacedpackage 1 component s an dth e hos t rock. ' Thes e interaction s fal l int o six general categories : (1 ) th e movemen t o f fluid s i n th e vicinit

y o f th e waste package , (2 ) th e corrosio n o

f th e caniste r

an dsleeve material s b

y thes e fluids , (3 ) th e dissolutio n o

f th e waste for

mby groundwate r

containin g th e adde dcorrosio n products , (4 ) th e sorptio n o f radionuclide s b y th e rock s an dth e engineere d component s o

f th e repository , (5 ) th e absorptio n o f radiatio n emitte dby th e waste , an d(6 ) th e alteratio n o f chemical phase s an dpropertie s in th e vicinit y , o f th e canisters . A study o

f thes e interaction s predict s th e kinds , amounts , an d chemical stat e o

f th e radionuclide s availabl e fo r entr y int o th e groundwater system(Jenn e 1979) . 4.2. 3 Groundwate

r Models fo

r Mill Tailings Wast

e Migratio

n

' i Th e modelin g o

f transport fro mmil l tailing s i s simila

r in many way s t p othe r type s o f groundwater

waste migratio n problem s suc h a s low-leve l radioactiv e wast e disposal . Th e uniqu e aspect s o f th e modelin g o

f mil l tailing s waste s ar e th e comple x chemistr

y o f th e waste s an dth e proces s o f neutralization , especially b y rock s in th e transport

pathway . Concentration s o f chemical s ma

y b e quit e hig h i

n amil l tailing s pond , whic h complicate s th e transport processes . Typical equilibriu mconcept s suc h a s th e retardatio n facto r an dKjwil

l no t wor k wel l fo r complicated , nonlinea

r phenomen asuch a s precipitation , whic

h ma y b e particularl y sever e a t high concentrations . Unsaturate dflo

w in som e o f th e pondsetting s an dth e transien t existenc e o

f th e millin g operation s ma y presen t specia l modelin g problem s (Shepar dan d Cherry 1980 ; USNR C 1979) . Radionuclide s i n Groundwate

r 4- 9



### . 4. 3 EQUATION S FO

RGROUNDWATE

RFLO W AN D RADIOACTIVIT

Y TRANSPOR

T Th e movemen

t o f radionuclide s in groundwate r can b e describe dby tw o equations : on e for

the.niovemen t

o f th e carrie r

flui d (water ) an don e fo r th e mas s transpor t

o f th e dissolvedconstituent s (radionuclides) . I n usin g thes e
, equations , th e movemen

t o f th e carrie r

in th e regio n unde r

consideratio n mus t b e known befor

e th e transport

equatio n ca

n b e solved . Th e following discussio n o f th e equation s associate dwit h th e movemen

t o f groundwater

an dth e transport

o f dissolve dradioactiv e substance s ca n b e used onl y a s ageneral guide . A perso n intendin g t o us e model s t o analyz e aspecifi c problemwoul dusuall y needthe.ai d o f a n experience dgroundwate r hydrologis t an dmodeler . A se t

o f highly simplifie dtransport model s i s presente di

n Sect . 4.5.3 , alon g wit

h example s o

f thei r use . 4.3. 1 Groundwate

r Flo w Radioactiv e release s ma

y trave l i

n th e unsaturate dregio n (e.g. , regio n aboveith e water

table ) befor

e enterin g th e zone o

f saturatio n (e.g . belo

w water table) . However,.th e releas e ca n als o b e directl y int o th e zone o f saturation . The predominan t

directio n o

f th e unsaturate dflo

w i s downwar

dunti l th e flo

w 1 reache s th e zon e o

f saturation . Withi n th e zon e o f saturatio n th e flo

w i s predominantl y lateral . Th e governin g equation s i n th e unsaturate dzon e consis t o f aset o f coupledequation s fo

r th e movemen

t o f ga s an dwater . T o date , only computer code s o f limite dapplicabilit

y ar e availabl e fo

r th e solutio n o

f thes e couple d gas-wate r equation s (Lappal a1981) . When th e assumption s ar e mad e tha t th e wate r move s a s asingl e phas e an dtha t n o trappe d ai r

pocket s exist , asingl e governing equatio n fo

r saturated-unsaturate dflo w i s obtaine d (AN S 1980) . n' dn ^= V-(*(/«)(VA+Vz)) , (4.1 ) wher e

n — i a — 

## 0- h = t = k = z = V = th e moistur e conten t (dimensionless) , tota l porosit

y (dimensionless) , th e modifie dcoefficien

t o f compressibilit

y o f th e medium(1/cm) , th e modifiedcoefficien

t o f compressibilit

y o f wate r (1/cm) , th e pressur e hea

d(cm) , th e tim e (s) , th e hydrauli c conductivit

y tenso r (cm/s) , th e elevatio n head(cm) , th e de l operator .4-1 0 Radiologica l Assessmen

t Radionuclides i n Groundwate

r 4-1 1 Equatio n 4. 1 i s nonlinea r because , fo

r unsaturate dflow , bot

h hydrauli c conductivit

y an d moisture conten t

ar e function s o

f pressure head . Th e solutio n o

f Eq . 4. 1 in thre e dimensjpn s i s generall y impractical : Simplification s mus t

usuall y b e fpund . Dependin g on th e natur e o f th e problem , analytical or

numerical method s lik

e th e one s describe di

n Reeve s an d Dugui d(1975 ) an d Reeve s e t al . (1977 ) ca

n b e usedt o analyz e saturated

unsaturate dflow . Th e hydrauli c conductivit y i s atenso r

tha t account s fo

r directional propertie s (anisotropy ) tha t aris e i n formation s such a s layere dsediment s (i.e. , hydrauli c conductivit

y i s different

in different

directions) . If

th e coordinate systemi s oriente dparalle l t o th e principal component s o f hydrauli c conductivity , only th e principal component s o f th e tenso r

ar e required . If

th e mediu mi s furthe r

assume dt o b e homogeneou s an d isotropi c (i.e. , propertie s p f th e mediu

mar e no t

directio n dependent) , hydrauli c conductivit y become s a scala r an d Eq . 4. 1 become s (AN S 1980 ) Ss dH ' ' (4.2 ) K dt ' wher e V 2 H =
H = th e tota l hea d= h + z (cm) , S , = p g ( a+ n'fl) = Specifi c Storag e Coefficien t (1/cm) , p = th e water

densit y (g/cm 3 ) , g = th e acceleratio n o

f gravit y (cm/s 2 ) , , a = th e coefficien

t o f compressibilit

y o f th e medium(cm-s 2 /g) , 0 = th e coefficien

t o f compressibilit

y o f wate r (cm-s 2 /g) . Thi s equatio n i s vali dfo r saturate dflo

w in confinedaquifers . Fo r a confine aaquife

r o f thicknes s b, th e storag e coefficien t an d transmissivit

y ar e respectively defineda s

an dEq . 4. 2 become s S = S,b an d T = Kb

V 

## - H 

## = A M T dt (4.3 ) (4.4 ) In simulation s usin g Eq . .4.4 , th e boundar y condition s o

f leakag e shoul db e usedwhe n appropriate . Fo r

problem s involvin

g leak y aquifers , method s lik

e thos e describe d by Bredehoef t an d Pinde r

(1970 ) ca

n be , used . Fo r unconfine daquifer s wher e compressibilit

y o f th e mediu man d th e wate r i s relativel y unimportan t compare dt o th e vertical ' movemen t o f th e free surfac e (wate r

table) , th e continuit

y equatio n ca

n b e writte n a s follow s (AN S 1980) : V 2 tf 2 = 2S

y dH K dt (4.5 ) wher e S y i s th e specifi c yiel do

f th e aquife

r (dimensionless) . Fo r specifi c yieldo

f steady flo

w i n eithe r confinedor

unconfine d aquifers , Eqs . 4. 4 an d 4. 5 reduc e respectively t o th e following : V 2 H = 0 V 2 # 2 = 0 . (4.6 ) (4.7 ) Fo r simplifie d cases , analytical solution s o f Eqs . 4.4 , 4.5 , 4.6 , an d 4. 7 suc h a s thos e give n i n Carslaw an d Jaege r

(1959 ) ca

n b e used . Fo r

mor e complex situations , numerical solution s suc h a s thos e describe di n Gray e t

al . (1977 ) shouldb e used ; An approximatio n o

f th e flu x (volume o

f flo w per uni t cross-sectiona

l area ) i n th e majo r

flo w directio n can b e obtaine d usin g Darcy' s law : 

### V x 

### = 

### -K^P- ~ 

### -K

dx AH

Ax (4.8 ) wher e AH/Ax i s th e hydrauli c gradien t i n th e directio n o

f flow . Thi s approximatio n i s crud e bu t in many case s i t

i s acceptabl e becaus e o

f th e inability t o accuratel y measur e spatia l variation s in th e hydrauli c conductivity . Us e o

f thi s equatio n assume s ahomogeneou s isotropi c mediu mi n whic h th e gradien t

i s constan t

ove r th e increment . Th e actua l velocit y o f a nonadsorbe dtrace r

wouldb e greate r

tha n th e flux sinc e wate r i s moving onl

y i n th e por e spaces . Th e pore velocit y (o r seepag e velocity) , U, ma y b e approx imate db y dividin g th e flu

x b y th e effectiv e porosity : U = V

x /n e . (4.8a )

• Example 4.1. Fo r

saturate dgroundwate r

flow , calculat e th e pore or seepag e velocit

y i n a n "average " sandston e under agradien t

o f 0.0 1 cm/cm . Us e arithmeti c mea

n value s in tables . Equatio n 4.8 aapplies . Th e arithmeti c mea n hydrauli c conductivity , K, i s 3.3 1 X IO - 4 cm/ s fro

mTabl e 4.5 . Th e arithmeti c mea n effectiv e porosity , n e , i s 0.2 1 fro mTabl e 4.4 . Therefore ,

.. ' U = VJn t - ~K~-/n e = 3.31X10 _4 cm/ s X 0.01/0.2 1

• '-.-= 1.58X10 _5 cm/s . [En do f Exampl e 4.1 ]4-1 2 Radiologica l Assessmen

t Th e difficult y associate dwit h th e solutio n o f th e flo w an dtranspor t

i n th e unsaturate dzone s lead s naturall y t o approximatio n methods . Th e tim e o f trave l ca n b e estimate db y assumin g tha t th e mea n downwar dvelocity v i s proportiona l t o th e rat e o f recharg e o f

wate r a t th e surface , r, an dinversel y proportiona l t o th e mea n volumetri c wate r content , 0 : v = r/8. (4.9 ) Th e recharg e rate , r, ca n b e estimate db y method s describe di n Sect . 4.3.4 . Th e volumetri c wate r

content , 6, ca n b e conservativel y assume dt o b e equa l t o th e fieldcapacity , whic h i s th e maximu mwate r conten t wher e moistur e ca n n o longe r b e hel dagains t

gravity . Fiel d capacit y i s equa l t o th e specifi c retentio n Sn whic h i s define da s S r— n e — n, (4.10 ) wher e n i s th e tota l porosit y an dn e i s th e effectiv e porosity . Representativ e value s o f n e an dn ar e tabulate di n Sect . 4.4.2 . . , , 4.3. 2 Mas s Transpor t Th e mos t

genera l for mo f

th e mas s transpor t

equatio n i s fo r

transpor t

i n saturated-unsaturate dmedia . I f loca l equilibriu mo f

mas s transfe r

an dfirstorde r chemica l reaction s ar e assumed , sorptio n ca n ib e represente da s alinea r relationship , an dth e genera l mas s transpor t equatio n ca n b e writte n a s (AN S 1980 ) RdO^-VA6DVc) + V-(Kc ) • +

dt R ^ + XOR,, c = 0 , (4.11 ) wher e £ = th e concentratio n o f dissolve dconstituen t

(g/cm 3 ) , D = th e dispersio n tenso r (cm 2 /s) , V = th e flu x (cm/s) , A . = th e radioactiv e deca y constant :

X = ln2/half-lif

e o f isotop e (1/s) . Th e ter mR d

i s th e retardatio n coefficien t

Rd = 1 Kd

> n t n. (4.12 ) (4.13 ) wher e n = th e tota l porosity , n e = th e effectiv e porosity , p b = th e bul k

densit y (g/cm 3 ) , K d = th e distributio n coefficien t (mL/g) . Radionuclide s in Groundwater 4-1 3 Mor e conservatively , b y assumin g n = n e , R d ca n b e estimate da s R d =l+^K

d . < 4 - 14 > n e . "A n equivalen t

retardatio n facto r

ma y b e define d fo r

fractur e flo w wher e th e expose dare ao f

th e fractur e i s use drathe r tha n th e porosit y (Freez e an d Cherr y 1978) . Example 4.2. Calculat e th e retardatio n coefficient , R d , fo r strontiu mi n a n "average " fine sandston e wit h abul k density , pj , of

2. 8 g/cm 3 an da distributio n coefficien t

o f 2 0 mL/g . Th e arithmeti c mea n value s o f n, n e ar e foun dfro mTable s 4. 3 an d4. 4 t o b e 0.3 4 an d0.2 1 respectively . Th e retardatio n coefficient , R d , calculate d fro m Eq . 4.1 3 i s therefor e -•

R d = - ^ + ^rX2 0 = 268. 3 0.2 1 0.2 1 Equatio n 4.1 4 give s R d = 1 + 2. 8

0.2 1 X2 0 = 267. 7 [En do f Exampl e 4.2 ] Fo r th e importan t

cas e whe n th e mediu mi s assume dt o b e full y saturated , th e mas s transpor t

equatio n become s R d ^f-VAD-Vc) + V' dt Vc + \R d C = 0 . (4.15 ) I f th e dispersio n • tenso r

i s assume dt o b e homogeneou s an disotropi c an dth e flu x i s assume dt o b e paralle l t o th e x-axis , Eq . 4.1 5 ca n b e writte n a s

R d —-V{DVc) + — Vc + \R d c = 0 dt n (4.16 ) Whe n th e flui dflu x i s assume dt o b e unifor man dsteady , Eq . 4.1 6 become s dc D x d

2 C Dy d

2 Dz d

2 U dc T-S—2—^- 2 o ^ T + ^ ^ + X c = 0 , (4.17 ) dt R d dx 2 R d

dy 2 R d

dz2 R d

dx wher e U i s th e por e velocit y define db y Eq . 4.8 aan dDx , D y, an dD zar e th e dispersio n coefficient s i n the.,* , y, an dz direction s respectivel y (cm 2 /s) , a s describe di n Sect . 4.4.1 .4-1 4 Radiologica l Assessmen t Radionuclide s in Groundwate r 4-1 5

Th e approximat e rat e o f movemen t

o f th e radionuclid e i s U/R d , whic h ma y b e use dt o estimat e th e trave l time . Th e abov e equation s ar e strictl y vali donl y fo r isotropi c medi a(i.e. , medi a whos e hydrauli c conductivit y i s unifor mi n al l directions ) bu t ma y b e applie d t o slightl y anisotropi c formation s whe n th e dispersivitie s ar e obtaine dfro m fieldstudies . 4.3. 3 Chai n Deca y o f

Radionuclide s Radionuclide s deca y eithe r t o stabl e product s o r

t o anothe r

radioactiv e specie s calle dadaughter . I n som e specie s severa l daughte r product s ma y b e produce dbefor e th e paren t specie s decay s t o astabl e element . Thi s proces s i s particularl y importan t

fo r modelin g actinide s an dtransuranics . I n considerin g thi s proces s ove r

th e transpor t

pat h o f radionuclides , on e transpor t equatio n mus t b e writte n fo r

eac h origina l specie s an deac h daughte r produc t t o yiel d th e concentratio n o f

eac h radionuclid e (origina l specie s an ddaughte r products ) a t point s o f interes t alon g th e flo w path . I n aconstan t one-dimensiona l velocit y field, th e genera l equation s ca n b e writte n a s (Burkholde r an dRosinge r

1980 ) dci dc\ 

# *«ir +v 

# -r7- D 

# >

_dc 2 , dc-, R 

## "2 

## i r + u 

## ~y = D 

## > at dx dc, dc, R 

# -- S 

# T +U 

# -A- D 

# ' d

2 c v dx 2 d 2 c 2 dx 2 d 2 c, dx 2 ' — Rdihci .

~ Rd2*2C2 + RdlhCl

. I " 7 Rdlh c l + RdJ-lh-lCi-t (4.18 )

wher e R dl = th e retardatio n facto r fo r specie s i , U = th e por e velocit y = V x /n e , c ( = th e concentratio n o f specie s i , Dx = th e dispersio n coefficient , X / = th e deca y coefficien t

fo r specie s i . , ' Equatio n 4.1 8 describe s th e materia l balance s o f th e /t h membe r

o f adeca y chai n an dal l precedin g chai n members . Analytica l model s incorporatin g • hai n deca y wit h differen t sorptio n propertie s fo r

eac h daughte r

ar e ayailabl e fo r

u p t o athree-componen t chai n (Burkholde r

an dRossinge r

1980) . A simple r

analytica l formulatio n applie s i f al l daughter s ar e assume dt o hav e equa l sorptio n properties . Th e concentratio n Cjo f th e rth daughte r

i n term s o f

th e paren t

concentratio n i s

c t = A l m- l y- i e '•



## IK**- 1 

## *;) ' k*j y • • (4.19 ) ; i Fo r lon g chai n decay s wit h sorptio n considerations , numerica l solution s ar e practicall y mandator y (Burkholde r an dRosinge r

1980 ; Dillo n e t

al . 1979) . ,4.3.4 , Percolatio n o f

Wate rint o th e Groun d A n importan t

par t o f th e analysi s o f

th e migratio n o f

contaminant s i n groundwate r

i s th e determinatio n o f

th e rat e o f

releas e o f

th e contaminan t

a t th e sourc e (e.g., , leachin g o f low-leve l waste ) an ddeterminin g th e spee do f transpor t o f th e groundwater . Bot h o f thes e aspect s o f

th e migratio n proble m frequentl y involv e knowin g th e rat e a t whic h wate r

infiltrate s th e groun deithe r fro masurfac e wate r

bod y suc h a s ariver

o r pon do r directl y fro mrainfal l (percolation) . Infiltratio n i s mos t importan t

fo r shallo w lan dburia l bu t les s importan t fo r adee p repositor y no t

affecte db y loca l recharge . Fo r example , th e sourc e o f

radioactiv e contaminatio n a t alow-leve l wast e sit e ma y b e limite db y th e amoun t

o f rainfal l tha t

penetrate s th e lan dsurfac e an dcome s i n contac t wit h th e burie dwaste . Th e flo w o f groundwate r

i n th e wate r

tabl e aquife r i s directl y relate dt o th e rat e a t whic h surfac e wate r

recharge s it . Thi s rat e o f infiltratio n o f

rainwate r ma y b e a n importan t

boundar y conditio n fo r shallo w lan dburia l problem s (Aiken s e t al . 1979) . Percolatio n o f

rainwate r i s frequentl y estimate db y calculatin g th e wate r budge t fo r th e roo t zone . Wate r

enter s th e roo t

zon e throug h infiltratio n fro m rainfal l an di s remove db y th e evaporatio n directl y fro mth e surface , b y transpiratio n fro mvegetation , an d b y seepag e verticall y t o th e wate r table . Bot h rigorou s (Gupt ae t

al . 1978 ) an d empirica l (Thornthwait e e t al . 1957 ) method s o f

performin g awate r

budge t ar e i n commo n use . Thes e method s ca n b e foun di n standar dhydrolog y textbook s alon g wit h coefficient s tha t appl y t o a variet y o f soi l an dvegetatio n type s an dclimate s (Cho w 1964) . 

### 4. 4 PARAMETERS FO

RTRANSPOR

T AN D FLO

W EQUATION S 4.4. 1 Dispersio n an d Diffusio n i n Porou s Medi a 4.4.1. 1 Molecula rDiffusio n Dispersio n i n Eq . 4.1 1 i s actuall y acombinatio n o f molecula r diffusio n an d mechanica l dispersion , whic h ar e processe s tha t irreversibl y distribut e dissolve d constituent s withi n porou s media . Molecula r diffusio n result s fro mth e rando m movemen t o f molecule s a t

aver y smal l scale . Diffusio n withi n fluid s depend s o n flui dpropertie s suc h a s temperature , concentration , an dviscosit y a s wel l a s temperatur e an dconcentratio n gradients . I n aone-dimensional , nonflowin g diffusio n process , transpor t du e t o diffusio n i s usuall y relate dt o Fick' s law :

• i : , i £ = -L D ,l£ t (4.20 ) dt dx dx4-1 6 Radiologica

l Assessmen

t Radionuclide s i n Groundwate

r 4-1 7 wher e D' i s th e effectiv e diffusion coefficien t fo r porou s media , which typicall y varie s fromabou t

IO - 5 t p IO - 8 cm 2 /s . Th e effective diffusion coefficien t £>' will b e lowe

r tha n th e molecula

r diffusion coefficien

t in afre e liqui dbecaus e diffusion will b e inhibite dby th e por e structur e (Evanso n an dDettinge r 1980) . 4.4.1. 2 Dispersio

n Dispersio n describe s th e mechanical mixin g o f dissolve dconstituent s b y th e complex flo

w path s th e flui dmust , tak e i n th e porou s medium . Variabilit y o f pat h lengt

h an dvelocit

y fromth e mea

n result s i n longitudinal an dlatera l spreadin g of

th e dissolve dconstituents . , ' Laborator y investigation s hav e shdw n tha t in porou s media , longitudinal dispersio n i s relate dt o th e seepag e velocity . Fo r a n isotropi c medium , th e dispersio n coefficient s D tJ can b e describe din term s o f th e longitudinal an d transvers e dispersivit

y (Scheidegger

1961) : where 0D,j= a r K«y'+.(ojr-«r^,FyK , (4.21 ) b u = 1 fo

r i = ; , by = 0 fo

r i * ; (Kronecker

delt afunction) , 6 = th e volumetri c wate r

content , a T — th e transvers e dispersivit

y (cm) ,

• ' • ' ' ,' • a, = th e longitudinal dispersivit y (cm) , * ! , ' V = th e magnitud e o

f th e flux (cm/s) , • V h Vj= th e component s o f th e flu x (cm/s) . Eve n in small-scal e laboratory experiment s in , unifor mporou s media , dispersio n processe s usuall y dominat e th e diffusion ' processes' . Dispersio n depend s o n flow , however . Fo r ver y lo w flo w rates , molecula

r diffusion , which i s independen t

o f flow , ma y dominat e th e diffusion

^ 4.4.1. 3 Macrodispersio

n Experiment s wit

h packe dlaboratory column s generall y yiel d dispersivitie s havin g dimension s o n th e orde r o f th e median grai n diameter , rangin g from , millimeter s t o centimeters . I f th e dispersivitie s measure di n th e laboratory wer e use din atranspor t

mode l fo

r alarg e aquifer , dispersio n wouldb e grossly underpredicted . A t

th e aquifer

scale , i t appear s tha t

th e heterogeneitie s i

n permeability , fracturing , stratificatio n an dothe r propertie s o

f th e medium , samplin g error s an dmode l approximations , ar e more importan t t o producin g dispersiv e behavior

tha n mixin g aroun dindividua l grain s an dpore s i

n th e laboratory-scal e experiment s (Evenso n an dDettinger 1980 ; Anderso n 1979) . Fiel dstudie s ten dt o suppor t th e hypothesi s tha t

macrodispersio n i s largel y aresul t o f heterogeneit

y in hydrauli c conductivity . Dispersivit y als o apparentl y increase s wit

h th e lengt

h scal e o f th e experiment . Ther e i s atendenc y fo r larg e dispersivitie s t o coincid e wit h experiment s involving larg e distances . Th e Fickian analog y fo

r dispersio n (Eq . 4.20 ) doe s no t alway s behav e satisfactorily , an dth e dispersio n canno t b e characterize dwit

h aparamete r

a s simple a s th e dispersivity . Th e assumption s o f homogeneit

y o f th e mediu m brea k down if th e heterogeneitie s ar e no t rando mo r

if the y ar e larg e in compariso n t o th e aquifer

bein g modele d(Winograd e an dPearso n 1976) . Th e case s in whic h th e simpl e dispersio n model s ar e likely t o fai l ar e 1 . Medi ai n whic h afe w extensiv e conductivit

y variation s dominat e th e transport process . 2 . Medi ai n whic h conductivit y variation s ar e abrupt , severe , an dten d t o fblto w well-define dpaths . 3 . Observation s tha t

ar e mad e on ascal e tha t i s smal l compare dt o th e scal e o f th e variation . 4 . Medi atha t

sho w variation s i n conductivit y tha t canno t b e modele d a s a rando mfieldwit

h apparentl y rando mvalues , spatia l extents , an dorientatio n assume db y th e aquifer

propertie s (Evenso n e t

al . 1980) . Thes e phenomen ahav e been describe dgenericall y a s "channeling. " "Evenso n e t al . (1980 ) sugges t

tha t medi ain which thes e phenomen aar e likely t o occu r (fractur e systems , karst , etc. ) shoul dno t b e modele d accordin g t o Fick' s la w withou t extensiv e justification . Model s capabl e o f dealin g wit

h thes e problem s ar e extremel y complex an dbeyondth e scopi e o f thi s book (Evenso n e t al . 1980) . 4.4.1. 4 Determinatio

n o f Dispersio n It , i s frequentl y th e cas e tha t th e onl y wa y th e value s o

f dispersio n coefficient s ca

n b e determine dfo

r agive n site i s b y direc t

observatio n o

f either man-mad e or

naturall y occurrin g tracers . Tracers,tha t hav e been deliberatel y introduce dar e use di

n groundwate r

studie s i n single

- or

double-wel l pumpin g experiment s ove

r relativel y shor

t distance s an dtimes . Direc t trace r method s have severa

l disadvantage s in groundwate r studies : 1 . Becaus e groundwate r

velocitie s ar e rarel y larg e unde r natura l conditions , undesirabl y lon g time s ar e normall y require dfo r tracer s t o move significant distance s throug h the 1 ,flow system . Fo r thi s reason , onl

y small , nonrepresentativ e portion s o f th e flo w fieldca n b e measured . 2 . Becaus e geologica

l material s ar e typicall y quit e heterogeneous , numerou s
, observation s ar e usuall y require dt o adequatel y monito r th e passag e o

f th e trace r throug h th e portio n o f th e flo w fieldunde r

investigation . Th e measurement s themselve s ma y actuall y distur

b th e flo w field significantly .4-1 8 Radiologica

l Assessmen

t Value s o f dispersivitie s obtaine dfromawid e rang e o f trace r experiment s an d als o thos e base don numerical model s o f observe dgroundwate r

solute transport case s ar e presente din Table s 4. 1 an d4.2 , respectively . (Anderso n 1979 ; Evenson an dDettinge r

1980) . Thes e value s represen t site-specifi c case s an d shoul db e extrapolate dt o othe r case s only wit

h extrem e caution . Furthermore , th e dispersivitie s reporte din Tabl e 4. 2 probabl y reflec t processes , suc h a s numerical dispersion , tha t ar e inaccuracie s o

f th e mathematica l mode l an d ar e no t measure di

n nature . 4.4. 2 Porosity an

d Effective Porosity ' Th e parameter s porosity an d effective pordsity (or specific yield) ar e necessar y fo r th e solutio n o

f th e flo w an d solute transport ' equations . Th e porosit y o f asoi l or rock i s ameasure o

f th e interstitia l spac e relativ e t o th e spac e occupie dby soli dmateria l an di s expresse dquantitativel y a s th e percentag e o

f th e tota l volum

e occupie dby th e interstices . Th e porosit

y o f asedimentar

y deposi t depend s chiefl

y o h th e shap e an d arrangemen t

o f it s constituen t

particles , th e degre e o f assortmen t

o f it s particles , th e cementatio n an dcompactio n t o which i t ha s bee n subjected , th e dissolutio n o

f mineral matte r

b y Water , an dth e fracturin g resultin g in ope n joint s othe r

tha n interstices . Th e porosit y o f man y sedimentar

y deposit s i s increase db y th e irregula

r angula r .shape s o f it s grains . Porosit

y decrease s wit

h increase s in th e variet

y o f siz e o f grain s becaus e smal l grain s fill interstice s between large r

grains . Tabl e 4. 3 give s representativ e value s o f porosit y fo r a wid e rang e o f soil s an drocks . Th e effective porosit

y i s th e portio n o

f th e porosit

y tha t ca n b e considere d t o b e availabl e fo

r th e flo w o f groundwater

throug h aporou s medium . No t al l o f th e wate r

i n th e interstice s o

f saturate drock

or soil i s availabl e fo

r flow. Par t o f th e wate r

i s retaine d in th e interstice s b y th e force s o f molecula

r attractio n o r

i s trappe di

n dead-en dpores . Th e amoun t o f wate r trappe di s greates t i n medi ahavin g smal l interstices . Tabl e 4. 4 give s representativ e value s o f effective porosit

y for awide rang e o

f soil s an drocks . 4.4. 3 Hydrauli c Conductivit y fo r Saturate dFlow Th e hydrauli c conductivity , K, fo r a n isotropic , homogeneou s saturate d mediu mdetermine s th e rat e at

whic h water move s throug h th e porou s medium fo r agive n hydrauli c gradient . Hydrauli c conductivit y i s apropert

y tha t depend s o n th e propertie s o

f bot h th e fluidan dth e mediuman dha s unit s o f velocit y (cm/s) . A measure o

f th e hydrauli c conductivity , whic h i s apropert

y o f th e porou s mediumalone , i s th e intrinsi c permeabilit y k, tha t ha s unit s o

f lengt h square dan di s usually expresse di n darcy s (on e darc y = 9.8 7 X IO - 9 cm 2 ) . Radionuclide s i n Groundwate

r 4-1 9
Tabl e 4.1 . Dispersivit

y value s a t an d a T obtaine ddirectl y throug h measurement

s o f trace r breakthroug

h curve s i n groundwate

r solut e transpor t • Settin g . Chal k River , Ontari o alluvia l aquife r Chal k River , strat a o f hig h velocit y Alluvia l aquife r Alluvial , strat a of

. hig h velocit y Lyons , Franc e alluvia l aquife r Lyon s (ful l aquifer ) Lyon s (ful l aquifer )

' Lyon s (ful l aquifer ) Lyon s (ful l aquifer ) Lyon s (ful l aquifer ) Alsace , Franc e alluvia l sediment s Carlsbad , N . Mex . fracture d dolomite . Savanna h River , S.C . fracture d schistgneis s Bestow , Calif

. alluvia l sediment s Dorset , Englan d chal k (fractured ) (intact ) Berkeley , Calif

. sand/grave l Mississipp i limeston e NTS , carbonat e aquife r Pensacola , Fla . ' ' • limeston e « i (m ) 0.034 , 0.034^-0. 1 ,0. 5 0. 1 • , 0.1-0. 5 , ' 5 , 12. 0

.8 | 5 . 7 ' , 1 2 38. 1 134. 1 15. 2 3. 1 1. 0 2- 3 11. 6 1 5 1 0 « r

(m )

3.1-1 4 Q.015- 1 0.145-14. 5 0.009- 1 4 . Ax"

(m )

,

i

38.1 , 53 8 6. 4»00oo

31 2 

## . o * (m/d )

7.2 9. 6 1 3 9

0.1 5 0. 4 Metho d

Single-wel l trace r Single-wel l Two-wel l Two-wel l Single-wel l Single-wel l Single-wel l tes t with resistivit y Single-wel l tes t with resistivit y Single-wel l tes t wit h resistivit y Single-wel l tes t wit h resistivit y Environmenta l Two-wel l trace r Two-wel l Two-wel l Two-wel l Two-wel l 311-138 2 Multiwel l trac e

0 6 tes t

Single-wel l Two-wel l trace r Two-wel l

a Ax ' = distance betwee

n well s i n two-wel l test . b D = groundwate

r seepag e velocity . . Source : Evenson

, D . E. , an dDettinger, M . D . 1980 . Dispersive Processes in Models of Regional

Radionuclide Migration, Universit y o f California, Lawrenc

e L.vermor

e 1 Laboratory , Livermore .

# I .

4-2 0 Radiologica

l Assessmen

t Radionuclide s i n Groundwate

r 4-2 1

Tabl e 4.2 . Dispersivit

y value s a L and . a T obtaine d b y calibratio

n o f numerica l transpor tmodel s against,observe

d groundwate

r solut e transpor t Settin g Rock y Mtn . Arsena

l alluvia l sediment s Arkansa s Rive r Valle y coalluvia l sediment s Californi a alluvia l sediment s Lon g Islan d glacia l deposit s Brunswick , Ga. limeston e Snak e River , Idah o fracture d basal t Idaho, fracture

d basal t Hanfor d site , Wash. fracture d basal t Barstow , Calif . alluvia l deposit s Roswel l Basin, N . Mex. limeston e Idah o Falls , lav

a flow s an d sediment

s Barstow , Calif .

. alluvia l sediment s Alsace , Franc

e alluvia l sediment s Florid a (SE ) limeston e Sutte r Basin , Calif . alluvia l sediment s « £ (m ) 30. 5 30. 5 30. 5 21. 3 6 1 9 1

• 9 1 30. 5

• 6 1 21. 3 9 1 6 1 1 5 6- 7 ; , 80-2Q0 Olf

, (m )

, 30. 5

, 9. 1 9. 1 4. 3 2 0 136. 5 9 1 1 8 1 8
13 7 0.1 8 1 0. 7 8-2 0 : Ax a (m ) 305 660X132 0 30 5 , y Variabl e (50-300 ) Variabl e 64 0 64 0
30 5
Variabl e 3X15 2 1 ' Variabl e (2-2 0 km ) 1 • , , J S Method ' (m/d ) Area

l (moc ) Area l (moc )

• • Area l 0. 4 Area

l (fe ) Area l (moc ) Area l Area l (fe ) Area l (rw ) Area l (fe ) Area l Area l Profil e (fe ) Profil e Profil e 3-D(fe ) "A x = gri d size i n program

. b U = groundwate

r seepag e velocity . ' , c (fe ) indicate

s use o f a finite-elemen

tmodel ; (moc) indicate

s metho d o f characteris tics ; an d (rw) indicate

s a random-wal

k model. Source : Evenson

, D . E. , an

d Dettinger, M . D . 1980 . Dispersive Processes in Models of Regional

Radionuclide Migration, ' Universit y o f California

, Lawrenc

e Livermor

e Laboratory

, Livermore

. Tabl e 4.3 . Typica

l value s o f porosit y o f aquife r material s Aquife r materia l Igneou s Rock s ' Weathere d granit e Weathere d gabbr o Basal t Sedimentar

y Materials Sandston e 1 1 Siltston e

San d (fine ) San d (coarse ) Grave l (fine ) Grave l (coarse ) Sil t Cla y Limeston e Metamorphi

c Rock s Schis t Numbe r o f analyse s 8 4 9 4 6 5 7 245 2 6 3 8 • 1 5 28 1 74 74 1 8 . Rang

e 0.34-0.5 7 0.42-O.4 5 , 0.03-0.3 5 0.14-0.4 9 0.21-0.4 1 0.25-0.5 3 0.31-0.4 6 0.25-0.3 8 0.24-0.3 6 0.34-0.5 1 0.34-0.5 7 0.07-0.5 6 0.04-0.4 9 Arithmetic

mea n 0.4 5 0.4 3 0.1 7 0.3 4 0.3 5 0.4 3 0.3 9 0.3 4 0.2 8 0.4 5 0.4 2 0.3 0 0.3 8

, Source : McWhorter

, D . B. , an

d Sunada, D . K

. 1977 . Ground-WaterHydrology and Hydraulics, Wate

r Resource

s Publications , For

tCollins , Colo . Reprinte

d wit h permission. Hydrauli c conductivity , K, and intrinsi c permeability , k, ar e generall ; relate d b y th e equatio n

r>- (4.2 2 wher e g i s th e acceleratio n o f gravity , p i s th e densit y o f th e fluid, an d p. i s th viscosity o f

th e fluid . Tabl e 4. 5 give s representativ e value s o f hydraul i conductivity i n centimeter s pe r secon d fo r a sampl e o f commo n porou

i jnaterial s (McWhorte r an d Sunad a 1977) . Environmenta l factor s ma y affec tth e hydrauli c conductivit y o f a give i porou s medium . Fo r example , io n exchang e o n cla y an d colloi d surface s wi ] caus e change s i n minera l volum e an d por e siz e an d shape . Change s i n pressu n ma y caus e compactio n o f th e materia l o r ma y caus e gase s t o com e ou to solution , whic h woul d reduc e th e hydrauli c conductivit y (Davi s an d D e Wie s i965) . 4.4. 4 Adsorptio n and Retardatio n Coefficient s A n importan tmechanis m i n retardin g th e migratio n o f radionuclide s i groundwate r i s sorption , whic h i s define d t o includ e al l rock-wate r interaction -4-2

2 Radiologica

l Assessmen

t Tabl e 4.4 . Typica

l value s o f effectiv e porosit y (o r specifi c yield ) o f aquife r material s Aquife r materia l Sedimentar

y Materials Sandston e (fine ) Sandston e (medium) Siltston e San d (fine ) San d (medium

) San d (coarse ) Grave l (fine ) Grave l (medium) Grave l (coarse ) SU t Cla y Limeston e Wind-Lai d Materials Loess Eolia n San d Tuf f Metamorphi

c Roc k Schis t Numbe r o f analyse s 4 7 1 0 1 3 287 29 7 14 3 33 1 3 9 1 29 9 2 7 3 2 1 5 " 14 9 0 1 1 Rang

e 0.02-0.4 0 0.12-0.4 1 0.01-0.3 3 0.01-0.4 6 0.16-0.4 6 0.18-0.4 3 0.13-&4 0 0.17-0.4 4 0.1,3-0.2 5 0.01-0.3 9 ,0.01-0.1 8 --0-0.3 6 , 0.14-0.2 2 0.32-0'4 7 0.02-0.4 7 0.22^.3 3 Arithmetic

mea n 0.2 1 0.2 7 0.1 2 0.3 3 0.3 2 0.3 0 0.2 8 0.2 4 0.2 1 0.2 0 0.0 6 0.1 4 • 0.1 8 0.3 8 0.2 1 ' 0.2 6 Source : McWhorter

, D i B. , an dSunada, D

- K . 1977 . Ground-Water

Hydrology and

Hydraulics, Wate

r Resources Publications , For

t Collins , Colo . Reprinte

dwit h permission . tha t caus e th e radionuclide s t o migrat e a t aslowe r rat e tha n th e groundwate r itself . Th e amoun t

o f sorptio n i s dependent

o n bot h th e chemistr

y o f th e water an do f th e rocks ; and , becaus e som e o f th e chemical reaction s ar e slow , i t i s a functio n o f tim e a s well . Value s o f sorptio n coefficient s ar e require dt o calculat e th e trave l tim e o f ke y radionuclide s fromth e sourc e t o th e biosphere . Th e sorptio n coefficient s ar e usuall y obtaine dusin g astandar dbatc h tes t wher e rock s ar e pu t

in contac t wit h groundwate r

i n whic h smal l amount s o f radionuclide s hav e been mixed . Th e proble mwit

h thi s typ e o

f approac h i s tha t

more detaile dgeochemica l dat a ar e necessar

y t o suppor

t th e validit

y o f th e sorptio n measuremen t ove r th e expecte dtrave l tim e o

f th e radionuclide s (whic h ma y b e o f th e order

o f thousand s o

f years) . T o provide th e justificatio n fo r usin g simpl e sorptio n coefficients , adetaile dunderstandin g of th e geochemica

l mechanism s pf

rock

water interaction s mus t

b e attained . Suc h mechanism s a s dissolution / precipitation , complexing , adsorptibn/desorption , phas e transformations , an d solubilit y shoul db e understoo dfo r radionuclide s o

f interes t i n th e geochemica

l Radionuclide s i n Groundwate

r 4-23 Tabl e 4.5 . Typica

l value s o f hydrauli c conductivit

y o f porous material s Materia l Numbe

r o f analyse s Rang e (cm/s ) Arithmetic

mea n (cm/s ) Igneou s rock s Weathere d granit e Weathere d gabbr o Basal t Sedimentar

y material

s Sandston e (fine ) Siltstone San d (fine ) San d (medium) San d (coarse ) Grave l SU t Cla y Metamorphi

c rock s Schist 7 4 9 3 2 0 8 15 9 25 5 15 8 4 0 3 9 1 9 1 7 (3.3-52 ) X IO- "

(0.5-3.8 ) X IO - 4 (0.2-4250 ) X 10 - 8 (0.5-2270 ) X 10" 6 (0.1-142 ) X 10 - 8 (0.2-189 ) X IO - 4 (0.9-567 ) X IO" 4 (0.3-6610 ) X IO - 4 (0.3-31.2 ) X 10 - 1 (0.09-7090 ) X 10 - 7 (0.1-47 ) X 10 - 8 (0.002-1130 ) X IO - 6 1.6 5 X 10 - 3 1.8 9 X IO - 4 9.4 5 X 10 - 6 3.3 1 X IO - 4 1. 9 X 10~ 7 2.88 X 10 - 3 1.4 2 X 10 - 2 5.20 X 10" 2 4.03 X 10" ' 2.8 3 X 10" 5 9 X 10" 8 1. 9 X IO" 4 Source : McWhorter

, D . B. , an dSunada, D . K

. 1977 . Ground-Water

Hydrology and Hydraulics, Wate

r Resources Publications , For t Collins , Colo . Reprinte

dwit h permission . environment . Th e effec

t o f heat , radiation , or

high concentration s o

f chemical s wil l b e particularl y importan t clos e t o th e sourc e o

f releas e i n som e situations . Much o f thi s understandin g fo

r shorte r period s o

f tim e an d clos e t o th e point s o f releas e ca

n b e obtaine dthroug h acombinatio n o f laborator y an dfieldexperi

ment s combine dwit

h dat afro

mnatura l system s tha t

ca n b e useda s analogs . However , ove

r longe r tim e period s or

fa r fro mth e point s o

f release , al l o

f th e dat amus t b e obtaine dfro

mstudie s o

f th e natura l system . Natura l analog s o

f interes t fo r applicatio n t o radionuclid e migratio n include hydrotherma l ore deposits , intrusiv e magma s int o generi c hos t rocks , uraniu mor e bodies , rich thoriu mdeposits , natura l fission reactors , an d undergroun dnuclea

r explosion s (Klingsber g an dDugui d 1979) . Also , th e behavior o f natura l radionuclide s an dthei r deca y product s in hos t

roc k formation s can provid

e th e dat anecessar

y t o choos e conservativ e sorptio n coef

ficient s fo r us e i n th e transport

model s ar e conservativ e ove r th e rang e o

f geo chemica l condition s andth e transport trave l time s expected . Table s 4. 6 an d4. 7 giv e typical range s o f distributio n coefficient s (K d ) fo r severa l significant' , randionuclide s in a n assortmen t

o f rock s an dsoil s (Isherwoo d1981) . Thes e table s illustrat e som e o f th e sensitivit

y o f Kd t o4-2

4 Radiologica

l Assessmen

t Radionuclides i n Groundwate

r 4-2 5 Tabl e 4.6 . Distributio

n coefficients : strontiu

man dcesiu m K d {mL/g) Tabl e 4.7 . Distributio

n coefficients : thoriu

m an duraniu m ' Basalt , 32-8 0 mesh Basalt , 0.5- 4 mm , 300 pp

mTD S Basalt—0.5- 4 mm , seawater Basalt-fracture di

n sit u measurement Sand , quartz—p H 7. 7 Sand s Carbonate , greate r

than 4 m m Dolomite , 400 0 pp

mTD S Granite , greate r

than 4 m m Granodiorite , 100-20 0 mesh Granodiorite , 0.5- 1 m

m Hanfor dsediment s Tuf f Soil s Shaley siltston e greate r

than 4 m m Shaley siltston e greate r

than . 4 m m Alluvium , 0.5- 4 m

m Salt , greate r

than 4 m m saturate dbrin e "S r i 16-13 5 . 220-122 0 1. 1 3 ' ' • ' 1.7-3^8 13-4 3 0.1 9 5-1 4 1/ 7 J ' 4- 9 11-2 3 5 0 45-400 0 19-28 2 8 1. 4 48-245 4 0.1 9 C s

792-952 0 39-28 0 6. 5

• • 22-31 4 10 0 13. 5 34. 3 8- 9 1030-181 0 300 800-1780 0 189-105 3 309 10 2 121-316 5 0.02 7 Source : Isherwood

, D . 1981 . Geoscience Data Base Handbook for Modeling a Nuclear, Wdste Repository, NUREG/CR-0912 , vols . 1 an d2 , U.S . Nuclea

r Regulator y Commission . factor s suc h a s particl e siz e an dchemistr y of th e wate r

phase . Value s o

f K d shoul db e extrapolate dt o situation s othe r tha n thos e fo

r which the y were 1 determine donly wit

h extrem e caution . 

### 4. 5 METHOD S O

F SOLUTIO

N FO R GROUNDWATE

RMOVEMEN

T 4.5. 1 Introductio

n Ove r th e pas t severa l years, 1 numerou s mathematica l model s hav e bee n develope dt o simulat e th e flow of groundwate r

an dth e transport

o f radioactiv e an dchemical substances , particularl y in , th e fieldd f waste management . Discussio n o

f th e virtuall y hundred s o

f groundwate r

model s i s beyondth e scop e o f thi s chapter , bu t

tw o excellent

compilation s o

f groundwater

model s ar e availabl e (Bredehoef

t 1978 ; SA I 1981) . , Groundwate r

mathematica l model s can b e broadl y classifie d a s either numerical o r

analytical . Numerica l technique s ar e usuall y direc t solution s o

f AT , (mL/g )

• " ' • ' . 160,000 400,00 0 160,000 40-13 0 310-47 0 270-10,00 0 8 6 0 120 1000 < 100,000 62,00 0 440 0 300 200 0 27 0 4. 5 2. 9 Condition s

Thorium Sil t loam , Ca-saturate dclay , p H 6. 5 Montmorillonite , Ca-saturate dclay , p H 6. 5 Clay soil , 5 mM

Ca(N0 3 ) 2 , p H 6. 5 Mediumsand , p H 8.1 5 Ver y fin e sand , p H 8.1 5 Silt/clay , p H 8.1 5 Schis t soil , 1 g/ L Th , p H 3. 2 Schis t soil , 0. 1 g/ L Th,"p H 3. 2 Illite , 1 g/ L Th , p H 3. 2 Illite , 0. 1 g/LTh , p H 3. 2 Illite , 0. 1 g/LTh,pH> 6 Uranium Sil t loam , U(VI) , Ca-saturated , p H 6. 5 Cla y soil , U(VI) , 5 mM

Ca(N0 3 ) 2 , p H 6. 5 Clay soil , 1 ppmUO +2 , p H 5. 5 Cla y soil , 1 pp

mUO +2 , p H 1 0 Cla y soil , 1 pp

mUO +2 , p H 1 2 Dolomite , 100-32 5 mesh , brine , p H 6. 9 Limestone , 100-17 0 mesh , brine , p H 6. 9 Source : Isherwood

, D . 1981 . Geoscience Data Base Handbook

for Modeling a Nuclear

Waste Repository, NUREG/CR-0912 , vols . 1 an d2 , U.S . Nuclea

r Regulator y Commission . th e differential equation s describin g wate r movemen

t an dsolute transport , usin g methpd s suc h a s finite difference s or finite elements . Thes e method s alway s requir e adigita l computer , alarg e quantity o f data , an da

n experience d modeler-hydrologist . Th e validit y o f th e result s fro

mnumerical model s depend s strongly o

n th e quality an dquantity o f th e inpu t

parameters . Analytical model s ar e usuall y approximat e or exac t solution s t o simplifie dform s o f th e differentia l equation s fo

r wate r movemen

t an dsolute transport . Suc h model s ar e simpler

t o us e tha n numerical model s an d ca n generall y b e solve

dwit h th e ai d o f acalculator , althoug h computer s ar e als o used . Analytica l model s ar e much more severely limite dt o simplifie drepresentation s o f th e physica

l situations . However, , the y ar e extremel y useful fo r scopin g th e proble mt o determin e dat aneed s or

th e applicabilit

y o f more detaile dnumerical models . Severa l o f th e more importan t

type s o f numerica l an danalytica l model s ar e discusse dbelow .4-26 Radiologica l Assessmen

t Radionuclide s i n Groundwate

r 4-2 7 4.5. 2 Numerica

l Method s 4.5.2. 1 Finit

e Differenc

e On e approac h tha t

ha s beep applie dt o th e solutio n o f groundwater equation s involve s finite-difference approximations . T o apply thes e approximations , th e regio n unde r consideratio n i s usuall y divide dint o a rectangular

grid . Th e intersection s o f th e gri dar e calle dnoda l point s an d represen t th e positio n a t

which th e solutio n fo

r unknown value s suc h a s hydrauli c headar e obtained . When differenc e equation s ar e writte n fo r al l node s an dboundar

y condition s ar e applied , asyste mo f n algebrai c equation s i

n « variable s can b e solve

dfo r th e variable s a t

eac h node fo

r eac h tim e increment (Faus t an dMercer

1980) . ' 4.5.2. 2 Finit

e Elemen t Th e finite-element

metho di s anumerical metho dwher e aregio n i s divide dint o subregions , calle delements , whos e shape s ar e determine dby ase t of point s calle dnode s (simila r t o th e finite-difference grid) . Th e first ste p i s t o deriv e a n integra l representatio n o f th e partia l differentia

l equations . Thi s i s commonly don e b y usin g th e metho do f weighte dresidual s o r

th e variational method . Th e next

step i s t p approximate th e dependen t variable s (hea do

r concentration ) in term s o

f interpolatio n fuhctibn s calle d basi s functions . Onc e th e basi s function s ar e specifiedan dth e element s defined , th e integra l relationshi p mus t b e expresse dfo

r eac h elernen

f a s afunction o

f th e coordinate s o

f al l nodal point s o

f th e element ; Then ' th e value s o f th e integral s ar e calculate dfo

r eac h element . Th e values'fo r al l element s ar e combine dand boundar y condition s applie dto , yiel dasystemo f first-order

linea r differentia

l equation s i n tim e (Faus t

an dMerce r

1980) . 4.5.2. 3 Metho

d o f Characteristic

s Th e metho do

f characteristic s i s use din convection-dominate dtransport problem s where finite-difference an dfinite-element : approache s suffe r fro m "numerical dispersion " or solution s tha t

oscillate . Th e approac h i s no t t o solv e th e transport

equation s directl y bu t

t o solv e a n equivalent

systemo f ordinar y , differentia l equation s tha t

ar e obtaine db

y rewritin g th e transport equatio n usin g th e flui dparticle s a s areferenc e point . Thi s i s accomplishe dnumericall y by introducin g aset

o f moving point s (referenc e particles ) tha t ca n b e trace d withi n th e stationary coordinate s o f afinite-difference gri dblock an dallowed t o move adistanc e proportional ,t o th e velocit

y an delapse dtime . Th e movin g particle s simulat e th e convectiv e transport becaus e concentratio n i s afunction o f spreadin g o r

convectin g o

f th e particles . Onc e th e convectiv e effect s ar e known , th e remainin g part s o f th e transport

equatio n ar e solvedusin g finitedifferenc e approximation s (Faust'an d Merce r 1980) . 4.5.2. 4 Random-Wal k Metho d A metho dsimila

r i n man y way s t o th e metho do f characteristic s i s th e random-walk

method . In thi s approach , aparticle-tracin g advectio n model i s usedt o simulat e advection . A t th e en do

f eac h advectio n tim e step , th e particle s ar e disperse dby bein g displace darando mdistanc e in arando m direction . Concentration s ar e calculate dby countin g th e resultin g number o f element s in eac h cel l an dcomparin g thi s t o th e initia l conditions . Thi s solutio n techniqu e i s base do

n th e realizatio n tha t

anormal probabilit

y distributio n i s a solutio n t o Fick' s La

w of'diffusion . Thi s metho do f transport modelin g i s easil y implemente dan dprovide s simulation s whos e accurac y ar e limite donl y by th e number of particle s tha t

can b e traced : Th e mai n disadvantag e o f thi s metho d an dth e metho do

f characteristic s i s th e difficult y an dexpens e o

f keepin g trac k o f larg e number s o

f particle s (Evenson . an dDettinge r 1980) . , I I 4.5.2. 5 Flo

w Networ k Models Th e numerical simulatio n b y finite difference s or finite element s o

f groundwater

flow an dsolute transpor t problem s in tw o an d thre e dimension s ca n b e costly in term s o

f computationa l resources . Flow networ k model s suc h a s th e networ

k flow and i transpor t mode l (NWFT ) (Campbel l e t al . 1980 ) ar e thos e tha t ca n b e use dt o describ e two - or three-dimensional fields i n amuc h mor e efficien t wa y by anetwor

k o f interconnectin g one-dimensiona l flow segments . Flui ddischarg e an dvelocit y ar e determine dby requirin g conservatio n o

f mas s at th e segment

junctions . Radionuclid e migratio n fro m the , point s o

f releas e i s calculate dby assumin g tha t transport occur s alon g a single one-dimensiona

l path havin g alengt

h equal t o th e tota l migratio n pat h length . Th e networ

k model i s particularl y usefu l whe n i t i s use di n conjunctio n wit h amore complicate dtwo - o r three-dimensional model t o first define th e flow, an dconcentratio n fieldfo r aparticular

example . Th e networ

k model i s first matche dor

tune d t o th e result s o

f th e complicate dmodel . Th e tune d networ k mode l ma

y the n b e use dfo

r furthe r computation s wit

h amuc h smalle r commitmen t

o f computationa l resource s the n th e original model fo r furthe r run s an dsensitivit

y experiments . 4.5.2. 6 Advectio

n Models Ther e ar e groundwate r

solute modelin g situation s wher e th e phenomeno n ' o f dispersion , togethe r

wty h it s man y uncertainties , i s onl y amino r

facto r in describin g th e transport

o f contaminant s in groundwater . Fo r example , th e flu

x o f contaminan t

enterin g ariver

tha t i s recharge dfro

macontaminate daquife r i s much les s sensitiv e t o dispersio n tha n th e concentratio n in aparticular well . In th e former

case , th e contaminate dgroundwate r would ente r

ove r awid e area , which wouldten dt o smea r ou t th e effect

o f dispersion . Fo r

simila r Reasons , th e transport

fro mnonpoint

source s o

f contaminatio n such a s larg e4-2 8 Radiologica l Assessmen

t low-leve l radioactiv e wast e landfill s woul ddiminis h th e sensitivit y o f modele d result s t o dispersion . A flo w mode l ca n b e use dt o generat e avelocit y potentia l fiel dan dstrea mlines . Th e flow pattern s fro mthe.source s t o th e sink s ca n the n b e use dt o formulat e th e arriva l tim e distribution , whic h ca n b e use dt o calculat e th e concentratio n o f flu x o f acontaminan t

a t th e poin t

o f use . Eithe r numerica l o r

analytica l solution s o f th e flow equation s ar e use dt o estimat e groundwate r

velocities , th e lengt h o f th e pat h o f

acontaminant , an dth e arriva l tim e distributio n (Nelso n 1978) . 4.5 3 Analytica l Model s Analytica l groundwate r transpor t

model s ca n b e use dfo r certai n type s o f analyse s wher e availabl e dat ad o no t warrant , amor e complicate dstudy . Suc h model s ar e usefu l fo r

scopin g th e transpor t proble man dma y frequentl y b e adequat e fo r

regulator y need s i f

mode l coefficient s are'ch,6se h conservatively . , A serie s o f

simpl e analytica l model s tha t have ' bee n use da t

th e U.S . Nuclea r Regulator y Commissio n (NRC ) i s presente dbelow . Man y of . thes e model s hav e bee n computerize dan dar e availabl e fro mth e NR C (Codel l e t al . 1982) . I n thei r

simples t forms , however , the y ma y b e use dwith th e ai do f onl y acalculator . Th e model s ar e develope d fo r th e limitin g cas e o f

undirectiona l saturate d convectiv e transpor t

o f asingl e dissolve d substanc e wit h three-dimensiona l dispersio n i n a n isotropi c aquife r a s discusse di n Sect . 4.3.2 : A . d 2 c dc j U dc _ dt R d dx R d

dx 2 ' R d

dy 2 ' R d

dz2 

## + \ d

2 c , Dz d

2 c , r -r ~ ~^ r ~ A c , (4.23 ) wher e c i s th e concentratio n i n th e liqui dphas e (Ci/cnr) , Dx , D y, D 2 ar e th e dispersio n coefficient s i n th e x, y, an dzdirection s respectivel y (cm 2 /s) , \i s th e deca y coefficien t = I n 2/half-lif

e (1/s) , U i s th e x componen t

groundwate r

por e velocit y (cm/s) , R d i s th e retardatio n coefficien t (dimensionless) . Th e dispersio n coefficien t ca n b e approximate dfro mEq . 4.21 . I n thi s cas e y 2 = V-i = 0 , V\ = V, an d 9ca n b e approximate dfo r saturate dflow b y th e effectiv e porosity , n e . Also , sinc e U = V/n e , D x = a L U D y = a T U ,D Z = a T U

, (4.23a ) (4.23b ) (4.23c ) wher e a L an da T ar e th e longitudina l an dtransvers e dispersivitie s respectively . Radionuclide s i n Groundwater

4-2 9 4.5.3. 1 Poin tConcentratio n Mode l Th e firs t mode l develope di s use dfo r calculatin g th e concentratio n i n th e aquife r a t som e poin t

downgradien t

o f areleas e (e.g. , wate r

suppl y well) . Equatio n 4.2 3 i s solve di n term s o f Green' s functions :

c t = 1 n eR, -X(x,t)Y(y,t)Z(z,t) , (4.24 ) wher e c , i s th e concentratio n a t an y poin t

i n spac e fo r

a n instantaneou s one curi e release , ri e i s th e effectiv e porosit y o f th e medium , an d X, Y, Z ar e th e Green' s function s i n th e x, y, zcoordinat e directions , respectively . Equatio n 4.2 4 ha s bee n develope dfo r avariet y o f

boundar y an dsourc e configurations : 1 . Fo r th e cas e o f

apoin t sourc e a t

(0 , 0 , z s ) i n a n aquife r o f infinit e latera l (x, y) exten t

an ddept h b, a s illustrate di n Fig . 4.2 ,

d = 1 n eR, wher e X{

= y/AitD x t/R d ex p Yi ** -Xi Y\Z\ , (x-Ut/R d ) 2 AD x t/R d

„ 2 - \ t tjAltDyt/R d ex p • 4D y t/R d Zx = 4 - 1 + 2 % ex p

m - l m X T?D zt

b 2R d co s m ir—

co s mi r — b b (4.25 )

(4.26 ) (4.27 ) (4.28 ) 2 . Fo r th e verticall y average dconcentratio n i n cas e 1 abov e (equivalen t t o a vertica l lin e sourc e o f

lengt h b),

Ci = 1 n eR, e it d -X x Y\Zi, wher e 

## z2 

## = 

## -r ; (4.29 ) (4.30 ) 3 . Fo r ahorizonta l lin e sourc e o f lengt h w centere da t

(0 , 0 , z s ), a s illustrate d i n Fig . 4.3 ,4-3 0 Radiologica l Assessmen

t

' ORNL-DW G 82-1440 6 

### POIN T SOURC E A T x = 0 , y = 0 , z = z . 

# •?>?« - TO P O F AQUIFE

R OR

BOTTO M OF AQUIFE

R Figur e 4.2 . Idealized groundwate r syste m fo r poin tconcentratio

n model, poin

tsource.


### HORIZONTA L LIN E SOURC E CENTERE D A T x = 0 , y = 0 , z = Zt ORNL-DW G 82-1440 7

->=^-^ — 

### TOPO F AQUIFE R O R

BOTTO M O F AQUIFE R Figur e 4.3 . Idealized groundwate r syste m fo r poin tconcentratio

n model, horizonta

l lin e source . Radionuclide s i n Groundwate

r 4-3 1
c, - = n eR< X\Y 2 Z\ , wher e Y 2 = -L(erf^±^ + erf^^ = i-2w \ y/AD y t/R d -J*Dyt/R d (4.31 ) (4.32 )

> , ^n d erf i s th e erro r function . Table s o f th e erro r functio n ar e availabl e in • standar d mathematica l text s (Abramowit z 1970) ; 4 . Fo r th e verticall y average d concentratio n in cas e 3 abov e (equivalen tt o a n are a sourc e o f

width w an d depth b),

1 n t R d X\Y

2 Z2 \ (4.33 ) 5 . Fo r a poin tsourc e a t(0 , 0 , z s ) i n a n aquife r o f

infinit e latera l exten tan d depth , c,- = 1 n t R d X\ Y\Z3 > wher e Z 3 = 1 •tfixDj/Ri ex p (z~z s ) 2 4D zt/R d + ex p (z+z,) 2

4D zt/R d (4.34 )

(4.35 ) 6 . Fo r a horizonta l lin e sourc e o f width xv centere d a t(0 , 0 , z s ) in a n aquife r ' o f infinit e latera l exten tan d depth , Ci = 1 n eR< X

X Y 2 Z 3 ', (4.36 ) 7 . Fo r a horizonta l are a sourc e o f lengt h / an d width w centere d a t(0 , 0 , 0 ) in a n aquife r o f

constan tdepth b, a s illustrate d i n Fig . 4.4 , Eq . 4.2 4 become s

Cl = 1 n eR ( •X

2 Y 2 Z2 , (4.37 ) wher e X, = 

## J_ 1 2f i erf X+

2 ut R d y/4D x t/R d -er f . 8 ut Rd y/4D x t/R d exp ( —X t). (4.38 )4-3 2 Radiologica l Assessmen

t ORNL-DW G 82-1440 8

SOURC E ARE A

x

Figur e 4.4 . VerticaU

y average dgroundwate

r dispersio n model . Example 4.3. Concentratio n i n a n aquifer o f limite dthickness . On e curi e o

f aradioactiv e pollutan t

leak s quickly int o awater tabl e aquife r throug h ahighly permeabl e groun dcove r ove r asquare surfac e are a5 0 mon aside . Th e pollutan t ha s ahalf-lif

e d f 3 0 y . A well trace r

tes t indicate s tha t th e groundwater

i s movin g i n th e directio n o

f tw o well s a t

aspeed , U, o

f 1. 5 m/ dan dtha t

th e longitudinal an dtransvers e dispersivities , ar. an da T , ar e 2 0 t o 1 0 m , respectively . Th e saturate dthicknes s o f th e wate r tabl e aquifer , b, i s 5 0 man dha s a n effective porosity , n e , o

f 0.2 . Th e pollutan t

ha s bee n determine dt o hav e a retardatio n coefficient , R d

, o f 2 0 j n th e aquifer . Calculat e th e concentratio n o f th e pollutan t

i n well s whos e downgradient coordinate s wit

h respec t t o th e cente r

o f th e sourc e are aar e Radionuclide s i n Groundwate

r 4-3 3 (a ) x = 20 0 m , y = 0 m (b) x = 40 0 m , y = 5 0 m Th e well s ar e screene dove r th e entir e dept

h o f th e aquifer . Cas e 7 i n Sect . 4.5.3. 1 applie s t o thi s example , sinc e th e sourc e i s a horizonta l are atyp e andth e well s ar e screene dove r th e tota l depth , which woul dverticall y averag e th e concentration . Equatio n 4.3 7 i s therefore evaluate dwit h Green' s function : A ^ determine dby Eq . 4.38 , ^determine d by Eq . 4.32 , an d Z 2 determine dby Eq . 4.30 , Th e dispersio n coefficient s ar e calculate db y Eqs . 4.23 aan d4.23b . Dx = a L U = 2 0 X 1.

5 = 3 0 m

2 D y = a T U

= 1 0 X 1.

5 = 1 5 m

2 Figure 4. 5 show s th e concentratio n a s afunction o f tim e calculate dfo

r th e tw o wells . [En do

f Exampl e 4.3 ]
ORNL-DW G 82C-2075 1 1 5

1 0 O O o E o X=20 0 m , Y= 0

X=40 0 m , Y=5 0 m

I 1 5 i • • 5 . • • 1 0 TIM E (100 0 d )
. Figur e 4.5 . Concentratio

n i n downgradien

t Well s fo r Exampl e 4.3 . 2 0

C O4-3 4 Radiologica l Assessmen t

4.5.3. 2 Flu x Model s Th e flu x mode l i s use dt o calculat e th e discharg e rat e o f aradionuclid e enterin g asurfac e wate r bod y tha t

ha s intercepte dthe'aquife r containin g th e transporte dmateria l a s depicte di n Fig . 4.6 . It'i s assume dtha t al l materia l

ORNL-DW G 82-1440 9

SOURC E A T x = 0 APPROXIMAT E DISTANC E T O RIVE R i < x . » 

# ^A/AAAAAM. SURFAC E WATE R

^STREAMLINES ^ Figur e 4.6 . Groundwater'-surfac e wate r interface , flu

x model . enterin g th e aquife r

eventuall y enter s th e surfac e wate r excep t fo r tha t whic h ha s bee n los t

throug h radioactive 1 decay . Th e assumption s tha t appl y t o the' • poin t concentratio n mode l als o appl y t o thi s model . Th e mode l provide s onl y th e rat e o f

inpu t t o th e surface , wate r

a t a n averag e distanc e x downgradien t fro mth e surface . Actually , th e contaminatio n woul d ente r th e surfac e wate r

a s adiffus e patch , bu t

th e mode l describe dher e give s n o informatio n abou t th e spatia l distributio n o f

thi s patch . ' . , " I n th e unidirectiona l flo w fiel dassumed , th e flux, F (Ci/s ) o f

materia l crossin g a n are adA = dy. dzperpendicula r t o th e x

axi s i s describe dby , th e equatio n dF dA 

# *-* £ n e , (4.39 ) wher e c i s th e concentratio n i n th e dissplve d phase . Th e tota l flu x across , th e plan e woul db e Radionuclide s i n Groundwater

4-3 5 

# J . b nCO 

## j-r Uc-D, dc_

dx dydz (4.40 ) 4.5.3. 3 Sourc e Release d fro m a Vertica l Plan e (x = 0 ) I f Q i s th e concentratio n fro ma n instantaneou s releas e o f 1 C i a t x = 0 an d tim e t

— 0 , a s describe db y Eq . 4.25 , the n th e resultin g flux a t distanc e x downgradien t

woul db e F, = , Ut\

Kd\ 4y/D x Tt

3 /R d ex p Ut

Rd 4D x t/R d ~\t (4.41 ) 4.5.3. 4 Horizonta l Are a Sourc e Fo r condition s expresse db y Eq . 4.37 , th e correspondin g flux woul d b e

1 F, = : [ -j^rJ*Dx t/R d [ erf(z, ) - erf(z 2 ) ] 2i-JirD x t/R d -|^iexp(-z 2 )-exp(-z 2 2 )]]exp(-XO , (442 ) wher e u • . 2 . u 2. x

— trt + -z x ——t — — Rd 2 .

. • . R d

2 z\ = ~ j===-=-- , an d z2 = ,AJ4D x t/R d y/4D x t/R d Example 4.4. Fo r

th e sam e condition s i n th e previou s exampl e (Sect . 4.5.3.1) , calculate , th e flux o f th e pollutan t

int o ariver

interceptin g th e groundwate r

flow , whic h i s adistanc e x o f 200 0 mdowngradien t fro mth e
, cente r o f th e source . ' Equatio n 4.4 2 applie s i n thi s case . Figur e 4. 7 show s th e flux int o th e rive r a s afunctio n o f

time . [En do f

Exampl e 4.4 ] 4.5.3. 5 Generalizatio n o f Instantaneou s Model s i Equation s 4.8 . an d4.2 3 ar e formulate donl y i n term s o f instantaneou s releases . The y . ca n b e generalize dfo r arbitrar y release s b y us e o f th e convolutio n integral : 6 = ••••IjM8,(t-T)dT , •' n (4.43 )

r o4-3 6 Radiologica l Assessmen

t ORNL-DW G 82C-2075 0

O 3 . x

.1 2 1 6 20 ' . 24 ; TIM E (100 0 d ) ,2 8 3 2 3 6 Figur e 4.7 . Flu x o f pollutan t int o river fo r Exampl e 4.4/ wher e 0 i s th e solutio n a t

tim e t fo r th e arbitrary release , 0,( f —T ) i s th e solutio n a t

tim e ( f —T ) for

a n instantaneou s releas e at (t —T ) =

T 0 , an d/(T ) i s th e sourc e releas e rat e a t T in curies/s . Certai n analytical solution s ca n b e foun

dt o Eq . 4.4 3 fo

r simpl e sourc e releas e rat e functions . Fo r example , Wilson develop s th e solutio n t o Eq . 4.4 3 fo r acontinuou s releas e i

n term s of

, th e "well function " (Wilso n an dMiller 1978) . Mos t usefu l solution s t o Eq . 4.4 3 us e numerical integration , generall y involvin g adigital computer . Severa l specia l precaution s mus t

b e taken , however , t o Deserv e computationa l accuracy , becaus e th e term s withi n th e integra l o f Eq . 4.4 3 can b e ver y nearl y zero ove

r par t o f th e integratio n range . Computer program s fo

r solvin g th e equation s i

n thi s sectio n ar e describe db y Codel l e t

al . (1982) . Pro

gra mlisting s in BASI C an dFORTRA N ar e give n in thi s reference . A compute r tap e o f th e program s i s als o availabl e fro mNRC . An alternat e metho dtha t coul db e use dt o simulate acontinuou s sourc e functio n i s t o presen t

th e continuou s sourc e a s aserie s of instantaneou s ones . Th e analytica l solution s ar e the n linearl y summed . Complicate dare asourc e Radionuclide s i n Groundwate

r 4-3 7 term s ca n als o b e solve

di n a n analogou s fashion b

y representin g th e sourc e are ab y aserie s o

f poin t source s an dlinearl y summin g th e solutions . 4.5. 4 Simplifie

d Analytica

l Methods for Minimu

m Dilutions Simplifie dform s o

f th e equation s o

f Sect . 4.5. 3 hav e been develope dfo r calculatin g th e minimu mdilution s (i.e. , maximu mconcentration ) o f volum e V T o f asubstanc e instantaneousl y release dfro mapoint sourc e int o a

n aquifer . 4.5.4. 1 Dilution atDowngradien tWell s i n Confine d Aquifier s fo r a n Instantaneou

s PointSourc

e atth e Surfac

e A tsom e distanc e downgradien t

fro mareleas e a t

th e surfac e o

f aconfine d aquifer , th e concentratio n can b e considere d t o b e mixe di n th e vertical direction . Clos e t o th e poin t o f release , p r

i n an unconfine daquifer , th e vertical dispersio n will no t

b e influence d by th e vertical boundarie s o f th e aquifer . Betwee n thes e region s ther e i s aregio n where th e concentratio n canno t b e considere dmixed , bu t

th e boundarie s (to p an dbottom ) affec t th e dispersion . The degre e o

f vertical mixing ca

n b e characterize di

n aconfinedaquife

r o f con stant thicknes s an dunifor

mtransport

propertie s by th e factor <*>. = b 2 a T x (4.44 ) wher e aj '••' :b

• ' x th e vertical (transverse ) dispersivity , th e thicknes s ofth e aquife r (ft) , th e distanc e downgradient o f th e release . Th e factor

<f> ca n b e usedt o characteriz e th e aquife r i n thre e approximate regions : (a) I f <t> < 3.3 , ,th

e releas e ma

y b e considere dt o b e withi n 10 % o f bein g verticall y mixedin th e aquifer ; (b ) I f 0 > 12 , th e releas e ma y b e considere dt o b e withi n 10 % o f bein g ' unaffecte dby th e vertical boundarie s o f th e aquifer ; (c ) I f 3. 3 < <t> < 12 , th e releas e i s neithe r completel y mixedno r

unaffecte dby th e boundaries . Differen t method s apply t o eac h o f th e thre e regions .

T J h o4-3 8 Radiologica l Assessmen

t Vertically Mixed

Region (<f> < 3.3) . Fo r a n instantaneou s releas e a tx = 0 , th e minimu m dilutio n correcte d fo r deca y directl y dpwngradien to f a sourc e woul d b e i '

D L = Rd4rn e ' —exp(Xf), (4.45 ) wher e D L = minimu m dilutio n =rCQ/C, R d = retardatio n coefficient * n e = effectiv e porosity , V T = volum e o f

liqui d sourc e ter m (cm 3 ) , a L ,a T = dispersivitie s (cm ) i n th e indicate d direction , x = distanc e downgradien t(cm) , b = aquife r thicknes s (cm) , t = trave l tim e (y) , X = deca y constan t= I n 2/11/2( 1 y) . Th e trave l time , t, ca n b e approximate d a s



## " ' - TJ

R 

## «> wher e U i s th e pore velocit y define d b y Eq . 4.8a . (4.46 )

Unmixed Region (<t> > 12). Fo r a n instantaneou s releas e a tx = 0 o n th e surfac e o f

th e aquifer , th e minimu m dilutio n o f th e surfac e o f

th e aquife r directl y downgradien tfro m th e sourc e woul d b e n eRd{.4irx) 3r2

yla L a\ D L = — exp(A<) , (4.47 ) wher e a L , a T are dispersivitie s i n th e indicate d directio n and th e othe r term s ar e a s previousl y defined . 1 Intermediate Region (3.3 < d > < 12). Fo r a

n instantaneou s releas e a tx = 0 o n th e surfac e o f

a n aquifer , th e minimum , dilutio n o n th e surfac e o f th e aquife r directl y downgradien tfrom th e sourc e woul d b e

D, = R d

4irn e y/aiaj-xb V T FC*V ', exp(Af) , (4.48 ) Radionuclide s i n Groundwate

r 4-3 9 wher e

. F(<f>) =1+2 2 « P n - l "-It V (4.49 )

and th e othe r term s ar e a s previousl y defined . , Th e functio n F(0 ) i s convenientl y plotte d i n Fig . 4.8 . I tca n b e easil y see n tha tfo r smal l value' s o f 0 , Fapproache s th e valu e o f 1.0 , whic h yield s th e verticall y mixe d case . Fo r larg e value s o f <t>, th e slop e o f

Fi s 1/2 , an d th e unmixe d cas e prevails . Thi s metho d ma y b e use d fo r an y valu e o f 4> tha tca n b e read o n Fig . 4.6 .

5 — ORNL-DW G 82-1441 0

2 —

1 0 2 0 5 0 10 0

a T x Figur e 4.8 . Mixin

g factor fo

r confine d aquifers .

4.5.4. 2 Groundwater-Surfac e Wate r Interface—Instantaneou s Sourc e Fo r a n instantaneou s releas e t o th e groundwate r a tx =0 , th e minimu m dilutio n i n a n interceptin g river, correcte d fo r decay , ca n b e foun d t o b e D, = 2RdQrJira L

x UV T exp(Xr ) , (4.50 )

ro ro4-4 0 Radiologica l Assessmen

t wher e Q <*L U flow rat e o

f river (cm 3 /s) , th e longitudina

l dispersivit

y o f th e aquife

r (cm) , th e volum e o f releas e (e.g. , tan k

volume ) (cm 3 ) , pore velocit

y o f groundwate r

(cm/s) . 4.5. 5 Models fo

r Populatio

n Dose s If interdictiv e o

r mitigativ e method s t o restric t water us e ar e no t

take n int o account , populatio n dos e .fro mcontaminate dgroundwate r i s proportional t o th e time-average dconcentration . Relativel y simpl e equation s ca n b e use dfo

r estimatin g averag e concentratio n i n groun dwate r or i n surfac e water

supplie s contaminate dby groundwater . Tw o equation s ar e presente dbelow tha t ar e usefu l fo r populatio n dos e estimates . 4.5.5. 1 Quantit

y o f Release d Radioactivit

y Crossin g a Vertica

l Plane I n th e cas e o

f groundwate r

flo w t o a n interceptin g river, th e tota l quantity M (curies ) o

f th e dissolvedsubstanc e enterin g th e river wouldb e M 

# J *0 0 Fdt,

o (4.51 ) wher e F i s th e flux definedfo

r eithe r a n instantaneou s poin

t o r vertical plan e sourc e b y Eq . 4.4 0 or

ahorizontal are asourc e b y Eq . 4.42 . Equatio n 4.5 1 ca n b e integrate dgraphicall y or numericall y an d i

n som e case s ma

y hav e a n analytical solution . : , If dispersio n i s relativel y small (e.g. , yx « £) , th e followin g approximatio n ma

y b e used : M

= M$e w curie s (4.52)
.

' • i wher e M 0i s th e quantity o

f radioactivit

y release dihstantaneousl y fro mth e sourc e (curies) , t

i s th e trave l tim e (y) , an dX i s th e deca y coefficien

t (1/y) . If th e substanc e i s bein g release dfro mth e sourc e a t

arat e proportional t o th e quantity remainin g (e.g. , aleachin g sourc e term) , M - Mr, X'+ X e-» (4.53 ) wher e X ' i s th e releas e rat e fro

mth e sourc e (1/y ) an dM 0i s th e initial quantit

y o f materia l i

n th e sourc e ter

m(curies) . Radionuclide s i n Groundwate

r 4-4 1 4.5.5. 2 Direc

tGroundwate

r Usag e A mode l fo r calculatin g th e quantity o f aradionuclid e ingeste db y a populatio n usin g th e contaminate dgroundwater was developedb

y th e U.S . Nuclea r Regulator

y Commission (USNR C 1978) . Groundwater usag e wa s considere dt o b e spatiall y continuou s instea do f bein g fro mdiscret e wel

l points . Th e tota l amoun t

o f th e release dradionuclid e ingeste db y th e populatio n i s

wher e 

# J *O O /.O O • /»0 0 n 

## LJyjQi dx 

## *y*. (4.54 )

/ = th e ultimat e numbe r

o f curie s ingeste dfro

mth e release , c = th e groundwate r

concentratio n (Ci/L) , Q g = th e groundwate r

withdrawa l rat e fo

r drinkin g wate r purpose s (m 3 / d km 2 ) . If al l usag e i s restricte dbeyonddowngradient distanc e C fro

mth e releas e point , Eq . 4.5 4 ma

y b e integrate di

n closedfor

mt o give MoQ g US. R d

P(\ + y) 2n eR db CXp 2DX

Dx 1/2

1 (4.55 ) 

### y/X+yiJZ+A-^) wher e 7 = U 2 4R d D x Mb'i s th e tota l quantity o f th e radionuclid e discharge dt o th e point source , an d the othe r term s ar e a s previousl

y defined : I f usag e o f th e groundwater

i s restricte dbetwee

n tw o downgradient distances , J?

i an d2 2 , th e curie s ingeste dwoul db e defined : / = /(fii)-/(c 2 ) , (4.56 ) wher e /(fij ) an d7(fi2) ar e evaluation s o f Eq . 4.5 5 fo

r fi, an dfi 2 respectively . Example 4.5. Th e us e o

f severa l o f th e simple

r analytical model s i

n Sects . 4.5. 4 and4.5. 5 wil

l b e demonstrate d by wa

y o f ahypothetical example : Leakag e int o th e groun drapidl y emptie s a1000-ft 3 tan k containin g 400 0 uCi/m L o f 3 H , 200 0 /*Ci/mL o

f ^Sr , an d300 0 /iCi/m L o f 137 C s a t anuclea r 

### y-4-4 2 Radiologica l Assessmen

t sit e an d remain s undetected . Th e sit e i s 5 0 f tabov e th e mea n leve l an d 300 0 f t upgradien tfro m a rivertha tha s representativ e lo w flow o f 500 0 ft 3 / s an d i s th e sin k fo r al l surficia l groundwate ri n th e area . Ther e ar e tw o shallo w well s locate d 40 0 an d 250 0 f tdirectl y downgradien tfrom th e sit e o f th e spill . Groundwate rexist s i n a homogeneou s alluvia l san d laye r10 0 f tthic k unde r wate rtabl e conditions . Dispersivitie s fo r th e san d hav e bee n determine d i n th e nea rfiel d fro m single-wel l trace rtest s t o b e 0. 5 f tfo r a.j an d 1. 0 f

tfo r o^ . Th e bul k densit y pj o f

th e san d i s 2. 6 g/cm 3 . It s tota l porosity n an d effectiv e porosity n e ar e 0. 4 an d 0.25 , respectively . Th e permeability K i s 0.02 ; cm/s . Distributio n coefficient s K d fo r th e san d hav e bee h determine d t o b e 0 , 2.0 , an d 20. 0 mL/ g fo r dilut e solution s o f 3 H , ^Sr , an d 137 Cs , respectively . Fro m th e abov e informatio n calculate'th e following : (a ) th e maximu m concentration s o f th e radioactiv e component s i n th e river, i • . (b ) th e maximu m concentration s o f th e component s i n th e nea rwell , (c ) th e maximu m concentration s p f th e component s i n th e fa r well , an d (d ) th e tota l quantit y o f

eac h radionuclid e escapin g t o th e river. Solution, (a ) I f

i ti s assume d tha tth e sourc e i s release d ove ra short period , Eq . 4.5 0 fo r instantaneou s release s ma y b e used , t o calculat e th e maximu m riverconcentration s o f 3 H , ^Sr , an d 137 Cs . Firs tdetermin e th e pore velocit y U fro m Eq . 4.8a an d th e effectiv e porosity n e : K U = AH

AA Th e gradien t

therefore , AH = -5 0 f

t A x 300 0 f

t = -0.0167 ;
TJ - 2 X 10~ 2 cm/ s X -0.016 7 „ 86,40 0 s/ d , , „ f ' U dr " —X 30.4 8 cm/f t= 378ft/d - Th e retardatio n coefficients'fo r 3 H , ^Sr , an d 137C s ca n b e determine d fro m Eq . 4.14 : , ' ,



# '* * " 1 + 1 4 x 

# °- ° " ! 

# • 4 0. 4 

### *J~ l 

### + 1 4 X2.0 = 14 , * 0. 4 ,3T 

### C s R d 

### -> 1 + | 4 x 20 = 131



### -

. 0. 4 3 H "S r Radionuclide s i n Groundwate

r4-4 3 Th e trave l time s fo r th e thre e component s ar e calculate d b y Eq . 4.46 :

3 R/ = A^L = 300 0 f

tX 1 "S r U

300 0 f tX 1 4 y 3.78 ft/ d X 365 d = 2A1 y •

137 , C s 3.7 8 ft/ d X 3^d " = 304y '

300 0 f tX 13 1 „ y _ 3.7 8 ft/ d X 365 d ~ 284 ' 8 ' -

, Th e half-live s o f 3 H , ^Sr , an d 137 C s ar e 12. 3 y , 2 9 y , an d 30. 1 y , respectively . Th e decay-correcte d minimu m dilution s i n th e rive rar e foun d b y applyin g Eq . 4.50 : 2 X 1. 0 X 3 H DL= 5{mix X V T X 1. 0 f tX 300 0 f t 3.7 8 ft/ d X 100

0 f tX 86,40 0 s Xexp | ln 2 12.3 y
= 2.5 1 X10 7 , X2.17 y

90 . S r 2X1 4 X 500 ° ft3 X Vir X 1. 0 f tX 300 0 f t D L= 3.78 ft/d X 100 0 f tX 86,40 0 s Xex p l n 2 29 y 6.42X10 8 , 2 X 13 1 X 137 , C s Z> x = X 30. 4 y

500 0 f tX AT

X 1. 0 f tX 300 0 f t 3.7 8 ft/ d X 1000 f

tX 86,40 0 s XCXp| 

### 30l7 ><284



### - 8 y

. = 2.05X10 12 . Th e pea k concentration s i n th e riverar e determine d b y dividin g th e tan k concentration s b y th e dilutio n factors :4-4 4 Radiologica l Assessmen

t c( 3 H ) = 400 0 /iCi/mL/2.5 1 XlO 7 = 1.5 9 X 1<F 4 ^Ci/m L , i • i c( 90 Sr ) = 200 0 /iCi/mL/6.42 X 10 8 = 3..12 X IO" 5 nCi/m L , c( 137 Cs ) = 300 0 MCi/mL/2.05X10 1 2 = 1.46X10~ 9 /iCi/m L . (b ) Minimu m dilutio n in wel l (40 0 f tdowngradient) . Firs tdetermin e whethe r o r no tthicknes s o f th e aquife r woul d affec tth e resul tb y calculatin g th e facto r <t> fro m Eq . 4.44 :

* b 1 = (10 0 ft) 2 arx 0. 5 f

tX 40 0 f t = 50 . Therefore , i n thi s regio n th e releas e wil l b e relativel y unaffecte d b y th e thicknes s o f

th e aquifer , and ' Eq . 4.4 8 applies . Th e trave l time s ar e estimate d usin g th e retardatio n factor s an d pore velocit y calculate d above : 3 H "S r ,37 C s 40 0 f

tX 1 x 3.78 ft/ d y • -

36 5 d 40 0 f tX 14 x y 3.7 8 ft/ d 36 5 d 40 0 f tX 13 1 • 3.78 d 

### x y

X 365 d 0.2 9 y ,

= 4.0 6 y ,

= 3 8 y . Applyin g equatio n 4.51 : 3H D 0.2 5 X 1 X (4 T X 40 0 ft) 3/2 V l f tX 0. 5 f tX 0. 5 f t L ' 2 X 100 0 ft 3 Xexp|-^X0.29 y 22.6 , »°S r „ = 0.25 X 1 4 X (4 x X 40 0 ft) 3/ V l f tX 6. 5 f tX 0. 5 ft " L = 2 X 1000 ft 3
XeXP (297 X 4 - 0 6 y

= 343.6 , 137 , C s

_= 0.25 X J3 1 X (4 r X 40 0 ft) 3/2 V l f tX 0. 5 f tX 0. 5 ft L 

### ~ '

2 x ioo o ft 3 

### y : 

### ~

= 6999^9 . Radionuclide s i n Groundwate

r 4-4 ! Th e pea k wel l concentration s ar e therefor e 177 /*Ci/m L fo r 3 H , 5 . M Ci/m L fo r 90 Sr , and 0.4 3 piCi/m L fo r l37 Cs . (c ) Wel l 250 0 f

tdowngradient . Calculat e <t> fo r thi s regio n fro m Eq . 4.4 4 <t> = (10 0 ft) 2 0. 5 f tX 200 0 f

t = 8.0 . Therefore ^ thi s wel l i s in th e intermediat e region , and Eq . 4.4 8 applies . T l facto r F (0 ) ca n b e read from Fig . 4. 7 t o b e 1.6 . Trave l time s fo r ea c componen tcalculate d from Eq . 4.4 6 ar e

3 H _250 0 f

tX I _JL

__g 3.78 ft/ d 36

5 d ' * a* , _250 0 f

tX 14 v y _

, , 1

. " • 3.78 ft/ d "365 d X y ' •"P. , _

250 0 f tX 13 1 „ y _^ . ' 3.78 ft/ d '^365 d "' ' y ' Applyin g Eq . 4.48 : 3 H

90o 1 X 4i

r X 0.25V0. 5 f

tX 0. 5 f tX 250 0 f

tX 10 0 f t f 1000ft 3 X 1. 6 Xex p lh 2 X.1.81 y ll2.3 y ' J

= 271. 8 , 1 4 X 4B - X 0.25V0. 5 f

tX 0. 5 f tX 250 0 f tX 10 0 f t 100 0 ft 3 X 1. 6 Xexpl-! ^ X25.2y ] 29 y

= 6275 , l37 C s

i _13 1 X4xX0.25V0.5f tX 0.5f tX 2500f tX 100f t L 1000ft 3 X 1. 6 

### Xexp 

# [3o 1 

# f V X237 

# - 4 y V : = 7.6 1 X IO

6 . Th e peak wel l concentration s ar e therefor e 14. 7 /*Ci/m L fo r 3 H , 0.3 2 pC m L fo r 90 Sr , an d 3. 9 X 10" 4 MCi/m L fo r 137 Cs .
r o O l4-4 6 Radiologica l Assessmen

t (d ) Quantit y Q o f

eac h radionuclid e eventuall y reachin g river. Equatio n 4.5 1 applie s t o thi s cas e becaus e a L « £ (i.e. , 1 f

t v s 100 0 ft) . Trave l time s ar e estimate di n par t (a ) above . Th e quantit y o f eac h radionuclid e initiall y i n th e tan k

i s th e concentratio n multiplie db y th e volume . Therefore , 3 H Q = 4000/tCi/m L X 1000 ft 3 X 28,30 0 mL/ft 3 -ln 2 Xex p 12.3 v X2.17 y X10-«Ci//iC i
"S r = 1.002X10*0 , Q = 200 0 /»Ci/m

L X 100 0 ft3 X 28,30 0 mL/ft 3 -I n 2 XeX p 2 9 y X30.4 y XlO-'Ci/ziC i

137 , C s = 27,37 0 Ci , '•.•'''.'. ' Q = 3bOO,MCi/m

L X 100 0 ft3 X 28,30 0 mL/ft 3 i I .-I n 2 ' Xex p 30. 1 y X 284. 7 y XlO-«Ci/ W C i 120. 7 Ci . 

### 4. 6 MODE

L VALIDATIO

N Hydrauli c flo w an dtranspor t model s ca n b e validate d(tested ) onl y b y compariso n wit h fielddat aand , i n th e 'cas e o f numerica l models ^ b y compariso n wit h th e analytica l solutio n o f asimplifie dse t

o f equations . I n general , th e mode l i s ase t o f equations , an dvalidatio n consist s o f compariso n o f th e solutio n o f

thes e equation s wit h field-measure ddata . Regardles s o f whethe r th e solutio n i s obtaine db y analytica l o r numerica l techniques , tru e validatio n ca n b e don e onl y throug h compariso n wit h fieldmeasurements . Agreemen t

o f anumerica l solutio n w ; h a n analytica l solutio n o f th e sam e equation s show s onl y tha t th e numerica l technique s wor k an dtha t

n o error s exis t i n th e compute r

code . 4.6. 1 Mode l Calibratio n Model s ar e calibrate dfo r aspecifi c proble mb y startin g wit h a n initia l se t o f paramete r

estimate s (field

- o r

laboratory-measured) , runnin g th e mode l fo r th e problem , an dcomparin g th e result s wit h observe dvalues . I f th e compariso n i s poor , th e paramete r

estimate s ar e modified , th e mode l i s rerun , an dresult s Radionuclide s in Groundwater 4- 4

ar e agai n compare dwit h observe ddata . Thi s proces s i s continue dunti l t l desire dleve l o f

agreemen t

betwee n observe ddat aan dth e simulatio n obtaine d(Merce r

an dFaus t

1980) . Th e modificatio n o f boundar y conditio i an dparameter s is ' subjectiv e an drequire s aconsiderabl e amoun t o f knowled f o f th e regio n bein g simulate dan dexperienc e o n th e par t o f th e modeler . T r boundar y condition s an dparameter s use di n th e fina l simulatio n mus t stil l t i n agreemen t

wit h th e knowledg e an dunderstandin g o f th e geology an hydrolog y o f

th e site i Throug h thi s proces s th e equation s o f groundwate r

fl o i n porou s medi ahav e bee n wel l teste dan dverified . Th e equation s o f fl o throug h fracture dmedi ahav e bee n teste dt o som e extent , bu t th e equation s < transpor t throug h fracture dmedi aremai n largel y untested . Wher e enoug h dat ao n hydrauli c head s an dvariatio n o f hea dove r

Io n tim e period s ar e available , th e invers e o f th e equation s fo r

hea dca n b e solv e fo r th e spatia l distributio n o f permeability . Th e solutio n o f th e invers e proble i (Newma n 1973 ) i s usefu l becaus e i t yield s aspatia l distributio n o f paramete i tha t ar e consisten t

wit h th e hydrolog y o f th e sit e unde r

consideration . I n t h process , field-measure dparameter s ar e usefu l fo r compariso n wit h compute i

, generate dparameter s t o ensur e tha t th e generate dvalue s ar e realistic . Fo r th e transpor t

equations , th e invers e proble mha s no t bee n solv e becaus e th e result s ar e no t unique . 4.6. 2 Misus e o f

Model s Th e thre e mos t

commo n misuse s o f

model s ar e overkill , inappropria t ' prediction , an dmisinterpretation . j Overkil l i s define da s usin g amor e sophisticate dmode l tha n i s appropria t fo r th e availabl e dat ao r

th e leve l o f

resul t desired . Th e temptatio n t o app l th e mos t sophisticate dcomputationa l too l t o aproble mi s difficul t t o resist . , questio n tha t

ofte n arise s is : whe n shoul dthree-dimensiona l model s b e use da oppose dt o twordimensipna l o r one-dimensiona l models ? Inclusio n o f flow i n t l thir ddimension , usuall y vertical , i s recommende donl y i n thic k aquifer s o r permeabilit y changes , drasticall y acros s th e thicknes s o f th e aquifers . Inclusi o o f th e thir ddimension ' require s substantiall y mor e dat atha n one - an dtw c dimensona l models . Fo r example , saturated-unsaturate dflow throug h ashallo i lan dburia l sit e i s trul y athree-dimensiona l problem . However , th e dat aa r seldo mavailabl e t o conside r mor e tha n on e dimensio n abov e th e wate r table . I n man y cases , sophisticate d model s ar e use dto o earl y i n analysi s o f problem . On e shoul dbegi n wit h th e simples t mode l appropriat e t o th e proble r an dprogra mtowar dth e mor e sophisticate d model s unti l th e desire d leve l c result s i s achieved . I n transpor t problem s th e flow modelin g shoul db complete dan dchecke dagains t th e understandin g o f

sit e hydrolog y befor e transpor t mode l i s applied . Misinterpretation s usuall y aris e becaus e inappropriat e boundar y conditio n wer e selecte do r

th e hydrologi c histor y o f asit e ha s bee n misread . Unde r eith eT l

h o o >4-4 8 Radiologica l Assessmen t Radionuclide s in Groundwate r 4

of thes e condition s th e simulate ddat awii l no t matc h th e hydrologi c histor y o f th e site . • ' . y Perhap s th e wors t

misus e o

f a'mode l i s blin dfait

h in model results . Simulate ddat atha t

contradic t

hydrologi c intuitio n almos t alway s aris e fro

ma mistake in som

e dat aentry ; a

n erro r in th e computer ; code , or applicatio n o

f a mode l t o aproble mfo

r which i t wa s no t designed . Th e latte r

cas e can occur

ir i applicatio n o

f a n analytical solutio n tha t was , obtaine dusin g boundar y condition s tha t

ar e differen

t fro mthos e t o whic h th e solutio n i s bein g applie d (Merce r an dFaus t

1980) . , ' High-qualit

y fiel ddat aon contaminan t or radionuclid e transport

i n groundwate r

ar e scarce . The , collectio n o f dat anecessar

y fo r ver y detaile d modelin g effort s i s extremel y costly sinc e th e aquife r in which th e transpprt

i s takin g plac e can b e measure donl y indirectl y fro mwells . Several well-known validatio n effort s ar e discusse db y Robersto n (1974b) , Codel l (1978) , Pinde r (1973) , Wilson (1978) , Evenson an dDettinge r (1980) , Anderso n (1979) , an d Isherwoo d(1981) . 

### 4. 7 PROBLEM S 1 . Saturate dgroundwater flow—Determine th e seepag e velocit y fo r a n averag e coars e gravel fo

r agroundwate r

gradien t o f 0.002 . 2 . Unsaturate dflow—

A water balanc e show s tha t

1 0 cm/ y o

f water infiltrat e th e groun dan drecharg e th e wate r table . Calculat e th e averag e downwar d velocit y o f th e water i n th e unsaturate dzone , which i s afine sand . 3 . Retardatio n coefficient—Calculat e th e retardatio n coefficien t fo r cesiumi n a n averag e san dwit

h abulk densit y o f 2. 8 g/cm 3 an dK d o f 5 0 mL/g . 4 . Groundwater

concentration—Calculate th e concentratio n a s afunction o f tim e 1500 mdirectl y downgradient o f a1-C i instantaneou s point

sourc e i n a n infinitely dee

p aquifer . Th e seepag e velocit y i s 1 m/d . Th e x, y, an dz dispersivitie s are 5 0 m , 2 0 m , an d1 m , respectively . Th e effective porosit y i s 0.2 . Th e retardatio n coefficien t i s 20 . Th e half-lif

e o f th e substanc e i s 1 0 y5 . Dilutio n i

n groundwater—Fo r

th e same condition s i

n problem4 , conside

r tha t th e 1 C i wa s dissolvedin 1000 L o f water . Calculat e th e minimum dilutioni n th e wel

l usin g th e equation s o

f Sect . 4.5.4.1 . 6 . Dilutio n i

n river—For

th e sam e condition s a s problem4 , calculat e th e minimumdilutio n i

n a n interceptin g river

hayin g a n averag e flow rat e o

f 10m 3 /s . .'''' . ' • . . . • 7 . Populatio n dos e for

averag e usage'—Wast

e i s bein g discharge dt o a n aquife r ingeste db y downgradient user s a t arat e o f 0. 1 m 3 /( d km 2 ) . Al l user s ar e greate r tha n 500 0 man d les s than 10,00 0 mdowngradient . propertie s o

f th e radionuclid e an dth e aquife r ar e U = 1 m/d , a x

= m , n e = 0.2 , R d

= 3 , b = 10

0 m , r 1/ 2 = 3 0 y . Calculat e th e c u ingeste dfo

r eac h curi e released . 8 . Groundwater

flux—Equatio n 4.5 1 i s a n approximatio n only . Th e tru e t quantity o

f flu x passin g aplan e i s mor e accuratel y determine d integratin g th e flux expresse db y Eqs . 4.4 1 o

r 4.4 2 fro mt = 0 t o t = Graphicall y integrat e th e flu x fro mEqs . 4.4 1 or

4.4 2 fo r arang e parameter s an ddetermin e ho w wel l Eq , 4.5 4 agrees .

r o ^ i4-5 0 Radiologica l Assessmen t 

### REFERENCE S

Abramowitz , M. , an dStegun , I. , eds . Novembe r 1970 . Handbook

of Mathematical

Functions, Applie dMathematic s Serie s 55 , Nat . Bur . Stan

dards . Aikens , A . E. , Berlin , R . E. , Clancy , J. , an dOztunali , O . I . 1979 . Generic Methodology forAssessment of Radiation Doses from GroundwaterMigration of Radionuclides in LWR Wastes in Shallow Land Burial Trenches, Atomi c Industria l Forum , Washington , D.C . America n Nuclear Societ

y 1980 . Evaluation of

Radionuclide Transport in Ground Water for NuclearPowerSites, Repor t2.1 7 L a Grang e Park , 111 . Anderson , M . P . November 1979 . "Usin g Model s t o Simulat e th e Movemen t o f Contaminant s Throug h Groundwater Flow Systems, " pp . 97-15 6 i n CRC Critical Reviews in Environmental Control. Barraclough , J . T. , Teasdale , W . E. , Robertson , J . B. , an dJenson , R . G . 1967 . Hydrology of

the National

ReactorTesting Station Idaho 1966, Ope n File Report TID-4500 , U.S . Geol . Surv, , Wate r Resour . Div. , Idah o Falls , Idaho . Bredehoeft , J. , an dPinder , G . 1970 . "Digital Analysi s o f Areal Flo

w ir i Multilayer

Groundwater

Systems : Aquati c Thre e Dimensiona l Model, " WaterResour. Res. G(3) , 883-88 . Bredehoeft

, J . O . 1978 . Utilization of Numerical

GroundwaterModels for Water Resources Management, EPA 60018-012 , U.S . Environ . Protectio n Administration . . ; , Burkholder , H . C , an dRosinger E . L . J . Jun e 1,980

, " A Mode l fo

r th e Trans

port o f Radionuclide s an dThei r Deca y Product s Throug h Geologi c Media, " . Nucl. Technol. 49 , 150-58. , , ' Campbell , J . E. , Kaestner , P . C , Langkopf,,P . S. , an dLantz , R . B . Februar y 1980 . Risk Methodology forGeologic Disposal of Radioactive Waste: The Network Flow and

Transport

(NWFT) Afbdel, NUREG/CR-H90 , U.S . • Nuclear Regulatory Commission . Carslaw , H. , an dJaeger , J . 1959 . Conduction of Heat in Solids, Oxford , Univ . Press , London . Chow , V . T. , ed . Handbook of Applied Hydrology, McGraw-Hill , Ne w York , 1964 . Codell , R . B. , an dSchreiber , D . L . 1978 . "NR C Model s fo r Evaluatin g th e Transport of Radionuclide s in Groundwater ^ in Management of Low-Level Radioactive Waste, ed . M . W . Carter , A . A. , Moghissi , an dB . Kahn , Per

gamon , Ne w York . Codell , R . B. , Key , K . T. , ari dWhalen , G . February 1982 . A Collection of Mathematical

Models forDispersion in Surface and Ground Water, NUREG-0868 , U.S . Nuclear Regulator

y Commission . Davis , S . M. , an dD e Wiest , R . M . M . 1965 , Hydrogeology. Wiley , Ne w York . Radionuclide s in Groundwate r 4 - Dillon , R . T. , Lantz , R . B. , an dPahwa , S . B . 1978 . Risk Methodology Geologic Disposal

of Radioactive Waste: The Sandia Waste Isolation F and Transport

(SWIFT) Model, SAN D 78-1267 , Sandi aNatl . Lab. , A l querque , N.Mex . Duguid , J. , an dLee , R . C . Y . 1977 . "Flo w in Fracture dPorou s Media, " Wt Resour. Res. 13(3) , 558-66 . Evenson , D . E. , an dDettinger , M . D . Ma y 1980 . Dispersive Processes Models of

Regional

Radionuclide Migration, Univ . Calif. , Lawr e Livermor e Lab. , Livermore . Faust , C . R. , an dMercer , J . W . July-Augus t 1980 . "Groun dWate r

Model i Numerica l Models, " Ground Water, 18(4) . .Freeze , R . A. , an dCherry , J . 1979 . Groundwater. Prentice-Hall , Engle w Cliffs , N.J . Gray , W. * Pinder , G. , an dBrebbia , C . 1977 . Finite Elements in Wt Resources, Penlac h Press , London . Gupta , S . K. , Tanji , K. , Nielsen , D. , Biggar , J. , Simmons , C , an dMaclnt ; , J. 1978 . Field

Simulation of

Soil-WaterMovement with Crop Wt Extraction, Wate r

Scienc e an d Engineerin g Pape r 4013 , Departmen t

L a Air , an dWate r

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